CFA
LEVEL 2
(Personal Notes)
1
Ethics
PAGE NOS. 13
CHAPTER 1
VOL. 1
CHAPTERS. 2
Code of Ethics and Standards of Professional Conduct
DEF Ethical vs. Legal: Ethical behavior is often distinguished from legal conduct by describing Legal behavior
as 'what is required' and Ethical behavior as 'conduct that is morally correct'. Ethical principles go
beyond that which is legally sufficient and encompasses what is the right thing to do.
The Standard tells you 'What to do?'
The Code tells you 'How to do it?'
Technique: IRAC
Issue, Rule, Application and Conclusion
2
3
Code of Ethics
1. Act with integrity, competence, diligence, respect and in an ethical manner.
2. Place integrity of investment profession & client above personal interests.
3. Use reasonable care and exercise independent professional judgement when conducting IRA (Investment
Recommendation & Analysis) and engaging in other professional activities.
4. Practice and encourage others to practice in professional and ethical manner.
5. Promote the integrity and viability of the global capital markets for the ultimate benefit of society.
6. Maintain and improve their professional competence and strive to maintain and improve the competence
of other investment professionals.
Standards of Professional Conduct
I
A.
B.
C.
D.
II
PROFESSIONALISM
Knowledge of the Law
Independence and Objectivity
Misrepresentation
Misconduct
INTEGRITY OF CAPITAL MARKETS
A. Material Non-public Information
B. Market Manipulation
III
A.
B.
C.
D.
E.
DUTIES TO CLIENTS
Loyalty, Prudence and Care
Fair Dealing
Suitability
Performance Presentation
Preservation of Confidentiality
IV
A.
B.
C.
DUTIES TO EMPLOYERS
Loyalty
Additional Compensation Arrangements
Responsibilities of Supervisors
V
INVESTMENT ANALYSIS, RECOMMENDATIONS AND ACTIONS
A. Diligence and Reasonable Basis
B. Communication with Clients and Prospective Clients
C. Record Retention
VI
A.
B.
C.
CONFLICTS OF INTEREST
Disclosure of Conflicts
Priority of Transactions
Referral Fees
VII RESPONSIBILITIES AS A CFA INSTITUTE MEMBER or CFA CANDIDATE
A. Conduct as Participants in CFA Institute Programs
B. Reference to CFA Institute, the CFA Designation and the CFA Program
CFA Institute Code & Standards are an example of 'Principle-Based' Standards.
4
CHAPTER 2
Standard I
Guidance for Standards I - VII
PROFESSIONALISM
A. Knowledge of the Law
1. Comply with laws, rules and regulations that apply to your professional activities and also
comply in which the member resides. In the event of conflict: follow stricter law and
regulations.
2. CFA member must not knowingly participate in any violation of law and must disassociate
themselves from it. The first step should be to attempt to stop the behavior by bringing it
to the attention of the employer through a supervisor or the firm's compliance
department. If this attempt is unsuccessful, then members can step away and
disassociate from the activity. Inaction with continued association may be construed as
knowing participation.
3. Members should document any violations when they disassociate themselves from
prohibited activity.
4. Members must know the laws and regulations relating to their professional activities in all
countries in which they conduct business. During times of changing regulations members
and candidates must remain vigilant in maintaining their knowledge of the requirements
for their professional activities.
B. Independence and Objectivity
1. CFA members must use reasonable care and judgement to maintain independence and
objectivity.
2. Best practice dictates that you reject any offer of gift or entertainment that could be
expected to threaten your independence and objectivity, e.g. lavish functions, favors and
job referrals. When possible prior to accepting 'bonus' or gifts from clients, members must
disclose to their employer such benefits offers by clients. Firms should impose clear value
limits on gifts.
3. Analyst must distinguish between fact and opinion in their reports. (Personal vs. firms)
4. Restrict employee participation in IPOs and private placements. Require pre-approval of
IPO purchases. Allocation of shares in oversubscribed IPOs to personal accounts is NOT
permitted.
5
5. If the firm is unwilling to permit dissemination of adverse opinions about a corporate
client, members and candidates should encourage the firm to remove the controversial
company from the research universe and put it on a 'Restricted List', so that the firm
disseminates only factual information about the company.
6. Always use commercial transportation at your expense or at the expense of your firm
rather than accept paid travel arrangements from an outside company. Should
commercial transportation be unavailable, you may accept modestly arranged travel to
participate in appropriate information-gathering events, such as a property tour.
7. Analysts may face pressure to issue credit ratings at a specific level because of other
services the agency offers - namely, advising on the development of structured products.
The rating agencies need to develop the necessary firewalls and protections to allow the
independent operations of their different business lines.
8. Firms should appoint a compliance officer and provide clear procedures for employee
reporting of unethical behavior and violations of applicable regulations.
9. 'Pay-to-Play': Managers looking to gain lucrative allocations from the large funds made
requested donations to the political campaigns of individuals directly responsible for the
hiring decisions. This has led to bans on hiring or hiring delays for managers that made
campaign contributions to representatives associated with the decision-making process.
C. Misrepresentation
1. Member must not knowingly omit or misrepresent information or give a false impression
of a firm, organization or security in the member's or candidate's oral representations,
advertising, electronic communications or written materials.
2. Members must not misrepresent any aspect of their practice including their qualification
or credentials or services provided by the firm, their performance record and record of
their firm & the characteristics of an investment.
3. Members should encourage their firms to develop strict policies for composite
development to prevent 'Cherry Picking' - situations in which selected accounts are
presented as representation of the firm's abilities. The omission of any accounts
appropriate for the defined composite may misrepresent to clients the success of the
managers implementation of its strategy.
4. Guaranteeing a specific return on securities that don't have an explicit guarantee from a
government body or financial institution is prohibited.
5. Must disclose intended use of external managers and must not represent those manager's
investment practices as your own.
6. When communicating through social media channels: members should provide only the
same information they are allowed to distribute to clients and potential clients through
other traditional forms of communication. E.g. Article's webpage should be the same as
the social media.
7. Crediting the source is not important only when it is from recognized financial and
statistical reporting services.
8. Copying verbatim (Plagiarism) any material without acknowledgment including plainlanguage descriptions in Wikipedia, Investopedia etc. is a violation. Even though these
concepts are general, best practice is to describe them in your own words and cite the
sources.
9. Must not take credit for the work done by others, but can omit names of the
presenters/colleagues who are no longer working in the company. The firm retains the
right to continue using the work completed after a member or candidate has left the
organization.
6
D. Misconduct
1. Not engage in any professional conduct involving dishonesty, fraud, deceit or commit any
act that adversely effects their professional reputation, integrity or competence. Must not
use this standard for personal revenge or issues not related to professional ethics or
competence.
2. Personal bankruptcy may not be a violation, but if the circumstances of the bankruptcy
involve fraud/deceit, the bankruptcy may be a violation.
Standard II
INTEGRITY OF CAPITAL MARKETS
A. Material Non-Public Information
1. It is material only if the disclosure would affect the price of the security before making an
investment decision. Must not act or cause others to act on the information. It is not a
violation if you are not aware that the information your receiving is material non-public
information and you trade & pass it on.
2. Mosaic Theory: A financial analyst gathers and interprets large quantities of information
from many sources. The analyst may use significant conclusions derived from the analysis
of public and non-material non-public information as the basis for IRA, even if those
conclusions would have been material inside information, has they been communicated
directly. They should save and document the research.
3. Some social media platforms require membership in specific groups in order to access the
published content. Members participating in groups with membership limitations should
verify that material information obtained from these sources can also be accessed from a
source that would be considered available to the public. Members should also complete all
appropriate regulatory filings related to information distributed through social media
platforms.
4. Members may provide compensation to individuals for their insights without violating this
standard. Members are ultimately responsible for ensuring that they are not requesting or
acting on confidential information received from external experts. However, they may not
act or cause others to act on any material non-public information obtained from these
experts until that information has been publicly disseminated.
5. When a well known analyst issues a report or makes changes to recommendation, it may
have an effect on the market and thus may be considered material. Theoretically, such a
report would have to be made public at the time it was distributed to clients.
6. Issue press releases prior to analyst meetings and conference calls & script those
meetings and calls to decrease the chance that further information will be disclosed. If
material non-public information is disclosed for the first time in an analyst meeting or call,
the company should promptly issue a press release or otherwise make the information
publicly available.
7. Securities should be placed on a restricted list when a firm has or may have material nonpublic information. Therefore, an information barrier 'Firewall' is most widely used
approach for preventing the communication of material non-public information within
firms. If sharing such material non-public information is necessary, the compliance officer
should coordinate the process of 'Looking over the Wall' so that the necessary information
will be shared and the integrity of the procedure will be maintained.
8. The restriction and review of a firm's proprietary trading, while the firm possesses
material non-public information will necessarily depend on the types of proprietary
trading in which the firm may engage: (a) If involved in market making, it can continue
trading provided it remains passive i.e. taking the contra-side of unsolicited requests.
7
(b) The most prudent course for firms is to suspend arbitrage activity when a security is
placed on the 'Watch List'.
B. Market Manipulation
1. Must not engage in distorting prices or artificially inflate trading volume to mislead the
market participants.
2. Liquidity Pumping Strategy: The exchanges or firms attempt to demonstrate that it has the
best liquidity. It may engage in LPS, but the strategy must be disclosed.
Standard III
DUTIES TO CLIENTS
A. Loyalty, Prudence and Care
1. One should inform clients that the advice provided will be limited to the propriety
products of the firm and not include other products available in the market.
2. When the manager is responsible for the portfolios of pension plans or trusts, it's the
client is not the person or entity who hires the manager but rather, the beneficiaries of the
plan or trust. The duty of loyalty is owed to the ultimate beneficiaries.
3. Investment decisions must be judged in the context if the total portfolio rather than by
individual investment within the portfolio.
4. Client Brokerage or Soft Dollars or Soft Commissions must be used to benefit the client. In
addition the member should disclose to the client, that the client may not be getting best
execution from the 'Directed Brokerage'.
5. The members should vote in an informed and responsible manner. They should disclose
to clients their proxy voting policies.
6. If members control client assets, one should:
(a) Submit to each client at least quarterly, an itemized statement showing the
funds/securities in your custody plus all transactions that occurred during the period.
(b) Disclose to the client where the assets are to be maintained as well as where or where
they are moved.
(c) Separate the client's assets from any other party's assets including your own assets.
7. If a member is uncertain about the appropriate course of action with respect to a client, if
in doubt, a member should disclose the questionable matter in writing to the client and
obtain approval from the client.
B. Fair Dealing
1. Treat both individual and institutional clients in a fair and impartial manner (fairly doesn't
mean equally). Disclose the different service levels to all clients.
2. Give all clients a fair opportunity to act upon every recommendation. Shorten the time
frame between decision and dissemination.
3. Clients who are unaware of a change in recommendation should be advised before the
order is accepted.
4. Disclose trade allocation procedures and how procedures would affect the client or
prospect.
5. Members are prohibited from withholding such securities for their own benefit and must
not use such securities as a reward or incentive to gain benefit e.g. holding oversubscribed
IPOs for their own benefit a.k.a. 'Hot Issue' securities.
6. Best practice includes allocating pro-rata on the basis of order size.
C. Suitability
1. Updating the IPS should be reported at least annually and also prior to material changes
8
to any specific investment recommendations or decisions on behalf of the client.
2. Members can be responsible for assessing the suitability of an investment only on the
basis of the information and criteria actually provided by the client. Portfolios should be
diversified.
3. In the case of unsolicited trade requests that a member knows are unsuitable for a client,
the member should refrain from making the trade until he/she discusses the concerns
with the client.
4. If the unsolicited request be expected to have a material impact on the portfolio; the
member should update the IPS.
5. You may have clients who decline to modify their IPS while insisting an unsolicited trade
be made. In such instances:
(a) Allow for the trade to be executed in a new unmanaged account.
(b) If this is not possible, you need to determine whether to continue the advisory
arrangement with the client.
D. Performance Presentation
1. Including terminated accounts as part of historical performance and clearly stating when
they were terminated.
2. Maintaining data and records used to calculate the performance being presented.
3. Presenting performance of weighted average composite of similar portfolios rather than a
single account.
4. Members should encourage their firms to comply with the GIPS standards. Claiming
compliance with GIPS while not being fully compliant with GIPS is a violation.
5. Include all appropriate disclosures to fully explain results e.g. model results, including the
gross or net of fees etc.
6. No prohibition as long as member discloses that the results are stimulated and not actual
firm or member performance.
7. No prohibition from showing past performance of funds managed at a prior firm as part of
a performance track record as long as that record is accompanied by appropriate
disclosures about where the performance took place and the person's specific role in
achieving that performance.
8. Modifying a performance attribution methodology without proper notifications to client is
a violation.
E. Preservation of Confidentiality
1. Members must keep information about current, former and prospective clients
confidential unless:
(a) The information concerns illegal activities on the part of the client or prospective
clients.
(b) Disclosure is required by Law or
(c) The client or prospective client permit disclosure of the information.
2. Members should avoid disclosing information received from a client to authorized coworkers who are also working for the client.
3. Members should follow firm procedures for storage and communication via. electronic
data and recommend adoption of procedures if they are not in place.
9
Standard IV
DUTIES TO EMPLOYERS
A. Loyalty
1. Members must not engage in any activities which would injure the firm, deprive it of profit
or deprive it of the advantage of employees' skills and abilities. Must consider the effects
of their actions on firm integrity and sustainability.
2. Members are encouraged to give their employer a copy of the code and standards.
3. Independent practice for compensation is allowed if a notification is provided to the
employer fully describing all aspects of the services including compensation, duration and
the nature of the activities and if the employer consents to all terms of the proposed
independent practice before it begins.
4. Whistle Blowing: If an employer is engaged in illegal or unethical activity, it is acceptable
for the employee to violate member's or candidate's duty to his/her employer (such as
contradicting employers instructions, violating certain policies and procedures or
preserving a record by copying employers record) only if the intent is clearly aimed at
protecting clients or the integrity of capital markets.
5. Members must continue to act in their employer's best interests until resignation is
effective. Activities which may constitute a violation:
(a) Misappropriation of trade secrets.
(b) Misuse of confidential information.
(c) Soliciting employer's clients prior to leaving.
(d) Self-dealing.
(e) Misappropriation of clients and client lists.
6. Employers often require employees to sign 'Non-Compete Agreements' that preclude a
departing employee from engaging in certain conduct. This Non-Compete provisions
within employment contracts that essentially require employees refrain from working with
competitors for a certain period of time after the end of the present employment
relationship.
7. Former employer's documents and fees: Except with the consent of their employer,
departing member may not take employees property i.e. books, records, reports and other
materials. Taking any employers' records, even those the member prepared violates the
standards.
8. The standard doesn't prohibit former employees from contacting clients of their previous
firm as long as the contact information does not come from the records of the former
employer. Members are free to use public information after departing, to contact former
clients without violating the standard as long as there is no specific 'Non-Compete
Agreements' not to do so.
9. If an agreement exists among employers (e.g. U.S. Protocol for Broker Recruiting) that
permits brokers to take certain client information when leaving a firm, hence a member
many act within the terms of the agreement without violating the standard. To be
protected, a copy of the information must be provided to the local management team for
review. Additionally, the specific client information may only be used by the departing
employee and not others employed by the firm.
10. When planning to resign, the employee must ensure that their social media use complies
with their employers' policies for notifying clients about employee separations.
B. Additional Compensation Arrangements
1. Members must not accept gifts (monetary and non-monetary), benefits, compensation
(direct and indirect) or consideration that competes with or might reasonably be expected
to create a conflict of interest with their employer's interest unless they obtain written
consent from all parties involved.
10
C. Responsibilities of Supervisors
1. Members must make reasonable efforts to prevent employees from violating laws, rules,
regulations or the code and standards as well as make reasonable efforts to detect
violations.
2. A member faced with no compliance procedures or with procedures he believes is
inadequate must decline supervisory responsibility in writing until adequate procedures
are adopted by the firm.
3. If there is a violation, respond promptly and conduct a thorough investigation while
increasing supervision or placing limitations on the wrongdoer's activities.
Standard V
INVESTMENT ANALYSIS, RECOMMENDATIONS AND ACTIONS
A. Diligence and Reasonable Basis
1. Have a reasonable and adequate basis supported by appropriate research and
investigation for any analysis. Must consider prior to making a recommendation or taking
investment action includes: macro-conditions, firm's history, financial data, ratios,
calculations, quality of assets and level of fees etc.
2. You can rely on others in your firm to determine whether secondary (advisors with your
firm) or third-party research is sound and use the information in good faith (e.g.
Bloomberg) unless you have reason to question its validity or the processes end
procedures used by those responsible for the research. Criteria on whether research is
sound, include:
(a) Assumptions used,
(b) Rigor of the analysis performed,
(c) Evaluation of the objectivity and independence of the recommendation.
3. Members must be able to explain the basic nature of the quantitative research and how it
is used to make investment decisions. Members should consider scenarios outside those
typically used to assess downside risk and the time horizon of the data used for model
evaluation to ensure that both positive and negative cycle results have been considered.
(Using Quantitative Research)
4. The standard requires greater diligence of members who create and understand
quantitative techniques than of those who use techniques developed by others. A member
who has created a quantitative strategy must test it thoroughly including extreme
scenarios with inputs that fall outside the range of historical data before offering it to
clients. (Developing Quantitative Techniques)
5. Members should make sure their firms have standardized criteria for reviewing these
selected external advisers and managers:
(a) Reviewing the adviser's established code of ethics.
(b) Understanding the adviser's compliance and internal controls.
(c) Assessing the quality of the published return information.
(d) Reviewing the adviser's investment process and adherence to its stated strategy.
6. Even if a member doesn't agree with the independence and objective view of the group, he
doesn't necessarily have to decline to be identified with the report as long as there is a
reasonable and adequate basis.
B. Communication with Clients and Prospective Clients
1. Disclose basic format and general principles of the investment process. Must promptly
disclose any changes that might materially effect those processes.
2. Disclose significant limitations and risks associated with investment processes. E.g.
models, liquidity and capacity.
11
3. Use reasonable judgement in identifying factors important to their IRA and please
communicate.
4. Distinguish between fact and opinion in presentation of research reports.
5. There is no obligation to make 'Buy' and 'Sell' recommendations on securities that are
covered in research reports.
C. Record Retention
1. If no other regulatory standards or firm policies are in place, the standard recommends a
7-year minimum holding period of records.
2. A member who changes firm must recreate the analysis documentation supporting her
recommendation using publicly available information or information obtained from the
company and must not rely on memory or materials created at her previous firm.
Standard VI
CONFLICTS OF INTEREST
A. Disclosure of Conflicts
1. Members must fully disclose to clients, prospects and their employers all actual and
potential conflicts of interest in order to protect investors and employers. E.g. broker or
dealer-market making activities, board service, actual ownership of stock in companies
that the member recommends or that client hold, any special compensation
arrangements, bonus programs, commissions and incentives should be disclosed. If
member's firm doesn't permit such disclosure, the member should document the request
and may consider disassociating from the activity.
B. Priority of Transactions
1. Establish Blackout or Restricted Periods: Investment personnel involved in the investment
decision-making process should establish blackout period prior to trade for clients so that
managers cannot take advantage of their knowledge of client activity by 'Front-Running'
(trading for one's personal account before trading for client's accounts) client trades.
C. Referral Fees
1. Members must disclose to their employer, clients and prospective clients as appropriate
any compensation, consideration (includes all fees, whether paid in cash, soft dollars or
in-kind) or benefit received from or paid to, other for the recommendation of products or
services. Such disclosures allow clients or employers to evaluate:
(a) Any partiality shown in any recommendation of services.
(b) The full cost of the services.
Standard VII
RESPONSIBILITIES AS A CFA INSTITUTE MEMBER OR CFA CANDIDATE
A. Conduct as Participants in a CFA Institute Programs
1. Expressing an Opinion: members are free to disagree and express their disagreement with
CFA Institute on its policies, procedures or any advocacy positions taken by the
organization. When expressing a personal opinion, a candidate is prohibited from
disclosing content-specific information including any actual exam question, subject matter
covered or not covered in the exam. E.g. you can express that an exam was tough or easy.
2. Members who volunteer in the CFA Program may not solicit or reveal information about
questions considered or included on a CFA exam about the grading process or about
scoring of questions.
12
B. Reference to CFA Institute, the CFA Designation and the CFA Program
1. Once accepted as CFA Institute Member, the member must satisfy the following
requirements to maintain his/her status:
(a) Remit annually to CFA Institute a CPC Statement.
(b) Pay applicable CFA Institute membership dues on annual basis.
If a CFA Institute Member fails to meet any of the requirements, the individual is no longer
considered an active member.
2. Candidacy in CFA Program: If an individual is registered for CFA Program but declines to sit
for the exam or otherwise, doesn't meet the definition of a candidate as described in CFA
Institute bylaw. Once the person is enrolled to sit for future examination, his/her CFA
Candidacy resumes.
3. CFA candidate must never state or imply that they have partial designation or cite an
expected completion date of any level. If a candidate passes each level of exam in
consecutive years and wants to state that he/she did so, that is not considered a violation,
since its a 'Statement of Fact'.
4. Members must not make promotional promises or guarantee tied to the CFA designation.
Do not:
(a) Over promise individual competence. E.g. superiority.
(b) Over promise investment results in the future. E.g. higher performance, less risk.
5. The CFA logo certification mark must be used only to directly refer to an individual
charterholder or group of charterholders. The appropriate use of the CFA logo is on the
business card or letter head of each individual CFA charterholder. Acceptable: Chartered
Financial Analyst®, CFA® or CFA logo.
13
1.
2.
3.
4.
5.
6.
7.
8.
1
Quantitative Methods
PAGE NOS. 55
VOL. 2
CHAPTER 4
CHAPTERS. 6
Introduction to Linear Regression
Linear Regression a.k.a. Linear Least Squares assumes a linear relationship between the dependent and the
independent variables. Linear Regression computes a straight line that best fits the observations; it chooses
values for the intercept 'b0' and slope 'b1', that minimize the sum of the squared vertical distances between
the observations and the regression line.
Y =b +b X +ε
ACTUAL
1
0
i
i
i
i
= 1, 2, 3 ... n
Whereas,
th
Y : i observation of the dependent variable Y. The dependent variable is also referred to as the
i
'Explained Variable', 'Endogenous Variable' or 'Predicted Variable'.
th
X : i observation of the independent variable X. The independent variable is also referred to as
i
the 'Explanatory Variable', 'Exogenous Variable' or 'Predicting Variable'.
b : Regression Intercept. It is the value of dependent variable, if value of the independent
0
variable is '0'. The intercept term in this regression is called the stock's Ex-post Alpha. It is a
Regression
measure of excess risk-adjusted returns a.k.a. Jensen's L p = R p - [R f + (R m- R f )]
Coefficients
b : Regression Slope Coefficient. It is the change in the dependent variable for a 1-unit change
1
in the independent variable. The slope coefficient in a regression like this is called Stock's
Beta and it measures the relative amount of systematic risk in returns.
th
ε : Residual for the i observation, also referred to as the 'Disturbance Term', 'Error Term' or
i
'Unexplained Deviation'. It represents the portion of the dependent variable that cannot be
explained by the independent variable.
^
^
^
Y =b +b X
PREDICTED
i
0
1 i
i = 1, 2, 3 ... n
I
I
I
Whereas,
^
Y : Estimated (Fitted Parameters) value of Yi given X i
^
^
^
b : Estimated Intercept. b0 = Y - b1 X ; The intercept equation highlights the fact that the
0
regression line passes through a point with coordinates equal to the mean of the
independent and dependent variables.
2
^
^
b : Estimated Slope Coefficient. b1 = Cov XY / o X
1
The sum of the squared vertical distances between the estimated and actual Y-values is referred to as the
'Sum of Squared Errors' (SSE). Thus, regression line is the line that minimizes the SSE. This explains why
simple linear regression is frequently referred to as 'Ordinary Least Squares' (OLS) regression and the values
^
estimated by the estimated regression equation Yi is called Least Squares estimates. Here are some
assumptions below:
2
Assumption 1: A linear relationship exists between the dependent and the independent variable. This
requirement means that b and b are raised to the first power only and that neither b nor b is multiplied or
divided by another regression parameter (as in b /b ). The requirement doesn't exclude X from being raised
to a power other than 1. If the relationship between the independent and dependent variables is non-linear
in the parameters, then estimating that relation with a linear regression model will produce invalid results.
b x
Y = b e 1 i+ ε
0
i
Non-Linear (Not Allowed)
i
2
Y =b +b X +ε
i
0
1
i
i
x
Linear (Allowed)
Assumption 2: The independent variable X is not random. If the independent variable is random, we can still
rely on the results of regression models given the crucial assumption that the error term is uncorrelated with
the independent variable.
Assumption 3: The expected value of the error term is 0; E(ε) = 0.
2
2
I
Assumption 4: The variance of the error term is the same for all observations; E(ε i ) = o ε . It is also known as
the 'Homoscedasticity' assumption i.e. violation.
Assumption 5: The error term is independently distributed, that the error for one observation is not
correlated with that of another observation. Related to 'Serial Correlation' i.e. violation.
Assumption 6: The error term is normally distributed. If the regression errors are not normally distributed,
we can still use regression analysis. Econometricians who dispense with the normality assumption use ChiSquare tests of hypothesis rather than F-tests.
An unbiased forecast can be expressed as E(Actual Change - Predicted Change) = 0. If the forecasts are
unbiased, the intercept 'b 0' should be 0 and the slope 'b 1' should be 1, then the error term
[Actual Change - b 0- b1 (Predicted Change)] will have an expected value of 0, as required by Assumption 3 of
the linear regression model.
lllll
ANOVA
Analysis of Variance (ANOVA) is a statistical procedure for analyzing the total variability of the
dependent variable. In regression analysis, we use ANOVA to determine the usefulness of the
independent variable or variables in explaining variation in the dependent variable.
1. Total Sum of Squares (TSS): measures the total variation in the dependent variable.
n
I
TSS = Σ (Yi - Y)
2
i=1
2. Regression Sum of Squares (RSS): measures the explained variation in the dependent variable.
n
^
I
RSS = Σ (Yi - Y)
2
i=1
3. Sum of Squared Errors (SSE): measures the unexplained variation in the dependent variable. It is also
known as the Sum of Squared Residuals or the Residual Sum of Squares.
n
^ 2
SSE = Σ (Yi - Yi )
i=1
∴
Total Variation = Explained Variation + Unexplained Variation
(TSS)
(RSS)
(SSE)
3
Y
Yi
.
^
^
^
Y i = b0 + b1 X i
(Y - Y ) = SSE
.
..
.
. .
.
.
.
.
I
Y
^
b0
i
i
(Y i - Y) = TSS
I
^
.
.
^
(Y - Y) = RSS
I
Fig 1: Components of
Total Variation
i
. Actual
Fig 2: ANOVA Table
Source of Variation
Predicted
I
X
df
ss
ss / df
Degrees of
Freedom
Sum of
Squares
Mean Sum of Squares
k
RSS
Regression
(Explained)
MSR = RSS
k
(Mean Regression
Sum of Squares)
Error
(Unexplained)
n-k-1
SSE
MSE = SSE
n-k-1
(Mean Squared
Error)
TOTAL
n-1
TSS
n: no. of observations
k: no. of independent variables
lllll
Variations Explained
lllll
T-Statistic
T-value measures the size of the difference relative to the variation in your sample data. When
T-Stat is very small it indicates that none of the autocorrelations are significantly different than 0.
^
t = b1 - b1
s ^b
df = n - k - 1
1
Fig 3: Null and Alternative Hypothesis
a.
H0 : μ = 0
IIlll
Ha : μ ≠ 0
accept
reject
lllII
reject
you don't believe in (wants to reject)
{ HH :: What
Something you believe in (wants to accept)
0
1
b.
H0 : μ ≤ 0
Ha : μ > 0
accept
lllII
reject
c.
H0 : μ ≥ 0
IIlll
Ha : μ < 0
accept
reject
4
ρ-value: is the smallest level of significance for which the null hypothesis can be rejected.
ρ < L : Reject H
0
ρ > L : Accept H0
Example 1:
The estimated slope coefficient of the ABC plc. is 0.64 with standard error equal to 0.26. Assuming that the
sample has 36 observations, determine if the estimated slope coefficient is significantly different than 0 at a
5% level of significance.
H0 : b 1 = 0
Reject H 0
Ha : b1 ≠ 0
^
IIlll
T-test = b1 - b1 = 0.64 - 0 = 2.46
s ^b
0.26
2.03
1
.
lllII
-2.03 -2.46
The critical two-tailed t-values are ± 2.03 (df = n - k -1 = 36 - 1 - 1 = 34). Because t > t c i.e. 2.46 > 2.03, we
reject the null hypothesis and conclude the slope is different from 0.
If none of the independent variables in a regression model helps explain the dependent variable,
the slope coefficients should all equal '0'. In a multiple regression however, we cannot test the null
hypothesis that all slope coefficients equal '0' based on T-tests that each individual slope coefficient
equals '0', because the individual tests don't account for the effects of interactions among the
independent variables. To test the null hypothesis that all of the slope coefficients in the multiple
regression model are jointly equal to '0' (H 0 : b1 = b2 = ... b k = 0) against the alternative hypothesis
that at least one slope coefficient is not equal to '0'; we must use an F-test. The F-test is viewed as a
test of the regression's overall significant.
lllll
F-Statistic
F-test assesses how well the set of independent variables as a group explains the variation in the
dependent variable. That is, the F-Stat is used to test whether at least one of the independent
variables explains a significant portion of the variation of the dependent variable. This is always a
one-tailed test, despite the fact that it looks like it should be a two-tailed test because there is an
equal sign in the null hypothesis. If the regression model does a good job of explaining variation in
the dependent variable, then this ratio should be high.
F = MSR =
RSS / k
MSE = SSE / n - k - 1
df
=k
denominator
df numerator = n - k - 1
For simple linear regression, F = t2b is true
1
H0 : b1 = 0
Ha : b1 ≠ 0
For multiple regression, F = t2b is not true
1
H0 : b1 = b2 = b3 = b4 = 0
Ha : At least one b i ≠ 0
Example 2:
An analyst runs a regression of monthly value stock returns on five independent variables over 60 months.
The TSS is 460 and the SSE is 170. Test the null hypothesis at the 5% significance level that all five
independent variables are equal to 0.
5
H0 : b1 = b2 = b3 = b4 = b5 = 0
Ha : At least one b i ≠ 0
Reject H
0
F-test = MSR = 58 = 18.41
MSE 3.15
llIII
0
.
2.4 18.41
The critical F-value for 5 and 54 degrees of freedom at a 5% significance level is significantly 2.4. Therefore,
we can reject the null hypothesis and conclude that at least one of the five independent variables is
significantly different than 0.
Standard Error of Estimate
The Standard Error of Estimate (SEE) measures the degree of variability of the actual Y-values
relative to the estimates Y-values from a regression equation. The SEE gauges the 'fit' of the
regression line. The smaller the standard error, the better fit. The SEE is the standard deviation of
the error terms in the regression. As such, SEE is also referred to as the Standard Error of the
Residual or Standard Error of the Regression. In regressions, the relationship between the
independent and dependent variables can be strong or weak; relative to total variability, SEE will be
low if the relationship is very strong and high if the relationship is weak. Any violation of an
assumption that affects the error term will ultimately affect the coefficient standard error, cause
coefficient standard error is calculated using SEE.
SEE = MSE =
lllll
2
SSE =
n-k-1
]
lllll
n
2
Σ (ε^ i )
n-k-1
1/2
]
2
Coefficient of Determination: R and Adjusted: R a
2
1. R : It is defined as the percentage of the total variation in the dependent variable explained by
2
the independent variable. R explains the correlation between predicted and actual values of
2
dependent variable. For example, an R of 0.63 indicates that the variation of the independent
variable explains 63% of the variation in the dependent variable.
2
R = TSS - SSE = RSS = 1 - SSE
TSS
TSS
TSS
↑Higher the better; k↑ R2↑
Regression output often includes multiple R (correlation coefficient), which is the correlation
between actual values of Y and forecasted values of Y. (Multiple R is the square root of R2 ).
2
For simple linear regression (i.e. one independent variable), the coefficient of determination R
2
may be computed by simply squaring the correlation coefficient 'r' i.e. R = r 2. This approach is
not appropriate when more than one independent variable is used in the regression.
2
2
For multiple regression, R by itself may not be reliable, this is because R almost always increases
as variables are added to the model, even if the marginal contribution of the new variables is not
statistically significant (if we add regression variables to the model, the amount of unexplained
variation will decrease and RSS will increase, if the new independent variable explains any of the
unexplained variation in the model. Such a reduction occurs when the new independent variable
is even slightly correlated with the dependent variable and is not a linear combination of other
2
independent variables in the regression). Consequently, a relatively high R may reflect the impact
of a large set of independent variables rather than how well the set explains the dependent
variable. This problem is often referred to as overestimating the regression.
2
2
2. Ra : Some financial analysts use an alternative measure of goodness of fit called Ra . To overcome
6
the problem of overestimating the impact of additional variables on the explanatory power of a
2
regression model, many researchers recommend adjusting R for the number of independent
variables 'k'.
2
]
n - 1 x (1 - R )
n-k-1
↑ 2↓
%
R
2
2
Ra
0
]
2
Ra = 1 -
k*
k
As k overly R a ; should not add independent variables beyond
2
k*. When a new independent variable is added, Ra can decrease if
adding that variable results in only a small increase in R 2 . When
2
2
2
k ≥ 1, then R ≥ R a. In fact R a can be negative (effectively consider
its value 0), although R2 is always non-negative. In addition, R2a may
be less than 0, if R2 is low enough. Furthermore, we must be aware
2
that a high R a doesn't necessarily indicate that the regression is
well specified in the sense of including the correct set of variables.
One reason for caution is that a high R 2a may reflect peculiarities
i.e. weirdness of the dataset used to estimate the regression.
Goodness of 'fit' (SEE) and Coefficient of Determination are different values for the same concept.
The Coefficient of Variation is not directly part of regression model.
Example 3:
Part a: An analyst runs a regression of monthly value stock returns on five independent variables over 60
2
2
months. The TSS is 460 and the SSE is 170. Calculate the R and Ra .
2
Part b: Suppose the analyst now adds four more independent variables to the regression and the R
increases to 65%. Identify which model the analyst should most likely prefer.
Ra = 1 -
2
2
Ra = 1 -
2
b. R = 65% (given)
]
]
2
a. R = 460 - 170 = 0.63 or 63%
460
60 - 1 x (1 - 0.63)
60 - 5 - 1
]
]
60 - 1 x (1 - 0.65)
60 - 9 - 1
2
2
= 59.6%
= 58.7%
With nine independent variables, even though the R has increased from 63% to 65%, the R a has decreased
2
from 59.6% to 58.7%. The analyst would prefer the first model because the Ra is higher and the model has
five independent variables as opposed to nine.
Confidence Intervals
Estimated Regression Coefficient ± (t c ) . (Coefficient Standard Error)
SEE
SEE
↑
↓
Standard Error
Standard Error
CI widens
CI tightens
tc
tc
↑
↓
Standard Error of Slope Coefficient
⤵
^
↑
↓
b1± (t c x S ^b )
1
^
Y ± (t c x S f )
⤴
lllll
Standard Error of Forecast
7
Whereas,
2
2
2
S f : SEE 1 + 1 + (X - X)
2
n (n - 1) Sx
]
I
]
2
; S x is variance of the independent variable.
2
SY = TSS is variance dependent variable.
n-1
Rule: H0 : b1 = 0 and Ha : b1 ≠ 0. If the confidence interval (CI) at the desired level of significance doesn't
include 0, the H0 is rejected and the coefficient is said to be statistically different from 0.
Example 4:
Coldplay forecasts the excess return on the S&P 500 for June 2017 to be 5% and the 95% CI for the predicted
value of the excess return on VIGRX for June 2017 to be 3.9% to 7.7% (b 0 : 0.0023 and b 1 : 1.1163). The
standard error of the forecast is closest to?
^
^
^
Y=b +b X
0
1
Y = 0.0023 + 1.1163 (0.05) = 0.058115
^
0.039 = 0.058115 + 2.03 (S f )
to
0.058115 - 2.03 (S f ) = 0.077
∴ S = 0.0093
f
NOTES
1. The distinction between SSE and SEE: SSE is the sum of the squared residuals, while SEE is the standard
deviation of the residuals.
2. Limitations of regression analysis:
(a) Linear relationships can change overtime. This is referred to as 'Parameter Instability'.
(b) Public knowledge of regression relationship may negate their future usefulness.
(c) If the assumptions underlying regression analysis don't hold, the interpretation and tests of hypothesis
may not be valid.
^
^
3. Prediction must be based on the parameters' estimated values (b0 and b1 ) in relation to the hypothesized
population values.
I
I
4. Cov = Σ (X i - X) (Yi - Y)
XY
n-1
I
I
Cov XY= Σ P (X i - X) (Yi - Y)
Past Sample
Future Sample
The covariance between two random variables is a statistical measure of the degree to which the two
variables move together. The covariance captures the linear relationship between two variables.
I
I
5. rXY= Cov XY
ox oY
The correlation coefficient is a measure of the strength of the linear relationship between two variables.
6. Regression analysis uses two principle types of data:
(a) Cross-Sectional: data involve many observations on X and Y for the same time period.
(b) Time Series: data use many observations from different time periods for the same company or country.
A mix of time series and cross sectional data is also known as 'Panel Data'.
8
CHAPTER 5
Multiple Regression
Multiple Regression is regression analysis with more than one independent variable. It is used to quantify the
influence of two or more independent variables on a dependent variable.
ACTUAL
Yi = b0 + b1 X 1i + b2 X2 i+ ..... + bk X k i+ ε
i
^
^
^
^
i
= 1, 2, 3 ... k
^
Yi = b0 + b1 X 1i + b2 X2 i+ ..... + bk X k i
PREDICTED
Assumptions for multiple regression are almost exactly the same as those for the single variable linear
regression model, except assumption 2 and 3. Here are some changes under multiple regression model:
Assumption 2: The independent variables (X 1 , X2 , ... , X k) are not random. Also, no exact linear elation exists
between two or more of the independent variables. If this part of assumption 2 is violated, then we cannot
compute linear regression, may encounter problems if two or more of the independent variables or
combinations thereof are highly correlated. Such a high correlation is known as 'Multicollinearity' i.e. violation
Assumption 3: The expected value of the error term, conditional on the independent variables is 0.
E ( ε| X , X , ... , X ) = 0.
1
lllll
2
k
Qualitative Factors
Qualitative Independent Variable i.e. Dummy Variables capture the effect of binary independent
variables. Whereas, Qualitative Dependent Variables (Categorical Dependent Variables) require
methods other than OLS i.e. Probit, Logit or Discriminant Analysis.
lllll
Dummy Variables
There are occasions when the independent variable is binary in nature, it is either 'on' or 'off'. One
type of qualitative variable is called a Dummy Variable, takes on a value of 1 if a particular condition
is true and 0 if that condition is false. Not all qualitative variables are simply dummy variables. For
example, in a trinomial choice model i.e. a model with three choices; a qualitative variable might
have value of 0, 1 or 2. Whenever we want to distinguish between n classes, we must use n -1
dummy variables to avoid 'Multicollinearity'. Otherwise, the regression assumption of no exact
linear relationship between independent variables would be violated.
Consider the following regression equation for explaining quarterly EPS in terms of the quarter of
their occurrence:
EPS = b + b Q + b Q + b Q + ε
t
0
1
1t
2
2t
3
3t
t
df = n - 1
Whereas,
EPS t : Quarterly observation of Earning Per Share.
Q1t : 1 if period t is the first quarter, Q1t : 0 otherwise.
Q2t : 1 if period t is the second quarter, Q2t : 0 otherwise.
Q3t : 1 if period t is the third quarter, Q 3t : 0 otherwise.
b0 : Average value of EPS for the fourth quarter.
b1 , b2 , b3 : Estimate the difference in EPS on average between the respective quarter
(i.e. quarter 1, 2 or 3) and the omitted quarter (the fourth quarter in this case).
Think of the omitted class as the reference point.
9
Example 1:
Some developing nations are hesitant to open their equity markets to foreign investments because they fear
that rapid inflows and outflows of foreign funds will increase volatility. You want to test whether the volatility
of returns of stocks traded on the Bombay Stock Exchange (BSE) increased after July 1993, when foreign
institutional investors were first allowed to invest in India. Your dependent variable is a measure of return
volatility of stocks traded on the BSE; your independent variable is a dummy variable that is coded 1 if foreign
investment was allowed during the month and 0 otherwise.
Coefficient
Standard Error
T-test
0.0133
-0.0075
0.002
-0.0027
6.5351
-2.7604
Intercept
Dummy
n = 95
a. State null and alternative hypothesis for the slope coefficient of the dummy variables that are consistent
with testing your stated belief about the effect of opening the equity markets on stock return volatility.
H 0: b 1≥ 0
Ha : b 1 < 0
b. Determine whether you can reject the null hypothesis at the 5% significance level in a one-sided test of
significance.
Reject H
0
.
df = n - k - 1 = 95 - 1 - 1 = 93
IIII
-2.7604 -1.661
We reject the null hypothesis because the dummy variable takes on a value of 1 when foreign investment is
allowed, we can conclude that the volatility was lower with foreign investment.
c. According to the estimated regression equation, what is the level of return volatility before and after the
market-opening event?
Before: Y = 0.0133 - 0.0075 (0) = 0.0133
After: Y = 0.0133 - 0.0075 (1) = 0.0058
lllll
Probit Model
Logit Model
0
= f (x) = Φ (1 + e-x)
= f (x) =
Y = In
1
(1 + e-x)
Event Happens
⤵
-1
Y = Φ (P)
Y = f (b0 + b 1 X1 + b 2X2 ) + ε
P
(1 - P)
⤴
Y = f (b0 + b1 X 1+ b 2 X 2 ) + ε
]
0
lllll
]
Event Doesn't Happen
10
Probit Model i.e. Probit regression, which is
based on the normal distribution, estimates
the probability that Y = 1 (a condition is
fulfilled) given the values of the
independent variables.
Logit Model i.e. Logistic regression is based
on the logistic distribution also called a log
adds ratio. Logistic regression is widely
used in machine learning, where the
objective is classification. Logistic
regression assumes a logistic distribution
for the error term; this distribution is
similar in shape to the normal distribution
but has heavier tails.
In most cases it makes no difference which one is used (probit or logit). Both functions increase
relatively quickly at x = 0 and relatively slowly at extreme values of x. Both functions lie between 0 and 1.
In econometrics, probit and logit models are traditionally viewed as models suitable when the
dependent variable is not fully observed.
lllll
Discriminant Model
Discriminant Analysis yields a linear function similar to a regression equation, which can then be
used to create an overall score. Based on the score, an observation can be classified into bankrupt
or not bankrupt category. Discriminant model makes use of financial ratios as the independent
variables to predict the qualitative dependent variable bankruptcy.
Examine the individual coefficients using T-tests, determine the validity of the model with the F-test, the R2
and look out for Heteroskedasticity, Serial Correlation and Multicollinearity.
Fig 3: Violations of Assumptions
A. Heteroskedasticity
B. Serial Correlation
Heteroskedasticity occurs when
the variance of the residuals is
not the same across all
observations in the sample.
This happens when there are
sub-samples that are more
spread out than the rest of the
sample.
I
(a)
Homoskedastic
2
[Var. (ε i) = o ]
x
.
.... .
.. . ....
.. ... .. .
. . .. .
.. .
High residual
x
Y
No relationship
between, value of
independent
variables and
regression
residuals
(b)
Heteroskedastic
[With errors]
value
. .. .
..... .. .
. .. . . . .
.. . .. .. . . .
.
...... .. Low residual
value
Y
On average,
regression
residuals grow
larger as the size
of the independent
variable increases
Serial Correlation a.k.a.
Autocorrelation refers to the
situation in which the residual
terms or regression errors are
correlated with one another. It
is a relatively common problem
with time series data. Any effect
of serial correlation appears
only in the regression
coefficient standard errors. If
one of the independent
variables is a lagged value of
the dependent variable, then
serial correlation in the error
term will cause all the
parameter estimates from
linear regression to be
inconsistent and they will not
be valid estimates of the true
parameter.
C. Multicollinearity
When one of the independent
variables is an exact linear
combination of other
independent variables, it
becomes mechanically
impossible to estimate the
regression. That case, known as
'Perfect Collinearity' is much
less of a practical concern than
multicollinearity.
Multicollinearity occurs when
two or more independent
variables (or combinations of
independent variables) are
highly (but not perfectly)
correlated with each other.
Multicollinearity if a serious
practical concern because
approximate linear
relationships among financial
variables are common.
11
I
(ii)
Conditional
[Var. (ε i /x) ≠
const.]
Unconditional
Heteroskedasticity
occurs when the
heteroskedasticity
is not related to
the level of
independent
variables; which
means that it
doesn't
systematically
or
with changes
↓
The type of
heteroskedasticity
that causes the
most problems for
statistical inference
is Conditional
Heteroskedasticity.
It is the error
variance that is
↑
correlated with
(conditional on)
the values of the
independent
variables in the
regression. It
identifies nonconstant volatility
related to prior
period's volatility.
Conditional
heteroskedasticity
is not predictable
by nature.
in the value of the
independent
variables. While
this is a violation of
the equal variance
assumption, it
usually causes no
major problems
with the
regression. It
refers to general
structural changes
in volatility that are
not related to prior
period's volatility,
Unconditional
heteroskedasticity
is predictable and
can relate to
variables that are
cyclical by nature.
^
T-test = b i = Reliable
(unreliable) S ^
Unreliable
b
i
(Type II Error)
(a) Overestimated S^b i :
T-test
(b) Underestimated S^ :
T-test
b
F-test = MSR
MSE
(unreliable)
i
↓
↑
(a)
Positive Sc.
x
-
(a)
Negative Sc.
x
+
. . +.
. +.
+
+
...
. .
(i)
Unconditional
[Var. (εi ) ≠ o 2 ]
-
-
-
. .+ +.
.- .-
+
Time
Y
Positive Sc. is serial
correlation in
which a positive
error for one
observation
increases the
chance of positive
error for another
observation. It also
means that a
negative error for
one observation
increases the
chance of a
1. Examining scatter plots of
the residuals.
Y
Negative Sc. occurs
when a positive
error in one period
increases the
probability of
observing a
negative error in
the next period and
vice versa.
negative error for
another
observation. First
Order Sc. means
the sign of the
error term tends to
persist from one
period to the next.
Positive Sc.
^
T-test = bi = Reliable
(unreliable) S^
Unreliable
bi
(b) Underestimated Sb^ :
i
(Type I Error)
MSE is a biased
estimate of the true
population variance
Time
F-test = MSR
MSE
↑
↑ T-test
When one of the independent
variables is an exact linear
combination of other
independent variables, it
becomes mechanically
impossible to estimate the
regression. That case, known as
'Perfect Collinearity' is much
less of a practical concern than
multicollinearity.
Multicollinearity occurs when
two or more independent
variables (or combinations of
independent variables) are
highly (but not perfectly)
correlated with each other.
Multicollinearity if a serious
practical concern because
approximate linear
relationships among financial
variables are common.
^
T-test = b i = Unreliable
(unreliable) S ^
Unreliable
b
i
(Type II Error)
(a) Overestimated Sb^ :
i
↓ T-test
(Type I Error)
MSE will be
underestimated,
F-test (Type I Error)
Negative Sc.
OLS standard errors need not
be underestimates of actual
standard errors, if negative Sc.
is present in the regression.
1. Durbin-Watson: is used to
detect the presence of serial
correlation.
Type I Error: Reject H 0
when it is true.
Type II Error: Fail to Reject H 0
when it is false.
1. The most common way to
detect multicollinearity is the
situation where T-tests
12
df = k
2
R from a second
regression of the
squared residuals from
the first regression on
the independent
variables.
H0 : No Conditional Heteroskedasticity
H1 : Conditional Heteroskedasticity
Conditional
Heteroskedasticity is only a
2
problem if the R and the BP
Test statistic are too large.
^
2
df = k
^
^
^
= [Var (ε t ) - 2 Cov (εt, ε t-1 )
^
^
+Var (ε t-1 )] / Var (ε t )
I
I
If the variance of the error is constant
through time, then we expect
2
Var (ε^t ) = o^ ε for all 't', where we use o^ 2ε
to represent the estimate of the constant
error variance. If in addition, the errors
are also not serially correlated, then we
expect Cov (ε^ t , ε^t-1) = 0. In that case DW is
approximately equal to:
^ 2
^ 2
o ε- 0 + o
2
o^ ε
ε
I
(one-tailed test)
2
residuals
^
DW = Σ (ε t - ε t-1 )
Σ (ε^ t2 )
=2
I
2
BP X Test = n x R
(a) Small Sample
I
2. Breusch-Pagan Chi-Square
Test: which calls for the
regression of the squared
residuals on the independent
variables. If conditional
heteroskedasticity is present,
the independent variables
will significantly contribute to
the explanation of the
squared residuals.
Therefore we can test the null hypothesis
that the errors are not serially correlated
by testing whether the DW test differs
significantly from 2
(a) Large Sample
DW ≈ 2 (1 - r)
df = k
'r' correlation coefficient
between residuals from
one period and those
from the previous period.
H0 : No Positive Serial Correlation
H1 : Positive Serial Correlation
Reject H0
Positive Sc.
0
Negative Sc.
Inconclusive
dL
(Lower)
Corrections:
1. Calculate 'Robust Standard
Errors' i.e. 'White Corrected
Standard Errors' or
'Heteroskedasticity Consistent Standard Errors'.
These robust standard errors
Accept H0
x
dU
0
(Upper)
No Autocorrelation
DW ≈ 2 (1 - 0) = 2
(r = 0)
(DW = 2)
Positive Serial Correlation
DW ≈ 2 (1 - 1) = 0
(r > 0)
(DW < 2)
Negative Serial Correlation
DW ≈ 2 (1 - (- 1) = 4
(r < 0)
(DW > 2)
Corrections:
1. Adjust the coefficient
standard errors, using the
'Hansen Method' i.e. 'Newey
& West Method'. These adjust
standard errors upwards
using the Hansen method,
indicate that none of the
individual coefficient is
significantly different than 0,
while the F-test is statistically
significant and the R2 is high.
This suggests that the
variables together explain
much of the variation in the
dependent variable, but the
individual independent
variables don't. The only way
this can happen is when the
independent variables are
highly correlated with each
other. If the absolute value of
the sample correlation
between any two
independent variables in the
regression is greater than
0.7, multicollinearity is a
potential problem. However,
this only works if there are
exactly two independent
variables. If there are more
than two independent
variables, while individual
variables may not be highly
correlated, linear
combinations might lead to
multicollinearity (conflicting
T-test and F-test statistics).
High pairwise correlations
among the independent
variables are not a necessary
condition for multicollinearity
and low pairwise correlations
don't mean that
multicollinearity is not a
problem.
Corrections:
1. The most common method
to correct for
multicollinearity is to omit
one or more of the
correlated independent
variables.
13
are then used to recalculate
the T-statistic using the
original regression
coefficients, if there is an
evidence of
heteroskedasticity.
2. Use 'Generalized Least
Squares', which attempts to
eliminate the
heteroskedasticity by
modifying the original
equation.
which are then used in
hypothesis testing of the
regression coefficients. Only
use the 'Hansen Method' if
serial correlation is a
problem. The 'White
Corrected Standard Errors'
are preferred, if only
heteroskedasticity is a
problem. If both conditions
are present. use the 'Hansen
Method'.
2. It is to explicitly incorporate
the time series nature of the
data e.g. include a seasonal
term (this can be tricky).
a. Heteroskedasticity: variance of error term is constant.
b. Serial Correlation: errors are not serially correlated.
c. Multicollinearity: no exact linear relationship among X's.
Example 2:
A variable is regressed against three other variables X, Y, Z. Which of the following would not be an indication
of multicollinearity? X is closely related to:
A. 3y + 2z
B. 9y - 4z + 3
2
C. y
✓
lllll
Model Specification
Principles of Model Specification:
(a) The model should be grounded in cogent economic reasoning.
(b) The functional form chosen for the variables in the regression should be appropriate given the
nature of the variables.
(c) The model should be parsimonious i.e. accomplishing a lot with little.
(d) The model should be examined for violations of regression assumptions before being accepted.
(e) The model should be tested and be found useful out of sample before being accepted.
There are three broad categories of 'Model Misspecification' or ways in which the regression model can be
specified incorrectly, each with several subcategories:
1. Misspecification of Functional Form
- Important variables are omitted: If the omitted independent variable (X 2) is correlated with the remaining
independent variable (X 1 ), then the error term in the model will be correlated with (X 1) and the estimated
values of the regression coefficients a 0 and a 1 would be biased and inconsistent, so will the standard error
of those coefficients.
Y = b 0+ b 1 X 1+ b 2X 2+ ε
Y = a 0 + a1 X 1+ ε
b 0≠ a 0
14
- Variables should be transformed: Sometimes analysts fail to account for curvature or non-linearity in the
relationship between the dependent variable and one or more of the independent variables, instead
specifying a linear relation among variables. We should also consider whether economic theory suggests a
non-linear relation. We may be able to correct the misspecification by taking the natural logarithm (In) of
the variable we want to represent as a proportional change.
- Data is improperly pooled: Suppose the relationship between returns and the independent variables
during the first three-years is actually different than relationship in the second three-year period i.e.
regression coefficients are different from one period to the next. By pooling the data and estimating one
regression over the entire period, rather estimate two separate regressions over each of the subperiods.
If we have misspecified the model, the predictions of portfolio returns will be misleading.
2. Time Series Misspecification
- Lagged dependent variable is used as an independent variable: If the error term in the regression model
is serially correlated as a result of the lagged dependent variable (which is common in time series
regression), then this model misspecification will result in biased and inconsistent regression estimates
and unreliable hypothesis tests. If lagged dependent variables help explain and not cause serial
correlation in residuals, they are okay!
- A function of the dependent variable is used as an independent variable - 'Forecasting the Past':
Sometimes as a result of the incorrect dating of variables.
- Independent variables are measured with error: Common example being when an independent variable
is measured with error is when we want to use 'Expected Inflation' in our regression but use 'Actual
Inflation' as a proxy.
3. Other Time Series Misspecifications that result in Non-Stationary
Means that a variable's properties such as mean and variance are not constant through time e.g.
consumption and GDP, random walks or exchange rates.
NOTES
1. If expected value of the sample mean is equal to the population mean, the sample mean is therefore an
unbiased estimator of the population mean. A consistent estimator is one for which the accuracy of the
parameter estimate increases as the sample size increases.
2. Predictions in multiple regression model are subject to both parameter estimate uncertainty and
regression model uncertainty.
3. Y = 5 + 4.2 (Beta) - 0.05 (Alpha) + ε. One unit increase in Beta risk is associated with a 4.2% increase in
return, while a $1 bn. increase in Alpha implies a 0.05% decrease in return.
15
CHAPTER 6
Time Series Analysis
Time Series is a set of observations on a variable's outcomes in different time periods e.g. quarterly sales for
a particular company during the past 5 years.
Fig 1: Linear Vs. Log-Linear Trend Models
Linear Trend Model
Yt
Log-Linear Trend Model
Linear TM
In (Y t )
Yt
Raw Data
0
Time
A Linear Trend is a time series pattern that can
be graphed using a straight line. A downward
sloping line indicates a negative trend, while an
upward sloping line indicates a positive trend.
↱Trend Coefficient
Yt = b 0 + b1 (t) + ε t
^
^
t = 1, 2, ... T
^
Yt = b0 + b1 (t)
When the variable increases over time by a
constant amount, a Linear Trend Model is most
appropriate.
0
.. . .
.
Transformed
..
Data
..
.
.
.
... . . . .
.
. . . . ..
Log-Linear TM
In (Y t )
Time
A Log-Linear Trend works well in fitting time
series that have exponential growth. Positive
exponential growth means that the time series
tend to increase at some constant rate of growth
i.e. the observations will form a convex curve ().
Negative exponential growth means that the
data tends to decrease at some constant rate of
decay i.e. the plotted time series will be concave
curve )(.
b0 + b1 (t) + ε t
Y =e
t = 1, 2, ... T
In (Yt ) = In (eb0 + b1 (t) )
Yt = e In (Yt )
^
Yt = e
^
In (Yt )
When a variable grows at a constant rate, a LogLinear Trend Model is most appropriate.
If on the other hand, the data plots with a non-linear curved shape, then the residuals from a Linear Trend
Model will be persistently positive or negative for a period of time. In this case, the Log-Linear Trend Model
may be more suitable. In other words, when the residuals from a Linear Trend Model are serially correlated, a
Log-Linear Trend Model maybe more appropriate. However, it may be the case that even a Log-Linear Trend
Model is not appropriate in the presence of serial correlation. In this case, we will want to turn to an
Autoregressive Model.
If Linear Model exhibits Autocorrelation, try Log-Linear
If Log-Linear Model exhibits Autocorrelation, try Autoregression
16
Fig 2: Covariance Stationary Vs. Non-Stationary
Stationary
Non-Stationary
A time series is Covariance Stationary if its mean,
variances and covariances with lagged and
leading values don't change over time.
Data points are often Non-Stationary or have
means, variances and covariances that change
over time.
E.g. Strict Stationary, Second-Order Stationary
(Weak-Stationary), Trend Stationary and
Difference Stationary Models.
E.g. Trends, Cycles, Random Walks or
combinations of three.
In order to receive consistent-reliable results the non-stationary data needs to be transformed into stationary
data.
Time series is Covariance Stationary if it satisfies the following three conditions:
1. Constant and Finite Expected Value: The expected value of the time series is constant over time. E(y t ) = μ
and |μ| < ∞, t = 1, 2, 3, ... T. We refer to this value as the 'Mean Reverting Level'. All covariance stationary
AR(1) time series have a finite mean reverting level, when the absolute value of the lag coefficient is less
than 1 i.e. |b 1|< 1.
↘
↗
(a) If time series
(b) If time series
(c) If time series —
AR (1): x t = b 0 + b1 x t-1
Current Value > Mean; Current Value above Mean
As per (c): x t = b 0 + b1 x t
Current Value < Mean; Current value below Mean
^
Current Value = Mean; Next value of the time series will equal its current value x t = x t-1
Decline: x t > x t+1 ; x t > b0
(1 - b 1 )
Rise: x t < x t+1 ; x t < b 0
(1 - b1 )
Same: x t = x t+1 ; x t = b0
(1 - b1 )
2. Constant and Finite Variance: The time series' volatility around its mean doesn't change over time.
3. Constant and Finite Covariance between values at any given lag: The covariance of the time series with
itself for a fixed no. of periods in the past or future must be constant and finite in all periods.
Both 2 and 3: The covariance of a random variable with itself is its variance
Covariance (yt , y t ) = Var (y t )
Stationary in the past doesn't guarantee stationary in the future. There is always the possibility that a well
specified model will fail with the state of change in time. Models estimated with shorter time series are
usually more stable than those with longer time series because a longer sample period increases the chance
that the underlying economic process has changed. Thus, there is a trade off between the increased
statistical reliability when using longer time periods and the increased stability of the estimates when using
shorter periods.
17
lllll
Autoregressive Model
When the dependent variable is regressed against one or more lagged values of itself, the resultant
model is called as an Autoregressive Model (AR).
BACKWARD
↷
To calculate
the successive
forecasts
FORWARD
referred to as
the 'Chain Rule
of Forecasting'
AR (1)
AR (2)
AR (p)
x t = b 0+ b 1x t-1+ ε t
No. of lagged values include
x t = b 0 + b 1x t-1+ b 2 x t-2+ ε t
↱as independent variable
x t = b 0+ b 1 x t-1+ b2 x t-2+ ... + bp x t-p+ ε t
↳Time
AR (1)
x t+1 = b0 + b1 x t
^
^
x^ t+2 = b 0 + b 1 x t+1
^
^
t = 1. 2, ... T
^
This implies that multi-period forecasts are more uncertain than single period forecasts.
Testing Autoregressive Model (AR)
Autocorrelation [Stationary]
If the residuals have significant autocorrelation, the AR model that produced the residuals is not the
best model for the time series being analyzed. We can estimate an AR model using ordinary least
squares (OLS) if the time series is covariance stationary and the errors are uncorrelated.
Unfortunately, our previous DW test statistic is invalid when the independent variables include past
values of the dependent variable. Residual autocorrelations drag to '0' as the no. of lags increases.
Step 1: Start with AR (1): x t = b0 + b 1x t-1+ ε t
Step 2: Calculate the autocorrelations of the model's residuals i.e. the level of correlation between
the forecast errors from one period to the next.
When k = 1
I
I
ρ k = Cov (x t , x t-1) = E [(x t - μ) (x t-k - μ)]
o 2x
o 2x
Where 'E' stands for the expected value. Note that we have the relationship
2
Cov (x t , xt-k ) ≤ o x with equality holding when k = 0. This means that the absolute value of
ρ k ≤ 1.
I
I
I
I
ρ k = Σ [(xt - x) (x t-k - x)]
t=k+1
Σ (x t - x) 2
t=1
Whenever we refer to autocorrelation without qualification, we mean autocorrelation of the
time series itself rather than autocorrelation of the error term.
I
I
ρε,k = Cov (ε t , εt-k ) = E [(ε t - 0) (ε t-k- 0)] = E (ε t . εt-k )
o 2ε
o 2ε
o 2ε
I
lllll
Step 3: Test whether the autocorrelations are significantly different from 'o': If the model is
correctly specified, none of the autocorrelations will be statistically significant. To test for
significance, a T-test is used to test the hypothesis that the correlations of residuals are 0.
t = ρ εt , εt-k
1/ n
H 0 : |Autocorrelations| = 0
→ Autocorrelation
→ Standard Error
H1 : |Autocorrelations| > 0
df = n - k -1
18
lllll
Unit Root: Dickey and Fuller
Remember, if an AR (1) model has a coefficient of 1, it has a 'unit root' and no finite mean reverting
level i.e. it is not covariance stationary. By definition, all Random Walks with or without a drift term
have unit roots. Dickey and Fuller (DF) transform the AR (1) model to run a simple regression.
x t = b 0+ b1 x t-1+ ε
x t - x t-1= b 0 + b1 x t-1- x t-1+ ε
x t - x t-1= b 0+ (b 1 - 1) x t-1+ ε
↷
AR (1)
Substract x t-1from both sides
Rather than directly testing whether the original coefficient is different from 1, they test whether
the new transformed coefficient (b1 - 1) is different from 0 using a modified T-test. In their actual
test, Dickey and Fuller use the variable 'g' = b 1 - 1.
;Time series has Unit Root
Not Stationary
H1 : g < 0
;Time series doesn't have a Unit Root
Stationary
Autoregressive Conditional Heteroskedasticity
Autoregressive Conditional Heteroskedasticity (ARCH) exists if the variance of the residuals in one
period is dependent on the variance of the residuals in a previous period. At times, however, this
assumption is violated and the variance of the error term is not constant. In such a situation, ARMA
models will be incorrect and our hypothesis tests would be invalid.
ARCH (1)
↱ Error Term
ε^t = a 0 + a1 ε^t-1 + μ t
^
^ ^2
2
o^t+1
=a
0+ a 1 ε t
2
2
I
lllll
H0 : g = 0
H0 : a1 = 0
;The variance is constant from period to period
H1 : a1 ≥ or ≤ 0
;The variance increases (decreases) over time i.e. the error terms
exhibit heteroskedasticity
If a time series model has been determined to contain ARCH errors, regression procedures that correct
for heteroskedasticity such as Generalized Least Squares (GLS) must be used in order to develop a
predictive model. Otherwise, the standard error of the model's coefficients will be incorrect, leading to
invalid conclusions. Engle and other researchers have suggested many generalizations of the ARCH (1)
model which include ARCH (p) and generalized autoregressive conditional heteroskedasticity (GARCH)
models. GARCH models are similar to ARMA models of the error variance in a time series. Just like
ARMA models, GARCH models can be finicky and unstable.
In-Sample Forecasts: are within the range of data i.e. time period.
^
Errors: (Yt - Yt )
Out-of-Sample Forecasts: are made outside of the sample period.
Out-of-sample forecast accuracy is important
because the future is always out-of-sample.
Errors: RMSE i.e. Root Mean Squared Error (Square root
of the average of the squared errors). The model
with the smallest RMSE is judged most accurate.
19
Example 1:
To qualify as a covariance stationary process, which of the following doesn't have to be true?
The covariance between any two
A. Covariance (x t , x t-2) = Covariance (x t , xt+2)
observations equal distance apart will be
B. E (x t ) = E (xt+1)
equal e.g. t and t-2 observations with
C. Covariance (x t , x t-1) = Covariance (x t , x t-2)
t and t+2 observations.
→
✓
Example 2:
Suppose the following model describes changes in the unemployment rate:
UER t = - 0.0405 - 0.4674
UER t-1
The current change (first difference) in the unemployment rate is 0.03. Assume that the mean reverting level
for changes in the unemployment rate is -0.0276.
△
△
a. What is the best prediction for the next change? - 0.0405 - 0.4674 (0.03) = - 0.0545 (y
)
t+1
b. What is the prediction of the change following the next change?
Using chain rule of forecasting: - 0.0405 - 0.4674 (-0.0545) = - 0.0150 (yt+2)
c. Explain your answer to Part 'b' in terms of equilibrium?
The answer to Part 'b' is quite close to the mean reverting level of - 0.0276 (-0.0405/0.4674). A stationary
time series may need many periods to return to its equilibrium, mean reverting level.
Example 3:
Table below gives actual sales, log of sales and changes in the log of sales of Cisco Systems for the period
1Q: 2001 to 4Q: 2001.
Quarters / Yr.
Actual Sales
1Q: 2001
2Q: 2001
3Q: 2001
4Q: 2001
1Q: 2002
2Q: 2002
6519
6748
4728
4298
x (4391)
x (4738)
Log of Sales
8.7825
8.8170
8.4613
8.3659
x (8.3872)
x (8.4633)
Actual Values
In (Sales t )
△
△
△
Step 1: Calculate forecast values of
In (Sales t ) with the help of above regression.
1Q 2002:
In Sales 1Q 2002 = 0.0661 + 0.4698 (-0.0954) = 0.0213
2Q 2002:
In Sales 2Q 2002 = 0.0661 +0.4698 (0.0213) = 0.0761
△
△
Step 2: Calculate Log of Sales with the help of Step 1
1Q 2002: In Sales 1Q 2002 = 8.3659 + 0.0213 = 8.3872
2Q 2002: In Sales 2Q 2002 = 8.3872 + 0.0761 = 8.4633
Step 3: Calculate Actual Sales
8.3872
1Q 2002: e
= 4391
2Q 2002: e8.4633 = 4738
△
0.1308
0.0345
- 0.3557
- 0.0954
Forecast the first and second quarter sales of Cisco Systems for 2002 using the regression
In (Sales t ) = 0.0661 + 0.4698 .
In (Sales t-1)
△
Forecast Values
In (Sales t-1)
x (0.0213)
x (0.0761)
20
Example 4:
Table below gives the actual change in the Log of Sales of Cisco Systems from 1Q: 2001 to 4Q: 2001, along
with the forecasts from the regression model
In (Sales t ) = 0.0661 + 0.4698 .
In (Sales t-1) estimated using
data from 3Q: 1991 to 4Q: 2000. Note the observations after the fourth quarter of 2000 are out-of-sample.
△
Date
Actual Values:
1Q: 2001
2Q: 2001
3Q: 2001
4Q: 2001
△
△ In Sales t
Forecast Values:
0.1308
0.0345
- 0.3557
- 0.0954
△ In Sales t
0.1357
0.1299
0.1271
0.1259
a. Calculate the RMSE for the out-of-sample forecast errors.
Error
1Q: 2001 - 0.0049
2Q: 2001 - 0.0954
3Q: 2001 - 0.4828
4Q: 2001 - 0.2213
Squared Error
0
0.0091
0.2331
0.0490
0.2912
Sum
0.0728
Average;
RMSE = 0.0728 = 0.2698
b. Compare the forecasting performance of the model given with that of another model having an
out-of-sample RMSE of 20%
The model with the RMSE of 20% has greater accuracy in forecasting than model in Part a, which has an
RMSE of 27%
Example 5:
Based on the regression output below, the forecasted value of quarterly sales for March 2016 for PoweredUP
is closest to?
AR (1) Regression Output
Coefficient
Intercept
In S t-1- In S t-2
In S t-4- In S t-5
0.0092
- 0.1279
0.7239
Quarterly Sales Data
Bn
Dec 2015 (S t-1)
Sept 2015 (S t-2 )
June 2015 (St-3 )
Mar 2014 (S t-4 )
Dec 2014 (St-5 )
$ 3.868
$ 3.780
$ 3.692
$ 3.836
$ 3.418
The quarterly sales for March 2016 is calculated as follows:
In S t - In St-1 = b 0+ b1 (In S t-1- In St-2 ) + b 2(In St-4 - In S t-5 )
In S t - In 3.868 = 0.0092 - 0.1279 (In 3.868 - In 3.780) + 0.7239 (In 3.836 - In 3.418)
In S t = 1.35274 + 0.0092 - 0.1279 (0.02301) + 0.7239 (0.11538)
In S t = 1.44251
S t = e 1.44251
= 4.231
21
Example 6:
David Brice, CFA has used AR (1) model to forecast the next period's interest rate to be 0.08. The AR (1) has a
positive slope coefficient. If the interest rate is the mean reverting process with an unconditional mean a.k.a
mean reverting level, equal to 0.09, then which of the following could be his forecast for two periods ahead?
Brice makes more distant forecast, each forecast will be closer to the
A. 0.081
unconditional mean, so the two period forecast would be between
B. 0.072
0.08 and 0.09; therefore 0.081 is the only possible outcome.
C. 0.113
✓
lllll
→
Random Walks [Non-Stationary]
Random Walks
Random Walks with a Drift
AR (1): x t = b0 + b1 x t-1+ ε t
Since b0 ≠ 0;
b1 = 1
x t = b0 + x t-1+ ε t
↶
x t = x t-1+ ε t
↶
Since b0 = 0;
b1 = 1
AR (1): x t = b 0 + b1 x t-1+ ε t
(Unit Root)
(Unit Root)
No Mean Reverting Level for Random Walk because,
x t = b0 = 0 = 0 = 0
1 - b1 1- 1 0
Undefined Mean Reverting Level for Random Walk
with Drift because, x t = b0 = b 0
1 - b1 0
I
(i) E (ε t ) = 0: The expected value of each error term is 0.
2
(ii) E (ε 2t ) = o : The variance of the error term is constant.
(iii) E (εi ε j ) = 0; If i ≠ j: There is no serial correlation in the error terms.
For a time series that is not covariance stationary, the least squares regression procedure that we have been
using to estimate an AR (1) model will not work without transforming the data.
b1 < 1: Stationary
b1 = 1: Non-Stationary 'Unit Root'
b1 > 1: 'Explosive Root'
Testing Random Walk Model
lllll
First Differencing
If we believe a time series is a random walk i.e. has a unit root, we can transform the data to a
covariance stationary time series using a procedure called 'First Differencing'. It is because, it
substracts the value of the time series in the first prior period from the current value of the time
series. Note that by taking first differences, you model the change in the value of the dependent
variable.
AR (1)
y t = b0 + b1 y t-1+ ε t
If b0 = b 1 = 0, then yt = ε t and ε t = x t - x t-1
△x
yt = xt - x t-1 = ε t
This transformed time series has a finite mean reverting level of b 0 = 0 = 0 and is therefore
covariance stationary.
1 - b1 1 - 0
22
This is how first differencing
removes the upward trend.
Linear
Trend
After 1st
Differencing
y =x -x
t
t t-1
lllll
Seasonality
Seasonality in a time series is a pattern that tends to repeat from year to year. One example is monthly
sales data for a retailer. Given that sales data normally vary accordingly to the time of year, we might
expect this month's sales (x t ) to be related to sales for the same month last year (x t-12 ). To adjust for
seasonality in an AR model, an additional lag of the dependent variable (corresponding to the same
period in the previous year) is added to the original model as another independent variable. For
example, if quarterly data are used, the seasonal lag is 4; if monthly data is used, the seasonal lag is 12.
Suppose for example, we model a particular quarterly time series using an AR (1) model,
x t = b 0 + b1 x t-1+ ε t . If the time series had significant seasonality, this model would not be correctly
specified. The seasonality would be easy to detect because the seasonal autocorrelation in the case of
quarterly data, the 4th autocorrelation of the error term would differ significantly from 0. Suppose this
quarterly model has significant seasonality. In this case, we might include a seasonal lag in the
autoregressive model and estimate x t = b 0 + b1 x t-1+ b2 xt-2 + ε t , to test whether including the seasonal
lag in the autoregressive model would eliminate statistically significant autocorrelation in the error
term.
lllll
Moving Average Time Series Model
Suppose you are analyzing the long-term trend in the past sales of a company. In order to focus on the
trend, you may find it useful to remove short-term fluctuations or noise by smoothing out the time
series of sales. One technique to smooth out period-to-period fluctuations is by:
n-period moving average = x t + x t-1 + ... + x t-n+1
n
↑
Moving Average
Moving Average always lags large movements in the
actual data i.e. slowly rises and slowly falls. Though often
useful in smoothing out a time series, it may not be the
best predictor of the future. A main reason for this is that
a simple moving average gives equal weight to all the
periods in the moving average.
Example 7:
Suppose we want to compute the four-quarter moving average of AstraZeneca's sales as of the beginning of
the first quarter of 2012. AstraZeneca's sales in the previous four quarters were 1Q 2011: $ 8490 m, 2Q 2011:
$8601 m, 3Q 2011: $ 8405 m and 4Q 2011: $ 8872 m.
23
The 4 Quarter Moving Average = 8490 + 8601 + 8405 + 8872 = $ 8592 m
of 1st Quarter of 2012
4
x t = ε t + θ εt-1
E (εt) = 0
2
2
E (εt) = o
Cov (εt, εs) = E (εt. εs) = 0
t≠s
I
MA (1)
MA (1) i.e. moving average model places different weights on the two terms
in the moving average ( 1 on ε t and θ on εt-1 ). Because the expected value of
x t is 0 in all periods and ε t is uncorrelated with its own part values, the first
autocorrelation is not equal to 0, but the second and higher autocorrelations
are equal to 0. Further analysis shows all autocorrelations except for the first
will be equal to 0 in an MA (1) model. Thus for an MA (1) process, any value x t
is correlated with x t-1 and xt+1 but with no other time series values; we could
say that an MA (1) model has a memory of one period.
x t = εt + θ 1 ε t-1+ ... + θp ε t-q
E (εt) = 0
2
2
E (εt) = o
Cov (εt, εs) = E (εt. εs) = 0
t≠s
I
MA (p)
For an MA (q) model, the first 'q' autocorrelations will be significantly
different from 0 and all autocorrelations beyond that will be equal to 0; an
MA (q) model has a memory of 'q' periods.
x t= μ + ε t
E (εt) = 0
2
2
E (εt) = o
Cov (εt, εs) = E (εt. εs) = 0
t≠s
I
MA (0)
MA (0) time series in which we allow the mean to be 'non-zero' which also
means that the time series is not predictable.
AR
ARMA (p, q)
MA
x t = b 0+ b1 x t-1+ ... + bp xt-p + ε t + θ1 ε t-1 + ... + θ qε t-q
I
E (εt) = 0
E (ε2t) = o 2
'b1 , b2 ... b p' are the autoregressive parameters and 'θ 1 , θ 2 ... θ q' are the
Cov (εt, εs) = E (εt. εs) = 0
moving average parameters. Estimating and using ARMA models have several
t≠s
limitations. First, the parameters in ARMA models can be very unstable.
Second, choosing the right ARMA model is more of an art than a science. The
criteria for deciding on 'p' and 'q' for a particular time series are far from
perfect. Moreover, even after a model is selected, that model may not
forecast well. Thirdly, even some of the strongest advocates of ARMA models
admit that these models should not be used with fewer than 80 observations
and they don't recommend using ARMA models for predicting quarterly sales
or gross margins for a company using even 15 years of quarterly data.
lllll
Cointegration
Occasionally an analyst will run a regression using two time series i.e. time series utilizing two different
variables. For example, using the market model to estimate the equity beta for a stock, an analyst
regresses a time series of the stock's returns (y t ) on a time series of returns for the market (x t ).
Cointegration means that two time series are economically linked (related to the same macro variable)
or follow the same trend and that relationship is not expected to change.
y t = b0 + b 1 x t + ε
whereas,
y t : Value of time series y at time t
xt : Value of time series x at time t
24
If neither data series has a unit root : VALID
If only one data series has a unit root: INVALID
If both data series have unit root & they are cointegrated : VALID
If both data series have unit root & they are not cointegrated : INVALID
Fig 3: Time Series Analysis Summary
25
CHAPTER 7
Machine Learning
Machine Learning (ML) refers to computer programs that learn from their errors and refine predictive models
to improve their predictive accuracy over time. Machine Learning is one method used to extract useful
information from Big Data. An elementary way to think of ML algorithms is to 'find the pattern, apply the
pattern'. ML techniques are better able than statistical approaches to handle problems with many variables
i.e. high dimensionality or with a high degree of non-linearity. ML algorithms are particularly good at
detecting change, even in high non-linear systems because they can detect the preconditions of a model's
break or anticipate the probability of a regime switch. ML is broadly divided into 3 distinct classes of
techniques: (a) Supervised Learnings, (b) Unsupervised Learning and (c) Deep Learning. Machine Learning is
the science of making computers learn and act like humans by feeding data and information without being
explicitly programmed.
The data set is typically divided into three non overlapping samples:
(i) Training Sample: used to train the model. } In-Sample Data
(ii) Validation Sample: for validating and tuning the model.
(iii) Test Sample: for testing the model's ability to predict well on new data.
}
Out-of-Sample Data
To be valid and useful, any supervised machine learning model must generalize well beyond the training data.
The model should retain its explanatory power when tested out-of-sample.
Fig 1: Types of Data Fits
Underfitting
Y
o
▲
o
▲
o
o
o
▲
o
o
o
o
▲
Overfitting
Y
▲
o
▲
o
o
▲
o
▲ ▲
o
x
Underfitting is similar to making a
baggy suit that fits no one. It also
means that the model doesn't
capture the relationships in the
data. The graph shows four
errors in this underfit model (3
misclassified circles and 1
misclassified triangle).
o
▲
▲
o
▲
Y
▲
▲
o
o
Good Fit
▲
o
o
▲
▲
o
o
▲
x
Think of Overfitting as tailoring a
custom suit that fits only one
person. An ML algorithm that fits
the training data too well, will
typically not predict well using
new data. The model begins to
incorporate noise coming from
quirks or spurious correlations; it
mistakes randomness for
patterns and relationships. The
algorithm may have memorized
the data rather than learn from it,
so it has perfect hindsight but no
foresight. The graph shows no
errors in this overfit model. As
models become more complex,
overfitting risk increases.
▲
o
▲
o
o
▲
▲
o
o
▲
▲
▲
▲
▲
x
Robust filling, the desired result
is similar to fashioning a
universal suit that fits all similar
people. A Good Fit or Robust
Model fits the training in-sample
data well and generalize well to
out-of-sample data, both within
acceptable degrees of error. The
graph shows that the good fitting
model has only 1 error, the
misclassified circle.
26
∴ The evaluation of any ML algorithm thus focuses on its prediction error on new data rather than on its
goodness of fit on the data in which the algorithm was fitted i.e. trained.
A Learning Curve plots the accuracy rate = 1 - Error Rate, in the validation or test sample i.e. out-of-sample
against the amount of data in the training sample i.e. in sample, so it is useful for describing under and
overfitting as a function of bias and variance errors. Low or no in-sample error but large out-of-sample error
are indicative of poor generalization. Data scientists decompose the total out-of-sample error into 3 sources:
(i) Bias Error, (ii) Variance Error and (iii) Base Error.
Fig 2: Types of Errors
Bias Error
Variance Error
Bias Error or the degree to which
a model fits the training data.
Algorithms with erroneous
assumptions produce high bias
with poor approximation, causing
underfitting and
in-sample
error. (Adding more training
samples will not improve the
model)
Variance Error or how much the
model's results change in
response to new data from
validation and test samples.
Unstable models pick up noise
and produce high variance
causing overfitting and
outof-sample error.
↑
Accuracy
Rate
Base Error
Base Error due to randomness in
the data. (Out-of-sample
accuracy increases as the
training sample size increases)
↑
Accuracy
Rate
0
Accuracy
Rate
0
No. of Training Samples
0
No. of Training Samples
Linear functions are more
susceptible to bias error and
underfitting.
No. of Training Samples
Non-Linear functions are more
prone to variance error and
overfitting.
Desired Accuracy Rate
Training Accuracy Rate
Validation Accuracy Rate
Fig 3: Fitting Curve shows Trade-off between Bias and Variance Errors and Model Complexity
Model Error
(E-in, E-out)
Optimal
Complexity
Total Error
E-in: In
Sample
Error
E-out: Out
of Sample
Error
'Finding the optimal point (managing overfitting
risk) - the smart spot just before the total error
rate starts to rise due to increasing variance error
- is a core part of the machine learning process
and the key to successful generalization'
↑
Variance
Error
As Complexity
in Training sets, E-in
Bias Error shrinks.
Bias Error
0
Model Complexity
(Out of Sample Error
rates are also a function
of model complexity)
↑
As Complexity
in Test sets, E-out
Variance Error rises.
↓ and
↑ and
27
Preventing Overfitting in Supervised Machine Learning:
1. Ocean's Razor: The problem solving principle that the simplest solution tends to be the correct one. In
supervised ML, it means preventing the algorithm from getting too complex during selection and training
by limiting the no. of features and penalizing algorithms that are too complex or too flexible by
constraining them to include only parameters that reduce out-of-sample error.
2. K-Fold Cross Validation: This strategy comes from the principle of avoiding sampling bias. The challenge is
having a large enough data set to make both training and testing possible on representative samples.
①
A
B
75%
C
D
25%
For example, imagine that this
column represents all of the data
we have collected about people
with or without heart disease.
③
②
Using ML lingo, we used the data to:
(a) Train the ML methods
(b) Test the ML methods
Reusing same data for both training & testing is a bad
idea because we need to know how the method will
work on data it wasn't trained on. A slightly better idea
would be to use the first 75% of data for training & test
25% of the data for testing. But how do we know that
using the first 75% of the data for training and the last
25% of the data for testing is the best way to divide up
the data? Rather than worry too much about which
block would be best for testing, cross validation uses
them all, one at a time and summarizes the results at
the end.
.
0
x
x
x
. ..
x
... ..
.
x
Training (Tr)
x Testing (T)
(ABC Tr, D T) (BCD Tr, A T) (CDA Tr, B T) (DAB Tr, CT)
4 Times = K
Note that K is typically set at 5 or 10; In this case it is 4. This process is then repeated 'K' times, which helps
minimize both bias and variance by insuring that each data point is used in the training set 'K-1' times and
in the validation (or testing) set once. The average of the K validation errors (E-val) is then taken as a
reasonable estimate of the model's out-of-sample error (E-out).
∴
Target Variable or Tag Variable i.e. Dependent Variable (Y-variable)
Feature i.e. Independent Variable (X-variable)
lllll
Tag variable can
be continuous,
categorical or
ordinal.
Supervised Learning
Supervised Learning uses labelled training data to guide the ML program in achieving superior
forecasting accuracy. To forecast earnings manipulators, for example, a large collection of attributes
could be provided for known manipulators and for known non-manipulators. A computer program
could then be used to identify patterns that identify manipulators in another dataset. Typical data
analytics tasks for supervised learning that include prediction (i.e. regression) and classification. When
the Y-variable is continuous, the appropriate approach is that of regression. When the Y-variable is
categorical (i.e. belonging to a category or classification) or ordinal (i.e. ordered or ranked), a
classification is used.
lllll
Penalized Regression
A special case of generalized linear model (GLM) is Penalized Regression. Penalized Regression
models seek to minimize forecasting errors by reducing the problem of overfitting. To reduce the
problem of overfitting, researchers may impose a penalty based on the no. of features used by the
model. Penalized Regression includes a constraint such that the regression coefficients are chosen
28
to minimize the SSE plus a penalty term that increases in size with the no. of included features. The
greater the no. of included features (i.e. variables with non-zero coefficients), the larger the penalty
term. Therefore, penalized regression ensures that a feature is included only if the SSE declines by
more than the penalty term increases. All types of penalty regression involve a trade-off of this
type. Imposing such a penalty can exclude features that are not meaningfully contributing to
out-of-sample prediction accuracy i.e. it makes the model more parsimonious. Therefore, only the
more important features for explaining Y will remain in the penalized regression model.
n
k
2
^
^
LASSO Penalized Regression = Σ (Yi - Y i ) + λ Σ | b k |
i=1
K=1
SSE (OLS)
Penalty Term
λ (Lamda) a.k.a.
hyperparameter, determines
how severe the penalty is
λ > 0. It also determines the
balance between fitting the
model vs. keeping the model
parsimonious.
In one popular type of penalized regression, LASSO (Least Absolute Shrinkage and Selection
Operator) aims to minimize the SSE and the sum of the absolute value of the regression
coefficients. When using LASSO or other penalized regression techniques, the penalty term is added
only during the model building process and not once the model has been built.
Support Vector Machine
Support Vector Machine (SVM) is a powerful supervised algorithm used for classifications,
regressions and outlier detection. SVM is a linear classifier that determines the hyperplane that
optimally separates the observations into two sets of data points. The intuitive idea behind the SVM
algorithm is maximizing the probability of making a correct prediction (here, that an observation is
a triangle or a cross) by determining the boundary that is furthest away from all the observations.
The general term for a n-dimensional hyperplane, with n = 1D is called Line, n = 2D is called Plane
and n = 3D is called space.
Y
Threshold
↶
lllll
0
▲
}
▲
▲
}
▲
Support
Vectors
x
x
x
x
Margin (a.k.a Maximal Margin Classifier)
▲
x
▲
▲
x
▲
x
x
x
x
x
The shortest distance between the observations and the threshold is called 'Margin'. When we use
the threshold that gives us the largest margin to make classification, we call that a 'Maximal Margin
Classifier' (MMC). MMC are super sensitive to outliers in the training data, which makes them pretty
lame! The margin is determined by the observations are called Support Vectors. Some observations
may fall on the wrong side of the boundary and be misclassified by the SVM algorithm. Choosing a
threshold that allows misclassification is an example of the 'Bias/Variance Trade-off' that plagues all
of ML. When we allow misclassification, the distance between the observation and the threshold is
called 'Soft Margin Classification', which adds a penalty to the objective function for observations in
the training that are misclassified. In essence, the SVM algorithm will choose a discriminant
boundary that optimizes the trade-off between a wider margin and a lower total error penalty. As
an alternative to soft margin classification, a non-linear SVM algorithm can be run by introducing
more advanced non-linear separation boundaries. These algorithms will reduce the no. of
misclassified instances in the training datasets but will have more features, thus adding to the
model's complexity.
29
lllll
K-Nearest Neighbor
K-Nearest Neighbor (KNN) is a supervised learning technique most often used for classification and
sometimes for regression. The idea is to classify a new observation by finding similarities 'nearness'
between this new observation and the existing data.
KNN with new observation, K = 1
x
Y
△
△
0
△
△
◆
△
△
KNN with new observation, K = 5
Y
x
△
x
x
△
x
x
△
0
x
△
x
x
◆
x
△
△
△
x
x
x
△
x
x
The diamond (observation) needs to be classified as belonging to either the cross or the triangle
category. If K = 1, the diamond will be classified into the same category as its nearest neighbor (i.e.
triangle in the left panel), whereas if K = 5, the algorithm will look at the diamond's 5 nearest
neighbors, which are 3 triangles and 2 crosses. The decision rule is to choose the classification with
the largest no. of nearest neighbors, out of 5 being considered. So, the diamond is again classified
as belonging to the triangle category.
KNN is a straightforward intuitive model that is still very powerful because it is non-parametric; the
model makes no assumption about the distribution of the data. Moreover, it can be used directly
for multi-class classification. A critical challenge of KNN however, is defining what it means to be
'similar' (or near). The choice of a correct distance measure may be even more subjective for ordinal
or categorical data. KNN results can be sensitive to inclusion of irrelevant or correlated features, so
it may be necessary to select features manually. If done correctly, this process should generate a
more representative distance measure. KNN algorithms tend to work better with a small no. of
features.
Finally, the number K, the hyperparameters of the model, must be chosen with the understanding
that different values of K can lead to different conclusions. If K is an even number, there may be ties
and no class classification. Choosing a value for K that is too small would result in a high error rate
and sensitivity to local outliers, but choosing a value for K that is too large would dilute the concept
of nearest neighbors by averaging too many outcomes. For K, one must take into account the no. of
categories and their partitioning of the feature space.
lllll
Classification and Regression Tree
Yes
Yes
Yes
No
No Yes
No
No
Yes
No
Regression trees are appropriate
when the target is continuous and
Classification trees are appropriate
when the target variable is
categorical. Most commonly,
Classification and Regression
Trees (CART) are applied to binary
classification or regression. Such a
classification requires a binary tree
a combination of an initial root
node, decision nodes and terminal
nodes. The root node and each
30
decision node represent a single feature (f) and a cut-off value (c) (e.g. X1 > 10%) for that feature.
The CART algorithm chooses the feature and the cut-off value at each node that generates the
widest separation of the labeled data to minimize Classification error (e.g. by a criterion such as
MSE, Mean or Average) or Regression error (e.g. by a criterion such as Mode or Class). Every
successive classification should result in a lower estimation error than the nodes that predicted it.
The tree stops when the error can no longer be reduced further resulting in a terminal node.
E.g: Classifying companies by whether or not they increase their dividends to shareholders:
↷ Partitioning of the
Decision Tree
↷
Feature Space
X2
Initial Root
Node
No
No
X1≤5% X1>5%
+
Yes
Decision
Nodes
Yes
No
—
+
+
—
+
X2>10%
feature (X1, X2)
—
Yes
+
+
—
+
—
+
+
+
—
Yes
↷
No
—
Terminal
Nodes
X2≤10%
—
—
X2>20%
X2≤20%
—
X1
0
X1≤10%
X1>10%
X1: Investment Opportunities Growth (IG)
X2: Free Cashflow Growth (FCFG)
If the goal is regression, then prediction at each terminal node is the mean of the labeled values.
If the goal is classification, then prediction of the algorithm at each terminal node will be mode.
For example, in the feature space, representing IG (X1 > 10%) and FCFG (X2 > 20%) contains 5
crosses. So a new company with similar features will also belong to the cross (dividend increase)
category.
CART makes no assumptions about the characteristics of the training data, so if left unconstrained,
potentially it can perfectly learn the training data. To avoid such overfitting, regularization
parameters can be added such as the maximum depth of the tree, the minimum population at a
node or the maximum no. of decision nodes. Alternatively, regularization can occur via a 'pruning'
technique that can be used later on, to reduce the size of the tree i.e. sections of the tree that
provide little classifying power are pruned or removed. By its iterative structure, CART is a powerful
tool to build expert systems for decision-making processes. It can induce robust rules despite noisy
data and complex relationships between high no. of features.
lllll
Ensemble Learning
Instead of basing predictions on the results of a single model, why not use the predictions of a
group or an ensemble of models? Every single model will have a certain error rate and will make
noisy predictions. But by taking the average result of many predictions from many model, we can
expect to achieve a reduction in noise as the average result converges towards a more accurate
prediction. This technique of combining the predictions from a collection of models is called
Ensemble Learning and the combination of multiple learning algorithm is known as the Ensemble
Method.
31
Ensemble learning typically produces more accurate and more stable predictions than one best
model. Ensemble learning can be divided into two main categories:
Voting Classifiers
An ensemble method can be an aggregation of heterogeneous learners i.e. different types of
algorithms combined together with a voting classifier.
There is an optimal no. of models
beyond which performance would be
expected to deteriorate from overfitting.
A majority-vote classifier will assign to a new data point the predicted label with most votes. The
more individual models you have trained, the higher the accuracy of predictions.
Bootstrap Aggregating
An ensemble method can be an aggregation of homogeneous learners i.e. a combination of the
same algorithm, using different training data that are based on a bootstrap aggregating i.e. bagging
technique.
Random with replacement
↷
n: no. of training instances
n': no. of instances in a 'bag'
m: no. of bags
Usually, n' < n
1
(Train)
2
(Train)
m
(Train)
Alternatively, one can use the same machine learning algorithm but with different training data.
Bootstrap aggregating or Bagging is a technique whereby the original training dataset is used to
generate 'n' new training datasets or bags of data. Each new bag of data is generated by random
sampling with replacement from the initial training set. The algorithm can now be trained on 'n'
independent datasets that will generate 'n' new models. Then for each new observation, we can
aggregate the 'n' predictions using a majority vote classifier for a classification or an average for a
regression. Bagging is a very useful technique because it helps to improve the stability of
predictions and protects against overfitting the model.
lllll
Random Forest
A Random Forest classifier is a collection of a large no. of decision trees trained via a bagging
method or by randomly reducing the no. of features available during training. A random forest is a
collection of randomly generated classification trees from the same dataset. A randomly selected
32
subset of features is used in creating each tree and each tree is slightly different from the others.
The process of using multiple classification trees uses crowdsourcing (majority wins) in determining
the final classification. Because each tree only uses a subset of features, random forests can
mitigate the problem of overfitting. Using random forests can increase the signal-to-noise ratio
because errors across different trees tend to cancel each other out.
Step 1: Create a 'Bootstrapped' dataset.
Original Dataset
Bootstrapped Dataset
↷
It's the same
as the 3rd
Step 2: Create a decision tree using the bootstrapped dataset, but only use a random subset of
variables or columns at each step.
We will consider X2
or X3 as initial
node (random
choice)
Bootstrapped Dataset
↷
We'll focus on
other variables as
decision nodes, like
X1 or X4
Step 3: Now go back to Step 1 and repeat: Make a new bootstrapped dataset and build a tree
considering a subset of variables at each step. Ideally, you'd do this 100s of times, using a
bootstrapped sample and considering only a subset of the variables at each step results in
a wide variety of trees (100 trees). The variety is what makes random forests more effective
than individual decision trees.
Step 4: Now that we've created a random forest, how do we use it? Well, first we get a new patient
with measurements (X1, X2, X3 and X4) and now we want to know if they have a 'Heart
Disease or No' (Y). So we take the data and run it down the first tree we made. The 1st tree
says 'Yes'. Now we run the data down the second tree; the 2nd tree also says 'Yes'. Then we
repeat for all 100 trees we made. After running the data down of all the trees in the random
forest, we see which option received more votes. In this case, 'Yes' received the most votes,
so we will conclude that this patient has Heart Disease.
However, an important drawback of random forest is that it lacks the case of interpretability of
individual trees, as a result it is considered a relatively 'Black-Box' type algorithm i.e. difficult to
understand the reasoning behind their outcomes and thus to foster in them.
33
Unsupervised Learning
Unsupervised Learning is machine learning that doesn't make use of labelled data. In Unsupervised
learning, we have inputs (X's) that are used for analysis without any target being (Y) supplied. In the
absence of any tagged data, the program seeks out structure of interrelationships in the data.
lllll
Principal Component Analysis
Principal Component Analysis (PCA) or Dimension Reduction (seeks to remove the noise i.e.
attributes that don't contain much information) focuses on reducing the no. of features while
retaining variation across observations to preserve the information contained in that variation. PCA
is used to summarize or reduce highly correlated features of data into few main uncorrelated
composite variables (composite variable is a variable that combines two or more variables that are
statistically strongly related to each other).
Step 1: For example, the samples could be 'Blood Samples' in a lab and the variables could be 'DNA'
With 1 DNA
With DNA 1 and DNA 2
DNA 2
(ii) The average
measurement
(x2 ) for DNA 2
↶
Sample 4, 5, 6
are similar
DNA 1
⑥⑤④
Low Values
③
Sample 1, 2, 3
are similar
③ ①②
x2
⑤
⑥
High Values
0
x*
④
↶
lllll
(i) We'll calculate
the average (x1 )
measurement for
DNA 1
①
(iii) With average
volumes, we can
calculate the
center of the (x*)
②
DNA 1
x1
We'll also talk about how PCA can tell us which DNA or variable is the most valuable for
clustering the data.
Step 2: Now we'll shift the data so that the center (x*) is on top of origin (0,0) in the graph. Note,
shifting the data did not change how the data points are positioned relative to each other.
DNA 2
PC 1
x*
(0,0)
DNA 1
How do we
determine if this is a
good fit or not?
34
Step 3: We need to talk about how PCA decides if a fit is good or no? If best fit, then the PCA can
either minimize ('b') the distance to the line or maximize ('c') the distance from the projected
point to the origin.
PC 1 (not a good fit)
x
x c
a2 =
DNA 1
x*
x
The distance between
each data point in the
direction that is parallel
to PC 1 represents the
spread or variation of
the data along PC 1
c2
But, it's actually
easier to calculate 'c',
the distance from
the projected point
to the origin. So PCA
finds the best fitting
by maximizing the
sum of the squared
distance
[SS (Distances)] from
the projected points
to the origin.
x
x
↶
b2 +
Intuitively, it makes
sense to minimize 'b',
the distance from the
point to line. 'b' is the
vertical distance
between the data
point and PC1
representing a
'Projection Error'.
PC 1 (good fit)
refer. Step 2
a
↶
△ Note
↷
b
Pythagoras Theorem
↶
x
DNA 2
↶
We can pick 'b' or 'c'. For now as per 'c's' principles; PCA measures 6 'c' distances.
(Conclusions
derived from
both 'b' or 'c'
will be same)
△ Note
d 21 + d 22 + d23 + d24 + d 25 + d 26 = Sum of Squared Distances
or SS (Distances)
We keep rotating the line until we end up with the line with the largest SS (Distances)
between the projected points and the origin. PCA calls such a line with largest SS (Distance)
as the 'best fit' line a.k.a. Eigenvalue for PC 1
SS (Distances for PC1 ) =
=
Eigenvalue for PC 1
Eigenvalue for PC 1 = Singular Value for PC 1
Step 4: Linear Combinations
↶
DNA 2
PC 1
x*
To make PC 1 :
Take 4 parts of DNA 1 and
1 part of DNA 2
( DNA 1 is more important than DNA 2)
∴
2
4.1x
DNA 1
Mathematicians call
this cocktail recipe a
'Linear Combination'
of DNA 1 & 2
1
4
2
Propositions of each
DNA is called
'Loading Score'
2
2
a = b + c
x 2 = 42 + 1 2 = 4.12
We can scale the length of PC 1 to 1 by
dividing by 4.12
This 1 unit long vector is called
Eigenvector for PC1 or Singular for PC 1
4.12 = 1
4.12
x
4 = 0.97
Informally, PCA involves transforming the
After standardizing:
4.12
Take 0.97 parts of DNA 1 and
covariance matrix of the features and involves
0.242 parts of DNA 2
two key concepts: Eigenvectors and Eigenvalues.
( DNA 1 is more important than DNA 2)
The Eigenvectors define new mutually uncorrelated
composite variables that are linear combinations of the original features. As a vector, an
Eigenvector also represents a direction. Associated with each Eigenvector is an Eigenvalue.
PCA selects as the first principal component the Eigenvector that explains the largest
proportion of variation in the dataset (the Eigenvector with the largest Eigenvalue). An
Eigenvalue gives the proportion of total variance in the initial data that is explained by each
Eigenvector.
∴
1 =
4.12
0.242
35
Step 5: The next largest portion of the remaining variance is best explained by PC 2 , which is at right
angles to PC 1 and thus is uncorrelated with PC1 . Since it is a 2D graph, PC 2 is simply a line
through the origin that is perpendicular to PC1 , without any further optimization that has to
be done.
PC 2
DNA 2
PC 1
Now calculate the
Eigenvector and Eigenvalue for PC 2
DNA 1
- 0.242 for DNA 1
0.97 for DNA 2
(
∴ DNA 2 is 4 times more important than DNA 1)
Step 6: We can convert the Eigenvalues or SS (Distances) into variation around the origin (0,0) by
dividing by the sample size minus 1 i.e. (n - 1).
SS (Distances for PC1 ) = Variation for PC 1
n-1
SS (Distances for PC2 ) = Variation for PC 2
n-1
For the sake of example, imagine that Variation for PC 1 15
Variation for PC 2 3
Total Variation for PC 18
15 / 18 = 83%
3 / 18 = 17%
= 100%
Step 7: Scree Plot i.e. It shows the proportion of total variance in the data explained by each
principal component (PC).
The first factor in PCA would be the
most important factor in explaining
the variation across observations.
The second factor would be the
second most important and so, up
to the no. of uncorrelated factors
specified by the researcher.
80
83%
PC 1 and PC 2 account for more than
90% of variation, so we can just use
these to draw a 2D graph.
40
0
17%
PC 1
If the scree plot looked like this,
where PC 3 and PC4 account for a
substantial amount of variation then,
using just the first 2 PCs would not
create a very accurate representation
of the data.
PC2
40
20
0
PC1
PC 2
PC3
PC 4
The main drawback of PCA is that since the PCs are combinations of the dataset's initial features,
they typically cannot be easily labeled or directly interpreted by the analyst. Compared to modelling
36
data with variables that represent well-defined concepts, the end user of PCA may perceive PCA as
something of a 'Black-Box'. Reducing the no. of features to the most relevant predictors is very
useful.
lllll
Clustering
Given a dataset, Clustering is the process of grouping observations into categories based on
similarities in their attributes i.e. meaning that the observations inside each cluster are similar or
close to each other, a property known as 'cohesion' and the observations in two different clusters
are as far way from one another or are as dissimilar as possible, a property known as 'separation'.
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K-Means Clustering
K-Means is a relatively old algorithm that repeatedly partitions observations into a fixed number 'K',
of non-overlapping clusters. The number of clusters 'K' is a model hyperparameter - a parameter
whose value must be set by the researcher before learning begins. Each cluster is characterized by
its 'Centroid' i.e. center and each observation is assigned by the algorithm to the cluster with the
centroid to which that observation is closest. The K-means algorithm follows an iterative process.
S1
S2
1
2
S3
3
C
C
1
C
1
1
C
C
C
3
C2
C
S4
3
C
2
S5
C
3
2
S6
1
C
C
2
3
C
1
C
C
2
3
C
1
C
C
2
3
37
The K-means algorithm will continue to iterate until no observation is reassigned to a new cluster
i.e. no need to recalculate new centroids. The algorithm will then be converged and reveal the final
K clusters with their member observations. The K-means algorithm has minimized inter-cluster
distance (thereby maximizing cohesion) and has maximized inter-cluster distance (thereby
maximizing separation) under the constraint that K = 3. The K-means algorithm is fast and works
well on very large datasets with hundreds of millions of observations. However, the final
assignment of observations to clusters can depend on the initial location of the centroids. One
limitation of this technique is that the hyperparameters 'K', the no. of clusters in which to partition
the data, must be decided before K-means can be run.
Hierarchical Clustering
Hierarchical Clustering is an iterative procedure used to build a hierarchy of clusters. In Hierarchical
Clustering, the algorithms create intermediate rounds of clusters of increasing (agglomerative) or
decreasing (divisive) size until the final clustering is reached. This process creates relationships
among the rounds of clusters. It has the advantage of allowing the investment analyst to examine
alternative segmentations of data of different granularity before deciding which one to use.
(a) Agglomerative Clustering
We start with one observation
as it's own cluster and add other
similar observations to that
cluster.
I
G
5
H
J
C
4
K
6
8
9
D
2
7
B
(b) Divisive Clustering
It starts with one giant cluster
and then partitions that cluster
into smaller and smaller
clusters.
E
A
F
3
1
5
4
6
8
9
2
7
1
Agglomerative Clustering
[Bottom-Up]
3
Divisive Clustering
[Top-Down]
- Agglomerative method is well
suited for identifying small
clusters.
- Divisive method is better suited
for identifying large clusters.
- Agglomerative method makes
decision based on local patterns
without initially accounting for the
global structure of the data.
- Divisive method starts with a
holistic representation of the data.
Example 1:
A category of general linear regression (GLS) models that focuses on reduction in the total no. of features
used is best described as?
A. Clustering Model
B. Dimension Reduction Model
C. Penalized Regression Model
✓
38
Example 2:
We apply ML techniques to a model including fundamental and technical variables (features) to predict next
quarter's return for each of the 100 stocks currently in our portfolio. Then, the 20 stocks with the lowest
estimated return are identified for replacement. The ML techniques appropriate for executing Step 1 are
most likely to be based on:
A. Regression
Because target variable (quarterly return) is continuous.
B. Classification
C. Clustering
✓
Dendrogram
A type of diagram for visualizing a hierarchical cluster analysis known as a Dendrogram, highlights
the hierarchical relationships among the clusters.
Distance
Arch
Dendrite
9
0.07
7
8
0.05
0.03
1
3
0.01
0
2
A
B
C
6
4
D
2 Clusters
lllll
5
E
F
G
H
I
6 Clusters
J
K
Clusters
Refer Agglomerative Clustering:
Clusters are represented by a
horizontal line - the 'Arch',
which connects two vertical
lines called 'Dendrites', where
the height of each arch
represents the distance
between the two clusters being
considered. Shorter dendrites
represent a shorter distance
(and greater similarity) between
clusters. The horizontal dashed
lines cutting across the
dendrites show the no. of
clusters into which the data are
split at each stage.
11 Clusters
Neural Networks, Deep Learning Nets and Reinforcement Learning
These sophisticated algorithms can address highly complex machine learning tasks such as Image
Classification, Face Recognition, Speech Recognition and Natural Language Processing. These
complicated tasks are characterized by non-linearities and interactions between large no. of feature
inputs.
lllll
Neural Networks
Neural Networks also called Artificial Neural Networks (ANNs) are highly flexible type of ML
algorithm that have been successfully applied to a variety of tasks characterized by non-linearities
and complex interactions among features.
Input Layer
(4 Features)
Hidden Layer Output Layer
(1 Target)
Input 1
Input 2
Links
Input 3
Input 4
Nodes
Neurons
>
Linking the information in the input layer to multiple
nodes in the hidden layers, each with its own activation
function, allows the neural network to model complex
non-linear functions to use the information in the input
variables well. The nodes in the hidden layer transform
the inputs in a non-linear fashion into new values that
are then combined into the target value. This structure
4, 5 and 1 is set by researcher and referred to as the
hyperparameters of the neural network.
39
0
①
0
Softplus
x
x
y
x will be given;
Sigmoid
ReLU
(Rectified Linear Unit)
x
y = f (x) = log (1+e x )
y
y = f (x) =
(Here, x and y are coordinates)
y
x
y = f (x) = max (0 , x)
e
ex + 1
If the process of adjustment works forward through the layers of network this process is called
'Forward Propagation', where as, if the process of adjustment works backward through the layers of
network, this process is called 'Backward Propagation'.
Learning Rate hyperparameter controls the rate or speed at which the model learns. Learning takes
place through this process of adjustment to network weights with the aim of reducing the total
error. When learning is complete, all the network weights have assigned values.
Partial Derivative or Rate of Change
↱ of the total error with respect to
New Weight = Old Weight - (Learning Rate) (Gradient)
the change in the old weight.
When the learning rate is too large, gradient descent can inadvertently increase rather than
decrease the training error. When the learning rate is too small, training is not only slower but may
become permanently stuck with a high training error.
E.g: ReLU function. F (x) = max (0 , x), y will be equal to β1 times z1 , where z 1is the maximum of
(x1 + x2 + x3 ) or 0, plus β2 times z2 , the maximum of (x2 + x4) or 0, plus β3 times z 3, the
maximum of (x1 + x3 + x4) or 0, plus an error term.
Input
Output
Input
Hidden
Output
x1
x1
z1
x2
x2
z2
Y
x3
> Y
x3
z3
x4
↷
1
Each node has conceptually two functional parts:
(i) Summation Operator: Once the node receives the four input values, the summation operator
multiplies each value by a weight and sums the weighted values to form the total net input. The
total net input is then passed to the activation function, which transforms this input into the final
output of the node.
(ii) Activation Function: Informally, the activation function operates like a light dimmer switch that
decreases or increases the strength of the input. The activation function is characteristically
non-linear.
x4
Weights
Y = 1 (x1) + 2 (x2) + 3 (x3) + 4 (x4)
Y = 1 . Max (0, x1 + x2 + x3) +
2 . Max (0, x2 + x4) +
3 . Max (0, x2 + x3 + x4)
= 1 (z1) + 2 (z2) + 3 (z3)
40
When more nodes and more hidden layers are specified, a neural network's ability to handle
complexity tends to increase, but so does the risk of overfitting. However, the tradeoffs in using
them are the lack of interpretability and the amount of data needed to train such models.
lllll
Deep Learning Nets
Neural networks with many hidden layers - atleast 3 but often more than 20 hidden layers are
known as Deep Learning Nets (DLNs) is the backbone of the artificial intelligence revolution.
Advances in DLNs have driven developments in many complex activities such as image, pattern and
speech recognition.
Input
Hidden
Output
Bias
x1
E.g: 0.2 (x1) + 0.8 (x2) + 2.14 = x-axis
B1
x2
B1
B2
1
Y1
x3
The function
gets activated
when it hits 1.
B2
B3
Y2
x4
B3
B4
x5
↷
0
Bias
The information is fed to the input layer and the information is transferred from one layer to
another over connecting channels, each of these has a value attached to it and hence is called a
weighted channel 'w ij ' (for neuron i and input j), each of which usually produces a scaled number in
the range (0,1) or (-1,1). All neurons have unique numbers associated with it called Bias. This bias is
added to the weighted sum of inputs reaching the neurons, which is then applied to the function
known as the Activation Function. The result of the activation function determines if the neurons
get activated. Every activated neuron passes on information to the following layer; this continues
uptil the second last year (a layer before the output layer). The one neuron activated in the output
layer corresponds to the input digit. [Note: The weights, biases or numbers are passed to another
layer of functions and into another and so on until the final year produces a set of probabilities of
the observation being in any of the target categories (each represented by a node in the output
layer). The DLN assigns the category based on the category with the highest probability]. The weight
and bias are continuously adjusted to produce a well trained network. The DLN is trained on large
datasets; during training the weights w i are determined to minimize a specified loss function.
Unfortunately, DLNs require substantial time to train and systematically varying the
hyperparameters, may not be feasible.
lllll
Reinforcement Learning
Reinforcement Learning (RL) is an algorithm that involves an agent that should perform actions that
will maximize its rewards over time, taking into consideration the constraints of its environment.
Gamer
② Action
③ Reward
State of
① Environment
or Situation
Video Game
41
In the case of AlphaGo. a virtual gamer (the agent) uses his/her console to command (the actions)
with the help of the information on the screen (the environment) to maximize his/her score (the
reward). Unlike supervised learning, RL gets instantaneous feedback i.e. the learning subsequently
occurs through millions of trail and errors. For example, an agent could be a virtual trader who
follows certain trading rules (the action) in a specific market (the environment) to maximize its
profits (its rewards). The success of RL is still an open question in financial markets.
Fig 4: Guide to ML Algorithms
Variables
Supervised ML
CONTINUOUS
CATEGORICAL
CONTINUOUS or
CATEGORICAL
Unsupervised ML
REGRESSION
- Penalized Regression (LASSO)
- Classification and Regression Tree (CART)
- Random Forest
DIMENSIONALITY REDUCTION
- Principal Component Analysis (PCA)
CLASSIFICATION
- Support Vector Machine (SVM)
- K-Nearest Neighbor (KNN)
- Classification and Regression Tree (CART)
DIMENTIONALITY REDUCTION
- Principal Component Analysis (PCA)
NEURAL NETWORKS
DEEP LEARNING
REINFORCEMENT LEARNING
CLUSTERING
- K-Means
- Hierarchical
CLUSTERING
- K-Means
- Hierarchical
NEURAL NETWORKS
DEEP LEARNING
REINFORCEMENT LEARNING
Fig 5: Decision Flowchart for Choosing ML Algorithms
NO
YES
YES
YES
NO
YES
NO
NO
NO
YES
YES
YES
NO
NO
YES
42
CHAPTER 8
Big Data Projects
Big Data differs from traditional data sources based on the presence of a set of characteristics commonly
referred to as the 4 V's: Volume, Variety, Velocity and Veracity.
Volume: refers to the quantity of data.
Variety: pertains to the array of available data sources.
Velocity: is the speed at which data is created (data in motion is hard to analyze compared to data at rest).
Veracity: related to the credibility and reliability of different data sources.
Big Data also affords opportunities for enhanced fraud detection and risk management.
E.g: One study conducted in the U.S. found that positive sentiment on Twitter could predict the trend for the
Dow Jones Industrial Average up to 3 days later with nearly 87% accuracy.
Fig 1: Model Building for Financial Forecasting using Big Data
②
①
②
①
⑤
2
③
④
⑤
1
②
③
④
Exploratory Data Analysis (EDA): is the preliminary step in data exploration. Exploratory graphs, charts and other visualizations
such as heat maps and word clouds are designed to summarize and observe data.
Feature Selection: is a process whereby only pertinent features from the dataset are selected for ML model training.
3 Feature Engineering: is a process of creating new features by changing or transforming existing features. Feature Engineering
techniques systematically alter, decompose or combine existing features to produce more meaningful
features.
(Feature Selection is a key factor in minimizing model overfitting & Feature Engineering tends to prevent model underfitting)
43
lllll
Data Preparation and Wrangling
Structured Data
*
Data Preparation (Cleansing): Data Cleansing is
the process of examining, identifying and
mitigating errors in raw data.
1. Incompleteness Error: where the data is not
present, resulting in missing data.
2. Invalidity Error: where the data is outside of
a meaningful range.
3. Inaccuracy Error: the data is not a measure
of true value.
4. Inconsistency Error: data conflicts with the
corresponding data points or reality.
5. Non-Uniformity Error: data is not present in
an identical framework.
6. Duplication Error: where duplicate
observations are present.
In addition to a manual inspection and verification
of the data, analysis software such as SPSS can be
used to understand 'MetaData' (data that
describes and gives information about other data)
about the data properties to use as a starting
point to investigate any errors in the data.
Unstructured Data
*
Text Preparation (Cleansing): Raw text data are a
sequence of characters and contain other
non-useful elements including html tags,
punctuation and white spaces. The initial step in
text processing is cleansing, which involves to clean
the text by removing unnecessary elements from
the raw text. A 'Regular Expression' (Regex) is a
series that contains characters in a particular order.
Regex is used to search for patterns of interest in a
given text. It can be used to find all the html tags
that are present in the form of < ... > in text. Once a
pattern is found, it can be removed or replaced.
1. Remove Html Tags: Most of the text data are
acquired from web pages and the text inherits
html mark-up tags with the actual content. The
initial task is to remove (or strip) the html tags
with the help of regex.
2. Remove Punctuations: Most punctuations are
not necessary for text analysis and should be
removed. However, some punctuations such as
period (dots), percentage signs, currency symbols
and question marks may be useful for ML model
training. These punctuations should be
substituted with such annotations as
/percentagesign/, /dollarsign/ and
/questionmark/ to presume their grammatical
meaning. The periods (dots) must be
appropriately replaced or removed i.e. period
(dots) after abbreviations, but the periods
separating the sentences should be replaced by
the annotation. Regex is often used to remove or
replace punctuations.
3. Remove Numbers: The numbers or digits should
be removed or substituted with an annotation
'number/. However, the number and any
decimals must be retained where the outputs of
interest are the actual values of the number. One
such text application is Information Extraction
(IE), where the goal is to extract relevant
information from a given text.
4. Remove White Spaces: Extra white spaces, tab
spaces, leading and ending spaces in the text
should be identified and removed to keep the
text intact and clean. For example, text mining
package in R offers a 'StripWhiteSpace' function.
Web Scrapping is a technique to extract raw
content from a source typically webpages.
44
*
Data Wrangling (Preprocessing): Data
Preprocessing primarily includes transformations
and scaling of the data. These processes are
exercised on the cleansed dataset.
1. Extraction: A new variable can be extracted
from the current variable for the ease of
analyzing and to use for training the ML model.
2. Aggregation: Two or more variables can be
aggregated into one variable to consolidate
similar variables.
3. Filtration: The data rows that are not needed
for the project must be identified and filtered.
4. Selection: The data columns that are intuitively
not needed for the project can be removed.
5. Conversion: The variables can be of different
types i.e. nominal, ordinal, continuous and
categorical. The variables in the dataset must be
converted into appropriate types to further
process and analyze them correctly. Before
converting, values must be stripped out with
prefixes and suffixes such as currency symbols.
Outliers may be present in the data and several
techniques can be used to detect outliers in the
data. In normally distributed data, data values
outside of 1.5 IQR (Interquartile range) are
considered as outliers and values outside of 3 IQR
as extreme values i.e. a data value that is outside
of 3 standard deviations from the mean may be
considered as an outlier. When extreme values
and outliers are simply removed from the dataset,
it is known as Trimming also called 'Truncation'.
When extreme values and outliers are replaced
with the maximum (for large outliers) and
minimum (for small outliers) values of data points
that are not outliers, this process is known as
Winsorization. Scaling is a process of adjusting the
range of a feature by shifting and changing the
scale of data. Scaling helps SVMs and ANNs
models. It is most important to remove outliers
before scaling is performed. There are two of the
most common ways of scaling:
(a) Normalization: is the process of rescaling
numeric variables in the range of (0, 1).
Normalization is sensitive to outliers.
*
Text Wrangling (Preprocessing): A token is
equivalent to a word and tokenization is the
process of splitting a given text into separate
tokens. In other words, a text is considered to be a
collection of tokens. Similar to structured data, text
data also requires normalization. The normalization
process in text processing involves the following:
1. Lower-casing: the alphabet removes distinctions
among the same words due to upper and lower
cases. This actions help the computers to process
the same words appropriately e.g. 'The' and 'the'.
2. Stop Words: are such commonly used words as
'the', 'is' and 'a'. For ML training purposes, stop
words typically are removed to reduce the no. of
tokens involved in the training set. In some cases,
additional stop words can be added to the list
based on the content. For example, the word
'exhibit' may occur often in financial filings, which
is general, is not a stop word but in the context of
the filings, it can be treated as a stop words.
3. Stemming: is the process of converting inflected
forms of a word into its base word, known as
stems. For example, the stem of the words
'analyzed' and 'analyzing' is 'analyz'.
4. Lemmatization: is the process of converting
inflected forms of a word into its morphological
root, known as Lemma. Lemmatization is an
algorithmic approach and depends on the
knowledge of the word and language structure.
For example, the lemma of the words 'analyzed'
and 'analyzing' is 'analyze'. Lemmatization is
computationally more expensive and advanced.
Stemming or Lemmatization will reduce the
repetition of words occurring in various forms and
maintain the semantic structure of the text data.
Stemming is more simpler to perform compared to
lemmatization. In text data, Data Sparseness refers
to words that appear very infrequently, resulting in
data consisting of many unique low frequency
tokens. Both techniques decrease data sparseness
by aggregating many sparsely occurring words in
relatively less sparse stems or lemmes, thereby
aiding in training less complex ML models. After the
cleansed text is normalized, a Bag-of-Words is
created. BOW is simply a set of words and doesn't
capture the position or sequence of words present
in the text. Note that the no. of words decreases as
the normalizing steps are applied, making the
resulting BOW smaller and simpler.
45
Normalization can be used when the
distribution of the data is not known.
Xi
(normalized)
= X i - X min
X max - X min
(b) Standardization: is the process of both
centering and scaling the variables. The
resultant standardized variable will have an
arithmetic mean of 0 and standard deviation of
1. Standardization is relatively less sensitive to
outliers as it depends on the mean and
standard deviation of the data. However, the
data must be normally distributed to use
standardization.
X i (standardized) = X i - μ
ox
The last step of text preprocessing is using the final
BOW after normalizing to build a Document Term
Matrix (DTM). DTM of four texts and using
normalized BOW filled with counts of occurrence.
BOW doesn't represent the word sequences or
positions, which limits its use for some advanced
ML training applications. In the example, the word
'no' is treated as a simple token and has been
removed during the normalization because it is a
stop word. Consequently, this fails to signify the
negative meaning ('no market') of the text (i.e. Text
4). To overcome such problems, a technique called
N-grams can be employed. N-grams is a
representation of word sequences. The length of a
sequence can vary from 1 to 0. Stemming can be
applied on the cleansed text before building
n-grams and BOW.
Even after removing isolated stop words, stop
words tend to persist when they are attached to
their adjacent words.
46
Data Exploration
Domain Knowledge plays a vital role in exploratory analysis as this stage should involve cooperation
between analysts, model designers and experts in this particular data domain. In Data exploration,
without Domain Knowledge can result in spurious relationships i.e. mislead analysis.
Data Exploration
Structured Data
1D: Summary statistics such as mean, median, quartiles, ranges , standard deviations, skewness
and kurtosis.
Box Plots
Histograms represent
equal bins of data and
their respective
frequencies. They can
be used to understand
the high-level
distribution of the data.
Bar Charts summarize
the frequencies of
categorical variables.
Box Plots shows the
distribution of
continuous data.
Frequency
↷
Maximum
→ 3rd Quartile
→
Frequency
Density Plots
Median
→ 1st Quartile
Data
Minimum
Data
Density Plots are
another effective way
to understand the
distribution of
continuous data.
Density Plots are
smoothed histograms.
Frequency
Bar Charts
Units
Histograms
Data
Salary
2D: Summary statistics of relationships such as a correlation matrix.
Line Graphs
A Scatter Plot provides
a starting point where
relationships can be
examined visually, but it
may not be a
statistically significant
relationship.
Line Graph is a type of
chart used to visualize
the value of something
over time.
..... .
........ .
.
.
..................
.
.
...... .
Temparature
Scatter Plots
Salary
lllll
↶
lllll
...
.
.
...
.
Time
Age
For Multivariate Data, commonly utilized exploratory visualization designs include stacked bar and
line charts, multiple box plots and scatter plots showing multivariate data that use different colors
or shapes for each feature.
Common Parametric Statistical Tests: ANOVA, T-test and Pearson Correlation
Common Non-Parametric Statistical Tests: Chi-Square and Spearman Rank-Order Correlation
Central Tendency helps measure minimum and maximum values for continuous data. Counts and
Frequencies for categorical data are commonly employed to gain insight regarding the distribution
of possible values.
47
Unstructured Data
The most common applications are,
(a) Text Classification: uses supervised ML approaches to classify texts into different classes. Text
Classification involves dividing text documents into assigned classes (a class is a category,
examples include 'relevant' and 'irrelevant' text documents or 'Bearish' and 'Bullish' sentences).
(b) Topic Modeling: uses unsupervised ML approaches to group the texts in the dataset into topic
clusters. Topic modeling is a text data application in which the words that are most informative
are identified by calculating the term frequency of each word. For example, the word 'soccer' can
be informative for the topic 'sports'. The words with high term frequency values are eliminated
as they are likely to be stop words or other common vocabulary words, making the resulting
BOW compact and more likely to be relevant to topics within the texts.
(c) Fraud Detection
(d) Sentiment Analysis: predicts the sentiment i.e. negative, neutral or positive of the texts in a
dataset using both supervised and unsupervised approaches.
(In Sentiment Analysis and Text Classification applications, the Chi-square measure of word
association can be useful for understanding the significant word appearances in negative and
positive sentences in the text or in different documents).
↶
Text data includes a collection of texts, also known as a Corpus, that are sequences of tokens.
Word Clouds are common visualizations when working with text data as they can be made to
visualize the most informative words and their term frequency values.
Document Frequency (DF): defined as the no.
of documents i.e. sentences that contain a given
word divided by the total no. of sentences.
DF = Sentence Count with Word
Total no. of Sentences
Inverse Document Frequency (IDF): a relative
measure of how unique a term is across the
entire corpus.
IDF = Log ( 1 / DF )
Term Frequency (TF): is the ratio of the no. of
times a given token occurs in all texts in the
dataset to the total no. of tokens in the dataset
e.g. word associations, average word and
sentence length & word and syllable counts. TF
at corpus level is known as 'Collection
Frequency' (CF).
Term Frequency - Inverse Document
Frequency (TF-IDF): Higher TF-IDF value indicate
words that appear more frequently within a
small no. of documents. This signifies relatively
more unique terms that are important.
Conversely, a low TF-IDF value indicate terms
that appear in many documents.
lllll
TF =
Total Word Count
Total no. of Words in Collection
TF-IDF = TF x IDF
Feature Selection
Structured Data
Typically, structured data even after the data preparation can contain features that don't contribute
to the accuracy of an ML model or that negatively effect the quality of ML training. Feature Selection
on structural data is a methodical and iterative process. Statistical measures can be used to assign a
score gauging the importance of each feature. These features can then be ranked using the score
and can either be retained or eliminated from the dataset. Methods include Chi-Square Test,
2
Correlation Coefficient and information-gain measures i.e. R
48
Feature Selection is different from Dimensionality Reduction, but both methods seek to reduce the
no. of features in the dataset. The Dimensionality Reduction method creates new combinations of
features that are uncorrelated, whereas Feature Selection includes and excludes features present in
the data without altering them.
Unstructured Data
For text data, Feature Selection involves selecting a subset of the terms or tokens occurring in the
dataset. The token serve as features for ML model training. Feature Selection in text data effectively
decreases the size of the vocabulary or BOW. This helps the ML model be more efficient and less
complex. Another benefit is to eliminate noisy features from the dataset. Noisy features are tokens
that don't contribute to ML model training and actually might detract from the ML model accuracy.
The frequent tokens strain the ML model to choose a decision boundary among the texts as the
terms are present across all the texts, an example of model underfitting. The rare tokens mislead
the ML model into classifying texts containing the rare terms into a specific class, an example of
model overfitting.
The general Feature Selection methods in text data are as follows,
(a) Frequency: measures can be used for vocabulary pruning to remove noisy features by filtering
the tokens with very high and low TF values across all texts. DF is another frequency measure
that helps to discard the noise features and often perform well when many thousands of tokens
are present.
(b) Chi-Square Test: is applied to test the independence of two events: occurrence of the token and
occurrence of the class. The test ranks the tokens by their usefulness to each class in text
classification problems. Tokens with the highest Chi-Square Test statistic values occur more
frequently in texts associated with a particular class and therefore can be selected for use as
features for ML model training.
(c) Mutual Information (MI): measures how much information is contributed by a token to a class of
texts. If MI = 0, then the token's distribution in all text classes is the same. The MI approaches to
1 as the token in any one class tends to occur more often in only that particular class of text.
lllll
Feature Engineering
Structured Data
For Continuous Data, a new feature may be created. For example, by taking the logarithm of the
product of two or more features. As another example, when considering a salary or income feature,
it may be important to recognize that different salary brackets impose a different taxation rate.
Domain Knowledge can be used to decompose an income feature into different tax brackets,
resulting in a new feature: 'income_above_100K' with possible values 0 and 1.
For Categorical Data, a new feature can be a combination. For example, sum or product of two
features or a decomposition of one feature into many. If a single categorical feature represents
education level with 5 possible values i.e. high school, associates, bachelor's, master's and
doctorate then these values can be decomposed into 5 new features, one for each possible value
(e.g. is, high, school, is, doctorate) filled with 0s (for false) and 1s (for true). The process in which
categorical variables are converted into binary form (0 or 1) for ML is called One Hot Encoding.
Unstructured Data
The following are some feature engineering techniques which may overlap with text processing
techniques:
(a) Numbers: In text processing, numbers are converted into a token such as '/number/'.
(b) N-grams
49
(c) Name Entity Recognition (NER): The NER algorithm analyzes the individual tokens and their
surrounding semantics while referring to its dictionary to tag an object class to the token. For
example, NER tags of the text 'CFA Institute was formed in 1947 and is headquartered in Virginia';
NER tags can also help identify critical tokens on which such operations as lowercasing and
stemming then can be avoided e.g. Institute here refers to an organization rather than a verb.
Additional object classes are for example MONEY, TIME and PERCENT which are not present in
the example text.
(d) Parts of Speech (POS): uses language structure and dictionaries to tag every token in the text
with a corresponding part of speech. Some common POS tags are noun, verb, adjective and
proper noun.
lllll
Model Training
lllll
Method Selection
(a) Supervised and Unsupervised learning models
(b) Type of Data:
For Numerical Data (e.g. predicting stock prices using historical stock market
values), CART methods may be suitable.
For Text Data (e.g. predicting the topic of a financial news article by reading the headline of the
article), methods as GLMs and SVMs are used.
For Image Data (e.g. identifying objects in a satellite image such as tanker ships moving in and
out of port), NNs and deep learning methods tend to perform better than others.
For Speech Data (e.g. predicting financial sentiment from quarterly earnings' conference call
recordings), deep learning methods can offer promising results.
(c) Size of Data: A typical dataset has two basic characteristics, no. of instances (observations) and
no. of features. For instance, SVMs have been found to work well on 'wider' datasets with 10,000
to 100,000 features and with fewer instances. Conversely, NNs often work better on 'longer'
datasets where the no. of instances is much larger than the no. of features.
Before model training begins,
Supervised Learning: The data are split using a random sampling technique such as K-Fold.
Unsupervised Learning: No splitting is needed due to the absence of labeled training data.
Class Imbalance: where the no. of instances for a particular class is significantly larger than for
other classes. For example, say for corporate issuers in the BB+/Ba1 to B+/B1
credit quality range, issuers who defaulted (positive or '1' class) would be very few
compared to issuers who did not default (negative or '0' class). Hence, on such
training data, a naïve model that simply assumes no corporate issuer will default
may achieve good accuracy, albeit with all default cases misclassified. Balancing
the training data can help alleviate such problems. In cases of unbalanced data,
the '0' class (majority class) can be randomly undersampled or the '1' class
(minority class) randomly oversampled.
50
lllll
Performance Evaluation
(a) Error Analysis: For classification problems, error analysis involves computing four basic
evaluation matrices TP, FP, TN and FN. Here's a Confusion Matrix for error analysis:
Precision: is the ratio of correctly predictive positive classes to all predictive positive classes.
Precision is useful in situations where the cost of FP or Type I Error is high. For
example, when an expensive product fails quality inspection (predicted class 1) and is
scrapped, but it is actually perfectly good (actual class 0).
P =
TP
TP + FP
Recall: also known as sensitivity i.e. is the ratio of correctly predicted positive classes to all actual
positive classes. Recall is useful in situations where the cost of FN or Type II Error is high.
For example, when an expensive product passes quality inspection (predicted class 0) and
is sent to the valued customer, but it is actually quite defective (actual class 1).
R =
TP
TP + FN
Accuracy: is the percentage of correctly predicted classes out of total predictions.
Accuracy =
TP + TN
TP + FP + TN + FN
F1 Score: is the harmonic mean of precision and recall.
F1 Score =
2. P. R
P+R
F1 Score is more appropriate than Accuracy when unequal class distribution is in the dataset and
it is necessary to measure the equilibrium of Precision and Recall. High scores on both of these
metrices suggest good model performance.
Example 1:
Calculate Precision, Recall, Accuracy and F1 Score with the help of table below:
51
Observation
Actual Training
Labels
Predicted
Results
Classification
1
2
3
4
5
6
7
8
9
10
1
0
1
1
1
1
0
0
0
0
1
0
1
0
1
0
0
0
0
1
TP
TN
TP
FN
TP
FN
TN
TN
TN
FP
Precision = 3 = 0.75
(3 + 1)
Recall = 3 = 0.6
(3 + 2)
Accuracy = (3 + 4) = 0.7
(3 + 1 + 4 + 2)
F1 Score = 2 (0.75) (0.6)
(0.75 + 0.6)
= 0.67
If the no. of classes in a dataset is unequal; however, then F1 score and Accuracy should be used as the
overall performance measure for the model.
(b) Receiver Operating Characteristic (ROC)
True Positive Rate
(TPR)
1
TPR = TP
TP + FN
AUC = 0.9
A
AUC = 0.75
B
AUC = 0.5
C
0
ROC: This technique for assessing
model performance involves the
plot of curve showing the
trade-off between TPR and FPR
for various threshold points
(cut off). The ROC curve
summarizes all of the Confusion
Matrices that each threshold
False Positive Rate
produced.
1
(FPR)
FPR =
The shape of the ROC curve provides
insight into the model's performance.
A more convex curve indicates better
model performance. It is clear that
Model A with the most convex ROC
curve with AUC (Area Under the Curve)
of more than 0.9 (90%) is the best
performing among the three models.
?
FP
FP + TN
1
0.75
Predicted
Probability
(P)
0
Threshold
Defective
If P from a logistics regression model
for a given observation is greater than
the threshold, then the observation is
classified as '1', otherwise the
observation will be classified '0'.
P > Threshold = 1
P < Threshold = 0
(c) Root Mean Squared Error (RMSE): is appropriate for continuous data prediction and is mostly
used for regression models. It is a simple matrix that captures all the prediction errors in the
data (n). A small RMSE indicates potentially better model performance.
RMSE =
Σ (Predicted i - Actual i ) 2 =
n
Σ (Y i - Y i ) 2
n
^
52
lllll
Tuning
↶
↑
(Bias Error is high when a model is
oversimplified and doesn't sufficiently learn
from the patterns in the training data)
↑
↷
If Prediction Error on Training Set
= Underfitting i.e. Bias Error
If Prediction Error on Cross Validation Set
= Overfitting i.e. Variance Error
(Variance Error is high when a model is
overlsimplified and memorizes the training
data so much so that it will likely perform
poorly on new data)
It is not possible to completely eliminate both types of error. The Bias-Variance trade-off is critical in
finding the optimal balance where a model neither underfits or overfits.
(a) Parameters: are critical for a model and are dependent on the training data. Parameters are
learned from the training data as part of the training process by a optimization technique.
Examples of parameters include: Coefficients in regression, weights in NN and Support Vectors in
SVM.
(b) Hyperparameters: are used for estimating model parameters and are not dependent on the
training data. Examples of hyperparameters include the Regularization Term (λ) in supervised
model, Activation Function and No. of Hidden Layers in NN, No. of Trees and Tree Depth in
Ensemble Methods, K in KNN and K-means Clustering & P-Threshold in Logistic Regression.
Hyperparameters are manually set and timed. Thus, timing heuristics and such techniques such
as Grid Search is used to obtain the optimum values of hyperparameters. Grid Search is a
method of systematically training an ML model by using various combinations of
hyperparameter values, cross validating each model and determining which combination of
hyperparameter values ensure the best model performance. The plot of training error for each
value of a hyperparameter (i.e. changing model complexity) is called a Fitting Curve.
Large Error
[Error CV > Error Train ]
High Variance
[Error CV < Error Train ]
High Bias
Error CV
Slight Regularization: highly penalized model complexity,
thereby allowing most or all of the features to be included in
the model and thus potentially enabling the model to
memorize the data. When high variance error and low bias
error exist, the model performs well on the training dataset but
generates many FP and FN errors on the CV dataset in other
words, the model is overfitted and doesn't generalize well to
new data.
Optimum Regularization: minimizes both variance and bias
error in a balanced fashion. The range of optimum
regularization values can be found heuristically using such
techniques as Grid Search.
Error Train
Error
Small Error
Slight
Regularization
Optimum
Regularization
λ
Large
Regularization
Large Regularization: excessively penalizes model complexity,
thereby allowing too few of the features to be included in the
model to learn less from the data. Typically, with large
regularizations, the prediction errors on the training and CV
datasets are both large. When high bias error exists, the model
doesn't perform well on either training or CV datasets because
it is typically lacking important predictor variables.
Regularization: describes methods that reduce statistical
variability in high dimensional data estimation problems,
reducing regression coefficient estimates toward '0' and
thereby avoiding complex models and the risk of overfitting.
Regularization models can also be applied to non-linear
models. For example: Asset returns typically exhibit strong
multicollinearity, making the estimation of the covariance
matrix highly sensitive to noise and outliers, so the resulting
optimized asset weights are highly unstable. Regularization
methods have been used to address this problem.
53
If high bias or variance error exists after tuning of hyperparameters, either a larger no. of training
instances may be needed or the no. of features included in the model may need to be decreased
(in the case of high variance) or increased (in the case of high bias). The model then needs to be
retrained and retuned using the new training dataset. In the case of a complex model, where a large
model is comprised of sub-model(s), Ceiling Analysis can be performed. Ceiling Analysis can help
determine which sub-model needs to be tuned to improve the overall accuracy of the larger model
i.e. is a systematic process of evaluating components in the pipeline of model building.
54
CHAPTER 9
Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations
The steps associated with running a simulation are as follows:
Step 1: Determine the Probabilistic Variables
It makes sense to focus attention on few
variables that have significant impact on value.
Step 2: Define Probability Distributions for these Variables: Generically, there are 3 ways in which we can go
about defining probability distributions:
- Historical Data: This method assumes that the future values of the variable will be similar to its past
e.g. long term treasury bond rate.
- Cross-Sectional Data: When past data is unavailable or reliable, we may estimate the distribution of
the variable based on the values of the variable for peers.
- Pick a Distribution and Estimate the Parameters: When neither historical nor cross-sectional data
provide adequate insight, subjective specification of a distribution along with related parameters is
the appropriate approach.
Step 3: Check for Correlation across Variables: When there is a strong correlation between variables, we can
either (a) Allow only one of the variables to vary i.e. it makes sense to focus on the impact that has
the bigger impact on value or (b) build the rules of correlation into the simulation (this necessities
more sophisticated simulation packages). As with the distribution, the correlations can be estimated
by looking at the past.
Step 4: Run the Simulation: Means randomly drawing variables from their underlying distributions and then
using them as inputs to generate estimated values. This process may be repeated to yield thousands
of estimates of value, giving a distribution of the investment's value, though the marginal
contribution of each simulation drops off as the no. of simulations increases. The no. of simulations
needed for a good output is driven by:
- No. of Probabilistic Inputs: The larger the no. of inputs that have probability distributions attached
to them, the greater will be the required no. of distributions.
- Characteristics of Probability Distributions: The greater the variability in types of distributions, the
greater the no. of simulations needed. Conversely, if all variables are specified by one distribution
(e.g. normal), then the no. of simulations needed would be lower.
- Range of Outcomes: The greater the potential range of outcomes on each input, the greater will be
the no. of simulations.
Advantages of Simulations
1. Better Input Quality: Superior inputs are likely to result when an analyst goes through the process of
selecting a proper distribution for critical inputs rather than relying on single best estimate.
2. Provides a Distribution of Expected Value rather than a Point Estimate: The distribution of an investments
expected value provides an indication of risk in the investment.
Disadvantages of Simulations
1. Garbage In, Garbage Out: Regardless of the complexities employed in running simulations, if the
underlying inputs are poorly specified, the output will be low quality. It is also worth noting that simulations
require more than a passing knowledge of statistical distributions and their characteristics; analysts who
cannot assess the difference between normal and lognormal distributions shouldn't be doing simulations.
55
2. Real data may not fit the requirements of statistical distributions which may yield misleading results.
3. Non-Stationary Distributions: Input variable distributions may change over time, so the distribution and
parameters specified for a particular simulation may not be valid anymore. For example, the mean and
variance estimated from historical data for an input that is normally distributed may change for the next
period.
4. Changing Correlation across Inputs: In the third simulation step, we noted that correlation across input
variables can be modeled into simulations. However, this works only if the correlations remain stable and
predictable. To the extent that correlations between input variables change over time, it becomes far more
difficult to model them.
Simulations with Constraints
1. Book Value Constraints: There are two types of restrictions on book value of equity that may call for risk
hedging:
- Regulatory Capital Requirements: Banks and insurance companies are required to main adequate levels
of capital. Violations of minimum capital requirements are considered serious and could threaten the very
existence of the firm.
- Negative Book Value for Equity: In some countries, negative book value of equity may have serious
consequences like in the European countries.
2. Earnings and Cashflow Constraints: Earnings or cashflow constraints can be imposed internally to meet
analyst expectations or to achieve bonus targets. Earnings constraints can also be imposed externally, such
as a loan covenant. Violating such a constraint could be very expensive for the firm.
3. Market Value Constrains: Market value constraints seek to minimize the likelihood of financial distress or
bankruptcy for the firm, by incorporating the costs of financial distress in a valuation model for the firm
e.g. stress testing and VaR.
Fig 1: Risk Types
Simulations and Decision Trees consider
all possible states of the outcome and
hence the sum of probabilities is 1.
Scenario Analysis doesn't consider full
spectrum of outcomes and hence the
combined probability of the outcomes is
less than 1.
Decision Trees and Simulation can be used as Complements or as Substitutes for risk-adjusted valuation.
Scenario Analysis doesn't include the full spectrum of outcomes and therefore can only be used as a
Complement to risk-adjusted valuation. If used as a substitute, the cashflows in an investment are discounted
at R f rate and then the expected value obtained is evaluated in conjunction with the variability obtained from
the analysis.
1
Corporate Finance
PAGE NOS. 62
CHAPTER 19
VOL. 3
CHAPTERS. 5
Capital Budgeting
In the United States, most companies use Straight-Line Depreciation (SL) for financial reporting and the
Modified Accelerated Cost Recovery System (MACRS) for tax purpose. For capital budgeting purposes, we
should use the same depreciation method used for tax reporting since capital budgeting analysis is based on
after-tax cashflows and not accounting income.
Fig 1: Recovery Allowance Percentage for Personal Property
*
↶
HALF
*
*
↶
HALF
↶
HALF
(3-Year, 5-Year, 7-Year, 10-Year): The Depreciation is Double-Declining Balance
with a switch to SL when optimal* and with a half-year convention.
(15-Year, 20-Year): The Depreciation is 150%-Declining Balance with a switch to SL
when optimal* and with a half-year convention.
(27.5-Year, 39-Year): SL Depreciation.
The effect pf this is to extend the recovery period of a 3-Year class asset to four
calendar years and a 5-year asset to six calendar years. However, with a half-year
convention, the asset is assumed to be in service for only 6 months during the first
year (only one-half of the depreciation is allowed the first year) and 6 months
during the last year of the service (e.g. 3-Year: 1/2, 1, 1, 1/2).
The depreciable basis is equal to the purchase price plus any shipping or handling and installation costs. The
basis is not adjusted for salvage value regardless of whether the accelerated or SL method is used. However,
the formula for computing depreciation expense differs between SL and accelerated depreciation.
∴ Accelerated Depreciation generally improves the NPV of a capital project compared to SL Depreciation.
Accelerated Depreciation will create higher after-tax cashflows for the project earlier in the project's life as
compared to SL Depreciation.
2
Example 1:
Suppose a piece of equipment costs $1000,000 and has a salvage value of $200,000. Find the depreciation
each year assuming the equipment has a life of a 5 years. Let's also compute the book value for each year
and the after-tax salvage value, if you sell the equipment in Year 3 for (a) $425,000 and (b) $200,000. Assume
that the tax rate is 35%.
1 Year
2 Year
3 Year
4 Year
5 Year
6 Year
1000,000
- 200,000
800,000
- 320,000
480,000
- 190,000
290,000
- 120,000
170,000
- 110,000
60,000
- 60,000
0
(1000,000 x 0.2)
(1000,000 x 0.32)
(1000,000 x 0.19)
(1000,000 x 0.12)
(1000,000 x 0.11)
(1000,000 x 0.06)
(a) Sales Price - (Sales Price - Bv) x Tax Value
= 425,000 - (425,000 - 290,000) x 0.35 = 425,000 - 47,250 = 377,750
Capital Gain
Tax Loss
(b) Sales Price - (Sales Price - Bv) x Tax Value
= 200,000 - (200,000 - 290,000) x 0.35 = 200,000 - ( -31,500) = 231,500
Capital Gain
Tax Savings
Example 2:
A switch from SL to Accelerated Depreciation would:
A. Increase the NPV and decrease the first year operating income after taxes.
B. Increase the first year operating income after taxes and decrease the NPV.
C. Increase both the NPV and first year operating income after taxes.
✓
Example 3:
For the Sailboat Project, what will be the effects of using accelerated depreciation instead of SL depreciation:
✓
A.
B.
C.
NPV
Increase
Increase
Decrease
Total Net CF in Terminal Year
Increase
Decrease
Increase
Capital Budgeting Pitfalls
1. Failing to Incorporate Economic Responses into the Analysis: For example, if a profitable project is in an
industry with low barriers to entry, competitors may undertake a similar project, lowering profitability.
2. Misusing Capital Budgeting Templates: Since hundreds or even thousands of projects need to be analyzed
over time, corporations have standardized capital budgeting templates for managers to use in evaluating
projects. However, the template may not be an exact match for the project, resulting in estimation errors.
3
3. Pet Projects of Senior Management: Projects that have the personnel backing of influential member of
senior management may contain only optimistic projections that make the project appear more profitable
than it really is. In addition, the project may not be subjected to the same level of analysis as other projects.
4. Basing Investment Decisions on EPS Income or ROE: Managers whose incentive compensation is tied to
increasing EPS or ROE may avoid positive long-term NPV investments that have the short-run effect of
reducing EPS or ROE.
5. Using the IRR Criterion for Project Decisions: Using IRR may result in conflicts with the NPV approach for
mutually exclusive projects.
6. Bad Accounting for Cashflows: In analyzing a complicated project, it is easy to omit relevant cashflows,
double count cashflows and mishandle taxes.
7. Misestimation of Overhead Costs
8. Using an Incorrect Discount Rate
9. Politics involved with Spending the entire Capital Budget: Many managers try to spend their entire capital
budget each year and ask for an increase for the following year. In a company with a culture of maximizing
shareholder value, managers will return excess funds whenever there is a lack of positive NPV projects and
make a cash for expanding the budget when there are multiple positive NPV opportunities.
10. Failure to generate Alternative Investment Ideas
11. Improper handling of Sunk and Opportunity Costs: Managers should not consider sunk costs in the
evaluation of a project because they are not incremental cashflows. Managers should always consider
opportunity costs because they are incremental. However, in reality, many managers do this incorrectly.
Fig 2: Expansion vs. Replacement Project
0
t
①
②
Outlay
OCF
Expansion Project
Terminal Year's CF
③
TNOCF
Replacement Project
Def: Expansion Project is an independent
investment that doesn't affect the CFs for
the rest of the company.
1. Initial Investment Outlay
Includes Shipping & Installation ↶
Outlay = FCInv + NWCInv ↷
NWCInv =
T
△ Non-Cash CA - △ Non-Cash CL = △ NWCInv
Cash is excluded because it is generally
assumed not to be an operating asset.
[Additional financing
If +ve NWCInv : Cash Outflow : If > 0
is required]
If -ve NWCInv : Cash Inflow : If < 0 [Project frees-up cash]
Def: Replacement Project must deal with the
difference between the CFs that occurs
with the new investment and the CFs that
would have occurred for the investment
being replaced.
1. Initial Investment Outlay
Outlay = FCInv + NWCInv - Sal 0 + T (Sal 0 - B0 )
4
2. After-Tax Operating Cashflow
OCF = (S - C - D) (1 - T) + D
EBIT
2. After-tax Operating Cashflow
[New-Old]
△ OCF = (△S - △C) (1 - T) + △DT
= (S - C) (1 - T) + TD
EBITD
We can account for depreciation either by
adding it back to net income from the project
(as in the first CF formula) or by adding the
tax savings caused by depreciation back to
the project's after-tax gross profit (as in the
second formula). In general, a higher
depreciation expense will result in greater
tax savings and high cashflows. How? Cause
we add (+) depreciation to CFO.
3. Terminal Year After-Tax Non-Operating
Cashflow
After-tax Salvage Value
TNOCF = Sal T+ NWCInv - T (SalT - B T)
Pre-tax cash proceeds
from sale of fixed capital
on termination date
3. Terminal Year After-Tax Non-Operating
Cashflow
TNOCF = (Sal T(New) - Sal T(Old) ) + NWCInv
- T [(Sal T(New) - B T(New) )
- (Sal T(Old) - B T(Old) )]
∴ Interest is not included in OCFs for capital budgeting purposes because it is incorporated into the project's
cost of capital.
Example 4: [Expansion Project]
Mayco Inc. would like to set up a new plant i.e. expand. Currently, Mayco has an option to buy an existing
building at a cost of $24,000. Necessary equipment for the plant will cost $16,000, including installation costs.
The equipment falls into a MACRS 5-Year class. The building falls into a MACRS 39-Year class. The project
would also require an initial investment of $12,000 in net working capital. The initial working capital
investment will be made at the time of the purchase of the building and equipment. The project's estimated
economic life is 4 years. At the end of that time, the building is expected to have a market value of $15,000
and a book value of $21,816, whereas the equipment is expected to have a market value $4000 and a book
value of $2720. Under MACRS, the pre-tax depreciation for the building and equipment is, Year 1: $3512,
Year 2: $5744, Year 3: $3664 and Year 4: $2544.
Annual sales will be $80,000. The production department has estimated that variable manufacturing costs will
total 60% of sales and that fixed overhead costs, excluding depreciation will be $10,000 a year [Costs: (0.6)
(80,000) + 10,000 = 58,000]. Depreciation expense will be determined for the year in accordance with the
MACRS rate. Mayco's marginal federal-plus-state tax rate is 40%, its cost of capital is 12% and for capital
budgeting purposes, the company's policy is to assume that OCFs occur at the end of each year. The plant will
begin operations immediately after the investment is made, and the first OCFs will occur exactly one year
later. Determine whether the project should be accepted using NPV analysis.
1. Initial Outlay = Price of Building + Price of Equipment + NWCInv
= $24,000 + $16,000 + 12,000
= $52,000
5
2. OCF = (S - C) (1 - T) + DT
OCF 1 = [(80,000 - 58,000) (0.6)] + (3512) (0.4) = $ 14,605
OCF2 = [(80,000 - 58,000) (0.6)] + (5744) (0.4) = $ 15,498
OCF3 = [(80,000 - 58,000) (0.6)] + (3664) (0.4) = $ 14,666
OCF4 = [(80,000 - 58,000) (0.6)] + (2544) (0.4) = $ 14,218
3. First calculate the after-tax terminal CFs associated with the building and the equipment separately:
CF for Building = 15,000 - 0.4 (15,000 - 21,816) = $ 17,726
CF for Equipment = 4000 - 0.4 (4000 - 2720) = $ 3488
Then include the return of NWCInv = TNOCF = 17,726 + 3488 + 12,000 = $ 38,214
NPV = - 52,000 + 14,605 + 15,498 + 14,666 + 14,218 + 33,214 = $ 13,978
(1.12)1 (1.12) 2 (1.12) 3
(1.12) 4
IRR = 21.9%
Decision: Since NPV > 0 and IRR > 12%, Mayco should accept the expansion project.
Example 5: [Replacement Project]
Suppose Mayco wants to replace an existing printer with a new high-speed copier. The existing printer was
purchased 10 years ago at a cost of $15,000. The printer is being depreciated using SL basis assuming a
useful life of 15 years and no salvage value (i.e. its annual depreciation is $1000). If the existing printer is not
replaced it will have zero market value at the end of its useful life. The old printer's current market value is
$2000, which is below its $5000 book value.
The new high-speed copier can be purchased for $24,000 (including freight and installation). Over its 5-Year
life, it will reduce labor and raw materials usage sufficiently to cut annual operating costs from $14,000 to
$8000. It is estimated that the new copier can be sold for $4000 at the end of 5 years; this is its estimated
salvage value. If the new copier is acquired, the old printer will be sold to another company. The company's
marginal federal-plus-state tax is 40% and the replacement copier is of slightly below-average risk. Net
working capital requirements will also increase by $3000 at the time of replacement. By an IRS ruling, the new
copier falls into a 3-Year MACRS class. The project's cost of capital is set at 11.5%. Under the MACRS system,
the pre-tax depreciation for the equipment is, Year 1: $7920, Year 2: 10,800, Year 3: $3600, Year 4: $1680 and
Year 5: $0. Determine whether the project should be accepted using NPV analysis.
1. Initial Outlay = $24,000 + $3000 - $2000 +0.4 ($2000 - $5000)
= $ 23,800
△ △
△
2. OCF = ( S - C) (1 - T) + DT
OCF 1 = [0 - (-6000)] (1 - 0.4) + (7920 - 1000) (0.4) = $6368
OCF 2 = [0 - (-6000)] (1 - 0.4) + (10,800 - 1000) (0.4) = $7520
OCF 3 = [0 - (-6000)] (1 - 0.4) + (3600 - 1000) (0.4) = $4640
OCF 4 = [0 - (-6000)] (1 - 0.4) + (0 - 1000) (0.4) = $3200
3. TNOCF = (4000 - 0) + 3000 - 0.4 [(4000 - 0) - 0] = $5400
NPV = - 23,800 + 6368 + 7520 + 4640 + 3872 + 3200 + 5400 = - $ 1197.28
(1.115)1 (1.115) 2 (1.115) 3 (1.115) 4
(1.115) 5
IRR = 9.46%
Decision: Since NPV < 0 and IRR < 11.5%, Mayco shouldn't explore the printer with the new copier.
6
Example 6:
Maria Hernandez's financial predictions for the project are, fixed capital equipment outlay is $2,750,000. At
the beginning of the project, a required increase in current assets of $200,000 and a required increase in
current liabilities of $125,000. SL depreciation to zero over a 5-Year life. Incremental annual unit sales for
3000 at a unit price of $600. Annual fixed cash expenses of $125,000; variable cash expenses of $125 per unit.
The capital equipment is expected to be sold for $450,000 at the end of Year 5. At the end of the project, the
net working capital investment will be recovered. Tax rate of 40%. Based on the CAPM, the required rate of
return is 12%. Calculate OCF, TNOCF and what would be the effect on the project's NPV if the initial fixed
capital equipment outlay increased from $2,750,000 to $3000,000 everything else held constant?
↶
1. OCF = (S - C - D) (1 - T) + D
= (1,800,000 - 125,000 - 375,000 - 550,000) (1 - 0.4) + 550,000
2,750,000
= $ 1000,000
5
2. TNOCF = Sal T + NWCInv - T (Sal T - B T)
= 450,000 + (200,000 - 125,000) - 0.4 (450,000 - 0)
= 450,000 + 75,000 - 180,000 = $ 345,000
NPV = - 250,000 + 20,000 = - $ 177,904;
(1.12) 5
Example 7:
Year End Projections
Capital Investment Required
Additional NWCInv
Depreciation
2013
10.36
2.2
0.21
Since 3000,000 = $ 600,000
5
2014
2015
2016
2017
2018
0.62
0.21
0.28
0.21
0.28
0.21
0.19
0.21
0.12
0.21
Weinberger suggests calculating the projects NPV with an assumption of selling the division for $12,000,000
at the end of 2018 (note that capital gains are fully taxed at 32%) and recapturing all working capital. If
Weinberger is correct about the selling price of the division, the after-tax non-operating cashflow from the
sale at the end of 2018 will be?
Bv of Assets = Initial Capital Investment - Accumulated Depreciation = 10.36 - (5 x 0.21) = $ 9.31
Capital Tax Gains = (12 - 9.31) x 0.32 = $ 0.8608
Net CF Inflow from Sale = 12 - 0.8608 = $ 11.14
After-tax Non-Operating Cashflow = 11.14 + (2.2 + 0.62 + 0.43 + 0.28 + 0.19 + 0.12) = $ 14.98
Example 8:
When assembling the cashflows to calculate an NPV or IRR, the project's after-tax interest expense:
A. IRR calculation, but not the NPV calculation
B. Both the NPV calculation and the IRR calculation
C. Neither the NPV calculation nor the IRR calculation
✓
Example 9:
An analyst has collected the following information on a replacement project:
7
Purchase Price of the New Machine
Shipping and Installation Charge
Sale Price of Old Machine
Book Value of Old Machine
Inventory , if the New Machine is Installed
Accounts Payable , if the New Machine is Installed
Marginal Tax Rate
↑
↑
$
8000
2000
6000
2000
3000
1000
25%
The initial cashflow is closest to: 10,000 + (3000 - 1000) - 6000 + 0.25 (6000 - 2000) = $ 7000
Effects of Inflation on Capital Budgeting Analysis
1. Analyzing normal or real cashflows: Nominal cashflows should be discounted at a nominal discount rate,
while real cashflows should be discounted at a real discount rate.
(1 + Nominal Rate) = (1 + Real Rate) (1 + Inflation Rate)
2. Changes in inflation affect project profitability: If inflation is higher than expected, future project cashflows
are worth less and the value of the project will be lower than expected. If inflation is lower than expected,
future cashflows from the project will be worth more, effectively increasing the project's value.
3. (i) If inflation is higher than expected, increases corporation's real taxes and reduces the value of the
depreciation tax shelter; unless the tax system adjusts depreciation for inflation.
(ii) If inflation is lower than expected, decreases corporation's real taxes and increases the value of the
depreciation tax shelter.
For Corporations
4. (i) Higher than expected inflation shifts wealth from Bondholders
Corporations [Real Interest Expense ]
(ii) Lower than expected inflation shifts wealth from Corporations
Bondholders [Real Interest Expense ]
→
→
↓
↓
5. Finally, inflation doesn't effect all revenues and costs uniformly. The company's after-tax cashflows will be
better or worse than expected depending on how particular sales outputs or cost inputs are affected.
lllll
Project Analysis and Evaluation
Upto this point, we have largely ignored the issue of accounting for risk. We will introduce risk analysis
in two ways. The first is accounting for risk on a standalone basis and the second is accounting for risk
on a systematic basis.
lllll
Mutually Exclusive Projects with Unequal Lives
When two projects are mutually exclusive, the firm may choose one project or the other, but not
both. If mutually exclusive projects have different lives and the projects are expected to be replaced
indefinitely as they wear out, an adjustment needs to be made in the decision-making process and
is called a 'Replacement Chain'. The key is to analyze the entire chain and not just the first link in the
chain. There are two procedures to make this adjustment:
(i) Least Common Multiple of Linear Approach (LCM)
(ii) Equivalent Annual Annuity Approach (EAA)
Example 10:
Mayco Inc. is planning to modernize its production facilities. Mayco is considering the purchase of either
8
(i) a book press with a useful life of 6 years or (ii) an offset printer, which has a useful life of 3 years. NPVs and
IRRs are given for both of these mutually exclusive projects. Calculate both LCM and EAA approaches.
1
2
3
4
5
6
)
)
)
) )
) )
) )
0
CF0
CF 1
CF 2
CF3
CF4
CF 0
(i) Book Press
(ii) Offset Printer
CF 5
CF 1
Book Press
CF 6
CF 2
Offset Printer
CF 3
0
1
2
3
4
5
6
-20,000
4000
7000
6500
6000
5500
5000
0
1
2
3
-10,000
3500
6500
6000
NPVBP @12% = $3245.47
IRR BP
= 17.5%
NPVOP @12% = $2577.44
IRR OP
= 25.2%
Least Common Multiple (LCM): Under LCM, the analyst extends the time horizon of analysis so that the lines
of both projects will divide exactly into the horizon. For example, if two
projects have lives of 8 and 10 years, the LCM of lives is 40 years, not 80. Both
8 and 10 are exactly divisible into 40.
NPVBP = $3245.47 > NPVOP = $2577.44
However, the IRRs recommend the opposite decision because the IRR of the
offset printer is larger than the IRR of the book press. To make the
comparison meaningful, we can find the NPVs for the two projects over the
LCM of lives.
a. Replacement Chain for Offset Printer
0
1
2
3
4
5
6
-10,000
3500
6500
6000
- 10,000
- 4000
3500
6500
6000
NPVOP @12% = $ 4412.01
IRR OP
= 25.2%
Since the $ 4412 extended NPV of two chained-together 3-Year printer
(6 years total) is greater than the $ 3245.5 NPV of the book press, the
printer should be selected.
b. Replacement Chain NPVs
0
2577.44
1
2
3
2577.44
4
5
6
NPVOP = 2577.44 + 2577.44
(1.12) 3
= $ 4412.01
We again come to the conclusion that the printer should be selected.
9
Equivalent Annual Annuity (EAA): For an investment project with an outlay and variable cashflows in the
future, the project NPV summarizes the equivalent value at time '0'. For this
same project, the EAA is the annuity payment (series of equal annual
payments over the project's life) that is equivalent in value to the NPV.
Step 1: Find each project's NPV
NPV BP = $ 3245
NPV OP = $ 2577
Step 2: Find an annuity (EAA) with a present value equal to the project's NPV
over its individual life at the WACC.
EAA BP
EAA OP
Pv = -3245
Pv = -2577
Fv = 0
Fv = 0
n=6
n=3
I/Y = 12%
I/Y = 12%
Compute PMT = 789
Compute PMT = 1073
Step 3: Select the project with the highest EAA. In this example, the printer
should be accepted since,
EAA OP > EAA BP
∴ The two methods will lead to the same conclusion.
lllll
Capital Rationing
Ideally, firms will continue to invest in positive return NPV projects until the marginal returns equal
the marginal cost of capital. Should a firm have insufficient capital to do this, it must ration its
capital i.e. Capital Rationing is the allocation of a fixed amount of capital among the set of available
projects that will maximize shareholder wealth. A firm with less capital than profitable (i.e. positive
NPV) projects should choose the combination of projects it can afford to find that has the greatest
total NPV. Note that capital rationing is not the optimal decision from the firm's perspective. More
value would be created by investing in all positive NPV projects. Therefore, capital rationing violates
market efficiency because society's resources are not allocated to their best use (i.e. to generate the
highest return).
(i) Hard Capital Rationing: Occurs when the funds allocated to managers under the capital budget
cannot be increased.
(ii) Soft Capital Rationing: Occurs when managers are allowed to increase their allocated capital
budget if they can justify to senior management that the additional funds
will create shareholder value.
Example 11:
Mayco has a $2000 capital budget and has the opportunity to invest in 5 different projects. The initial
investment and NPV of the projects are described in the following figure. Determine in which projects Mayco
should invest.
F
G
H
I
J
Investment Outlay
- $1200
- $1000
- $800
- $450
- $200
NPV
$500
$480
$300
$150
$40
10
All of the projects are profitable, but with a capital budget of only $ 2000, Mayco should choose projects G, H
and I that have a combined NPV of $ 820. Remember, the goal with capital rationing is to maximize the overall
NPV within the capital budget, not necessarily to select the individual projects with the highest NPV.
Concepts
(Underestimated)
SML
C
R
1. R project = R f + β project [E(R m) - R f ]
Unsystematic
Risk
R
Systematic
Risk
f
.
.B
A
(Overestimated)
.
β
Beta risk is based on the equation of the CAPM or SML (Security Market Line), which defines a project's
required rate of return (discount rate) using the above equation. Here, the required rate of return
sometimes is called 'Hurdle Rate' is specific to the risk of the project and assumes the project is 100%
equity financed. Hurdle rates vary from project to project.
Example 12:
Compute the NPV for a 3-Year project that has a beta of 1.2. The initial investment is $1000 and the project
will generate annual cashflows of $400. Use a risk-free interest rate of 8% and an expected market return of
13%.
R i = R f + β i [E(R m ) - R f ] = 8% + 1.2 (13% - 8%) = 14%
NPV = -1000 + 400 + 400 + 400 = - $ 71.35
(1.14)1 (1.14) 2 (1.14) 3
Example 13:
A company is analyzing two projects. Project A has a project beta of 1.2 and Project B has a beta of 0.6. The
company's WACC is 10%. The risk-free rate is 5% and market risk premium 9%. If the company incorrectly
uses the company's WACC to calculate NPV of both projects, will it overestimate or underestimate NPV?
R A = 5% + 1.2 (9%) = 15.8%
RB = 5% + 0.6 (9%) = 10.4%
If the company uses the overall WACC of 10%, it will overestimate the value of both projects because the
WACC is too low to reflect the higher risk of each project.
2. Real Options: allow managers to make future decisions that change the value of capital budgeting
decisions made today. Real options are similar to financial call and put options in that they give the option
holder the right, but not the obligation to make a decision. The difference is that real options are based on
real assets rather than financial assets and are contingent on future events. Real options offer managers
flexibility that can improve the NPV estimates for individual projects. A combination of optimal current and
future decisions is what will maximize company value. Real option analysis tried to incorporate rational
future decisions into the assessment for evaluating the profitability of an investment with real options;
examples of different approaches include (i) Determine NPV without the option, (ii) Calculate the NPV
without the option and the estimated value if the real option i.e. overall NPV = NPV (based on DCF) - Option
Cost + Option Value, (iii) Use Decision Trees and (iv) Use Option Pricing Models.
11
Types of Real Options include the following:
a. Timing Options: allow a company to delay making an investment with the hope of having better
information in the future.
b. Abandonment Options: are similar to put options. They allow management to abandon a project if the
present value of the incremental cashflows from exiting a project exceeds the present value of the
incremental cashflows from continuing a project.
c. Expansion Options: are similar to call options. Expansion options allows a company to make additional
investments in a project, if doing so creates value.
[Sizing Option includes both Abandonment Options and Expansion Options]
d. Price-Setting Options: allow the company to change the price of a product. For example, the company
may raise price if demand for a product is high in order to benefit from that demand without increasing
production.
e. Production-Flexibility Options: may include paying workers overtime, using different materials as inputs
or producing a different variety of product.
[Flexibility Options includes both Price-Setting Options and Production-Flexibility Options. It gives
managers choices regarding the operational aspects of a project].
f. Fundamental Options: are projects that are options themselves because the payoffs depend on the price
of an underlying asset. For example, the payoff for a copper mine is dependent on the market price for
copper.
Example 14: [Production-Flexibility Option]
Black Pearl Yachts estimated that the NPV of the expected cashflows from a new production facility to
produce classic yachts is negative $8000,000. Black Pearl's production manager is evaluating an additional
investment of $5000,000 in equipment that would give management the flexibility to switch between classic,
deluxe and elite models of yachts depending on demand. The option to switch production among models of
yachts is estimated to have a value of $15,000,000. Evaluate the profitability of the project including the real
option.
Overall NPV = Project NPV - Option Cost + Option Value
= - 8000,000 - 5000,000 + 15,000,000
= $ 2000,000
Without option, the NPV of the production facility is negative. However, the real option adds enough value to
make the overall project profitable.
Example 15: [Abandonment Option]
Nybeeg Systems is considering a capital project with the following characteristics: The initial outlay is
$200,000 and project life is 4 years. Annual after-tax operating cashflows have a 50% probability of being
$40,000 for the four years and a 50% probability of being $80,000. Salvage value at project termination is
zero. The required rate of return is 10%. In one year, after realizing the first-year cashflow, the company has
the option to abandon the project and receive the salvage value of $150,000.
a. Compute the NPV assuming no abandonment: 0.5 (40,000) + 0.5 (80,000) = $60,000
Expected
NPV = - 200,000 + 60,000 + 60,000 + 60,000 + 60,000 = - $ 9808
(1.1)1
(1.1) 2
(1.1) 3
(1.1) 4
The project should be rejected because it has a negative NPV.
12
b. What is the optimal abandonment strategy? Compute the project NPV using that strategy.
Success
NPV = - 200,000 + 80,000 + 80,000 + 80,000 + 80,000 = $ 53,589
(1.1) 1
(1.1) 2
(1.1) 3
(1.1) 4
Fail
NPV = - 200,000 + 40,000 + 150,000 = - $ 27,273
(1.1)
Now the expected NPV is then, NPV = 0.5 (53,589) + 0.5 (27,273) = $ 13,158
Optimal abandonment raises the NPV by 13,158 - (-9808) = $ 22,966. The abandonment option has made the
project viable; we should now accept it because the NPV is greater than 0.
3. Accounting and Economic Income: measures are alternatives to the basic discounted incremental
cashflows approach used in the standard capital budgeting model.
Economic Income: is equal to the after-tax cashflow plus the change in the investment's market value. As
with the basic capital budgeting model, interest (finance charge) is ignored for cashflow
calculations and is instead included as a component of the discount rate.
Economic Income = Cashflow + (Ending Mv - Beginning Mv)
Economic Income = Cashflow - Economic Depreciation
(Beginning Mv - Ending Mv)
↶
or
↷
Loss in Mv of an asset
Accounting Income: is the reported net income on a company's financial statements that results from an
investment in a project. Accounting depreciation is based on the original cost (not Mv)
of the investment. Financing costs e.g. interest expense are considered as a separate
line item and substracted out to arrive at net income.
Example 16:
Blue Wave is a startup company that uses a brushless machine to wash cars. Suppose Blue Wave makes an
initial investment in equipment of $400,000. The equipment is depreciated on a SL basis over 4 years to a '0'
book value. At the end of four years, the equipment will have a salvage value of $10,000. Blue Wave's
marginal tax rate is 30%. The company is financed with 50% debt and 50% equity. The debt carries an interest
rate of 6% and the cost of equity is 19.8%. Blue Wave only expects to operate for the four years duration of
the project, so all income is distributed to bondholders and stockholders. The company plans to maintain a
50% debt to value ratio.
Sales
Variable Expenses
Fixed Expenses
Depreciation
Operating Income (EBIT)
Taxes at 30%
Operating Income after Taxes
Add Back Depreciation
After-tax Operating Cashflow
Salvage Value
Tax on Salvage Value
After-tax Salvage Value
Total After-tax Cashflow
Year 1
Year 2
Year 3
400,000
(150,000)
(40,000)
(100,000)
110,000
(33,000)
77,000
100,000
177,000
450,000
(175,000)
(40,000)
(100,000)
135,000
(40,500)
94,500
100,000
194,500
450,000
(175,000)
(40,000)
(100,000)
135,000
(40,500)
94,500
100,000
194,500
177,000
194,500
194,500
Year 4
400,000
(150,000)
(40,000)
(100,000)
110,000
(33,500)
77,000
100,000
177,000
10,000
(3000)
7000
184,000
13
The project's required rate of return (i.e. its cost of capital) is the WACC. Calculate the project's NPV, Economic
Income, Accounting income and discuss the differences between the various income measures.
WACC = (0.198) (0.5) + (0.06) (1 - 0.3) (0.5)
= 0.12 or 12%
158,035
155,054
138,441
116,935
NPV = - 400,000 + 177,000 + 194,500 + 194,500 + 184,500 = $ 168,467
(1.12)1
(1.12) 2
(1.12)3
(1.12) 4
568,467
Economic Income
e.g. Year 1
Beginning Mv (158,035 + 155,054 + 138,441 + 116,935)
+ 194,500 + 194,500)
Ending Mv (194,500
(1.12)1
(1.12) 2 (1.12) 3
Change in Mv
After-tax Cashflow
Economic Income
Economic Rate of Return (68,215 / 568,467)
Year 1
Year 2
Year 3
Year 4
568,467
459,682
(108,785)
177,000
68,215
12%
459,682
320,344
(139,338)
194,500
55,162
12%
320,344
164,286
(156,058)
194,500
38,442
12%
164,286
0
(164,286)
184,000
19,714
12%
The economic Income rate of return for each year is precisely equal to the project's WACC. This makes sense
because the WACC is the discount rate used to determine the value of the company.
Accounting Income
e.g. Year 1
Sales
Variable Expenses
Fixed Expenses
Depreciation
Operating Income (EBIT)
Interest Expense (568,467 x 0.5 x 0.06)
Earnings before Tax (EBT)
Taxes at 30%
Net Income before Salvage
After-tax Salvage Value
Accounting Income
Year 1
Year 2
Year 3
Year 4
400,000
(150,000)
(40,000)
(100,000)
110,000
(17,054)
92,946
(27,884)
65,062
450,000
(175,000)
(40,000)
(100,000)
135,000
(13,790)
121,210
(36,363)
84,847
450,000
(175,000)
(40,000)
(100,000)
135,000
(9610)
125,390
(37,617)
87,773
65,062
84,847
87,773
400,000
(150,000)
(40,000)
(100,000)
110,000
(4928)
105,072
(31,522)
73,550
7000
80,550
- Accounting Depreciation is based on the original cost of the investment, while Economic Depreciation is
based on the market value of the asset.
- Interest Expense is deducted from the Accounting Income figure. Interest Expense is ignored when
computing Economic Income because it is reflected in the WACC.
- The Economic Depreciation for the project is much larger than the Accounting Depreciation, resulting in an
Economic Income amount that is much smaller than Accounting Income.
4. Economic Profit: or EVA is a measure of profit in excess of the dollar cost of capital invested in a project.
The economic profit (EP) approach focuses on returns to all suppliers of capital, which includes both debt
and equity holders and therefore all appropriate discount rate is the WACC.
14
Before depreciation is added back,
but include the salvage value
EP = NOPAT - $ WACC
The NPV based on Economic Profit is called the Market Value Added (MVA).
↶
∞
MVA = Σ
EPt
(1 + WACC) t
t=1
Mv - Capital Invested
Example 17:
Using data from Example 16, calculate the economic profit and NPV.
Capital
NOPAT
$ WACC
Economic Profit
Year 1
Year 2
Year 3
Year 4
400,000
77,000
48,000
29,000
300,000
94,500
36,000
58,500
200,000
94,500
24,000
70,500
100,000
84,000
12,000
72,000
NOPAT includes after-tax
gain from sale
MVA = 29,000 + 58,500 + 70,500 + 72,000 = $ 168,467
(1.12)1 (1.12) 2 (1.12) 3 (1.12) 4
Value of the Company = NPV + Initial Investment
= 168,467 + 400,000
= $ 568,467
5. Residual Income: Net Income - Equity Charge
RI t = NI t - (re . B t-1 )
∞
NPV = Σ
RI t
(1 + re ) t
t=1
Residual Income approach focuses only on returns to equity holders, therefore, the appropriate discount
rate is the required return on equity.
Example 18:
Recall that the WACC for the Blue Wave project is 12% and the required return on equity is 19.8%. Beginning
book value of assets on the balance sheet is equal to the initial outlay in the project of $400,000. The book
value is depreciated using SL to 0 over 4 years, so assets decline by $100,000 each year. Liabilities each year
are equal to 50% of the market value of the project. Balance sheet is provided in the following table.
Assets
Liabilities
Book Value
Year 0
Year 1
Year 2
Year 3
Year 4
400,000
284,233
115,767
300,000
229,841
70,159
200,000
160,172
39,828
100,000
82,143
17,857
0
0
0
Calculate the NPV of the Blue Wave project and the company value.
Net Income
Equity Charge
Residual Income
Year 1
Year 2
Year 3
Year 4
65,062
22,922
42,140
84,847
13,891
70,956
87,773
7886
79,887
80,550
3536
77,014
15
NPV = 42,140 + 70,956 + 79,887 + 77,014 = $ 168,467
(1.198)1 (1.198) 2 (1.198) 3 (1.198) 4
Value of the Company = Pv of Residual Income + Equity Investment + Debt Instrument
= 168,467 + 115,767 + 284,233
= $ 568,467
Example 19:
A company has provided the following financial data:
Target Capital Structure is 50% Debt and 50% Equity
After-tax Cost of Debt is 8%
Cost of R/E is estimated to be 13.5%
Cost of Equity is estimated to be 14.5%, if the company issues new common stock
Net Income is $2500
The company is considering the following investment projects
Project
Size of Project
IRR of Project
Project A
Project B
Project C
Project D
Project E
$ 1000
$ 1200
$ 1200
$ 1200
$ 1000
12 %
11.5 %
11 %
10.5 %
10 %
If the company follows a residual dividend policy, its payout ratio will be?
The cost of new equity is always higher than the cost of R/E. Under Pecking Order, internally generated equity
(i.e. R/E) is most favored and external equity (i.e. newly issued shares) is least favored. In this case, the equity
half of these projects can be financed using R/E, so new equity doesn't need to be issued. The cost of R/E is
thus the appropriate rate to use.
WACC = (0.5) (0.08) + (0.5) (0.135)
= 0.1075 or 10.75%
Since projects A, B and C have an IRR greater than WACC, this results in a total capital budget of
$ 1000 + $ 1200 + $ 1200 = $ 3400
Residual Income = NI - Equity Portion
= 2500 - 0.5 (3400)
= 2500 - 1700
= $ 800
Residual Dividend Policy = Residual Income = $ 800 = 0.32 or 32%
NI
$ 2500
6. Claims Valuation Approach: divides operating cashflows based on the claims of debt and equity holders
that provide capital to the company. These debt and equity cashflows are valued separately and then
added together to determine the value of the company. The Claims Valuation method calculates the value
of the company, not the project. This is different from the economic profit and residual income
16
approaches, which calculate both project and company value. The Claims Valuation approach is based on
the balance sheet concept that A = L + E.
Cashflows to Debtholders = Interest and Principle Payments (Discounted at Cost of Debt)
Cashflows to Equityholders = Dividends and Share Repurchases (Discounted at Cost of Equity)
The sum of present value of each stream of cashflows will equal to the value of the company.
Example 20:
Year 1
Year 2
Year 3
Year 4
Principle Payments
Interest Expense
CF to Bondholders
54,392
17,054
71,446
69,669
13,790
83,459
78,029
9610
87,639
82,143
4928
87,071
CF to Equityholders
110,670
115,178
109,744
98,407
Calculate the value of the company, assuming cost of debt is 6% and cost of equity is 19.8%
Pv of CF to Bondholders = 71,446 + 83,459 + 87,639 + 87,071 = $ 284,233
(1.06) 1 (1.06) 2 (1.06) 3 (1.06) 4
Pv of CF to Equityholders = 110,670 + 115,178 + 109,744 + 98,407 = $ 284,234
(1.198) 1 (1.198) 2 (1.198) 3 (1.198) 4
Value of the Company = Pv of CF to Bondholders + Pv of CF to Equityholders
= Mv of Debt + Mv of Equity
= 284,233 + 284,234
= $ 568,467
Whether the EP, RI or Claims Valuation method for determining income is used, the key is that in theory, any
of the three methods should result in the same value for the company. In practice, however, accounting
complications such as Pension Adjustments, Goodwill and Deferred Taxes may complicate the calculation of
income.
17
CHAPTER 20
Capital Structure
Modigliani and Merton Miller (MM)'s study is based on the following simplifying assumptions:
- Capital markets are perfectly competitive: no transaction costs, taxes or bankruptcy costs.
- Investors have homogeneous expectations i.e. same expectations.
- Riskless borrowing and lending at R f rate as long as no bankruptcy costs.
- No agency costs: no conflict on interest between managers and shareholders.
- Investment decisions are unaffected by financing decisions.
Fig 1: MM Propositions
MM Proposition I (Without Taxes)
MM Proposition I (With Taxes)
"The market value of a company is not affected by
the capital structure of the company".
(A company's capital structure is irrelevant in perfect
markets, which assume no taxes)
"The value of the company increases with increasing
levels of debt and the optimal capital structure is
100% debt".
↷
(Here, Equity is more risky)
↷
(Here, Equity is less risky)
Unlevered Firm
D
50%
D
30%
E
50%
E
50%
A = 100%
Marginal
Tax Rate
↑)
VL = VU
(
Value of
Unlevered Firm
MM Proposition II (Without Taxes)
Cost of
Capital
Constant
rE
rE = rA + D (rA - rD ) (1 - t)
increases with
E decrease with
↷
increase in D/E
WACC L = wD . rD + w E . rE
Constant
rA
% Debt in
Cap. Structure
Return on Assets (r A )
WACC L = w D . r D (1 - t) + w E . rE
WACC U = rE
increase in D/E
WACC
rD (1-t)
↷
↷
Return on Assets (rA )
WACC U = rE
increase in D/E
WACC
% Debt in
Cap. Structure
Constant
↷
rE = rA + D (rA - rD )
increases with
E
rD
(compared to
taxes on equity)
"Tax shield provided by debt causes the WACC to
decline as leverage increases".
(As tax increases, WACC decreases. The value of firm
is maximized at the point where the WACC is
minimized at which is 100% debt)
rE
Constant
↑)
Tax Shield (
MM Proposition II (With Taxes)
"The cost of equity increases linearly as the company
increases its proportion of debt financing".
(Since debtholders have prior claim to assets and
income relative to equityholders, K D < K E ; capital
structure is irrelevant, assuming no taxes)
Cost of
Capital
↑)
↷(
More Debt = Larger Interest = Less Taxes = Larger
Tax Shield
Firm Value
(D x Interest Rate
x Tax Rate)
rA
A = 100%
V L = VU + (t x D)
↷
↶
A = 100%
Value of
Leverage Firm
D
50%
E
100%
E
70%
A = 100%
Levered Firm
Constant
18
∴ The cost of equity doesn't rise as fast as it does in the no-tax case. The slope coefficient is (rA - rD ) (1 - t) is
smaller than the slope coefficient (rA - rD ) in the case of no taxes. Interest payments are a pre-tax expense
and are tax deductible, while dividends are paid on after-tax basis.
MM Proposition I and II (Without Taxes) = Firm Value = EBIT
Constant (VU) WACC
Constant
MM Proposition I and II (With Taxes) = Firm Value = EBIT (1 - t)
WACC
Increase with
Constant
Debt (VU)
Constant
Lower cost of debt are
offset by the increased
cost of equity. It now
doesn't effect the overall
value of the firm.
Decrease with
more debt
The risk of the equity depends on two factors: the risk of the company's operations (business risk) and the
degree of financial leverage (financial risk). Business risk determines the cost of capital, whereas the capital
structure determines financial risk. According to MM, the company's cost of capital doesn't depend on its
capital structure but rather is determined by the business risk of the company.
β E = β A + (β A - β D ) D
E
As the proportion of debt rises, β E also rises, the risk of the company defaulting on its debt increases. These
costs are born by the equityholders.
Example 1: [No Tax]
Leverkin Company currently has an all-equity capital structure. Leverkin has an expected operating income of
$5000 and a cost of equity, which is also its WACC 10%. Let us suppose that Leverkin is planning to issue
$15,000 in debt at a cost of 5% in order to buyback $15,000 worth of its equity.
VU = EBIT = 5000 = 50,000 = V L
WACC 0.1 15,000
35,000
D
E
Value of Firm = D + E = rD . D + EBIT - rD . D
rD
rE
= (0.05) 15,000 + 5000 - (0.05) 15,000 = 50,000
0.05
0.12143↷
rE = 0.1 + 15,000/35,000 (0.1 - 0.05)
rA or WACC = wD . rD + wE . rE
= (15,000 / 50,600) (0.05) + (35,000 / 50,000) (0.12143)
= 0.1 or 10%
Example 2: [Tax]
Leverkin Company currently has an all-equity, has an EBIT of $5000 and a WACC, which is also its cost of
equity of 10%. As before, Leverkin is planning to issue $15,000 of debt in order to buyback an equivalent
amount of equity. Now, however, Leverkin pays corporate taxes at a rate of 25%.
VU = EBIT (1 - t) = 5000 (1 - 0.25) = 37,500 ≠ V L
WACC
0.1
19
so, V L = V U + (t x D) = 37,500 + (0.25) (15,000) = 41,250
15,000
D
26,250
E
Value of Firm = D + E = rD . D + (EBIT - rD . D) (1 - t)
rD
rE
= (0.05) 15,000 + [5000 - (0.05) 15,000] (1 - 0.25) = 41,250
0.05
0.12143 ↷
rE = 0.1 + 15,000/26,250 (0.1 - 0.05) (1 - 0.25)
rA or WACC = w D . rD (1 - t) + w E . rE
= (15,000 / 41,250) (0.05) (1 - 0.25) + (26,250 / 41,250) (0.12143)
= 0.09091 or 0.091%
Example 3: [Tax]
EBIT is $400,000 and WACC is 10%. Garth has announced that it plans to abandon the prior policy of all-equity
financing by the issuance of $1000,000 in debt in order to buyback an equivalent amount of equity. Garth's
before-tax cost of debt is 6% and Tax is 30%.
VU = 400,000 (1 - 0.3) = 2,800,000
0.1
VL = VU + (t x D) = 2,800,000 + (0.3) 1000,000 = 3,100,000
VL = D + E
E = VL - D = 3,100,000 - 1000,000 = 2,100,000
rE = 0.1 + (0.1 - 0.06) (1 - 0.3) 1 = 0.1133 or 11.33%
2.1
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Factors Affecting Capital Structure
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Costs of Financial Distress
Costs of Financial Distress are the increased costs a company faces when earnings decline and the
firm has trouble paying its fixed financing costs i.e. interest on debt. The expected costs of financial
distress has two components:
1. Costs of Financial Distress and Bankruptcy: can be direct or indirect. Direct costs of financial
distress include the cash expenses associated with the bankruptcy, such as legal fees and
administrative fees. Indirect costs include foregone investment opportunities and the costs
that result from losing the trust of customers, creditors, suppliers and employers
(i.e. agency costs).
2. Probability of Financial Distress: D , Tax Shield and Probability of Financial Distress .
Lower quality management and corporate governance leads to a higher probability of
financial distress.
Higher expected costs of financial distress tend to discourage companies from using large amounts
of debt in their capital structure, all else equal.
↑
↑
↑
20
Static Trade-Off Theory: seeks to balance the costs of financial distress with the tax shield benefits
from using debt.
VL = VU + (t x D) - Pv (Costs of Financial Distress)
↑
↑
As the rD , the rE , because some of the costs of financial distress are effectively borne by
equityholders. The optimal proportion of debt is reached at the point when the marginal benefit
provided by the tax shield of taking on additional debt is equal to the marginal costs of financial
distress incurred from the additional debt. This point also represents the firm's optimal capital
structure because it is the point that minimizes the firm's WACC and therefore maximizes the value
of the firm.
rE
Cost of
Capital
WACC
rD (1 - t)
0
As the proportion of debt in a business
rises, the cost of debt and equity are
likely to rise to offset the higher risks
associated with higher levels of debt.
These cost increases, negate the cost
savings due to greater use of debt. The
result is U-shaped WACC.
% Debt
Optimal Capital Structure
Every firm will have a different optimal capital structure that depends on each firm's operating risk,
sales risk, tax situation, corporate governance, industry influences and other factors.
Firm Value
Maximum
Firm Value
Costs of Financial Distress
Pv of Tax
Shield
VL
VU
0
% Debt
↶
Optimal Capital Structure
(Not fixed, Continues to Change)
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Agency Costs of Equity
Agency Costs of Equity refer to the costs associated with the conflicts of interest between managers
and owners. The smaller the stake that managers have in the company, the less is their share in
bearing the cost of excessive perquisite consumption or not giving their best efforts in running the
company. Because shareholders are aware of this conflict, they will take steps to minimize these
costs and the net result is called the net 'agency cost of equity'. The better a company is governed,
the lower the agency costs. Good governance practices translate into higher shareholder value
reflecting the fact that managers' interests are better aligned with those of shareholders. Similarly,
the more financially leveraged a company is, the less freedom managers have to either take on
more debt or unwisely spend cash. This is the foundation of Michael Jensen's FCF hypothesis. It
follows that, greater amounts of financial leverage tend to reduce agency costs. The net agency
costs of equity therefore have 3 components:
21
1. Monitoring Costs: These are the costs borne by owners to monitor the management of the
company and include the expenses of the annual report, board of director expenses and
the cost of the annual meeting.
2. Bonding Costs: These are the costs borne by management to assure owners that they are
working in the owner's best interest. These include the implicit cost of non-compete
employment contracts and the explicit cost of insurance to guarantee performance.
3. Residual Loss: may occur even with adequate monitoring and bonding provisions because
such provisions don't provide a perfect guarantee.
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Costs of Asymmetric Information
Asymmetric Information i.e. an unequal distribution of information, arises from the fact that
managers have more information about a company's performance and prospects (including future
investment opportunities) than do outsiders such as owners and creditors. Providers of both debt
and equity capital demand higher returns from companies with higher asymmetry in information
because they have a greater likelihood of agency costs. Being aware of this scrutiny, these investors
will look for management behavior that 'signals' what knowledge management may have.
1. Taking on the commitment to make fixed interest payments through debt financing sends a
signal that management has confidence in the firm's ability to make payments in future.
2. Issuing equity is typically viewed as a negative signal that managers believe a firm’s stock is
overvalued and that the management is looking to generate financing by diluting shares in
the company. The cost of asymmetric information increases as the proportion of equity in
the capital structure increases.
Pecking Order Theory: based on asymmetric information is related to the signals management
sends to investors through its financing choices. It is developed by Myers and
Majluf (1984), suggests that managers choose methods of financing
according to a hierarchy that gives first preference to methods with the least
potential information content (internally generated funds) and lowest
preference to the form with the greatest potential information content
(public equity offerings). In brief, managers prefer internal financing
(e.g. retained earnings); if internal financing is insufficient, managers next
prefer debt and finally equity (e.g. newly issues shares). Therefore, the
Pecking Order Theory predicts that the capital structure is a by-product of
the individual financing decisions.
Miller (1977) argued that if investors face different tax rates on dividend and interest income for their
personal taxes, this may reduce the advantage of debt financing somewhat. If investors face a higher
personal rate of tax on income from debt investments relative to stock investments, they will demand a
higher return on debt, driving up the cost of debt to the company. Why? It can be argued that there is a
higher personal tax on debt income because debt instruments typically provide investors with taxable
interest periodically, whereas taxable income from stock investments could, conceivably be lower because
the tax consequences of investing in non-dividend paying stocks are deferred until the stock is sold.
lllll
International Differences
- Total Debt: Companies in Japan, Italy and France tend to have more total debt in their capital
structure than firms in the US and UK.
- Debt Maturity: Companies in North America tend to use longer maturity debt than companies in Japan
- Emerging Market Differences: Companies in developed markets typically use more long-term debt
and tend to have higher long-term debt to total debt ratios compared to their emerging market peers.
22
1. Institutional and Legal Environment
These factors include taxation, accounting standards, financial reporting and even the presence or lack of
corruption - may effect a company's optimal capital structure.
Strength of Legal System : Weak LS : Greater Agency Costs :
Strong LS : Lower Agency Costs :
↑ Debt (ST Debt)
↓ Debt (LT Debt)
Type of Legal System : Common Law : Better Cap. Protection : Market-Based :
Civil Law : Lower Cap. Protection : ↶Bank-Based :
↓ Debt (LT Debt)
↑ Debt (ST Debt)
(Efficient Information Asymmetry)
↶
Information Asymmetry : High IA : Reduced Transparency :
Low IA : High Transparency :
↑ Debt (ST Debt)
↓ Debt (LT Debt)
In countries where auditors and financial analysts have greater presence
Presence of Auditors is more important in Emerging Markets
Presence of Analysts is more important in Developed Markets
(Lower Tax on Dividend Income)
Taxes :
↓ Tax Rate : Favorable Tax Rate : ↓ r : ↓ Debt
↑ Tax Rate : Unfavorable Tax Rate : ↑ r : ↑ Debt
The benefit from the tax
deductibility of interest encourages
companies to use debt financing
instead of equity financing.
(Higher Tax on Dividend Income)
2. Financial Markets and Banking Sector
These factors include characteristics of the banking sector, as well as the size and activity of the financial
markets.
Liquidity of Capital Markets : High Liquidity : (LT Debt)
Low Liquidity : (ST Debt)
Reliance on Banking System : Market-Based : Low Reliant :
Bank-Based : More Reliant :
Institutional Investor Practice : Greater IP :
(Preferred Habitat)
Lower IP :
↓ Debt
↑ Debt
↓ Debt (LT Debt)
↑ Debt (ST Debt)
3. Macroeconomic Environment
These factors capture the general economic and business environment, addressing the influence of
economic growth and inflation on the capital structure.
↑ I : ↓ Debt (ST Debt)
↓ I : ↑ Debt (LT Debt)
Developed (LT Debt)
GDP Growth : ↑ g : Developing (Equity) : ↓ Debt
(ST Debt) : ↑ Debt
↓g :
Inflation :
↶
Stable Company :
E.g. Utility
: More Tangible Assets : Low Information Asymmetry :
Unstable Company : E.g. Technology/ : Few Tangible Assets : High Information Asymmetry :
Secretive about their products
Pharmaceutical
↑ Debt
↓ Debt
23
CHAPTER 21
Analysis of Dividends and Share Repurchases
Cash Dividends can be distributed to shareholders through regular, extra or liquidating dividends. Other
forms of dividends include stock dividends and stock splits.
Fig 1: Types of Dividends
1. Regular Cash Dividends: Periodic dividend payments made in cash are known as 'Regular Cash Dividends'. Stable
or increasing dividends paid regularly are perceived as a sign of consistent (and growing) profitability. In some
countries, companies can have 'Dividend Reinvestment Plans' (DRP or DRIP) that, at shareholders' requests
automatically invest all or part of regular cash dividend by purchasing shares. The additional shares can be
purchased in the open market DRP or can be newly issues (as in a new issue DRP or Scrip Dividend Scheme in
the UK) by the company.
Advantages
- DRPs may encourage a diverse shareholder base by providing small shareholders an easy means to accumulate
additional shares. They stimulate long-term investment in the company because shareholders might value the
ability to acquire additional shares.
- New-Issue DRPs allow the company to raise new equity capital without the flotation costs associated with
secondary equity issuance using investment bankers.
- DRPs allow the accumulation of shares using 'Cost Averaging' (i.e. you determine a fixed dollar amount to
invest regardless of the market price).
- Participating shareholders typically have no transaction cost in obtaining the additional shares through a DRP.
- Some companies, typically new-issue DRPs, offer the additional benefit to DRP participants of purchasing
shares at a discount (usually 2.5%) to the market price. Note that such discounts dilute the holdings of
shareholders who don't participate in the DRP.
Disadvantages
- A disadvantage to the shareholder is the extra record keeping involved in jurisdictions where capital gains are
taxed. Shares purchased through DRPs change the average cost basis for capital gains tax purposes. If the
share price for the reinvested dividend is higher (lower) than the original purchase, reinvesting the dividend will
increase (decrease) the average cost basis. Either way, detailed records must be kept to accurately compute
gains or losses when shares are sold.
- Cash dividends are fully taxed in the year received even when reinvested, which means the shareholders is
paying tax on cash that is not in hand. Therefore, it makes sense to hold DRPs in tax-sheltered accounts
e.g. retirement accounts.
2. Extra or Special (Irregular) Dividends: Special Dividends are paid under unusual circumstances (e.g. when the
company sells of a division) under the expectation that the dividend is not recurring. Extra Dividends may be
paid if the company had an especially profitable year but doesn't want to commit to a higher ongoing regular
dividend payment.
3. Liquidating Dividends: This is paid by a company when the whole firm or part of the firm is sold, or when
dividends in excess of cumulative retained earnings are paid (resulting in a reduction of stated capital). A
Liquidating Dividend is considered to be a return of capital as opposed to a return on capital i.e. it is a return of
capital rather than a distribution from earnings or retained earnings.
4. Stock Dividends: A non-cash dividend paid in the form of additional shares in known as a stock dividend a.k.a.
'Bonus Issue of Shares'. After payment of a stock dividend, shareholders have more shares and the cost per
share will be lower (while shareholder's proportionate ownership of the company doesn't change). A stock
dividend is accounted for as a transfer of R/E to contributed capital.
5. Stock Splits: These are similar to stock dividends (non-cash) but generally larger in size. A two-for-one stock split
is the same as a 100% stock dividend. Reverse Stock Split is one-for-two stock split is the same as a 100% stock
dividend.
EPS, DPS, Share Price
Stock Split
Reverse Stock Split
↓
↑
Outstanding Shares
↑
↓
P/E, Mv, DPR, Dividend Yield
=
=
24
Fig 2: Stock Split Vs. Reverse Stock Split
Stock Split
Reverse Stock Split
- Typically a stock split is announced after a period in
which the stock price has risen.
- Typically a reverse stock split is announced if the
stock price is low.
- Many investors view the announcement of a stock
split as a positive sign pointing to future stock price
increases.
- Many investors view the announcement of a
reverse stock split as a negative sign (not
necessarily always) pointing to future stock price
decreases. Reverse stock splits are perhaps most
common for companies in or coming out of
financial distress.
- Stock Split is used to bring the stock price down.
- Reverse Stock Split is used to raise the stock price
up.
Fig 3: Cash Dividends Vs. Stock Dividends
Same applies for Stock Split
Cash Dividend
[Cash
and R/E ]
A
E
Affects Capital Structure
↓
Quick Ratio
Current Ratio
D/E
D/A
↑
↓
↓
↑
↑
Stock Dividend
[R/E
and Contributed Capital
E
E
Unaffected Capital Structure
↓
↑]
=
=
=
=
Example 1:
If the reverse split were to take place when the share price was $2.9, find the expected stock price after a
1-for-30 reverse split, assuming no other factors affect the split.
If price was $ 2.9 (before) for every 30 shares:
Now a shareholder would have 1 share priced at 30 x $ 2.9 = $ 87.
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Theories of Dividend Policy
lllll
Dividend Irrelevance
In a 1961, MM argued that in a world without taxes, transaction costs, brokerage costs, infinitely
divisible shares and equal 'symmetric' information among all investors i.e. under perfect capital
market assumptions - a company's dividend policy should have no impact on its cost of capital or an
shareholder wealth. MM's argument of dividend irrelevance is based on their concept of
'Homemade Dividends'. Assume, that you are a shareholder and you don't like the firm's dividend
policy. If the firm's cash dividend is too big, you can just take the excess cash received and use it to
little bit of your stock in the firm to get the cashflow you want. Note that by reducing shares
holding, second-period dividend income is reduced; higher dividend income in one period, is at the
expense of exactly offsetting lower dividend income in subsequent periods. The irrelevance
argument doesn't state that dividends per se are irrelevant to share value but that dividend policy is
25
irrelevant. Note that under MM assumptions, there is no meaningful distinction between dividends
and share repurchases (repurchases of outstanding common shares by the issuing company). In
either case above, the combination of the value of your investment in the firm and your cash in
hand will be the same.
In real world, market imperfections create some problems for MMs dividend policy irrelevance
propositions. Firstly, both companies and individuals incur transaction costs like flotation costs
(i.e. costs in selling shares to the public that include underwriter's fees, legal costs, registration
expenses and possible negative price effects). Also, shareholders selling shares to create a
'Homemade Dividend' would incur transaction costs like in some countries i.e. capital gains taxes.
lllll
Bird-in-the-Hand Argument
Mayron Gordon and John Lintner, argue that the required rate of return on equity capital (re )
decreases as the dividend payout increases (i.e. therefore, higher share price). why? Because
investors are less certain of receiving future capital gains from the reinvested retained earnings
than they are of receiving current (certain) dividend payments. The main argument of Gordon and
Lintner is that investors place a higher value on a dollar of dividends that they are certain to receive
than on a dollar of expected capital gains i.e. money today has more value, due to uncertainties in
the future. They base this argument on the fact, when measuring total return, the dividend yield
component 'D1 / P0 ', has less risk than the growth component 'g' i.e. amount of dividends is less
risky than the same amount of capital gains. The Gordon-Lintner Argument is called the
'Bird-in-the-Hand' theory based on the old expression: a 'Bird-in-the-Hand' (dividends) is worth two
in the bush (expected capital gains). But, MM contends that this argument is incorrect because
paying or increasing the dividend today doesn't affect the risk of future cashflows, such actions only
lower the 'Ex-Dividend' price of the share as per their assumptions.
Example 2:
Sophie Chan owns 100,000 shares of PAT Company. PAT is selling for $40 per share, so Chan's investment is
worth $4000,000. Chan reinvests the gross amount of all dividends received to purchase additional shares.
Assume that the cliental for PAT shares consist of tax-exempt investors. If PAT pays a $1.5 dividend, Chan's
new share ownership after reinvesting dividends at the Ex-Dividend price is most likely to be closest to:
A. 103,600
Dividend = 40 - 1.5 = 38.5
B. 103,750
Chan will purchase additional shares of 150,000/38.5 = 3896 shares.
Therefore, 100,000 + 3896 = 103,896
C. 103,900
✓
Example 3:
Gordon's Bird-in-the-Hand argument suggests that:
A. Dividends are irrelevant
B. Firms should have a 100% payout policy
C. Shareholders are generally risk averse and attach less risk to current dividends
✓
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Tax Aversion
According to the Tax-Aversion Theory, Tax CG > Tax D : Investors Prefer Dividends
Tax CG < Tax D : Investors Don't Prefer Dividends
Real world market considerations may complicate the picture. For example, in some jurisdictions
governmental regulation may require companies to distribute excess earnings as dividends or to
classify share repurchases as dividends if the repurchases appear to be ongoing in lieu of dividend
payments. For example, if tax on dividends and capital gains are same (5%), it is likely to make the
tax aversion theory irrelevant.
26
lllll
Clientele Effect
It refers to the varying dividend preferences of different groups of investors, such as individuals,
institutions and corporations. The Dividend Clientele Effect states that different groups desire
different levels of dividend. The Clientele Effect does exist and that equity market participants can
be sorted by those who prefer to receive returns in the form of dividends and those who prefer
capital gains returns.
↓
↑
Share price
mostly after
ex-dividend date
Share price
due
to high demand.
①
Declaration Date
The company declares the
dividend on the Declaration
Date. The board of directors
announce how much each
shareholder receives per
share.
③
1 or 2 Days
Ex-Dividend
The exchange the
stock trades on
sets a cut-off date
called Ex-Dividend
Date, prior to a
record date.
②
④
Weeks or Months
Record Date
Is the date on
which must be a
shareholder in
order to receive a
stock's dividend.
Payment Date
1. Dividends will be paid only if you buy a share; and if you don't own a share, you have to buy
shares before ex-dividend date and not on ex-dividend date.
2. If you sell your shares on ex-dividend or later, you will still receive your dividend, even if you
are no longer a shareholder on the payment date.
(a) Sell just before the share goes Ex-Dividend: CF = Pw - (Pw - Pb ) (TCG )
Sale Price
Purchase Price
(b) Sell just after the share goes Ex-Dividend: CF = Px - (Px - Pb ) (TCG ) + D (1 - TD)
Ex-Dividend Price
(c) Indifferent to risk, selling before or on Ex-Dividend:
△ P = Pw - Px + D (1 - TD ) ↶
Change in Price
when stock goes
from with Dividend
to Ex-Dividend
Indifference
Relationship
(1 - TCG )
TD = TCG : (1 - TD ) = 1 ; The share's price should drop by the
(1 - TCG )
amount of the dividend when the
share goes ex-dividend.
TD > TCG : (1 - TD ) < 1 ; The share's price should drop by less
(1 - TCG )
than the amount of the dividend when
the share goes ex-dividend.
TD < TCG : (1 - TD ) > 1 ; The share's price should drop by more
(1 - TCG )
than the amount of the dividend when
the share goes ex-dividend.
It should be noted that the existence of dividend clienteles doesn't contradict dividend irrelevance
theory. A firm's dividend policy would attract a certain clientele. After that, changes in the policy
would have no impact as the firm would simply be swapping one clientele for other. For example,
high tax bracket investors (like some individuals) tend to prefer low dividend payout, while low tax
bracket investors (like corporations and pension funds) may prefer high dividend payouts i.e. net
effect is irrelevant.
Example 4:
Consider a firm that is planning to declare $12 in dividends. The tax rates for a marginal investor are:
27
T CG = 15% and TD = 30%. Compute the expected drop in share price when the stock goes ex-dividend?
Expected Drop in Stock Price = 12 (1 - 0.3) = $ 9.88
(1 - 0.15)
Since TD > TCG , investors would prefer capital gains over dividends. Hence, a $ 1 dividend is worth less than
$ 1 capital gain.
Example 5:
Suppose the TCG = 25%. If the stock price of a company falls by 85% of the dividend amount on average when
the stock goes ex-dividend, what is the tax rate on dividends for a marginal investor in that stock?
△ P = D (1 - TD)
(1 - TCG )
0.85 = 1 (1 - TD)
(1 - 0.25)
T D = 0.3625
Example 6:
An individual investor pays taxes of 28% on the next dollar of dividend income and taxes of 15% on the next
dollar of capital gains. Which would she prefer, $1 in dividend or $0.87 in capital gains?
$ 1 in dividends results in $ 0.72 [1 (1 - 0.28) = $ 0.72] after taxes and
$ 0.87 in capital gains results in $ 0.74 [0.87 (1 - 0.15) = $ 0.74] after taxes.
Thus, the investor would prefer $ 0.87 in capital gains to $ 1 in dividends.
∴
Example 7:
Consider a US Corporation with a corporate income tax rate of 40%. The corporation needs to report as
taxable income only 30% of dividends received from other corporations i.e. it takes a 70% deduction on that
type of dividend income in calculating taxes owed. Assume that both capital gains and reported dividends
(dividends net of any deductible amount) are taxed at 40%. What is $1 of dividends worth in terms of capital
gains for such a corporate investor?
Because 70% of the dividend is excluded from taxation, the effective tax rate on dividends is 0.4 (1 - 0.7) =
0.12. Thus, a $ 1 dividend is worth (1 - 0.12) = $ 1.47 of capital gains for the corporate investor described.
(1 - 0.4)
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Signaling
Information asymmetry refers to differences in information available to a company's board and
management (insiders) as compared to the investors (outsiders). Dividends convey more credible
information to the investors as compared to plain statements. This is so because dividends entail
actual CF (cash) and are expected to be 'sticky' (i.e. continued in the future). Companies refrain from
increasing dividends unless they expect to continue to pay out the higher levels in the future.
Similarly, companies loathe cutting dividends unless they expect that the lower level of dividends
reflect long-run poorer prospects of the company in the future. But, dividend increases are costly to
mimic because a company that doesn't expect its cashflow to increase will not be able to maintain
28
the dividend at increasingly high levels in the long-run; In short-run, a company might be able to
borrow to fund dividends.
The information conveyed by 'Dividend Initiation' is ambiguous. On one hand, a dividend initiation
could mean that a company is optimistic about the future and is sharing its wealth with
stockholders i.e. a positive signal. On the other hand, initiating a dividend could mean that a
company has lack of profitable reinvestment opportunities i.e. a negative signal.
Some researchers have argued that a company's dividend initiation or increase tends to be
associated with share price increases because it attracts more attention to the company. Managers
have an incentive to increase the company's dividend if they believe the company to be
undervalued because the increased scrutiny will lead to a positive price adjustment. In contrast,
according to this line of reasoning, managers of overvalued companies have little reason to mimic
such a signal because increased scrutiny would presumably result in a downward price adjustment
to their shares.
Unexpected Dividend Increase : Business Prospects are Strong i.e. Positive Signal
Unexpected Dividend Decrease : Business is in Trouble i.e. Negative Signal
Companies that consistently increase their dividends seem to share certain characteristics:
- Dominant or Niche Positions in the Industry
- Global Operations
- Relatively Less Volatile Earnings
- High ROA and Low Debt Ratios
lllll
Agency Costs
(i) Between Shareholders and Managers: Agency costs reflect the inefficiencies due to divergence of
interests between managers and stockholders. One aspect of agency issue is that managers may
have an incentive to overinvest 'Empire Building'. This may lead to investment in some negative
NPV projects, which reduces stockholder wealth. The potential overinvestment agency problem
might be alleviated by the payment of dividends. In general, it makes sense for growing
companies in industries characterized by rapid change to hold cash and pay low or no dividends,
but it doesn't make sense for large mature companies in relatively non-cyclical industries.
(ii) Between Shareholders and Bondholders: When there is risky debt outstanding, shareholders can
pay themselves a large dividend, leaving the bondholders with a lower asset base as collateral.
This way there could be a transfer of wealth from bondholders to stockholders. Typically, agency
conflict between stockholders and bondholders is resolves via provisions in the bond indenture.
These provisions may include restrictions on dividend payment, maintenance of certain balance
sheet ratios and so on. Covenants, often don't really restrict the level of dividends as long as
those dividends come from new earnings or from new issues of stock. What the covenants
attempts to do is prevent the payment of dividends financed by the sale of the company's
existing assets or by the issuance of new debt.
Factors Affecting Dividend Policy
1. Investment Opportunities: A company with many profitable investment opportunities will tend to payout
less in dividends than a company with fewer opportunities because the former company will have more
uses for internally generated cashflows. A company with the ability to delay the initiations of projects
without penalty may be willing to payout more in dividends than a company that needs to act immediately
29
to exploit profitable investment opportunities. Technology companies have much lower average dividend
yields than utilities which have higher dividend payouts.
2. Expected Volatility of Future Earnings: Managers are very reluctant to cut dividends and tend to smooth
dividends. Smoothing takes the form of relating dividend increases to the long-term earnings growth rate,
even if short-term earnings are volatile. Thus, when earnings are volatile, we expect companies to be more
cautious in the size and frequency of dividend increases.
3. Financial Flexibility: Firms with excess cash and a desire to maintain financial flexibility may resort to stock
repurchases instead of dividends as a way to pay out excess cash.
4. Tax Considerations: Generally, in countries where capital gains are taxed at a favorable rate compared to
dividends, high-tax bracket investors (like some individuals) prefer low dividends payouts and low-tax
bracket investors (like corporations and pension funds) prefer high dividend payouts. Stockholders may not
prefer a higher dividend payout, even if the tax rate of dividends is more favorable, for multiple reasons:
- Taxes on dividends are paid when the dividend is received, while capital gains taxes are paid only when
shares are sold.
- The cost basis of shares may receive a step-up in valuation at the shareholders death. This means that
taxes on capital gains may not have to be paid at all.
- Tax-exempt institutions such as pension funds and endowments will be indifferent between dividends or
capital gains.
a. Double-Taxation System: Earnings are taxed at Corporate Level and
Dividends are taxed at Shareholder Level
Effective Tax Rate = Corporate Tax Rate + (1 - Corporate Tax Rate) (Individual Tax Rate)
b. Split-Rate System: Earnings distributed as Dividends are taxed at Corporate Level and
Dividends are taxed at Shareholder Level
Effective Tax Rate on Income = Corporate Tax Rate + (1 - Corporate Tax Rate) (1 - Individual)
Distributed as Dividends
on Earnings paid
on Earnings paid
Tax Rate
out as Dividends
out as Dividends
Corporate earnings that are distributed as dividends are taxed at a lower rate at the corporate level than
earnings that are retained. Under a Split-Rate System, earnings that are distributed as dividends are still
taxed twice, but at a lower corporate tax rate (the corporate rate for distributed income).
c. Imputation Tax System: Taxes are paid at the corporate level but are attributed to the shareholder, so
that all taxes are effectively paid at the shareholder rate. Shareholders deduct their portion of the taxes
paid by the corporation from their tax return.
Effective Tax Rate = Shareholder's Tax Rate
Under this system, a corporation's earnings are first taxed at the corporate level. When those earnings
are distributed to shareholders in the form of dividends, however shareholders receive a tax credit
known as a 'Franking Credit', for the taxes that the corporation paid on those distributed earnings i.e.
corporate taxes paid are imputed to the individual shareholder.
Tax Credit (Shareholder's Perspective)
Tax S < Tax C : Receive
Differences between
the two rates.
Tax S > Tax C : Pay
30
5. Flotation Costs: Flotation costs include (i) Fees that the company pays (to investment bankers, attorneys,
securities regulators, auditors and others) to issue shares and (ii) Possible adverse market price impact
from a rise in the supply of shares outstanding. The cost of new equity capital is always higher than the
cost of retained earnings.
Flotation Costs
Larger Companies
Smaller Companies
↓
↑
Dividend Payout
↑
↓
6. Contractual and Legal Restrictions: Common legal and contractual restrictions on dividend payments
include (i) Impairment of capital rule i.e. a legal requirement in some countries mandates that dividends
paid cannot be in excess of retained earnings and (ii) Debt Covenants.
Example 8: [Double-Taxation System]
A US company's annual earnings are $300 and the corporate tax rate is 35%. Assume that the company pays
out 100% of its earnings as dividends. Calculate the effective tax rate on a dollar of corporate earnings paid
out as dividends assuming a 15% tax rate on dividend income.
0.35 + (1 - 0.35) (0.15) = 0.4475 or 44.75%
After Tax Dividend to Investor = 0.4475 x 300 = $ 165.75
or
Earnings
- Tax @ 35%
Earnings after Tax
Dividends (100% Payout)
- Tax on Dividends @ 15%
After Tax Dividend to Investor
$
300
(105)
195
195
(29.25)
165.75
Effective Double Tax Rate = 300 - 165.75 = 44.75%
300
Example 9: [Split-Rate]
A German Company's annual pre-tax earnings are $300. The corporate tax rate on retained earnings is 35%
and the corporate tax rate that applies to earnings paid out as dividends is 20%. Assuming that the company
pays out 50% of its earnings as dividends and the individual tax rate that applies to dividend is 30%. Calculate
the effective tax rate on one dollar of corporate earnings paid out as a dividend.
0.2 + (1 - 0.2) (0.3) = 0.44 or 44%
Example 10: [Imputation Tax System]
Phil Cornelius and Todd Ian both own 100 shares of stock in a British Corporation that makes $1 per share in
pre-tax income. The corporation pays out all of its income as dividends. Cornelius is in the 20% individual tax
bracket, while Todd is in the 40% individual tax bracket. The tax rate applicable to the corporation is 30%.
Calculate the effective tax rate on the dividend for each shareholder.
31
TS is equal to
Effective Tax Rate
TS
TC
↶
Tax due from
Shareholder
Cornelius
20%
10%
30%
($ 10)
Todd
40%
30%
10%
($ 10)
Therefore, Effective Tax Rate for Cornelius is (20/100) 20% and Todd is (40/100) 40%.
lllll
Dividend Payout Policies
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Stable Dividend Policy
Companies that use a stable dividend policy typically look at a forecast of their long-run earnings to
determine the appropriate level for the stable dividend, even though earnings may be volatile from
year to year. A stable dividend policy could be gradually moving towards a target dividend payout
ratio. A model of gradual adjustment is called a 'Target Payout Adjustment Model'. If company
earnings are expected to increase and the current payout ratio is below the target payout ratio an
investor can estimate future dividends through the following formula:
Expected Increase = [(Expected Earnings x Target Payout Ratio) - Previous Dividend] x AF
in Dividends
Whereas,
AF : Adjustment Factor =
1
No. of Years over which the Adjustment
in Dividends will take place.
Example 11:
Suppose that the current dividend is $0.4, the target payout ratio is 50%, the adjustment factor is 0.2 (i.e. the
adjustment is to occur over 5 years) and expected earnings are $1.5 for the year ahead (an increase from the
$1 value of last year). What is the expected increase in dividends?
Expected Increase in Dividends = [(1.5) (0.5) - 0.4] x 0.2 = $ 0.07
Therefore, even though earnings increased 50% from $ 1 to $ 1.5, the dividend would only incrementally
increase by about 17.5% from $ 0.4 to $ 0.47.
Now, what if in the following year earnings temporarily fall from $ 1.5 to $ 1.34, the dividend will now be:
Expected Increase in Dividend = [(1.34) (0.5) - 0.47] x 0.2 = $ 0.04
Because the implied new dividend of $ 0.51 would still be moving the company toward its target payout ratio
of 50%. Even if earnings were to fall further or even experience a loss, the company would be reluctant to cut
or eliminate the dividend (unless its estimate of sustainable earnings or target payout ratio were lowered);
instead it would rather opt to maintain the current dividend until future earning increases justified an
increase in the dividend.
lllll
Constant Dividend Payout Policy
A payout ratio is the percentage of total earnings paid out as dividends. The amount of those
dividends would fluctuate directly with earnings.
32
lllll
Residual Dividend Policy
Residual Dividend Policy is rarely used in practice because it typically results in highly volatile
dividend payments. The residual dividend policy is based on payout as dividends the full amount of
any internally generated funds remaining after financing the current period's capital expenditures
(investment in positive NPV projects) consistent with the target capital structure. A residual dividend
policy presumes that equity financing comes from retained earnings rather than new share
issuance, which is more expensive. The model is based on the firm's (i) Investment Opportunity
Schedule (IOS), (ii) Target Capital Structure and (iii) Access to and Cost of External Capital.
Residual Earnings = Earnings - (we . Capital)
To overcome the problem of volatile dividends, companies may use a long-term residual dividend
approach to smooth their dividend payments. The leftover earnings over this longer time frame are
allocated as special dividends or distributed in the form of share repurchases.
Example 12:
Suppose that the Larson Company has $1000 in earnings and $900 in planned capital spending. Larson has a
target debt-to-equity of 0.5. Calculate the company's dividend under a residual dividend policy.
D/E = 0.5
,
D/A = 0.33
,
E/A = 0.66
Residual Earnings = $ 1000 - (0.66 x $ 900) = $ 1000 - $ 600 = $ 400
Dividend Safety
Dividend Safety is the metric used to evaluate the probability of dividends continuing at the current rate for a
company. Higher coverage ratios lead to higher dividend sustainability.
1. Dividend Payout Ratio = Dividends
NI
2. Dividend Coverage Ratio =
Higher than normal (DPR) tends to typically
indicate a higher probability of a dividend cut.
When a (DCR) drops to 1, the dividend is considered to be in jeopardy unless non-recurring
events such an employee strike or typhoon are responsible for a temporary decline in
earnings.
If ratio = 1, the company is returning all available cash to shareholders.
If ratio > 1, Improving liquidity by using funds to increase cash.
FCFE
If ratio < 1, Decreasing liquidity i.e. not sustainable because the
Share Repurchases company is paying out more than it can afford. At some point the
company will have to raise new equity or cut back on capital spending.
NI
Dividends
3. FCFE Coverage Ratio =
Dividends +
Extremely high dividend yields compared with a company's past record and current bond yields is often
another warning signal that investors are predicting a dividend cut. This over-representation of very high
dividend yielding stocks in the bottom decile of performance is likely attributable to deteriorating corporate
fundamentals resulting in non-sustainable dividends.
lllll
Share Repurchases
A Share Repurchase or Buyback is a transaction in which a company buyback its own shares. Usually,
share repurchases can be viewed as an alternative to cash dividends. Shares that have been issued and
subsequently repurchased are classified as 'Treasury Shares' or 'Treasury Stock', if they may be
reissued or 'cancelled shares' if they will be retired; in either case, they are not then considered for
dividends, voting or computing EPS. Share repurchases in many markets remain subject to more
restrictions than in US.
33
Following are the four main ways that companies repurchase shares, listed in order of importance:
1. Open Market Transactions: are the most flexible approach, allowing a company to buyback its shares in the
open market at the most favorable terms. There is no obligation on the part of the company to complete
an announced buyback program. Unlike their European counterparts, American counterparts don't need
shareholder approval for Open Market Transactions.
2. Fixed Price Tender Offer: is an approach where the firm buys a pre-determined no. of shares at a fixed
price, typically at a premium over the current market price from current shareholders. Although the
company forgoes flexibility (the firm cannot execute its purchases at an exact opportune time), it allows a
company to buyback its shares rather quickly. If shareholders are willing to sell more than certain no. of
shares, the company will typically buyback a pro-rated no. of shares from each shareholder responding to
the offer.
3. Dutch Auction: is a tender offer in which the company specifies not a single fixed price but rather a range
of prices. Dutch auctions identify the minimum clearing price for the desired no. of shares that needs to be
repurchased. Each participating shareholder indicates the price and the no. of shares tendered. Bids are
accepted based on lowest price first until the desired quantity is filled. The price of the last offer accepted
(the highest accepted bid price) will however be the price paid for all shares tendered. Hence, a
shareholder can increase the change of having their tender accepted by offering shares at a low price.
Dutch auctions also can be accomplished rather quickly, though not as quickly as tender offer.
4. Repurchase by Direct Negotiation: entails negotiating or purchasing shares from a major shareholder,
often at a premium over market price e.g. greenmail scenario. Surprisingly, many negotiated transactions
occur at a discount to market price, indicating the urgent liquidity needs of large investors who are in a
weak negotiating position.
Fig 4: Cost, Flexibility, Speed Rating
Open Market
Fixed Price
Dutch
Cost
Flexibility
Speed
✻✻
✻✻✻
✻
✻
✻✻
✻✻✻
✻✻✻
✻
✻✻
✻ ✻ ✻ : Best
✻ ✻ : Okay
✻ : Bad
Example 13:
Which type of stock repurchase often takes place at a price below the current market price of the stock?
A. Fixed price tender
B. Dutch auction tender
C. Targeted stock repurchase
✓
Example 14:
Which type of stock repurchase allows management to set the repurchase price at the lowest level necessary
to repurchase the desired number of shares?
A. Open-market repurchase
B. Fixed-price tender offer repurchase
C. Dutch auction tender offer repurchase
✓
34
Effect of a Share Repurchase on: EPS and BVPS
A=L+E
A=L+E
↟
Cash
Debt
↓↑↓
a. EPS
When the purchase is financed with the company's
If the repurchase was financed with additional debt
surplus cash; assuming a company's NI doesn't
offerings, the resulting NI can offset the effect of the
change, a smaller no. of shares o/s after the
reduced shares o/s producing lower (depends) EPS.
buyback will produce higher EPS.
(After Repurchase)
(After Repurchase)
EPS
Sp @ Premium
Sp @ Discount
Sp @ Par
EPS
E 0 / P1 > rD (1 - t)
↑
E 0 / P1 < rD (1 - t)
E 0 / P1 = rD (1 - t)
↑
↓
(higher leverage,
higher cost of capital)
=
b. BVPS
After a stock repurchase, the no. of o/s shares will decrease and the book value per share (pre-purchase) is
likely to change as well.
(After Repurchase)
BVPS
Sp > BVPS
Sp < BVPS
↓
↑
Example 15: [EPS by Cash]
Takemiya has 10,000,000 shares outstanding and its net income is $100,000,000. Takemiya's share price is
$120. Cash not needed for operations total $240,000,000 and is invested in Japanese government short-term
securities that can virtually zero interest. Calculate the impact on EPS, if Takemiya uses the surplus cash to
repurchase shares at (a) Premium: $140, (b) Discount: $100 and (c) Par: $120?
Given: NI = $ 100 m
O/S = 10 m
Sp = $ 120
Excess Cash = $ 240 m
Current EPS = NI = $ 100 m = $ 10
O/S
a. New EPS = NI = $ 100 m = $ 100 m = $ 12
(Premium)
O/S 10 m - 1.7 m
8.29 m
b. New EPS = NI = $ 100 m = $ 100 m = $ 13.15
(Discount)
O/S 10 m - 2.4 m
7.6 m
c. New EPS = NI = $ 100 m = $ 100 m = $ 12.5
(Par)
O/S 10 m - 2 m
8m
140 x x = 240 m
x = 240 m = 1.7 m
140
100 x x = 240 m
x = 240 m = 2.4 m
100
120 x x = 240 m
x = 240 m = 2 m
120
Example 16: [EPS by Debt]
Jet Fun Inc. has 10,000,000 shares outstanding and has just reported net income of $50,000,000. Because the
company has $100,000,000 of excess cash currently earning no return, Jet Fun's directors are considering
35
repurchasing Jet Fun shares at a premium of 25% over the current market price of $40. Calculate and
compare the effect of repurchase on EPS when the repurchase is financed using new debt borrowed at an
after-tax interest rate of 3%.
Given: NI = $ 50 m
O/S = 10 m
Sp = $ 40
Excess Cash = $ 100 m
Premium Sp = 40 (1.25) = $ 50
Current EPS = NI = $ 50 m = $ 5
O/S
10 m
New EPS = NI = $ 50 m - $ 3 m = $ 47 m = $ 5.875
O/S
10 m - 2 m
8m
If 10% E0 / P1 > 3% After-tax Cost of Debt =
After-tax Cost of Funds
= 100 m x (0.03) = 3 m
↑ EPS
50 x x = 100 m
x = 100 m = 2 m
These relationships should be viewed with caution so far as any valuation
50
implications are concerned. Notably, not always will increase in EPS, indicate
an increase in shareholders wealth.
Example 17: [BVPS]
Consider two companies Alpha and Beta, which are both considering the repurchase of $5000,000 worth of
shares. Additional details are given below:
Alpha
Beta
Stock Price
$ 20
$ 20
O/S
1m
1m
Book Value
$ 15 m
$ 25 m
Calculate the impact of the repurchase transactions on BVPS.
Alpha: BVPS (Before Buyback) = Bv / O/S = 15 / 1 = $ 15
BVPS (After Buyback) = Bv / O/S = 15 m - 5 m = 10 m = $ 13.33
1 m - 0.25 m 750,000
Beta: BVPS (Before Buyback) = Bv / O/S = 25 / 1 = $ 25
BVPS (After Buyback) = Bv / O/S = 25 m - 5 m = 20 m = $26.67
1m - 0.25 m
750,000
∴ Alpha : 15 < 20 : After BVPS ↓
Beta : 25 > 20 : After BVPS ↑
Before Sp
BVPS
20 x x = 5 m
x = 5 m = 0.25 m
20
"Theory concerning the dividend-share repurchase decision generally concludes that share repurchases are
equivalent to cash dividends of equal amount in their effect on shareholders wealth, all other things equal" (If
tax treatment is same, no difference in effect on shareholder wealth). Further discussion about the choice
revolves around what might 'not' be equal is:
36
There are 5 common rationales for Share Repurchases Vs. Dividends:
1. Potential Tax Advantages: When the TCG < TD i.e. share repurchases have tax advantage over cash
dividends.
2. Share Price Support/Signaling: An unexpected announcement of a meaningful share repurchase program
can often have the same positive impact on share price as would a better-than-expected earnings report
or similar positive event.
3. Added Managerial Flexibility: Unlike regular cash dividends, share repurchase program appear not to
create the expectation among investors of continuance in the future. Additionally, the timing of share
repurchases via. open market activity is at managers discretion. Share repurchases also afford
shareholders flexibility because participation is optional, which is not the case with the receipt of cash
dividends.
4. Offsetting Dilution from Employee Stock Options: Repurchases offset EPS dilution that results from the
exercise of employee stock options.
5. Increasing Financial Leverage: When funded by new debt, share repurchases increase leverage.
Management can change the company's capital structure (and perhaps move towards the company's
optimal capital structure) by decreasing the percentage of equity. Share repurchases funded by newly
issued debt increase leverage more than those funded by surplus cash.
When Economy is Strong : Share Repurchases increase in volume
When Economy is Weak : Share Repurchases are abandoned and
then sometimes even eliminate or
curtain dividends (if worse).
Example 18:
Assume that a company is based in a country that has no taxes on dividends or capital gains. The company is
considering either paying a special dividend or repurchasing its own shares. Shareholders of the company
would have:
A. Greater wealth if the company paid a special cash dividend.
B. Greater wealth if the company repurchased its shares
C. The same wealth under either a cash dividend or share repurchase program
✓
Example 19:
Waynesboro Chemical Industries Inc. (WCII) has 10 m shares outstanding with a current market value of $20
per share. WCII's board of directors is considering two ways of distributing WCII's current $50,000,000 FCFE.
Pay a cash dividend of $ 50 m / 10 m = $ 5 per share.
Repurchase $ 50 m worth of shares.
For simplicity, we make the assumptions that dividends are received when the shares go Ex-dividend and that
any quantity of shares can be bought at the market price of $20 per share. We also assume that the taxation
and information content of cash dividends and share repurchases, if any, don't differ. How would the wealth
of a shareholder be affected by WCII's choice of method in distributing the $50,000,000?
(i) Cash Dividend
After the shares go Ex-Dividend, a shareholder of a single share would have $5 in cash and a share worth
PDiv - Div = PEx-Div i.e. $ 20 - $ 5 = $ 15;
Total wealth from ownership of one share PEx-Div + Div = PDiv i.e. $ 15 + $ 5 = $ 20.
Ex-Dividend = (10 m) (20) = $ 15
10 m
37
(ii) Share Repurchase
Share Price after Repurchase = (10 m) (20) - 50 m = $ 20
10 m - 25 m
Assuming the tax treatment of the two alternatives is the same, a share repurchase has the same impact on
shareholder wealth as a cash dividend payment of an equal amount.
Example 20:
Florida Citrus (FC) common shares sell at $20 and there are 10,000,000 shares outstanding. Management
becomes aware that Kirk Parent recently purchased a major position in its outstanding shares with the
intention of influencing the business operations of FC in ways the current board doesn't approve. An advisor
to the board has suggested approaching parent privately with an offer to buyback $50,000,000 worth of
shares from him at $25 per share, which is a $5 premium over the current market price. The board of FC
declines to do so because of the effect of such a repurchase on FC's other shareholders. Determine the effect
of the proposed share repurchases on the wealth of shareholders other than parent.
Post Repurchase Share Price = (20) (10 m) - 50 m = $ 18.75
10 m - 2 m
Shareholders other than parent would lose = $ 20 - $ 18.75 = $ 1.25. If a company repurchases share from an
individual shareholder at premium, the remaining shareholder's wealth is reduced.
Example 21:
Canadian Holdings Inc. (CHI) with debt and a debt ratio of $30,000,000 and 30%, respectively, plans a share
repurchase program involving $7000,000 or 10% of the market value of its common shares:
a. Assuming nothing else changes, what debt ratio
would result from financing the repurchase using
cash on hand?
b. Assuming nothing else changes, what debt ratio
would result from financing the repurchases
If CHI uses cash on hand, the debt ratio would
increase to 32% i.e. $ 30 m = 0.3226
(100 m - 7 m) $ 93 m
If CHI uses debt, the debt ratio would increase to
37% i.e. (30 m + 7 m) $ 37 m = 0.37
using new debt?
c. Discuss the effect on value of equity from
$ 100 m
After repurchase, CHI's equity stands at $ 63 m.
financing the repurchases using cash on hand,
Then, with the same P/E, CHI's market value of
assuming CHI's net income and P/E remain the
equity would be expected to increase above $ 63 m.
same.
d. Discuss the effect on value of equity from
After repurchase, CHI's equity stands at $ 63 m.
financing the repurchases using new debt,
Then, with the same P/E, CHI's market value of
assuming CHI's after-tax cost of debt is greater
equity would be expected to decrease above
than its E/P, which remains the same.
$ 63 m.
e. Discuss the effect on value of debt from financing
After repurchase, CHI's debt stands at $ 37 m. The
the repurchases using new debt, assuming the
post repurchase Mv of both equity and debt would
conditions in question (d) and knowing that CHI is
be expected to decrease. Therefore, CHI's post
in imminent danger of a credit rating downgrade.
repurchase capital structure is indeterminate based
on the information given.
38
The initial ratio of net debt to capital is (30 - 7) / (23 + 70) = 25%. Using cash for the share repurchase, this
ratio would become 30 / (30 + 63) = 32% and using debt in the transaction, it would also be (37 - 7) / (30 + 63)
= 32%.
Before
Buyback
$
%
After Buyback
All Cash
All Debt
$
%
$
%
Debt
30
30
30
32
37
37
Equity
70
70
63
68
63
63
Total Capital
100
100
93
100
100
100
Canadian Holdings beginning debt ratio was 30%. If Canadian Holdings uses borrowed funds to repurchase
equity, the debt ratio at market value will increase to 37%, which is significantly more than if it used excess
cash 32%.
39
CHAPTER 22
Corporate Governance and Other ESG Considerations in Investment Analysis
Corporate Governance is the system of principles, policies, procedures and clearly defined responsibilities
and accountabilities used by stakeholders to overcome conflicts of interest inherent in the corporate form.
Benefits of effective corporate governance may include higher profitability, growth in ROE, better access to
credit, higher and sustainable dividends, favorable long-term share performance and a lower cost of capital.
In contrast, companies with ineffective corporate governance may experience reputational damage, reduced
competitiveness, potential share price weakness/volatility, reduced profitability and a higher cost of capital. In
practice, analysts typically adjust the risk premium (cost of capital) or credit spread of a company to reflect
their assessment of corporate governance considerations.
When managers and
The percentage of independent board of
directors tends to be higher in jurisdictions
with generally dispersed ownership structure
relative to concentrated ownership structure.
Independent directors have better
control in dispersed structures and
lesser control in concentrated
structure.
↶
board of directors are
also shareholders of a
company, they are
known as insiders.
↷
↶
Supervisory board inspects
management board's
corporations books and records,
reviewing the annual report,
overseeing the work of external
auditors, analyzing information
provided by the management
board and setting or influencing
management compensation.
Approaches used to identify a company's or industry's ESG (Environmental, Social and Governance) factors
include (i) Proprietary Methods: analysts use their own judgement or their firm's proprietary tools to identify
ESG information by researching companies news reports, industry associations, environmental groups and
governmental organizations, (ii) ESG Data provides such as MSCI or sustainalytics e.g. Individual ESG analyses,
scores and/or rankings for each company in the vendor's universe and (iii) Not-for-Profit Industry
Organizations and Initiatives. In Equity analysis, ESG integration is used to both identify potential
opportunities and mitigate downside risk, whereas in fixed income analysis, ESG integration is generally
focused on mitigating downside risk.
Dispersed Ownership: reflects the
existence of many shareholders, none of
which have the ability to individually
exercise control over the corporation.
E.g. Australia, UK, US and Ireland.
Concentrated Ownership: reflects an
individual shareholder or a group called
controlling shareholders with the ability to
exercise control over the corporation.
E.g. China, India, Russia, Singapore and
South Africa.
Hybrid Ownership: is a mix of dispersed
and concentrated ownership.
E.g. Canada, Japan and Germany.
[Controlling shareholders may be either
'Majority Shareholder' i.e. own more than
50% shares or 'Minority Shareholder' i.e.
own less than 50% shares]
Horizontal Ownership: involves
companies with mutual business interests
(e.g. key customers or suppliers) that have
cross-holding share arrangements with
each other. This structure can help
facilitate strategic alliance and foster
long-term relationships among such
companies.
Vertical Ownership or Pyramid
Ownership: involves a company or group
that has a controlling interest in two or
more holding companies, which in turn
have controlling interests in various
operating companies.
[In many ownership structures,
shareholders may have disproportionately
high control of a corporation relative to
their ownership stakes as a result of
horizontal and/or vertical ownership
arrangements].
Straight Voting Share: shareholders are
granted the right of one vote for each
share owned.
Dual-Class Share: have more voting power
than the class of shares available to the
general public. It can benefit one group of
shareholders over another. The existence
of dual-class shares can also serve to
disconnect the degree of share ownership
from actual control. When used in
connection with vertical ownership
arrangements, the company or group a the
top of the pyramid can issue to itself all or
a disproportionately high no. of shares
with superior voting rights and thus
maintain control of the operating
companies with relatively fewer total
shares of the company owned.
40
Fig 1: Conflicts within Different Ownership Structures
1 Dispersed Ownership & Dispersed Voting
(Takeovers are Possible)
2 Concentrated Ownership & Concentrated Voting
(Takeovers are Possible)
3 Dispersed Ownership & Concentrated Voting
(Takeovers are Impossible)
Weak Shareholders
'Principle-Agent'
Strong Managers
Problem
(Controlling Sh.)
Strong Shareholders
'Principle-Principle'
Weak Managers
Problem
(Minority Sh.)
(Controlling Sh.)
Strong Shareholders
Strong controlling
Weak Managers
shareholders with
(Minority Sh.)
less than majority
ownership can exert
control over other
minority owners
through certain
mechanisms, such
as dual-class and
pyramid structures
and can also monitor
management owning
to their outsized
voting power.
4 Concentrated Ownership & Dispersed Voting
(Takeovers are Different)
Weak Shareholders
Arises when there
Strong Managers
are legal restrictions
on the voting rights
of large share
positions known as
'Voting Caps'. Voting
Caps have been
imposed by a
no. of sovereign
governments to
deter foreign
investors from
obtaining controlling
ownership positions
in strategically
important local
companies.
NOTES
1. Shareholder Activism refers to strategies used by shareholders to attempt to compel a company to act in a
desired manner.
2. Stewardship Code is a part of UK company law concerning principles that institutional investors are
expected to follow. It encourages investors to exercise their legal rights and increase their level of
engagement in corporate governance.
41
3. When a corporation has a 'Say-on-Pay' provision, shareholders can vote and/or provide feedback on
remuneration issues. A claw-back policy allows a company to recover previously paid remuneration if
certain events such as financial restatements, misconduct, breach of the law or risk management
deficiencies are uncovered.
4. Green Bonds are designated bonds intended to encourage sustainability and to support climate related or
other types of special environmental projects. Green Bonds can command a premium over comparable
conventional bonds. Lower cost of capital due to green bond premium. Additional costs related to the
monitoring and reporting of the use of the bond's proceeds. Lack of liquidity of green bonds when
purchased and held by buy and hold investors.
5. In some cases, also commonly in Latin America, individuals serve on the board of directors in multiple
corporations. This situation known as 'Interlocking Directorates' typically results in the same family or the
same member of a corporate group controlling several corporations. A benefit of family control is lower
risks associated with principle-agent problems as a result of families having concentrated ownership and
management responsibility.
6. Stated Owned Enterprises (SOEs) are partially owned by sovereign governments but also have shares
traded on public stock markets. This structure is called a mixed ownership model. This model tends to have
lower market scrutiny of management than that of corporate ownership models, where there are implicit
or explicit state guarantees to prevent corporate bankruptcy. In some cases, SOEs may pursue policies that
enhance social or public policy considerations at the expense of maximizing shareholder value.
42
CHAPTER 23
Mergers and Acquisitions
An Acquisition refers to one company buying only part of another company. If the acquirer absorbs the entire
target company, the transaction is considered a Merger. Once a merger is completed, only one company will
remain and the other will cease to exist.
Classifying M&A activities based on how the companies physically come together (i.e. Forms of Integration)
and on how the companies' business activities relate to one another (i.e. Types of Mergers).
Forms of Integration
2. Subsidiary Merger
↶
1. Statutory Merger
3. Consolidation
Well-known
Brand
A&L
(Becomes a
Subsidiary)
+
=
Similar Size
Companies
Types of Mergers
1. Horizontal Merger: the two businesses operate in the same or similar industries and may often be
competitors. Therefore, if Burger World and World of Burgers were to merge, the basic operations of the
new firm would be very similar to those of the separate entities. One of the great motivators behind
horizontal mergers is the pursuit of 'Economies of Scale' and increase in market power.
2. Vertical Merger: the acquiring company seeks to move up or down the product supply chain.
Forward Integration: where the acquirer is moving up the supply chain
towards the ultimate consumer. For example, an ice-cream manufacturer
decides to acquirer a restaurant chain.
Backward Integration: company is moving down the supply chain toward
the raw material inputs. For example, ice-cream manufacturer purchases
a farm, so it can supply its own milk.
APPLE
↓
Retailer
Chip Manufacturer
↑
APPLE
3. Conglomerate Merger: the two companies operate in completely separate industries. As such, there are
expected to be few, if any, synergies from combining the two companies e.g. Pandora and Carrefour. By
investing in companies from a variety of industries, companies hope to reduce the volatility of the
conglomerate's total cashflow.
Motives for M&A
1. Synergies: The whole of the combined company will be worth more than the sum of its parts. Generally
speaking, synergies created through a merger will either reduce costs or enhance revenues. Cost Synergy is
exactly the strategy behind a pure horizontal merger. Cost synergy is typically achieved through economies
43
of scale in R&D, procurement, manufacturing, sales and marketing, distribution and administration.
Revenue Synergy is created through the cross selling of products, expanded market share or higher prices
arising from reduced competition.
2. Growth: Companies can grow either by making investments internally (i.e. organic growth) or by buying the
necessary resources externally (i.e. external growth). External growth via. M&A activity is usually a much
faster way for managers to increase revenues than making investments internally and a less risky way too.
Growth through M&A is especially common in mature industries where organic growth opportunities are
limited.
3. Increasing Market Power: When a horizontal merger occurs in an industry with few competitors, the newly
combined company will typically come away with increased market share and a greater ability to influence
market prices. Taken to an extreme, horizontal integration results in a monopoly. Vertical integration may
also result in increased market power. Vertical mergers can lock in a company's sources of critical supplies
or create captive markets for its products. By controlling critical supply inputs, the firm can influence
industry output and market prices. Regulators closely scrutinize both horizontal and vertical mergers to
make sure the combined company doesn't gain too much market power, which could potentially harm
consumers. In 1982, the Herfindahl-Herschman Index (HHI) replaced market share as the key measure of
market power for determining potential anti-trust violations.
n
HHI = Σ
(
or
n
2
)
Sales or Output of Firm i
x 100
i=1
Total Sales or Output of Market
HHI = Σ (MS i x 100)
2
i=1
Whereas,
MS i : Market share of firm i
n : No. of firms in the industry
Post Merger HHI
Less than 1000
Between 1000 - 1800
Greater than 1800
Industry Concentration
Change in Pre
and Post Merger HHI
Anti-trust Action
Not Concentrated
Any Amount
No Action
Moderately Concentrated
100 or more
Possible Antitrust Ch.
Highly Concentrated
50 or more
Antitrust Ch.
Virtually Certain
Example 1:
Given an industry with 10 competitors and the following market shares, calculate the pre-merger HHI. How
would the HHI change if companies 2 and 3 merged? How would it change if companies 9 and 10 merged
instead? Would either set of mergers be likely to evoke an anti-trust challenge?
Company
Market Share %
1
2
3
4
5
25 20 10 10 10
Calculate the post-mergers using the table above.
6
7
8
9
10
5
5
5
5
5
44
Pre-Merger
Post Merger: Company 2 & 3 Post-Merger: Company 9 & 10
Company
MS%
(MS%) 2 Company
MS%
(MS%) 2 Company
MS%
(MS%) 2
HHI
1
25
625
1
25
625
1
25
625
2
20
400
2+3
30
900
2
20
400
3
10
100
4
10
100
3
10
100
4
10
100
5
10
100
4
10
100
5
10
100
6
5
25
5
10
100
6
5
25
7
5
25
6
5
25
7
5
25
8
5
25
7
5
25
8
5
25
9
5
25
8
5
25
9
5
25
10
5
25
9+10
10
100
10
5
25
1450
HHI
HHI Change
1850
HHI
400
HHI Change
Would likely invoke
antitrust challenge
1500
50
Unlikely to raise
antitrust concerns
Example 2:
Imagine that there are five firms in the industry, each with a 20% market share. Also suppose that firms 4 and
5 decide to merge, calculate the pre-merger and post-merger HHI and discuss the likelihood of an antitrust
challenge for the merger.
2
Pre-Merger HHI = (0.2 x 100) x 5 = 2000
Post-Merger HHI = (0.2 x 100) 2 x 3 + (0.4 x 100) 2 = 2800
Since the Post-Merger HHI > 1800, the market is considered concentrated and an antitrust challenge is likely.
Although the introduction of the HHI was an improvement, regulators still found it to be too mechanical
and inflexible. Thus, by 1984, the Department of Justice sought to increase the flexibility of its policies
through the inclusion of additional information such as market power measured by the responsiveness of
consumers to price changes, as well as qualitative information such as the efficiency of companies in the
industry, the financial viability of potential merger candidates, and the ability of US companies to compete
in foreign markets.
4. Acquiring Unique Capabilities and Resources: Many companies undertake a merger or an acquisition either
to pursue competitive advantages or to shore up lacking resources. When a company cannot
cost-effectively create internally the capabilities needed to sustain its future success, it may seek to acquire
them elsewhere.
5. Diversification: This makes no sense for shareholders but may be rational for the managers. It is much
easier and cheaper for the shareholders to diversify simply by investing in the shares of unrelated
companies themselves rather than having one company go through the long, expensive process of
acquiring and merging the two firm's operations and corporate cultures. Findings demonstrates that
mergers are not likely to increase value purely for diversification reasons.
6. Bootstrapping EPS: Is a way of packaging the combined earnings from two companies after a merger so
that the merger generates an increase in the EPS of the acquirer, even when no real economic gains have
been achieved. The 'Bootstrap Effect' occurs when a high P/E (generally a firm with high growth prospects)
acquires a low P/E firm (generally a firm with low growth prospects) in a stock transaction. Post merger, the
45
earnings of the combined firm are simply the sum of the respective earnings prior to the merger. However,
by purchasing the firm with a lower P/E, the acquiring firm is essentially exchanging higher-priced shares
for lower-priced shares. As a result, the no. of shares outstanding for the acquiring firm increases, but at a
ratio that is less than 1-for-1. When we compute the EPS for the combined firm, the numerator (total
earnings) is equal to the sum of the combined firms, but the denominator (total shares outstanding) is less
than the sum of the combined firms. The result is a higher reported EPS, even when the merger creates no
additional synergistic value.
Example 3:
Fastgro Inc. is planning to acquire Slowgro Inc. in a merger transaction. Financial information for the two
companies both prior to and after the merger are shown in the following table. Calculate Fastgro's
post-merger EPS and determine whether the merger created economic gains.
Stock Price
EPS
P/E Ratio
Total Shares O/S
Total Earnings
Market Capitalization
Fastgro Inc.
$ 80
$3
26.7
200,000
$ 600,000
$ 16,000,000
Slowgro Inc.
$ 40
$2
20
100,000
$ 200,000
$ 4000,000
Fastgro Post-Merger
$ 80
250,000
$ 800,000
$ 20,000,000 ↷
However, no
economic value
was created by
the merger.
Newly Issued Shares = $ 4000,000 = 50,000 Shares
$ 80
EPS = NI =
$ 800,000
= 3.2
O/S 200,000 + 50,000
P/E Ratio = $ 80 = 25; this would mean that post-merger P/E would be about 25, which would imply that
3.2
Fastgro's stock price would remain at $ 80 after the merger.
For Bootstrapping to work, the acquirer's P/E must be higher than the target's P/E.
Acquirer EPS Post-Merger > Acquirer EPS Pre-Merger
In practice, the market tends to recognize the bootstrapping effect and post-merger P/E's adjust
accordingly.
Example 4:
Tresell Candies, Claire's Confections and Cindy's Sweets following details:
Revenues
Earnings
Shares O/S
Dividends
EPS
Share Price
Tresell Candies
87,484
4454.6
4000
2450
1.114
22.5
Claire's Confections
21,871
1113.7
1200
600
0.928
19.95
Cindy's Sweets
26,245.2
1336.4
800
276.3
1.671
23.05
46
Nelson's observation that Tresell Candies will be able to increase its EPS by executing an all stock acquisition
of either Cindy's Sweets or Claire's Confections is best characterized by?
A. Only when considering Cindy's Sweets
B. For both companies
C. Only when considering Claire's Confections
✓
Tresell Shares needed to Acquire
No. of Tresell Shares
after Acquisition
Tresell Earnings
after Acquisition
EPS after
Acquisition
Claire's Confections
(19.95 x 1200) / 22.5
= 1064
4000 + 1064
= 5064
4454.6 + 1113.7
= 5568.3
1.1
Cindy's Sweet
(23.05 x 800) / 22.5
= 819.55
4000 + 819.55
= 4819.55
4454.6 + 1336.4
= 5791
1.2
or
Calculate P/E Ratios: Tresell = 22.5 / 1.114 = 20.2
Claire = 19.95 / 0.928 = 21.5
Cindy = 23.05 / 1.671 = 13.79
Tresell Candies's EPS will raise with acquiring Cindy's Sweets but fall with acquiring Claire's Confections based
on bootstrapping effect.
7. Managers' Personnel Incentives: Studies concerning executive compensation find there is a high
correlation between the size of a company and how much a manager is paid. This means that there is a
strong financial incentive for managers to maximize the size of the firm rather than shareholder value.
8. Tax Considerations: Consider the case of two companies where one has large amounts of taxable income
and the other has accumulated large tax loss carryforwards. By merging with the company that has tax
losses, the acquirer can use the losses to lower its tax liability. Regulators typically don't approve mergers
that are undertaken purely for tax reasons, but with the many potential motivations to enter into a merger,
proving that tax reasons are the key factor is difficult.
9. Unlocking Hidden Value: When a potential target is underperforming, an acquirer may believe it can
acquire the company cheaply and then unlock hidden value through reorganization, better management or
synergy i.e. obtain the company for less than its breakup value (a company's breakup value is the value
that can be achieved if a company's assets are divided and sold separately). Sometimes mergers are
conducted because the acquirer believe that it is purchasing assets for below their replacement cost.
10. Cross-Border Motivations: Since the 1990s, international M&A deals have become an important way for
multinational companies to achieve cross-border business goals.
- Taking advantage of market inefficiencies e.g. acquiring a manufacturing plant in a country where labor
costs are less expensive.
- Working around disadvantageous government policies.
- Use technology in new markets.
- Product differentiation.
- Provide support to existing multinational clients.
47
Studies on the performance of mergers fall into two categories,
Short-Term (ST) performance studies which examine stock returns surrounding merger announcement dates.
Long-Term (LT) performance studies of post-merger companies.
The following are characteristics of M&A deals that create value:
(i) Strong Buyer: Acquirers that have exhibited strong performance (earnings & stock price) in prior 3 years,
(ii) Low Premium,
(iii) Few Bidders,
(iv) Favorable Market Reaction: Positive market price reaction to the acquisition announcement.
Fig 1: Industry Life Cycle
Output
(Industry)
.
Rapid
Growth
.
.
.
Stabilization
Mature
Growth
.
Decline
Pioneer
0
Industry Life Cycle
Stage
1
- High development costs
Development
- Low and slowly increasing sales
Rapid Accelerating
Growth
Conglomerate
Horizontal
3
4
Industry
Characteristics
Pioneering
Conglomerate
Horizontal
2
Time
Mature Growth
Horizontal
Vertical
Stabilization and
Market Maturity
growth
Merger
Motivations
- Gain access to capital from more
mature businesses
- Share management talent
- Large capital requirements
- High profit margins caused by few
- Explosive growth in sales may
participants in market
require large capital requirements
(Low Competition)
to expand existing capacity
- Industry experiences a drop in the
entry of new competitors, but
- Economies of scale or Synergies
- Increase operational efficiencies
growth potential remains
- Faces capacity constraints
(High Competition)
Horizontal
- Economies of scale or reduce costs
- Large companies may acquire
smaller companies to improve
management and provide a broader
financial base
5
Deceleration of
- Consumer tastes have shifted
Growth and Decline
- Overcapacity or Shrinking profit
Conglomerate
Horizontal
Vertical
margins
- To exploit synergy in related
industries
- Companies in this industry may
acquire companies in young
industries.
48
Fig 2: Forms of Acquisition
Stock Purchase
Payment
Asset Purchase
Made directly to target company
Made directly to target company.
shareholders in some
Easy and quick process.
combination of cash and
securities, in exchange for their
shares. Difficult and time
consuming process.
Approval
Majorly shareholder approval
No shareholder approval needed
required (atleast 50% of
unless asset sale is substantial
shareholder approval)
(more than 50% of the company
assets)
Corporate Taxes
None
Target company pays capital gains
taxes
Shareholder
Shareholders pay capital gains tax
None
Most stock purchases involve
Asset purchase acquisitions
purchasing the entire company
usually focus on specific parts of
and not just a portion of it. This
the company that are of particular
means that not only will the
interest to the acquirer, rather
acquirer gain the target
than the entire company, which
company's assets, but it will also
means that the acquirer generally
assume the target's liabilities.
avoids assuming any of the target
Taxes*
Liabilities
company's liabilities. However, an
asset purchase for the sole
purpose of avoiding the
assumption of liabilities is
generally not allowed from a legal
standpoint.
* If the target company has accumulated tax losses, a stock purchase benefits the shareholders because under US
rules, the use of a target's tax losses is allowable for stock purchases, but not for asset purchases.
Fig 3: Method of Payment
|
Securities
Cash
Mixed
Securities Offerings
*
Cash Offerings
The target shareholders receive shares of
Here, the acquirer simply pays an agreed
the acquirer's common stock in exchange
upon amount of cash for the target
for their shares in the target company. The
company's shares.
no. of the acquirer's shares received for
each target company share is based on the
'Exchange Ratio'. For e.g. shareholders in
the target company may receive 1.3 shares
49
of the acquirer's stock for every 1 share
they own in the target company. In
practice, exchange ratios are negotiated in
advance of a merger due to the daily
fluctuations that can occur in stock prices.
The total compensation ultimately paid by
the acquirer in a stock offering is based on
three factors: (i) Exchange Ratio, (ii) No. of
shares outstanding of the target company
and (iii) The value of the acquirer's stock on
the day the deal is completed.
*
Since the target company's shareholders
All of the risk related to the value of the
receive new shares in the post-merger
post-merger company is borne by the
company, they share in the risk related to
acquirer.
the ultimate value that is realized from the
merger.
*
Changes in Capital Structure: Issuing a new
Change in Capital Structure: If the acquirer
stock for a securities offering can dilute the
borrows money to raise cash for a cash
ownership interest for the acquirer's
offering, the associated debt will increase
existing shareholders.
the acquirer's financial leverage and risk.
When an acquirer's shares are considered overvalued by the market relative to the target's company shares, stock
financing is more appropriate.
Example 5:
Discount Books, a Canadian bookseller, has announced its intended acquisition of Premier Marketing
Corporation, a small marketing company specializing in print media. In a press release, Discount Books
outlines the terms of the merger, which specify that Premier Marketing's shareholders will each receive 0.9
shares of Discount Books for every share of Premier Marketing owned. Premier Marketing has 1000,000
shares outstanding. On the day of the merger announcement, Discount Book's stock closed at C$20 and
Premier Marketing's stock closed at C$15. Catherine Willie is an individual investor who owns 500 shares of
Premier Marketing, currently worth C$7500 (500 x C$15).
a. Based on the current share price, what is the cost of the acquisition for Discount Books?
b. How many shares of Discount Books will Catherine Willie receive and what is the value of those shares
(based on current share price)?
(1000,000 O/S)
D
0.9 Sh D/P
C$ 20
P
C$ 15
a. 0.9 x 1000,000 = 900,000 Shares of Discount Book
C$ 20 x 900,000 = C$ 18,000,000
The pre-merger value of Premier Marketing was C$ 15,000,000 but Discount Book's total cost to purchase
the company was C$ 18,000,000. The 20% or C$ 3000,000 (18,000,000 - 15,000,000) difference is the total
control premium paid by Discount Books to the target company.
50
b. 0.9 x 500 = 450 Shares of Discount Book
C$ 20 x 450 = C$ 9000
Note that the value of Willie's Premier Marketing shares was C$ 7500. The C$ 1500 (9000 - 7500) difference
in value is a premium paid by Discount Books to Willie for control of Premier Marketing.
The more confident both parties are that synergies will be realized, the more the acquirer will prefer to pay
cash and the more the target will prefer to receive stock. Conversely, if estimates of synergies are uncertain,
the acquirer may be willing to shift some of the risk (and potential reward) to the target by paying for the
merger with stock, but the target may prefer the guaranteed gain that comes from a cash deal.
Fig 4: Attitude of Target Management
Friendly Merger
Hostile Merger
(a) Friendly Merger Offers
If both parties
like or accept
the idea
Each of the parties
examine the other's
books and records
PRIVATE
Attorney's of both
companies
After DMA is signed,
it is then announced
to the public
Target shareholders'
are given
After the
shareholders and
regulators approval
PUBLIC
51
(b) Hostile Merger Offers
to Board of Directors
(Bypassing the CEO)
The Williams Act Sought to remedy these problems in two ways: Disclosure requirements and Formal process
for tender offers.
S13 (d) : Requires public disclosure whenever a party acquires 5% or more of a target's outstanding
common stock.
S14 : The tender offer period be at least 20 business days, that the acquirer must accept all shares
tendered, that all tendered shares must receive the same price and that target shareholders can
withdraw tendered shares during the offer period. It also gives target management the time and
opportunity to adequately respond to a hostile tender offer.
52
lllll
Defense Mechanisms
lllll
Pre-Offer Defense Mechanisms
1. Poison Pills: When a company becomes the target of a hostile takeover, it may use the poison pill
strategy to make itself less attractive to the potential acquirer. The mechanism protects minority
shareholders and avoids the change of control of company management. In its most basic form, a
poison pill gives current shareholders the right to purchase additional shares of stock at
extremely attractive prices i.e. at a discount to current market value, which causes dilution and
effectively increases the cost to the potential acquirer. It was designed as a way to prevent an
acquiring company from buying a majority share in the potential target. There are two types of
poison pill strategies:
a. Flip-In Pill: Where the target company's shareholders have the right to buy the target's share at
a discount. The acquirer is subject to a significant level of dilution. This right to purchase is
given to the shareholders before the takeover is finalized and is often triggered when the
acquirer amasses a certain threshold percentage of shares of the target's company. For
example, let's say a Flip-In poison pill plan is triggered when the acquirer purchases 30% of the
target company's shares. Once triggered, every shareholder excluding the acquirer, is entitled
to buy new shares at a discounted rate. The greater the no. of shareholders who buy additional
shares, the more diluted the acquiring company's interest becomes. This makes the cost of the
bid much higher. As new shares make way to the market, the value of shares held by the
acquirer reduces, thereby making the takeover attempt more expensive and more difficult. The
board of the target generally returns the right to redeem the pill should the transaction
become friendly.
b. Flip-Over Pill: Where the target's shareholders have the right to buy the acquirer's shares at a
discount, which has the effect of causing dilution to all existing acquiring company
shareholders. For example, a target company shareholder may gain the right to buy the stock
of its acquirer at a two-for-one rate, thereby diluting the equity in the acquiring company. The
acquirer may avoid going ahead with such acquisitions, if it perceives a dilution of value post
acquisition. Most plans give the target's board of directors the right to redeem the pill prior to
any triggering event.
2. Dead Hand Poison Pill: Is an anti-takeover strategy that involves issuing new shares to everyone
but the hostile bidder seeking to buy the company. Like other poison pills, the job of a dead hand
provision is to make the hostile takeover prohibitively expensive. Once a hostile bidder acquires a
designated amount of the target company's shares, typically between 10% to 20%, rights allowing
all other stockholders to buy newly issued shares at reduced prices automatically kicks into
action. Suddenly, the stock held by the acquirer becomes less influential. Flooding the market
with new shares dilutes the value of the shares it already purchased, reducing its percentage of
ownership and making it harder and more costly to gain control. A hostile bidder can overcome a
regular poison pill by launching a proxy contest and then electing a new board of directors to
redeem it. That's not the case with dead hand provisions. In dead hand provision, existing
directors can prevent the acceptance of an unsolicited offer, regardless of the shareholders'
wishes or the views of the newly elected directors. Placing all this power in the hands of the
current board of directors has perhaps, understandably, generated a lot of controversies. The
dead hand provision can serve as a way to prolong the tenure of unfit and unwanted directors, as
well as prevent the majority of voting shareholders from having a say in whether they want an
acquisition to go ahead or not.
3. Poison Put: These puts give target's bondholders the option to demand immediate repayment of
their bonds if there is a hostile takeover. This additional cash burden may fend off a 'would-be'
acquirer and raises the cost of the acquisition.
53
4. Restrictive Takeover Laws: Companies in the US are incorporated in specific states and the rules
of that state apply to the corporation. Some states are more target friendly than others when it
comes to having rules to protect against hostile takeover attempts. Companies that anticipate the
possibility of a hostile takeover attempt may find it attractive to reincorporate in a jurisdiction
that has enacted strict anti-takeover laws. Historically, Ohio and Pennsylvania have been
considered to provide target companies with the most protection.
5. Staggered Board of Directors: In this strategy, the board of directors is split into roughly 3 equal
sized groups. Each group is elected for a 3-year term in a staggered system; in the first year the
first group is elected, the following year the next group is elected and in the final year the third
group is selected. The implications are straight-forward. To any particular year, a bidder can win
at most one-third of the board seats. It would take a potential acquirer atleast two years to gain
majority control of the board since the terms are overlapping for the remaining board members.
This is usually longer than a bidder would want to wait and can deter a potential acquirer.
6. Restricted Voting Rights: Equity ownership above some threshold level (e.g. 15% or 20%) triggers
a loss of voting rights of target's shareholders unless approved by the board of directors. This
greatly reduces the effectiveness of a tender offer and forces the bidder to negotiate with the
board of directors directly.
7. Supermajority Voting Provisions: A supermajority provision in the corporate charter requires
shareholder support in excess of a simple majority. For example, a supermajority provision may
require 66.7%, 75% or 80% of votes in favor of a merger. Therefore, a simple majority
shareholder vote of 51% would still fail under these supermajority limits. Thus, even if an acquirer
is able to accumulate a substantial portion of the target's shares, it may have great difficulty
accumulating enough votes to approve a merger.
8. Fair Price Amendments: Restricts a merger offer unless a fair price is offered to current
shareholders. This fair price is usually determined by some formula or independent appraisal.
Fair price amendments protect targets against temporary declines in their share prices by setting
a floor value bid. Additionally, fair price amendments protect against two-tiered tender offers
where the acquirer offers a higher bid in a first step tender offer with the threat of a lower bid in
a second step tender offer for those who don't tender right away.
9. Golden Parachutes: Are compensation agreements between the target and its senior
management that give the managers lucrative cash payouts if they should be dismissed as a
result of a merger or takeover. In practice, payouts to managers are generally not big enough to
stop a large merger deal, but they do ease the target management's concern about loosing their
jobs.
lllll
Post-Offer Defense Mechanisms
1. 'Just Say No' Defense: Simply say 'no' to the acquirer's hostile takeover. If the acquirer attempts a
'Bear Hug' or Tender Offer, then target management typically lobbies the board of directors and
shareholders to decline and build a case for why the offering price is inadequate or why the offer
is otherwise not in the shareholder's best interest. This strategy forces the hopeful acquirer to
adjust its bid or further reveal its own strategy in order to advance the takeover attempt.
2. Litigation: The basic idea is to file a lawsuit against the acquirer that will require expensive and
time consuming legal efforts to fight. The typical process is to attack the merger on anti-trust
grounds or for some violation of securities law.
54
3. Greenmail: Is a payoff to the potential acquirer to terminate the hostile takeover attempt.
Greenmail is an agreement that allows the target to repurchase its shares from the acquiring
company at a premium to the market price. The agreement is usually accompanied by a second
agreement that the acquirer will not make another takeover attempt for a defined period of time.
Greenmail used to be popular in the US in the 1980s, but it has been rarely used after a 1986
change in tax laws added a 50% tax on profits realized by acquirers through greenmail.
4. Share Repurchase: Here a target might use a share repurchase to acquire shares from any
shareholder i.e. a target may initial a cash tender offer for its own outstanding shares. An
effective repurchases can increase the potential cost for an acquirer by either increasing the
stock's price outright or by causing the acquirer to increase its bid to remain competitive with the
target company's tender offer for its own shares. Additionally, a share repurchase often has the
effect of increasing the target company's use of leverage because borrowing is typically required
to purchase the shares. This additional debt makes the target less attractive as a takeover
candidate. In some cases, a target company buys all of its shares and converts to a privately held
company in a transaction called a Leverage Buyout (LBO). This strategy may allow the target to
defend against a hostile bid provided that the LBO provides target shareholders with a level of
value that exceeds the would-be acquirer's offer.
5. Leveraged Recapitalization: Involves the assumption of a large amount of debt that is then used
to finance share repurchases (but in contrast to a LBO, in a recapitalization, some shares remain
in public hands). Like the share repurchase, the effect is to create a significant change in capital
structure that makes the target less attractive while delivering value to shareholders.
6. 'Crown Jewel' Defense: After a hostile takeover is announced, a target may decide to sell off a
subsidiary or asset to a third party. If the acquisition of this subsidiary or asset was one of the
acquirer's major motivations for the proposed merger, then this strategy could cause the
acquirer to abandon its takeover effort when a target initiates such a sale after a hostile takeover
bid is announced, there is a good change that the courts will declare this strategy illegal.
7. 'Pac-Man' Defense: In the video game Pac-Man, electronic ghosts would try to each the main
character, but after eating a power pill, Pac-Man would then turn around and try eat the ghosts.
Likewise, the analogy applies here, after a hostile takeover offer, the target can defend itself by
making a counteroffer to acquire the acquirer. In practice, the Pac-Man defense is rarely used
because it means a smaller company would have to acquire a larger company, and the target may
also loose the use of other defense tactics as a result of its counteroffer.
8. White Knight Defense: A white knight is a friendly third party that comes to the rescue of the
target company. In many cases, the while knight defense can start a bidding war between the
hostile acquirer and the third party, resulting in the target receiving a very good price when a deal
is ultimately completed. This tendency for the winner to overpay in a competitive bidding
situation is called the 'Winner's Curse'.
9. White Squire Defense: In medieval times, a squire was a 'Junior Knight'. In today's M&A world, the
squire analogy means that target seeks a friendly party to buy a substantial minority stake in the
target, enough to block the hostile takeover without selling the entire company. Although the
white squire may pay a significant premium for a substantial no. of the target's shares, these
shares may be purchased directly from the target company and the target shareholders may not
receive any of the proceeds, for e.g. the white squire defense may purchase shares of convertible
preferred stock instead of common stock. The use of the white squire defense may carry a high
litigation risk depending on the details of the transaction and local regulations. Additionally, stock
55
exchange listing requirements sometimes require that target shareholders vote to approve these
types of transactions and shareholders may not endorse any transaction that doesn't provide an
adequate premium to them directly.
In US, most hostile takeover attempts result in litigation. The courts generally bless legal pre-offer
defense mechanisms but tend to scrutinize post-offer defense very closely. The target usually
assumes the burden of proof in showing that the recently enacted defenses are simply intended to
perpetuate management's tenure at the target company.
lllll
Target Company Valuation
An analyst is likely to use some combination of these primary techniques and possibly others, when
gauging a company's fair value.
lllll
Discounted Cashflow Analysis
Free Cash Flow (FCF) or (FCFF) is the relevant measure in this context because it represents the
actual cash that would be available to the company's investors after making all investments
necessary to maintain the company as an ongoing enterprise. FCFs are the internally generated
funds that can be distributed to the company's investors e.g. shareholders and bondholders,
without impairing the value of the company.
Step 1: Calculate FCF or FCFF
Step 2: Determining Terminal Value
Constant Growth Formula: Use, only if terminal growth rate (g T ) < Adjusted WACC (WACC a)
TV T = FCF T (1 + g T )
(WACCa - g T )
or
But in context of evaluating potential merger
target, we want to adjust the target's WACC to
reflect any changes in the target's risk or capital
structure that may result from the merger.
Terminal growth rate is often lower than the
growth rate applied during the first stage
because any advantages from synergies, new
opportunities are adjusted.
Market Multiple: applies when analyst believes the firm will trade at the end of first stage.
Projected Price
↷
TV T = FCF T x (P / FCF)
Times
∴
Constant
↷
TV T =
TV T Price
(1 + WACC a ) T
Price
Multiple
Step 3: Now discount first stage FCF values & Terminal Value to determine the value of target firm.
Enterprise Value (EV) used to describe the sum of FCF
values of the first stage and the terminal value.
Discounted FCF =
FCF 1
+
FCF 2
+ ..... + FCF n+ TV T
1
2
(1 + WACC a ) (1 + WACCa )
(1 + WACC a ) n
56
lllll
Comparable Company Analysis
Comparable Company Analysis uses relative valuation metrics for similar firms to estimate market
value and then adds a takeover premium to determine a fair price for the acquirer to pay for target.
Step 1: Identify the set of Comparable Firms
Step 2: Calculate various relative value measures based on the current market price of companies
in the sample.
↶
EV = Mv of Debt + Mv of Equity - Cash and Investments
EV/FCF, EV/EBITDA, EV/Sales
↶
Equity Multiples
P/E, P/B, P/CF, P/S
Step 4: Estimate a Takeover Premium (TP)
↶
Deal
Price
↷
Step 3: Calculate Mean, Median and Range for the chosen relative value measures and apply those
to the estimates for the target to determine the target's value e.g. Value = EPS x (P/E).
(SP should be the price of the target
Target's
stock before any market speculation
Stock Price
causes the target's stock price to jump)
↶TP = (DP - SP) ↷
TP is the amount that the takeover price of each of the target's
SP
Analysts usually look at premiums paid in recent takeovers of
companies most similar to the target firm. Preferably, the
calculations will be from the recent past because acquisition
values and premium tend to vary over time and economic cycles.
shares must exceed the market price in order to persuade the
target shareholders to approve the merger deal.
Step 5: Calculate the estimated takeover price for the target as the sum of estimated stock value
based on comparables and the takeover premium. Once the takeover price is computed,
the acquirer should compare it to the estimated synergies from the merger to make sure
the price makes economic sense.
lllll
Comparable Transaction Analysis
Comparable Transaction Analysis uses details from recent takeover transactions of similar
companies to estimate the target's takeover value. The methodology behind the approach is very
similar to the comparable company approach we just showed you, except that all of the
comparables are firms that have recently been taken over. However, using recent transaction data
means that the takeover premium is already included in the price, so there is no need to calculate it
separately.
Step 1: Identify a set of recent Takeover Transactions
Step 2: Calculate various relative value measures based on completed deal prices for the
companies in the sample. They are based on prices for completed M&A deals rather than
current market prices.
Step 3: Calculate Mean, Median and Range for the chosen relative value measures and apply those
to the estimates for the target to determine the target's value e.g. Value = EPS x (P/E). No
need to calculate a separate transaction premium because it is already incorporated into
the price.
57
Example 6:
Comparable company analysis and comparable transaction analysis are similar in many ways. However, the
key differentiator is most likely that:
A. Comparable company analysis compares companies that are similar to the target while comparable
transaction analysis does not.
B. Comparable transaction analysis derives valuation from details of recent takeover transactions for
comparable companies, while comparable company analysis use relative valuation metrics for similar
companies.
C. Comparable company analysis discounts free cash flows estimated with pro forma financial statements.
✓
Fig 5: Advantages and Disadvantages of Target Company Valuations
Advantages
[Future]
Discounted Cashflow
(DCF)
Disadvantages
1. Easy to model expected changes in
the target company's CF resulting
from operating synergies and cost
structure changes.
1. Difficult to apply when FCFs
are negative.
2. Easy to customize.
Comparable
[Past]
Company Analysis
(CCA)
1. Debt is easy to access.
2. Assumption: similar assets should
have similar values.
1. Difficult to incorporate merger
synergies & capital structure into
analysis.
2. Fair stock price but not fair takeover
price (TP should be determined
separately).
3. Data available for past premium may
not be accurate for evaluating the
target company.
Comparable
[Present]
Transaction Analysis
(CTA)
1. Value derived directly from recent
prices for actual deals.
2. Due to recent transactions, it reduces
litigation risk for both companies'
board of directors and managers
regarding the merger transaction's
1. Difficult to incorporate merger
synergies & capital structure into
analysis.
2. Inadequate data available on no. of
comparable transactions.
pricing.
Example 7: [Comparable Company Analysis]
Mia Yost, an investment banker has been retained by the Gracio Corporation to estimate the price that
should be paid to acquirer Albert Garage Systems Inc. (AGSI). Calculate the appropriate valuation metrics and
using the mean of those metrics, estimate the price that Gracio should pay AGSI?
Comparable Company Data
and Comparable Firms (CF)
Takeover Prices in Recent
M&A Transactions
Company Statistics
Current Stock Price
EPS
BVPS
CF per Share
Sales per Share
Target Company
Target 1
Target 2
Target 3
Target 4
AGSI
2.95
15.2
3.8
46
CF1
$ 25
1.5
8.8
2
21.6
CF2
$ 33
2.25
10.5
2.9
28.7
CF3
$ 19
1.2
6
1.8
19.5
Stock Price Prior to Takeover
$ 22
$ 18.25
$ 108.9
$ 48.5
Takeover Price
$ 27.25
$ 21
$ 130
$ 57
58
Relative Value Measure
Current Stock Price
P/E
P/B
P/CF
P/Sales
Target Company
Statistics
EPS
BVPS
CF per Share
Sales per Share
CF1
$ 25
25/1.5 = 16.67
25/8.8 = 2.84
25/2 = 12.5
25/21.6 = 1.16
a
AGSI
Statistics
2.95
15.2
3.8
46
Estimate a Takeover Premium
CF2
$ 33
33/2.25 = 14.67
33/10.5 = 3.14
33/2.9 = 11.38
33/28.7 = 1.15
CF3
$ 19
19/1.2 = 15.83
19/6 = 3.17
19/1.8 = 10.56
19/19.5 = 0.97
b
Mean
15.72
3.05
11.48
1.09
(a x b)
Relative Value
Mean Relative
Estimated Stock Value
Measure
Value Measure
for AGSI
P/E
15.72
$ 46.37
P/B
3.05
$ 46.36
P/CF
11.48
$ 43.62
P/Sales
1.09
$ 50.14
Mean Estimated Stock Value
$ 46.62
Target 1:
27.25 - 22 / 22 = 23.9%
Target 2: 21 - 18.25 / 18.25 = 15.1%
Target 3: 130 - 108.9 / 108.9 = 19.4%
Target 4:
57 - 48.5 / 48.5 = 17.5%
Mean Premium 19%
∴ Estimated Takeover Price for AGSI = $46.62 (1.19) = $55.48
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Bid Evaluation
V AT = V A + V T + S - C
Whereas,
VAT : Post-Merger value of the combined company (Acquirer + Target)
VA : Pre-Merger value of acquirer
V T : Pre-Merger value of target (Price of target stock before any market speculation)
PAT : Price per share of combined firm after the merger announcement
N : No. of new shares the target receives
S : Synergies
C : Cash paid to target shareholders
P T : Price paid for target
Target's Gain
Acquirer's Gain
Gain T = TP = PT - V T
Gain A = S - TP = S - (PT - V T )
The gains for the acquirer and the gains for the target leave us with 'S' or the synergies
from the deal. It's the gain resulting from the estimated value of cost reduction synergies
or revenue enhancement synergies that the acquirer and the target are dividing.
Cash Deal : PT = C
Stock Deal : PT = N x PAT
Mixed Deal : PT = [C + (N x PAT )]
Acquirer will want to pay the lowest possible price i.e. VT (Pre-Merge Value of the Target)
Target will want to pay the highest possible price i.e. VT + S (Pre-Merger Value of the Target + Expected Synergies)
59
Fig 6: Effect of Payment Method
Cash
Stock
Actual Synergies > Expected Synergies
Target: TP Unchanged
Acquirer: reaps additional reward
Target will share the upside of
the shares
Actual Synergies < Expected Synergies
Target: TP Unchanged
Acquirer's gain may evaporate
Target will share the downside of
the shares
Example 8:
Adagio Software Inc. and Tantalus Software Solutions Inc. are negotiating a friendly acquisition of Tantalus by
Adagio. The management teams at both companies have informally agreed upon a transaction value of about
€12 per share of Tantalus Software Solutions stock but are presently negotiating alternative forms of
payment. Sunil Agrawal CFA works for Tantalus Software Solutions' investment banking term and is
evaluating three alternative offers presented by Adagio Software:
a. Cash offer: Adagio will pay €12 per share of Tantalus stock
b. Stock offer: Adagio will give Tantalus shareholders 0.8 shares of Adagio stock per share of Tantalus stock
c. Mixed offer: Adagio will pay €6 plus 0.4 shares of Adagio stock per share of Tantalus stock
Agrawal estimates that the merger of the two companies will result in economies of scale with a NPV of
€90,000,000. To aid in the analysis, Agrawal has also compiled the following data:
Pre-Merger Stock Price
No. of Shares Outstanding
Pre-Merge Market Value
Adagio
€ 15
75,000,000
€ 1,125,000,000
Tantalus
€ 10
30,000,000
€ 300,000,000
Based only on the information given, which of the three offers should Agarwal recommend to the Tantalus
Software Solutions management term?
Cash = V AT = 1125 + 300 + 90 - (12 x 30)
= 1125 + 300 + 90 - 360 = 1155
Stock = V AT = 1125 + 300 + 90 = 1515
0.8 A / 1 T = 0.8 x 30 = 24
Total = 24 + 75 = 99
Mixed = V AT = 1125 + 300 + 90 - (6 x 30)
= 1125 +300 + 90 - 180 = 1335
0.4 A / 1 T = 0.4 x 30 = 12
Total = 12 + 75 = 87
Target's Gain
Cash
TP = C - V T
S - TP = S - (C - V T )
TP = (N x PAT ) - V T
S - TP = S - [(N x PAT ) - V T ]
TP = [C + (N x PAT )] - V T
S - TP = S - {[C + (N x P AT )] - V T }
= 360 - 300 = 60
Stock
= (24 x 1515 / 99) - 300 = 67.2727
Mixed
Acquirer's Gain
= [180 + (12 x 1335 / 87)] - 300 = 64.137
= 90 - 60 = 30
= 90 - 67.2727 = 22.7273
= 90 - 64.137 = 25.863
60
Agarwal should recommend that the Tantalus Software Solutions management team opt for all stock offer
because that alternative provides Tantalus shareholders the most value i.e. highest premium. The dilution
from the stock offer effectively reduced the acquirer's gains because the target was able to share in the risk
and reward of the deal as a result of receiving shares.
Example 9:
Modern Auto, an automobile parts supplier, has made an offer to acquire Sky Systems, creator of software
for the airline industry. The offer is to pay Sky Systems's shareholders the current market value of their stock
in Modern Auto's stock.
Share Price
No. of Shares Outstanding
Earnings
Modern Auto
$ 40
40,000,000
$ 100,000,000
Sky Systems
$ 25
15,000,000
$ 30,000,000
If Sky Systems were to be acquired by Modern Auto under the terms of the original offer, the post-merger
EPS of the new company would be closest to?
Acquires
M
→ S by exchanging 1 of its shares for $ 40 / $ 25 = 1.6 shares of Sky Systems: 1 M / 1.6 S or 0.625 M / 1 S
Now, 0.625 x 15
= 9.375
V AT = 100 + 30 = 130
EPS after Merger =
130
= $ 2.63
40 + 9.375
Example 10:
Fedora is prepared to offer to buy Ubuntu by directly issuing to the shareholders of Debian a total of
3000,000, $10 par value shares that will rank equally with its existing shares (Ubuntu is a subsidiary of
Debian).
Value of Ubuntu as a standalone business
Value of Ubuntu to Debian
Value of Fedora (5000,000 shares, $10 par)
Value of Fedora and Ubuntu as a combined entity
$ 78
$ 85
$ 132
$135
(135 = 132 + 85 + x - 90)
Value of Fedora and Ubuntu post stock acquisition: V AT = 132 + 85 + 8 = 225
a. If Debian shareholders accept the stock offer by Fedora, the economic impact on them would be closest to:
n x PAT = 3000,000 x
225 = $ 84,375,000
8000,000
Loss to Debian shareholders = $ 84,375,000 - $ 85,000,000 = - $ 625,000
b. Under Fedora's stock offer, the economic impact on the current shareholders of Fedora is closest to:
n x PAT = 5000,000 x
225 = $ 140,625,000
8000,000
61
Gain to Fedora shareholders = $ 140,625,000 - $ 132,000,000 = $ 8,625,000
On average, both acquirer and target tend to see higher stock return surrounding cash acquisition offers than
around share offers.
ST: On average target shareholders reap 30% premiums over the stock's pre-announcement market price and
the acquirer's stock price falls on average between 1% and 3%. The high average premiums paid to target
shareholders may be attributed to, at least partly, to the 'Winner's Curse' i.e. the tendency for competitive
bidding to result in overpayment. Managers also may overstate the synergies and expected benefits of the
merger. This tendency is called 'Management Hubris'. Even if there are no synergies from a merger,
Managerial Hubris would lead to higher than market-bids and a transfer of wealth from the acquiring
company's shareholders to the target's shareholders. The empirical evidence is consistent with Roll's
Hubris Hypothesis.
LT: Empirical evidence shows that acquirers tend to underperform comparable companies during 3 years
following an acquisition. This implies a general post-merger operational failure to capture synergies.
Average returns to acquiring companies subsequent merger transactions are - 4.3% with about 61% of
acquirers lagging their industry peers.
lllll
Corporate Restructuring
Just as mergers and acquisitions are a means by which companies get bigger, a corporate restructuring
is usually used in reference to ways that companies get smaller by selling, liquidating, splitting off or
spin off a division or a subsidiary, it is referred to as a Divestiture. Most Divestitures involve a direct
sale of a portion of a firm to an outside buyer. The selling firm is typically paid in cash and gives up
control of the portion of the firm sold.
Equity Carve-Outs: Create a new independent company by giving an equity
Shares issued
to outside
shareholders
i.e. public.
interest in a subsidiary to outside shareholders i.e. public (management and
operations are separate from the parent company).
Spin-Offs: Similar to Carve-Outs in that they create a new independent company
that is distinct from the parent company. The primary difference is that shares are
not issued to public, but are instead distributed proportionately to the parent
company's shareholders. Since shares of new company are simply distributed to
Shares issued
to existing
shareholders.
existing shareholders, the parent company doesn't receive any cash in the
transaction. A Spin-Off simply results in shareholders owning stock in two different
companies where there used to be one (management team and operations are
completely separate).
Split-Offs: Allow shareholders to receive new shares of a newly created entity i.e.
division of the parent company in exchange for a portion of their shares in the
Giving up parent shares,
in exchange for shares
in new division.
parent company. The key being shareholders are giving up a portion of their
ownership in the parent company to receive the new shares of stock in the division.
Liquidation: Breaking up a company, division or subsidiary and selling off its assets
piecemeal. Most liquidations are associated with bankruptcy.
62
Previous mergers that didn't work out as planned is the reason (mostly) for divestitures. Indeed,
periods of intense merger activity are often followed by periods of heightened restructuring activity.
Some of the common reasons for restructuring follow: (i) Divisions no longer fits into managements'
long-term strategy, (ii) Poor Fit i.e. Lack of Profitability, (iii) Reverse Synergy i.e. Individual parts are
worth more than the whole and (iv) Financial or CF needs i.e. due to financial difficulty.
1
Economics
PAGE NOS. 26
CHAPTER 10
VOL. 4
CHAPTERS. 3
Currency Exchange Rates: Understanding Equilibrium Value
Price
Base
1.5 $ / €
1 € = 1.5 $
Spot Exchange Rate: is the currency exchange rate for immediate delivery i.e. the exchange of currencies
takes place two days after the trade. (T+2)
Forward Exchange Rate: is a currency exchange rate for an exchange to be done in the future i.e. 30 days,
60 days, 90 days or one year.
Bid
Ask
Interbank: 1.3649 - 1.3651 $ / €
Dealer: 1.3648 - 1.3652 $ / €
or
or
1.3649 / 1.3651 $ / €
1.3648 / 1.3652 $ / €
Interbank
6
Interbank sells at 1.3651
5
Interbank buys for 1.3649
Dealer
7
Dealer sells at 1.3652
4
Dealer buys for 1.3648
Client
Interbank and Dealer: Buy cheap and sell expensive
Client or Investor: Buy at ask expense and sell at buy cheap
The bid-offer spread a dealer provides to most client typically is slightly wider than the bid-offer spread
observed in the interbank market. Dealers buy and sell foreign exchange among themselves in what is called
the interbank market. The interbank market is typically for dealing sizes of at least 1000,000 units of the base
currency. The spread quote by a dealer depends on: (i) The spread in a interbank market for the same
currency pair, (ii) The size of the transaction and (iii) The relationship between the dealer and client. Typically
the larger the transaction, the further away from the current spot exchange rate the dealing price will be.
Hence, a client who asks a dealer for a two-sided spot $ / € price on, for example, €50,000,000 will be shown a
wider bid-offer spread than a client who asks for a price on €1000,000. Now, the interbank spread on a
currency pair depends on 'Liquidity'. Liquidity is influenced by several factors like Currencies involved (highvolume currency pairs command lower spreads than do lower-volume currency pairs), Time of the Day (when
both the New York and London currency markets are open is considered the most liquid and have narrow
spreads), Market Volatility (higher volatility leads to higher spreads to compensate market makers for the
2
increased risk of holding the currencies), Geopolitical Events (war, civil strife), Market Crashes and Major Data
Releases. Spreads in forward exchange rate quotes increase with maturity. Counterparty risk and interest
rate risk in forward contracts increase with maturity; reason being longer maturity contracts tend to be less
liquid.
1. Spot Currency Conversion
Rule: Up the bid and Multiply; Down the ask and divide
Example 1:
A dealer is quoting the A / $ spot rate as 1.5060 - 1.5067. How would we:
(a) Convert 1000,000 $ to A: A ↶
/ $ Up: 1.5060 x 1000,000 = A 1,506,000
↷
(b) Convert 1000,000 A to $: A / $ Down: 1000,000 / 1.5067 = $ 66,702.13
Example 2:
JPY / GBP
Bid
187.39
Ask
187.43
Midpoint
187.41
Using quotes, the amount received by Goldsworthy from converting JPY 225,000,000 will be closest to:
Because she is buying the base currency i.e. GBP, she
A. GBP 1,200,448
must pay the offer price of JPY 187.43 per GBP.
B. GBP 1,200,576
Therefore, JPY 225,000,000 / 187.43 = GBP 1,200,448
C. GBP 1,200,704
✓
2. Cross Rates
The Cross Rate is the exchange rate between two currencies implied by their exchange rates with common
third currency. It is necessary to use cross rates when there is no active foreign exchange (Fx) market in the
currency pair being considered. The cross rates that we personally calculate is called Implied Cross Rates.
Whereas,
(B / C) Bid =
(B / C) Ask =
(A / C) Bid = (A / B) Bid x
(B / C) Bid
(A / C) Ask = (A / B) Ask x
(B / C) Ask
1
(C / B) Ask
1
(C / B) Bid
3. Triangular Arbitrage
Rule: We begin with three pairs of currencies, each with bid and ask quotes and construct a triangle where
each node in the triangle represents one currency. To check for arbitrage opportunities, we go around
clockwise until we reach our starting point.
The bid-offer quotes a dealer shows in the interbank Fx market must respect two arbitrage constraints;
otherwise the dealer creates riskless arbitrage opportunities for other interbank marker participants.
(i) The bid shown by a dealer in the interbank market cannot be higher than the current interbank offer
(4 < 6), and the offer shown by a dealer cannot be lower than the current interbank bid (7 > 5).
(ii) The cross rate bids (offers) posted by a dealer must be lower (higher) than the implied cross rate offers
(bids) available in the interbank market.
3
Example 3:
The bid-ask quotes for $, G and € are:
€ / $ = 0.7 - 0.7010
$ / G = 1.7 - 1.7010
€ / G = 1.2 - 1.2010
The potential arbitrage profit from a triangular arbitrage based on initial position of $1000,000 is?
$
→ ②
€
$
Down-Divide-Ask
= 705,467.372 = $ 1,006,372.856
0.7010
€
①$→G
Down-Divide-Ask
= 1000,000 = G 587,889.476
1.7010
③€→$
G
Up-Multiply-Bid
= 587,889.476 x 1.2 = € 705,467.372
Therefore, $ 1,006,372 - $ 1000,000 = $ 6372.
Example 4:
€/$
D/$
D/€
Bid
0.8045
1.2050
1.514
Ask
0.8065
1.21
1.519
S1
0.82
1.228
1-Year LIBOR Rate
€ 0.8%
$ 0.9%
D 3%
For European investor who attempts to exploit the DNR currency market with a normal 1-year carry trade
based on a €100,000 position will most likely achieve a net profit of € of?
⑤
153,893.37 x 0.82 = € 102,763
1.228
①
④
€ 100,000
Invest @ 3% = 149,411 (1.03)
= D 153,893.37
③
D 149,411
$ 123,992.56
$
D
Up-Multiply-Bid
= 123,992.56 x 1.2050
€
$
Down-Divide-Ask
= 100,000
0.8065
→
→
②
↷ Borrowing Cost = 100,000 x 0.8%
Profit = € 102,763 - € 100,000 - € 800
= € 1963
✓
↷
Example 5:
The cross rate implied by the interbank market is 85.76 / 85.8 in $ / C and a dealer quoted a bid-offer rate of
85.73 / 85.75 in $ / C, then a triangular arbitrage would involve buying:
A. CAD in the interbank market and selling it to the dealer for a profit of $ 0.01 per CAD.
B. $ from the dealer and selling it in the interbank market for a profit of CAD 0.01 per $.
C. CAD from the dealer and selling it in the interbank market for a profit of $ 0.01 per CAD.
Why? The dealer is offering to sell the CAD (the base currency in the quote) too cheaply, at an offer
rate that is below the interbank bid rate (85.75 vs. 85.76, respectively). Triangular arbitrage would
involve buying CAD from the dealer (paying the dealer's offer) and selling CAD in the interbank
market (hitting the interbank bid), for a profit of $ 0.01 (85.76 - 85.75) per CAD translated.
4
4. Forward Discount and Premium
In professional Fx markets, forward exchange rates are typically quoted in terms of points i.e. the
difference between the forward exchange rate quote and the spot exchange rate quote.
Forward Premium (Discount) = F - S 0
F > S 0 : Premium
F < S 0 : Discount
The premium or
discount is for the
base currency.
Example 6:
Given the following quotes for A / $, compute the bid-offer rates for a 30-day forward contract.
S0
30-day
1.0511 / 1.0519
+3.9 / +4.1
Since the forward quote is positive, the $ (base currency) is trading at a forward premium.
30-day Bid = 1.0511 + 3.9 = 1.05149
10,000
30-day Ask = 1,0519 +
4.1 = 1.05231
10,000
Therefore, 1.05149 - 1.05231 A / $
5. Mark-to-Market Value
The value of the contract at initiation is zero to both parties. After initiation, the value of the forward
contract will change as forward quotes for the currency pair change in the market. The value of a forward
currency contract prior to expiration is also known as the mark-to-market value. However, the
mark-to-market value of the forward contract will change as the spot exchange rate changes and as
interest rates change in either of the two currencies.
Vt = (F t - F) x Contract Size
1 + r Days T-t
360
(
)
Whereas,
Vt : Value of the forward contract at time 't' (to the party buying the base currency),
(t < T) denominated in price currency.
F t : Forward price (to sell base currency) at time 't' in the market for a new contract
maturing at time 'T'.
F : Forward price specified in the contract at inception (to buy the base currency).
r : Interest rate of price currency.
Create a offsetting forward position that is equal to the original forward position. E.g. If the market
participant was long $10,000,000 forward, so the offsetting forward contract would be to sell $10,000,000.
While offsetting, if the base currency of the exchange rate quote is being sold (bought), then we use the bid
(ask) side of the market.
5
Example 7:
The following are the current spot rates and forward points being quoted for the $ / GBP pair:
S0
1.4939 / 1.4941
1-month
+3.9 / +4.1
2-month -17.4 / -16.8
3-month -25.4 / -24.6
↶
a. The current all-in bid rate for delivery of GBP against the $ in 3 months is? $ / GBP : Up-Bid
1.4939 + (-25.4) = 1.49136
10,000
b. The all-in rate that the dealer will be quote today by another dealer to sell the $ in 3 months forward
↷
against the GBP is? $ / GBP : Down-Ask
1.4941 + (-24.6) = 1.4916
10,000
The dealer will sell $ against GBP, which is equivalent to buying GBP (the base currency) against the $.
Hence, the offer side of the market will be used for forward points. The all-in forward price is 1.4916.
Example 8:
Yew Mun Yip has entered into a 90-day forward contract long CAD1000,000 against $ at a forward rate of
1.05358 $ / CAD. 30 days after initiation, the following $ / CAD quotes are:
S0
30-day
60-day
1.0612 / 1.0614
+4.9 / +5.2
+8.6 / +9
✓
The following information is available (at t=30) for $ interest rates:
30-day
60-day
1.12%
1.16%
✓
What is the mark-to-market value in $ of Yip's forward contract? Offsetting is sell, so buy quote.
1.0612 + 8.6
10,000
Vt = (1.06206 - 1.05358) x 1000,000 = $ 8463.64
1 + 0.0116 60
360
( )
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International Parity Relations
These parity conditions typically make simplifying assumptions, such as zero transaction costs, perfect
information is available to all market participants, risk neutrality and freely adjustable market prices.
lllll
Covered Interest Rate Parity
The word 'Covered' in the context of covered interest parity means bound by arbitrage. Covered
interest rate parity holds when any forward premium or discount exactly offsets differences in
6
interest rates, so that an investor would earn the same return investing in either currency. E.g. If
covered interest rate parity holds; assuming euro interest rates are higher than dollar interest rates,
then depreciation of the euro relative to the dollar will just offset the higher euro interest rate.
Mostly, covered (currency-hedged) interest rate differential between the two markets is 0. For
covered interest rate parity to hold exactly, it must be assumed that there are zero transaction
costs and the market to be absolutely perfect in everyway; which is technically not possible.
[
[
]
]
1 + RA
F =
S0
1 + RB
(Days
)
360
(Days
360 )
Whereas,
R A : R f interest rate for Currency A i.e. Price Currency
R B : R f interest rate for Currency B i.e. Base Currency
F - S0
Domestic Currency trades at a Forward Premium
Foreign Currency trades at a Forward Discount
If F > S 0 and R A > R B
Domestic Currency trades at a Forward Discount
Foreign Currency trades at a Forward Premium
If F > S 0 and R A > R B
If it is possible to earn more interest in the foreign market than in the domestic market, then the
forward discount for the foreign currency will offset the higher foreign interest rate.
Example 9:
The US dollar interest rate is 8% and the euro interest rate is 6%. The spot exchange rate is 1.3 $ / € and the
1-year forward rate us 1.35 $ / €. Determine whether a profitable arbitrage opportunity exists and illustrate
such an arbitrage if it does.
Implied Forward Rate = 1.08 x 1.3 = 1.3245
1.06
Because, the market forward rate F m > F, we should sell euros in the forward market and do the opposite i.e.
buy euros in the spot market.
Initially Step 1: Borrow $ 1000 at 8%
Step 2: Purchase $ 1000 / 1.3 = € 769.23 in the spot market.
Step 3: Invest the € at 6%
Step 4: Enter into a forward market to sell the expected proceeds at the end of one year i.e.
€ 769.23 x 1.06 = € 815.38 at $ 1.35 each.
After 1 Year Step 1: Sell the € 815.38 under the terms of the forward contract at $ 1.35 to get $ 1100.76
Step 2: Repay the $ 1000, 8% loan which requires $ 1080
Step 3: Keep the difference of $ 20.76 as an arbitrage profit.
If F m > F : Arbitrage Profit
If F m < F : Arbitrage Loss
7
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Uncovered Interest Rate Parity
Uncovered in this context means not bound by arbitrage. Uncovered interest parity is the condition
in which the difference in interest rates between two nations is equal to the expected change in
exchange rates between those nation's currencies. Uncovered interest rate parity derives the
expected future spot rate (which is not market traded), due to which the investor's currency
position in the future is not hedged i.e. uncovered.
Consider Country A's interest rate as 4% and Country B's interest rate as 9%. Under uncovered
interest rate parity, currency B is expected to depreciate by 5% annually relative to currency A, so
that an investor should be indifferent between investing in country A or B. When RA - RB is negative,
currency B is expected to depreciate. E.g. If the foreign interest rate is higher by 2%, the foreign
currency is expected to depreciate by 2%, so the investor should be indifferent between investing in
foreign currency or in their own domestic currency.
%
△ S0e = R A - R B
Whereas,
e
S0 : Change in the spot rate expected for future periods.
R A : Nominal interest rate for currency A i.e. price currency.
R B : Nominal interest rate for currency B i.e. base currency.
△
An investor that chooses to invest in the foreign currency without any additional return (since
interest rate differential is offset by currency value changes) is not demanding a risk premium for
the foreign currency risk. Hence, uncovered interest rate parity assumes that the investor is risk
neutral i.e. risk neutral investors base their decisions solely on the expected return and are
indifferent to the investment's risk. Longer-term expected future spot rates based on uncovered
interest rate parity are often used as forecasts of future exchange rates. Most studies find that over
short and medium-term periods, the rate of depreciation of the high yield currency is less than that
what would be implied by uncovered interest rate parity; therefore, uncovered interest rate parity
works better over very long-term horizons.
Example 10:
Spot exchange rate quote is 8.385 Z / €. The 1-year nominal rate in the Eurozone is 10% and 1-year nominal
rate in South Africa is 8%. Calculate the expected percentage change in the exchange rate over the coming
year using uncovered interest rate parity.
R A - R B = 8% - 10% = -2%
Therefore, the euro is expected to depreciate by 2% relative to the rand, leading to a change in exchange rate
from 8.385 Z / € to 8.385 (1 - 0.02) = 8.2173 Z / € over coming years.
Example 11:
1-year interest rate are 7.5% in the US and 6% in New Zealand. The current spot exchange rate is 0.55 $ /
NZD. If uncovered interest rate parity holds, the expected spot rate in 1-year is?
R A - R B = 7.5% - 6% = 1.5%
Therefore, $ interest rate is 1.5% higher, hence NZD will appreciate by 1.5% under uncovered interest rate
parity, leading to a change from 0.55 $ / NZD to 0.55 (1 + 0.015) = 0.55825 $ / NZD.
8
lllll
Forward Rate Parity
Forward Rate Parity states that forward exchange rate will be an unbiased predictor of the future
spot exchange rate. It doesn't state that forward rate is 'perfect', but just an unbiased one; the
forward rate may overstate or understate the future spot rate from time to time, but on average it
will equal the future spot rate.
Covered Interest Parity
derives the no-arbitrage
forward rate.
F - S0 = %
S0
△ S 0e = R A - R B
Uncovered Interest Parity
derives the expected future
spot rate.
In theory, then the forward exchange rate will be an unbiased forecast of the future spot exchange
rate if both covered and uncovered interest rate parity hold. If the forward rate is equal to the
expected future spot rate, we say that the forward rate is an unbiased predictor of the future spot
rate i.e. this is called Forward Rate Parity.
e
F = S0
Stated differently, if uncovered interest rate interest parity holds, forward rate parity also holds. If
the forward rate is above (below) speculator's expectations of the future spot rate, then risk-neutral
speculators will buy the domestic currency in the spot (forward) market and simultaneously sell it in
the forward (spot) market. These transactions would push the forward premium into alignment
with the consensus expectation of the future spot rate. If the speculator's expectations are correct,
they will make a profit. Because speculators are rarely risk neutral, due to which uncovered interest
rate parity is often violated. As a result, we can conclude that forward exchange rates are typically
poor predictors of future spot exchange rates in the short run.
lllll
Purchasing Power Parity
We now turn to examining the relationship between exchange rates and inflation differentials. The
basis of their relationship is known as Purchasing Power Parity. The law of one price states that
identical goods should have the same price in all locations, after adjusting for the exchange rate. In
x
other words, the law of one price asserts that the foreign price of good 'x ', Pf should equal the
x
exchange rate-adjusted price of the identical goods in the domestic country Pd .
x
x
Pf = Ex x Pd
* Absolute Purchasing Power Parity
Instead of focusing on individual products, Absolute Purchasing Power Parity (Absolute PPP)
compares the average price of a representative basket of consumption goods between countries.
B f = Ex x Bd
Rearranging this equation and solving for the nominal exchange rate 'E x ', the Absolute PPP states
that the nominal exchange rate will be determined by the ratio of the foreign and domestic broad
price indexes.
Ex = Bf
Bd
The Absolute PPP asserts that the equilibrium exchange rate between two countries is determined
entirely by the ratio of their national price levels. However, it is highly unlike that their relationship
actually holds because of the transaction costs and the weights (consumption patterns) of the
various goods in the two economies may not be the same (e.g. people eat more potatoes in Russia
and more rice in Japan).
9
* Relative Purchasing Power Parity
Relative Purchasing Power Parity (Relative PPP) states that changes in exchange rates should exactly
offset the price effects of any inflation differential between two countries. E.g. If Country A has a 6%
inflation rate and Country B has 4% inflation rate, then Country A's currency should depreciate by
approximately 2% relative to Country B's currency over the period.
%
△ S0 = I A - I B
If the spot exchange rate 'S 0 ' remained unchanged, the higher foreign inflation rate would erode
the competitiveness of foreign companies relative to domestic companies. The Relative PPP focuses
on actual changes in the exchange rates being driven by actual differences in national inflation
rates.
* Ex-Ante Purchasing Power Parity
Ex-Ante Purchasing Power Parity (Ex-Ante PPP) asserts that the expected changes in the spot
e
exchange rate 'S 0 ' are entirely driven by expected differences in national inflation rates.
%
△ S0e = IAe - IeB
Ex-Ante PPP tells us that countries that are expected to run persistently high inflation rates should
expect to see their currencies depreciate over time, while countries that are expected to run
relatively low inflation rates on a sustainable basis should expect to see their currencies appreciate
over time.
Therefore, while over shorter horizons nominal exchange rate movements may appear random,
over long time horizons nominal exchange rates tend to gravitate towards their long-run PPP
equilibrium value.
lllll
Domestic Fisher Relation
Professor Irving Fisher originated the idea that the nominal rate of return is approximately the sum
of the real rate and the expected rate of inflation.
R
Nominal
=R
Real
+I
e
If the Fisher effect holds, the nominal interest rates in both countries will equal the sum of their
respective real interest rates and expected inflation rates.
Nominal
RA
Nominal
RB
lllll
Real
= RA
=
e
+ IA
①
Real
e
RB + IB
Real Interest Rate Parity
If Uncovered Interest Rate Parity and Ex-Ante PPP hold:
Real
RA
Real
- RB
Real
= 0 ; RA
Real
= RB
②
The real yield spread between the domestic and foreign countries will be zero and the level of real
interest rates in the domestic country will be identical to the level of real interest rates in the
foreign country. The proposition that real interest rates will converge to the same level across
different markets is known as Real Interest Rate Parity condition. It is based on the idea that with
free capital flows, funds will move to the country with a higher real rate until real rates are
equalized.
10
lllll
International Fisher Relation
Taking the Fisher Relation and Real Interest Rate Parity together gives us the International Fisher
Effect. This tells us that the difference between two countries' nominal interest rates should be
equal to the difference between their expected inflation rates. Substituting
in :
① ②
Nominal
(R A
Nominal
RA
e
e
Nominal
- IA = R B
Nominal
RA
Nominal
- I A ) - (R B
Nominal
- RB
Fig 1: Relations among the
International Parity
Conditions
e
- IB )
e
- IB
e
e
= IA - I B
Uncovered Interest
Rate Parity
Relative PPP
Forward Rate
Parity
International
Fisher Effect
Covered Interest
Rate Parity
Various parity relationships usually hold over long time horizons, not short term:
1. Since there is no true arbitrage available to force Relative PPP to hold, violations of Relative PPP in the short run are common.
However, because the evidence suggests that the Relative PPP holds approximately in the long run, it remains a useful
method for estimating the relationship between exchange rates and inflation rates.
2. If the Ex-Ante PPP as well as the International Fisher relation both hold, Uncovered Interest Rate Parity will also hold.
3. If both Covered and Uncovered Interest Rate Parity hold, then the forward exchange rate will be an unbiased predictor of the
future spot exchange rate.
4. If all the key international parity conditions held at all times, then the expected percentage change in the spot exchange rate
would equal the forward premium or discount; the nominal spread between countries and the difference between expected
national inflation rates. All investors will earn the same expected return in their own currency on any investment
denominated in a foreign currency. Exchange rate risk is simply inflation risk, so investors interested in real returns will not
face exchange rate risk. The real return will be the same in all countries. If all parity conditions held, the real yield spread
would equal 0, regardless of expected changes in the spot exchange rate.
5. If Relative PPP holds at any point in time, the real exchange rate (the exchange rate adjusted for relative historical inflation
between currency pairs) would be constant. However, since Relative PPP doesn't hold over the short term, the real exchange
rate fluctuates around its mean-reverting equilibrium value.
6. Uncovered Interest Rate Parity and PPP are not bound by arbitrage.
Fx Carry Trade
According to Uncovered Interest Rate Parity, high yield currencies are expected to depreciate in value, while
low-yield currencies are expected to appreciate in value. If Uncovered Interest Rate Parity held at all times,
investors would not be able to profit from a strategy that undertook long positions in high yield currencies
11
and short positions in low yield currencies. Since Uncovered Interest Rate Parity doesn't hold over short and
medium time periods, one can profit by investing in higher yielding currency and borrowing in lower yielding
currency. In a Fx Carry Trade, an investor invests in a higher yielding currency using funds borrowed in a
lower yielding currency. The lower yielding currency is called the funding currency.
Fx Carry Trade = %
△ Interest Rates - % △ Currency Spot Rate
= Interest Earned on Investment - Funding Cost - Currency Depreciation
Carry Trades typically perform well during low volatility periods. Sometimes, higher yields attract larger
capital flows, which in turn lead to an economic boom and appreciation (instead of depreciation) of the
higher yielding currency. This could make the Carry Trade even more profitable, because the investor earns a
return from currency appreciation in addition to the return from the interest rate spread.
The Carry Trade is profitable only if Uncovered Interest Rate Parity doesn't hold over the investment horizon.
The risk is that the funding currency may appreciate significantly against the currency of the investment,
which would reduce a trader's profit or even lead to a loss. Furthermore, the return distribution of the Carry
Trade is not normal; it is characterized by negative skewness and excess kurtosis i.e. fat tails meaning that
the probability of a large loss is higher than the probability implied under a normal distribution (leptokurtic
and positive excess kurtosis). We call this high probability of large loss the crash risk of the Carry Trade. The
'Flight to Quality' during turbulent times and the leverage inherent in the Carry Trade further compound the
losses. Risk management for Carry Trade include (i) Volatility filter using options market and (ii) Valuation
filter using PPP bands.
[
]
Fx Carry Trade =
S T (1 + R B ) - 1
S0
- RA
Example 12:
A Tokyo based asset asset manager enters into a Carry Trade position based on borrowing Yen and investing
in 1-year Australian LIBOR:
JPY
A
Currency Pair
JPY / $
$/A
0.1%
4.5%
S0
S1
81.3
80
1.075 1.0803
After 1-year, the all-in return to this trade, measured in JPY terms would be closest to:
]
Fx Carry Trade =
S1
80 x 1.0803 = 86.42
[
JPY / A
S0
81.3 x 1.075 = 87.39
86.42 (1 + 0.045) - 1
87.39
- 0.001 = 3.23%
Example 13:
'Fortunately, the International Parity condition most relevant for Fx Carry Trades doesn't always hold'. The
International Parity condition referred to is?
A. Purchasing Power Parity
B. Covered Interest Rate Parity
Carry Trade strategy is dependent upon the fact that Uncovered
C. Uncovered Interest Rate Parity
Interest Rate Parity doesn't hold in the short or medium term.
✓
12
Balance of Payments
Balance of Payments (BOP) accounting is a method used to keep track of transactions between a
country and its international trading partners (BOP accounts reflect all payments and liabilities to
foreigners as well as all payments and obligations received from foreigners). It includes government
transactions, consumer transactions and business transactions.
lllll
Current Account
The current account measures the exchange of goods, the exchange of services, the exchange of
investment income and unilateral transfers (gifts to and from other nations). It refers to that part of
the economy engaged in the actual production of goods and services. The current account balance
summaries whether we are selling goods and services to the rest of the world than we are buying
from them and vice versa.
Current a/c Surplus : Exports > Imports : Currency Appreciates
Deficit : Exports < Imports : Currency Depreciates
The amount by which exchange rates must adjust to restore current accounts to balanced positions
depends on a no. of factors: (a) The initial gap between imports and exports, (b) The response of
import and export prices to changes in the exchange rate and (c) The response of import and
export demand to changes in import and export prices. Many studies find that the response of
import and export demand to changes in traded goods prices is often quite sluggish and as a result
relatively long lags lasting several years. The second mechanism, the Portfolio Balance Channel i.e.
countries with current account surpluses actually have capital account deficits, which typically take
the form of investments in countries with current account deficits. As a result of these flows of
capital, investor countries may find their portfolios' composition being dominated by few investee
currencies. When investor countries decide to rebalance their investment portfolios, it can have a
significant negative impact on the value of those investee country currencies. Here's an example
below:
(Investor)
Surplus
lllll
Deficit
Deficit
Deficit
Investee
Investee
Investee
In long run, if China wishes
to diversify its portfolio by
selling '$', then this will
depreciate the '$'.
Capital Account
Capital Account a.k.a. financial account (i.e. both investment and financials) measures the flow of
funds for debt and equity investment into and out of the country. Capital flows tend to be the
dominant factor influencing exchange rates in the short to intermediate term, as capital flows tend
to be larger and more rapidly changing than good flows.
↷ Domestic Interest Rates > Foreign Interest Rates
Capital a/c Surplus : Capital Inflow > Capital Outflow : Currency Appreciates
Deficit : Capital Inflow < Capital Outflow : Currency Depreciates
Domestic Interest Rates < Foreign Interest Rates
↶
lllll
However, capital flows in excess of needed investment capital pose several problems; this is
especially problematic for emerging markets, such as: (a) Excessive real appreciation of the
domestic currency, (b) Financial asset and/or real estate bubbles, (c) Increases in external debt by
13
businesses or government, (d) Excessive consumption in the domestic market fueled by credit and
(e) Overinvestment in risky projects and questionable activities. Governments in emerging markets
often counteract excessive capital inflows by imposing capital controls or by direct intervention in
the Fx markets. Correlations between equity returns and exchange rates are unstable in the short
term and tend towards zero in the long run.
The objectives of capital controls or central bank intervention in Fx market are to: (a) Ensure that
the domestic currency doesn't appreciate excessively, (b) Allow the pursuit of independent
monetary policies without being hindered by their impact on currency values and (c) Reduce the
aggregate volume of inflow of foreign capital. Evidence has shown that for developed markets,
central banks are relatively ineffective at intervening in the foreign exchange markets due to lack of
sufficient resources.
A combination of 'Push' and 'Pull' factors determine the flow of capital into a country:
Pull factors represent a favorable set of developments that encourage foreign capital inflows. These
include relative price stability, a flexible exchange rate regime, improved fiscal position,
privatization of state owned enterprises, liberalization of financial markets and lifting of foreign
exchange regulations and controls.
Push factors are largely driven by mobile international capital seeking high returns from a
diversified portfolio.
Example 14:
Monique Kwan, examines two countries: Developed Market (DM) and Emerging Market (EM) countries and
notes that the DM country has what is considered a low yield safe haven currency (safe haven is an
investment that is expected to retain or increase in value during times of market turbulence) while EM
country has a high yield currency whose value is more exposed to fluctuations in the global economic growth
rate. Kwan is trying to form an opinion about movements in exchange rate for the EM currency.
a. All else equal, the exchange rate for the EM currency will most likely depreciate if the:
A. Long run equilibrium value of the high yield currency is revised upward.
B. Nominal yield spread between the EM and DM countries increase over time.
C. Expected inflation differential between the EM and DM countries is revised upward.
✓
b. An increase in safe haven demand would most likely:
A. Increase the risk premium demanded by international investors to hold assets denominated in the EM
currency.
B. Raise the return earned on Carry Trade strategies.
C. Exert upward pressure on the value of the EM currency.
✓
Example 15:
'A trade surplus will tend to cause the currency of the country in surplus to appreciate while a deficit will
cause currency depreciation. Exchange rate changes will result in immediate adjustments in the prices of
traded goods as well as in the demand for imports and exports. These changes will immediately correct the
trade imbalance'. This statement is most likely failing to consider:
A. Initial gap between the country's imports and exports.
B. Effect of an initial trade deficit on a countries' exchange rates.
C. Lag in the response of import and export demand to price changes.
✓
14
Example 16:
'Some studies have found the EM central banks tend to be more effective in using exchange rate intervention
than DM central banks, primarily because of one important factor'. The factor referred to is:
A. Fx reserve levels
B. Domestic demand
C. The level of capital flows
✓
Example 17:
'I mentioned that capital inflows could cause a currency crisis, leaving fund managers with significant losses.
In the period leading up to a currency crisis, I would predict that an affected country's':
Prediction 1: Foreign exchange reserves will increase.
Prediction 2: Broad money growth will increase.
Prediction 3: The exchange rate will be substantially
higher than its mean reverting level
during tranquil periods.
Which of her predictions is least likely to be correct?
A. Prediction 1
B. Prediction 2
C. Prediction 3
✓
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Monetary and Fiscal Policy Models
lllll
Mundell Fleming Model
Developed in early 1960s, the Mundell-Fleming model evaluates the impact of monetary and fiscal
policies on interest rates and consequently on exchange rates. The model assumes that there is
sufficient slack in the economy to handle changes in aggregate demand, and changes in inflation or
inflation is not a concern and are not explicitly modeled by this model.
Floating (flexible) exchange rate system, are determined by supply and demand in the Fx. Fixed
exchange rate regime, the government fixes the rate of exchange of its currency relative to one of
the major currencies. Under a fixed rate regime, the government would then have to purchase (sell)
its own currency in the Fx market. This action essentially reverses the expansionary (restrictive)
stance.
Fig 2: High & Low Capital Mobility
International capital flows are relatively unrestricted. It
is more relevant for the G-10 countries because capital
mobility tends to be high in developed economies.
International capital flows are relatively restricted. It is more relevant
for emerging market economies that restrict capital movement. In this
case, the impact of trade imbalance on exchange rates (good flow
effect) is greater than the impact of interest rates (financial flow effect)
15
Uncovered Interest Rate Parity : r
Mundell-Fleming Model : r
↑ : Currency Depreciation (Considers Inflation)
↑ : Currency Appreciation (Doesn't consider inflation)
lllll
Pure Monetary Model
Under a Pure Monetary Model, the PPP holds at any point in time and output is held constant. An
expansionary (restrictive) monetary policy leads to an increase (decrease) in prices and a decrease
(increase) in the value of the domestic currency. Because PPP rarely holds in either the short or
medium run, the pure monetary model may not provide a realistic explanation of the impact of
monetary forces on the exchange rate. The pure monetary approach doesn't take into account
expectations about future monetary expansion or contraction.
lllll
Dornbusch Overshooting Model
Dornbusch in 1967 constructed a modified monetary model that assumes prices have limited
flexibility in the short run (don't immediately reflect changes in monetary policy) but are fully
flexible in the long run. The model concludes that exchange rates will overshoot the long run PPP
value in the short term. Expansionary monetary policy leads to a decrease in interest rates and a
larger than PPP implied depreciation of the domestic currency due to capital outflows. In long term,
exchange rates gradually increase toward their PPP implied values. Similarly, a Restrictive monetary
policy leads to excessive appreciation of the domestic currency in the short term and then a slow
depreciation toward the long term PPP value.
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Portfolio Balance Approach
The Portfolio Balance Approach or Asset Market Approach focuses only on the effects of fiscal
policy (not monetary policy). While the Mundell-Fleming model focuses on the short term
implications of fiscal policy, the Portfolio Balance approach takes a long term view and evaluates
the effects of a sustained fiscal deficit or surplus on currency values. When the government runs a
fiscal deficit, it borrows money from investors. Under this approach, investors evaluate the debt
based on expected risk and return. Such a return could come from (a) higher interest rates and or
higher risk premium, (b) immediate depreciation of the currency to a level sufficient to generate
anticipation of gains from subsequent currency appreciation or (c) some combination of these two
factors. The currency adjustments required in the second mechanism is the core of the portfolio
balance approach.
* If the government
doesn't reverse
course, it will have
to monetize its
debt (i.e. print
money monetary
expansion), which
would also lead to
depreciation of the
domestic currency.
*
Example 18:
Over the medium term, as the Domestic Country (DM) government debt becomes harder to finance, Kwan
would most likely expect that:
A. Fiscal policy will turn more accommodative.
B. The mark-to-market value of the debt will increase.
C. Monetary policy will become more accommodative. (Printing Money)
✓
16
Example 19:
Early warning system is least likely to include an impending crisis when there is:
A. An expansionary monetary policy.
B. An overly appreciated exchange rate.
C. A rising level of foreign exchange reserves at the central bank.
✓
Example 20:
'Assume a developed market (EM) country has an expansive fiscal policy under high capital mobility
conditions. Why is its currency most likely to depreciate in the long run under an integrated Mundell-Fleming
and Portfolio Balance approach?' The most likely response is:
A. Increase in the price level.
B. Decrease in risk premiums.
C. Increase in government debt.
✓
Example 21:
Current Inflation Rate
Current Level of Output
Current Policy Rate (Nominal)
Neutral Real Policy Rate
Potential Level of Output
Target Inflation Rate
2.67%
3.5%
3%
2.75%
4%
3.27%
Taylor Rule = i = rn + p + a (p - p * ) + b (y - y *)
If the country follows the monetary policy approach, its policy rate should be closest to?
i = 2.75% + 2.67% + 0.5 (2.67% - 3.27%) + 0.5 (3.5% - 4%) = 4.87%
17
CHAPTER 11
Economic Growth and the Investment Decision
GDP and Per Capita GDP are the best indicators economists have for measuring a country's standard of living
and its level of economic development. Economic growth is calculated as the annual percentage change in
real GDP (growth in real GDP measures how rapidly the total economy is expanding) or in real per capita GDP
(real per capita GDP reflects the average standard of living in each country, essentially the average level of
material well being).
GDP measured by using current exchange rate is not appropriate. Using market exchange rates has two
problems. First, market exchange rates are very volatile. Second, market exchange rates are determined by
financial flows and flows in tradable goods and services. This ignores the fact that much of global
consumption is for non-tradable goods and services. Non-tradable goods are generally less expensive in
developing countries than in developed countries e.g. Labor is cheap in Mexico than in London. Thus, cross
country comparisons of GDP should be based on PPP rather than current market exchange rates.
Equity prices are positively related to earnings growth. Therefore, the potential GDP of a country, the upper
limit of real growth for an economy is an important factor in predicting returns on aggregate equity markets.
%
△ P = % △ GDP + % △ (
%
△ P = % △ GDP
E
GDP
)+ % △ (PE )
Short Term
Long Term
Whereas,
P : Aggregate value or price of equities
E / GDP : Earnings relative to GDP
P / E : Price to Earnings ratio
Over the long term, we have to recognize that %
△ ( E )= % △ (P )= 0, because labor will be unwilling to
GDP
E
to accept an ever decreasing share of GDP and investors will not continue to pay an ever increasing price for
the same level of earnings forever. Hence, over a sufficiently long time horizon, the potential GDP growth rate
equals the growth rate of aggregate equity valuation.
Actual GDP > Potential GDP : Inflationary (Follow restrictive monetary policy)
Actual GDP < Potential GDP : Deflationary (Follow expansionary monetary policy)
↑ Potential GDP growth, ↑ Real Interest Rates and ↑ Expected Real Asset Returns
Determinants of Economic Growth: What are the forces during long run economic growth? Labor, Physical
and Human Capital, Technology and other factors such as Natural Resources and Public Infrastructure as
inputs to economic growth and production functions and how changes in such inputs affect growth. Here's a
Cobb-Douglas Production function:
L
(1- L )
Y=A.K .L
Whereas,
Y : Output
A : A scale factor that represents that technological progress of the economy, often
referred to as 'Total Factor Productivity' (TFP)
L and (1- L ) : The share of output allocated to capital (K) and labor (L) respectively ( L and 1- L are
also referred to as capital's and labor's share of total factor cost, where L < 1). Also
work as elasticities.
18
Firstly, the Cobb-Douglas production function exhibits 'Constant Returns to Scale' i.e. increasing both inputs
by a fixed percentage leads to the same percentage increase in output.
Y / L = A . (K / L)
L
Labor Productivity = TFP x Capital Deepening
Assuming the no. of workers and L remain constant, increases in output can be gained by increasing capital
per worker (capital deepening) or by improving technology (TFP). The lower the value of L , the lower the
benefit of capital deepening. For Developed countries, the capital per worker ratio is relatively high, so those
countries gain little from capital deepening and must rely on technological progress for growth in
productivity. Moreover, if labor is added, it has higher impact on GDP. In contrast, Developing countries often
have low capital per worker ratios, so capital deepening can lead to atleast a short term increase in
productivity.
Per Capita GDP
Growth in GDP
Population
Developed
Developing
Slower
Faster
↑
↓
↓
↑
Labor Productivity Growth Rate (%) = Growth due to Technological + Growth due to Capital
Change (%)
Deepening (%)
Secondly, Cobb-Douglas production function exhibits 'Diminishing Marginal Productivity' with respect to each
individual input.
MPK = L . (Y / K) = r
L
= r . (K / Y)
Whereas,
MPK : Marginal Product of Capital is the additional output for one additional unit of
capital. Marginal Productivity of Capital is the increase in output per worker for
one additional unit of capital per labor i.e. increasing capital while keeping
labor constant.
r . K : Amount of return to providers of capital.
MC : L . (Y / K) i.e. Marginal Cost of Capital.
= 0 ; means diminishing marginal returns to capital are very significant and the extra output made possible
by additional capital declines quickly as capital increases.
L = 1 ; means that the next unit of capital increases output as much as the previous unit of capital.
L
MPK = MC ; Profit maximizing producers will stop adding capital. Once the economy reaches this steady state,
capital deepening cannot be a source of sustained growth in the economy.
MPK > MC ; Only when the economy is operating below the steady state and when the MPK > MC, can capital
deepening raise Per Capita growth.
Thirdly, Technological progress can lead to continued increases in output despite diminishing marginal
productivity of capital. Here's the Solow's growth accounting equation:
△ Y / Y = △ A / A + L . △ K / K + (1- L ) . △ L / L
19
Growth Rate in Potential GDP = Long term growth rate of Technology
+ L (Long term growth rate of Capital)
+ (1- L ) (Long term growth rate of Labor)
or
Growth Rate in Potential GDP = Long term growth rate in Labor Productivity
+ Long term growth rate of Labor Force
For the US, the relative shares of Labor and Capital are approximately 0.7 and 0.3 respectively. This means
that an increase in the growth rate of labor will have a significant larger impact i.e. roughly double, on
potential GDP growth than will an equivalent increase in the growth rate of capital, holding all else equal. For
example, because capital's share in GDP in the US economy is 0.3, a 1% increase in the amount of capital
available for each worker increases output by only 0.3%. An equivalent increase in the labor input would
boost growth by 0.7%. Also, the change in TFP is not directly observable, therefore it must be estimated as a
residual: the ex-post (realized) change in output minus the output implied by ex-post changes in labor and
capital. Even though access to natural resources e.g. via trade is important, ownership and production of
natural resources is not necessary for a country to achieve a high level of income.
Increase in quantity of labor will increase output, but not per capita output. Labor Force Participation can
increase as more women enter the workforce.
Labor = Labor Force x Avg. Hours
or
Labor = Population x Labor Force Participation (%) x Avg. Hours
(Working Age 16 - 64)
Employed and Unemployed
Example 1:
GDP Growth Rate
Labor Cost / Total Factor Cost
Growth rate of Labor
Growth rate of Capital
1.8%
0.36
1.2%
1.67%
The rate of technological change will be lower by 0.1% going forward. The growth rate of capital will increase
by 0.1% going forward. Using the Cobb-Douglas production function, France's growth rate of potential GDP
is?
1.8% = Rate of Technological Change + (0.64)(1.67%) + (0.36)(1.2%)
Rate of Technological Change = 0.3%
E (Rate of Technological Change) = 0.3% - 0.1% = 0.2%
E (Growth in Capital) = 1.67% + 0.1% = 1.77%
Growth in Potential GDP = E (GDP Growth Rate)
= 0.2% + (0.64)(1.77%) + (0.36)(1.2%)
= 1.76%
20
Example 2:
△ in Labor Productivity
Country A
Country B
Country C
2.4 %
1.6 %
0.8 %
△ in TFP
0.6 %
0.8 %
- 0.3 %
Capital deepening as a source of growth was most important for:
A. Country A
Pure Capital Deepening =
in Labor Productivity in TFP
B. Country B
(Country A)
1.8% = 2.4% - 0.6%
(Country B)
0.8% = 1.6% - 0.8%
C. Country C
✓
△
(Country C)
1.1% = 0.8% + 0.3%
Theories of Growth
A. Classical Model: Per capita economic growth is only temporary because
an exploding population with limited resources brings growth to end.
B. Neoclassical Model: Long run per capita growth depends solely on
exogeneous technological process.
C. Endogenous Model: The final model of growth attempts to explain
technology within the model itself. This theory states that economic
growth is generated internally in the economy. Investing can increase
growth. 'Endo' means from within so think investing in yourself can
increase growth.
lllll
Classical Model
The key assumption underlying the classical model (Malthusian theory) is that population growth
accelerates when the level of per capita income rises above the subsistence income, which is the
minimum income needed to maintain life. Population growth leads to diminishing marginal returns
to labor, which reduces productivity and drives GDP per capita back to the subsistence level. This
mechanism would prevent long term growth in per capita income. As a result of this gloomy
forecast, economics was labeled the 'dismal science'. The prediction from the Malthusian model
failed for two reasons: (a) The link between per capita income and population broke down. In fact,
as the growth of per capita income increased, population growth slowed rather than accelerating as
predicted by the classical growth model and (b) Growth in per capital income has been possible
because technological progress has been rapid enough to more than offset the impact of
diminishing marginal returns. Classical model's pessimistic prediction never materialized.
lllll
Neoclassical Model
The objective of the neoclassical growth model (Solow Growth Model) is to determine the long run
growth rate of output per capita and relate it to (a) the savings/investment rate, (b) the rate of
technological change and (c) population growth. Steady state rate of growth occurs when the output
to capital ratio is constant.
Physical capital stock in an economy will increase because of gross investment (I) and decline
because of depreciation. In closed economy, investment must be funded by domestic saving (S).
I=Sy
Income
↶
lllll
△
Assuming physical capital stock depreciates at a constant rate 'g'; Also K / L = C.
21
△K=Sy-gK
△C=Sy/K-g-△L
In the steady state, growth rate of capital per worker is equal to growth rate of output per worker.
△C=△Y=△A+L .△C
* Sustainable growth of output per capita (output per worker) =
* Sustainable growth of output =
Y/K=
△A
1-
L
Steady state growth rate
of labor productivity.
△A+△L
1-L
(1S )(1△- A + g + △ L )= ε
L
(△1 - A)in the steady state, the increase in 'C' has no impact on MPK.
Even though 'C' is rising at rate
L
Capital deepening is occurring, but it has no effect on the growth rate of the economy or on the
MPK once the steady state is reached.
SY=
(△1 -A + g + △ L) C
L
△ Y = (△ A )+ L S (Y - ε )
or
△ C = (△ A )+ S(Y - ε)
or
1-L
1-
Because of corresponding
increase in the steady
growth, rate of capital is
required.
L
K
K
(△1 -A )+ S(Cy - ε )
L
L
(△1 -A )+ S(Cy - ε )
L
But it has no impact
on the steady growth
rate of output per
capita or output
Overtime, the growth rates of output per capita and the capital to labor ratio decline to the steady
state rate. Overtime, output grows faster than capital, the output to capital ratio rises and growth
converges to the trend rate. Capital deepening affects the level of output but not the growth rate in
the long run. Capital deepening may temporarily increase the growth rate, but the growth rate will
revert back to the sustainable level if there is no technological progress. Neoclassical model ignores
the very factor driving growth in the economy i.e. Technology. Technology is simply the residual or
the part of growth that cannot be explained by other inputs, such as capital and labor. This lack of
an explanation for technology led to gaining dissatisfaction with the neoclassical model. The other
source of criticism of the neoclassical model is the prediction that the steady state rate of economic
growth is unrelated to the rate of saving and investment. Long run growth of output in Solow model
22
depends only on the rates of growth of the labor force and technology. Higher rates of investment
and savings have only a transitory impact on growth. Finally, the neoclassical model predicts that in
an economy where the stock of capital is rising faster than labor productivity, the return to
investment should decline with time. For the advanced countries, the evidence doesn't support this
argument because returns have not fallen over time.
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Endogenous Growth Model
Endogenous growth model contends that technological growth emerges as a result of investment in
both physical and human capital. Technological progress enhances productivity of both labor and
capital. Unlike the neoclassical model, there is no steady state growth rate, so that increased
investment can permanently increase the rate of growth. Endogenous model implies that an
increase in savings will permanently increase the growth rate. It also assumes that capital
investment i.e. R&D expenditures may actually improve TFP. As per this theory, there is no reason
why the incomes of developed and developing countries should converge.
MPK = Y / L = K / L = C
;C=Y/K
△Y=△K=SC-g-△L
Convergence
Will less developed countries experience productivity growth to match the productivity of developed nations?
1. Absolute Convergence: neoclassical absolute convergence hypothesis states that less developed countries
will achieve equal living standards overtime. It assumes that every country has access to the same
technology. This leads to countries having the same growth rates but not the same per capita income.
2. Conditional Convergence: states that convergence in living standards will only occur for countries with the
same savings rates, population growth rates and production function. Under this hypothesis, the growth
rates will be higher for less developed countries until they catch up. Under neoclassical model, once a
developing country's standard of living converges with that of developed countries, the growth rate will
then stabilize to the same steady state growth rate as that of developed countries.
3. Club Convergence: under this hypothesis, countries may be part of a 'club' i.e. countries with similar
institutional features such as financial markets, health and educational services etc. Under this method,
poorer countries that are part of the club will grow rapidly to catch up with their richer peers. Countries
can 'join' the club by making appropriate institutional changes.
4. Non-Convergence Trap: Those countries that are not part of the club or don't implement necessary
institutional reforms never achieve the higher standard of living.
Countries that follow outward oriented policies of integrating their industries with the world economy and
increasing exports, their standard of living tends to converge to that of more developed countries. Countries
following inward oriented policies by protecting domestic industries, can expect slower GDP growth and
convergence may not occur.
NOTES
1. Empirical research has found a strong positive correlation between investment in physical capital and GDP
growth rates.
23
2. Under neoclassical theory, the benefit of open market is temporary. Higher growth rates are possible
because foreign investment can provide capital to less developed countries. Under endogenous theory,
open markets lead to higher rate of growth permanently for all markets. The larger markets and greater
opportunity to take advantage of innovation will also increase the growth rate in open economies.
Ultimately, convergence of living standards is likely to be quicker in an open economy.
3. Dutch Disease refers to a situation where global demand for a country's natural resources drives up the
country's currency values making all exports more expensive and rendering other domestic industries
uncompetitive in the global markets.
4. Credit rating agencies consider the growth rate of potential GDP when evaluating the credit risk of
sovereign debt. A higher potential GDP rate reduces expected credit risk and generally increase the credit
quality of all debt issues.
24
CHAPTER 12
Economics of Regulation
Regulations can be classified as, Statutes are laws made by legislative bodies. Administrative Regulations
rules issued by government agencies or other bodies authorized by the government or Judicial Law is the
findings of the court. Substantive Law focuses on the rights and responsibilities of entities and relationships
among entities and Procedural Law focuses on the protection and enforcement of the substantive laws.
Regulators typically have responsibility for both substantive and procedural aspects of their regulations.
Fig 1: Types of Regulators
1
2
3
1 Independent Regulators are given recognition by government agencies and have power to make rules and enforce them.
However, independent regulators are usually not funded by the government and hence are politically independent. They are
immune from political influence and pressure.
2
3
SROs (Self Regulating Organizations) differ from standard industry self-regulatory bodies in that they are given recognition and
authority, including enforcement power, by a government body or agency. SROs are funded independently rather than by the
government. SROs are subject to pressure from their members. Regulators are concerned with the corporate governance of SROs
and the management of their conflict of interest. The extent of the concern is a factor in deciding the regulatory role. Not all
independent regulators are SROs. FINRA (Financial Industry Regulatory Authority) is an SRO recognized by the SEC in the US.
FINRA's primary objective is to protect investors by maintaining the fairness of the US capital markets. FINRA has the authority to
enforce security laws and regulations. IOSCO (International Organization of Securities Commissions) is a SRO but not a regulatory
authority. This organization has established objectives and principles to guide securities and capital market regulation. It is a
standard setter too.
Non-SROs, SROs without government recognition are not considered regulators. Some independent regulators such as the
PCAOB (Public Company Accounting Oversight Board) is a non-profit corporation, established by the US Congress to oversee the
audits of public companies. The PCAOB is funded primarily through annual fees paid by public companies, brokers and dealers.
The SEC oversees the PCAOB. PCAOB's are not SROs.
Use of independent SROs is more prevalent in common law countries than in civil law countries. In these civil law countries, nonindependent SROs may support the regulatory framework via. guidelines, codes of conduct and continuing education. Securities
regulators focus on protecting investors, creating confidence in market and encouraging capital formation. They also help reduce
systematic risk and ensure fairness and integrity of markets; and transparency (disclosures).
Regulatory Intervention
1. Informational Frictions (are market inefficiencies that lead to sub-optimal outcomes)
a. Adverse Selection (private information in the hands of some, but not all. This allows
the holder of the information to gain at the expense of others)
b. Moral Hazard
c. Asymmetrical Information (may give one entity an inherent advantage over another entity)
2. Externalities (spill-over or unintended consequences of production and consumption. Can be either positive or negative)
a. Positive Externalities (spill-over benefits. E.g. Those living nearby may benefit from your garden i.e. planting trees)
b. Negative Externalities (spill-over costs. E.g. Environment pollution)
3. Weak Competition (leads to high prices, less choices and lack of innovation)
4. Social Objectives (E.g. Placing regulatory obligations on energy companies to give discounts
on energy bills to vulnerable customers. Usually funded by government)
25
1. Regulatory Capture: regardless of the original purpose behind its establishment, a regulatory body will, at
some point in time, be influenced or even possibly controlled by the industry that is being regulated. The
rationale behind the theory is that regulators often have experience in the industry and this affects the
regulator's ability to render impartial decisions. Regulatory capture is often cited as a concern with the
commercialization of financial exchanges.
2. Regulatory Competition: regulatory differences between jurisdictions can lead to regulatory competition in
which regulators compete to provide the most business-friendly regulatory environment. Regulators may
compete to provide a regulatory environment designed to attract certain entities. Regulatory competition
can lead to what is sometimes referred to as a 'Race to the Bottom', where countries continually reduce
their regulatory standards to attract as many companies as possible to their jurisdiction.
3. Regulatory Arbitrage: occurs when businesses shop for a country that allows a specific behavior rather
than changing the behavior. Regulatory arbitrage also entails exploiting the difference between the
economic substance and interpretation of a regulation. To avoid regulatory arbitrage, cooperation at a
global level to achieve a cohesive regulatory framework is necessary. For example, regulations limiting
greenhouse gas emissions should be consistent globally, otherwise polluters would simply relocate to less
restrictive jurisdictions and the objectives of the regulators will not be achieved.
4. Prudential Supervision: is regulation and monitoring of the safety and soundness of financial institutions in
order to promote financial stability, reduce system-wide risks and protect customers of financial
institutions. It is important because the failure of one financial institution can have a far-reaching impact
and may result in a loss of confidence. Types of prudential supervision include those that focus on
diversifying assets managing and monitoring risk taking and ensuring adequate capitalization. For example,
some regulators, such as those in the European Union, may also require that designated investment firms
have in place appropriate recovery plans and resolution plans to be applied if they encounter financial
distress.
5. Financial Contagion: is a situation in which financial shocks spread from their place of origin to other
region; in essence, a faltering economy infects other healthier economies.
6. Bail-In Tool: is seen as improving the toolkit for dealing with the failure of large, globally systematic banks.
Bail-in policies are intended to ensure that shareholders and creditors of the failed institution, rather than
taxpayers, pay the costs of failures. It focuses on mitigating the systematic risk.
Regulatory Tools
1. Price Mechanisms: Price mechanisms such as taxes and subsidies can be used to further specify
regulatory objectives e.g. sin taxes are often used to deter consumption of alcohol.
2. Restricting or Requiring Certain Activities: Regulators may ban certain activities (e.g. use of specific
chemicals) or require that certain activities be performed (e.g. filing of 10K reports by publicly listed
companies).
3. Provision of Public Goods or Financing of Private Projects: Regulators may provide public goods (e.g.
national defense) or fund private projects (e.g. small business loans) depending on their political
priorities and objectives.
Example 1:
The regulatory tools least likely to be used by self-regulating organizations are:
A. Price Mechanisms
B. Restrictions on Behaviors
C. Provision of Public Goods
✓
26
Example 2:
The tools least likely to be used by regulators to intervene in financial markets owing to informational
frictions are:
A. Blackout Periods
B. Capital Requirements
C. Insider Trading Restrictions
✓
Regulatory costs and benefits are especially difficult to assess on a prospective basis relative to retrospective
basis. Regulatory costs are difficult to assess before a regulation is put in place. For this, many regulatory
provisions include a 'Sunset Clause' that requires regulators to revisit the cost-benefit analysis based on
actual outcomes before reviewing the regulation.
Private Costs of Regulation (Government Burden or Regulatory Burden)
+ Indirect Costs (Unanticipated Costs and Costs due to Changes in Economic Behavior)
- Private Benefits of Regulation
= Net Regulatory Burden
NOTES
1. Antitrust laws work to promote domestic competition by monitoring and restricting activities that reduce or
distort competition. Regulators often block a merger that leads to excessive concentration of market share.
Anti-competitive behavior such as discriminatory pricing, bundling and exclusive dealing is often also
prohibited.
2. Analysts need to understand not just how regulation affects companies and industries at present; they
should also be able to understand and anticipate the impact of proposed new or changing regulations in
the future prospects for companies and industries: (a) Assessment of the likelihood of regulatory change,
(b) Assessment of the impact of regulatory change on a sector, (c) Impact of regulatory interventions,
(d) Pricing Transparency and provide a detailed breakdown of their fees e.g. client fees, (e) Additional costs
related to minimum wages, increased information disclosures and data protection requirements and
(f) Evaluating business risk.
3. The Coase Theorem states that if an externality can be traded and there are not transaction costs, then the
allocation of property rights will be efficient and the resource allocation will not depend on the initial
assignment of property rights. According to the Coase theorem, in the face of market inefficiencies
resulting from externalities, private citizens (or firms) are able to negotiate a mutually beneficial, socially
desirable solution as long as there are no costs associated with the negotiation process. The result is
expected to hold regardless of whether the polluter has the right to pollute or the average affected
bystander has a right to a clean environment. Which states that where there are complete competitive
markets with no transaction costs and an efficient set of inputs and outputs, an optimal decision will be
selected.
Financial Statement
Analysis
PAGE NOS. 62
CHAPTER 13
VOL. 5
CHAPTERS. 6
Intercorporate Investments
The ownership percentage is only a guideline, ultimately the category is based on the investor's ability to
influence or control the investee.
lllll
Financial Assets
Financial Assets
Amortized Cost
E.g. Held-to-Maturity
[Criteria: (a) Business Model
Test: The financial assets are
being held to collect
contractual cashflows and (b)
Cashflow Characteristic Test:
The contractual cashflows are
solely payments of principle
and interest on principle]
FVPL
E.g. Held for Trading and
Derivatives (except for hedging
instruments)
[If the financial assets meets the
amortized cost criteria, but may be
sold, a 'hold-to collect and sell'
business model, it may be
measured at FVOCI. However,
management may choose the FVPL
option to avoid an accounting
mismatch (accounting mismatch
refers to inconsistency resulting
from different measurement basis
for assets and liabilities i.e. some
are measured at amortized cost
and some at fair value)].
Debt: Amortized Cost, FVPL and FVOCI
Equity: FVPL and FVOCI
1
FVOCI
E.g. Available for Sale
2
Fig 1: Allocation of Financial Assets
(US GAAP)
(IFRS)
(IFRS & US GAAP)
1
2
(Par)
(Mv)
+
Amortization (Discount) / Premium = Coupon - Interest Income
Unrealised G/L = Fv - Cost of original debt security - Discount or + Premium
(R): Remove, not deduct (just don't
show the amount).
(A): Add
IFRS and US GAAP: Require disclosure of Fair value of each class of investment in Financial Asset.
Reclassification of debt instruments is only permitted if the business model for the financial assets
(objective for holding the financial assets) has changed in a way that significantly affects operations.
The choice to measure equity instruments at FVPL or FVOCI is irrevocable. When reclassification is
deemed appropriate, there is no restatement of prior periods at the reclassification date.
Amortized Cost
FVPL
→
→ FVPL : Asset is then measured at Fair value with any gain or loss
immediately recognized in profit or loss.
Amortized Cost : Fair value at the reclassification date becomes the carrying
amount.
The new standard moves the recognition criteria from an 'incurred loss' model to 'expected loss'
model. Under the new criteria, there is an earlier recognition of impairment i.e. 12-month expected
losses for performing assets and lifetime expected losses for non-performing assets, to be captured
upfront.
3
Example 1:
At the beginning of the year, Midland Co. purchased a 9% bond with a fair value of $100,000 for $96,209 to
yield 10%. The coupon payments are made annually at year end. Let's suppose the fair value of the bond at
the end of the year is $98,500. Now let's imagine that the bonds are called on the first day of the next year for
$101,000.
(9% of100,000 - 10% of 96,209)
Held-to-Maturity = B/S : Amortized Cost = 96,209 + 621 = 96,830
(HTM)
I/S : Interest Income = 10% of 96,209 = 9621
Realized G/L = 101,000 - 96,830 = 4170 (gain)
FVPL = B/S : Fair Value = 98,500
I/S : Interest Income = 10% of 96,209 = 9621
Realized G/L = 101,000 - 98,500 = 2500 (gain)
Unrealized G/L = 98,500 - 96,209 = 1670
FVOCI = B/S : Fair Value = 98,500
I/S : Interest Income = 10% of 96,209 = 9621
Realized G/L = 101,000 - 96,830 = 4170 (gain)
Equity : Unrealized G/L = 98,500 - 96,209 - 621 = 1670
Example 2:
White is considering the securities shown below, all of which are currently held by GLI.
Security
ABC
HIJ
XYZ
Cost
$80
$20
$40
2005
$75
$30
$20
2006
$85
$35
$45
GLI holds 100,000 shares of each security.
a. If securities are classified as trading securities, balance sheet value for the portfolio at year end 2005 is?
A. $ 14,000,000 and record no gains or losses.
B. $ 12,500,000 and record no gains or losses.
C. $ 12,500,000 and record an unrealized loss of $ 1,500,000
✓
Original Portfolio Cost = $ 8000,000 + $ 2000,000 + $ 4000,000 = $ 14,000,000
2005 = $ 7,500,000 + $ 3000,000 + $ 2000,000 = $ 12,500,000
Thus, we write the portfolio down by $ 1,500,000 and take an unrealized loss.
b. If securities are classified as trading securities, balance sheet value for the portfolio at year end 2006 is?
A. $ 16,500,000 and record an unrealized gain over the past year of $ 4000,000
B. $ 14,000,000 and record an unrealized gain over the past year of $ 2,500,000
C. $ 16,500,000 and record an unrealized gain over the past year of $ 2,500,000
✓
Original Portfolio Cost = $ 8000,000 + $ 2000,000 + $ 4000,000 = $ 14,000,000
2005 = $ 7,500,000 + $ 3000,000 + $ 2000,000 = $ 12,500,000
2006 = $ 8,500,000 + $3,500,000 + $ 4,500,000 = $ 16,500,000
Thus, we write the balance sheet value up to current value and recognize an unrealized gain of $ 4000,000.
4
lllll
Associates
The ability to exert significant influence means that the financial and operating performance of the
investee is partly influenced by management decisions and operational skills of the investor. The
equity method of accounting for the investment reflects the economic reality of this relationship
and provides a more objective basis for reporting investment income than the accounting
treatment for investments in financial assets because the investor can potentially influence the
timing of dividend distributions.
(% share in Investee's company)
↷
B/S = Cost + x % (Earnings - Dividends)
I/S =
x % x Earnings
Purchase Price = Book Value of Investee
Under the equity method, the initial investment is recorded at cost and reported on the balance
sheet as a non-current asset. The equity method is often referred to as 'one-line consolidation'
because the investor's proportionate ownership interest in the assets and liabilities of the investee
is disclosed as a single line item (net assets) on its balance sheet and the investor's share of the
revenues and expenses of the investee is disclosed as a single line item on its income statement.
The investor's share of the profit or loss of equity method investments, and the carrying amount of
those investments, must be separately disclosed on the income statement and balance sheet. The
recorded investment value can decline as a result of investee losses or a permanent decline in the
investee's market value. If the investment value is reduced to 0, the investor usually discontinues
the equity method and doesn't record further losses. If the investee subsequently reports profits,
the equity method is resumed after the investor's share of the profits equals the share of losses
that were not recognized during the suspension period of the equity method.
Example 3:
Brown purchases a 20% interest in Williams for $200,000 on 1st January 2016. Calculate the investment in
Williams that appears on Brown's balance sheet as of the end of 2018. William reports the following:
2016
2017
2018
Income
200,000
300,000
400,000
900,000
Dividends
50,000
100,000
200,000
350,000
Balance Sheet
200,000 + 0.2 (200,000 - 50,000)
= 230,000
Income Statement
0.2 (200,000)
= 40,000
2017 :
230,000 + 0.2 (300,000 - 100,000)
= 270,000
0.2 (300,000)
= 60,000
2018 :
270,000 + 0.2 (400,000 - 200,000)
= 310,000
0.2 (400,000)
= 80,000
2016 :
Purchase Price > Book Value of Investee
The cost (purchase price) to acquire shares of investee is often greater than the book value of those
shares. This is because among other things, many of the investee's assets and liabilities reflect
historical cost rather than fair value.
5
IFRS: Allows a company to measure its PPE using either historical cost or fair value less accumulated
depreciation.
US GAAP: Requires the use of historical cost less accumulated depreciation to measure PPE.
At the acquisition date, the excess of the purchase price over the proportionate share of the
investee's book value is allocated to the investee's identifiable assets and liabilities based on their
fair value. Any remainder is considered goodwill. In subsequent periods, the investor recognizes
expense based on the excess amounts assigned to the investee's assets and liabilities. The expense
is recognized consistent with the investee's recognition of expense. For example, the investor might
recognize additional depreciation expense as a result of the fair value allocation of the purchase
price to the investee's fixed assets. It is important to note that the purchase price allocation to the
investee's assets and liabilities is included in the investor's balance sheet, not the investee's. In
addition, the additional expense that results from the assigned amounts is not recognized in the
investee's income statement. Under the equity method, the investor must adjust its balance sheet
investment account and proportionate share of the income reported from the investee for this
additional expense. The investor reports the investment (proportionate share of the investee's net
assets at fair value and the goodwill) in one line on its balance sheet. Goodwill is not amortized,
instead it is reviewed for impairment on a regular basis and written down for any identified
impairment.
IFRS and US GAAP: Both treat the difference between the cost of the acquisition and investor's
share of the fair value of the net identifiable assets as goodwill.
Example 4:
Blake Co. acquires 30% of the outstanding shares of Brown Co. for $100,000. At the acquisition date, book
values and fair values of Brown's recorded assets and liabilities are as follows. Calculate goodwill.
Current Assets
Plant & Equipment
Land
Liabilities
Net Assets
B
30%
R
30% of 220,000
Book Value
10,000
190,000
120,000
320,000
100,000
220,000
Fair Value
10,000
220,000
140,000
370,000
100,000
270,000
$ 100,000
(Fair Value)
$ 66,000
$ 34,000
(Book Value)
(Includes Net Asset differences & Goodwill)
Attributable to Net Assets
Plant & Equipment (30% x 30,000)
Land (30% x 20,000)
Goodwill (Residual)
$ 9000
$ 6000
$ 19,000
$ 34,000
Fv - Bv = 270,000 - 220,000 = 50,000
} 0.3 (50,000) = 15,000
The investment carries as a non-current asset on the Blake's book as a single line item (Investment om Brown
$ 100,000) on the acquisition date.
6
Example 5:
At the beginning of the year, Red Co. purchased 30% of Blue Co. for $80,000. On the acquisition date, the
book value of Blue's identifiable net assets was $200,000. Also, the fair value and book value of Blue's assets
and liabilities were the same except for Blue's equipment, which has a book value of $25,000 and a fair value
of $75,000 on the acquisition date. Blue's equipment is depreciated over 10 years using the straight line
method. At the end of the year, Blue reported net income of $100,000 and paid dividends of $60,000.
a. Calculate goodwill
Blake Co. acquires 30% of the outstanding shares of Brown Co. for $100,000. At the acquisition date, book
values and fair values of Brown's recorded assets and liabilities are as follows. Calculate goodwill.
R
30%
B
$ 80,000
(Fair Value)
$ 60,000
$ 20,000
(Book Value)
(Includes Net Asset differences & Goodwill)
30% of 200,000
Attributable to Net Assets
Equipment (30% x 50,000)
Goodwill (Residual)
b. I/S : 30% of $ 100,000
- Additional depreciation
from excess of purchase
price allocated to Blue's
equipment. (15,000/10)
$ 15,000
$ 5000
$ 20,000
$ 30,000
$ 1500
$ 28,500
c. B/S : 80,000 + 28,500 - 0.3 (60,000) = $ 90,500
(Equity Income)
(Investment Balance)
IFRS: The fair value option is only available to venture capital firms, mutual funds, unit trusts and
similar entities including investment linked insurance funds.
US GAAP: Allows equity method investments to be recorded at fair value.
(Both standards require that the election to use the fair value option occur at the time of initial
recognition and is irrevocable).
IFRS: There must be objective evidence of impairment as a result of one or more (loss) events that
occurred after the initial recognition of the investment and that loss event has an impact on
the investment's future cashflows, which can reliably estimated. Because goodwill is included
in the carrying amount of the investment and is not recognized or separately tested for
impairment. Instead, the entire carrying amount is tested for impairment by comparing its
recoverable amount with its carrying amount. The impairment loss is recognized on the
income statement and the carrying amount of the investment on the balance sheet is either
reduced directly or through the use of an allowance account.
Carrying Value > Recoverable Amount
Value in Use
(Pv of FCF)
or
Selling Price
(Fv - Selling Price)
(Highest of Both)
7
US GAAP: If the fair value of the investment falls below the carrying value (investment account on
the balance sheet) and the decline is considered permanent, the investment is written
down to fair value on the balance sheet and a loss is recognized on the income
statement.
IFRS and US GAAP: If there is a recovery in fair value in the future, the assets cannot be written-up.
(Prohibits reversal of impairment losses)
Because of its ownership interest, the investor may be able to influence transactions with the
investee. Thus, profit from these transactions must be deferred until the profit is confirmed
through use or sale to a third party. Transactions can be described as upstream (investee to the
investor) or downstream (investor to the investee).
* Upstream Sale: The investee has recognized all of the profit in the income statement. However, for
profit that is unconfirmed (goods have not been used or sold by the investor), the investor must
eliminate its proportionate share of the profit from the equity income of the investee. E.g. Suppose
that investor owns 30% of investee. During the year, investee sold goods to investor and recognized
$15,000 of profit from sale. At year end, half of the goods purchased from investee remained in
investor's inventory. Now all of the profit is included in investee's net income. Investor must reduce
its equity income of investee by investor's proportionate share of the unconfirmed profit. Since half
of the goods remain, half of the profit is unconfirmed. Thus, investor must reduce its equity income
by $2250 ($15,000 total profit x 50% unconfirmed x 30% ownership interest). Once the inventory is
sold by investor $2250 of equity income will be recognized.
Investor
Investee
A (Upstream) B
C
* Downstream Sale: The investor has recognized all of the profit in its income statement. Like the
upstream sale, the investor must eliminate the proportionate share of the profit that is
unconfirmed. E.g. Investor owns 30% of investee. During the year, investor sold $40,000 of goods to
investee for $50,000. Investee sold 90% of the goods by year end. In this case, investor's profit is
$10,000 ($50,000 sales - $40,000 COGS). Investee has sold 90% of the goods, thus, 10% of the profit
remains in investee's inventory. Investor must reduce its equity income by the proportionate share
of the unconfirmed profit by $300 ($10,000 profit x 10% unconfirmed x 30% ownership interest).
Once investee sells the remaining inventory, investor can recognize $300 of profit.
Investor
Investee
A (Downstream) B
C
(Upstream & Downstream: Adjustments are made in the books of investor)
Example 6: [Upstream]
On 1st January 2018, Wicker Co. acquired a 25% interest in Foxworth Co. for $1000,000 and used the equity
method to amount for its investment. The book value of Foxworth's net assets on that date was $3,800,000.
An analysis of fair value revealed that all fair value of assets and liabilities were equal to book values except
for a building. The building was undervalued by $40,000 and has a 20-year remaining life. The company used
straight-line depreciation for the building. Foxworth paid $3200 in dividends in 2018. During 2018, Foxworth
reported net income of $20,000. During the year, Foxworth sold inventory to Wicker. At the end of the year,
8
there was $8000 profit from the upstream sale in Foxworth's net income. The inventory sold to Wicker by
Foxworth had not been sold to an outside party. Calculate equity income and balance sheet investment for
the year 2018.
W
25%
F
25% of 3,800,000
Attributable to Net Assets
Building (25% x 40,000)
Goodwill (Residual)
$ 100,000
(Fair Value)
$ 950,000
$ 50,000
(Book Value)
(Includes Net Asset differences & Goodwill)
$ 10,000
$ 40,000
$ 50,000
I/S : 25% of $ 20,000
- Amortization of excess
purchase price
attributable to building
(10,000/20)
- Unrealized Profit
(25% x 8000)
$ 30,000
$ 500
$ 2000
$ 2500
(Equity Income)
B/S : 1000,000 + 2500 + 0.25 (3200) = $ 1,001,700
(Investment Balance)
Example 7: [Downstream]
Jones Co. owns 25% of Black Co. and approximately applies the equity method of accounting. Amortization of
excess purchase price, related to undervalued assets at the time of the investment is $8000 per year. During
2017 Jones sold $96,000 of inventory to Black for $160,000. Black resold $120,000 of their inventory during
2017. The remainder was sold in 2018. Black reports income from its operations of $800,000 in 2017 and
$820,000 in 2018. Calculate equity income for the year 2017 and 2018.
(2017)
(2018)
I/S : 25% of $ 800,000
- Amortization of excess
purchase price
- Unrealized Profit
(25% x 16000)
I/S : 25% of $ 820,000
- Amortization of excess
purchase price
+ Realized Profit
(25% x 16000)
$ 200,000
75% = 120,000
160,000
$ 8000
Jones share of
unrealized profit
$ 4000
$ 188,000
= 160,000 - 96,000
= 64000 x 25%
= 16000 (Total Unrealized
Profit)
(Equity Income)
$ 205,000
$ 8000
$ 4000
$ 201,000
(Equity Income)
When an investee is profitable and its dividend payout ratio is less than 100%, the equity method
usually results in higher earnings as compared to the accounting methods. Equity method
accounting presents several challenges for analysts. First, analysts should question whether equity
9
method is appropriate. For example, an investor holding 19% of an associate may in fact exert
significant influence but may attempt to avoid using the equity method to avoid reporting associate
losses. On the other hand, an investor holding 25% of an associate may be unable to exert
significant influence and may be unable to access cashflows, and yet may prefer the equity method
to capture associate income. Second, there can be significant assets and liabilities of the investee
that are not reflected on the investor's balance sheet. By ignoring the investee's debt, leverage is
lowest (net margin ratios could be overstated because income for the associate is included in
investor net income but is not specifically included in sales).
lllll
Business Combinations
Business combinations involve the combination of two or more entities into a larger economic
entity. Business combinations are typically motivated by expectations of added value through
synergies, including potential for increased revenues, elimination of duplicate costs, tax advantages,
coordination of the production process and efficiency gains in the management of assets.
IFRS: Business combinations are not differentiated based on the structure of surviving entity.
US GAAP: Business combinations are categorized into:
Merger: 100% of target's assets and liabilities are absorbed into the acquiring company
and target ceases to exist. The acquiring firm is the surviving entity.
A + B = A Company
Acquisition: Both entities continue to exist in a parent-subsidiary relationship. Each
entity is an individual that maintains separate financial records, but the
parent (the acquirer) provides consolidated financial statements in each
reporting period. If the acquiring company acquires less than 100%,
non-controlling (minority) shareholder's interest are reported on the
consolidated financial statements. A + B = (A + B) Company
Consolidation: A new entity is formed that absorbs both of the combining companies.
A + B = C Company
IFRS and US GAAP: In the past, business combinations could be accounted for either as a
(a) purchase method and (b) the pooling (uniting) of interest method. However,
the pooling method has been eliminated. Now, the acquisition method (replaces
the purchase method) is required. Pooling of interests method a.k.a. uniting of
interests method under IFRS, combined the ownership interests of the two firms
and viewed the participants as equals i.e. neither firm acquired the other. The
assets and liabilities of the two firms are simply combined. Key attributes of the
pooling method include the following:
(i) The two firms are combined using historical book values.
(ii) Operating results for prior periods are restated as though the two firms were
always combined.
(iii) Ownership interest continue and former accounting bases are maintained.
IFRS and US GAAP: Require that the acquirer measure the identifiable tangible and intangible assets
and liabilities of the target at fair value as of the date of acquisition. The
acquirer must also recognize any assets and liabilities that the target has not
previously recognized as assets and liabilities in its financial statements.
IFRS: Include contingent liabilities if their fair values can be reliably measured.
US GAAP: Includes only those contingent liabilities that are probable and can be reasonably
estimated.
(Contingent liability is a potential liability that may occur, depending on the outcome of an
uncertain future event)
10
IFRS: Full Goodwill or Partial Goodwill
US GAAP: Full Goodwill
Full Goodwill = Fv of Equity of Whole Subsidiary - Fv of Net Identifiable Assets
of the Subsidiary
Partial Goodwill = Purchase Price - (% Owned x Fv of Net Identifiable Assets
or
of the Subsidiary)
= % Owned x Full Goodwill
IFRS: The parent can measure the NCI at either its fair value (full goodwill) or at the partial goodwill.
US GAAP: The parent must use the full goodwill and measure the NCI at fair value (Non-controlling
'minority' interest is the portion of the subsidiary's equity i.e. 'residual interest' that is held
by third parties i.e. not owned by the parent. NCI are created when the parent acquires
less than 100% controlling interest in a subsidiary.
(IFRS and US GAAP have similar treatment for how NCI are classified or presented but differ in
measurements and calculations).
NCI (Full Goodwill) = (100 - % Owned) x Fv of Equity of Whole Subsidiary
NCI (Partial Goodwill) = (100 - % Owned) x Fv of Net Identifiable Assets of Subsidiary
IFRS: If Carrying Value (including goodwill) > Recoverable Amount
Impairment loss is recognized (I/S)
Value in Use
or
(Pv of FCF)
Selling Price
(Fv - Selling Price)
US GAAP: (a) If Carrying Value (including goodwill) > Fv
(b) Carrying Goodwill - Implied Goodwill ↷
(Fv of Reporting Unit - Fv of Net Identifiable Assets)
Impairment loss is recognized (I/S)
Example 8: [Goodwill Impairment - IFRA]
The cash generating unit of French company has a carrying value of $1,400,000 which includes $300,000 of
allocated goodwill. The recoverable amount of the cash generated unit is determined to be $1,300,000 and
the estimated fair value of its identifiable net assets is $1,200,000. Calculate impairment loss.
Carrying Value
Recoverable Amount
1,400,000 > 1,300,000 = $ 100,000
Impairment Loss (I/S)
The goodwill allocated to the cash generating unit would be reduced by $ 100,000 to $ 200,000. If the
recoverable amount of the cash generating unit had been $ 800,000 instead of $ 1,300,000, the impairment
loss recognized would be $ 600,000. This would first be absorbed by the goodwill allocated to the unit ($
300,000). Once this has been reduced to 0, the remaining amount of the impairment loss ($ 300,000) would
then be allocated on a pro-rata basis to the other non-cash assets within the unit.
Example 9: [Goodwill Impairment - US GAAP]
Last year, Parent company acquired Sub company for $1000,000. On the date of acquisition, the fair value of
Sub's net assets was $800,000. Thus, Parent reported acquisition goodwill of $200,000 ($1000,000 purchase
price - $800,000 fair value of Sub's net assets). At the end of this year, the fair value of Sub is $950,000 and
11
the fair value of Sub's net assets is $775,000. Assuming the carrying value of Sub is $980,000, determine if an
impairment exists and calculate the loss if applicable under US GAAP.
Carrying Value
Fv
(a) 980,000 > 950,000
Impairment exists
(b) 200,000 - (950,000 - 775,000) = $ 25,000
Impairment loss of $ 25,000 is recognized, thereby reducing goodwill to $ 175,000.
Example 10:
On 1st January 2018, Apple Co. (Parent) acquired 90% of the outstanding shares of the Brown Co. in exchange
for shares of Apple Co. no par common stock with a fair value of $180,000. The fair market value of the
subsidiary's shares on the date of the exchange was $200,000.
Cash and Receivables
Inventory
PPE
Payables
Long-term Debt
Net Assets
Capital Stock
Retained Earnings
Parent Book
Value
40,000
125,000
235,000
400,000
55,000
120,000
175,000
225,000
87,000
138,000
Subsidiary
Bv
Fv
15,000
15,000
80,000
80,000
95,000
155,000
190,000
250,000
20,000
20,000
70,000
70,000
90,000
90,000
100,000
160,000
34,000
66,000
a. Calculate value of goodwill and NCI under the full goodwill method.
b. Calculate value of goodwill and NCI under the partial goodwill method.
Full Goodwill = 200,000 - 160,000 = 40,000
Partial Goodwill = 180,000 - (0.9 x 160,000) = 36,000
NCI (Full Goodwill) = 0.1 x 200,000 = 20,000
NCI (Partial Goodwill) = 0.1 x 160,000 = 16,000
Cash and Receivables
Inventory
PPE
Goodwill
Total Assets
Payables
Long-term Debt
Total Liabilities
NCI
(0.9)(200) + 87 = 267 Capital Stock (No Par)
Retained Earnings
Total Equity
Full Goodwill
55,000
205,000
390,000
40,000
690,000
75,000
190,000
265,000
20,000
267,000
138,000
425,000
Partial Goodwill
55,000
205,000
390,000
36,000
686,000
75,000
190,000
265,000
16,000
267,000
138,000
421,000
12
Fig 2: Impact on
Ratios
Full Goodwill
Goodwill
Assets
NCI
Equity
ROA & ROE
D/E
Asset Turnover
Impairment
↑
↑
↑
↑
↓
↓
↓
↑
1. Company looks less profitable
under full goodwill and less
leverage in full goodwill.
Partial Goodwill
↓
↓
↓
↓
↑
↑
↑
↓
2. Management deliberately
increase goodwill to reduce
depreciation (overstating profits)
3. More intangible assets, more
issues in accounting.
4. Net income will remain same
under with the methods.
Example 11:
Company P has acquired 80% of Company S. Income Statement for the year ended December 31st, 2010.
Revenue
Expenses
Net Income
Company P
10,000
40,000
20,000
Company S
20,000
16,000
4000
Now, let's compare income statement effects of the acquisition method and equity method.
Revenue
Expense
Operating Income
(0.8)(4000) Equity in Income 'S'
(0.2)(4000) Minority (NCI)
Net Income
Acquisition
80,000
56,000
24,000
Equity
60,000
40,000
20,000
3,200
(800)
23,200
23,200
Example 12:
Franklin Co. acquired 100% of the outstanding shares of Jefferson Inc. by issuing 1000,000 shares of its £1 par
common stock (£15 market value). Immediately before the transaction, the two companies complied the
following information:
Cash and Receivables
Inventory
PPE
Current Payables
Long-term Debt
Net Assets
Capital Stock
Additional Paid in Capital
Retained Earnings
Franklin Book
Value
10,000
12,000
27,000
49,000
8000
16,000
24,000
25,000
Bv
300
1700
2500
4500
600
2000
2600
1900
Jefferson
5000
6000
14,000
400
700
800
Fv
300
3000
2500
7800
600
1800
2400
5400
Show the balances in the post combination balance sheet using the acquisition method.
13
100%
J
$ 15,000,000
(Fair Value)
100% of 19,000,000
$ 1,900,000
$ 13,100,000
(Book Value)
(Includes Net Asset differences &
Goodwill)
F
Attributable to Net Assets
Inventory (100% x 1300)
PPE (100% x 2000)
Long-term Debt (100% x 200)
Goodwill (Residual)
$ 10,000
$ 2000,000
$ 200,000
$ 9,600,000
$ 13,100,000
Consolidated balance sheet of the combined entity:
Cash and Receivables
Inventory
PPE
Goodwill
Total Assets
Current Payables
Long-term Debt
Total Liabilities
5000 + 1000 = 6000 Capital Stock (£1 Par)
6000 + (15000 - 1000) = 20,000 Additional Paid in Capital
Retained Earnings
Total Equity
Total Liabilities and Equity
£
10,300
15,000
31,500
9,600
66,400
8,600
17,800
26,400
6000
20,000
14,000
40,000
66,400
Recognition and Measurement of Indemnification Assets: Indemnity means security or protection
against a loss or other financial burden. On the acquisition date, the acquirer must recognize an
indemnification asset if the target contractually indemnifies the acquirer for the outcome of a
contingency or an uncertainty related to all or part of a specific asset or liability of the target. The
seller (target) may also indemnify the acquirer against losses above a specified amount on a liability
arising from a particular contingency. If the indemnification relates to an asset or a liability that is
recognized at the acquisition date and measured at its acquisition date fair value, the acquirer will
also recognize the indemnification asset at the acquisition date at its acquisition date fair value.
IFRS and US GAAP: When the purchase price is less than fair value of target's net assets, the
acquisition is considered to be a 'Bargain Purchase' acquisition. Both the
standards require that the difference between fair value of net assets and
purchase price to be recognized as a gain in the income statement. Any
contingent consideration must be measured and recognized at fair value at the
time of the business combination. Any subsequent changes in value of the
contingent consideration are recognized in profit or loss.
14
Issues in Business Combinations that weaken comparability:
1. Contingent Assets and Liabilities
IFRS: Only contingent liabilities whose fair value can be measured reliably are recognized at the time of
acquisition. Contingent assets are never recognized. In subsequent periods, contingent liabilities are
measured at the higher or the value initially recognized, or the best estimate of the amount needed
to settle the liabilities.
US GAAP: Dividends contingent assets and liabilities into contractual and non-contractual. Contractual
contingent assets and liabilities are recognized and recorded at their fair value at the time to
acquisition. Non-contractual contingent assets and liabilities must also be recognized and
recorded only if it is 'more likely than not' they meet the definition of an asset or a liability at the
acquisition date. Subsequently, a contingent liability is measured at the higher of the amount
initially recognized or the best estimate of the amount of the loss. A contingent asset, however is
measured at the lower of the acquisition date fair value or the best estimate of the future
settlement amount.
2. Contingent Consideration
Contingent consideration may be negotiated as part of the acquisition price. For example, the acquiring
company (parent) may agree to pay additional money to the target's former shareholders if certain agreed
upon event occurs. These can include achieving specified sales or profit levels for the target and/or the
combined entity.
IFRS and US GAAP: Such contingent consideration is recognized at fair value and classified as an asset,
liability or equity. Subsequent changes in value are recognized in the income statement, unless the value
was originally classified in equity (any changes then settle within equity and not via. the income statement).
3. In Process R&D
IFRS and US GAAP: In Process R&D is classified or capitalized as an intangible asset and is measured at fair
value, if it can be measured reliably. In Process R&D is subsequently amortized (if successful) or impaired (if
unsuccessful).
4. Restructuring Costs
IFRS and US GAAP: Restructuring costs are expensed when incurred and not capitalized as part of the
acquisition cost.
Example 13:
Recognizing an impairment loss and restructuring charges in a single period, although consistent with most
GAAP, is most likely to overstate:
Impairment and restructuring were
A. Prior period's net income
likely the result of past activities.
B. Current period's net income
C. Future period's net income
→
✓
lllll
Joint Ventures
Ventures undertaken and controlled by two or more parties can be a convenient way to enter
foreign markets, conduct specialized activities and engage in risky projects. Some join ventures are
primarily contractual relationships, where as others have common ownership of assets,
e.g. partnerships, limited liability companies like corporations, or other legal forms i.e.
unincorporated associations.
IFRS and US GAAP: Requires equity method of accounting for join ventures. In rare circumstances,
the proportion consolidation method may be allowed. Since only proportionate
share is reported by the investor, no minority owner's interest is necessary.
15
Example 14:
Company P acquired 80% of Company S on January 1st 2010 for $8000 cash. Compute balance sheet and
income statement under proportionate consolidation method.
(Not Given)
lllll
Current Assets
Other Assets
Total
Current Liabilities
Common Stock
Retained Earnings
Total
Company P
48,000
32,000
80,000
40,000
28,000
12,000
80,000
Company S
16,000
8000
24,000
14,000
6000
4000
24,000
Proportionate Consolidation
52,800 (0.8)(16000) + 48000 - 8000
38,400 (0.8)(8000) + 32000
91,200
51,200
28,000
12,000
91,200
Revenue
Expenses
Net Income
Company P
48,000
32,000
80,000
Company S
16,000
8000
24,000
Proportionate Consolidation
52,800
38,400
91,200
Special Purpose Entity and Variable Interest Entity
Special Purpose Entity (SPE) is a legal structure created to isolate certain assets and liabilities of the
sponsor. An SPE can take the form of a corporation, partnership, joint venture or trust. The typical
motivation is to reduce risk and thereby lower the cost of financing. SPEs are often structured such
that the sponsor company has control over the SPE's finances or operating activities while third
parties have controlling interest in the SPE's equity. In most cases, the creator/sponsor of the entity
retains a significant beneficial interest in the SPE even though it may own little or none of the SPE's
voting equity. Under IFRS, the sponsor of a SPE must consolidate the SPE if their economic
relationship indicates that the sponsor controls the SPE.
Variable Interest Entity (VIE), FASB uses VIE to more broadly define an entity that is financially
controlled by one or more parties that don't hold a majority voting interest. The primary beneficiary
of a VIE must consolidate it as a subsidiary regardless of how much of an equity investment the
beneficiary has in the VIE. The primary beneficiary is the entity absorbs the majority of the risks and
receives the majority of the rewards. If there are NCI in the VIE, these would also be shown in the
consolidated balance sheet and consolidated income statement of the primary beneficiary. It also
requires entities to disclose information about their relationships with VIEs, even if they are not
considered the primary beneficiary. Common examples of VIE are entities created to lease real
estate or other property, entities created for the securitization of financial assets (debt, subordinate
debt or senior debt) or entities created for research and development activity. FASB provides
guidance for US GAAP which classifies SPEs as VIEs if:
1. Total equity at risk is insufficient to finance activities without financial support from other parties/
2. Equity investors lack any of the following:
a. The ability to make decisions
b. The obligation to absorb losses
c. The right to receive returns
IFRS: uses SPE
US GAAP: uses SPE and VIE
Example 15:
Odewa, an Italian auto manufacturer, wants to raise £55,000,000 in capital by borrowing against its financial
16
receivables. To accomplish this objective, Odewa can choose between two alternatives:
Alternative 1: Borrow directly against the receivables (Borrow £55,000,000 from bank)
Alternative 2: Create a SPE, invest £5,000,000 in the SPE, have the SPE borrow £55,000,000 and then use
the funds to purchase £60,000,000 of receivables from Odewa.
Cash
Accounts Receivable
Other Assets
Total Assets
Current Liabilities
Non-Current Liabilities
Equity
Total Liabilities and Equity
Alternative 1:
Cash
Accounts Receivable
Other Assets
Total Assets
Current Liabilities
(20 + 55) Non-Current Liabilities
Equity
Total Liabilities and Equity
(30 + 55)
£
30,000,000
60,000,000
40,000,000
130,000,000
27,000,000
20,000,000
83,000,000
130,000,000
Alternative 2:
£
85,000,000
60,000,000
40,000,000
185,000,000
27,000,000
75,000,000
83,000,000
185,000,000
Cash
Accounts Receivable
Other Assets
Total Assets
Current Liabilities
Non-Current Liabilities
Equity
Total Liabilities and Equity
B/S
£
85,000,000 -5 +55
60,000,000 -60
40,000,000
185,000,000
27,000,000
75,000,000 +55
83,000,000
185,000,000
Example 16:
Blanca Co. wants to raise $40,000,000 in capital by borrowing against its financial statements. Blanca plans to
create a SPE, invests $10,000,000 in the SPE, have the SPE borrow $40,000,000 and then use the funds to
purchase $50,000,000 of receivables from Blanca. Blanca meets the definition of control and plans to
consolidate the SPE. Blanca's plan to borrow against its financial receivables, the consolidated balance sheet
will show total assets of:
Blanca Co. Balance Sheet (Given)
Cash
20,000,000
Accounts Receivable
50,000,000
Other Assets
30,000,000
Total Assets
100,000,000
Current Liabilities
Non-Current Liabilities
Equity
Total Liabilities and Equity
25,000,000
30,000,000
45,000,000
100,000,000
50,000,000
50,000,000
Long-term Debt
Equity
Total Liabilities and Equity
45,000,000
10,000,000
50,000,000
Blanca Co. Consolidated Balance Sheet
(20 + 40) Cash
60,000,000
Accounts Receivable
50,000,000
Other Assets
30,000,000
Total Assets
140,000,000
Current Liabilities
Non-Current Liabilities
Equity
Total Liabilities and Equity
25,000,000
70,000,000 (30 + 40)
45,000,000
140,000,000
SPE Balance Sheet
Accounts Receivable
Total Assets
SPE
+5
+60
17
Example 17:
On January 1st 2010, Company P acquires 80% of the common stock of Company S by paying $8000 in cash to
the shareholders of Company S. Compare all 3 methods.
Current Assets
Other Assets
Total Assets
Current Liabilities
Common Stock
Retained Earnings
Total Liabilities and Equity
Company P
48,000
32,000
80,000
40,000
28,000
12,000
80,000
Company S
16,000
8000
24,000
14,000
6000
4000
24,000
Revenue
Expenses
Net Income
Company P
60,000
40,000
20,000
Company S
20,000
16,000
4000
Acquisition
Current Assets
Investment in S
Other Assets
Total Assets
56,000
Proportionate
Consolidation
52,800
40,000
96,000
38,400
91,200
40,000
8000
32,000
80,000
Current Liabilities
Minority Interest
Common Stock
Retained Earnings
Total Liabilities and Equity
54,000
2000
28,000
12,000
96,000
51,200
40,000
28,000
12,000
91,200
28,000
12,000
80,000
Acquisition
Equity
80,000
56,000
24,000
Proportionate
Consolidation
76,000
52,800
23,200
(800)
23,200
23,200
Revenue
Expenses
Operating Income
Equity in Income of S
Minority Interest
Net Income
Equity
60,000
40,000
20,000
3200
23,200
- All three methods report the same net income (I/S).
- Equity method and Proportionate Consolidation report the same equity. Acquisition method will be higher
by the amount of minority interest.
- Acquisition is highest, Equity lowest and Proportionate Consolidation in between (B/S).
18
CHAPTER 14
Employee Compensation: Post Employment and Share Based
A pension is a form of deferred compensation earned over time through employee service. The most
common forms are Defined Contribution Plans (DCP), Defined Benefit Plans (DBP) and Share Based
Compensations.
lllll
Defined Contribution Plan
Defined Contribution Plan (DCP) is a retirement plan whereby the firm contributes a certain sum
each period to the employee's retirement account. The firm's contribution can be based on any no.
of factors including years of service, the employee's age, compensation, profitability or even a
percentage of the employee's contribution. Employee accounts are generally invested through a
financial intermediary, such as an investment management company or an insurance company.
After the employer makes its agreed upon contribution to the plan on behalf of an employee,
generally in the same period in which the employee provides the service i.e. the employer has no
obligation to make payments beyond this amount. In any event, the firm makes no promise to the
employee regarding the future value of the plan assets. The investment decisions are left to the
employee, who assumes all of the investment risk. Pension expense is simply equal to the
employer's contribution. There is no future obligation to report on the balance sheet beyond
required contributions. An accrual (current liability) is recognized at the end of the reporting period
only for any unpaid contributions.
lllll
Defined Benefit Plan
The firm promises to make periodic payments to the employee after retirement. The benefit is
usually based on the employee's years of service and the employee's compensation at or near
retirement. For example, an employee with 20 years of service might earn a retirement benefit of
2% of her final salary of $100,000 for each year of service, would receive $40,000 ($100,000 x 2% x
20 Years) each year upon retirement until death. Since the employee's future benefit is defined, the
employer assumes the investment risk. The complexity of pension reporting causes differences
between the timing of cashflows (contributions into the plan ad payments from the plan) and the
timing of accrual-basis reporting. Accrual-basis reporting is based on when the services are
rendered and the benefits are earned. Most Defined Benefit Plans (DBP) are funded through a
separate legal entity i.e. trust.
Key Terms:
1. Current Service Cost: It is the present value of benefits earned by the employees during the current period
or it is the increase in the present value of defined benefit obligation as a result of employee service in the
current period.
2. Interest Cost: It is the increase in the obligation due to the passage of time. Interest cost increases every
year regardless of whether the employee works another year or not. Interest cost is immediately
recognized as a component of pension expense.
Interest Cost = Beginning PBO x Discount Rate
3. Net Interest Expense/Income: A net interest expense represents the following cost of deferring payments
related to the plan i.e. if underfunded - liability, then an expense is reported. A net interest income
represents the following income from prepaying amounts related to the plan i.e. if overfunded - asset, then
an income is reported.
Net Interest Expense/Income = Beginning Funded Status x Discount Rate
19
4. Remeasurements: Remeasurements include (a) actuarial gains and losses and (b) any differences between
the actual return on plan assets and the amount included in the net interest expense/income calculation.
Actual Return on Plan Assets = Beginning Plan Assets x Discount Rate
5. Past (Prior) Service Cost: Is the amount by which a company's pension obligation relating to employee's
service in prior periods changes as a result of plan amendments or a plan curtailment. A curtailment occurs
when there is a significant reduction by the entity either in the no. of employees covered by a plan or in
benefits.
6. Actuarial Gains and Losses: Occur due to changes in PBO, due to change in actuarial assumptions. An
actuarial gain (e.g. when an employee dies earlier) will decrease benefit obligation and an actuarial loss
(e.g. when an employee lives longer) will increase the obligation. However, there are two components
within actuarial gains and losses; the first component is the gain (loss) due to decrease (increase) in PBO
occurring on account of changes in actuarial assumptions and the second component is the difference
between actual and expected return on plan assets.
7. Expected Return on Plan Assets: Is the assumed long-term rate of return on the plan's investment. Under
IFRS, the expected rate of return on plan assets is implicitly assumed to be the same as the discount rate
used for computation of PBO. Also, the expected return on plan assets requires estimating in which future
period the benefits will be paid.
Expected Return on Plan Assets = Beginning Plan Assets x Expected Return
8. Amortization of Actuarial Gains and Losses: Under IFRS, actuarial gains and losses are not amortized.
Under US GAAP, actuarial gains and losses are amortized using the 'Corridor Approach'.
If Actuarial Gains and Losses > 10% of Beginning PBO or Beginning Plan Assets
(Difference between actual and
expected return on plan assets)
(Pick the greater of two)
Amortization is required. The excess amount over the corridor is amortized as a component of periodic
pension cost in P&L over the remaining service life of the employees. Companies can choose to amortize
actuarial gains and losses more quickly than implied by the corridor method. However, the application has
to be consistent for gains as well as losses over time.
9. Discount Rate: It is not the risk-free rate. rather it is based on interest rates of high quality fixed income
investments with a maturity profile similar to the future obligation.
Example 1:
Assume the beginning balance of the DBO is $5000,000, the beginning balance of fair value of plan assets is
$4,850,000 and the beginning balance of unrecognized actuarial losses is $610,000. The expected average
remaining working lives of the plan employees is 10 years.
610,000 > 10% of 5000,000
610,000 > 500,000, Hence impaired.
Amortization = 610,000 - 500,000 = 110,000; 110,000 = $ 11000 (I/S)
10
End of Year = 610,000 - 11,000 = $ 559,000 (OCI)
20
Fig 1: Effects of Changes in these assumptions
↑ Discount Rate
Service Cost
Interest Cost
PBO
Funded Status
TPPC
PPC
Decrease
Increase
Decrease
Increase
Decrease *
Decrease *
↓ Rate of
↑ Expected
Compensation
Rate of Return
Decrease
Decrease
Decrease
Increase
Decrease
Decrease
Decrease
* For mature plans, a higher discount rate might increase interest costs. In rare cases, interest cost will
increase by enough to offset the decrease in the current service cost and periodic pension cost will
increase.
Example 2:
Under US GAAP, all else equal, which of the following statement best describes the impact of an increase in
the expected return on plan assets?
A. Increase in plan assets and decrease in PPC in P&L
B. Decrease in PBO and increase in service cost
C. Increase in net income
Lower pension expense results in higher net income
✓
IFRS and US GAAP: Measure the pension obligation as the present value of future benefits earned
by employees for service provided to date.
IFRS: The obligation is called Present Value of Defined Benefit Obligation (PVBO). This measure is
defined as the 'present value without deducting any plan assets of expected future payments
required to settle the obligation arising from employee service in the current and prior
periods'. The 'Projected Unit Credit Method' is the IFRS approach to measure the PBO or
PVBO. Under this method, for each period in which an employee provides service, they earn a
portion of the post employment benefits that the company has promised to pay. The objective
of this method is to allocate the entire expected retirement costs (benefits) for an employee
over the employee's service periods. This obligation is based on actuarial assumptions about
demographic variables such as future inflation and the discount rate.
US GAAP: The obligation is called Projected Benefit Obligation (PBO). This measure is defined as 'the
actuarial present value as of a date of all benefits attributed by the pension benefit
formula to employee service rendered prior to that date'. In addition to PBO, US GAAP
identified two other measures of the pension liability, (a) The Vested Benefit Obligation
(VBO) is the actuarial present value of vested benefits. 'Vesting' refers to a provision in
pension plans whereby an employee gains right to future benefits only after meeting
certain criteria, such as a pre-specified no. of years of service. If the employee leaves the
company before meeting the criteria, he or she may be entitled to none or a portion of
benefits earned up until that point and (b) The Accumulated Benefit Obligation (ABO) is
the actuarial present value of benefits (whether vested or non-vested) attributed,
generally by the pension benefit formula, to employee service rendered before a
specified date and based on employee service and compensation (if applicable) before
that date. The ABO differs from PBO in that it includes no assumption about future
compensation levels. Both VBO and ABO are based on the amounts promised as a result
of an employee's service up to a specific date. Thus VBO < ABO < PBO.
21
Fig 2: Funded Status of a Pension Plan
Plan Assets
PBO
Fv at Beginning of the year
Beginning PBO
+ Contributions (Employer & Employee) + Service Cost
+ Actual Return
+ Interest Cost
- Benefits Paid
+ Past Service Cost
= Fv at End of the year
-/+ Actuarial Gains/Losses
- Benefits Paid
= Ending PBO
Plan Assets > PBO = Overfunded Plan
Plan Assets < PBO = Underfunded Plan
Funded Status = Fv of Plan Assets - PBO
If the funded status is negative, it is reported as a liability. If the funded status is positive, it is reported as an
asset subject to a ceiling of present value of future economic benefits (such as future refunds or reduced
contributions) under IFRS and US GAAP. Always show net A/L not gross A/L.
Benefits Paid
193,000,000 + 38,000,000 - x = 220,000,000
x = 11,000,000
(PBO Formula)
159,000,000 + 32,000,000 - 11,000,000 = 180,000,000
Ending Fv of
Plan Assets
↶
↷
Example 3:
The net amount of the cost components of Heritage Bakery's pension plan for 2016 was $38,000,000. Cost
components include current service cost and interest cost. The fair value of plan assets on January 1st 2016
was $159,000,000. The PBO on January 1st 2016 was $193,000,000 and the PBO on December 31st 2016 was
$220,000,000. Actual return on assets in 2016 was $32,000,000. The expected return on plan assets for 2016
was 10%. The fair value of plan assets on December 31st 2016 is?
* Total Periodic Pension Cost
Total Periodic Pension Cost (TPPC) is the true pension cost as these two items are actually
smoothed items (items that ensure that volatility of net income is lower) that have no relation to the
true economic pension cost. TPPC is same under IFRS and US GAAP. TPPC will not appear anywhere
in accounts, only disclosed in notes to accounts and is true pension expense.
TPPC = Employer Contributions - (Ending Funded Status - Beginning Funded Status)
or
TPPC = Current Service Cost
+ Interest Cost
+ Past Service Cost
-/+ Actuarial Gains/Losses
- Actual Return on Plan Assets
IFRS and US GAAP: TPPC = PPC in Income Statement + PPC in OCI
22
* Periodic Pension Cost
Periodic Pension Cost (PPC) is not an economic concept; it's an accounting concept. What affects the
net income is the PPC not TPPC. In some cases, amount of pension costs may qualify for inclusion
as a part of ending inventory and thus be included in P&L as part of cost of goods sold when those
inventories are later sold.
IFRS: For pension costs that are not capitalized, IFRS don't specify where companies present the
various components of PPC beyond differentiating between components included in P&L and
in OCI.
US GAAP: For pension costs that are not capitalized, US GAAP requires all components of PPC that
are recognized in P&L to be aggregated and presented as a net amount within the same
line item on the income statement.
IFRS
+ Service Cost
-/+ Net Interest Income/Expense 1
+ Past Service Cost 2
= PPC (Income Statement)
1. Remeasurements are reflects in OCI and not subsequently amortized to P&L.
2. Past Service Costs reported in net income.
US GAAP
+ Service Cost
Recurring Costs (Actual)
+ Interest Cost
- Expected Return on Plan Assets
-/+ Amort. of Actuarial Gains/Losses 1
-/+ Amort. of Past Service Cost1
= PPC (Income Statement)
Amort. of Actuarial Gains or
Losses and Amort. of Past
Service Costs reduce the
volatility of PPC i.e. Smoothed
due to amortization
1. Unamortized Past Service Cost and Unamortized Actuarial Gains and Losses
in OCI. For Past Service Costs are reported in OCI and amortized to net
income over the service life of the employees. For Actuarial Gains and Losses,
it is subsequently amortized to P&L using the corridor or fast recognition
method. (Under US GAAP difference expected return and the actual return
on plan assets represent another source of actuarial gains or losses. All
actuarial gains and losses are included in the net pension liability or net
pension asset and can be reported in either P&L or in OCI. Typically
companies report actuarial gains or losses in OCI and recognize gains and
losses in P&L only when certain conditions are met under so called corridor
approach).
Example 4:
For ABC company, the present value of the company's defined benefit obligation is $5485 and the fair value
of the pension plan assets is $5998. In addition, the present value of available future refunds and reductions
in future contributions is $326.
Funded Status = Fv of Plan Assets - PBO
= 5998 - 5485
= $ 513
In this case the asset ceiling is given as $326, so the amount of ABC company's reported net pension asset
would be limited to $326.
23
Example 5:
Employer Contributions
Current Service Costs
Plan Amendments
Actuarial Gain
Plan Assets at Start of Year
Plan Assets at End of Year
Benefit Obligation at Start of Year
Benefit Obligation at End of Year
Actual Return on Plan Assets
Discount Rate
2017
75.5
57.4
(189)
274.7
4038
3307.5
3651.9
3431.7
-18.6%
6.6%
The loss reported in OCI in Austell's 2017 financial statements related to the pension plans is closest to:
Actuarial Gain
$ 274.7
A. $ 742.9
Net Return on Plan Assets ($ 1017.6)
B. $ 1017.6
(-18.6% - 6.6%) x 4038
C. $ 209.9
Total OCI Loss
($ 742.9)
✓
Fig 3: Projected Unit Credit Method
Estimated Annual Salary
x
Benefits attributed to:
+ Prior Years (Annual Unit Credit x Yrs. of Prior Service)
+ Current Year (Annual Unit Credit)
Total Benefits Earned
x
x
xx
Obligations attributed to:
+ Opening Obligations [Prior Year / (1 + r) Years until Retirement ]
+ Interest Cost (Opening Obligation x r)
+ Current Service Costs [Annual Unit Credit / (1 + r) Years until Retirement - 1 ]
Closing Obligation
x
x
x
xx
Example 6:
The following information applies to each of the four scenarios. The employee has a salary in the coming year
of $50,000 and is expected to work 5 more years before retiring. The assumed discount rate is 6% and the
assumed annual compensation increase in 4.75% percent. For simplicity, assume that there are no changes in
actuarial assumptions, all compensation increases are awarded on the first day of the service year and no
additional adjustments are made to reflect the possibility that the employee may leave the company at an
earlier date. The pension benefit is 1.5% of the employee's final salary for each year of service beyond the
date of establishment.
50,000
Retiring
Final Year estimated Salary = Current Year's Salary (1 + Annual Compensation Increase) Years until Retirement - 1
= 50,000 (1.0475) 5 - 1
= 50,000 (1.0475) 4
= 60,198.56
24
a. Benefit is paid as a lump sum amount upon retirement
Lump sum payment to be paid upon retirement = (Final Salary x Benefit Formula) x Years of Service
= (60,198.56 x 0.015) x 5
= 4514.89
Annual Unit Credit (Benefit) per service year = Value at Retirement = 4514.89 = 902.98
Years of Service
5
b. Prior 10 years of service and benefit paid as a lump sum upon retirement
Lump sum payment to be paid upon retirement = (Final Salary x Benefit Formula) x Years of Service
= (60,198.5 x 0.015) x 15
= 13,544.68
Annual Unit Credit (Benefit) per service year = Value at Retirement = 13,544.68 = 902.98
Years of Service
15
c. Employee to receive benefit payments for 20 years (no prior years of service)
Estimated Annual Payment (end of year) for each of the 20 years = (60,198.56 x 0.015) x 5
= 4514.89
Value at the end of year 5 (retirement date) of the estimated future payments
n = 20
PMT = 4514.89
I/Y = 6%
Fv = 0
Pv = 51,785.46
Annual Unit Credit = 51,785.46 = 10,357.09
5
d. Employee to receive benefit payments for 20 years and is given credit for 10 years of prior service with
immediate vesting.
Estimated Annual Payment (end of year) for each of the 20 years = (60,198.56 x 0.015) x 15
= 13,544.68
Value at the end of year 5 (retirement date) of the estimated future payments
n = 20
PMT = 13,544.68
I/Y = 6%
Fv = 0
Pv = 155,356.41
Question a.
52,375
50,000 (1.0475)1
↷
↶
Annual Unit Credit = 155,356.41 = 10,357.09
15
Yr. 1
Yr. 2
Yr. 3
Yr. 4
Yr. 5
Estimated Annual Salary
50,000
52,375
54,863
57,469
60,198
Prior Years
Current Year
Total Benefits Earned
0
902.98
902.98
902.98
902.98
1805.9
1805.9
902.98
2708.9
2708.9
902.98
3611.9
3611.9
902.98
4514.8
Opening Obligation
Interest Cost
Current Service Cost
Closing Obligation
0
0
715.24
715.24
715.24
42.91
758.16
1516.3
1516.3
90.98
803.65
2410.9
2410.9
144.66
851.87
3407.4
3407.4
204.45
902.97
4514.8
25
Example 7:
10 Years Prior Service
Current Annual Salary = $100,000
Years to Retirement = 17 Years
Retirement Life Expectancy = 20 Years
Annual Compensation = 6%
Pension Benefit = 1%
Discount Rate = 4%
Calculate Annual Unit Credit.
Retirement
100,000
10 Years
17 Years
to Retirement
20 Years
Life Expectancy
Step 1: Final year Estimated Salary = Current year's Salary (1 + Annual Compensation) Years until Retirement - 1
(Compensation)
= 100,000 (1.06) 17 - 1
= 254,035.168
Step 2: Estimated Annual Payment for each of 20 years = (Final Salary x Benefit Formula) x Years of Service
(Lump sum payment
= (254,035.168 x 0.01) x (10 + 17)
to be paid upon
= 68,589.5
retirement)
Step 3: Value at the end of year 17 (Retirement Date)
(Only if they have
n = 20
PMT = 68,589.5
I/Y = 4%
mentioned how
many years the
employee is going
to live in the future
after retirement)
Fv = 0
Pv = 932,153.688
No. of payments made
by the company to the
employee per year in the
future. (After Retirement)
Step 4: Annual Unit Credit = Value at Retirement = 932,153.688 = 34,524.21
(Benefit per Service year)
Years of Service
10 + 17
Analyst Adjustment
1. Differences in key assumptions can affect comparisons across companies.
2. Amounts disclosed in the balance sheet are net amounts (plan liabilities minus plan assets). Adjustments to
incorporate gross amounts would change certain financial ratios. For example, ROA would likely be lower if
the gross amounts were reported on the balance sheet (higher denominator).
3. PPC recognized in P&L (pension expense) may not be comparable. IFRS and US GAAP differ in their
provisions about costs recognized in P&L vs. in OCI. Alternatively, the analyst could simply use
comprehensive income (i.e. net income plus OCI) as the metric for comparison.
4. Classification of the components of PPC in P&L as operating/non-operating differs under US GAAP vs. IFRS.
Under US GAAP, the entire PPC in P&L (including interest) is shown as an operating expense. Under IFRS,
the components of PPC can be included in various line items. Analysts can adjust GAAP-reported income by
adding back the PPC in P&L and substracting only service cost in determining operating income. Interest
cost should be added to the firm's interest expense and the actual return on plan assets should be added
to non-operating income. Note that this adjustment excludes (ignores) any amortizations.
26
5. Cashflow information may not be comparable. Under IFRS, some portion of the amount of contributions
might be treated as a financing activity rather than an operating activity and under US GAAP the
contribution is treated as an operating activity.
Example 8:
Use the following information to reclassify the components of PPC between operating and non-operating:
Partial Income Statement
Operating Profit
Interest Expense
Other Income
Income before Tax
$
145,000
(12,000)
2000
135,000
Partial Income Statement
Other Data
Current Service Cost
Interest Cost
Expected Return on Assets
Actual Return on Assets
$
7000
5000
8000
9500
Reported
Adjustments
Adjusted
Operating Profit
145,000
142,000
Interest Expense
Other Income
Income before Tax
(12,000)
2000
135,000
+4000 (PPC)
-7000
-5000
+9500
(17,000)
11,500
136,500
Example 9:
Income statement lists the following: current service cost of $40,000,000, interest costs of $263,000,000,
expected return on plan assets of $299,000,000 and actual return on plan assets of $205,000,000. After
reclassifying pension components to reflect economic income or expense, the net adjustments to profit
before taxation is?
Current Service Costs
Interest Cost
Expected Return on Plan Assets
PPC
40,000,000
263,000,000
(299,000,000)
4000,000
Actual Return < Expected Return = Operating Income will be adjusted Downwards
= 299,000,000 - 205,000,000
= 94,000,000
Adjustments
Revenue
Net Operating Expense
+ 4000,000 - 40,000,000
Operating Profit
Interest Expense
- 263,000,000
Interest and Investment Income
205,000,000
Share of Post-tax results of Associates
Adjustment to Profit before Taxation
- 94,000,000
Actual Return > Expected Return = Deferred Gains
Actual Return < Expected Return = Deferred Losses
27
If the difference cashflow and TPPC is material, the analyst should consider reclassifying the
difference from operating activities to financing activities in the cashflow statement.
Contribution > TPPC : Principle PMT
(Contribution - TPPC)(1 - T);
CFF and
↑ CFO
Contribution < TPPC : Borrowing
(Contribution - TPPC)(1 - T);
CFF and
↓ CFO
↓
↑
Example 10:
Thakur notices that BC has a defined benefit pension plan in place. He finds out that the company made a
$340,000,000 contribution to the plan during the year. He collects the following additional information:
Beginning Funded Status $2530,000,000
Ending Funded Status
$2180,000,000
BC reported a net income during the year of $812,000,000. CFO and CFF were reported as $948,000,000 and
$112,000,000 respectively. BC's tax rate was 40% during the year.
TPPC = Contributions - (Ending Funded Status - Beginning Funded Status)
= 340,000,000 - (2180,000,000 - 2530,000,000)
= 690,000,000
Contribution < TPPC
340,000,000 < 690,000,000
After tax Shortfall = (690,000,000 - 340,000,000)(1 - 0.4)
= 350,000,000 (1 - 0.4)
= 210,000,000
Adjusted CFO = 948,000,000 - 210,000,000 = 738,000,000
Adjusted CFF = 112,000,000 + 210,000,000 = 322,000,000
Example 11:
Based on information from the company's 2009 annual report, she determines that the company's TPPC was
$437,000,000; however, the company also disclosed that it made a contribution of $504,000,000. GeoRace
reported cash inflow from CFO of $6,161,000,000 and cash outflow from CFF of $1,741,000,000. The
company's effective tax rate was 28.7%.
Contribution > TPPC
504,000,000 > 437,000,000
After the Shortfall = (504,000,000 - 437,000,000)(1 - 0.287)
= 67,000,000 (1 - 0.287)
= 48,000,000
Adjusted CFO = 6,161,000,000 + 48,000,000 = 6,209,000,000
Adjusted CFF = - 1,741,000,000 - 48,000,000 = 1,789,000,000
28
Example 12:
Total Assets
Total Liabilities
Total Equity
Benefit Obligation Change
Benefit Expense Change
24,130
17,560
6570
100 Bps
$ 93
$ 12
↑
Sensitivity of Benefit Obligation to 100 Bps
100 Bps
- $ 76
- $ 10
↓
↑ = $ 93
Debt to Equity ratio is?
Adjusted Liabilities = 17,560 + 93 = 17,653
Adjusted Equity = 6570 - 93 = 6477
Therefore, D/E = 17,653 / 6477 = 2.73
Funding is an area where other post-employment benefit plans differ from defined benefit pension
plans. Pension plans are typically funded at some level, while Other Post-employment Benefit Plans
(OPB) are usually unfunded. In the case of unfunded plan, the employer recognizes expense in the
income statement as the benefits are earned; however, the employer's cashflow is not affected until
the benefits are actually paid to the employee. The complexity in reporting for OPB may be even
greater than for defined benefit plans because of the need to estimate future increases in costs.
Unlike defined benefit plans, companies may not be required by regulation to fund an OPB in
advance to the same degree as defined benefit pension plans. The assumptions are similar for OPB
except the compensation growth rate is replaced by a health care inflation rate. Generally, the
presumption is the inflation rate will taper off and eventually become constant. This constant rate is
known as the 'Ultimate Healthcare Trend Rate'. Analyst must compare the pension and OPB
assumptions over time and across firms to assess the quality of earnings. In addition, analyst
should consider whether the assumptions are internally consistent. For example, discount rate and
compensation growth rate should reflect a consistent view of inflation. Under US GAAP, the
assumed expected rate of return should be consistent with plan's assets allocation. If the
assumptions are inconsistent, the firm may be manipulating the financial statements by using
aggressive assumptions.
↶
Manipulations: 1. Discount Rate (IFRS)
(High chances of 2. Expected Return (US GAAP)
manipulation)
3. Actuarial G/L (IFRS and US GAAP)
4. Amortization G/L (US GAAP)
lllll
Share Based Compensation
Share Based Compensation (SBC) is intended to align employee's interest with those of the
shareholders and is typically a form of deferred compensation. One disadvantage is that the
recipient of the SBC may have limited influence over the company's market value, so SBC doesn't
necessarily provide the desired incentives. Another disadvantage being the increased ownership,
may lead managers to be risk averse or risk taker. Finally, when SBC is granted to employees,
existing shareholder's ownership is diluted.
IFRS and US GAAP: Companies use the fair value of the SBC granted to measure the value of the
employee's services for purposes of reporting compensation expense. The usual
29
disclosures required for SBC include:
(i) The nature and extent of SBC arrangements during the period.
(ii) How fair value of SBC arrangement was determined.
(iii) The effect of SBC on the company's income for the period and on its
financial position.
* Stock Options
IFRS and US GAAP: Compensation expense related to option grants is reported at fair value. Both
standards require that fair value be estimated using an appropriate valuation model. Several
models are commonly used, such as the Black-Scholes Option Pricing Model or a Binomial Model.
Generally though, the valuation model should be consistent with fair value measurement, be based
on established principle of financial economic theory and reflect all substantive characteristics of
the award. Once a valuation model is selected, a company must determine the inputs.
Exercise Price
Stock Price
Expected Term
Expected Volatility
Expected Dividends
R f rate
↑
↑
↑
↑
↑
↑
Fv of Compensation Expense
↓
↑
↑
↑
↓
↑
Net Income
↑
↓
↓
↓
↑
↓
Recognition of compensation expense over the relevant vesting period will decrease net income
and retained earnings; however, paid-in-capital is increased by an identical amount. This results in
no change to total equity.
Grant Date
(Is the day that
options are
granted to
employees)
Service
Period
Vesting Date
Exercise Date
(Is the date that
employees can
first exercise the
stock options)
(Is the date when
employees actually
exercise the options and
convert them to stock)
Expiration Date
The vesting can be immediate or over a future period. (a) If the share based payments vest
immediately (i.e. no further period of service is required), then expense is recognized on grant date,
(b) If the share based awards don't vest until a specified service period is completed, compensation
expense is recognized and allocate over the service period and (c) If the share based awards are
conditional upon the achievement of a performance condition or a market condition (i.e. a target
share price), then compensation expense is recognized over the estimated service period.
Moreover, if the option go unexercised, they may expire at some pre-determined future date,
commonly 5 or 10 years from the grant date. If facts affecting the value of options granted depends
on events after the grant date, then compensation expense is measured at the exercise date.
* Stock Grants
A company can grant stock to employees outright, with restrictions or contingent on performance.
For an outright stock grant, restricted stock and performance shares are reported on fair value of
the stock on the grant date i.e. generally the market value at grant date and compensation expense
is allocated over the service period. 'Restricted Stock' which requires the employee to return
ownership of those shares to the company, if certain conditions are not met. Shares granted
contingent on meeting performance goals are called 'Performance Shares'. Performance stock is
contingent on performance goals such as accounting earnings or return on assets. Unfortunately,
30
performance to accounting earnings and other metrics may result in manipulation. Compensation
expense for stock grants is not affected by the stock's volatility.
* Stock Appreciation Rights
With Stock Appreciation Rights (SAR) an employee's compensation is based on increases in a
company's share price. The potential for risk aversion is limited because employees have limited
downside risk and unlimited upside potential similar to employee stock options and shareholder
ownership is not diluted since no shares are actually issued. A disadvantage of SAR is that they
require current period cash outlay. SARs are valued at fair value and compensation expense is
allocated over the service period of the employee.
* Phantom Stock
Phantom Stock is based on the performance of hypothetical stock rather than company's actual
stock. Phantom stock is valued at fair value and compensation expense is allocated over the service
period. Phantom stock is used in privately held firms and firms with highly illiquid stock.
Both stock grants and stock options allow the employees to obtain direct ownership in the
company. Other types of SBC such as SAR or phantom stock, compensate an employee on the basis
of changes in the value of shares without requiring the employee to hold the shares. These are
referred to as cash-settled SBC.
31
CHAPTER 15
Multinational Operations
Local Currency: is the currency of the country being referred to.
Functional Currency: determined by management, is the currency of the primary
environment in which the entity operates. It is usually the currency
in which the entity generates and expends cash. The functional
currency can be the local currency or some other currency.
Presentation Currency: is the currency in which the parent company prepares its
financial statements i.e. reporting currency.
Assume A Co., a Finland based company, imports goods from Mexico in January under 45-day credit terms
and the purchase is denominated in Mexican Pesos. By deferring payment until April, Fin Co. runs the risk
that from the date, the purchase is made until the date of payment , the value of the Mexican Peso might
increase relative to the Euro. In this case, Fin Co. is said to have an exposure to Foreign Exchange Risk or
Transaction Exposure.
IFRS and US GAAP: Requires the change in the value of the foreign currency assets or liability resulting from a
foreign currency translation to be treated as a gain or loss reported on the income
statement. If or otherwise, a balance sheet date falls between the initial transaction date
and the settlement date, then companies are required to include (recognize) a gain or loss
in income before it has been realized.
Example 1:
Fin Co. purchases goods from its Mexican supplier on 1st November, 2001; the purchase price is 100,000
Mexican Pesos. Credit terms allow payment in 45 days and Fin Co. makes payment of 100,000 Pesos on 15th
December, 2001. Fin Co.'s functional and presentation currency is the Euro. Spot Exchange rates between the
Euro (€) and Mexican Pesos (Mex$) are as follows:
1st November, 2001
15th December, 2001
1 Mex$ = 0.0684 €
1 Mex$ = 0.0703 €
Fin Co.'s fiscal year end is 31st December. How will Fin Co. account for this foreign currency transaction and
what effect will it have on the 2001 financial statements?
Mex$ 100,000
M
Transaction Date
1st Nov
0.0684
F
Settlement Date
Balance Sheet Date
15th Dec
0.0703
31st Dec
Foreign Exchange Loss = 100,000 (0.0703 - 0.0684)
= € 190
A loss of € 190 will be recognized as retained earnings under equity and as foreign exchange loss under
income statement.
32
Example 2:
Fin Co. sells goods to a customer in the UK for £ 10,000 on 15th November, 2001 with payment to be received
in British Pounds on 15th January, 2002. Fin Co.'s functional and presentation currency is the Euro. Spot
exchange rates between the Euro (€) and British Pound (£) are as follows:
15th November, 2001
31st December, 2001
15th January, 2002
1 £ = 1.460 €
1 £ = 1.480 €
1 £ = 1.475 €
Fin Co.'s fiscal year end is 31st December. How will Fin Co. account for this foreign currency translation and
what effect will it have on the 2001 and 2002 financial statements?
£ 10,000
F
B
Transaction Date
Balance Sheet Date
Settlement Date
1st Nov
1.460
31st Dec
1.480
15th Jan
1.475
①
②
Foreign Exchange Gain = 10,000 (1.430 - 1.460) = € 200
Foreign Exchange Loss = 10,000 (1.480 - 1.475) = - € 50
A gain of € 200 was recognized in 2001 and a loss of € 50 is recognized in 2002. Over the two months period,
the net gain recognized in the financial statements is equal to the actual realized gain on the foreign currency
transaction.
Fig 1: Transaction and Foreign Currency Exposure
Transaction
Type of Exposure
Export Sale
Import Purchase
Asset (Account Receivable)
Liability (Account Payable)
Foreign Currency
Strengthens
Weakens
Gain
Loss
Loss
Gain
Whether a change in exchange rate results in a foreign currency translation gain or loss (measured in local
currency) depends on (i) the nature of the exposure to foreign exchange risk (asset or liability) and (ii) the
direction of change in the value of the foreign currency (strengthens or weakens).
IFRS and US GAAP: Neither standard indicates where on the income statement these gains and losses should
be placed. The two most common treatments are either (i) as a component of other
operating income/expense or (ii) as a component of non-operating income/expense, in
some cases as a part of net financing cost. The calculation of operating profit margin is
affected by where foreign currency transaction gains or losses are placed on the income
statement. Gross profit and net profit are unaffected but operating differs under the two
alternatives.
IFRS and US GAAP: The procedures for translating foreign currency financial statements essentially requires
the use of either the current rate or the temporal method.
33
Fig 2: Types of Currency Translation Methods
Current Rate Method
Temporal Method
Current/Temporal Method
Presentation Currency
Presentation Currency
Presentation Currency
CR
and
Functional Currency
CR
Functional Currency
and
T
Local Currency
Local Currency
Translation usually
involves self-contained,
independent subsidiaries
whose operating,
investing and financing
activities are decentralized
from the parent.
Remeasurement usually
occurs when a subsidiary
is well integrated with the
parent i.e. the parent
makes the operating,
investing and financing
decisions.
IFRS: 'Reporting foreign
currency transactions in
the functional currency'
instead of
remeasurements.
US GAAP: Uses the term
remeasurements.
Fig 3: Current Rate and Temporal Method
↷
Liabilities
Monetary (AP, LTD, Accrued Expenses and Deferred Income Tax)
Non-Monetary
- Measured at Current Value
- Not Measured at Current Value
(Deferred Revenue or Unearned Revenue)
CTA
T
Local Currency
Current Rate: Exchange rate on the balance sheet date
Average Rate: Average Exchange rate over the reporting period.
Historical Rate: Actual rate that was in effect when the original transaction occurred.
Assets
Monetary (Cash, Receivables)
Non-Monetary
- Measured at Current Value
(Marketable Securities, Inventories)
- Measured at Historical Cost
(Inventory, PPE and Intangible Assets)
Equity
Other than R/E
R/E
Functional Currency
Current Rate
Temporal
(From I/S to B/S)
(From B/S to I/S)
CR
CR
CR
CR
CR
HR
CR
CR
CR
CR
CR
HR
HR
Formula1
HR
Formula (β)
Accumulated as
a component of
Equity
Included as
gain or loss in
Net Income
Dirty Surplus
Accounting
Clean Surplus
Accounting
34
Current Rate
Temporal
Revenue
AR
AR
Expenses
Most Expenses (Tax)
Expenses related to Assets translated at Historical Cost
(COGS, Depreciation and Amortization)
AR
AR
AR
HR
1. Derive Closing R/E
Opening R/E
NI
- Dividends Declared (HR)
Closing R/E
x
x
(x)
x
2. Derive NI (Temporal)
Opening R/E
NI (β)
- Dividends Declared
Closing R/E
x
x
(x)
x
3. Derive COGS (Temporal)
Beginning Inventory
Purchases (β)
- Ending Inventory
COGS
x
x
(x)
x
4. Ending Inventory and COGS under Temporal Method
FIFO
Ending Inventory: CR
COGS: HR
1
2
3
4
5
LIFO
Ending Inventory: HR
COGS: CR
Ending
Inventory
COGS
1
2
3
4
5
Weighted Average
Ending Inventory: AR
COGS: AR
COGS
Ending
Inventory
Firms operating in the same industry may use different methods for translation purposes thereby making
comparisons more difficult. One solution involves adding the
CTA to the firm's net income. By bringing the
translation gain or loss into the income statement, comparisons with a temporal method firm are improved.
△
A. Current Rate Method
The basic concept underlying the current rate method is that the entire investment in a foreign entity is
exposed to translation gain or loss. Therefore, all assets and liabilities must be revalued at each
successive balance sheet date. The net translation gain or loss that results from this procedure is
unrealized, however and will be realized only when the entity is sold. In the meantime, the unrealized
translation gain or loss that accumulates over time is deferred on the balance sheet as a separate
component of stockholder's equity. When a specific foreign entity is sold, the cumulative translation
adjustment (CTA) related to that entity is reported as a realized gain or loss in net income.
Total Assets > Total Liabilities : Net Asset Balance Sheet Exposure
Total Assets < Total Liabilities : Net Liability Balance Sheet Exposure
When foreign currency increases in value i.e. strengthens, application of the current rate method results
in an increase in the positive CTA (or decrease in the negative CTA) reflected in stockholder's equity.
When foreign currency decreases in value i.e. weakens, the current rate method results in a decrease in
the positive CTA (or an increase in the negative CTA) in stockholder's equity.
B. Temporal Method
Only monetary assets and liabilities are translated at the current exchange rate. This variation of the
monetary/non-monetary method sometimes is referred to as the temporal method.
Monetary Assets > Monetary Liabilities : Net Monetary Asset Balance Sheet Exposure
Monetary Assets < Monetary Liabilities : Net Monetary Liabilities Balance Sheet Exposure
Whereas the non-monetary assets and liabilities are remeasured at historical rates. Hence, they are not
prone to such exposures as monetary assets and liabilities.
35
When using the temporal method, companies can manage their exposure to translation gain (loss) more
easily than when using the current rate method. If a company can manage the balance sheet of a foreign
subsidiary such that monetary assets equal monetary liabilities, no balance sheet exposure exists. Elimination
of balance sheet exposure under the current rate method occurs only when total assets equal total liabilities.
This equality is difficult to achieve because it requires the foreign subsidiary to have no stockholder's equity.
Example 3:
a. Flex Co. International is a US company with a subsidiary Vibrant Co. located in the country of Martonia.
Vibrant was acquired by Flex Co. on 31st December, 2004. Flex Co. reports its financial results in US dollars.
The currency of Martonia is Loca (LC). The majority of Vibrant's operational, financial and investment
decisions are made locally in Martonia, although Vibrant does rely on Flex Co. for information technology
expertise.
b. Suppose instead that the majority of Vibrant's operating, financial and investment decisions are made by
the parent company, Flex Co. In this case, Vibrant's functional currency and Flex Co.'s presentation
currency are likely the same, thus, the temporal method is used to remeasured the loss if the $.
Vibrant's financial statements for 2015 are shown in the following two figures.
Balance Sheet (LC)
2014
2015
Cash
Accounts Receivable
Inventory
Current Assets
Fixed Assets
Accumulated Depreciation
Net Fixed Assets
Total Assets
100
500
1000
1600
800
(100)
700
2300
100
650
1200
1950
1600
(700)
900
2850
Accounts Payable
Current Debt
Long-term Debt
Total Liabilities
Common Stock
Retained Earnings*
Total Equity
Total Liabilities and Equity
400
100
1300
1800
400
100
500
2300
500
200
950
1650
400
800
1200
2850
* Retained Earnings on December 31st 2014 were $50.
Income Statement (LC)
Revenue
COGS
Gross Margin
Other Expenses
Depreciation Expenses
Net Income
December 31st, 2014: $ 0.5 and December 31st, 2014: $ 0.4545
Average for 2015: $ 0.4762
2015
5000
(3300)
1700
(400)
(600)
700
36
Historical rate for equity: $ 0.5
Historical rate for PPE and depreciation: $ 0.4881
Historical rate for beginning and ending inventory $ 0.52 and $ 0.456
Purchases were made evenly throughout the year.
Use appropriate method for both 'a' and 'b' to translate Vibrant's 2015 balance sheet and income statement
into $.
a.
Income Statement
2015 (LC)
Rate
2015 ($)
5000
(3800)
1700
(400)
(600)
700
0.4762
0.4762
2381
(1571.5)
809.5
(190.5)
(285.7)
333.3
2015 (LC)
Rate
2015 ($)
Cash
Accounts Receivable
Inventory
Current Assets
Fixed Assets
Accumulated Depreciation
Net Fixed Assets
Total Assets
100
650
1200
1950
1600
(700)
900
2850
0.4545
0.4545
0.4545
45.5
295.4
545.4
886.3
(318.2)
409
1295.3
Accounts Payable
Current Debt
Long-term Debt
Total Liabilities
Common Stock
Retained Earnings
CTA (Remeasurement Loss)
Total Equity
Total Liabilities and Equity
500
200
950
1650
400
800
0.4545
0.4545
0.4545
Revenue
COGS
Gross Margin
Other Expenses
Depreciation Expense
Net Income
Balance Sheet
a. Derive Closing R/E
Opening R/E
$ 50
NI
$ 333.3
- Dividends Declared
0
Closing R/E
$ 383.3
b.
Balance Sheet
Cash
Accounts Receivable
Inventory
Current Assets
Fixed Assets
Accumulated Depreciation
Net Fixed Assets
Total Assets
0.4762
0.4762
0.4545
0.4545
0.5
a
b
1200
2850
227.2
90.9
431.8
749.9
200
383.3
(37.9)
545.4
1295.3
b. CTA = 1295.3 - 749.9 - 200 - 383.3
= - 37.9 (Plug Figure)
2015 (LC)
Rate
2015 ($)
100
650
1200
1950
1600
(700)
900
2850
0.4545
0.4545
0.4560
45.5
295.4
547.2
888.1
781
(341.7)
439.3
1327.4
0.4881
0.4881
37
Accounts Payable
Current Debt
Long-term Debt
Total Liabilities
Common Stock
Retained Earnings
Total Equity
Total Liabilities and Equity
Income Statement
Revenue
COGS
Gross Margin
Other Expenses
Depreciation Expense
Income before CTA
CTA (Remeasurement Gain)
Net Income
500
200
950
1650
400
800
1200
2850
0.4545
0.4545
0.4560
2015 (LC)
Rate
2015 ($)
5000
(3300)
1700
(400)
(600)
700
0.4762
a c
700
b
2381
(1639.5)
741.5
(190.5)
(292.9)
258.1
69.3
327.4
a. Purchases = 300 + 1200 - 1000
= 3500
c. Derive COGS
Beginning Inventory (1000 x 0.52) $ 520
Purchases (3500 x 0.4762)
$ 1666.7
- Ending Inventory
($ 547.2)
COGS
$ 1639.5
227.3
90.9
431.8
750
200
377.4
577.4
1327.4
0.5
b
0.4762
0.4881
b. Derive NI
Opening R/E
NI
- Dividends Declared
Closing R/E
$ 50
x
0
$ 377.4
x = Net Income = $ 3274
or
R/E = 1327.4 - 750 - 200
= 377.4
The current rate method results in net asset balance sheet exposure (except in the rare case in which an
entity has negative stockholder's equity). Temporal method generates either a net asset or net liability
balance sheet exposure, depending on whether assets translated at the current exchange rate.
Fig 4: Effect of Currency Exchange Rate Movement on Financial Statements
Current Rate Method
Income Statement
Revenue
COGS
Other Expenses
Depreciation Expenses
Net Income
Balance Sheet
CA
NCA
Total Assets
Total Liabilities
Common Stock
R/E
CTA
Total Equity
Strength
Weaken
↑
↑
↑
↑
↑
+ CTA
↑
↓
↓
↓
↓
↓
- CTA
↓
↑
↑
↑
↑
↑
↓
↓
↓
↓
↓
Stable by same
amount due to HC
+ and - by the
same amount
38
Temporal Method
Income Statement
Revenue
COGS
Other Expenses
Depreciation Expenses
Net Income before CTA
CTA
Net Income
Strength
↑
↑
↑
↑
- CTA
↓
Balance Sheet
CA
NCA
Total Assets
Total Liabilities
Common Stock
R/E
Total Equity
↑
↑
↑
↓
↓
Weaken
↓
↓
↓
↓
+ CTA
↑
↓
↓
↓
↑
↑
Stable by same
amount due to HC
+ and - by the
same amount
Stable by same
amount due to HC
Stable by same
amount due to HC
- Common Stock is stable under both Current and Temporal Method due to Historical Cost (HC).
-
Current Method
App
Dep
Asset Exposure
Liability Exposure
(A > L)
(L > A)
Gain
Loss
Temporal Method
App
Dep
Gain
Loss
Loss
Gain
Usually Current rate method results in net asset balance sheet exposure
- Pure balance sheet and pure income statement ratios will be same under the current rate method. ('Pure' means all components of the
ratio are from balance sheet or all of the components are from the income statement and 'Mixed' ratios means it combines inputs from
both the income statement and balance sheet).
- If foreign currency is depreciating, translated mixed ratios (I / B) will be larger than the original ratio.
If foreign currency is appreciating, translated mixed ratios (I / B) will be smaller than the original ratio.
(I: Income statement item in the numerator and B: Balance sheet item in the denominator)
Example 4:
a. A US multinational firm has a Japanese subsidiary. The subsidiary's functional currency is the Japanese Yen
(¥). The subsidiary's books and records are maintained in Yen. The parent's presentation currency is the US
dollar. Determine which foreign currency transaction method is appropriate? Current Rate Method
b. Now imagine the Japanese subsidiary's functional currency is the US dollar. Determine which foreign
currency translation method is appropriate? Temporal Method
lllll
Hyper Inflation
In a hyperinflationary environment, the local currency will rapidly depreciate relative to the parent's
presentation currency because of a deterioration of purchasing power. In this case, using the current
rate to translate all of the balance sheet accounts will result in much lower assets and liabilities after
translation. Using the lower values, the subsidiary will most likely disappear in the parent's
consolidated financial statements.
IFRS: The IASB doesn't specifically define hyperinflation; however, it indicates that a cumulative inflation
rate approaching or exceeding 100% over the 3 years would be an indicator of hyperinflation.
39
If a country in which a foreign entity is located ceases to be classified as highly inflationary, the
functional currency of that entity must be identified to determine the appropriate method for
translating the entity's financial statements.
US GAAP: Defines a hyperinflationary environment is one where cumulative 3-year inflation rate
exceeds 100%. A cumulative 3-year inflation rate of 100% equates to average of
3
approximately 26% per year i.e. (1.26) - 1 is almost equal to 100%.
IFRS: Foreign currency financial statements are restated for inflation and translated using the 'current'
exchange rate. Restating for inflation involves the following procedures:
- Non-monetary assets and non-monetary liabilities are restated for inflation using price index i.e
Balance Sheet Date / Acquisition Date. It is not necessary to restate monetary assets and
monetary liabilities.
- The components of shareholder's equity (other than R/E) are restated by applying
Balance Sheet Date / Acquisition Date.
- R/E is the plug figure that balances the balance sheet.
- In the statement of R/E, net income is the plug figure.
- Net income statement items are restated by multiplying Balance Sheet Date / Average.
- The net purchasing power gain or loss is recognized in the income statement based on the net
monetary asset or liability exposure. Holding monetary assets during inflation results in
purchasing power loss. Conversely, holding monetary liabilities during inflation results in
purchasing power gain.
US GAAP: Temporal method is required when subsidiary is operating in a hyperinflationary
environment.
Example 5:
ABC Co. formed a subsidiary in a foreign country on 1st January, 2001, through a combination of debt and
equity financing. The foreign subsidiary acquired land on 1st January, 2001, which it rents to a local farmer.
Income Statement
Revenue
Expense
Net Income
Balance Sheet
Cash
Land
Total
Note Payable
Capital Stock
R/E
Total
2001 (FC)
1000
(250)
750
1st Jan, 2001
1000
9000
10,000
5000
5000
0
10,000
31st Dec, 2001
1750
9000
10,750
5000
5000
0
10,750
The foreign country experienced significant inflation in 2001, especially in the second half of the year. The
general price index during the year was as follows:
1st January, 2001
31st December, 2001
Average, 2001
100
200
125
As a result of the high inflation rate in the foreign country, the FC weakened substantially during the year
relative to other currencies. Relevant exchange rates are:
40
1st January, 2001
31st December, 2001
Average, 2001
1 FC = 1 $
1 FC = 0.5 $
1 FC = 0.8 $
Calculate the balance sheet and income statement for the ABC Co.
(From B/S to I/S)
Balance Sheet
Cash
Land
Total
Notes Payable
Capital Stock
R/E
Total
FC
Inflation
Adjusted
FC
Current
Exchange
Rate
$
1750
9000
10,750
5000
5000
750
10,750
1750
18,000
19,750
5000
10,000
4750 (β)
19,750
0.5
0.5
875
9000
9875
2500
5000
2375
9875
1600
(400)
3550
4750 (R/E)
0.5
0.5
0.5
200/100
200/100
0.5
0.5
0.5
Income Statement
Revenue
Expenses
Purchasing Power G/L
Net Income
1000
(250)
200/125
200/125
750
800
(200)
1775
2375
The net purchasing power gain of FC 3650 can be explained as follows:
Loss - Holding Beginning Cash 1000 x [(200-100)/100]
(1000)
Loss - Increase in Cash during the Year 750 x [(200-125)/125]
(450)
Gain - Holding Notes Payable 5000 x [(200-100)/100]
5000
3550
Note, that all inflation-adjusted FC accounts are translated at the current exchange rate and thus no
transaction adjustment is needed.
Example 6:
Turkey was one of the few remaining highly inflationary countries at the beginning of the 21st century.
Annual inflation rates and selected exchange rates between the Turkish Lira (TL) and US Dollar during the
2000-2002 period were as follows: Assume that a US based company established a subsidiary in Turkey on
1st January, 2000. The US parent sent the subsidiary $1000 on 1st January, 2000 to purchase a piece of land at
a cost of TL 542,700,000 (TL542,700 x $1000).
Date
1st Jan, 2000
31st Dec, 2000
31st Dec, 2001
31st Dec, 2002
Exchange Rates
1 $ = 542,700 TL
1 $ = 670,800 TL
1 $ = 1,474,525 TL
1 $ = 1,669,000 TL
Inflation Rates (%)
38
69
45
Assuming no other assets or liabilities, what are the annual and cumulative CTA gains or losses that would be
reported under temporal and inflation adjusted?
41
a. Temporal Method: No CTA
Date
Carrying Value
(TL)
Historical
Ex-Rate
Translated Amount
($)
Annual CTA
($)
Cumulative
CTA
1st Jan, 2000
31st Dec, 2000
31st Dec, 2001
31st Dec, 2002
542,700,000
542,700,000
542,700,000
542,700,000
542,700
542,700
542,700
542,700
1000
1000
1000
1000
-
-
b. Inflation Adjusted: $100 CTA (Unrealized Gain)
Date
1st Jan, 2000
31st Dec, 2000
31st Dec, 2001
31st Dec, 2002
Inflation
Rate
Carrying Value
(TL)
Current
Ex-Rate
Translated Amount
($)
Annual CTA
($)
Cumulative
CTA
38
69
45
542,700,000
748,926,000
1,265,684,940
1,835,243,163
542,700
670,800
1,474,525
1,669,000
1000
1116
858
1100
116
(258)
242
116
(142)
100
31st December, 2000: 542,700 (1.38) = 74,892,600
31st December, 2001: 74,892,600 (1.69) = 1,265,684,940
31st December, 2002: 1,265,684,940 (1.45) = 1,835,243,163
Although the CTA gain of $ 100 on 31st December, 2002, is unrealized, it could have been realized if (i) the
land had appreciated in TL value by the rate of local inflation, (ii) the Turkish subsidiary sold the land for
TL 1,835,243,163 and (iii) the sale proceeds were converted into $ 1100 at the current exchange rate on
31st December, 2002.
Fig 5: Difference between Temporal Method and Inflation-Adjusted Hyperinflation
Temporal Method
(US GAAP)
Inflation Adjusted
(IFRS)
- Monetary Assets and Liabilities are
exposed to changing exchange
rates.
- Monetary Assets and Liabilities are
exposed to risk of inflation.
- Net Monetary Liability exposure
when foreign currency is
depreciating will result in gain.
- Net Monetary Liability exposure in
hyperinflation will result in a
purchasing power gain.
- Gain or loss from changing
exchange rate is recognized in the
I/S.
- Purchasing Power gain or loss is
recognized in the I/S.
42
NOTES
1. Effective Tax Rate is the tax expense in the income statement divided by pre-tax profit. Statutory Tax Rate
is provided by the tax code of the home country. Accounting standards require companies to provide a
reconciliation between the effective tax rate and the statutory tax rate. This reconciliation disclosure can be
used by the analyst to project future tax expense. Changes in the effective tax rate impact of foreign taxes
could be caused by changes in the applicable tax rates and/or changes in the mix of profits earned in
different jurisdictions.
2. Sales growth owning to an increase in volumes or prices is considered more sustainable then sales growth
due to appreciation of the foreign currencies in which sales were made. 'Organic Growth' in sales is defined
as growth in sales excluding the effects of acquisitions or divestitures and currency effects. E.g. %
in
sales is 2% and the impact of currency was 3% therefore organic growth or %
in sales excluding currency
effect is 2% - 3% = - 1%
△
△
43
CHAPTER 16
Analysis of Financial Institutions
One of the most important global organizations is the Basel Committee on banking supervision, a standing
committee of the Bank of International Settlements. Basel Committee develops the regulatory framework for
banks (Based III) with the purpose of 'improving the banking sector's ability to absorb shocks arising from
financial and economic stress, whatever the source, improving risk management and governance, and
strengthen bank's transparency and disclosures'. The 3 pillars of the Basel III framework are:
- Minimum Capital Requirements: prevents a bank from assuming so much financial leverage that it is unable
to withstand loan losses (asset write downs). The riskier a bank's assets are, the higher its required capital.
- Liquidity: bank should hold enough liquid assets to mark demands under a 30-day liquidity stress scenario.
- Stable Funding: requires stable funding relative to a bank's liquidity needs over a 1-year time horizon.
Stability of funding is based on the tenor of deposits (e.g. longer-term deposits are more stable than
shorter-term deposits) and the type of depositor (e.g. funds from consumer's deposits are considered more
stable than funds raised in the interbank markets).
lllll
CAMELS Approach
A banker examines using CAMELS approach to evaluate a bank conducts an analysis and assigns a
numerical rating of 1 (best rating) through 5 (worst rating) to each component.
1. Capital Adequacy
To prevent financial insolvency, a bank must maintain adequate capital to sustain business losses. Capital
adequacy is based on Risk-Weighted Assets (RWA). Lower the risk weighting, lower the risk and higher the
risk weighting, higher the risk. Basel III defines a bank's capital in a hierarchical approach.
Total Tier 1 Capital:
a. Common Equity Tier 1 Capital: This is the most important component. It is widely recognized as
the most loss-absorbing form of capital. E.g. Common Stock, Additional Paid-in-Capital,
Retained Earnings and OCI - Intangible Assets and Deferred Tax Assets.
b. Other Tier 1 Capital: Subordinated instruments with no specified maturity and no contractual
dividends. E.g. Preferred Stock with discretionary dividends.
Tier 2 Capital: Subordinated instruments with original (i.e. when issued) maturity of more than 5 years.
Common Equity Tier 1 Capital: 4.5% of RWA
Total Tier 1 Capital: 6% of RWA
Total Capital (Tier 1 and Tier 2): 8% of RWA
Minimum
Example 1:
Mega Bank is regulated by the Central Bank of Zima. As per the Central Bank rules, the following risk
weightings are applied to the bank's assets. Calculate Mega Bank's RWA.
Asset Type
Risk Weight
B/S
RWA
Cash and Central Banks Deposits
Corporate Loans - Performing
Corporate Loans - Non Performing
Consumer Real Estate Loans
Total RWS
0%
100 %
200 %
90 %
$ 120
$ 1130
$ 920
$ 2450
$0
$ 1130
$ 1840
$ 2205
$5175
44
Example 2:
Mega Bank's RWA for 2007 and 2008 were $4,700,000 and $5,175,000 respectively. An analysis of bank's
capital for the two years reveals the following information. Determine the Common Equity Tier 1 Capital ratio,
Total Tier 1 Capital and Total Capital (Tier 1 and Tier 2).
Capital Type
Common Equity Tier 1
Subordinate Debt - No Maturity
Subordinate Debt - 5+ Year Maturity
2007
2008
$ 200
$ 85
$ 110
$ 190
$ 95
$ 120
Common Equity Tier 1 Capital = (190/1575) = 3.67%
Total Tier 1 Capital = (190 + 95)/1575 = 5.51%
Total Capital (Tier 1 and Tier 2) = (190 + 95 + 120)/1575 = 7.83%
All three have fallen below the minimum requirements threshold.
2. Asset Quality
In broad context, asset quality derives from the processes of generating assets, managing them and
controlling overall risk. Bank assets include loans (the largest component) and investments in securities;
while loans are generally carried on the balance sheet at amortized cost, net of allowances.
- Loans and Advances to Banks
- Loans and Advances to Customers
- Reverse Repurchase Agreement (REPO) is a form of Collateralized Loan made by a Bank to Client.
(Lender): Refers as Reverse Repurchase Agreement
①
Bank
Buy back at a
future date
②
Sells financial
assets
Client
(Borrower): Refers as Repurchase Agreement
- Assets Held for Sale is related to discontinued operations and specifically refers to long-term assets
whose value is driven mainly by their intended disposition rather than their continued use.
↷
Credit risk is present in debt securities that the bank invests in, loans the bank makes as well as in the
bank's off-balance sheet liabilities i.e. guarantees, used committed credit lines and letters of credit
represent potential assets (as well as potential liabilities) to the bank.
Actual Loan Losses (Charge-Offs - Net of Recoveries)
reduce the amount of the allowance for Loan Losses
Ratio of Allowance for Loan Losses to Non-Performing Loans
Bad Loans eliminated
Ratio of Allowance for Loan Losses to Net Loan Charge-Offs
from B/S and expensed to
I/S (Net of Recoveries).
Ratio of Provision for Loan Losses to Net Loan Charge-Offs
↶
→
Bad Debt Expense in I/S = Net Charge-Offs +
△ Allowance for Loan Losses
Culture Evaluation is the speed with which a bank adjusts its loan loss provisions relative to actual loss
behavior. A slower response rate generally indicates aggressive accounting practices and risk-taking
culture.
3. Management Capabilities
A strong governance structure with an independent board that avoids excessive compensation or
self-dealing is also critically important. Sound internal controls, transparent management communication
and financial reporting quality are indicators of management effectiveness.
45
4. Earnings
A major source of earnings of a bank is from investment in securities. Estimates used in the valuation of
these securities may lead to biased earnings. Both IFRS and US GAAP use the concept of 'Fair Value
Hierarchy' based on types of inputs used in determining the fair value of financial assets and liabilities.
Level 1 inputs are quoted market prices of identical assets or liabilities in active markets. Level 2 inputs
include quoted prices of similar assets, quoted prices of identical assets in non-active markets, observable
interest rates, credit spreads and implied volatility. Level 3 inputs are non-observable and hence subjective.
For example, fair value may be derived from models or based on estimated future cashflows discounted at
an estimated discount rate. Similar to other companies, other subjective estimates (e.g. goodwill
impairment, recognition of deferred tax assets and recognition of contingent liabilities) affects the quality
of a bank's earnings. For a typical bank, major sources of earnings are (i) net interest income (the difference
between interest earned in loans minus interest paid on the deposits supporting those loans), (ii) service
income and (iii) trading income; of these trading income is the most volatile year to year and hence on a
relative basis banks with proportionally higher net interest income and service income would have more
sustainable earnings.
5. Liquidity Position
The impact of a bank's failure to honor a current liability could affect an entire economy. Deposits in most
banks are insured upto some specified amount by government insurers.
Liquidity Coverage Ratio (LCR) =
Highly Liquid Assets
Expected Cash Outflows
1-month liquidity
→ Estimate
needs in a stress scenario.
↷
(Minimum 100% of LCR)
Available Stable Funding includes sources
↶
Net Stable Funding Ratio (NSFR) = Available Stable Funding like capital deposits and other liabilities.
(Minimum 100% of NSFR)
Required Stable Funding Required Stable Funding is a function of
the composition of the bank's asset base.
Under Basel III, the Available Stable Funding is determining by assigning a bank's capital and liabilities to
one of 5 categories show below: The amount assigned to each category is then multiplied by an Available
Stable Funding (ASF) Factor and the total Available Stable Funding is the sum of the weighted amounts.
ASF Factor
Regulatory capital minus Tier 2 instruments maturing in a
year and other capital instruments and liabilities with
maturity > 1 year.
100 %
Stable demand deposits and term deposits (maturity < 1
year) from retail and small business customers.
95 %
Less-Stable demand deposits and term deposits
(maturity < 1 year) from retail and small business
customers.
90 %
Funding from non-financial corporates (maturity < 1
year), operational deposits, funding from sovereigns,
public sector (maturity < 1 year), multilateral and national
development banks.
50 %
All other liabilities including 'Trade Date' payables arising
from purchases of financial instruments, foreign
currencies and commodities.
0%
46
- Concentration of Funding: Relatively concentrated funding indicates a bank's reliance on relatively few
funding sources. This lack of diversification can result in heightened liquidity risk for the bank.
- Contractual Maturity Mismatch: Occurs when the asset maturities differ meaningfully from maturity of the
liabilities (funding sources). The higher the mismatch, the higher the liquidity risk for the bank.
6. Sensitivity to Market Risk
Bank earnings are affected by various market risks e.g. volatility of security prices, currency values and
interest rates. The most critical of these is interest rate risk. A bank's interest rate risk is the result of
differences in maturity, rates and repricing frequency between the bank's assets and its liabilities. For
example, in a rising interest rate scenario, if the assets are repriced more frequently than liabilities, the
bank's net interest income would increase. The impact of market risk can be captured by Value at Risk
(VaR). In the MD&A section, banks disclose information about the sensitivity of earnings to different market
conditions, namely, the earnings impact of a shift up or down in some market.
7. Government Support (Not Part of CAMELS from now on)
Apart from providing deposit insurance, governments often serve as a backdrop against bank failure.
Usually the larger the bank and more inter-linked it is, the more likely that its failure will have a contagion
affect i.e. spread of economic crisis globally. As a deposit-taking institution, banks are especially prone to
the risk of a bank run i.e. a large sudden withdrawal of deposits. Hence, public ownership of bank increases
faith (of implicit government backing) in a bank.
8. Competitive Environment
It affects a bank's culture and risk-taking behavior e.g. managers of a global bank may not be satisfied with
following the lead of other banks and may pursue ambitious goals of growing market share.
9. Off-Balance Sheet Items
Off-Balance Sheet assets or liabilities may seriously affect an entity, if the underlying risks turn out to be
larger than the available resources. Operating leases and benefit plans are a low risk example of
Off-Balance Sheet and whereas, VIEs and Assets Under Management (AUM) may pose a significant
Off-Balance Sheet risk not reflected in the financial statements. Also at inception, many derivative contracts
don't give risk to an asset or liability on the balance sheet or to a gain or loss in the income statement.
If each component receives the same rating, the weighting of the components is irrelevant. The arithmetic
mean approach however, fails to take into account the fact that some components of the CAMELS approach
are more important to some analysis than others.
lllll
Insurance Companies
Insurance companies earn revenues from premiums (amount paid by the purchaser of insurance
products) and from investment income earned on the float (amount collected as premium and not yet
paid out as benefits).
Property & Casualty
1. Profile: P&C Insurers' policies are usually short
term. P&C insurer's claims are more variable and
lumpier because they arise from accidents and
other unpredictable events.
Life & Health
1. Profile: L&H insurer's policies are usually long
term. L&H insurer's claims are more predictable
because they correlate closely with relatively
stable actuarially based mortality rates when
applied to large populations.
47
2. Earnings: Keys to the profitability of an insurer are
prudence in underwriting, pricing of adequate
premiums for bearing risk and diversifying risk by
transferring policies in whole or in part to
reinsurers.
2. Earnings: L&H companies can be diversified
across revenue sources, product offerings,
geographic coverage, distribution channels and
investment assets.
- Causality Insurance (Liability Insurance) protects
against a legal liability related to an insured
event.
- Multiple Peril Policy may cover both P&C losses
occurring during a covered event.
- During periods of heightened competition, price
cutting to obtain new business leads to slim or
negative margins. This Soft Pricing Period leads
to losses and a shrinking capital base for many
insurers, who either leave the industry or stop
underwriting new policies. The resulting
reduction in competition leads to a healthier
pricing environment - Hard Pricing Period, which
results in fatter margins.
3. Costs: Major expenses include claims expense
and the expense of obtaining new policy business.
The Loss Reserve is an estimated value of unpaid
claims (based on estimated losses incurred during
the reporting period). It is subject to management
discretion. Insurers revise their estimate of the
loss reserve as more information becomes
available. Downward revisions indicate the
company was conservative in estimating their
losses in the first place. Upward revisions indicate
aggressive profit booking, a warning sign for
analysts.
3. Costs: If policyholder were to cancel the contract
before its contractual maturity and receive the
accumulated cash value. Such early cancellation is
known as a Contract Surrender. This may result in
additional expenses.
4. Ratios
4. Ratios: Other common profitability measures
including ROA, ROE, growth and volatility of
capital, BVPS, pre and post tax operating margin.
* Combined Ratio = Total Insurance Expenses
Net Premium Earned
Combined Ratio = Underwriting Loss Ratio +
Expense Ratio
If the ratio is Low: Hard Market
High: Soft Market
For a single insurer, a Combined Ratio in excess of
100% indicates an underwriting loss.
△
Loss Reserves
* Underwriting = Claims Paid +
Loss Ratio
Net Premium Earned
*
*
Total Benefits Paid
Net Premiums Written and Deposits
Commissions and Expenses
Net Premiums Written and Deposits
48
Underwriting Loss Ratio is also called the 'Loss
and Loss Adjustment Expense' whether the
policies are appropriately priced.
* Expense Ratio = Underwriting Expenses including
Commissions and Employee
Expenses
Net Premium Writing
Expense Ratio measures the efficiency of the
company's operations. It is also called the
Underwriting Expense Ratio.
Notice that the denominator in the two ratios is
different. For reporting purposes, sometimes
issuers use US GAAP which calls for Net Premium
Earned as the denominator for both ratios.
* Loss and Loss = Loss Expense + Loss Adjustment
Adjustment
Expense
Expense Ratio
Net Premium Earned
* Dividends to Policyholders =
Dividends to
Policyholders
Net Premium Earned
It is a liquidity measure measuring cash outflow
on account of dividends relative to premium
income.
* Combined Ratio = Combined Ratio after Dividends Dividends to Policyholders
(CRAD)
Ratio
Combined Ratio after Dividends measures total
efficiency and is more comprehensive than the
Combined Ratio.
5. Investment Returns: After premium income,
investment return is an important source of P&C
insurers' profitability. To counteract the
environment of uncertainty, P&C insurers
conservatively invest the collected premiums i.e.
Short Float Period.
* Total Investment = Total Investment Income
Return Ratio
Invested Assets
Instead of Total Investment Income, computing
the ratio after excluding unrealized capital gains
from income provides information about the
importance of unrealized gains and losses to the
insurer's total income.
5. Investment Returns: L&H insurers have a Longer
Float Period than P&C insurers.
* Total Investment = Investment Income
Income Ratio
Invested Assets
Analysts often recalculate this ratio after removing
unrealized gains and losses from Investment
Income to remove the impact of estimated fair
values for some of the Investment Assets.
49
6. Liquidity: The uncertainty of the payouts involved
in the P&C business requires a high degree of
liquidity so loss obligations can be met.
Level 1: Most Liquid
Level 2: Lower Liquidity
Level 3: Least Liquid
6. Liquidity: Keeping excess liquidity is not much of a
concern for L&H insurers compared to P&C
insurers. Such investments as non-investment
grade bonds and equity real estate are typically
less liquid than investment grade fixed income
investments.
Adjusted based on
↶
↷
* = Adjusted Investment Assets convertibility into cash
Adjusted Obligations Adjusted based on assumptions
about withdrawals
The typical Current Ratio is not directly applicable
to L&H companies because their balance sheets
often don't include the classifications 'Current'
and 'Non-Current'.
7. Capitalization: No risk-based global insurance
capital standard exists, capital standards do exist
in various other jurisdictions.
Qualifying Capital
* Minimum Capital =
Adequacy Ratio
Risk-based Capital Required
7. Capitalization: No global minimum capitalization
standards. Domestic regulators often do specify
risk adjusted minimum capital requirements.
L&H companies' earnings can also be distorted by
the accounting treatment of certain items.
E.g. Mismatches between the valuation approach
for assets and liabilities can introduce distortion
when interest rate changes occur. Accordingly, the
calculation of risk-based capital for an L&H
company incorporates interest rate risk.
There are two methods of distributing insurance: (a) Direct Writing and (b) Agency Writing. Direct insurance
have their own sales and marketing staff. Direct Writers also may sell insurance policies via. the internet i.e.
through direct response channels such as mail and through groups with a shared interest or bond, such as
membership in a profession. Direct Writers have higher fixed cost because of the in-house nature of their
distribution method. The sales and marketing staff are salaried employees. Agency Writers use independent
agents, exclusive agents and insurance brokers to sell policies. Agency Writers don't have this fixed cost,
instead the commissions paid to agents and brokers are a variable cost.
Example 3:
Which of the following industries most likely has the highest level of global systematic risk?
A. International Property and Casualty Insurance
B. Credit Unions
C. Global Commercial Banks
✓
NOTES
1. Other global organizations that coordinate regulations include the:
- Financial Stability Board (FSB): Its overall goal is to strengthen financial stability. It seeks to coordinate
actions of participating jurisdictions in identifying and managing systematic risks.
- International Association of Deposit Insurers (IADI): Seeks to improve the effectiveness of deposit
insurance systems.
Part of
- International Organization of Securities Commissions (IOSCO): Seeks to promote fair and efficient security
joint
markets.
forum
with the
- International Association of Insurance Supervisors (IAIS): Seeks to improve supervision of the insurance
Basel
industry.
Committee
50
CHAPTER 17
Evaluating Quality of Financial Reports
Decision Useful Information
1. Comparability
2. Verifiability
3. Timeliness
4. Understandability
(Relevance and Faithful Representation)
Fig 1: Earnings Quality
Vs. Financial
Reporting Quality
Value Enhancing
1. Recurring Earnings
2. Earnings Persistence
3. Accruals
(High proportion of
non-recurring items are less
likely to be sustainable and
considered lower quality).
Biased accounting can be aggressive (recognizing future revenues/earnings in the current period) or
conservative (postponing current earnings to the future). A related bias is 'Earnings Management' e.g.
Earnings Smoothing.
Fig 2: Quality Spectrum
1
2
3
4
5
IFRS and US GAAP: Both accept Quality Spectrum
6
51
1. Potential problems that affect the quality of financial reports may arise from:
a. Measurement and Timing Issues: A - L = E. E.g. Conservative revenue recognition such as deferred
recognition of revenue results in understated net income, understated equity and assets.
b. Classification Issues: E.g. Reclassifying expense as non-operating (wrong) will cause the analysts to
treat recurring expense as one-time costs. Secondly, treating investing cashflows (e.g. sale of
long-term assets) as operating cashflows (wrong) will wrongly consider operating cashflow to be
recurring and may lead to increase in operating cashflow and higher equity valuation.
2. Acquiring companies often underestimate the value of identifiable net assets, thereby overestimating
goodwill on acquisition. Fair value adjustments for identifiable assets typically result in excess depreciation
which reduces profits for future reporting periods. Since goodwill is not amortized, the effect of
overestimating goodwill (and underestimating the value of identifiable assets) is to increase future
reported profits. Such inflated goodwill will eventually have to be written down as part of impairment
testing but such losses can be timed, In addition, impairment losses can be downplayed as a one-off,
non-recurring event.
3. Beneish Model: It is probit regression quantitative model that estimates the probability of earnings
manipulation using 8 independent variables. The M-Score probability of earnings manipulation i.e. higher
values indicate higher probabilities.
M-Score = - 4.84 + 0.92 (DSRI) + 0.528 (GMI) + 0.404 (AQI) + 0.892 (SGI)
+ 0.115 (DEPI) - 0.172 (SGAI) + 4.679 (Accruals) - 0.327 (LEVI)
Whereas,
M-Score : If M-Score > - 1.78 (i.e. less negative) indicates a higher than acceptable probability
of earnings manipulation.
DSRI : A large increase in DSRI (Days Sales Receivable Index) could be indicative of
inappropriate revenue recognition or revenue inflation.
GMI : When the GMI ratio (Gross Margin Index) > 1, the gross margin has deteriorated. A
firm with declining margins is more likely to manipulate earnings.
AQI : [(NCA - PPE) / TA]. Increases in AQI (Asset Quality Index) could indicate excessive
capitalization of expenses.
SGI : While not a measure of manipulation by itself, growth companies tend to find
themselves under pressure to manipulate earnings to meet ongoing expectations
by observing the Sakes Growth Index.
DEPI : The depreciation rate is [Depreciation Expense / (Depreciation Expense + PPE)]. A
DEPI (Depreciation Index) > 1 suggests that assets are being depreciated at a slower
rate in order to manipulate earnings.
SGAI : Increases in SGA (Sales, General and Administrative Expenses Index) expenses
might predispose companies to manipulate.
Accruals : As per IFRS, [(Income before Extraordinary Items - CFO) / TA], since US GAAP doesn't
include the concept of extraordinary items.
LEVI : Leverage Index (Total Debt / Total Assets).
△
△
△
△
△
△
△
Beneish considered that the likely relevant cutoff for investors is a probability of earnings manipulation of
3.8% (an M-Score exceeding -1.78). The Beneish model relies on accounting data, which may not reflect
economic reality. A study found that the predictive power if Beneish model is decreasing over time.
4. Altman Model: Relies on discriminant analysis to generate a Z-Score using 5 variables. It was developed to
assess the probability that a firm will file for bankruptcy.
52
Z-Score = 1.2
x 1 + 1.4 x 2 + 3.3 x 3 + 0.6 x 4 + x 5
Whereas,
x 1 : Net Working Capital / TA
x 2 : R/E / TA
x 3 : EBIT or Operating Profit / TA
x 4 : Mv of Equity / Bv of Liabilities
x 5 : Sales / TA
(Liquidity Measure)
(Profitability Measure)
(Profitability Measure)
(Leverage Measure)
(Activity Ratio)
Each variable is positively related to the Z-Score and a higher Z-Score means less likelihood of bankruptcy.
Hence, higher values of any the 5 variables reduce the probability of bankruptcy under this model.
Limitation being Altman model, it is a simple-period static model and hence doesn't capture the change in
key variables over time. Also Altman model mostly uses accounting data.
5. Earnings comprised of a high proportion of non-recurring items are considered to be non-sustainable and
hence low quality. Classification of non-recurring items is highly subjective and is open for gaming.
Classification shifting doesn't affect the total net income but rather is an attempt to mislead the user of the
financial statements into believing that the 'core' or 'recurring' portion of earnings is higher than it actually
is. E.g. Misclassifying normal operating expenses as expenses from discontinued operations. Analysts
should be wary of large special items or when the company is reporting usually large operating income for
a period. Companies may include non-GAAP metrics such as pro forma income which excludes
non-recurring elements. Residual income model of valuation is most closely linked to this concept of high
earnings quality. Earnings persistence can be expressed as the coefficient on current earnings.
Earnings (t + 1) = L + β1 . Earnings t + ε
↓
Higher β1 represents more
persistent earnings
Earnings can be viewed as being composed of a cash component and an accruals component. Among the
two, cash component is more persistent. Hence β1 > β2 . Earnings with a larger component of accruals
would be less persistent and this of lower quality.
Earnings (t + 1) = L + β1 . Cashflow t + β2 . Accruals t + ε
It is important to recognize that some accruals occur as part of normal business. Such accruals are called
as 'non-discretionary accruals'. Discretionary Accruals result from non-normal transactions or non-normal
accounting choices and are sometimes used to manipulate earnings. One mechanism to separate
discretionary and non-discretionary accruals is to model total accruals using regression. The residuals from
such a model would be an indicator of discretionary accruals. Finally a major red flag about earnings
quality is raised when a company reports positive net income while reporting negative operating cashflow.
6. When examining net income, analysts should be aware that earnings at extreme levels tend to revert back
to normal levels over time. This phenomenon is known as mean reversion. Because of mean reversion,
analysts should not expect extreme earnings (high or low) to continue indefinitely. When earnings are
largely comprised of accruals, mean reversion will occur faster and even more so when the accruals are
largely discretionary.
7. a. Revenue Recognition Issues
Revenues generated via deliberate channel-stuffing or as a result of bill and hold arrangements should
be considered spurious and inferior. Even (relatively) genuine revenues, when secured via the use of
heavy discounting practices, come at the expense of deteriorating margins. A higher growth rate of
53
receivable relative to the growth rate of revenues is a red flag. Similarly, an increasing day's sales
outstanding (DSO) over time is an indication of poor revenue quality. Check for related part transactions,
a company might artificially boost fourth-quarter revenues by recognizing a large scale to an affiliated
entity. Multiple Element Contracts i.e. vendors often provide multiple products or services to their
customers as part of a single arrangement or a series of related arrangements. These deliverables may
be provided at different points in time or over different time periods.
b. Expense Capitalization Issues
One way to boost reported performance is to under-report an operating expense by capitalizing it.
Capitalizing an expense does however show up on the balance sheet as an asset. Stable or improving
profit margins coupled with a buildup of non-current assets would be a warning sign. Steady or rising
revenue coupled with declining asset turnover ratios is another warning sign of cost capitalization.
Compare depreciation expense as a proportion of asset size over time and with peers. Finally compare
capital expenditures to gross PPE over time and with peers. Check for related party transactions like
'Tunneling' is a problematic transaction if the company is shifting resources to a privately held company
that is owned by senior managers i.e. the transfer of assets and profits out of firms for the benefit of
those who control them and 'Propping', when managers-owned entity could transfer resources to the
public company to ensure its economic viability and thus preserve the option to misappropriate or to
participate in profits in the future.
8. To Evaluate the cashflow quality of a company; an analyst must (a) Check for any unusual items or items
that have not shown up in prior years, (b) Checking revenue quality: aggressive revenue recognition
practices typically result in an increase in receivables which reduces CFO. Another common indicator of
aggressive revenue recognition is an increase in inventories (and hence a cash outflow) when sham sales
are reversed i.e. treated as return from customers, (c) Checking for strategic provisioning: provisions for
restructuring charges show up as an inflow i.e. non-cash expense in the year of the provision and then as
an outflow when ordinary operating expenses are channeled through such reserves and (d) Classification
of cashflow items reduce comparability across companies: under US GAAP, interest paid, interest received
and dividends are classified under CFO. Whereas under IFRS, interest paid is classified under CFO or CFF.
On the other hand, 'Sale of Available for Sale Securities' appear under CFI and 'Sale of Trading Securities'
appear under CFO. But managers have significant flexibility in designating investments.
9. High-quality financial balance sheet reporting is evidenced by completeness, unbiased measurement and
clarity of presentation.
a. Completeness: Analysts should restate the reported balance sheet by capitalizing operating leases
(off-balance sheet items) and recording purchase contract obligations, if significant. One common
source of off-balance sheet obligation is purchase contracts which may be structured as 'take or pay
contracts' i.e. take or pay contractual provisions obligate a party to either take delivery of goods or
pay a specified amount (penalty). Also, a company operating with numerous or material
unconsolidated subsidiaries for which ownership levels approach 50% could be a warning sign of
accounting issues.
b. Unbiased Measurement: The balance sheet reflects subjectivity in the measurement of several assets
and liabilities e.g. value of the pension liability, value of investment in debt or equity of other
companies for which a market value is not readily available, goodwill valuation - testing impairment,
inventory valuation and impairment of PPE and other assets.
c. Clear Presentation: While accounting standards specify which items should be included in balance
sheet, they don't typically specify how such items must be presented. Clarity should be evaluated in
conjunction with information found in the notes to financial statement and supplementary
disclosures.
10. Sources of Information about risk are financial statement, auditor's report, notes to financial statements
i.e. both IFRS and US GAAP requires companies to disclose risks related to pension benefits, contingent
54
obligations and financial instruments, MDA, SEC form 'NT' (Notification of inability to Timely file), such
filings typically signal problems in reporting quality and financial press i.e. often the initial information
about accounting irregularities at a company is obtained from the financial press. Analyst should do their
own due diligence to ensure that the information revealed has merit.
Example 1:
The presence of significant off-balance sheet financing most likely indicates:
A. A lack of completeness, which reduces financial reporting quality
B. A decrease in leverage, which reduces financial results quality
C. A lack of clear presentation, which reduces financial reporting quality
✓
Example 2:
Webster's Notes to Financial Statements.
Note 1: Operating income has been must lower than CFO.
Note 2: Accounts payable has increased, while accounts receivable and inventory has substantially decreased.
Which of Webster's notes about BIG Industrial provides an accounting warning sign of a potential reporting
problem?
A. Only Note 1
Operating Income greater than CFO is a warning sign.
Operating Income lower than CFO is not a warning sign.
B. Only Note 2
C. Both Note 1 and Note 2
✓
→
Example 3:
Andre Bursch is analyzing large retailers and has collected the following information on three companies
based on the most recent financial statements.
Total Earnings
Cash Element
Accrual Element
Allied Stores
$ 2.8
$ 1.9
$ 0.9
Beta Mart
$ 1.33
$ 0.78
$ 0.55
Cash N Carry
$ 0.75
$ 0.25
$ 0.5
Brusch notes that all three companies have reported stellar earnings this past year. Which company's
earnings will revert to its mean fastest?
A. Allied Stores
(Accruals/Total Earnings)
B. Cash N Carry
A: (0.9/2.8) = 32%
B: (0.55/1.33) = 41%
C: (0.5/0.75) = 67%
C. Beta Mart
✓
→
55
CHAPTER 18
Integration of Financial Statement Analysis Techniques
Fig 1: Financial Statement Analysis Framework
56
lllll
Du Pont
The Du Pont decomposition allows us to identify the firm's performance drivers, allowing us to expose
effects of weaker areas of business that are being masked by the effects of other stronger areas. For
example, a firm could offset a declining EBIT Margin by increasing asset turnover or increasing
leverage.
We must also consider the firm's sources of income and whether the income is generated internally
from operations or externally. If equity income from associates or joint ventures is a significant sources
of earnings, we should isolate these effects by removing the equity income from our Du Pont analysis
to eliminate any bias. After isolating, both the investor firm's earnings and NP margin decrease. Since,
under the equity method, the firm's investment is reported as a balance sheet asset, total assets
should be reduced by the carrying value of investment. This will increase Total Asset Turnover.
Another common adjustment made by analysts is to remove the effects of any unusual items e.g.
provisions for restructuring and litigation, goodwill impairment etc., from reported operating earnings
(EBIT) before computing the EBIT Margin and the tax burden ratios.
Ratios after
isolation of
equity income
Tax Burden = NI - Equity Income
EBT
Total Asset Turnover =
Sales
(Beg. TA + End. TA) - (Beg. Equity Inv. + End. Equity Inv.)
2
Step 1: ROE = NI
Equity
Step 2: ROE = NI
Sales
x
Sales
Equity
or
NI
x
Avg. Total Assets
[NP Margin] [Equity Turnover]
Step 3: ROE = NI
Sales
x
[NP Margin]
Step 4: ROE = NI
EBT
x
[Tax Burden or
Tax Retention
Rate] (1-t)
[ROA]
Sales
Avg. Total Assets
x
[Asset Turnover]
EBT
EBIT
x
[Interest
Burden]
Step 5: ROE = Earnings after Tax
Sales
[After Tax Profit Margin]
[Leverage or Equity Multiplier]
Avg. Total Assets
Equity
[Leverage or Equity Multiplier]
EBIT
Sales
x
[EBIT Margin]
x
Avg. Total Assets
Equity
Sales
Avg. Total Assets
[Asset Turnover]
Sales
Avg. Total Assets
[Asset Turnover]
x
x
Avg. Total Assets
Equity
[Leverage or Equity Multiplier]
Avg. Total Assets
Equity
[Leverage or Equity Multiplier]
High profit margins, leverage and asset turnover will lead to high levels of ROE. However, this version of
the formula shows that more leverage doesn't always lead to higher ROE. As leverage rises, so does the
interest burden. Hence, the positive effects of leverage can be offset by the higher interest payments
that accompany more debt. Note, higher taxes will always lead to lower levels of ROE.
57
1. Capital Allocation Decisions: Consolidated financial statements can hide the individual characteristics of
dissimilar subsidiaries. As a result, firms are required to disaggregate financial information by segments to
assist users. Recall that a business segment is a portion of a larger company that accounts for more than
10% of the company's revenue or assets and is distinguishable from the company's other lines of business
in terms of risk and return characteristics. By comparing the EBIT Margin contributed by each segment to
its ratio of 'Capital Expenditures to Assets' proportion, we can determine if the firm is investing its capital in
its most profitable segments. Accrual-based measures such as EBIT may not be a good indicator of an
entity's ability to generate cashflow. We would rather evaluate segmental capital allocation decisions based
on cashflows generated by cash segment. However, segmental cashflow data is generally not reported. We
can however, approximate Cashflow as EBIT + Depreciation + Amortization. Therefore, we can now
compute approximate Cashflow (CFO) on Average Total Assets. We can also calculate EBIT divided by
Average Assets.
2. Earnings: Earnings quality refers to the persistence and sustainability of a firm's earnings. Earnings that are
closer to operating cashflow are considered higher quality. The ratio of 'Accruals to Average Net Operating
Assets' can be used to measure earnings quality. The lower the ratio, the higher the earnings quality.
e: Ending
b: Beginning
B/S Accruals = NOA e- NOA b
B/S Accruals Ratio* =
NOA e - NOA b
(NOA e+ NOA b)/2
Cashflow Accruals = NI - CFO - CFI
Cashflow Accruals Ratio* = NI - CFO - CFI
(NOA e+ NOA b)/2
Whereas,
NOA : Net Operating Assets i.e. Operating Assets - Operating Liabilities
Operating Assets : TA - Cash Equivalents to Cash and Marketable Securities
Operating Liabilities : TL - Short-term Debt and Long-term Debt
* Just like ROA and ROE, the measure can be distorted if a firm is growing or contracting quickly. Scaling the
measure also allows for comparison with other firms. Scaling is done by dividing the accrual measure by
the average NOA for the period. If fluctuations happen to continue years on end, it is an indication of
earnings manipulation. Because of the potential for earnings manipulation by increasing accruals, we
decide to compare cashflows to its operating income. However, we cannot directly use the cashflow from
operating activities (CFO) as proxy for cashflow for this purpose. Therefore, we make adjustments i.e.
Cash Generated from Operations (CGO).
CGO = EBIT + Non-Cash Charges - Increase in Working Capital
or
CGO = CFO + Interest Paid + Taxes Paid
Firms that follow IFRS have the choice of reporting cash paid for interest as an CFO or CFF. If a firm reports
the interest as a CFF, no interest adjustment is necessary.
CGO
Operating Income
CGO
Avg. Total Assets
(Useful for
Acquisition)
58
CGO
Capital Expenditures
(Cashflow to
Reinvestment Ratio)
CGO
Total Debt
CGO
Cash Interest
(Cashflow to Interest
Coverage Ratio)
Example 1:
$
2271
1347
11,268
20,097
2449
81
532
824
73
Current Debt
Long-term Debt
Total Shareholder Equity
Total Assets
CFO
Cash Interest Paid
Cash Taxes Paid
Capital Expenditures
Expenditures on Intangible Assets
Using data above, if the company wished to pay off all of its debt while maintaining its current reinvestment
policy in 2016, the no. of years it would take to do so is closest to:
Total Debt = Current Debt + LT Debt = 2271 + 1347 = 3618
A. 2.3
CFO - Reinvestment = 2449 - (824 + 73) = 1552
B. 1.5
C. 1.7
Years to Repay Debt from CFO = 3618 = 2.3 Years
✓
1552
3. Market Value Decomposition: When a parent company has an ownership interest in an associate
(subsidiary or affiliate), it may be beneficial to determine the standalone value of the parent i.e. the implied
value of the parent without regard to the value of the associate.
Implied Value = Parent's Mv - Parent's Pro-rata Share of the Associate's Mv
Example 2:
Big Co. owns 25% of Small Co. Recent financial data for both the firms follow below:
Current Exchange Rate
Average Exchange Rate
1 € = 0.85 £
1 € = 0.8 £
Big
£ 16,000
£ 275,000
Net Income
Market Capitalization
Small
€ 6000
€ 150,000
25%
B
S
£ 275,000
€ 150,000
59
Implied Value = Parent's Mv - Parent's Pro-rata Share of the Associate's Mv
= £ 275,000 - (€ 150,000 x 0.25 x 0.85)
= £ 275,000 - £ 31,875
= £ 243,125
(Net Income)
Implied
P = 275,000 = 17.18
E
P
=
243,125
= 16.4
E - Equity Income 16,000 - (6000 x 0.25 x 0.8)
If the method S&P multiple P/E is 20.1. Then Big's stock is undervalued relative to the market. This can also be
effected by differences in accounting methods used by two firms i.e. US GAAP (Big) vs. IFRS (Small).
NOTES
1. Some liabilities are more burdensome than others. Financial liabilities and bond liabilities for example can
be placed in default if not paid in time or in the event of non-compliance with the lending covenants i.e.
technical default. On the other hand, liabilities such as employee benefit obligations, deferred taxes and
restructuring provisions are less burdensome and may or may not require a cash outflow in the future.
60
RATIOS
A | Activity Ratios
1. Receivables Turnover =
Sales
Avg. Receivables
COGS
Avg. Inventory
3. Payables Turnover =
Purchases
Avg. Trade Payables
↷
2. Inventory Turnover =
and
Days of Sales Outstanding =
and
Days of Inventory on Hand =
and
Sales
Avg. Total Assets
5. Fixed Asset Turnover =
Sales
Avg. Net Fixed Assets
6. Working Capital Turnover =
(CA - CL)
(DOH)
365
Receivables Turnover
365
Inventory Turnover
Credit Purchase = End. Inventory - Beg. Inventory + COGS
4. Total Asset Turnover =
(Labor vs. Capital Intensive)
(DSO)
Days of Payables Outstanding =
(DOP)
365
Payables Turnover
Sales
Avg. Working Capital
B | Liquidity Ratios
1. Current Ratio = CA
CL
2. Quick Ratio = Cash + Marketable Securities + Receivables
CL
or
CA - Inventory
CL
3. Cash Ratio = Cash + Marketable Securities
CL
4. Defensive Interval = Cash + Marketable Securities + Receivables
Avg. Daily Expenditures
Company is at the verge of
collapsing. E.g. Start-up
companies 36 days i.e. the
company only survives for
36 days.
5. Cash Conversion Cycle = DSO + DOH + DOP
C | Solvency Ratios
1. Debt to Equity =
Total Debt
Total Shareholder's Equity
2. Debt to Capital =
or
Long-term Debt + Interest Bearing Short-term Debt
Total Shareholder's Equity
Excludes Non-Interest Bearing Liabilities
Total Debt
Total Debt + Total Shareholder's Equity
3. Debt to Assets = Total Debt
Total Assets
4. Financial Leverage = Avg. Total Assets
Avg. Total Equity
61
5. Interest Coverage =
EBIT
Interest Payments
6. Fixed Charge Coverage =
This is more meaningful
measure for companies
that lease a large portion
of their assets, such as
some airlines.
EBIT + Lease Payments
Interest Payments + Lease Payments
D | Profitability Ratios
1. Net Profit Margin = NI
Sales
NI from Continuing Operations i.e. Earnings After Tax before Dividends
2. Gross Profit Margin = GP
Sales
3. Operating Profit Margin = Operating Income
Sales
or
EBIT
Sales
4. Pre-Tax Margin = EBT
Sales
5. Return on Assets =
(ROA)
6. Return on Equity =
(ROE)
NI
Avg. Total Assets
or
NI + Interest Expense (1-t)
Avg. Total Assets
NI
Avg. Total Equity
7. Operating Return on Assets = Operating Income
Avg. Total Assets
8. Return on Total Capital =
or
EBIT
Avg. Total Capital
EBIT
Avg. Total Capital
9. Return on Common Equity = NI - Preferred Dividends
Avg. Common Equity
E | Valuation Ratios
1. g = RR x ROE
↳
Retention Rate = NI to Common Shareholders - Dividends Declared
NI to Common Shareholders
= 1 - Dividend Payout Ratios
= 1 - DPS
EPS
2. Dividend Payout Ratio =
3. Earnings Per Share =
Dividends Declared
NI to Common Shareholders
or
NI - Preferred Dividends
Weighted Avg. No. of Ordinary Shares O/S
Dividends Declared
NI - Preferred Dividends
62
F | Cashflow Ratios
1. Interest Coverage = CFO + Interest Paid + Taxes Paid
Interest Paid
1
Equity
PAGE NOS. 67
CHAPTER 24
VOL. 6
CHAPTERS. 8
Equity Valuation: Applications and Processes
Intrinsic Value (IV) refers to the valuation of an asset or security by someone who has complete
understanding of the characteristics of the asset or issuing firm. To the extent that stock prices are not
perfectly (informationally) effective, they may diverge from the intrinsic values.
IVanalyst - Price = (IV actual - Price) + (IV analyst - IV actual )
①
②
Actual Mispricing
Valuation Error
This attributes to the
abnormal returns i.e. Alpha
The error in the estimate
of the intrinsic value
Equity Valuation Process
1. Understand the Business
2. Forecast Company Performance
3. Select the Appropriate Valuation Model
4. Convert the Forecasts into a Valuation
5. Apply the Valuation Conclusions
Absolute Valuation Model: is a model that specifies an asset's intrinsic value, which arises from its investment
characteristics without regard to the value of other firms. In theory, present value models are considered the
fundamental approach to equity valuation, e.g. DDM, DCF, FCF (FCFE and FCFF), RI (accrual accounting) and
Asset Based Valuation (values a company on the basis of the market value of the assets or resources it
controls).
Relative Valuation Model: estimates an asset's value relative to that of another asset. It is also a type of going
concern valuation models. The application of relative valuation to equity is often called the method of
comparables. The method of comparables has the advantages of being simple, related to market prices and
grounded in sound economic principle i.e. similar assets should sell at similar prices e.g. EPS, P/E and
Enterprise Multiples.
NOTES
1. Conglomerate Discount is based on the idea that investors apply a markdown to the value of a company
that operates multiple unrelated business or industries, compared to the value a company that has a single
industry focus. Conglomerate Discount is thus the amount by which market value under represents
sum-of-the-parts value. Three Explanations for Conglomerate Discounts are: (i) Internal Capital Inefficiency:
The company's allocation of capital to different divisions may not have been based on sound decisions,
2
(ii) Endogenous Internal Factors: For example, the company may have pursued unrelated business
acquisitions to hide poor operating performance and (iii) Research Measurement Errors: Some hypothesize
that conglomerate discounts don't exist, but rather are a result of incorrect measurement.
Company A
Division 1
$ 60,000,000
Division 2
$ 30,000,000
Division 3
$ 10,000,000
$ 100,000,000
Hence, Conglomerate Discount is 10%. Now they trade at a discount $ 90,000,000 due to inefficiency.
2. Blockage Factor is the discounted price or value the market gives stocks when a block of shares is sold. A
block is characterized by $200,000 worth or more of securities.
3
CHAPTER 25
Return Concepts
Holding Period Return: is the return earned from investing in an asset over a specified time period.
HPR = P1 - P0 + D1 = D1 + P1 - P0
P0
P0
P0
Cashflow Yield or
Dividend Yield
Price Appreciation Return
or Capital Gains Yield
In most cases, HPR is annualized. For example, if the return for one month is 1% or 0.01, then the analyst
12
might report an annualized HPR of (1 + 0.01) - 1 = 0.1268 or 12.68%.
Required Return: is the minimum level of expected return that an investor requires in order to invest in the
asset over a specified time period, given the asset's riskiness. It also represents the
opportunity cost for investing in the asset; the highest level of expected return available
elsewhere from investments of similar risk.
RR = R f + β (R m - R f )
Risk Premium
Expected Return: is based on forecasts of future prices and cashflows. Such expected returns can be derived
from elaborate models or subjective opinions.
'L ' Abnormal Return
ER = RR + (V0 - P0 )
P0
Active investors essentially 'second guess' the market price. The risks of that activity include the risks that (i)
their value estimates are not more accurate than the market price and that (ii) even if they are more accurate,
the value discrepancy may not narrow over the investor's time horizon. (iii) It is possible that there are
chronic inefficiencies that impede price convergence. Therefore, even if an analyst feels that V0 ≠ P0 for a given
asset, the convergence yield may not be realized.
Fig 1: Required Return vs. Expected Return
A
B
C
RR
15 %
13.4 %
16.6 %
.
.
Undervalued
B
ER
12 %
17.5 %
16.6 %
13.4
.
.
.
Fairly Valued
C
16.6
15
A
Overvalued
0
Example 1:
The Novo-Gemini shares were purchased for $20.75 per share. At the time of purchase, research by Chen
suggested that Novo-Gemini shares were expected to sell for $29 per share at the end of a 3-year holding
period. Exactly 3 years after the purchase date, the shares were sold for $30.05 per share. No dividends were
4
paid by Novo-Gemini. What is the (a) Expected 3-year HPR and (b) Realized 3-year HPR?
a. Expected HPR = 29 - 20.75 = 39.76 %
20.75
b. Realized HPR = 30.05 - 20.75 = 44.82%
20.75
Example 2:
In May 2013, the current price for Microsoft was $33.31. A 1-year target price for Microsoft is $37.5. The most
recent quarterly dividend was $0.23 per share. Over the coming year, two more quarterly dividends of $0.23
per share are expected, followed by two quarterly dividends of $0.25 per share. Microsoft's required return
on equity is 7%.
a. What is the analyst's 1-year expected return?
Dividends = $ 0.23 + $ 0.23 + $ 0.25 + $ 0.25 = $ 0.96
ER = 0.96 + (37.5 - 33.31) = 0.029 + 0.126 = 0.155 or 15.5%
33.31
33.31
b. What is a target price that is most consistent with Microsoft being fairly valued?
= RR - Dividend Yield
= 7% - 2.9%
= 4.1%
Thus, (1.041)(33.31) = $ 34.68
lllll
Equity Risk Premium
The Equity Risk Premium (ERP) is the incremental return (premium) that investors require for holding
equities rather than Rf . The ERP like the required return depends strictly on expectations for the future
because the investor's returns depend only on the investment's future cashflow.
ERP = Required Return on Equity - R f
lllll
Historical Estimates
A Historical ERP estimate is usually calculated as the mean value of the differences between
broad-based equity market index returns and government debt returns over some selected sample
period. If investors don't make systematic errors in forming expectations, then over long-term,
average returns should be unbiased estimate of what investors expected. A weakness of the
approach is the assumption that the mean and variance of the returns are constant i.e. they are
stationary. In fact, the premium actually appears to be counter-cyclical i.e. it is low during good
times and high during bad times. Thus, an analyst using this method to estimate the current equity
premium must choose the sample period carefully.
Historical Estimates
Arithmetic Mean
An Arithmetic Mean ERP estimate
equals to the sum of the annual
return differences divided by the
no. of observations in the sample
Geometric Mean
A Geometric Mean ERP estimate
equals to the compound annual
excess return on equities over the
R f return
5
Geometric Mean ≤ Arithmetic Mean, if the yield curve is upward sloping, the use of longer term
bonds rather than shorter term bonds to estimate the R f rate will cause the estimated risk premium
to be smaller. The arithmetic mean return as the average one-period return best represents the
mean return in a single period. Whereas, the geometric mean return of a sample represents the
compound rate of growth that equates the beginning value to the ending value of one unit of
money initially invested in an asset. Discounting is just the reverse side of compounding in terms of
finding amounts of equivalent worth at different points in time, because the geometric mean is a
compound growth rate, it appears to be a logical choice for estimating a required return in a
multiperiod context, even when using a single period required return model. In contrast to the
sample arithmetic mean, using the sample geometric mean doesn't introduce bias in the calculated
expected terminal value of an investment. ERP estimates based on the geometric mean have
tended to be closer to supply-side and demand-size estimates from economic theory than
arithmetic mean estimates. For the above reasons, the geometric mean is increasingly preferred for
use in historical estimates of ERP. Survivorship Bias i.e. the bias arises when poorly performing or
defunct companies are removed from membership in an index, so that only relative winners
remain. One type of adjustment is made to offset the effect of biases in the data series being used
to estimate the ERP. A second type of adjustment is made to take account of an independent
estimate of ERP. In both cases, the adjustment could be upward or downward.
lllll
Forward Looking Estimates
Forward Looking or Ex-ante Estimates are likely to be less subject to an issue such as non-stationary
or data biases than historical estimates i.e. no survivorship bias. However, such estimates are often
subject to other potential errors related to financial and economic models and potential behavioral
issues or biases in forecasting.
Gordon Growth Model
The Constant Growth Model a.k.a. Gordon Growth Model is a popular method to generate forward
looking estimates. The assumptions of the model are reasonable when applied to developed
economies and markets.
(D1 / P)
g
Rf
GGM ERP = 1-year Forecasted + Consensus Long-term - Long-term Government
Dividend Yield on
Earnings Growth Rate
Bond Yield
Market Index
(Earnings growth rate based
on top-down forecasts a.k.a.
supply-side estimates)
V0 = D1
r-g
Assume market is fairly valued
V0 = P0
R m or r = D1 + g
P0
R m- R f = D1 + g - R f
P0
ERP = R m - R f
A weakness of the approach is that the forward looking estimates will change through time and
need to be updated. During a typical economic term, dividend yields are low and growth
expectations are high, while the opposite is generally true when the economy is less robust.
6
DY
2%
4%
Boom
Bust
g
6%
3%
LT Bond Yield
6%
3%
Another weakness is the assumption of a stable growth rate, which is often not appropriate in
rapidly growing economies. Such economies might have 3 or more stages of growth: the 'r' required
return on equity is derived from the IRR.
Equity Index Price = Pv Rapid Growth (r) + Pv Transition (r) + Pv Mature (r)
Whereas,
Pv Rapid Growth : Pv of projected cashflows during rapid growth stage IRR
Pv Transition : Pv of projected cashflows during transitional growth stage IRR
Pv Mature : Pv of projected cashflows during mature growth stage IRR
Using the IRR as an estimate of the required return on equities, substracting a government bond
yield 'R f ' gives an ERP estimate.
Macroeconomic Model
Macroeconomic Model estimates of the ERP are based on the relationships between
macroeconomic variables and financial variables. A strength of this approach is the use of proven
models and current information. Such models may be more reliable when public equities represent
a relatively large share of the economy as in many developed markets. Many such analysis focuses
on the supply-side variables that fuel GDP growth and are thus known as supply-side estimates.
Required Return
Rm
Capital Gains
Nominal EPS
[(1 + g)(1 + E(I))]
Real
EPS
Expected
Inflation
Income (Dividends)
P/E
(1 + PEG)
%
D 1 / P0
△ in P/E
Ibbotson-Chen ERP =
^
^
^
^
^
D1 + [(1 + g)(1 + E(I))(1 + PEG)] - 1 - R f
P0
Whereas,
^
D1 / P0 : Expected income component including return from reinvestment of income.
g^ : Expected real growth in EPC should be approximately equal to the real GDP 'g'
g^ = Real GDP growth
g^ = Labor Productivity Growth Rate + Labor Supply Growth Rate
g^ = Population growth rate +
Labor Force Participation Rate
^
E(I) : Expected inflation can be derived from the differences in the yields for T-bonds
and Treasury Inflation Protected Securities (TIPS) having comparable
maturities.
^
E(I) = 1 + YTM of 20 year T-Bonds - 1
1 + YTM of 20 year TIPS
^
^
PEG : Expected changes in P/E ratio. Overvalued
: PEG < 0
^
Undervalued
: PEG > 0
^
Correctly Priced : PEG = 0
↑△
(
)
7
Survey Estimates
Survey Estimates of the ERP uses the consensus of the opinions from a sample of people. If the
sample is restricted to people who are experts in the area of equity valuation, the results are likely
to be more reliable.
In Macroeconomic Factor Models, the factor are economic variables that effect the expected future
cashflows of companies and/or the discount rate that is appropriate to determine their present values.
In Statistical Factor Models, statistical models are applied to historical returns to determine portfolios
of securities that explain those returns on various senses.
lllll
Required Return on Equity
1. Capital Asset Pricing Model: The CAPM is an equation for required return that should hold in equilibrium
i.e. supply equals demand, if the model's assumptions are met, among the key assumptions are that
investors are risk averse and make investment decisions based on their total portfolio. Investors evaluate
the risk of an asset in terms of the assets contribution to the systematic risk of their total portfolio. If a
stock trades in more than one market, this can lead to more than one estimate of required return.
CAPM = R f + β (R m - R f )
2. Arbitrage Pricing Theory: Multifactor models have greater explanatory power than CAPM, but this is not
assured. Multifactor models (APT) have the weakness of being more complex and expensive. They include
FFM, PSM and Macroeconomic multifactor models.
APT = R f + β1 (Risk Premium)1 + β2 (Risk Premium) 2+ ... + βn (Risk Premium) n
These risk premiums have '0' sensitivity to all other factors.
3. Fama-French Model: Is a multifactor model that attempts to account for the higher returns generally
associated with small-cap stocks. Many developed economic and markets have sufficient data for
estimating the model.
FFM = R f + β market (R m - R f ) + β size (R small - R big ) + β value (R HBM - R LBM )
CAPM
Small company is risker
than large company
Value stocks are riskier
than growth stocks
Return of > Return of
Value
Growth
Stock
Stock
(Bv/Price ) (Bv/Price )
↑
↓
Whereas,
R m- R f : Return on a value weighted market index minus R f .
x<1
x>1
Baseline for β market = 1; not risky
more risky
1
R small - R big : A small-cap return premium equal to the average return on small-cap portfolios
minus the average return on large-cap portfolios.
- ve
+ ve
Baseline for β size = 0; large company
small company
0
R HBM - R LBM : A value return premium equal to the average return on high book-to-market
portfolios minus the average return on low book-to-market portfolio.
- ve
+ ve
Baseline for β value = 0;
LBM
HBM
0
8
The FFM views the returns premiums to small size and value as compensation for bearing types of
systematic risk. Many practitioners and researchers believe, however that those return premiums arise
from market inefficiencies rather than compensation for systematic risk.
4. Pastor-Stambaugh Model: Adds a liquidity factor to the FFM. This model has been applied to public security
investment as well as certain private security investments.
PSM = R f + β market (R m - R f ) + β size (R small - R big ) + β value(R HBM - R LBM ) + β Liquidity (R LLB - R HLS )
Whereas,
R LLB - R HLS : A liquidity return premium equal to average return on low liquidity stock minus
average return on high liquidity stock.
- ve
+ ve
Baseline for β Liquidity = 0;
High Liquidity
Low Liquidity
0
The concept of liquidity may be distinguished from marketability. Liquidity relates to the case and potential
price impact of the sale of an equity interest into the market. Liquidity is a function of several factors
including the size of the interest and the depth and breadth of the market and its ability to absorb a block
i.e. a large position without an adverse price impact. In the strictest sense, marketability related to the right
to sell an asset.
5. Burmeister, Roll and Ross Model: Macroeconomic multifactor model use factors associated with economic
variables that can be reasonably believed to affect cashflows and/or appropriate discount rates.
BIRR = R f + β confidence (Confidence Risk) + β time (Time Horizon Risk) + β inflation (Inflation Risk)
+ β business (Business Cycle Risk) + β market (Market Timing Risk) Residual
Income
↶
Whereas,
Confidence Risk : Unexpected change in the difference between the return of risky corporate
bonds and government bonds with similar maturities.
Time Horizon Risk : Unexpected change in the return difference between 20-year government
bonds and 30-day Treasury bills. This factor reflects investor's willingness to
invest for the long term.
Inflation Risk : Unexpected change in inflation rate. Nearly all stocks have negative
exposure to this factor since their returns decline with positive surprises in
inflation.
Business Cycle Risk : Unexpected change in the level of real business activity.
Market Timing Risk : The equity market return that is most explained by the other four factors.
6. Build-Up Method: The Build-Up method parallels the risk premium approach embodied in multifactor
models with the difference that specific beta adjustments are not applied to factor risk premiums. It is
typically used for valuation of private businesses where betas are not readily obtainable. It doesn't account
for systematic risk.
BUM = R f + (R m- R f ) + Size Premium + Specific Company Premium
Whereas,
Size Premium : The size premium would be scaled up or down based on the size of the
company. Smaller companies would have a larger premium.
The formula could have a factor for the level of controlling vs. minority interests and a factor for
marketability of the equity; however, these latter two factors are usually used to adjust the value of the
9
company directly rather than through the required return. The weakness with build-up model is that they
typically use historical values as estimates that may or may not be relevant to current market conditions.
Some representations use an estimated beta to scale the size of the company-specific ERP but typically not
for the other factors.
7. Bond Yield plus Risk Premium Method: It is a build-up method that is appropriate if the company has
publicly traded debt.
BYRP = YTM + Risk Premium
YTM of the company's bonds includes the
effects of inflation, leverage and the firm's
sensitivity to the business cycle.
lllll
Beta
The simplest estimate of beta results from an ordinary least squares regression of the return on the
stock on the return on the market. The result in often called an unadjusted or raw historical beta. When
making forecasts of the ERP, some analysis recommended adjusting the beta for 'Beta Drift'. Beta Drift
refers to the observed tendency of an estimated beta to revert to a value of 1 over time. The Blume
Method can be used to adjust the beta estimate.
Adjusted Beta = (2/3 x Regression Beta) + (1/3 x 1)
or Unadjusted Beta
or Historical Estimated Beta
Note: Derivation of Beta
Σ Correlation (R i , R m ) x o i
o m
I
or
I
I
β = Cov. (R i , R m )
o R2 m
↶
PUBLIC :
β E = βA x
↶
Fig 2: Beta for Private Company
β P = βE x
Unlevered Beta
NON-PUBLIC :
Beta Portfolio
1
1+D
E
( )
(1 + DE )
International Investment, if not hedged exposes the investor to exchange rate risk. To compensate for
anticipated changes in exchange rates, an analyst should compute the required return in the home
currency and then adjust it using forecasts for changes in the relevant exchange rate.
A. Country Spread Model: ERP for Developed Market + Country Premium
The country premium represents a premium associated with
the expected greater risk of the emerging market compared to
the benchmark developed market. The country premium is
often estimated as the yield on emerging market bonds
(denominated in the currency of the developed market) minus
the yield on developed market government bonds.
10
B. Country Risk Rating Model: provides a regression-based estimate of
the ERP based on the empirical relationship between developed
equity market returns and institutional investor's semi-annual risk
ratings for those markets. The model is then used for predicting the
ERP for emerging markets using the emerging markets risk-ratings.
The cost of capital is the overall required rate of return for those who supply a company with capital.
The suppliers of capital are equity investors and those who lend money to the company. Weighted
Average Cost of Capital (WACC) is appropriate for valuing a total firm.
WACC = rd . wd . (1 - t) + re . we
$ WACC = Mv. of Equity + Mv. of Debt
Fig 3: Choosing R f
ST
↓, ERP ↑
T-Bill
T-Bond
Fama-French
Pastor Stambaugh
5-Factor BIRR
GGM
Ibbotson-Chen
(ST)
(LT)
LT
↑, ERP ↓
11
CHAPTER 26
Industry and Company Analysis
*
Top-Down Approaches to modeling revenue: (i) Growth relative to GDP Growth: approach is the relationship
between GDP and company sales. For example, if we forecast that GDP will grow at 5% and we believe that
our company's revenue will grow at a 20% faster rate, then our forecast of increase in company revenue
would be: GDP growth % (1 + x%) = 5% (1 + 0.2) = 6% and (ii) Market Growth and Market Share: begins with an
estimate of Industry Sales (market growth) and then company revenue is estimated as a percentage of
Industry Sales (market share). Market Share x Estimated Industry Sales provides the estimate of company
revenues. Bottom-Up Approaches to modeling revenue: (i) Time-Series Analysis, (ii) Return on Capital (ROC)
and (iii) Capacity-based Measures.
*
Variable Costs are directly linked to revenue growth and Fixed Costs are not directly related to revenue,
rather related to future investment in PPE and total capacity growth. If the average cost of production
decreases as industry sales increase, we say that industry exhibits Economies of Scale. Economies of Scale in
GDP are evidenced by lower COGS (similarly lower SG&A) as a proportion of sales for larger companies.
Factors that can lead to economies of scale include, higher levels of production, greater bargaining power
with suppliers, lower cost of capital and lower per unit advertising expenses. Gross and operating margins
tend to be positively correlated with sales levels in an industry that enjoys economies of scale.
Example 1:
Candidate A: '2013 SG&A/Net Sales Ratio will be the same as the average ratio over the 2010-2012 time
period'.
Net Sales
SG&A
2010
46.8
19.3
2011
50.5
22.5
2012
53.9
25.1
Average SG&A/Net Sales (2010-2012) = 41.24% + 44.55% + 46.57% = 44.12%
3
Average Annual Growth Sales in Net Sales (2010-2012) = 7.91% + 6.73% = 7.32%
2
0.4412 =
*
SG&A
; SG&A = $ 25.52
53.9 (1 + 0.0732)
Future COGS is usually estimated as a percentage of future revenue i.e. a closer examination of the volume
and price of a firm's inputs may improve the quality of a forecast of COGS, especially in the short-run.
Forecast COGS = Historical COGS x Estimate of Future Revenue
Revenue
= (1 - Gross Margin) (Estimate of Future Revenue)
Gross Margin differences among companies within a sector should logically relate to differences in their
business operations. However, when differences in firm's business models are accounted for, any remaining
difference should be investigated.
12
Example 2:
French notes that for the year just ended (2014). Archway's COGS was 30% of sales. To forecast Archway's
income statement for 2015, French assumes that all companies in the industry will experience an inflation
rate of 8% on the COGS.
Avg. price increase per unit
Volume growth
5%
- 3%
Let's assume initial volume 10 pcs and initial price $1.
Volume
x
Price
10 x (1 - 0.03) x $1 (1.05)
10.185 (0.3)
Sales
COGS
10.185
(3.055)
7.1295
GP Margin = GP = 7.1295 = 0.7 or 70%
Sales 10.185
Example 3:
The price of an imported specialty metal used for engine parts increases by 20%. This metal constitutes 4% of
cost of sales. The company will not be able to pass on the higher metal expense to its customers. What's the
decrease in 2015 base gross profit margin?
Sales
COGS
152.38
(105.38)
47
COGS x 0.04 x 0.2 = 105.38 x 0.04 x 0.2 = 0.0055
Sales
152.39
*
Compared to COGS, SG&A operating expenses are less sensitive to changes in sales volume; SG&A's fixed
cost component is generally greater than its variable cost component. Expenses for corporate headquarters,
management salaries and IT operations are examples for fixed costs. R&D expense is another example of an
expense that tends to fluctuate less than sales. These costs tend to increase or decrease gradually over time
rather than being driven by changes in firm sales in the current period. Selling and distribution costs, on the
other hand, may be more directly related to sales volume, because it is likely that more sales people will be
hired to support higher firm sales.
*
Financing Cost (Interest Expense): The financial structure of a company includes both debt and equity
financing. The primary determinants of gross interest expense are the levels of (gross) debt and market
interest rates. Analysts should also use any planned debt issuance or retirement and the maturity structure
of existing debt (disclosed in footnotes to financial statements) to improve the forecasts of future financing
costs.
Net Debt = Gross Debt - Cash & Cash Equivalents - Short-term Securities
Net Interest Expense = Gross Interest Expense - Interest Income on Cash and
Short-term Debt Securities
13
Example 4:
Atwood Inc. a small manufacturer of knobs and switches, provided the following information:
2011
3200
800
2400
Gross Debt
Cash + Short-term Securities
Net Debt
Gross Interest Expense
Interest Income
Net Interest Expense
2012
3600
700
2900
Avg.
3400
750
2650
2012
220
8
212
Calculate Atwood's 2012 interest expense on average gross, average net debt and the yield on average cash
balances.
Net Interest Expense = 212 = 8%
2650
*
Gross Interest Expense = 220 = 6.47%
3400
Interest Income = 3 = 1.07%
750
Statutory Tax Rate is the percentage tax charged in the country where the firm is domiciled.
Effective Tax Rate is the reported tax amount on the income statement divided by the pre-tax income.
Cash Tax Rate is calculated as cash tax divided by pre-tax income.
Differences between cash taxes and reported taxes typically result from differences between accounting and
tax calculations; and are reflected as a DTA or DTL. The Statutory and Effective tax rates may differ because
there are expenses recognized in the income statement that are not deductible for tax purposes (a
permanent difference). If a company reports a high profit in a country with a high tax rate and a low profit in
a country with a low tax rate, the Effective tax rate will be weighted average of the rates and higher than the
simple average tax rate of both countries. An analyst should pay special attention to estimates of tax rates for
companies that consistently report an effective tax rate that is less than the Statutory rate (or consistently
less than that of comparable peer companies). If the income from equity method investees is a substantial
part of pre-tax income and also a volatile component of it, the Effective tax rate excluding this amount is likely
to be a better estimate for the future tax costs for a company. A good indication for the future tax expense is
a tax rate based on normalized operating income, before the results from associate and special items.
*
Forecasts may also be improved by analyzing capital expenditures for maintenance separately from capital
expenditures for growth. Maintenance capital expenditures, which are necessary to sustain the current
business and growth capital expenditures, which are needed to expand the business. Historical depreciation
should be increased by the inflation rate when estimating capital expenditures for maintenance because
replacement cost can be expected to increase with inflation. Maintenance capital expenditure forecasts
should normally be higher than depreciation because of inflation.
*
Return on Invested Capital (ROIC) is a return to both equity and debt. ROIC is a better measure of profitability
than ROE because it is not affected by a company's degree of financial leverage. Firms with higher ROIC
relative to their peers are likely exploiting some competitive advantage in the production and/or sale of their
products.
ROIC =
(Earnings before Interest Expense)↷
NOPLAT
= Net Operating Profit - Adjusted Taxes
Invested Capital Operating Assets - Operating Liabilities
14
Example 5:
Sophie Moreau, a buy-side analyst, is analyzing a French manufacturing company. Working capital and PPE
account for almost all of the company's assets. Moreau believes that the depreciation schedule used by the
company is not reflective of economic reality. Rather, she expects PPE to last twice as long as what is implied
by the depreciation schedule and as such she projects capita expenditure to be significantly less than
depreciation for the next 5 years. Moreau projects that both earnings and net working capital will grow at a
low single digit rate during this time. What do Moreau's assumptions most likely imply for returns on invested
capital during the next 5 years?
Earnings , Working Capital
i.e. RPIC is stable but since net PPE
A. ROIC will increase
is expected to decline because depreciation is expected to exceed
B. ROIC will decrease
capes. Total invested capital will thus grow ,pre slowly than
C. ROIC will stay the same
earnings or even shrink.
✓
*
↑
↑
Return on Capital Employed (ROCE) is similar to ROIC but uses pre-tax operating earnings i.e. it is essentially
ROIC before tax, in the numerator to facilitate comparison between companies that face different tax rates.
ROCE = Operating Profit = Operating Profit
Capital Employed
Debt + Equity
*
If intrinsic value differs markedly from market price, a common approach is for analysts to value such
companies by using probability-weighted average of the various scenarios, instead of using basic sensitivity
or scenario analysis.
*
Companies that are vertically integrated and are in effect their own suppliers will be less subject to the effects
of variations in input prices.
*
The impact of higher prices on volume depends on the price elasticity of demand. If demand is relatively price
inelastic, revenues will benefit from inflation. If demand is relatively price elastic, a company's efforts to pass
on inflation through higher prices can have a negative impact on volume.
*
Inflation: The first competitor to lower prices will usually benefit with an upside in volume, however resulting
in a short-term benefit. Firms that are too quick to increase prices will experience declining sale volumes,
whereas firms that are slow to increase prices will experience declining gross margins. Deflation: Lowering
prices too soon will result in a lower gross margin but waiting too long will result in volume losses.
*
Cannibalization Factor is the percentage of new product sales that will replace existing product sales. This
cannibalization factor can be different for different sales channels and is likely to be lower for business
customers than for direct purchases by consumers.
Cannibalization Rate = New Product Sales that Replace Existing Product Sales
or Externalities
Scenario or Sensitivity Analysis of New
Total New Product Sales
Product Offerings can be informative
Variable Cost (%) = %
△ (Cost of Revenue + Operating Expenses) = % △ Total Operating Expenses
% △ Revenue
% △ Revenue
Fixed Cost (%) = 1 - Variable Cost (%)
*
A stated investment time horizon of 4 years would imply average annual portfolio turnover of 25%. The
average holding period is calculated as 1 / Portfolio Turnover. Longer-term projections often provide a better
representation of the normalized earnings potential of a company than a short-term forecast, especially
when certain temporary factors are present. Normalized Earnings are adjusted to remove the effects of
15
seasonality and revenue & expenses that are unusual or one-time influences. It also helps to understand a
company's true earnings from its normal operations. Normalized free cashflow can be defined as the
expected level of mid-cycle cashflow from operations adjusted for unusual items (due to events such as M&A
and restructurings) less recurring capital expenditures.
*
Assuming that the growth in future profitability will be the same as average profitability growth in the past,
may not be justified. A difficult part of an analyst's job is recognizing inflection points i.e. those instances
when the future will not be like the past. Inflection Points occur due to changes in overall economic
environment like business cycle stage, government regulations or technology.
Fig 1: Porter's 5 Forces
Pricing Power
1
Threat of Substitute Products
Switching Costs
2
Intensity of Industry Rivalry
3
Bargaining Power of Suppliers
4
Bargaining Power of Buyers
5
Threat of New Entrants
HIGH
LOW
High Barriers
to Entry
Low Barriers
to Entry
↓
↑
↓
↓
↓
↓
↑
↓
↑
↑
↑
↑
Product
No Product
Differentiation Differentiation
Highly
Concentrated
Fixed Costs
↓
Fragmented
Industry
Fixed Costs
↑
16
CHAPTER 27
Discounted Dividend Valuation
The Dividend Discount Model (DDM) defines cashflows as dividends. Reinvested earnings should provide the
basis for increased future dividends. Therefore, the DDM accounts for reinvested earnings when it takes all
future dividends into account. Because dividends are less volatile than earnings; relative stability of dividends
may make DDM values less sensitive to short run fluctuations. DDM reflects the long-term earning potential
of the company. DDM is appropriate only for a company that has a history of dividend payments. The
disadvantage of measuring cashflow with dividends is that it takes the perspective of an investor who owns a
minority stake in the firm and cannot control the dividend policy. If the dividend policy dictated by the
controlling interests bears a meaningful relationship to the firm's underlying profitability, then dividends are
appropriate. However, if the dividend policy is not related to the firm's ability to create value, then dividends
are not an appropriate measure of expected future cashflow to shareholders. DDM is usually for mature
companies.
EPS
DPS - A Co.
B Co.
2001
2002
2003
2004
2005
2006
7.5
1
0.16
8.25
1
0.21
9.04
1
0.28
9.95
1
0.37
10
1
0.51
10.95
1
0.6
Applying DDM to A Co. is inappropriate because dividends don't appear to adjust to reflect changes in
profitability. Because B Co. is dividend-paying and dividends bear an understandable and consistent
relationship to earnings, using DDM to value B Co. is appropriate.
V0 =
D1 + P1 = D1 + P1
(1 + r)1 (1 + r)1 (1 + r)1
Two Period DDM:
V0 =
D 1 + D 2 + P2 = D 1 + D 2 + P2
(1 + r)1 (1 + r) 2 (1 + r) 2 (1 + r)1 (1 + r) 2
∞
Infinite Dividends: V0 = Σ
Dn
n
(1 + r)
t=1
lllll
Selling
D1 + D 2 + D 3 + ..... + D n + Pn Price
(1 + r) 1 (1 + r) 2 (1 + r) 3
(1 + r) n
↶
Multi Period DDM: V0 =
↷
One Period DDM:
Required
Return on
Equity
Gordon Growth Model
The Gordon Growth Model assumes that dividends grow indefinitely at a constant rate. It is
unrealistic to assume that any firm can continue to grow indefinitely at a rate higher than the
long-term growth rate in real GDP plus the long-term inflation rate. In general, a perpetual dividend
growth rate forecast above 5% is suspect. If the growth rate in dividends is too high, then it is best
replaced by a growth rate closer to that of GDP.
D1
D2
D3
Dn
1
2
3
n
P0 = D 0 (1 + g) + D0 (1 + g) + D 0 (1 + g) + ..... + D 0 (1 + g)
(1 + r) 1
(1 + r) 2
(1 + r) 3
(1 + r) n
(i) Pv of Perpetuity = P0 = D 1
r
and
Value of perpetual
preferred shares = Dp / rp
since growth rate is '0'
(ii) Infinite Growth of Dividends = P0 = D0 (1 + g) = D1
r-g
r-g ↷
Capital
Appreciation
or Capital
Gains Yield
17
The Present Value of Growth Opportunities
The value of a stock can be analyzed as the sum of (a) the value of the company without earnings
reinvestment and (b) the present value of growth opportunities (PVGO) also known as the value of growth,
since the expected value today of opportunities to profitably reinvest future earnings. Companies that have
good business opportunities and/or a high level of managerial flexibility in responding to changes in the
marketplace should tend to have higher values of PVGO than companies don't have such advantages. A firm
that has additional opportunities to earn returns in excess of the required rate of return would benefit from
retained earnings and investing in those growth opportunities rather than paying out dividends.
D1 = E 1
If a company has a payout ratio of 100%
No Reinvestment, g = 0
(a) V0 = E1 ; Value of a no growth company, g = 0
r
(b) V0 > E 1 ; If a company is expected to grow, g > 0
r
①
E
10
6
A
B
CMP
140
65
No Growth Value
100
60
↷
P0 = E 1 + PVGO
r
PVGO
40
5
Firms leading P/E ratio
attributable to PVGO
(PVGO/P)
PVGO % of Price
28.5%
7.69%
Company A is growing faster than Company B, assuming market is fairly valued.
②
P0 = 1 + PVGO
E1 r
E1
Growth P/E
40/10 = 4x
5/6 = 0.83x
P/E
140/10 = 14x
65/6 = 10.83x
Growth
Premium
A
B
No Growth P/E
1/0.1 = 10x
1/0.1 = 10x
Therefore,
↑ P/E, ↑ Growth. If ↑ P/E and No Growth i.e. It's overvalued.
[10x + 4x]
[10x + 0.83x]
A substantial portion of the value of growth companies is in their PVGO. In contrast, companies in slow
growth industries (e.g. utilities) have low PVGO and most of their value comes from their assets in place. What
determines PVGO? One determinant if the value of a company's options to invest, captured by the word
'opportunities'. Thus, a second determinant of PVGO is the value of the company's options to modify projects.
Share repurchases are generally harder to forecast than the cash dividends of companies with an
identifiable dividend policy. If correctly applied, the DDM is a valid approach to common stock
valuation even when the company being analyzed engages in share repurchases. Valuations are
very sensitive to estimates of growth rates and required rate of return (sensitive to small changes
in 'r' and 'g').
lllll
Two Stage DDM
The Two Stage fixed growth rate model is based on the assumption that the firm will enjoy an initial
period of high growth, followed by a mature or stable period in which growth will be lower but
18
sustainable. A possible limitation of the two stage model is that the transaction between the initial
abnormal growth period and the final steady state growth period is abrupt.
g
15%
3%
0
STAGE 1 4 Yrs
Time
STAGE 2
V0 = D0 (1 + g S )1 + D0 (1 + g S ) 2 + D0 (1 + g S ) 3 + ..... + D 0 (1 + g S ) n (1 + g L )
(1 + r)1
(1 + r) 2
(1 + r) 3
(1 + r) n (r - g L )
Whereas,
g S : Short-term growth rate
g L : Long-term growth rate
n
D0 (1 + g S ) (1 + g L ) : Terminal Value
(r - g L )
Example 1:
Sea Island Recreation currently pays a dividend of $1. An analyst forecasts growth of 10% for the next 3 years,
followed by 4% growth in perpetuity thereafter. The required return is 12%. Calculate the current value per
share.
Option 1:
0
1.1
1.21
1.33
1.38
1
2
3
4
5
V0 = 1 (1.1)1 + 1 (1.1) 2 + 1 (1.1)3 +
1 (1.1) 3 (1.04)
(1.12)1 (1.12) 2 (1.12) 3 (1.12) 3 (0.12 - 0.04)
V0 = 1.1 + 1.21 + 1.33 +
1.38
= $ 15.21
1
2
3
3
(1.12) (1.12)
(1.12) (1.12) (0.12 - 0.04)
Option 2:
0
1.1
1.21
17.3 + 1.33
1
2
3
V3 =
4
D4 = 1.3842 = $ 17.3
(r - g L ) 0.12 - 0.04
V0 = 1 (1.1)1 + 1 (1.1) 2 + 1 (1.1)3 + 17.3
(1.12)1 (1.12) 2
(1.12) 3
V0 = 1.1 + 1.21 + 1.33 + 17.3 = $ 15.21
(1.12)1 (1.12) 2
(1.12) 3
5
19
Example 2:
An analyst is reviewing the valuation of DuPont as of the beginning of July 2013 when DuPont is selling for
$52.72. In the previous year, DuPont paid a $1.7 dividend that the analyst expects to grow at a rate of 4%
annually for the next 4 years. At the end of Year 4, the analyst expects the dividend to equal 35% of EPS and
the trailing P/E for DuPont to be 13. If the required return on DuPont common stock is 9%, calculate the per
share value of DuPont common stock.
D0 = 1.7
D1 = 1.7 (1.04) = 1.768
D2 = 1.768 (1.04) = 1.8387
D3 = 1.8387 (1.04) = 1.9123
D4 = 1.9123 (1.04) = 1.9888
EPS 4 = Earnings = 1.9888 = 5.6822
DPS
0.35
Trailing P/E = EPS x P/E
= 5.6822 x 13
V4 = 73.8682
V0 = 1.768 + 1.8387 + 1.9123 + 1.9888 + 73.8682 = $ 58.3852
(1.09)1 (1.09) 2
(1.09)3
(1.09) 4
Example 3:
Arena Distributors is a new company and currently pays no dividends. The company recently reported
earnings of $1.5 per share and is expected to grow at a 15% rate for the next 4 years. Beginning in Year 5,
Arena is expected to distribute 20% of its earnings in the form of dividends and to have a constant growth
rate of 5%. The required rate of return is 12%. Calculate the value of Arena shares today.
E 0 = 1.5
E1 = 1.5 (1.15) = 1.725
E2 = 1.725 (1.15) = 1.98375
E3 = 1.98375 (1.15) = 2.28131
E4 = 2.28131 (1.15) = 2.6235
V4 =
0.55
= $ 7.86
0.12 - 0.05
V0 = 7.86 = $ 5
(1.12)4
D4 = 2.6235 x 0.2 = 0.5247
D5 = 0.5247 (1.05) = 0.5509
lllll
H-Model
The H-Model utilizes a more realistic assumption: the growth rate starts out high and then declines
linearly over the high-growth stage until it reaches the long-run average growth rate.
g
gS
15%
Triangle = 1/2 . L . B
= D0 . 1/2 . T . (g S - g L )
r - gL
②
gL
3%
0
STAGE 1
4 Yrs
STAGE 2
Time
①
D0 (1 + g L )
r - gL
T
The first term is what the shares would be worth if there were no high growth period and the
perpetual growth rate was g L . The second term is an approximation of the additional value that
results from the high growth period. In general, the H-value approximation is more accurate the
20
↷
shorter the high-growth period 'T', and/or the smaller the spread between the short-term and
long-term growth rates, g S - g L .
T/2
V0 = D0 (1 + g L ) + D 0 x H x (g S - g L )
r - gL
r - gL
Example 4:
The share price as of mid August 2013 was $41.7. The current dividend is 1.77. The initial dividend growth
rate is 7%, declining linearly during a 10-year period to a final and perpetual growth rate of 4%. Delacour
estimates Vinci's required return on equity as 9.5%. Using the H-model and the information given, estimate
the per share value of Vinci. Also estimate the value of Vinci shares if its normal growth period began
immediately.
V0 = 1.77 (1.04) + 1.77 (5) (0.07 - 0.04)
0.095 - 0.04
0.095 - 0.04
= 33.47 + 4.83 = $ 38.3
$ 33.47 is the value of Vinci shares if its normal growth period began immediately. $ 38.3 is approximately 8%
less than Vinci's current market price. This Vinci appears to be overvalued.
lllll
Three Stage DDM
Three Stage models are appropriate for firms that are expected to have 3 distinct stages of earnings
growth. Alternatively, in stage 2, the growth rate may also linearly decay to the stage 3 stable,
long-run growth rate.
g
25%
15%
3%
0
STAGE 1 3 Yrs STAGE 2
8 Yrs
STAGE 3
Time
Example 5:
R&M has a current dividend of $1 and a required rate of return of 12%. A dividend growth rate of 15% is
projected for the next 2 years, followed by a 10% growth rate for the next 4 years before settling down to a
constant 4% growth rate. Thereafter, calculate the current value of R&M.
D0 = 1
D 1 = 1 (1.15) = 1.15
D 2 = 1.15 (1.15) = 1.323
D 3 = 1.323 (1.1) = 1.455
D 4 = 1.455 (1.1) = 1.6
D 5 = 1.6 (1.1) = 1.76
D 6 = 1.76 (1.1) = 1.936
V6 = 1.936 (1.04) = $ 25.168
0.12 - 0.04
21
V0 = 1.15 + 1.323 + 1.455 + 1.6 + 1.76 + 1.936 + 25.168 = $ 18.864
(1.12)1 (1.12) 2 (1.12) 3 (1.12) 4 (1.12) 5
(1.12) 6
Example 6:
The current annual dividend is $0.75. Dividends are expected to grow at a rate of 12% over the next 3 years,
decline linearly to 4% over the next 6 years and then remain at a long-term equilibrium growth rate of 4% in
perpetuity. The required return is 9%. Calculate the value of the company.
D0 = 0.75
D1 = 0.75 (1.12) = 0.84
D2 = 0.84 (1.12) = 0.9408
D3 = 0.9408 (1.12) = 1.0536
V3 = D 3 (1 + g L ) + D3 x H x (g S - g L )
r - gL
r - gL
= 1.0536 (1.04) + 1.0536 (3) (0.12 - 0.04) = $ 26.9747
0.12 - 0.04
0.12 - 0.04
V0 = 0.84 + 0.9408 + 1.0536 + 26.9747 = $ 23.2056
(1.09) (1.09)
(1.09)
lllll
Spreadsheet Modelling
Spreadsheet modelling is applicable to firms that have a great deal of information and can project
different growth rates for differing periods, such as construction firms and defense contractors with
many long-term contracts. Spreadsheet is flexible and has computational accuracy. Spreadsheet
can be programmed in a virtually infinite series of combinations, any dividend pattern desired can
be achieved. Now the analyst can conduct detailed scenario analysis to see how changes in the
pattern of future dividends, interest rates and firm risk affect firm valuation estimates.
Fig 1: Growth, Transition and Maturity Phase
(But falling)
↶
↷
Sustainable growth rate is the rate of dividend or earnings growth that can be sustained for a given level of
return on equity, assuming that the capital structure is constant through time and that additional common
stock is not issued.
(1 - Dividend Payout Ratio) or Earnings Retention Rate
g S = b x ROE
A practical logic for defining sustainable in terms of growth through internally general funds (R/E) is that
external equity (secondary issues of stock) is considerably more costly than internal equity (reinvested
earnings).
22
Dividend Displacement of Earnings e.g. the positive effect on value from an increase in 'g' will be offset by
the negative effect from a decrease in dividend payouts in the expression for the value of the stock in any
DDM.
g = NI - Dividend x Net Income x
Sales
x Total Assets
S
NI
Sales
Total Assets
Equity
Retention Rate
ROA
This expression of the sustainable growth expression has been called PRAT model. Growth is a function of
Profit Margin (P), Retention Rate (R), Asset Turnover (A) and Financial Leverage (T). The profit margin and
asset turnover determine ROA. The other two factors, the retention rate and financial leverage reflect the
firm's financing decisions. The correct way to calculate 'g' is with ROE based on beginning shareholder's
equity. However, it is often done with average equity as an approximation. If g A > g S ; the actual growth rate is
forecasted to be greater than sustainable growth rate (SGR), the firm will have to issue equity unless the firm
increases its retention rate ratio, profit margin, total asset turnover or leverage.
Example 7:
'A colleague of Uto asserts that the greater the earnings retention ratio, the greater the sustainable growth
rate because 'g' is a positive function of 'b'. The colleague argues that Brother should decrease payout ratio'.
Explain the flaw in that argument.
As 'b' increases, growth rate increases, holding all else constant. However, all else may not be constant. In
particular, the return accruing to additional investments may be lower, leading to a lower overall ROE. If that
is the case and if it lowers the payout ratio to below 0.4 (thus increasing 'b' to above 0.6), ROE would be
expected to decline, which may lead to a lower growth rate.
NOTES
1. If a stock is trading at a price (market price) higher than the price implied by a DDM model i.e. model price,
the stock is considered to be overvalued. Similarly, if the market price is lower than the model price, the
stock is considered to be undervalued, and if the model price is equal to the market price, the stock is
considered to be fairly valued.
↶ ↷
MP (Overvalued)
DDM Price
MP (Undervalued)
23
CHAPTER 28
Free Cashflow Valuation
The concept of Free Cashflow (FCF) responds to the reality that of a going concern, some of the CFO is not
'free' but rather needs to be committed to reinvestment and new investment in assets. FCF models are not
appropriate for firms that don't have a dividend payment history or have a dividend history that is not clearly
and appropriately related to earnings. Also for firms that align with profitability within a reasonable forecast
period i.e. long-run. If a company is viewed as an acquisition target, FCF is a more appropriate measure
because the new owners will have discretion over its distribution i.e. valuation perspective is that of a
controlling shareholders. If investors are willing to pay a premium for control of the firm, there may be a
difference between the values of the same firm derived using the two models (DCF vs. FCF). FCF is also useful
to minority shareholders because the firm may be acquired for a market price equal to the value to the
controlling party. Firms that have significant capital requirements may have negative FCF for many years into
the future. This negative FCF complicates the cashflow forecast and makes the estimates less reliable.
Fig 1: Difference between FCFF and FCFE
FCFF
FCFE
Def: FCFF is the cash available to all of the
firm's investors, including stockholders
and bondholders, after cash operating
expenses (including taxes but excluding
interest expense), working capital and
fixed capital investments have been
made.
Def: FCFE is the cash available to common
shareholders after funding capital
requirements, working capital needs and
debt financing requirements.
What does the firm do with its FCFF? It first
takes care of its bondholders because
common shareholders are paid after all
creditors. However, making interest payments
to bondholders has one advantage for
common shareholders: it reduces the tax bill.
What does the firm do with its FCFE? It could
pay it all out in dividends to its common
shareholders, but it might decide to only pay
out some of it and put the rest in the bank to
save for next year.
The value of the firm is the present value of
the expected future FCFF discounted at WACC
(Weighted Average Cost of Capital).
The value of the firm's equity is the present
value of the expected future FCFE discounted
at the 're ' (Required Return on Equity).
∞
Firm Value = Σ
FCFF t
t=1
(1 + WACC) t
∞
Equity Value = Σ FCFE t
t=1
(1 + r) t
FCFF is pre-debt FCF concept.
FCFE is a post-debt FCF concept.
Dividends, Share Repurchases and Share
Issues have no effect on FCFF; changes in
leverage have no effect on FCFF.
Dividends, Share Repurchases and Share
Issues have no effect on FCFE; changes in
leverage have only a minor effect on FCFE. For
example, a decrease in leverage through a
repayment of debt will increase FCFE in the
current year and increase forecasted FCFE in
future years as interest expense is reduced.
24
Capital Structure: If historical data are used to
forecast FCF growth rates, FCFF growth might
reflect fundamentals more clearly than does
FCFE growth, which reflects fluctuating
amounts of net borrowing.
Capital Structure: FCFE is easier and more
straightforward to use in cases where the
company's capital structure is not particularly
volatile.
If a company has negative FCFE and significant debt outstanding, FCFF is generally the best choice.
Example 1:
Suppose an analyst estimates equity value by discounting FCFE at WACC in FCFE model and estimates firm
and equity value by discounting FCFF at the re in the FCFF model. The analyst would most likely:
A. Overestimate equity value with FCFE model and Underestimate firm value and equity value with FCFF
model.
B. Underestimate equity value with FCFE model and Overestimate firm value and equity value with FCFF
model.
C. Underestimate equity value with FCFE model and Underestimate firm value and equity value with FCFF
model.
✓
(Value of Operating Assets +
↶
Value of Non-Operating Assets)
Firm Value = Equity Value + Mv of Debt
Technically, what we've called 'Firm Value' is actually the value of operating assets (the assets that generate
cashflow) and significant non-operating assets such as excess cash (not total cash on the B/S), excess
marketable securities or land held for investment. Non-operating assets should be added to this estimate to
calculate total firm value. If you are asked to calculate the value of the firm using FCFF approach, calculate the
present value of the FCFFs and then look for any additional information in the problem that specifically says
'excess cash and marketable securities' or 'land held for investment'.
Sometimes, the CFO figures from the statement of cashflows may be contaminated by cashflows arising from
financing and/or investing activities, when analysts use CFO in a valuation context, ideally they should remove
such contaminations. Contaminations like acquisitions and divestitures and the presence of non-domestic
subsidiaries. Discrepancies, may also occur from currency translations of the earnings of non-domestic
subsidiaries may also occur from currency translations of the earnings of non-domestic subsidiaries. The
resulting analyst-adjusted CFO is then the starting point for FCF.
Specialized DCF approaches are also available to facilitate the equity valuation when the capital structure is
expected to change. In the Adjusted Present Value (APV) approach, firm value is calculated as the sum of
(i) the value of the company under the assumption that debt is not used (i.e. unlevered firm value) and (ii) the
NPV of any effects of debt on firm value (such as any tax benefits of using debt and any costs of financial
distress). In this approach, the analyst estimates unlevered company value by discounting FCFF (under the
assumption of no debt) at the unlevered cost of equity (the cost of equity given the firm doesn't use debt).
Example 2:
Clay Cooperman has valued the operating assets of Johnson Extension at $720,000,000. The company also
has short-term cash and securities with a market value of $60,000,000 that are not needed for Johnson's
operations. The non-current investments have a book value of $30,000,000 and a market value of
$45,000,000. The company also has an overfunded pension plan, with plan assets of $210,000,000 and plan
liabilities of $170,000,000. Johnson Extension has $215,000,000 of notes and bonds outstanding and
100,000,000 outstanding shares. What is the value per share of Johnson Extension Stock?
25
Non-Operating Assets = $ 60,000,000 + $ 45,000,000 + $ 40,000,000 = $ 145,000,000
Value of the Firm = Operating Assets + Non-Operating Assets
= $ 720,000,000 + $ 145,000,000 = $ 865,000,000
Value of the Firm = Value of Equity + Value of Debt
$ 865,000,000 = x + $ 215,000,000
x = $ 865,000,000 - $ 215,000,000
= $ 650,000,000
Value per Share = $ 650,000,000 = $ 6.5 per share
100,000,000
lllll
Free Cashflow to Firm (FCFF)
FCFF = NI + NCC + Interest (1 - t) - FCInv - WCInv + Preferred Dividends
Whereas,
NI : It represents income after depreciation, amortization, interest expense, income taxes
and payment of dividends to preferred shareholders (but not payments of dividends to
common shareholders) i.e. NI available to common shareholders.
NCC : Non-cash charges are added back to NI to arrive at FCFF because they represent
expenses that reduced reported NI but didn't actually result in an outflow of cash.
Make an adjustment,
only if unlikely to reverse.
DTL reserves : Debt
DTL doesn't : Equity
reverses
If these cashflows are not
expected to persist in the
future i.e. temporary,
analysts should not
include them in their
forecasts of cashflows.
(Based on no. of shares
expected to be o/s)
Depreciation and Amortization
+
Impairment / Write Downs
+
Gains and (Losses) of Operating Assets
(Long-term Assets)
- / (+)
DTL / (DTA)
+ / (-)
Provision for Restructuring
Expense / (Income)
+ / (-)
Amortization of Bond Discount / (Premium)
+ / (-)
Employee Share-based Compensation
or Stock Options
+
Interest : Interest is expensed on the income statement, but it represents a financing cashflow
Expense to bondholders that is available to the firm, before it makes any payments to its capital
suppliers. Therefore, we have to add it back. However, we don't add back entire interest
expense, only the after-tax interest cost because paying interest reduces the tax-bill.
Because interest is tax deductible, the after-tax interest expense reduces net income,
but we want to add it back to net income and then substract it out as a CFF.
FCInv : Investments in Fixed Capital represents the outflows of cash to purchase fixed capital
necessary to support the company's current and future operations. These investments
26
are capex for long-term assets, such as PPE necessary to support the company's
operations. Necessary capex may also include intangible assets such as trademarks. In
the case of cash acquisition (not in exchange for stock or debt) of another company
instead of direct acquisition of PPE, the cash purchase amount can also be treated as a
capex that reduces the company's FCF (note that this treatment if conservative because
it reduces FCFF).
FCInv = Capex*- Proceeds from Sales of Long-term Assets (Old)
↶
(Spend on
new assets)
(Carrying Value +/- Profit / Loss)
or Gain / Loss on Sale of Assets
Capex = Ending Net PPE - Beginning Net PPE + Depreciation
Investments in Long-term
Fixed Assets
WCInv : The investment in Net Working Capital is equal to the change in working capital
excluding cash, cash equivalents, notes payable and the current portion of long-term
debt. Notes payable and current portion of long-term debt are excluded because they
are liabilities with explicit interest costs that make them financing items rather than
operating items.
+
↓ CA
↑ CL
-
↑ CA
↓ CL
If the carryforward is '-10' as the value for WCInv, don't change the sign i.e.
not '-(-10)': wrong, but yes '-10': right.
Preferred : Under FCFF, preferred dividends paid would be added to the cashflows to obtain FCFF.
Dividends The WACC should also be revised to reflect the percent of total capital raised by
preferred stock and the cost of that capital source.
Fig 2: Summarizes IFRS vs. US GAAP Treatment of Interest and Dividends
FCFF = NI + NCC - WCInv + Interest (1 - t) - FCInv
FCFF =
CFO
+ Interest (1 - t) - FCInv
FCFF = EBIT (1 - t) + Dep - FCInv - WCInv
①
②
In the calculation of NI, many non-cash charges are made after computing EBIT or EBITDA, so they
don't need to be added back when calculating FCFF based on EBIT or EBITDA (We assume that the
only non-cash charge that appears above EBIT is depreciation in the equation 'FCFF from EBIT'. In
general however, the rule is to adjust for any non-cash charge that appears on the income
27
statement above the income statement item you are starting with). Another important
consideration is that some non-cash charges such as depreciation are tax deductible. A non-cash
charge that affects taxes must be accounted for.
FCFF = EBITDA (1 - t) + Dep (t) - FCInv - WCInv
③
EBITDA doesn't account for the investments a company makes in fixed capital or working capital.
Depreciation charges as a percentage of EBITDA differ substantially for different companies and
industries, as does the depreciation tax shield (depreciation times the tax rate). Although FCFF
captures this difference, EBITDA doesn't. Hence, EBITDA is a poor measure of the cashflow available
to the company's investors. EBITDA is a before-tax measure, so the discount rate applied to EBITDA
would be a before-tax rate. The WACC used to discount FCFF is an after-tax cost of capital. It is an
even poorer proxy for FCFE.
Uses of FCFF = +/- Increases or Decreases in Cash Balances
+ Net Payments to providers of Debt Capital
= + Interest Expense (1 - t)
+ Repayment of Principal in excess of New Borrowing or
- New Borrowing in excess of Debt Repayment if New Borrowing is greater
+ Net Payments to providers of Equity Capital
= + Cash Dividends
+ Share Repurchases in excess of Share Issuance or
- New Share Issuance in excess of Share Repurchases if Share Issuance is
greater
lllll
Free Cashflow to Equity (FCFE)
FCFE = NI + NCC - FCInv - WCInv + Net Borrowing
↶
Whereas,
Net Borrowing : Obtaining cash via net new borrowings and using the cash for dividends or
Take the differences
net share repurchases will increase the company's leverage, whereas
from the liabilities
obtaining cash from net new share issuances and using that cash to make
section under B/S
principal payments in excess of new borrowings will reduce leverage.
e.g. notes payable
and LT debt in B/S.
Net Borrowing = LT/ST New Debt Issues - LT/ST Debt Repayments or
in Debt Instruments +/(-) Amortized Premiums
(Discounts)
△
Preferred : If preferred dividends were already subtracted when arriving at NI, no
Dividends further adjustment for preferred dividends would be required. Issuing
(Redeeming) preferred stock increases (decreases) the cashflow available
to common stockholders, however so this term would have to be added in
like how we calculate new debt issuances and borrowings.
If all the inputs were known and mutually consistent, a DDM and a FCFE model would result in
identical valuations for a stock. One possibility would be that FCFE equals cash dividends each year.
Then, both cashflow streams would be discounted at the required return for equity 're ' and would
have the same present value. Generally, however FCFE and dividends will differ, but the same
economic forces that lead to low (high)) dividends lead to low (high) FCFE.
28
FCFE = NI + NCC - WCInv - FCInv + Net Borrowing
FCFE =
CFO
- FCInv + Net Borrowing
FCFE = EBIT (1 - t) + Dep - Interest (1 - t) - FCInv - WCInv + Net Borrowing
FCFE = EBITDA (1 - t) + Dep (t) - Interest (1 - t) - FCInv - WCInv + Net Borrowing
①
②
③
NI is a poor proxy for FCFE. On the other hand, EBITDA is an even poorer proxy for FCFE. From a
stockholders perspective, additional defects of EBITDA include its failure to account for the after-tax
interest costs or cashflows from new borrowing or debt repayments.
When depreciation is the only significant net non-cash charge, this method yields the same results
as the previous equations for estimating FCFF or FCFE.
FCFE = NI + Dep - FCInv - WCInv + Net Borrowing
FCFE = NI - (FCInv - Dep) - WCInv + Net Borrowing
If Net Borrowing = DR (FCInv - Dep) + DR (WCInv)
FCFE = NI - (1 - DR) (FCInv - Dep) - (1 - DR) WCInv
% of FCInv
④
% of WCInv
Whereas,
DR : Debt ratio also called as debt to asset ratio is debt as a percentage of debt plus equity.
DR indicates the percentage of the investment in fixed capital in excess of depreciation,
also called 'net new investment in fixed capital' and in working capital that will be
financed by debt.
Uses of FCFE = +/- Increases or Decreases in Cash Balances
+ Net Payments to providers of Equity Capital
= + Cash Dividends
+ Share Repurchases in excess of Share Issuance or
- New Share Issuance in excess of Share Repurchases if Share Issuance is
greater
FCFE = FCFF - Interest (1 - t) + Net Borrowing
Capital Expenditures have two dimensions: outlays that are needed to maintain existing capacity (fixed capital
replacement) and marginal or incremental expenditures necessary for growth. In forecasting, the
expenditures to maintain capacity are likely to be related to the current level of sales and the expenditures
for growth are likely to be forecast of sales growth. To estimate (future) FCInv and WCInv, we multiply their
past proportion to sales increases by the forecasted sales increases.
Incremental FC Expenditures = Capex - Dep
in Sales
↑
Incremental WC Expenditures = ↑ in WC
↑ in Sales
29
Example 3:
Carla Espinosa is an analyst following Pits Corporation at the end of 2012. She can see that the company sales
for 2012 were $3000,000,000 and she assumes that sales grew by $300,000,000 from 2011 to 2012. Espinosa
expects Pits Corporation's sales to increase by 10% per year thereafter. Pits Corporation is a fairly stable
company, so Espinosa expects it to maintain its historical EBIT margin and proportions of incremental
investments in fixed and working capital. Pits Corporation's EBIT for 2012 is $500,000,000; It's EBIT margin is
16.67% (500/3000) and its tax rate is 40%. Pits Corporation's 2012 statement of cashflows mentions the
amount for 'Purchases of Fixed Assets' i.e. Capex of $400,000,000 and depreciation of $300,000,000. She
expects the profit margin will remain at 8% (240/3000) and the company will finance incremental fixed and
working capital investments with 50% debt i.e. the target DR. Forecast 2013 FCFF and FCFE.
FCInv = Capex - Dep = 400 - 300 = 33.33%
in Sales
300
↑
FCFF (2013)
Sales
EBIT
EBIT (1 - t)
Incremental FC
Incremental WC
FCFF
$
3300
550
330
(100)
(45)
185
FCFE (2013)
Sales
EBIT
Incremental FC
Incremental WC
Net Borrowing
FCFE
$
3300
550
(100)
(45)
72.5
191.5
WCInv =
↑ in WC = 45 = 15%
↑ in Sales 300
↑ 10% of Sales
16.67% of Sales
40% Tax
33.33% of Sales Increase
15% of Sales Increase
↑ 10% of Sales
8% of Sales
33.33% of Sales Increase
15% of Sales Increase
(100 FCInv + 45 WCInv) x 50%
Example 4:
Welch Corporation uses bond, preferred stock and common stock financing. The market value of each of
these sources of financing and the before-tax required return for each are given below:
Bonds
Preferred Stock
Common Stock
NI available to common shareholders: $32
Interest Expense: $32
Preferred Dividends: $8
Depreciation: $40
Investment in Fixed Capital: $70
Investment in Working Capital: $20
Net Borrowing: $25
Tax Rate: 30%
Mv
400
100
500
1000
Required Return
8%
8%
12%
a. Calculate WACC
b. Calculate FCFF and FCFE
a. WACC = rd . wd (1 - t) - re . we + rp . wp = 400 (0.08) (1 - 0.3) + 500 (0.12) + 100 (0.08) = 9.04%
1000
1000
1000
use target
weights
30
b. FCFF = NI + NCC + Interest (1 - t) - FCInv - WCInv + Preferred Dividends
= 110 + 40 + 32 (1 - 0.3) - 70 - 20 + 8
Under FCFE approach,
= $ 90.4
c. FCFE = FCFF - Interest (1 - t) + Net Borrowing - Preferred
= 90.4 - 32 (1 - 0.3) + 25 - 8
= $ 85
preferred dividends were not a
↶ part of NI, but it is a part of
Dividends FCFF calculation, so we have to
substract $8 from the formula.
Example 5:
Singh and Ho Review key financial data from ABC's most recent annual report:
Income Statement
EBITDA
Depreciation Expense
Operating Income (EBIT)
Interest Expense
Tax Rate
$
4000
800
3200
440
35%
Cashflow Statement
NI
Depreciation
Accounts Receivable
Inventory
Accounts Payable
CFO
Purchases of PPE
CFI
Borrowing (Repayment)
CFF
Total Cashflow
$
1794
800
(2000)
(200)
1000
1394
(1000)
(1000)
500
500
894
↑
↑
↑
Calculate FCFF and FCFE.
FCFF = NI + NCC + Interest (1 - t) - FCInv - WCInv = 1794 + 800 + 440 (1 - 0.35) - 1000 - 1200 = $ 680
FCFF = EBIT (1 - t) + Dep - FCInv - WCInv = 3200 (1 - 0.35) + 800 - 1000 - 1200 = $ 680
FCFF = CFO + Interest (1 - t) - FCInv = 1394 + 440 (1 - 0.35) - 1000 = $ 680
FCFE = NI + NCC - FCInv - WCInv + Net Borrowing = 1794 + 800 - 1000 - 1200 + 500 = $ 894
FCFE = CFO - FCInv + Net Borrowing = 1394 - 1000 + 500 = $ 894
FCFE = FCFF - Interest (1 - t) + Net Borrowing = 680 - 440 (1 - 0.35) + 500 = $ 894
lllll
Single Stage FCF Model
FCFF
Value of Firm =
FCFE
FCFF = FCFF (1 + g)
WACC - g
WACC - g
Value of Firm = FCFE = FCFE (1 + g)
r-g
r-g
Constant expected
g rate in FCFF*
* It's quite likely that a firm's growth rate in FCFF
will be different than its FCFE growth rate
Constant expected
g rate in FCFE*
31
The single stage FCE model is often used in international valuation, especially for companies in
countries with high inflationary expectations when estimation of nominal growth rates and required
returns is different. In those cases, 'real' inflation-adjusted values are estimated for the inputs to the
single stage FCFE model. To estimate real discount rates, they use a modification of the build-up
method. The firms supply their analysis with estimates of the real economic growth rate for each
country, and each analyst chooses a real growth rate for the stock being analyzed that is benchmarked
against the real country growth rate. Whenever, real discount rates and real growth rates can be
estimated more reliably than nominal discount rates and nominal growth rates, this method is worth
using.
V0 = FCFE 0 (1 + g real )
r real - g real
Whereas,
r real : Country Return (Real) %
+/- Industry Adjustment %
+/- Size Adjustment %
+/- Leverage Adjustment %
Example 6:
YPF Sociedad Anonima is an integrated oil and gas company headquartered in Buenos Aires, Argentina.
Although the company's cashflows have been volatile, an analyst has estimated a per share normalized FCFE
of 7.05 Argentina Pesos (ARS) for the year just ended. The real country return for Argentina is 7.3%;
adjustments to the country return for YPF S.A. are an industry adjustment of 0.8%, a size adjustment of
-0.33% and a leverage adjustment of -0.12%. The long-term real growth rate for Argentina is estimated to be
3% and the real growth rate of YPF S.A. is expected to be 0.5% below the country rate.
rreal = 7.3% + 0.8% - 0.33% - 0.12% = 7.65%
g real = 3% - 0.5% = 2.5%
V0 = FCFE 0 (1 + g real ) = 7.05 (1 + 0.025) = ARS 140.32
r real - g real
0.0765 - 0.025
lllll
Two Stage FCF Model
Terminal Value in Year 'n' = Trailing P/E x Earnings in Year 'n'
Terminal Value in Year 'n' = Leading P/E x Forecasted Earnings in Year 'n + 1'
Example 7:
The Prentice Paint Company earned a net profit margin of 20% on revenues of $20,000,000 this year. Fixed
capital investment was $2000,000 and depreciation was $3000,000. Working capital investment equals 7.5%
of sales every year. Net income, FCinv, depreciation, interest expense and sales are expected to grow at 10%
per year for the next 5 years. After 5 years, the growth in sales, net income, FCInv, depreciation and interest
expense will decline to a stable 5% per year. The tax rate is 40% and Prentice has 1000,000 shares of common
stock outstanding and long-term debt paying 12.5% interest trading at its par value of $32,000,000. Calculate
the value of the firm and its equity using the FCFF model, if the WACC is 17% during the high growth stage
and 15% during the stable stage.
32
FCFF in Year 0: NI = 20 (0.2) = $ 4
Interest = 32 (0.125) = $ 4
= 4 (1 - 0.4) = $ 2.4
WCInv = 20 (0.075) = $ 1.5
FCFF 0 = $ 4 + $ 2.4 + $ 3 - $ 2 - $ 1.5 = $ 5.9
Sales
Net Income
Interest (1 - t)
Depreciation
- FCInv
- WCInv
FCFF
0
1
2
3
20
4
2.4
3
2
1.5
5.9
22
4.4
2.64
3.3
2.2
1.65
6.49
24.2
4.84
2.9
3.63
2.42
1.82
7.13
4
5
6
26.62 29.28 32.21 33.82
5.32 5.86 6.44 6.76
3.19 3.51 3.87 4.06
3.99 4.39 4.83 5.07
2.66 2.93 3.22 3.38
2
2.2
2.42 2.54
7.84 8.63
9.5
9.97
V5 = FCFF 5 (1 + g) = FCFF 6
WACC - g
WACC - g
=
9.97
= $ 99.7
0.15 - 0.05
Value of Firm = 6.47 + 7.13 + 7.84 + 8.63 + 9.5 + 99.7 = $ 70.06
(1.17) 1 (1.17) 2 (1.17) 3 (1.17) 4 (1.17) 5
Value of Equity = $ 70.06 - $ 32 = $ 38.06
Example 8:
Vishal Noronha needs to prepare a valuation of Sindhuh Enterprises. Noronha has assembled the following
information for his analysis. It is now the first day of 2013. EPS for 2012 is $2.4. For the next 5 years, the
growth rate in EPS is given in the following table. After 2017, the growth rate will be 7%.
Growth Rate for EPS
2013
30%
2014
18%
2015
12%
2016
9%
2017
7%
Net investments in fixed capital (net of depreciation) for the next 5 years are given in the following table.
After 2017, capex are expected to grow at 7% annually.
Net Capital Expenditures per Share
2013
3
2014
2.5
2015
2
2016
1.5
2017
1
The investment in working capital each year will equal 50% of the net investment in capital items. 30% of the
net investment in fixed capital and investment in working capital will be financed with new debt financing.
Current market conditions dictate a required return of equity of 10.4%.
a. What is the pre-share value of Sindhuh Enterprises on the first day of 2013.
b. What should be the trailing P/E on the first day of 2013 and the first day of 2017.
33
EPS
- Net FCInv per Share
- WCInv per Share
Debt Financing per Share*
FCFE
2013
2014
2015
2016
2017
3.12
-3
- 1.5
1.35
- 0.03
3.682
- 2.5
- 1.25
1.125
1.057
4.123
2
1
0.9
2.023
4.494
1.5
0.75
0.675
2.919
4.809
1
0.5
0.45
3.759
* 0.3 (Net FCInv + WCInv)
a.
V 2016 = FCFE 2016 (1 + g) = FCFE 2017
r-g
r-g
= 3.759 = $ 110.56
0.104 - 0.07
Value of Equity = - 0.03 + 1.057 + 2.023 + 2.919 + 110.56 = $ 78.73
2012
(1.104) 1 (1.104) 2 (1.104) 3
(1.104) 4
b. Trailing P/E on the first day of 2013 = 78.73 = 32.8
2.4
Trailing P/E on the first day of 2017 = 110.56 = 24.6
4.494
After its high-growth phase has ended, the P/E for the company declines, substantially.
Example 9:
Singh and Ho now analyze Bern:
Debt
Preferred Stock
Common Stock
Mv
15,400
4000
18,100
Required Return
6%
5.5%
11%
FCFF0 = 3226
Corporate Tax Rate = 26.9%
Singh notes that Bern has two new drugs that are currently in clinical trails awaiting regulatory approval. In
addition to its operating assets, Bern owns a parcel of lend from a decommissioned manufacturing facility
with a current market value of $50 that is being held for investment. Value Bern assuming the two drugs
don't receive regulatory approval. In this scenario, FCFF is forecast using a stable growth in FCFF of 1.75% for
the next 3 years and then 0.75% thereafter into perpetuity.
WACC = re . we + rd . wd (1 - t) + rp . wp
= (0.4826) (0.11) + (0.4106) (0.06) (1 - 0.269) + (0.10666) (0.055)
= 7.7 %
FCFF0 = 3226
FCFF1 = 3226 (1.015) = 3274.39
FCFF2 = 3274.39 (1.015) = 3323.5
FCFF3 = 3323.5 (1.015) = 3373.35
V3 = 3373.35 (1.0075) = $ 48,921.38
0.0695
34
V0 = 3274.39 + 3323.5 + 3373.35 + 48,921.38 = $ 47,770.2
(1.077)1 (1.077) 2
(1.077) 3
Value of Firm = Operating Assets + Non-Operating Assets
= 47,770.2 + 50
= $ 47,820.2
Value of Equity = Value of Firm - Value of Debt
= 47,820.2 - 15,400 - 4000
= $ 28,420.2
Since the current market value of Bern's common stock ($ 18,100) is less than the estimated value
($ 28,420.2), the shares are undervalued.
lllll
Three Stage FCF Model
Example 10:
Medina Classic Furniture Inc is expected to experience growth in three district stages in the future. It's most
recent FCFE is 0.9 C$ per share. The following information has been complied.
High Growth
Duration = 3 Years
FCFE growth rate = 30%
Transitional
Duration = 3 Years
FCFE growth will decline
by 9% per year down to
the indicated stable
growth rate.
re = 15%
re = 20%
Stable
FCFE growth rate = 3%
re = 10%
Calculate the value of the firm's equity using the 3-stage FCFE model.
Option A:
g
30%
21%
12%
3%
0
Time
3 Yrs 4 Yrs 5 Yrs 6 Yrs
20%
FCFE 0 = 0.9
FCFE 1 = 0.9 (1.3) = 1.17
FCFE 2 = 1.17 (1.3) = 1.521
FCFE 3 = 1.521 (1.3) = 1.9773
FCFE 4 = 1.9773 (1.21) = 2.3925
FCFE 5 = 2.3925 (1.12) = 2.6796
FCFE 6 = 2.6796 (1.03) = 2.76
15%
10%
35
Let's work backwards,
Step 1: V6 = 2.76 (1.03) = C$ 40.611
0.1 - 0.03
Step 2: V3 = 2.3925 + 2.6796 + 2.76 + 40.611 = C$ 32.6244
(1.15)1 (1.15) 2
(1.15) 3
Step 3: V0 = 1.17 + 1.521 + 1.9773 + 32.6244 = 22.05525
(1.2)1 (1.2) 2
(1.2) 3
Option B:
V0 = 1.17 + 1.521 + 1.977 + 2.393 + 2.680 + 2.760 + 40.611 = C$ 22.055
(1.2)1 (1.2) 2 (1.2)3 (1.2) 3 (1.15)1 (1.2) 3 (1.15) 2 (1.2) 3 (1.15) 3
NOTES
1. If a stock is trading at a price (market price) higher than the price implied by a FCF model i.e. model price,
the stock is considered to be overvalued. Similarly, if the market price is lower than the model price, the
stock is considered to be undervalued, and if the model price is equal to the market price, the stock is
considered to be fairly valued.
↶ ↷
MP (Overvalued)
FCF Value
MP (Undervalued)
36
CHAPTER 29
Market Based Valuation: Price and Enterprise Value Multiples
Price Multiples are amongst the most widely used tools for valuation of equities. Comparing stock's price
multiples can help an investor judge whether a particular stock is overvalued, undervalued or properly valued
in terms of measures such as earnings, sales, cashflow or book value per share. Enterprise Value Multiples
relate to the total value of a company, as reflected in as earnings before interest, taxes, depreciation and
amortization. Momentum Indicators compare a stock's price or a company's earnings to their values in earlier
periods or in some cases, to its expected values.
Method of Comparables: refers to the valuation of an asset based on multiples of comparable (similar)
assets i.e. valuation based on multiples benchmarked to the multiples of similar assets. The similar assets
may be referred to as the comparables or the 'guidelines companies'. The economic rationale underlying the
method of comparables is the 'Law of One Price' i.e. the economic principle that two identical assets should
sell at the same price a.k.a. 'Relative Valuation' approach.
Method of Forecasted Fundamentals: values a stock based on the ratio of its value from a DCF model to
some fundamental variable (e.g. EPS). It involves forecasting the company's fundamentals rather than making
comparisons with other companies. The price multiple of an asset should be related to its expected FCFs.
Example 1:
Shares of Countronics Inc. are selling for $30. The mean analyst EPS forecast for next year is $4 and the longrun growth rate is 5%. Countronics has a DPR of 60%. The required return is 14%. Calculate the fundamental
value of Countronics using GGM and determine whether Countronics shares are over or undervalued using
the method of forecasted fundamentals.
V0 = D1 = (0.6) 4 = $ 26.67
r - g 0.14 - 0.05
Company: P = 30 = 7.5 Times
E
4
Intrinsic Value: P = 26.67 = 6.67 Times
E
4
Countronics is overvalued because the observed P/E multiple of 7.5 is greater than the fair value P/E ratio of
6.67. Notice, that we would have come to the same conclusion by comparing market price $ 30 to intrinsic
value $ 26.67.
lllll
P/E Ratio
Trailing P/E =
P0
EPS over Previous 12 Months
Leading P/E =
P0
Forecasted EPS over Next 12 Months
When earnings are sufficiently volatile so that next year's earnings are not forecastable with any
degree of accuracy or when earnings are not readily predictable, a Trailing P/E may be more
appropriate than Forward P/E. During a major acquisition or divestiture or a significant change in
financial leverage may change a company's operating or financial risk so much that the Trailing P/E
based on past EPS is not informative. In such cases, the Forward P/E is the appropriate measure.
37
* Trailing P/E
- For comparative purposes, analysts generally prefer to use diluted EPS so that the EPS of
companies with differing amounts of dilutive securities are on a comparable basis. Dilution
refers to a reduction in proportional ownership interest as a result of the issuance of new shares.
- Estimating the appropriate earnings measure is crucial to successfully using the P/E ratio in
market-based valuation. The key focus of an analyst is estimating underlying earnings a.k.a.
persistent, continuing or core earnings, which are earnings that exclude non-recurring
components, such as gains and losses from asset sales, asset write-downs, provisions for future
losses and changes in accounting estimates. Earnings may be decomposed into cashflow and
accrual components. Some research indicates that the cashflow component of earnings should
receive a greater weight than the accrual component of earnings in valuation.
- In addition to company specific effects, such as restructuring costs, transitory effects on earnings
can come from business-cycle or industry-cycle influences. Because of cyclical effects, the most
recent four quarters of earnings may not accurately reflect the average or long-term earning
power of the business, particularly for cyclical businesses. Trailing EPS for such stocks are often
depressed or negative at the bottom of a cycle and unusually high at the top of a cycle. Whereas,
high P/Es on depressed EPS at the bottom of a cycle and low P/Es on unusually high EPS at the top
of the cycle reflect the countercyclical property of P/Es known as the 'Molodovsky Effect'. Analysts
address this problem by normalizing EPS i.e. estimating the level of EPS that the business could be
expected to achieve under mid-cyclical conditions (normalized EPS or normal EPS). The following
two methods are used to normalize earnings: (a) Method of Historical Average EPS: Normalized
EPS is calculated as average EPS over the most recent full cycle. Average EPS can be volatile
compared to average ROE. The method of historical average EPS ignores size effects, so the
method of average ROE is preferred. (b) Method of Average ROE: Normalized EPS is calculated as
the average ROE from the most recent full cycle, multiplied by current BVPS. The reliance on BVPS
reflects the effect of firm size changes more accurately than does the method of historical average
EPS. The analyst can also estimate normalized earnings by multiplying total assets by an estimate
of the long-term return on total assets or by multiplying shareholder's equity by an estimate of the
long-run return on total shareholder's equity. These methods are particularly useful for a period in
which a cyclical company has reported a loss.
- If an analyst is comparing a company that uses LIFO permitted by US GAAP vs. the company that
uses FIFO permitted by IFRA and US GAAP, the analyst should adjust earnings to provide
comparability in all ratio and valuation analyses.
- Negative earnings render P/E ratios meaningless. In such cases, it is common to use normalized
EPS and/or restate the ratio as the Earnings Yield (E/P) because price is never negative. A high E/P
suggests a cheap security and a low E/P suggests an expensive security, so securities can be
ranked from cheap to expensive based on E/P ratios.
- An extremely high P/E, an outlier P/E can overwhelm the effect of the other P/Es in the calculation
of the mean P/E. Although the use of media P/Es and other techniques can mitigate the problem of
skewness caused by outliers, the distribution of inverse price ratios is inherently less susceptible
to outlier induced skewness.
- Look Ahead Bias is the use of information that was not contemporaneously available in computing
a quantity. Investment analysts often use historical data to back test an investment strategy that
involves stock selection based on price multiples or other factors. To avoid this bias an analyst
would calculate the Trailing P/E based on the most recent four quarters of EPS.
38
Example 2:
Using the data in the following figure, calculate normalized earnings using the method of historical average
EPS and the method of average return of equity for Magnolia Enterprises.
EPS
BVPS
ROE
2012
4.2
26.02
14%
2013
3.75
27.78
12%
2014
4.75
29.25
16%
2015
4.8
32.29
14%
Historical Average EPS = 4.2 + 3.75 + 4.75 + 4.3 = $ 4.25
4
Average ROE = 14% + 12% + 16% + 14% = 0.14 or 14%
4
Average ROE EPS = Average ROE x Current BVPS
= 0.14 x 32.29
= $ 4.52
Normalized earnings are $ 4.25 based on the method of historical average EPS and $ 4.52 based on the
method of average ROE.
Example 3:
Adesivo's current stock price is $14.72 and diluted EPS of last four quarters of $0.81. On a per share basis,
Adesivo incurred in the last four quarters (i) from a lawsuit, a non-recurring gain of $0.04 and (ii) from factory
integration, a non-recurring cost of $0.03 and a recurring cost of $0.01 in increased depreciation. Calculate
the trailing P/E?
EPS = 0.81 - 0.04 + 0.03 = 0.8
P/E = 14.72 / 0.8 = 18.4
* Leading P/E
- The 'Next Twelve Method' (NTM) P/E, which corresponds in a forward looking sense to the 'Trailing
Twelve Method' (TTM) P/E concept of trailing P/E. NTM P/E is useful because it facilitates
comparison of companies with different fiscal year-ends without the need to use quarterly
estimates, which for many companies are not available. 'Forward P/E' contrasts with 'Current P/E',
which is based on the last reported annual EPS.
Example 4:
They present actual and forecasted EPS for Boyd Gaming Corp. that owns and operates 21 gaming
entertainment properties in Nevada, Mississippi, Illinois, New Jersey, Indiana, Kansas, Iowa and Louisiana.
Year
2013
2014
31st March
0.01
E 0.07
30th June
0
E 0.08
30th September
E(0.01)
E 0.03
31st December
E(0.05)
E(0.03)
Annual Estimate
(0.05)
0.15
On 9th August 2013, Boyd closed at $12.2. Boyd's fiscal year ends on 31st December. Calculate NTM P/E?
EPS = (5/12) (-0.05) + (7/12) (0.15) = 0.067
NTM P/E = 12.2 / 0.067 = 182.1
39
↷
Retention
Rate
Justified Trailing P0 / E 0 = D 0 (1 + g) / E 0 = (1 - b) (1+ g)
r-g
r-g
Justified Leading P0 / E 1 = D1 / E 1 = 1 - b
r-g
r-g
Dividend Payout Ratio
∴ Justified Trailing P/E = Justified Leading P/E x (1 + g)
Based on
Fundamentals
Predicted P/E can be estimated from linear regression of historical P/Es on its fundamental
variables, including expected growth and risk. There are 3 main limitations: (a) The predictive power
of the estimated P/E regression for a different time period and/or sample of stocks is uncertain,
(b) The relationships between P/E and the fundamental variables examined may change over time
and (c) Multicollinearity is often a problem in these time series regressions, which makes it difficult
to interpret individual regression.
Example 5:
An analyst is valuing a public utility with a DPR of 0.5, a beta of 0.95 and an expected earnings growth rate of
0.06. A regression on other public utilities produces the following regression equation:
Predicted P/E = 6.75 + (4 x Dividend Payout) + (12.35 x Growth) - (0.5 x Beta)
The firm's P/E ratio is 12. Calculate the predicted P/E on the basis of the values of the explanatory variables
for the company and determine whether the stock is over or underpriced?
Predicted P/E = 6.75 + (4) (0.5) + (12.35) (0.06) - (0.5) (0.95)
= 9.02
Actual P/E is greater than predicted P/E, so the firm is overpriced.
Based on
Comparables
The basic idea of the method of comparables is to compare a stock price multiple to that of a
benchmark portfolio. Firms with multiples below the benchmark are undervalued and firms with
multiples above the benchmark are overvalued.
Example 6:
An analyst has gathered P/E information on two stocks, AllBright Interiors and Basic Designs. Evaluate the
value and P/E of each stock based on the method of comparables.
AllBright
Basic Designs
Peer Median
Trailing P/E
10
14
13.3
Leading P/E
8.7
12.7
12.1
g%
11%
9%
11%
β
1.3
1.4
1.3
AllBright has a lower P/E than the Peer Median, despite the fact that it has a comparable growth rate and
beta. This indicates AllBright is undervalued. Basic Designs, on the other hand, has a higher P/E, despite lower
expected growth and a higher beta, which suggests it's overvalued relative to the benchmark.
No matter which ratio is used, terminal value is calculated as the product of the price multiple
(e.g. P/E ratio) and the fundamental variable (e.g. EPS). TV = Price Multiple (or Benchmark Price
Multiple) x Fundamental Variable.
40
Fundamental
TV = Justified Trailing P/E x Forecasted Earnings in Year n
TV = Justified Leading P/E x Forecasted Earnings in Year n+1
Comparables
TV = Benchmark Trailing P/E x Forecasted Earnings in Year n
TV = Benchmark Leading P/E x Forecasted Earnings in Year n+1
The strength of the comparables approach is that it uses market data exclusively. In contrast, the
fundamentals approach requires estimates of the growth rate, required rate of return and payout
ratio. One weakness of the comparables approach is that a benchmark marred by mispricing will
transfer that error to the estimated terminal value.
PEG Ratio
The relationship between earnings growth and P/E is captured by the P/E-to-Growth (PEG) ratio. The PEG ratio
in effect 'standardizes' the P/E ratio for stocks with different expected growth rates. The implied valuation
rule is that stocks with lower PEGs are more attractive than stocks with higher PEGs, assuming that risk is
similar. For traditional growth firms PEG ratios fall between 1 and 2. Anything above 2 is considered
expensive i.e. overvalued and anything below 2 is considered cheap i.e. undervalued.
PEG Ratio = P/E
g
There are a number of drawbacks to using the PEG ratio, (a) The relationship between P/E and 'g' is not linear,
which makes comparisons difficult. E.g. Firms with negative expected earnings growth will have a negative
PEG ratio, which is meaningless, (b) The PEG ratio still doesn't account risk and (c) The PEG ratio doesn't
reflect the duration of the high-growth period for a multi-stage valuation model, especially if the analyst uses
a short-term high growth forecast.
Fed Model
Fed Model predicts the return on the S&P 500 on the basis of the relationship between forecasted Earnings
Yields (E/P) and yields on bonds. It considers the overall market to be overvalued (undervalued) when the
earnings yield (E/P) on the S&P 500 Index is lower (higher) than the yield on 10 year US Treasury Bonds. A
concern in considering the Fed Model is that the expected growth rate is lacking in this model and ignores the
ERP. The model also inadequately reflects the effects of inflation and incorrectly incorporates the differential
effects of inflation on earnings and interest payments. Another drawback being that the relationship between
interest rates and earnings yields is not a linear one. The drawback is most noticeable at low interest rates.
Furthermore, small changes in interest rates and/or corporate profits can significantly alter the justified P/E
predicted by the model. Should avoid overreliance on the model as a predictive method, particularly in
periods of low inflation and low interest rates.
Justified P/E on S&P 500 = 1
rf
Yardeni Model
Yardeni (2000) developed a model that incorporates the expected growth rate in earnings.
CEY or E = CBY - Wm (LTEG) + ε i
P
P =
1
E CBY - W m (LTEG)
41
Whereas,
CEY : Current Earnings Yield on the market index
CBY : Current Moody's A-rated Corporate Bond Yield incorporates a default
risk premium relative to T-Bonds (not ERP).
LTEG : 5-Year consensus earnings growth rate
W m : Weight the market gives to 5-Year earnings projections.
Adjusted P/E Inflation
Differences in macroeconomic contexts may distort comparisons of benchmark P/E levels among companies
operating in different markets. Here we talk about differences in inflation rates and in the ability of
companies to pass through inflation in their costs in the form of higher prices to their customers. For two
companies with the same pass-through ability, the company operating in the environment with higher
inflation will have a lower justified P/E, if the inflation rates are equal but pass-through rates differ, the
justified P/E should be lower for the company with the lower pass-through rate.
P0 = E 0 (1 + I)
r-I
The company's only
growth is from inflation
↷
Let ' γ' represent the percentage of inflation in costs that the company can pass through to earnings.
P0 = E 0 (1 + γ I) = E 1
r - γI
r -γI
Now, introduce a real rate of return, defined as p = r - I.
P0 =
E1
p + (1 - γ ) I
or
P0 =
1
E1 p + (1 - γ ) I
100% : If a company can pass through all inflation: γ = 1, then P/E = 1/p
0% : If a company can pass through no inflation: γ = 0, then P/E = 1/(p + I) or 1/r
With less than 100% cost pass-through, the justified P/E is inversely related to the inflation rate and with
complete cost pass-through, the justified P/E should not be affected by inflation. The higher the inflation rate,
the greater the impact of incomplete cost pass-through on P/E. One can also infer that the higher the inflation
rate, the more serious the effect on justified P/E of a pass-through rate that is less than 100%.
Example 7:
The real rate of return 'p' required on the shares of Company M and Company P is 3% per year.
a. Suppose both Company M and Company P can pass through 75% of cost increases. Cost inflation is 6% for
Company M but only 2% for Company P. Estimate Justified P/E.
b. Suppose both Company M and Company P face 6% a year inflation. Company M can pass through 90% of
cost increase but Company P can pass through only 70%. Estimate Justified P/E.
a.
Company M
P/E =
1
= 22.2
0.03 + (1 - 0.75) 0.06
γ : 75%
Company P
I : 6%
P/E =
1
= 28.57
0.03 + (1 - 0.75) 0.02
With less than 100% cost pass-through, the justified P/E is inversely related to the inflation rate.
γ : 75%
I : 2%
42
b.
Company M
P/E =
1
= 27.77
0.03 + (1 - 0.9) 0.06
γ : 90%
Company P
I : 6%
P/E =
γ : 70%
I : 6%
1
= 20.83
0.03 + (1 - 0.7) 0.06
For equal inflation rates, the company with the higher pass-through rate has a higher justified P/E.
Advantages
1. Earnings power as measured by EPS is the primary determinant of investment value.
2. The P/E ratio is popular in the investment community.
3. Empirical research shows that P/E differences are significantly related to long-run average stock
returns.
Disadvantages
1. Earnings can be negative, which produces meaningless P/E ratio. However an analyst may base
the P/E calculation on a longer run expected average EPS value.
2. The ongoing or recurring components of earnings that are most important in determining
intrinsic value can be practically difficult to distinguish from transient components.
3. Management discretion within allowed accounting practices can distort reported earnings, and
thereby lessen the comparability of P/Es across firms. One reason being the differences in P/Es
calculated by different methods can be systematic as opposed to random. For example, for
companies with rising earnings, the forward P/E will be smaller than the trailing P/E because the
denominator in the forward P/E calculation will be larger.
lllll
P/B Ratio
P/B = Mv of Equity / Common Stock o/s Shares = Market Price per Share
Bv of Equity / Common Stock o/s Shares
Book Value per Share
Common Shareholder's
Equity
Some preferred stock issues have
the right to premiums (liquidation
premiums) if they are liquidated.
↷
Whereas,
BVPS : (Total Assets - Total Liabilities) - Preferred Stock - Dividends in Arrears in Preferred Stock
We often make adjustments to book value to create more useful comparisons of P/B ratios across
different stocks. A common adjustment is to use tangible book value, which is equal to book value
of equity less intangible assets. Examples of intangible assets include goodwill from acquisitions
and patents. Furthermore, balance sheets should be adjusted for significant off balance sheets and
liabilities and for differences between the fair and recorded value of assets and liabilities. Finally,
book values often need to be adjusted to ensure comparability. For example, companies using FIFO
for inventory valuation cannot be accurately compared with peers using LIFO. When assets are
reported at fair value, P/Bs become more comparable among companies; for this reason, P/Bs are
considered to be more comparable for companies with significant amounts of financial assets.
Example 8:
Edward Stamos is a junior analyst at a major US pension fund. Stamos is researching Barclays Plc. for his
fund's credit services portfolio.
43
Balance Sheet
2012
ASSETS
Cash and Balances at Central Banks
Items in the Course of Collection from other Banks
Trading Portfolio Assets
Financial Assets designated at Fair Value
Derivative Financial Instruments
Available for Sale Investments
Loans and Advances to Bank
Loans and Advances to Customers
Reverse Repurchase Agreements and other similar Secured Lending
Prepayments, Accrued Income and Other Assets
Investments in Associates and Joint Ventures
PPE
Goodwill and Intangible Assets
Current Tax Assets
Deferred Tax Assets
Retirement Benefits Assets
Total Assets
86,175
1,456
145,030
46,061
469,146
75,109
40,489
425,729
176,956
4,360
570
5,754
7,915
252
3,016
2,303
1,490,321
LIABILITIES
Deposits from Banks
Items in the Course of Collection due to other Banks
Customer Accounts
Repurchase Agreements and other similar Secured Borrowing
Trading Portfolio Liabilities
Financial Liabilities designated at Fair Value
Derivative Financial Instruments
Debt Securities in Issue
Subordinated Liabilities
Accrual, Deferred Income and Other Liabilities
Provisions
Current Tax Liabilities
Deferred Tax Liabilities
Retirement Benefits Liabilities
Total Liabilities
77,010
1,573
385,707
217,342
44,794
78,280
462,468
119,581
24,018
12,232
2,766
621
719
253
1,427,364
EQUITY
Shareholder's Equity excluding NCI
NCI
Total Shareholder's Equity
Footnotes
Financial Assets
Financial Liabilities
53,586
9,371
62,957
Carrying Amount
Fv
643,174
823,658
625,841
823,259
The 31st December, 2012 share price for Barclays was $ 2.4239 and the diluted weighted average number of
shares was 12,614. Calculate P/B and Adjusted P/B.
44
BVPS = 62,957 = $ 4.9910
12,614
P/B = 2.4239 = 0.49
4.991
Tangible BVPS = Common Equity - Goodwill and Intangible = 62,957 - 7915 = $ 4.3636
12,614
12,614
Tangible P/B = 2.4239 = 0.56
4.3636
Stamos notes Fv of Financial Assets is $ 17,333 less than their carrying amount ($ 643,174 - $ 625,841) and the
Fv of Financial Liabilities is $ 339 less than their Carrying Amount ($ 823,658 - $ 823,259).
Adjusted BVPS = 62,957 - 7915 - 17,333 + 399 = $ 3.0211
12,614
Adjusted P/B = 2.4239 = 0.8
3.0211
An analyst should also be aware of differences in accounting standards related to how assets and
liabilities are valued in financial statements.
P0 = D1
r-g
= EPS 1 x Payout Ratio
r-g
= EPS 1 (1 - b)
r-g
EPS1 = B 0 x ROE
P0 = EPS 1 (1 - b)
B0
B0
r-g
= ROE (1 - b)
r-g
= ROE - ROE x b
r-g
g = ROE x b
Justified P/B = ROE - g
r-g
The larger the spread between ROE and 'r', all else equal, the higher the P/B ratio. The larger the
spread, all else equal, the more value the firm is creating through its investment activities and the
higher its market value as represented by V0 or P0 .
ROE = r,
ROE > r,
ROE < r,
P/B = 1
P/B > 1
P/B < 1
45
If we are evaluating two stocks with the same P/B, the one with higher ROE is relatively
undervalued, all else equal. The expression for the Justified P/B based on the residual income
valuation.
Justified P0 = 1 + Pv of Expected Future Residual Earnings = 1 + Pv of EFRE
B0
B0
B0
Pv of EFRE = 0,
Pv of EFRE is +ve,
Pv of EFRE is -ve,
P/B = 1
P/B > 1
P/B < 1
Advantages
1. Book value is a cumulative amount that is usually positive, even when the firm reports a loss and
EPS is negative. Book value is more stable than EPS, so it may be more useful than P/E when EPS
is very volatile.
2. Book value is an appropriate measure of net asset value for firms that primarily hold liquid assets
e.g. finance, investment, insurance and banking firms. P/B can be useful in valuing companies
that are expected to go out of business.
3. Empirical research shows that P/Bs help explain differences in long-run average stock returns.
Disadvantages
1. P/Bs don't reflect the value of intangible economic assets such as human capital. Similarly, the
good reputation that a company develops by consistently providing high quality goods and
services is not reflected as an asset on the balance sheet. Book value may not mean much for
service firms without significant fixed costs.
2. P/B may be misleading as a valuation indicator when the levels of assets used by the companies
under examination differ significantly because in some cases the firm's business model dictates
the size of its assets base. A firm that outsources its production will have fewer assets, lower
book value and a higher P/B ratio than an otherwise similar firm in the industry that doesn't
outsource.
3. Different accounting conventions can obscure the true investment in the firm made by
shareholders, which reduces the comparability of P/Bs across firms and countries. For example,
treatment of R&D costs under IFRS and US GAAP.
4. Inflation and technological change can cause the book and market values of assets to differ
significantly, hence book value is not an accurate measure of the value of shareholder's
investment.
5. Share repurchases or issuances may distort historical comparisons. For example, as of 13th
September 2013, Colgate Palmolive's trailing P/E and P/B were, respectively 24.84 and 36.01. Five
years earlier, Colgate-Palmolive's trailing P/E and P/B were 23.55 and 15.94. In other words, the
company's P/E widened by 5.5%. While its P/B widened by 125.9%. This is due to substantial
amount of shares Colgate-Palmolive repurchased over those 5 years, as reflected by book value
(i.e. total common equity) declining from $2.48 as of 30th June 2008 to $1.53 as of 30th June 2013.
Because of those share repurchases, Colgate-Palmolive's book value declined at an annual rate of
9.2%. In summary, when a company repurchases shares at a price higher than current BVPS, it
lowers the overall BVPS for the company. All else being equal, the effect is to make the stock
appear more expensive if the current P/B is compared to its historical values.
6. Book value of equity can be made negative by a series of negative earnings, which limits the
usefulness of the variable.
46
lllll
P/S Ratio
P/S = Mv of Equity / Common Stock o/s Shares = Market Price per Share
Total Sales / Common Stock o/s Shares
Sales per Share
Whereas,
Total Sales : Sales - Returns and Customer Discounts
If a company is engaging in questionable revenue recognition practices and the amount being
manipulated is unknown, the analyst might do well to suggest avoiding investment in that
company's securities. At the very least, the analyst should be skeptical and assign the company a
higher risk premium than otherwise, which would result in a lower justified P/B.
P/S = P/E x Net Profit Margin
For two stocks with the same positive P/E, the stock with the higher P/S has a higher (actual or
forecasted) net profit margin, calculated as the ratio of P/B to P/E.
Justified Trailing P0 / S 0 = (E 0 / S 0 ) (1 - b) (1 + g)
r-g
= (E 0 / S 0 ) x (1 - b) (1 + g)
r-g
= Net Profit Margin x Justified Trailing P/E
Justified Leading P0 / S 1 = (E 0 / S1 ) (1 - b)
r-g
= (E 0 / S1 ) x (1 - b)
r-g
= Net Profit Margin x Justified Leading P/E
Advantages
1. P/S is meaningful even for distressed firms, since sales revenue is always positive unlike P/E and
P/B. P/S ratios are not as volatile as P/E multiples. This may make P/S ratios more reliable in
valuation analysis when earnings for a particular year are volatile relative to long-run.
2. Sales revenue is not as easy to manipulate or distort as EPS and book value which are
significantly affected by accounting conventions.
3. Like P/E and P/B ratios, empirical research finds that differences in P/S are significantly related to
differences in long-run average stock returns.
4. P/S ratios are particularly appropriate for valuing stocks in mature or cyclical industries and
start-up companies with no record of earnings. It is also often used to value investment
management companies and partnerships.
Disadvantages
1. High growth in sales doesn't necessarily indicate high operating profits as measured by earnings
and cashflows.
2. P/S ratios don't capture differences in cost structures across companies.
3. Share price reflects the effect of debt financing on profitability and risk. In the P/S multiple,
47
however, price is compared with sales, which is a pre-financing income measure - a logical
mismatch. For this reason, some experts use a ratio of enterprise value to sales because
enterprise value incorporates the value of debt.
4. Although P/S is relatively robust with respect to manipulation, revenue recognition practices have
the potential to distort P/S. Analysts should look for company practices that speed up revenue
recognition. Example, Bill and Hold basis, this practice accelerates sales into an earlier reporting
period and distorts the P/S ratio.
lllll
P/CF Ratio
P/CF = Mv of Equity / Common Stock o/s Shares = Market Price per Share
Cashflow / Common Stock o/s Shares
CF per Share
Whereas,
Cashflow : CF, CFO, Adjusted CFO (to reflect changes when comparing companies that use
different accounting standard), FCFE or EBITDA
P0 = FCFE 0 (1 + g)
r-g
Justified P/CF = FCFE 0 (1 + g) / r - g
CF
Advantages
1. Cashflow is harder for managers to manipulate than earnings.
2. P/CF is more stable than P/E. Using P/CF rather than P/E addresses the issue of differences in
accounting conservatism between companies (differences in the quality of earnings).
3. Empirical evidence indicates that differences in price to cashflow are significantly related to
differences in long-run average stock returns.
Disadvantages
1. Some companies have increased their use of accounting methods that enhance cashflow
measures. CFO can be enhanced by securitizing accounts receivable to speed up a company's
operating cash inflow or by outsourcing the payment of accounts payable to slowdown the
company's operating cash outflow. Companies make a number of opportunistic accounting
choices to increase their reported CFO.
2. CFO under IFRS may not be comparable to the CFO under US GAAP.
3.
Ignores
NI + NCC
FCInv and WCInv
P
CFO
FCInv
CF
Adjusted CFO
CFO + Interest (1 - t)
FCFE
(a) Can be -ve, if high Capex
(b) More volatile i.e. not informative
lllll
D/P Ratio
Trailing D0 / P0 = 4 x Most Recent Quarterly Dividend
Market Price per Share
Leading D0 / P0 = Forecasted Dividends over next 4 Years
Market Price per Share
48
Total return on an investment has two components: Dividend Yield (D/P) and Capital Appreciation.
The supposed lower risk of dividends relative to capital appreciation assumes that the market is
biased in its assessment of the components of return.
V0 = D0 (1 + g)
r-g
1= r-g
V0 D0 (1 + g)
D0 = D0 (r - g)
V0 D 0 (1 + g)
Justified D/P = r - g
1+g
D/P is positively related to 'r' and negatively related to the forecasted growth rate in dividends. This
implies that choosing high D/P stocks reflects a value rather than a growth investment strategy.
Example 9:
OnePrice Inc. just paid a dividend of $0.5 per share. The consensus forecasted dividends for OnePrice Inc.
over the next four quarters are $0.5, $0.55, $0.6 and $0.65. The current market price is $47.5. Calculate
Leading and Trailing D/P.
Trailing D/P = 4 x 0.5 = 0.042 or 4.2%
47.5
Leading D/P = 0.5 + 0.55 + 0.6 + 0.65 = 0.048 or 4.8%
47.5
Advantages
1. Dividends are not as risky as the capital appreciation component of total return. Dividends are
fairly stable.
Disadvantages
1. Investors may tradeoff future earnings growth to receive higher current dividends, which means,
dividends paid now displace earnings in all future periods, known as Dividend Displacement of
Earnings.
2. Another consideration used by some investors is the security of the dividend (the probability that
it will be reduced or eliminated). A useful metric in assuming the safety of the dividend is the
payout ratio. A high payout relative to other companies operating in the same industry may
indicate a less secure dividend because the dividend is less well covered by earnings. Relevant
ratios to consider include the interest coverage ratio and the ratio of net debt to EBITDA.
lllll
EV/EBITDA Ratio
Only for Value
Equity of Firm
✓
x
↷
EV/EBITDA = P or EV
EBITDA
Earnings
for both
Debt & Equity
Cash and Investments sometimes
termed as 'non-earnings assets' are
substracted because EV is designed to
measure the net price an acquirer
would pay for the company as a whole.
49
↷
Whereas,
EV : Mv of Common Stock + Mv of Debt + Mv of Preferred Equity + Minority Interest - Cash and
Investments
EBITDA : NI (recurring earnings from continuing operations) + Interest + Taxes + Depreciation +
Amortization or
: EBIT + Depreciation + Amortization
Pre-interest
Method
Return on Invested Capital (ROIC) is calculated as operating profit after tax divided by total invested
capital. In analyzing ratios such as EV/EBITDA, ROIC is the relevant measure of profitability because
EBITDA flows to all providers of capital. Total Invested Capital (TIC) a.k.a. 'Market Value of Invested
Capital', that is an alternative to EV. Similar to EV, TIC includes the Mv of Equity and Debt, but
doesn't deduct, instead includes cash and investments.
Example 10:
Western Digital Corporation (WDC) manufactures hard disk drives.
Balance Sheet
ASSETS
Cash and Cash Equivalents
Accounts Receivables
Inventories
Other Current Assets
Total Current Assets
PPE
Goodwill and Other Intangible Assets
Other Non-Current Assets
Total Non-Current Assets
Total Assets
$
4060
1700
1197
383
7340
3803
2610
174
6587
13,927
LIABILITIES
Accounts Payable
Accrued Expenses
Accrued Warranty
Current Long-term Debt
Total Current Liabilities
Long-term Debt
Other Liabilities
Total Non-Current Liabilities
Total Liabilities
2037
837
122
230
3226
1783
495
2273
5504
EQUITY
Common Stock ($0.01 par value) Outstanding Shares 238
Additional Paid-in-Capital
Accumulated Comprehensive Income
Retained Earnings
Treasury Stock - Common Stock at Cost
Total Shareholder's Equity
3
2232
20
7073
(905)
8423
From WDC's financial statements, the income statement and statement of cashflows for the year ended 29th
50
June 2012 and for the nine months ended 29th March 2013 and 30th March 2012 provided the following
items (in millions).
Source
Net Income
I/S
Income Tax Provision
I/S
Interest Expense
I/S
Depreciation & Amortization CF
Year Ended
29th June, 2012
9 Months Ended
29th March, 2013
9 Months Ended
30th March, 2012
1612
145
14
825
1245
207
35
931
867
88
8
486
The company's share price as of 1st July 2013 was $63.06. Calculate EV/EBITDA.
EV = Mv of Equity + Mv of Long-term Debt - Cash
= (238 x 63.06) + (1783 + 230) - 4060
= 15,008 + 2013 - 4060
= $ 12,961
EBITDA = NI + Interest + Taxes + Depreciation + Amortization
= (1612 + 1245 - 867) + (14 + 38 - 8) + (145 + 207 - 88) + (825 + 931 - 486)
= 1990 + 41 + 264 + 1270
= $ 3565
EV
= 12,961 = 3.6
EBITDA
3565
Advantages
1. EV/EBITDA is usually more appropriate than P/E alone for comparing companies with different
financial leverage (debt), because EBITDA is a pre-interest earnings figure, in contrast to EPS,
which is post-interest. EV is less sensitive to the effects of financial leverage than price multiple.
2. EBITDA is useful for valuing capital-intensive businesses with high levels of depreciation and
amortization.
3. EBITDA is usually positive even when EPS is not.
Disadvantages
1. If working capital is growing, EBITDA will overstate CFO. Further, the measure ignores how
different revenue recognition policies affect CFO.
2. Because FCFF captures the amount of capital expenditures, it is more strongly linked to the
valuation theory than EBITDA. EBITDA will be an adequate measure, if capital expenses equal
depreciation expenses, to reflect differences in businesses' capital programs.
lllll
EV/Sales Ratio
EV/S = EV
Sales
EV/Sales ratio can be used as an alternative to the P/Sales ratio. For example, a P/S for a company
with little or no debt would not be comparable to a P/S for a company that is largely financed with
debt. The EV/S ratio is appropriate for comparing companies with significantly different capital
structures.
The least affected is P/FCFE and P/CFO, while P/B, P/E, P/EBITDA and EV/EBITDA will be more seriously
affected because they are more influenced by management's choice of accounting methods and estimates.
51
lllll
Momentum Indicators
Momentum Indicators relate to either the market price or a fundamental variable like EPS to the time
series of historical or expected value. Common momentum indicators include Earnings Surprise
(Unexpected Earnings), Standardized Unexpected Earnings (Scaled Earnings Surprise) and Relative
Strength (Price Momentum). An analyst might be correct about the intrinsic value of a firm and the
momentum indicators might provide a clue about when the market price will converge with that
intrinsic value.
1. Earnings Surprise: Reported EPS - Expected EPS
This is usually scaled by a measure that expresses the variability if analysts' EPS forecasts. The economic
rationale for examining earnings surprises is that, positive surprises may lead to persistent positive
abnormal returns.
2. Standardized Unexpected Earnings:
Earnings Surprise
o of Earnings Surprise
The principle is that the smaller (larger) the historical size of forecast errors, the more (less) meaningful a
given size of EPS forecast error.
3. Relative Strength: compares a stock's performance during a particular period either with its own past
performance or with the performance of some group of stocks (stock's performance divided by the
performance of an equity index, if the value of this ratio increases, the stock price increases relative to the
index and displays positive relative strength). Often, the relative strength indicator is scaled to 1 at the
beginning of the study period. If the stock goes up at a higher (lower) rate than the index, then relative
strength will be above (below) 1.
lllll
Valuation Indicators
Consider two stocks: one priced at $10 with earnings of $1 per share (P/E = 10) and the other priced at
$16 with earnings of $2 per share (P/E = 8). Therefore, a portfolio P/E will be ($10 + $16) / ($1 + $2) =
$8.67.
n
1. Arithmetic Mean = Σ P/E i = 10 + 8 = 9
i=1
n
2
n
2. Weighted Mean = Σ w i x P/E i = 10 x 10 + 16 x 8 = 8.76
i=1
26
26
( )
3. Harmonic Mean =
( )
n
=
2
= 8.88
n
Σ (1 / P/E i ) (1/10) + (1/8)
i=1
4. Weighted Harmonic Mean =
1
=
1
= 8.67
Σ w i x (1 / P/E i ) (10/26) (1/10) + (16/26) (1/8)
n
i=1
When there are extreme high or low outliers, the Arithmetic mean will be the most affected. The
Harmonic mean inherently gives less weight to higher P/Es and more weight to lower P/Es. In general,
unless all the observations in a dataset have the same value, the Harmonic mean < Arithmetic mean.
The Harmonic mean tends to mitigate the impact of large outliers, but may aggravate the impact of
small outliers i.e. those close to '0'. In this case, Median or Weighted Harmonic Mean with the outliers
excluded may be the most appropriate measures of the P/E for a portfolio or index. For an equal
weighted portfolio or index, the Harmonic mean and Weighted Harmonic mean will be equal.
52
CHAPTER 11
Residual Income Valuation
↷
Residual Income (RI) or Economic Profit or Abnormal Earnings or EVA or Edwards-Bell-Ohlson Model (EBOM)
is the net income of a firm less a charge that measures stockholder's opportunity cost of capital. Residual
Income is the amount of earnings during the period that exceeds the investor's required return. The appeal
of residual income model stems from a shortcoming of traditional accounting. Specifically, although a
company's income statement includes a charge for the cost of debt capital in the form of interest expense, it
doesn't include a charge for the cost of equity capital or dividends. This means that accounting income may
overstate returns from the perspective of equity investors. Residual Income models explicitly recognize the
costs of all the capital used in generating income (RI explicitly deducts all capital costs). Suppose
shareholder's initial investment is $200 and the required rate of return on the stock is 8%. The company
earns $18 in the course of a year. How much value has the company added for shareholders? A return of
0.08 x 200 = $16 just meets the amount investors could have earned in an equivalent risk investment i.e.
opportunity cost. Only the residual or excess amount of $18 - $16 = $2 represents value added or an
economic gain to shareholders. The residual income approach attempts to match profits to the time period in
which they are earned, but not necessarily realized as cash.
Equity Charge
↷
RI = NI - (Equity Capital x re )
Capital Charge
RI = EBIT (1 - t) - (Total Capital x WACC)
The residual income approach can be applied to firms with negative FCF and to dividend and non-dividend
paying firms. Residual Income models can be more difficult to apply. Management discretion in establishing
accruals for both income and expense may obscure the true results for a period. If the accounting is not
transparent or if the quality of the firm's reporting is poor, the accurate estimation of residual income is likely
to be difficult.
Example 1:
Axis Manufacturing Company, Inc. (AXCI), a very small company in terms of market capitalization, has total
assets of $2000,000 financed 50% with debt and 50% with equity capital. The cost of debt is 7% before taxes;
this example assumes that interest is tax deductible, so the after-tax cost of debt is 4.9%. The cost of equity
capital is 12%. The company has EBIT of $200,000 and a tax rate of 30%. Net income is $91,000. Now calculate
Residual Income.
WACC = (0.5) (0.049) + (0.5) (0.12) = 0.0845 or 8.45%
RI = EBIT (1 - t) - (Total Capital x WACC)
= 200,000 (1 - 0.3) - (2000,000 x 0.0845)
= 140,000 - 169,000
= - 29,000
Economic
Sense
RI = NI - (Equity Capital x re )
= 91,000 - (1000,000 x 0.12)
= - 29,000
Equity Charge = Equity Capital x re
= 1000,000 x 0.12
= 120,000
Accounting
Sense
53
AXCI didn't earn enough to cover the cost of equity capital. As a result, it has negative residual income.
Although AXCI is profitable in an accounting sense, it is not profitable is an economic sense. In summary,
higher (lower) residual income is expected to be associated with higher (lower) market prices with higher P/Bs
(lower P/Bs). A company that is generating more income than its cost of obtaining capital i.e. positive residual
income is creating value. Conversely, a company that is not generating enough income to cover its cost of
capital i.e. negative residual income is destroying value.
Economic Value Added
Economic Value Added (EVA) measures the value added for shareholders by management during a given
year.
$WACC
EVA = EBIT (1 - t) - (Total Capital x WACC)
EVA = NOPAT - (Total Capital x WACC)
Whereas,
NOPAT : Net Operating Profit after Tax. (Interest expense is not deducted and leverage
is not considered).
Total Capital : Net Working Capital + Net Fixed Assets or
: Bv of Long-term Debt + Bv of Equity
(The analyst should make the following adjustments (if applicable) to the financial statements
before calculating NOPAT and Invested Capital:
- R&D expenses are capitalized and amortized rather than expensed i.e. R&D expense, net of
estimated amortization, is added back to earnings to compute NOPAT.
- In the case of strategic investments that are not expected to generate an immediate return,
a charge for capital is suspended until a later date.
- Eliminate deferred taxes and consider only cash taxes as an expense.
- Treat operating leases are capital leases and adjust non-recurring items.
- Add LIFO reserve to invested capital and add back change in LIFO reserve to NOPAT)
Market Value Added
Market Value Added (MVA) is the difference between the market value of a firm's long-term debt and equity
and the book value of invested capital supplied by investors. It measures the value created by management's
decisions since the firm's inception.
↶
Accounting Bv
MVA = Market Value - Total Capital
∴ For the purpose of EVA computation, we use beginning-of-year total capital. MVA uses end-of-year
(the same point at which market value is determined) total capital.
Example 2:
VBM Inc. reports NOPAT of $2100, a WACC of 14.2% and invested capital of $18,000 at the beginning of the
year and $21,000 at the end of the year. The market price (year-end) of the firm's stock is $25 per share and
VBM has 800 shares outstanding. The market value (year-end) of the firm's long-term debt is $4000. Calculate
EVA and MVA.
EVA = NOPAT - (Total Capital x WACC)
= 2100 - (18,000 x 0.142) = - 456
MVA = Market Value - Total Capital
= (25 x 800) + 4000 - 21,000 = 3000
54
In the long term, companies that earn more than the cost of capital should sell for more than book value and
companies that earn less than the cost of capital should sell for less than book value.
Per Share Cost of Equity
RI t = E t - (B t-1 x r)
Type I
RI t = (ROE - r) x B t-1
Type II
Whereas,
B t : B t-1 + E t - D t
The Residual Income Valuation Model breaks the intrinsic value of a stock into two elements: (a) Current Book
Value of Equity and (b) Present Value of Expected Future Residual Income.
{ (1RI+ r) + (1RI+ r) + ... + (1RI+ r) }
V0 = B 0 +
1
2
1
n
2
n
Example 3: [Type I]
Consolidated Pipe Products has a required rate of return of 14%, the current book value is $6.5. Earnings
forecasts for 2019, 2020 and 2021 are $1.1, $1 and $0.95 respectively. Dividends in 2019 and 2020 are
forecasted to be $0.5 and $0.6 respectively. The dividend in 2021 is a liquidating dividend, which means that
consolidated will pay out its entire book value in dividends and cease doing business at the end of 2021.
Calculate the value of consolidated stock using the residual income model.
↷
RI19 = 1.1 - (6.5 x 0.14) = 0.19
B 19 = 6.5 + 1.1 - 0.5 = 7.1
↷
RI 20 = 1 - (7.1 x 0.14) = 0.01
B 20 = 7.1 + 1 - 0.6 = 7.5
RI 21 = 0.95 - (7.5 x 0.14) = - 0.1
V0 = 6.5 + 0.19 + 0.01 - 0.1 = $ 6.61
(1.14)1 (1.14) 2 (1.14) 3
Example 4: [Type II]
Bugg Properties has expected earnings per share of $2, $2.5 and $4, and expected dividends per share of $1,
$1.25 and $12.25 for the next 3 years. Analysts expect that the last dividend will be a liquidating dividend and
that Bugg will cease operating after Year 3. Bugg's current book value per share is $6 and its estimated
required rate of return on equity is 10%.
(2/6)
(2.5/6)
↷
RI 1 = (0.333 - 0.1) x 6 = 1.393
B 1= 6 + 2 - 1 = 7
(4/8.25)
↷
RI 2 = (0.3571 - 0.1) x 7 = 1.7997
B2 = 7 + 2.5 - 1.25 = 8.25
RI 3 = (0.4848 - 0.1) x 8.25 = 3.17
V0 = 6 + 1.393 + 1.7997 + 3.17 = $ 11.15
(1.1)1 (1.1) 2 (1.1) 3
Value tends to be recognized earlier in the RI approach than in other present value based approaches. To see
55
this, recall that with a DDM or FCFE model, a large portion of the estimated intrinsic value comes from the
present value of the expected terminal value. Yet the uncertainty of the expected terminal value usually
greater than any of the other forecasted cashflows, because it occurs several years in the future. Valuation
with RI models, however, is relatively less sensitive to terminal value estimates which reduces forecast error.
This is because intrinsic values estimated with RI models include the firm's current book value and the
current book value usually represents a substantial percentage of the estimated intrinsic value.
If ROE > re
ROE < re
ROE = re
Pv of RI: Positive P0 > B 0
Pv of RI: Negative P0 < B 0
Pv of RI: 0
P0 = B 0
Let's begin with Justified P/B,
P0 / B 0 = ROE - g
r-g
Add and Substract 're '
in the numerator
P0 / B 0 = ROE - g + r - r
r-g
P/B > 1
P/B < 1
P/B = 1
= (ROE - r) + (r - g)
r- g
= 1 + (ROE - r)
r-g
Constant RI Model
[
P0 = B 0+ (ROE - r) . B 0
r-g
]
or
B 0 + RI 1
r-g
A drawback to the single-stage model is that it assumes the excess ROE above cost of equity will persist
indefinitely. More likely, a company's ROE will revert to a mean value of ROE over time and at some point, the
company's RI will be '0'.
Example 5:
Western Atlantic Railroad has a book value of $23 per share. The company's return on new investments (ROE)
is 14% and its required return on equity is 12%. The dividend payout ratio is 60%. Calculate the value of
shares using GGM.
g = (1 - b) x ROE
= (1 - 0.6) X 0.14
= 0.056 or 5.6%
V0 =
↷
E1 = Equity x ROE
= 23 x 0.14 = $ 3.22
0.6 (3.22)
1.932
= $ 30.19
0.12 - 0.056
Example 6:
You are considering the purchase of Tellis Communications Inc. which has a P/B ratio of 2.5. ROE is expected
to be 13%, current book value per share is $8 and the cost of equity is 11%. Calculate the growth rate implied
by the current P/B ratio.
56
P0 = 3 x 2.5 = 20
20 = 3 +
- 11% x 8
[ 13%
11% - g ]
↷
g = 0.0967 or 9.67%
Multi-Stage RI Model
It is RI after the
forecast horizon
V0 = B 0 + Pv of Interim High-growth RI + Pv of Continuing RI
RI will continue beyond a specified earnings horizon depending on the fortunes of the industry as well as on
the sustainability of a specific firm's competitive prospects over the longer term. The projected rate at which
RI is expected to fade over the lifecycle of the firm is captured by a 'Persistence Factor' 'ω', which is between 0
and 1; 0 ≤ ω ≤ 1. The strength of the persistence factor will depend partly on the sustainability of the firm's
competitive advantage and the structure of the industry. The more sustainable the competitive advantage
and the better the industry prospects, the higher the persistence factor.
ROE has been found to revert to mean levels over time and may decline to the cost of equity in a competitive
environment. As ROE approaches the cost of equity, RI income approaches zero. RI approach often model
ROE fading toward the cost of equity. If ROE = re , RI = 0.
↑ Higher Persistence Factors will be associated with the following:
- Low Dividend Payouts
- Historically High RI Persistence in the Industry
↓ Lower Persistence Factors will be associated with the following:
- High ROE
- Significant Levels of Non-recurring Items
- High Accounting Accruals
Assumption 1: Residual Income Persists at the Current Level Forever (ω = 1)
Pv of Continuing RI in Year T-1 =
RI T = RI T
1+r-ω r
Assumption 2: Residual Income Drops Immediately to Zero (ω = 0)
Pv of Continuing RI in Year T-1 =
RI T = RI T
1+r-ω 1+r
Assumption 3: Residual Income Declines over Time to Zero (0 ≤ ω ≤ 1)
Pv of Continuing RI in Year T-1 =
RI T
1+r-ω
Assumption 4: Residual Income Declines to Long-term Level in Mature Industry (No 'ω')
↶
Market Value
↷
Pv of Continuing RI in Year T-1 = (PT - B T ) + RI T
1+r
Book Value
Pv of Continuing RI in Year T = PT - B T
Premium over Book value at the
end of the forecast horizon.
Whereas,
PT : B T x Forecasted P/B ratio
57
The longer the forecast period, the greater the chance that the company's RI will converge to '0'. For long
forecast periods, this last term may be treated as '0'. For shorter forecast periods, a forecast of the premium
should be calculated.
Example 7:
Jane Metals is expecting a ROE of 15% over each of the next 5 years. Its current book value is $5 per share, it
pays no dividends and all earnings are reinvested. The required return on equity is 10%. Forecasted earnings
in Year 1 through 5 are equal to ROW times beginning book value. Calculate the intrinsic value of the
company using a RI model using four assumptions mentioned before. For Assumption 3, RI will decay over
time to '0' with a persistence factor of 0.4 and for Assumption 4, assume that Jane's ROE falls to a long-run
average level and P/B ratio falls to 1.2.
Ending Book
value
Year
0
1
2
3
4
5
ROE
0.15
0.15
0.15
0.15
0.15
Equity
Charge
RI
Et
B t-1
r x B t-1
E t - (B t-1 x r)
0.75
0.86
0.99
1.14
1.31
5
5.75
6.61
7.6
8.74
10.05
0.5
0.57
0.66
0.76
0.87
0.25
0.29
0.33
0.38
0.44
Assumption 1: Pv of Continuing RI in Year 4 =
RI 5 = 0.44 = $ 4.4
1 + r - 1 0.1
V0 = 5 + 0.25 + 0.29 + 0.33 + 0.38 + 4.4 = $ 8.98
(1.1) 1 (1.1) 2 (1.1) 3
(1.1) 4
Assumption 2: Pv of Continuing RI in Year 4 = RI 5 = 0.44 = $ 0.4
1 + r - 0 1 + 0.1
V0 = 5 + 0.25 + 0.29 + 0.33 + 0.38 + 0.4 = $ 6.25
(1.1) 1 (1.1) 2 (1.1) 3
(1.1) 4
Assumption 3: Pv of Continuing RI in Year 4 =
RI 5 = 0.44 = $ 0.63
1 + r - 0.4 0.7
V0 = 5 + 0.25 + 0.29 + 0.33 + 0.38 + 0.63 = $ 6.4
(1.1) 1 (1.1) 2 (1.1) 3
(1.1) 4
Assumption 4: Pv of Continuing RI in Year 5 = 12.06 - 10.05 = $ 2.01
Pv of Continuing RI in Year 4 = (P5 - B 5 ) + RI 5 = 2.45 = $ 2.23
1+r
V0 = 5 + 0.25 + 0.29 + 0.33 + 0.38 + 2.23 = $ 7.5
(1.1) 1 (1.1) 2 (1.1) 3
(1.1) 4
RI model is applicable even when cashflows are volatile. The model focuses on economic profitability rather
58
than just on accounting profitability. The weakness being, it assumes that the 'Clean Surplus' relation holds or
that its failure to hold has been properly taken into account. But if any accounting charges that are taken
directly to the equity accounts such as currency translation gains and losses will cause the 'Clean Surplus'
relation not to hold. RI models are not appropriate if (a) the clean surplus accounting relation is violated
significantly and (b) if there is significant uncertainty concerning the estimates of book value and ROE.
Fig 1: Related Concept
Tobin's Q =
Mv of Debt + Mv of Equity
Replacement Cost of Total Assets
Tobin's Q is expected to be higher, the greater
productivity of a company's assets.
lllll
or use market value of total
asset, if replacement cost is
not available. Replacement
costs take into account the
effects of inflation.
Accounting Issues
Two principle drivers of residual earnings are ROE and Book value.
1. Clean Surplus Violations
The Clean Surplus relationship (i.e. Ending Book Value = Beginning Book Value + NI - Dividends) may not
hold when items are charged directly to shareholder's equity and don't go through the income statement.
Therefore, we have to adjust net income to account for these items if they are not expected to reverse in
the future. Items that can bypass the income statement include:
- Foreign Currency Translation Gains and Losses that flow directly to Retained Earnings under the
Current Rate Method.
- Certain Pension Adjustments
- Gains/Losses on certain Hedging Instruments
- Changes in Revaluation Surplus for Long-Lived Assets or Intangible Assets (IFRS Only).
- Changes in the Value of certain Liabilities due to changes in the Liability's Credit Risk (IFRS Only).
- Changes in the Market Value of Debt and Equity Securities Unrealized, classified as Available for Sale.
The risk in applying RI model when the clean surplus relation doesn't hold is that the ROE forecast will not
be accurate if the clean surplus violations are not expected to offset in future years. The analyst should
incorporate explicit assumptions about future accounts of OCI. In practice, to apply the RI model most
accurately, the analyst may need to: (a) Adjust book value of equity for off balance sheet items,
discrepancies from fair value or the amortization of certain intangible assets, (b) Adjust reported net
income to reflect clean surplus accounting and (c) Adjust reported net income from non-recurring items
misclassified as recurring items.
Example 8:
The company's cost of equity capital is 10%. Answer the following questions with help of table.
a. Assuming the forecasted terminal price of Mannistore's share at the end of Year 5 (t = 5) is $68.4,
estimate the value per share of Mannistore using DDM.
b. Assuming the forecasted terminal price of Mannistore's share at the end of Year 5 (t = 5) is $68.4,
estimate the value of a share of Mannistore using RI model and calculate RI based on:
(i) NI without adjustment
(ii) NI plus OCI
59
Shareholders Equity t-1
+ NI
- Dividends
- OCI
Shareholders Equity t
a. DDM
b. RI
1
2
3
4
5
8.58
2
0.26
10.32
10.32
2.48
0.29
1
11.51
11.51
3.46
0.29
14.68
14.68
3.47
0.29
17.86
17.86
4.56
0.38
22.04
V0 = 0.26 + 0.29 + 0.29 + 0.29 + 0.38 + 68.4 = $ 43.59
(1.1)1 (1.1) 2 (1.1) 3 (1.1) 4
(1.1) 5
(i) RI = NI - (B t-1 x r)
V0 = 8.58 + 1.14 + 1.45 + 2.3 + 2 + 2.77 + 68.4 - 22.04 = $ 44.42
(1.1) 1 (1.1) 2 (1.1) 3 (1.1) 4
(1.1) 5
(ii) RI = NI + OCI - (B t-1x r)
V0 = 8.58 + 1.14 + 2.45 + 2.3 + 2 + 2.77 + 68.4 - 22.04 = $ 43.59
(1.1)1 (1.1) 2 (1.1) 3 (1.1) 4
(1.1) 5
The first calculation (Bi) incorrectly omits an adjustment for violation of the clean surplus relation. The second
calculation (Bii) includes an adjustment and yields the correct value, which is consistent with the DMM
estimate.
2. Balance Sheet Adjustments for Fair Value
The accrual method of accounting causes many balance sheet items to be reported at book values that are
significantly different than their market values.
- Operating Leases: Should be capitalized by increase A & L by the present value of the expected future
operating lease payments.
- SPE: Whose A & L are not reflected in the financial statements of the parent company should be
considered.
- Reserves and Allowances: Should be adjusted. For example, the allowance for bad debts, which is an
offset to accounts receivable, should reflect the expected loss experience.
- Inventory: For companies that use LIFO should be adjusted to FIFO by adding the LIFO reserve to
inventory and equity, assuming no deferred tax impact.
- Pension Asset or Liability: Should be adjusted to reflect the funded status of the plan.
- DTL: Should be eliminated and reported as equity if the liability is not expected to reverse. For
example, If the DTL results from different depreciation methods for tax and financial statement
reporting purposes and if the company is growing.
3. Intangible Asset effects on Book Value
Two intangible assets require special allocation: (a) Intangibles recognized at acquisition and (b) R&D
Expenditures. The amortization of such intangible assets reduces the combined ROE (when company A
acquires B) and hence results in lower valuation of the combined entity compared to the sum of the values
of individual entities prior to acquisition. To remove the distortion the amortization of intangibles
capitalized during acquisition should be removed prior to computing the ROE used for RI valuation. On the
other hand, productive R&D expenditures increase ROE and RI; unproductive expenditures reduce ROE and
RI.
60
4. Non-Recurring Items
Non-recurring items should be included in RI forecasts because they represent items that are not expected
to continue in the future. Items that may need adjustment in measuring recurring earnings include
discontinued operations, accounting changes, unusual items, extraordinary items (applicable under
US GAAP but not under IFRS) and restructuring charges. In some cases, management may record
restructuring or usual charges in every period. In these cases, the items may be considered on ordinary
operating expense and may not require adjustment.
5. Aggressive Accounting Practices
One such example called 'Cookie Jar' reserves (reserves saved for future use), in which excess losses or
expenses are recorded in an earlier period and then use, to reduce expenses and increase income in future
periods.
6. International Accounting Differences
Analysts should expect the RI model to work best only if forecasts are available, clean surplus violations are
limited and accounting rules don't result in delayed recognition. If not, it should consider a model less
dependent on accounting data, such as FCFE model.
NOTES
1. If a stock is trading at a price (market price) higher than the price implied by a RI model i.e. model price,
the stock is considered to be overvalued. Similarly, if the market price is lower than the model price, the
stock is considered to be undervalued, and if the model price is equal to the market price, the stock is
considered to be fairly valued.
↶↷
MP (Overvalued)
RI Model
MP (Undervalued)
61
CHAPTER 31
Private Company Valuation
Private Company Valuation
Income Approach
Market Approach
Asset Based Approach
Values a firm as the present
value of its expected future
income a.k.a. DCF or present
value analysis.
[Absolute Valuation]
Values a firm using the price
multiples based on recent
sales of comparable assets.
[Relative Valuation]
Values a firm's assets minus its
liabilities.
[Absolute Valuation]
The selection of an appropriate valuation approach depends on the firm's operations and its lifecycle stage.
Early in its life, a firm's future cashflows may be subject to so much uncertainty that an Asset Based Approach
would be most appropriate. As the firm moves to a high growth phase, it might be approximately valued
using an Income Approach, including a particular form of the income approach known as a FCF valuation
model. A mature firm might be more appropriately valued using Market Approach.
Overall, company specific factors can have positive or negative effects on private company valuations
whereas stock specific factors are usually a negative.
lllll
Income Based Valuation
1. Normalized Earnings are 'economic benefits adjusted or exclude non-recurring, non-economic or unusual
items to eliminate anomalies and facilitate comparisons' (firm earnings if the firm were acquired). In the
case of private firms with a concentrated control, these may be discretionary or tax motivated expenses
that need to be adjusted when calculating normalized earnings. These adjustments can be quite significant
when the firm is small. In private companies, the above-market compensation or other expenses would
reduce taxable upon the payment of dividends to the controlling shareholders and other shareholders.
Above-market expenses can also result in the controlling shareholder receiving a disproportionately high
return in relation to other shareholders.
2. If the company is using owned property in its business operations, adding a market rental charge for the
use of the real estate to the expenses of the company would produce a more accurate estimate of the
earnings of the business operations. The value of real estate is therefore separated from its operations and
treated as a non-operating asset. If real estate is leased to the private company by a related entity, the level
of expense may require an adjustment to a market rental rate. If real estate is leased from an unrelated
party but the rental charge is not at a market level, an adjustment to normalize this expense may also be
appropriate.
3. The normalized earnings for a strategic buyer incorporate acquisition synergies, whereas a non-strategic
buyer's (financial) transaction doesn't.
Example 1:
Given the following figure, calculate the normalized EBITDA for a financial and strategic buyer:
Reported EBITDA
Current Executive Compensation
Market-Based Executive Compensation
Current SG&A Expenses
SG&A Expenses after Synergistic Savings
Current Lease Rate
Market-Based Lease Rate
$ 6,700,000
$ 800,000
$ 650,000
$ 8,100,000
$ 7,300,000
$ 200,000
$ 250,000
62
Calculate normalized EBITDA for Strategic Buyer and Non-Strategic (Financial) Buyer?
Strategic Buyer = 6,700,000 + (300,000 - 650,000) + (200,000 - 250,000) + (8,100,000 - 7,300,000) = $ 7,600,000
Normalized Earnings
Synergistic Savings
Non-Strategic Buyer = 6,700,000 + (800,000 - 650,000) + (200,000 - 250,000) = $ 6,800,000
Normalized Earnings
4. CAPM, Expanded CAPM and Build-Up Approach are used to estimate the required rate of return for the
private company equity:
- CAPM = R f + β (R m - R f )
- Expanded CAPM = CAPM + Size Premium + Firm-Specific (Unsystematic) Risk Premium
- Build-Up = R f + Equity Risk Premium + Size Premium + Firm-Specific (Unsystematic) Risk Premium +
Industry Premium. (No Beta)
If firms have unusually high levels of unsystematic risk, the use of the CAPM may be inappropriate.
5. Three methods consistent with the income approach are: FCF Method, the Capitalized Cashflow Method
and RI or Excess Earnings Method. The Capitalized Cashflow Method (CCM) is most often used for small
private companies.
CCM
Value of Firm = FCFF 1
(WACC - g)
Value of Equity = FCFE 1
(r - g)
Capitalization Rate
Capitalization Rate
Because equity capital is usually more expensive than debt and because the higher operating risk of
smaller private companies results in a higher cost of debt as well, WACC will typically be higher for private
firms. The Excess Earnings Method (EEM) is used significantly but can be used for small firms when their
intangible assets are significant. This residual amount of excess earnings are capitalized by using the
growing perpetuity formula from the CCM to obtain an estimate of the value of intangible assets. For
valuing the entire business, the values of working capital and fixed assets are added to the capitalized
value of intangibles.
Example 2:
Given the following figures, calculate the value of the firm using the EEM.
Working Capital
Fixed Assets
Normalized Earnings
Required Return for Working Capital
Required Return for Fixed Assets
Growth Rate of RI
Discount Rate for Intangible Assets
$ 300,000
$1000,000
$ 130,000
6%
10%
5%
14%
Step 1: Calculate Required Return for WC and FA
Working Capital = 0.06 x 300,000 = $ 18,000
Fixed Assets = 0.1 x 1000,000 = $ 100,000
63
Step 2: Calculate Excess Earnings
Excess Earnings = $ 130,000 - $ 18,000 - $ 100,000 = $ 12,000
Step 3: Value the Intangible Assets
Value of Intangible Assets = 12,000 (1.05) = $ 140,000
0.14 - 0.05
Step 4: Calculate Total Firm Value
Firm Value = $ 300,000 + $ 1000,000 + $ 140,000 = $ 1,440,000
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Market Based Valuation
Market Approaches to valuing private firms use price multiples and data from previous public and
private transactions. The three methods discussed in the following are the Guideline Public
Company Method (GPCM), the Guideline Transactions Method (GTM) and the Prior Transaction
Method (PTM).
1. Guideline Public Company Method (GPCM)
The GPCM uses price multiples from trade data for public companies with adjustments to the multiples to
account for differences between the subject firm and the comparables. When evaluating a controlling
equity interest in a private firm, the control premium i.e. the value of control should be estimated. The
control premium equals the difference between the pro-rata value of a controlling interest and the pro-rata
value of a non-controlling interest. Most public share trades are for small, non-controlling interests,
therefore, the price multiple doesn't reflect a control premium. Compared with non-strategic buyer, control
premiums for an acquisition by a strategic buyers are typically larger because of the expected synergies.
Control premium adjustments are made only to the equity portion of the firm's value. There are two way to
incorporate control premium under a GPCM.
a. Use the raw multiple to estimate firm value (without control premium) and estimate the equity
portion (by substracting debt). Apply the control premium to the equity portion as estimated.
b. Beginning with an equity control premium, we adjust this control premium for valuation using a
Market Value of Invested Capital (MVIC) multiple.
Adjusted Control Premium = Control Premium on Equity (1 - DR)
↷
This Adjusted Control Premium is
then applied to a MVIC multiple
(MVIC, from which the debt could
be substracted to derive the
equity value) based value.
Example 3:
An analyst, Natalie Hoskins, is valuing a private firm, Rensselaer Components using the GPCM and MVIC to
EBITDA multiples. Hoskins has gathered data for comparable public firms; however they are larger in size
than Rensselaer. Hoskins decided to deflate the average public company multiple by 20% to account for the
higher risk of Rensselaer. A premium of 30% was paid for a firm by an acquiring firm in the same industry.
The acquirer exchanged stock for the target.
Mv of Debt
Normalized EBITDA
Average MVIC / EBITDA Multiple
$ 1,100,000
$ 12,800,000
8
64
Calculate the equity value of Rensselaer using the GPCM.
The control premium of 30% is probably not relevant for the valuation of Rensselaer.
Adjustment to MVIC / EBITDA = 8 (1 - 0.2) = 6.4
Firm Value = 6.4 x 12,800,000 = $ 81,920,000
Equity Value = 81,920,000 - 1,100,000 = $ 80,820,000
Example 4:
An analyst is valuing a private firm on the behalf of a strategic buyer and deflates the average public company
multiple by 30% to account for the higher risk of the private firm. The analyst plans to calculate the value of
firm equity using GPCM.
Mv of Debt
Normalized EBITDA
Average Public Company MVIC / EBITDA
Control Premium from Past Transaction
$ 1,100,000
$ 12,800,000
8
25%
Adjustment to MVIC / EBITDA = 9 (1 - 0.3) = 6.3
Value of Firm = 6.3 x 27,100,000 = $ 170,730,000
Value of Equity = 170,730,000 - 2,600,000 = $ 168,130,000
Strategic Buyer (with Control Premium) = 168,130,000 (1.25) = $ 210,162,500
2. Guideline Transactions Method (GTM)
When using the GTM, prior acquisition values for entire (public and private) companies that already reflect
any control premiums are used, so no additional adjustment for a controlling interest is necessary. If a
subject transaction is non-strategic the analyst may need to adjust the historical multiple, unlike for
strategic buyers. For example, let us find out previous M&A P/E transactions across mobile taxi
e.g. Careem.
↷
Uber China
↷
Uber India
EV / EBITDA
EV / EBITDA
14x
10x
12x
Average
If we happen to acquire Careem, we should target 12x as per the past M&A data.
3. Prior Transaction Method (PTM)
The PTM uses transactions data from stock of the actual subject company and is most appropriate when
valuing minority (non-controlling) interests. The valuation under this method can be based on the actual
transaction price or multiples derived from such transactions. If available, timely and arm's length, the PTM
would be expected to provide the most meaningful evidence of value. The PTM provides less reliable
valuation evidence if transactions are infrequent.
lllll
Asset Based Valuation
The Asset Based Approach estimates the value of firm equity as the fair value of its assets minus
65
the fair value of its liabilities. It is generally not used for going concerns. The asset based approach
might be appropriate in the following circumstances (a) Firms with minimal profits and little hope
for better prospects. In this situation, the firm might be valued more highly for its liquidation value
rather than as a going concern by a firm that can put the assets to better use, (b) Finance firms such
as banks, where their asset and liability values (loan and security values) can be based on market
prices and factors, (c) Investment companies such as REITs and closed-end investment companies
(CEICs), (d) Small companies or early stage companies with few intangible assets and (e) Natural
resource firms where assets can be valued using comparable sales. If we wish to value a minority
stake in private firm, we would need to apply discounts for both a Lack of Control and a Lack of
Marketability (Liquidity).
1. Discount for Lack of Control (DLOC)
DLOC is an amount or percentage deducted from the pro-rata share of 100% of the value of an equity
interest in a business to reflect the absence of some or all of the power of control.
DLOC = 1 -
1
[ Control Premium
]
Lack of Control discounts may be necessary for valuing non-controlling equity interests in private
companies if the value of total equity was developed on a controlling interest basis.
Fig 1: Application of DLOC
REITs / CEIC
Fig 2: Premiums (Discounts) for Controlling Interest
_
=
GTM
GPCM
2. Discount for Lack of Marketability (DLOM)
DLOM is an amount or percentage deducted from the value of an ownership interest to reflect the relative
absence of a ready market for company shares. Lack of Marketability discounts are frequently applied in
the valuation of non-controlling equity interests in private companies. It is often the case that if a DLOC is
applied, a DLOM will also be applied. For example, if a controlling shareholder believes that a private firm
should not be sold, then minority shareholders both lack control and lack the ability to sell their position.
To estimate DLOM, (a) the price of the restricted shares is compared to the price of the publicly traded
shares, (b) the price of pre-IPO shares is compared to that of post-IPO sharers (since post-IPO firms have
more certain cashflows and lower risk, the estimated DLOM may not purely reflect changes in
marketability) and (c) put option divided by the stock price, where the put used is 'at-the-money'. One
advantage of the put option analysis is the ability to directly address perceived risk of the private company
66
through volatility estimate (estimated risk of the firm can be factored into the option price). Put option only
provide price protection (the protection lasts for the life of the option) but doesn't provide liquidity for the
asset holding. It does not exactly model lack of marketability. In addition, a variety of other potential
valuation discounts exist like key person discounts, portfolio discounts (discounts for non-homogeneous
assets) and possible discounts for non-voting shares.
Total Discount = 1 - (1 - DLOC) (1 - DLOM)
Fig 3: Variations in DLOM
↓
↑
Example 5:
Suppose that a minority shareholder holds 15% of a private firm's equity and that the CEO holds the other
85%. There are two possible scenarios:
Scenario 1: The CEO will likely sell the firm very soon. A DLOM of 5% will be applied and a DLOC will not be
applied under the assumption that all selling shareholders will receive the same price. The value
of the firm's equity is estimated at $10,000,000.
Scenario 2: The CEO has no plans to sell the firm and the minority shareholder cannot sell its interest easily.
A DLOM of 20% will be applied. A DLOC will be estimated by using reported earnings instead of
normalized earnings to provide an estimated firm equity value of $9,000,000.
Calculate the value of minority shareholders equity.
Scenario 1
Scenario 2
Minority Interest = 10,000,000x 0.15 = 1,500,000
Minority Interest = 9,000,000x 0.15 = 1,350,000
1,500,000
- 75,000
1,425,000
1,350,000
- 270,000
1,080,000
1,500,000 (1 - 0.05)
1,350,000 (1 - 0.2)
The smaller value of the minority interest in Scenario 1 is due to the higher DLOM and the DLOC (as reflected
in the lower firm equity value of $ 9,000,000).
Of the three approaches, the Asset-Based Approach generally results in lowest valuation.
NOTES
1. There are 3 reasons for valuing the total capital or equity capital of private companies:
(a) Transaction related valuations are necessary when selling or financing a firm,
(b) Compliance related valuations are performed for legal or regulatory reasons and primarily focus on
financial reporting and tax issues
(c) Litigation related valuations may be required for shareholders suits, damage claims, lost profit claims or
divorce settlements.
67
2. Uniform Standards of Professional Appraisal Practice (USPAP) is a quasi-governmental entity. This standard
covers real estate, fixed income and private business valuation. The International Valuation Standards
Committee (IVSC) has created the International Valuation Standards, concerning businesses, business
interests, real estate and tangible as well as intangible assets.
3. Both market values and intrinsic values ignore the control premium and the value of specific synergies for
an acquisition, unlike investment value.
4. Large Private Firm: EBIT or EBITDA Multiples
Small Private Firm: Net Income Multiples
Extremely Small Firm: Revenue Multiples
1
Portfolio Management
PAGE NOS. 50
CHAPTER 43
VOL. 7
CHAPTERS. 6
Exchange Traded Funds: Mechanics and Applications
Mutual Fund
ETF
Mutual Fund (MF) shares must be purchased or sold
at the end of the day from the fund manager or via.
a broker at the closing NAV of the fund's holdings, in
a cash-for-shares or shares-for-cash swap.
Exchange Traded Funds (ETF) trade during the day,
just like a stock. ETF shares are created or redeemed
in kind, in a shares-for-shares swap; can be created
or redeemed continuously.
Primary Market
The Primary Market for ETF trading is that which exists on an over-the-counter basis between Authorized
Participants (APs), a special group of institutional investors, and the ETF issuer or sponsor. This process is
referred to as 'Creation/Redemption'. The 'Trade' in this market is in-kind i.e. a pre-specified basket of
securities (which can include cash) is exchanged for a certain number of shares in the ETF.
Creation/Redemption enables ETFs to operate at lower cost and with greater tax efficiency than mutual funds
and generally keeps ETF prices in line with their NAVs.
AP acquires the securities
in the CB using AP's
inventory or cash
1
4 a. Creation Units
b. Redemption
Baskets
2
①
AP delivers this
basket of
securities to
ETF manager
③
②
In exchange for an
equal value of ETF
shares
3
ETF Issuer or
Sponsor
1. Authorized Participants (AP) are large broker/dealers often market makers, who are authorized by the ETF issuer to participate in the
Creation/Redemption process. This in-kind swap happens off the exchange, in the primary market for the ETF shares.
2. Each business day, the ETF manager publishes a list of required in-kind securities for each ETF. For instance, an S&P 500 Index ETF will
typically list the index securities in quantities that reflect the index weighting. The list of securities specific to each ETF and disclosed publicly
each day is called the 'Creation Basket'. This basket determines the intrinsic NAV of the ETF based on prices during the trading day.
3. Importantly, the pricing of both the ETF and the basket is of minimal concern in this exchange: If the issuer receive 100 shares of certain
stock as part of the CB, the price the AP might have paid to acquire that stock or what its price happens to be at the end of the day is not
relevant to the exchange taking place. Because it is an in-kind transaction, all that matters is that 100 shares of the required stock move
from the AP's account to the ETF's account. Similarly, when the issuer delivers ETF shares to an AP, the ETF's closing NAV is not relevant.
This exchange of shares happens after markets are closed through the settlement process.
4. These transactions between the AP and the ETF manager are done in large blocks called 'Creation Units'. This in-kind exchange involves the
basket of underlying securities is exchange for a no. of ETF shares of equal value. The basket of securities the AP receives when it redeems
the ETF shares is called the 'Redemption Basket'. Although the actual process of exchanging baskets and blocks of ETF shares happens after
the markets are closed, the AP is able to execute ETF trades throughout the trading day because the AP knows the security composition of
the basket needed for ETF share creation or redemption, because of the fund's daily holdings disclosures to APs.
Securities in CB > ETF Shares: ETF Shares Undervalued i.e. ETF Shares Redemption [ETF Trading at Discount to NAV]
Securities in CB < ETF Shares: ETF Shares Overvalued i.e. ETF Shares Creation [ETF Trading at Premium to NAV]
2
For example, (a) If ETF is trading in the market at $25.1. The fair value of the ETF based on its underlying
securities, however is only $25. So an AP will step in to transact and buy the basket of securities (at ETF fair
value of $25) and simultaneously sell ETF shares on the open market for $25.1, realizing the $0.1 per share
difference. The AP may choose to create additional ETF shares by exchanging the basket securities for ETF
shares with the fund's issuer. This action puts downward pressure on the ETF price because the AP is selling
shares out into the market and puts upward pressure on the prices of the underlying shares. APs will repeat
this process until no further arbitrage opportunity exists [ETF Shares Creation] and (b) If ETF is trading in the
market at $ 24.9; the fair value of the underlying stock is $25. Here, the AP market maker steps in and
purchases ETF shares in the open market while simultaneously selling the stocks on exchange, realizing the
$0.1 per share price difference. The AP may choose to redeem ETF shares by exchanging them for the basket
securities with the fund's issuer. If the share price continues to be at a discount, the AP will continue this
process until no further arbitrage opportunity exists [ETF Shares Redemption].
Secondary Market
ETF shares trade in the secondary market on exchanges. For inventors, exchange trading is the only way to
buy or sell ETFs, through a brokerage account.
Placed buy
order
①
Broker submits the order
to the public market to
find a willing seller
(Investor 2)
②
③
Market maker,
Broker, Dealer or
opposite Transactor
Order executes
Investor 1's
and Investor 1
Account
receives ETF shares
ETF manager or issuer doesn't know that you have brought the shares from the secondary market nor does it
receive an inflow of money to invest. Shares simply transfer in the open market, the secondary market for
ETF shares from one investor to another and go through a settlement process based on the local exchange
where the transaction took place.
The arbitrage gap can be as small as the minimum tick size in the local market e.g. - $0.01 in the US markets,
whereas for other ETFs with underlying securities that are hard to trade e.g. high yield bonds can be more
than 1% wide. A significant advantage of the ETF Creation/Redemption process is that the AP absorbs all costs
of transacting the securities for the fund's portfolio. AP's pass these costs to investors in the ETF's bid-ask
spread, incurred by ETF buyers and sellers. Thus, non-transacting shareholders of an ETF are shielded from
the negative impact. In contrast, when investors enter or exit a traditional mutual fund, the mutual fund
manager incurs costs to buy or sell investments arising from this activity, which affect all fund shareholders.
This makes the ETF structure inherently more fair. Additionally, because creation and redemption happen inkind, they allow the ETF's portfolio managers to manage the cost basis of their holdings by selecting low basis
holdings for redemptions leading to greater tax efficiency.
US Settlement: In US, all trades that have been entered into a given business day are submitted at the end of
the day to the National Security Clearing Corporation and Depositary Trust Company (NSCC). As long as both
parties of a transaction agree that Party 1 sold to Party 2 'N' shares of XYZ stock, the NSCC becomes the
guarantor of that transaction - the entity that ensures all parties are immunized against the financial impact
of any operational problems. On the evenings of the trade; the trade is considered 'cleared'. After this point,
the buyer is guaranteed beneficial ownership in the stock or ETF as of the time, the trade was marked
'executed', even if something (i.e. bankruptcy) happens to the seller before the trade is settled. The
Depositary Trust Company (DTC) of which the NSCC is a subsidiary, holds the books of accounts - the actual
3
list of security holders and ownership. For example, suppose at the end of a trading day the following is true:
(i) E-Trade owes Schwab 1000 shares of SPY and (ii) Schwab owes Bank of America Merill Lynch 1000 shares of
SPY. Then, from the DTC's perspective, Schwab is 'whole' i.e. it is both owned and owes 1000 shares of SPY.
To settle the day's transactions. E-Trade's account will be debited the 1000 shares of SPY and Bank of America
Merill Lynch will be credited 1000 shares. The NSCC has T+2 days to complete this process and have each firm
review its records and correct any discrepancies. Market markers receive special treatment on settlement
requirements because of the time required to create or borrow ETF shares, market makers are given up to
T+6 days to settle their accounts.
European Settlement: In Europe, the majority of ETF owners are institutional investors. Additionally the
market is fragmented across multiples exchanges, jurisdictions and clearing houses.
∴ In US, trades are cleared and settled centrally. In Europe, they are cleared to one of 29 central securities
depositories (i.e. fragmented). This has no direct impact on investors other than the inherent complexity of
such a system, which may result in wider spreads and higher local market trading costs.
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Understanding ETF
Even though an ETF's expense ratio is useful, it doesn't fully reflect the cost of holding an ETF.
Index Tracking is often evaluated using the one-day difference in returns between the fund, as
measured by its NAV and its Index. Tracking Error is defined as the annualized standard deviation of
the daily differential returns of the ETF and its benchmark i.e. reports by how much ETF's returns
deviate from those of its benchmark over time. Tracking Error as a statistic reveals only ETF tracking
variability i.e. it doesn't reveal to investors whether the fund is over-or-underperforming its index or
whether that Tracking Error is concentrated over a few days or is more consistently experienced.
Therefore, Tracking Error should be assessed with the mean or median values. An alternative approach
is to look at Tracking Differences calculated over a longer holding period. A series of rolling holding
periods can be used to represent both central tendencies and variability. Such an analysis allows
investors to see the cumulative effect of portfolio management and expenses over an extended period.
It also allows for comparison with other annual matrices such as a fund's expense ratio.
ETFs include several unique risks like counterparty risk, settlement risk and investor related risk.
Although Exchange Traded Notes (ETNs) trade on exchanges and have a creation/redemption
mechanism, they are not truly funds because they don't hold underlying securities. In the US, ETNs are
registered under the Securities Act of 1993 because they are general obligation debt securities of a
bank and are not managed by an investment firm for a fee. Similar ETN structures exist in most
markets where ETFs are listed. ETNs have the largest potential counterparty risk of all exchange traded
products because they are unsecured, unsubordinated debt notes and therefore, are subject to default
by the ETN issuer. Theoretically, an ETN's counterparty risk is 100% in the event of an instantaneous
default by the underwriting bank, because ETNs and deposit-based ETFs are backed by banks, their
default risk can be monitored via. the issuing bank's credit default swap (CDS) pricing. A fund that uses
OTC derivatives has settlement risk; to minimize settlement risk, OTC contracts are typically settled
frequently that reduces the exposure the swap partners face if a company goes bankrupt, but there is a
theoretical risk of counterparty default between settlement periods. With ETFs, swap exposures are
somewhat transparent because these holdings are disclosed daily by the ETF provider, although full
information on counterparties and terms may not be disclosed. ETFs may introduce risks to investors
who don't fully understand them. For many investors, leveraged and inverse ETFs fall into this category
by failing to meet investor expectations.
E.g. 300% Exposure
(100 x 3x)
NAV = $ 100
$ 300
+15 (100 x 3 x 0.05)
Return 5% = $ 300 (1.05) = $ 315
End of Day NAV = $ 100 + $ 15 = $ 115
↑
4
In order to deliver 300% of the index's daily performance for the following day, the TEF now valued at
$115, requires notional exposure of $345 for 3 times exposure ($115 x 3), because at the end of the day
the ETF has only $315 in exposure, it must reset its expenses, in this case, increasing notional swap
exposure by $30 ($345 - $315).
Example 1:
lllll
Tracking Error
3.5%
9.375%
3%
According to the data, which portfolio mostly
likely exhibits the risk characteristics of an
aggressive active equity manager?
Pacific Rim
↶
Sector Region
Euro Zone
Pacific Rim
Japan
Executive Passive Investment Strategy:
Aggressive Active Equity Manager:
↓ Tracking Error
↑ Tracking Error
Fees and Expenses
Fund Operating Expenses vary by ETF, but all else equal, one would normally expect an index fund
to underperform its benchmark on on annual basis by the amount of its expense ratio.
ETF Bid-Ask Spreads are generally less than or equal to the combination of the following =
± Creation/Redemption Fees and other direct trading costs such as Brokerage and Exchange Fees
+ Bid-Ask Spreads of the underlying securities held in the ETF
+ Compensation to market maker or liquidity provider for the risk of hedging or carrying positions
for the remainder of the trading day
+ Market maker's desired Profit Spread, subject to competitive forces
- Discount related to the likelihood of receiving an offsetting ETF order in a short time frame
Round Trip Trading Cost % = (One-Way Commission % x 2) + (1/2 x Bid-Ask Spread x 2)
Holding Period Cost % = Round Trip Trading Cost % + Management Fee %
Example 2: [Trading Costs vs. Management Fees]
Consider an investor who pays a commission of $10 on a $20,000 trade (0.05% each way) combined with a
0.15% bid-ask spread on purchase and sale. Calculate holding period cost of a 3-month, 12-month and
3-year?
Round Trip Trading Cost % = (0.05% x 2) + (1/2 x 0.15% x 2) = 0.25%
For a round-trip trade that happens over a year, 0.25% can be larger than the annual expense ratios of many
ETFs. If held for less than a year, the trading costs may be far larger than the expense ratio paid on the ETF.
3 Month Holding Period Cost % = 0.25% + (3/12) x 0.15% = 0.29%
12 Month Holding Period Cost % = 0.25% + (12/12) x 0.15% = 0.4%
3 Year Holding Period Cost % = 0.25% + (36/12) x 0.15% = 0.7%
Commission
Bid-Ask Spread
Management Fee
TOTAL
W Trading Costs
W Management Fees
3 Months
12 Months
3 Years
0.1 %
0.15 %
0.0375 %
0.29 %
0.86 %
0.14 %
0.1 %
0.15 %
0.15 %
0.4 %
0.625 %
0.375 %
0.1 %
0.15 %
0.45 %
0.7 %
0.36 %
0.64 %
Over
Time
5
Fig 1: Cost Factor Comparison between ETFs and Mutual Funds
Fund Cost Factor
Holding
Period
Explicit/Implicit
ETFs
Management Fees
Tracking Error
Commissions
Bid-Ask Spread
Premium/Discount to NAV
Portfolio Turnover
Taxable Gains/Loss to Investors
Security Lending
Yes
Yes
No
No
No
Yes
Yes
Yes
Ex
I
Ex
I
I
I
Ex
I
x
x
x
x
x
x
x
x
(Often less)
(Often less than
MF)
Mutual Funds
x
Index Funds Only
(Some free)
(Often less)
(Often less)
(Often more)
x
x
x
* ETF trading costs such as commissions and bid-ask spreads are incurred only at purchase and sale and their return diminishes over
longer holding periods where as management fees and other ongoing costs (such as portfolio turnover and security lending proceeds)
become more significant or increase in proportion of total costs, as the holding period lengthens.
* The quoted bid-ask spread is generally for a specific, usually small trade size and doesn't always reflect ETF liquidity for larger
transactions (more than 10% of average daily volume). For larger trades, posted spreads may not reflect trading costs and these trades
may best be handled by negotiation. ETF bid-ask spreads on fixed income relative to equity tend to be wider because the underlying
bonds trade in dealer markets and hedging is more difficult.
* Tracking error can be considered a positive or negative implicit cost.
* Premiums and discounts may reflect a lag in the timing of the underlying security valuations relative to current market conditions and
can be considered positive costs (in case of premiums) or negative costs (in case of discounts). Premiums or Discounts are driven by a
no. of factors including (a) Timing Differences: NAV is often a poor fair value indicator for ETFs that hold foreign securities because of
differences in exchange closing times between the underlying (e.g. foreign stocks, bonds or commodities) and the exchange where the
ETF trades, Sometimes, bond pricing model inputs reflect the price at which a dealer is willing to buy the bonds and the risk and cost to a
dealer in carrying the bonds in inventory. In this case, the ETF's closing price is often higher than the bid prices of the underlying bond
holdings used to calculate NAV, making it appear that the ETF is at a premium. During times of market stress, few bonds may trade
leaving pricing services without updated inputs for their models. In this case, fixed income ETFs with sufficient trading volumes may
appear to be trading at discounts to NAV. In these cases, by reflecting the markets most current assessment of value, liquid ETFs become
'Price Discovery' vehicles. ETFs also provide 'Price Discovery' for after-hours markets. For example, US listed ETFs holding European
stocks trade until 4pm. ET hours after European markets have closed. In these cases, premiums or discounts resulting from closed
underlying markets are mispricing, rather, they are the market's best estimate as to where the fund holdings would trade if the
underlying markets were open; and (b) Stale Pricing: ETFs that trade infrequently may also have large premiums or discounts to NAV. ETF
prices may be more accurate reflection than NAVs or iNAVs in the following situations i.e. when the underlying securities are less actively
traded (less liquid or volatile markets) when the underlying market is closed and when the underlying market has time lags with respect
to the market where the ETF trades.
iNAV < ETF Share Price : ETF Trading at Premium
iNAV > ETF Share Price : ETF Trading at Discount
End-of-Day ETF Premium or Discount (%) = ETF Price - NAV per Share
NAV per Share
Intra-Day ETF Premium or Discount (%) = ETF Price - iNAV per Share
iNAV per Share
Whereas,
iNAV : 'Indicated' NAVs or iNAVs are intraday 'fair value' estimates of an ETF share
based on its creation basket composition for that day.
* Portfolio turnover costs are incurred within the fund as the portfolio manager buys and sells securities to execute the investment
strategy and manage fund cashflows. Portfolio turnover costs reduce returns and affect performance for all investors in the fund. Many
ETFs are based on indexes that have lower portfolio turnover than actively managed funds.
* Taxable gains incurred upon sale can be considered positive costs for the investors and taxable losses represent negative costs.
* Tactical traders will generally choose an ETF on the basis of its liquidity and trading costs (e.g. commissions, bid-ask spreads). In many
cases, shorter term tactical traders may use an ETF with a higher management fee but a tighter bid-ask spread and more active or
continuous two-way trading flow to avoid incurring the capital commitment cost of a market maker or the cost of arbitrage of the ETF vs.
the underlying securities. The size of the management fee is typically a more significant consideration for longer term buy and hold
investors.
6
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Representative Sampling or Optimization
Rather than fully replicate the index, funds may hold only a subset of index securities to track the
benchmark index. Sampling or Optimization can affect long term tracking in two ways: (a) It can
make the median value unpredictive of future median values, especially if market regimes shift and
(b) It dramatically expands the range of results. Representative Sampling or Optimization therefore
enhances or detracts from fund returns relative to the index depending on whether ETF portfolio
holdings outperform or underperform those in the index. Compared to a full replication approach,
representative sampling or optimization introduces a greater potential for tracking error.
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Depositary Receipts and ETFs
When local market shares are illiquid, the portfolio managers may hold securities that are different
from those in the index such as ADRs, GDRs etc. Differences in trading hours and security prices
create discrepancies between portfolio and index values. Similarly, ETF issuers or portfolio
managers may choose to hold ETFs as underlying holdings. This also creates discrepancies between
fund NAV and index value, because the ETFs' holdings are valued at their closing market prices and
not their NAV.
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Index Changes
Funds may trade index changes at times and prices that are different from those of the benchmark
tracked. ETF portfolio managers can use the creation/redemption process to manage rebalance
trades by cooperating with APs to ensure market-on-close-pricing on the rebalance date, this
minimizing this source of tracking error.
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Fund Accounting Practices
Fund accounting practices may differ from the index calculation methodology. Differences in
valuation practices between the fund and its index can create discrepancies that magnify daily
tracking differences. Valuation discrepancies can also occur for ETFs holding futures, foreign
securities, physical metals and currencies held in specie.
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Regulatory and Tax Requirements
Regulatory and tax requirements may cause a fund to mis-track its index i.e. (a) On average, ETFs
distribute less in capital gains than mutual funds for two primary reasons: Tax Fairness i.e. In a
traditional mutual fund, any securities sold at a profit incur a capital gains charge, which is
distributed to remaining shareholders. Let's put this another way, shareholders may have to pay tax
liabilities triggered by other shareholders redeeming out of the fund. In contrast, if an AP redeems
ETF shares, this redemption occurs in-kind and is not a taxable event. Thus, redemptions don't
trigger capital gain realizations. This aspect is why ETFs are considered 'Tax Fair'. The actions of
investors selling shares of the fund don't influence the tax liabilities for remaining fund
shareholders. Tax Efficiency i.e. By choosing shares with the largest unrealized capital gains i.e.
those acquired at the lowest cost basis, ETF manager can use the in-kind redemption process to
reduce potential capital gains in the fund. Rebalancing increases the potential for capital gains to
occur. Tax efficiency can be improved if redemptions occur on index rebalance dates, because this
allows the portfolio manager to remove appreciated securities via. in-kind exchanges, rather than
selling and realizing a capital gain. Tax lot management allows portfolio managers to limit the
unrealized gains in a portfolio. Certain types of ETF portfolios are particularly vulnerable to capital
gains, (b) Taxes on Sale: In most jurisdictions, ETFs are taxed according to their underlying holdings,
(c) Other Distributions: Other events such as security dividend distributions, can trigger tax
liabilities for investors. In most markets, ETFs distribute their accumulated dividends or choose to
re-invest dividends into the fund.
7
lllll
Asset Manager Operations
ETF issuers may engage in security lending or foreign dividend recapture to generate additional
income to offset fund expenses. These act as 'Negative Costs', which enhance fund performance
relative to the index. Since security lending income is not accounted for in the index calculation, it is
a source of tracking error.
Example 3:
Which of the following statements regarding distributions made by ETFs is correct?
A. Return of Capital (ROC) distributions are generally not taxable
B. ETFs generally reinvest any dividends received back into the ETFs' holding
C. A dividend distribution is a distribution paid to investors in excess of an ETFs' earnings
✓
Similar to mutual fund closures, ETF issuers may decide to close an ETF. Fund Closure: Primary reasons for a
fund to close include regulation, competition and corporate activity. Soft closures which don't involve an
actual fund closing include (a) Creation Halts: ETN issuers may halt creations and redemptions when the
issuer no longer wants to add debt to its balance sheet related to the index on which the ETN is based. When
creations are halted, the ETN can trade at a substantial premium over fair value, as the arbitrage mechanism
breaks down and (b) Changes in Investment Strategy: Some ETF issuers find it easier to repurpose a low asset
ETF from their existing lineup than to close one fund and open another.
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ETF Strategies
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Efficient Portfolio Management
The use of ETFs to better manage a portfolio for efficiency or operational purposes i.e. managing
portfolio activity necessitated by cashflows and changes in external managers. In addition, ETFs can
be used to easily accommodate portfolio rebalancing needs and unwanted gaps in portfolio
exposures.
1. Portfolio Liquidity Management: One of the primary institutional applications of ETFs is cashflow
management. ETFs can be used to invest excess cash balances quickly (known as cash
equitization), enabling investors to remain fully invested in target benchmark exposure, thereby
minimizing potential cash drag. 'Crag Drag' refers to a fund's mis-tracking relative to its index that
results from holding uninvested cash. Managers may also use ETFs to transact small cashflows
originating from dividends, income or shareholder activity e.g. Liquid ETFs benchmarked to asset
category.
2. Portfolio Rebalancing: Maintain exposure to target weights by frequent rebalancing. For
investors, who have the ability to sell short, reducing exposures associated with a rebalance can
be done quickly using an ETF and as the underlying securities are sold off, the short position can
be covered e.g. Domestic Equity, International Equity and Domestic Fixed Income (Asset Classes,
Sub-Asset Classes).
3. Portfolio Completion: ETFs can also be used for completion strategies to fill a temporary gap in
exposure to an asset class, sector or investment theme or factor. Gaps may arise with changes in
external managers or when an existing manager takes an active view that moves the portfolio out
of market segment to which the investor wishes to have continued exposure. The investor may
want to retain the manager but use a tactical ETF strategy to maintain exposure to the desired
market segment e.g. International Small Cap, Canada, Bank Loans, Technological, Quality and
ESG (Countries, Sectors and Theme).
8
4. Transition Management: Transition management refers to the process of hiring and firing
managers or making changes to allocations with existing managers while trying to keep target
allocations in place. For very large asset owners, there are three potential drawbacks to using
ETFs for portfolio management: (i) They may be able to negotiate lower fees for a dedicated
separately managed account (SMA) or find lower-cost comingled trust accounts that offer lower
fees for large investors, (ii) SMA can be customized to the investment goals and needs of the
investor and (iii) Many regulators require large ETFs holdings (as a percentage of ETF assets) to be
disclosed to the public. This can detract from the flexibility in managing the ETF position and
increase the cost of shifting investment holdings e.g. ETFs benchmarked to new manager's target
benchmark.
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Asset Exposure Management
The use of ETFs to achieve or maintain core exposure to key asset classes, market segments or
investment themes on a strategic, tactical or dynamic basis.
1. Tactical Strategies: Some financial advisers and institutional investors allocate a portion of their
portfolios for opportunistic trading based on their firm's or strategist's research or short-term
outlook. Thematic ETFs hold stocks passively but allow investors to take an active view on a
market segment they believe will deliver strong returns. These ETFs typically cover a narrow or
niche area of the market not well represented by an industry e.g. technology such as
cybersecurity and robotics. Thematic ETFs should be evaluated similarly to stocks because they
tend to have comparable levels of volatility and represent specialized active views. ETFs that have
the highest trading volumes in their asset class category are generally preferred for tactical
trading applications. Trading costs and liquidity, rather than management fees are the important
criteria in selecting an ETF for tactical adjustments. To identify the most commonly used ETFs for
tactical strategies, one can look at the ratio of average dollar volume to average assets for the
ETF.
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Active and Factor Investing
The use of ETFs to target specific active or factor exposures on the basis of an investment view or
risk management need.
1. Factor ETFs: Factor ETFs (Smart Beta) are usually benchmarked to an index created with
predefined rules for screening and/or weighting constituent holdings. The strategy index rules
are structured account return drivers or factors such as value, dividend yield, earnings or
dividend growth, quality, stock volatility or momentum. Global assets in Smart Beta equity funds,
including both single factor and multi-factor strategies, now represent approximately 20% of ETF
assets. In a multi-factor ETF, strategy design involves factor selection, factor strategy construction
and a weighting scheme across factors that is managed over time. A multi-factor approach
typically has lower return volatility than a single-factor approach over time but may also have less
return potential for investors who want to capitalize on factor timing.
2. Risk Management: ETFs are also used to manage other portfolio risks such as currency and
duration risk. With respect to interest rate risk management, several small smart beta fixed
income ETFs hold long positions in corporate or high yield bonds and hedge out the duration risk
of these bonds with futures or short positions in government bond. Active investors with a
negative macro view can use inverse asset class or factor ETF exposure to temporarily reduce
benchmark holding risk.
3. Alternatively Weighted ETFs: ETFs that weight their constituents by means other than market
capitalization such as equal weighting or weightings based on fundamentals.
9
4. Discretionary Active ETFs: The largest active ETFs are in fixed income where passive management
is much less prominent than in equities. 'Liquid Alternative' ETFs are rules-based strategy indexes
that attempt to deliver absolute return performance and/or risk diversification of stock and bond
holdings.
5. Dynamic Asset Allocation and Multi-Asset Strategies: Multi-asset and global asset allocation or
macro strategies that manage positions dynamically as market conditions change are areas
where ETFs are frequently used.
Fig 2: Active and Factor Investing
Factor Strategy ETFs are considered
longer term, buy-hold investment
options rather than tactical
Portfolio Application
Factor Exposure
Risk Management
Limit Losses on shorting to
invested funds
Leveraged and Inverse
Exposure
Alternative Hedging
Within asset class
Active Strategies
Dynamic Asset Allocation and
Multi-Asset Strategies
S: Strategic
D: Dynamic
T: Tactical
Category
S, D, T
D, T
T
S, D, T
S
D, T
10
CHAPTER 44
Using Multifactor Models
Multifactor models including the Arbitrage Pricing Theory (APT) and Carhart (4 Factor) Model are introduced
as alternatives to the Capital Asset Pricing Model (CAPM).
Arbitrage Pricing Theory: In the 1970s, Ross (1976) developed the APT as an framework that explains the
expected return of an asset or portfolio in equilibrium as a linear function of the risk of the asset or portfolio
with respect to a set of factors capturing systematic risk. The APT is similar to the CAPM, but the APT makes
less strong assumptions than the CAPM. The APT makes three assumptions: (i) Unsystematic risk can be
diversified away in a portfolio: There are many assets, so investors can form well-diversified portfolios that
eliminate asset-specific risk. This is a reasonable assumption and is supported by empirical evidence, (ii)
Returns are generated using a factor model; the lack of clarity for the risk factors is a major weakness of the
APT and (iii) No arbitrage opportunities exist among well diversified portfolios: APT assumes there are no
market imperfections preventing investors from exploiting arbitrage opportunities; as a result investors will
undertake infinitely long and short positions to exploit any perceived mispricing causing asset prices to adjust
immediately to their equilibrium values.
E (Rp ) = R f + βp (λ 1 ) + βp (λ 2 ) + ... + βp (λ n ) + ε p
1
2
n
Whereas,
λ : Factor Risk Premium or Factor Price represents the expected reward for bearing the risk of
the portfolio with a sensitivity of 1 to factor 'j' and a sensitivity of 0 to all other factors.
β : Sensitivity of the portfolio to factor 'j'.
n : No. of factors.
E (Rp ) : Expected return to portfolio 'p'. A portfolio with a sensitivity of 1 to factor 'j' and a sensitivity
of 0 to all other factors is called a 'Pure Factor Portfolio' for factor 'j'.
Each factor in the APT is 'priced' meaning that each risk premium is statistically and economically significant.
Unlike the CAPM, the APT doesn't require that one of the risk factors is the market portfolio. This is a major
advantage of the APT.
Example 1:
Suppose your investment firm uses a single factor model to evaluate assets. Consider the following data for
portfolios A, B, C.
Portfolio
A
B
C
Expected Return
10%
20%
13%
Beta
1
2
1.5
Calculate the arbitrage opportunity from the data provided.
By allocating 50% of our funds to portfolio A and 50% to portfolio B, we can obtain a portfolio C* with beta
equal to the portfolio β = 1.5.
C
Step 1: Find the weights for portfolios A & B.
β C = wA . β A+ (1 - wA ) . β B
1.5 = w A (1) + (1 - wA ) (2);
wA = 0.5 and w B= 0.5
11
Step 2: Calculate the return of new portfolio C*.
R*
= w A . R A+ w B . R B
C
= (0.5) (0.1) + (0.5) (0.2) = 15%
Step 3: Compare 'C' and 'C*'
RC
13%
R*C
15%
Profit 2%
β
1.5
1.5
Generally, we want to go long assets that have a high ratio of return-per-unit of factor exposure and short
assets that have a low return-to-factor exposure ratio.
Example 2:
Given a one factor model and the following information, calculate the R f and the factor 'λ' for portfolio 'C'.
Portfolio
A
B
Expected Return
7%
7.8%
Beta
1
1.2
Verify that portfolio 'C' with an expected return of 6.2% and factor sensitivity is priced correctly?
A : 7% = R f + 1 (λ)
B : 7.8% = R f + 1.2 (λ)
; R f = 7% - λ
①
or
λ = 7% - R f
②
Substitute
in B
7.8% = R f + 1.2 (λ)
7.8% = 7% - λ + 1.2 (λ)
0.8% = 0.2 λ
4% = λ
Substitute
in B
7.8% = R f + 1.2 (λ)
7.8% = R f + 1.2 (7% - R f )
- 0.6% = - 0.2 R f
3% = R f
E (R C ) = 0.03 + 0.8 (0.04) = 6.2%
Carhart Model (4 Factor): Presented in Carhart (1997), it is an extension of the three-factor model developed
by Fama and French (1992) to include a momentum factor. According to the model, there are three groups of
stocks that tend to have higher returns than those predicted by their sensitivity to the market returns.
Rp - R f =
L p + βp . RMRF + βp . SMB + βp . HML + β p . WML + ε p
1
2
3
4
E (Rp ) = R f + βp . RMRF + βp . SMB + βp . HML + βp . WML
1
2
3
4
Whereas,
L p : Expected value of Alpha is '0'.
E (Rp ) : Expected return on portfolio.
RMRF : Return on value weighted equity index i.e. R f .
SMB : Small minus Big, a size (market capitalization) factor i.e. average return on three small cap
portfolios - average return on three large portfolios.
HML : High minus Low i.e. Average return on two high book-to-market portfolios - average return
on two low book-to-market portfolios.
WML : Winners minus Losers, momentum factor i.e. average returns on past years winners average returns on past years losers.
12
From the perspective of the CAPM, there are size, value and momentum anomalies. From the perspective of
the Carhart Model, however size, value and momentum represent systematic risk factors; exposure to them
is expected to be compensated in the marketplace in the form of differences in mean return. All three factors
continue to have robust uses in active management risk decomposition and return attribution.
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Types of Multifactor Models
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Macroeconomic Factor Model
Macroeconomic Factor Models assume that asset returns are explained by surprises or 'shocks' in
macroeconomic risk factors e.g. GDP, interest rates and inflation. Factor surprises are defined as
the difference between the realized value of the factor and its consensus predicted value.
R i = E (R i ) + b i 1 . F 1 + b i 2 . F 2+ ... + b i n . F n+ ε i
Consider a factor model in which the returns to each asset are correlated with two factors i.e.
Inflation rates and GDP growth.
R i = E (R i ) + b i 1. F INFL+ b i 2. FGDP+ ε i
Risk Premium for the
GDP growth factor is
typically positive and
for Inflation factor is
typically negative.
Whereas,
R i : Return for asset i .
E (R i ) : Expected return for asset i , in the absence of any surprises.
F : Factor surprise i.e. Realized Value - Predicted Value.
b : Sensitivity of the stock to the surprise. The higher the sensitivity, the larger the
change in return for a given factor surprise. Example, retail stocks are very sensitive
to GDP growth and hence, have a large sensitivity to the GDP factor. The factor
sensitivities of the model can be estimated by regressing historical asset returns on
the corresponding historical macroeconomic factors. Factor sensitivities are
sometimes called factor bets or factor loadings.
ε i : Firm specific surprise i.e. unrelated to the two macro factors. Firm specific surprise
captures the part of the return that can't be explained by the model. It represents
unsystematic risk a.k.a. 'Not Priced' i.e. investors cannot expect to be rewarded for
being exposed to that type of risk. The factors in our example model, GDP and
inflation are systematic risk a.k.a. 'Priced' risk i.e. they will effect even well diversified
portfolios i.e. risk for which investors can expect compensation.
Example 3:
A portfolio manager is analyzing the returns on a portfolio of two stocks, Manumatic (MAN) and Nextech (NX).
One third of the portfolio is invested in Manumatic stock and two thirds is invested in Nextech stock.
R MAN = 0.09 - 1 . FINFL+ 1. FGDP+ ε MAN
R NX = 0.12 + 2 . F INFL+ 4 . F GDP+ ε NX
a. Formulate an expression for the return on the portfolio.
R p= 1 [0.09 - 1 . FINFL+ 1 . FGDP + ε MAN] + 2 [0.12 +2 . F INFL+ 4 . FGDP + ε NX ]
3
3
= 0.11 + 1 . F INFL+ 3 . FGDP + 0.333 . ε MAN+ 0.666 . ε NX
13
b. State the expected return on the portfolio. Expected return on the portfolio is 11%.
c. Calculate the return on the portfolio given that the surprises in inflation and GDP growth are 1% and 0%
respectively, assuming that the error terms for MAN and NX both equal 0.5%.
R p= 0.11 + 1 . F INFL+ 3 . FGDP+ 0.333 . ε MAN + 0.666 . ε NX
= 0.11 + 1 (0.01) + 3 (0) + 0.333 (0.005) + 0.666 (0.005)
= 0.125 or 12.5%
Example 4:
Last year the return on Harry Company stock was 5%. The portion of the return on the stock not explained by
a two-factor macroeconomic factor was 3%. Using the data given below, calculate Harry Company stock's
expected return.
Change in Interest Rate
Growth in GDP
Actual
Value
2%
1%
Expected
Value
0%
4%
Factor
Sensitivity
- 1.5
2
5% = Expected Return - 1.5 (Interest Rate Surprise) + 2 (GDP Surprise) + Error Term
5% = Expected Return - 1.5 (2% - 0%) + 2 (1% - 4%) + 3%
5% = Expected Return - 6%
Expected Return = 11%
Example 5:
RStone = 0.11 + 1 . FINT + 1.2 . F UN + ε Stone
Expected vs. Actual Interest Rates & Unemployment Rate Surprises.
Interest Rate
Unemployment Rate
Actual
0.053
0.072
Expected
0.051
0.063
Company-Specific Surprise Returns
0
0
What is the predicted return for Stonebrook, if the return unexplained by the model was - 1%?
F INT = 0.053 - 0.051 = 0.002
F UN = 0.072 - 0.068 = 0.004
Actual Return = 0.11 + 1 (0.002) + 1.2 (0.004) - 0.01 = 10.68%
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Fundamental Factor Model
Fundamental Factor Model assume asset returns are explained by multiple firm-specific factors
e.g. P/E ratio, Market cap, Leverage ratio and Earnings growth rate.
R i = a i + b i1 . Fr1 + b i 2 . Fr2 + ... + b i n . Frn+ ε i
Whereas,
R i : Return for stock i .
a i : Intercept term; the expected factor values are not zero.
14
I
b i n : Value of attribute 'n' for asset - Average value of attribute 'n' ; Fundamental factor
o (Values of Attribute 'n')
sensitivities are standardized attributes, similar to z-statistics from the standard
normal distribution. An asset's sensitivity to a factor is expressed using a
'Standardized Beta'. Also note that by standardizing the factor sensitivity, we measure
the no. of standard deviations that each sensitivity is from the average. For example,
a stock with a standardized P/E sensitivity of '2' has a P/E that is 2 standard deviations
above the mean; a stock with a sensitivity of - 1.5 has a P/E that is one and a half
standard deviations below the mean.
Fr : Factor returns i.e. rates of return associated with each factor e.g. FP/E is the
difference in rate of return between low and high P/E stocks, also known as the
return on a factor mimicking portfolio.
In practice, the values of the fundamental factors are estimated as slopes of cross-sectional
regressions in which the dependent variable is the set of returns for all stocks and the independent
variables are the standardized sensitivities. Financial analysts use fundamental factor models for a
variety of purposes including portfolio performance attribution and risk analysis.
Example 6:
The P/E for stock i is 15.2, the average P/E for all stocks is 11.9 and the standard deviation of P/E ratios is 6.3.
Calculate the standardized sensitivity of stock i to the P/E factor.
I
β i P/E = (P/E) i - P/E = 15.2 - 11.9 = 0.52
o P/E
6.3
Therefore, the P/E ratio for the stock is 0.52 standard deviations higher than the average P/E.
Example 7:
A portfolio manager uses a two-factor model to manage her portfolio. The two factors are confidence risk and
time-horizon risk. If she wants to bet on an unexpected increase in the confidence risk factor (which has a
positive risk premium), but hedge away her exposure to time-horizon risk (which has a negative risk
premium). She should create a portfolio with a sensitivity of:
A. 1 to the confidence risk factor and 0 to the time-horizon factor.
B. 1 to the confidence risk factor and - 1 to the time-horizon factor.
C. - 1 to the confidence risk factor and 1 to the time-horizon factor.
✓
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Statistical Factor Model
Statistical methods are applied to historical returns of a group of securities. Two major type of
factor models are (a) Factor Analysis and (b) Principal Components Model. In Factor Analysis, factors
are portfolios that explain historical return covariance and in Principal Components Model, factors
are portfolios that explain historical return variances. The weakness being, statistical factors don't
lend themselves well to economic interpretation. Advantage being minimal assumptions.
Uses of Multifactor Models
1. Passive Management
2. Active Management
Factor portfolios are useful for speculation and hedging purposes
3. Rules-based Active Management
Biases in portfolio related to value weighted benchmark indexes. Examples include
Alternative Indexes, Tilt Factor Exposures and Manager Skill 'L ' (Alpha). Rules-based Active
Management is low cost.
15
Fig 1: Differences between Macroeconomic Vs. Fundamental Model
Macroeconomic
Fundamental
Time series of surprises.
Cross-sectional asset
returns.
Regression based between
R i and factor surprises.
Factor sensitivities are
estimated last in
macroeconomics.
Standardized from
attribute data. Factor
sensitivities are
determined first in
fundamentals.
Factor 'F'
Factor surprises 'F i '.
Factor return 'F r '
computed from multiregression between R i and
β.
Intercept
Expected return.
Undefined: simply
regression intercept
necessary to make the
unsystematic risk of the
asset equal to '0'.
Less
More
No
Yes*
Regression
Factor Sensitivity 'β'
No. of Factors
Style of the Manager
* Style factors include those related to earnings, risk and valuation that define types of securities typical of
various styles of investing.
lllll
Return and Risk Attribution
1. Return Attribution
Active Return = RP - R B
①
②
Active Return = Factor Return + Security Selection Return
n
(RP - RB )
Σ (β i p - β i b ) . (λ i )
i=1
Whereas,
β i p : Factor sensitivity for the i th factor in the active portfolio.
β i b : Factor sensitivity for the i th factor in the benchmark portfolio.
λ i : Factor risk premium for factor i.
We can decompose active return into two components: (a) Factor Return: It is the product of the
portfolio manager's factor tilts (over or under weights relative to the benchmark factor sensitivities)
and the factor returns also known as return from factor tilts and (b) Security Selection: It reflects the
manager's skill in individual asset selection i.e. overnight securities that outperform the benchmark
or underweight securities that underperform the benchmark. It is also the residual differences
between active return and factor return.
16
2. Risk Attribution
I
Active Risk = Tracking Error = o (R p- Rb )
①
②
I
I
Active Risk Squared = Active Factor Risk + Active Specific Risk
n n
n
2
a
a
2
2
o (Rp - Rb )
= Σ Σ b i b j Cov. (F i , F j ) + Σ (w ip - w i b) o ε i
i=1 j=1
i=1
I
Whereas,
w ip : Weight of i th security in the active portfolio.
w ib : Weight of i th security in the benchmark portfolio.
o ε2 i : Residual i.e. unsystematic risk of the i th asset.
n
a
a
b j : Σ w i b j i , Portfolio's active factor exposure.
i=1
We can decompose active risk squared (or variance of active return) into two components: (a) Active
Factor Risk: Risk from active factor tilts attributable to deviations of the portfolio's factor sensitivities
from the benchmark's sensitivities to the same net of factors and (b) Active Specific Risk or Security
Selection Risk: Measures the active non-factor or residual risk assumed by the manager. Portfolio
managers attempt to provide a positive average return from security selection as compensation for
assuming active specific risk.
For example, imagine that Manager A earned a constant 0.5% active return on each of the last four
quarters and Manager B earned active returns of 8%, 5%, -3% and -8% over the same four quarters.
The average active returns for Managers A & B are both 0.5%, but Manager B experienced far more
volatility i.e. less consistency than Manager A. To demonstrate a manager's consistency in generating
active return, we use the Information Ratio i.e. It is the mean active returns per unit of active risk.
I
I
IR = Rp - Rb
o (Rp - Rb)
I
↶
Historical or
Ex-Post IR
17
CHAPTER 45
Measuring and Managing Market Risk
Value at Risk (VaR) is the minimum loss that would be expected a certain percentage of the time over a
certain period of time given the assumed market conditions. VaR measures downside risk of a portfolio. It has
3 components: the loss size, probability of a loss greater than equal to the specified loss size, and time frame.
'The 5% VaR of a Portfolio is $2.2m over a one-day period'
It is stating that $2.2m is the minimum loss we would expect 5% of the time
or 5% of the time losses would be atleast $2.2m. A 5% VaR is often expressed
as its complement i.e. a 95% level of confidence VaR can also be expressed in
percentage terms so that for a portfolio, we could state that the 5% monthly
VaR is 3% meaning that 5% of the time the monthly portfolio value will fall by
atleast 3%. We can also state VaR as a confidence level: we are 95% confident
that the portfolio will experience a loss of more than 3%. VaR is typically
expressed for 1%, 5% or 16% as 2.33, 1.65 or 1 standard deviations below the
mean for a normal distribution.
5% of VaR
x
μ
Types of VaR
1. Conditional VaR: The Conditional VaR (CVaR) is the expected loss given that the loss is equal to or greater
than the VaR. The CVaR is expected loss given that the loss in the left hand tail past the VaR. For this
reason, the CVaR is also referred as the 'Expected Tail Loss or Expected Shortfall'. CVaR is best derived
using the historical stimulation or Monte Carlo Methods, in which one can observe all of the returns
throughout the distribution and calculate the average of the losses beyond the VaR cutoff or have all the
values equal to or less than the VaR loss, so it is straightforward to take the average of these to get the
CVaR. With the parametric method, it uses a continuous distribution and obtaining the average loss beyond
the VaR cutoff is mathematically complex.
2. Incremental VaR: Incremental VaR (IVaR) is the change in VaR from a change in the portfolio allocation to a
security. We recalculate the VaR under the proposed allocation and the incremental VaR is the difference
between the 'before' and 'after' VaR. If a 2% increase in the weight of a security in the portfolio increases
the portfolio's VaR from $1,345,600 to $1,562,400, the IVaR for the 2% increases in the portfolio weight of
the security is $1,562,400 - $1,345,600 = $216,800.
3. Marginal VaR: The Marginal VaR (MVaR) is estimated as the slope of a curve that plots VaR as a fuction of a
security's weight in the portfolio. Some people interpret MVaR as a change in the VaR for a $1 or 1% change
in the portfolio, although that is not strictly correct. Nonetheless, this interpretation is a reasonable
approximation of the concept behind marginal VaR, which is to reflect the impact of a small change. In a
diversified portfolio, marginal VaR may be used to determine the contribution of each asset to the over
VaR; the marginal VaRs for all positions may be proportionately weighted to sum of the total VaR. Both can
be useful in evaluating the potential effect of a trade before the trade is done.
4. Relative VaR: Relative VaR (RVaR) a.k.a. Ex-Ante Tracking Error is a measure of the degree to which the
performance of a given investment portfolio might deviate from its benchmark. A 5% monthly relative VaR
of 2.5% implies that 5% of time, the portfolio's relative underperformance will be atleast 2.5%. RVaR is
portfolio holdings minus the holdings in the specified benchmark. The relative VaR can be calculated as the
VaR of a combination of a long position in the subject portfolio and a short position in the benchmark
portfolio.
18
lllll
Estimating VaR
It is to convert the set of holdings in the portfolio into a set of exposures to risk factors, a process
called 'Risk Decomposition'. These risk factors include market risk, interest rate risk or currency risk
among others.
lllll
Parametric Method
Parametric Method of estimating VaR is sometimes referred to as the analytical method and
sometimes the variance-covariance method. Often we assume that the risk factors are distributed
normally, but we could also assume other distributions. Assuming normality allows us to estimate
the risk of the portfolio based only on the means, variance and covariances or correlations of the
various risk factors. An assumption that risk factor probabilities are non-normal would increase the
complexity of the analysis and require that we estimate values for other parameters such as
skewness and kurtosis.
2
2
2
2
2
I
I
I
o p = wA o A + w B o B + 2 . wA w B Cov AB
Example 1:
Imagine that we are provided the following information about two assets, Security A and Security B.
Daily Mean
CovAB
0.0158
0.0112
0.0004
0.0003
0.000106
I
Daily o
A
B
How would we use this information to estimate the 5% annual VaR for a portfolio that is 60% invested in
Security A and 40% invested in Security B?
Mean Daily Portfolio Return = 0.6 (0.0004) + 0.4 (0.0003) = 0.00036
Variance of Portfolio Return = (0.6)2 (0.0158) 2 + (0.4) 2 (0.0112)2 + 2 (0.4) (0.6) (0.000106) = 0.000161
Standard Deviation of Portfolio Return =
0.0061 = 0.012682
5% Daily VaR = 0.00036 - 1.65 (0.012682) = - 0.0206
Assuming the distribution of daily returns is constant over the year, that there are 250 trading days in one
year and that daily returns are independently distributed, we can calculate the annual VaR:
Annual Mean Portfolio Return = 250 (0.00036) = 0.09
Standard Deviation of Annual Portfolio Return = 250 (0.012682) = 0.20052
5% Annual VaR = 0.09 - 1.65 (0.20052) = - 0.2409
For a portfolio with a value of $10,000,000, the 5% daily and annual VaR:
$ Daily VaR = 10,000,000 (0.0206) = $ 206,000
$ Annual VaR = 10,000,000 (0.2409) = $ 2,409,000
If you wish to covert from Annual VaR to Daily VaR, divide Annual VaR return by 250 days to derive Daily
Expected Return and divide Annual VaR Standard Deviation Return by square root of 250 days to derive Daily
standard deviation return.
19
The calculated VaR is also very sensitive to the covariance estimate. The length of the lookback
period will affect the parameter estimate and care must be taken to adjust estimates based on
recent results when they may not reflect the future distribution of returns. In cases where normality
cannot be reasonably assumed, such as when the portfolio contains options, the parametric
method has limited usefulness. On the other hand, the parametric method is relatively simple to
apply under the assumption of normally distributed returns.
lllll
Historical Simulation Method
Historical Simulation Method of VaR uses the current portfolio and reprises it using the actual
historical changes in the key factors experienced during the lookback period. Instead, we reprise
the current portfolio given the returns that occurred on each day of the historical lookback period
and sort the results from largest loss to greatest gain. By ordering the changes in portfolio value
from most positive to most negative, we can find the largest 5% of losses. The smallest of those
losses in our estimate of the 5% VaR of the current portfolio.
Plotted 100 monthly returns 'r' from lowest to highest
No.
1
2
3
4
5
Loss
- 15%
- 13%
- 12%
- 11%
- 9%
th
5 Largest Loss (VaR)
Under the historical simulation method, no adjustments are made for the difference between the
results for the lookback period and the results over a longer prior period. Historical simulation is
highly sensitive to the input parameters. Both the parametric and historical simulation methods
have a limitation i.e. all observations are weighted equally. The historical simulation method can
adjust for this problem by using a weighting methodology that gives more weight to more recent
observations and less weight to more distant observations. One positive aspect of the historical
simulation method is that we don't need the assumption of normality or any other distributional
assumption to estimate VaR. Because the historical results for a portfolio containing options include
the changes in option values, the historical simulation method can be used to estimate the VaR for
portfolios that include options. VaR based on an unusually volatile lookback period will yield
overestimates of VaR, just as VaR based on a lookback period with low volatility will likely
underestimate the true VaR over subsequent periods. The historical simulation method estimates
VaR based on what actually happened; yet therein also lies the primary weakness i.e. there can be
no certainty that a historical event will re-occur.
Example 2:
Which of the following statements about the historical simulation method of estimating VaR is most correct?
A. A 5% historical simulation VaR is the value that is 5% to the left of the expected value.
B. A 5% historical simulation VaR is the value that is 1.65 standard deviations to the left of the expected
value.
C. A 5% historical simulation VaR is the 5th percentile, meaning the point on the distribution beyond which
5% of the outcomes result in larger losses.
✓
lllll
Monte Carlo Simulation Method
Computer software is used to generate random values for each risk factor and pricing models are
used to calculate the change in portfolio value for that set of risk factor changes. This procedure is
20
repeated thousands of times. We then sort the results in order from worst to best. A 5% Monte
Carlo VaR would simply be the 5th percentile of the simulated values instead of the historical
values. Just like any other methods, the data used and the assumptions about the distributions of
the risk factors will have significant effects on the estimated VaR. Monte Carlo simulation can
accommodate virtually any distribution. The Monte Carlo and historical simulation methods are
much more capable than the parameter method of accurately incorporating the effects of option
positions or bond position with embedded options. Assuming a large sample size, the Monte Carlo
method will produce identical results as the parametric method if the distribution specified and the
parameters are the same.
Advantages of VaR
1. Simple Concept: The concept of VaR is simple and easy to explain, although the details of the methodology
can be complex.
2. Provides a Basis for Risk Comparison: VaR can be useful in comparing risks across asset classes, portfolios
and trading units giving the risk manager a better picture of which constituents are contributing the least
and the most to the overall risk i.e. performance evaluation. VaR also allows calculation of the ratio of
trading income to VaR.
3. Facilitates Capital Allocation Decisions: A firm's risk managers can also look at the allocation of VaR and
optimize the allocation of capital given the firm's determination of the maximum VaR that the organization
should be exposed to sometimes referred to as 'Risk Budgeting'.
4. Reliability of VaR: as a measure of risk can be verified by 'Backtesting'. To determine whether a VaR
estimate is reliable, one can determine over a historical period of time whether losses of atleast $5000,000
were incurred on 5% of trading days, subject to reasonable statistical variation.
5. Global Banking Regulators accept VaR as a measure of Financial Risk.
Disadvantages of VaR
1. Subjectivity: VaR estimation requires many choices like loss percentage, lookback period, distribution
assumptions and parameter estimates, and can be significantly affected by these choices.
2. Underestimating the Frequency of Extreme Events: VaRs based on an assumption of normality tend to
underestimate the probability of extreme outcomes. The assumption of normality leads to underestimates
of downside (tail) risk or left tail events because actual returns distribution frequently have 'fatter tails'
than a normal distribution.
3. Failures to take into Account Liquidity: Liquidity often falls significantly when asset prices fall. A VaR which
doesn't account for this will understate the actual losses incurred when liquidating positions that are under
extreme price pressure i.e. if some assets in a portfolio are relatively illiquid, VaR could be understated
even under normal market conditions. Additionally, liquidity squeezes are frequently associated with tail
events and major market downturns, thereby exacerbating the risk.
4. Disregard for Right Tail Events: VaR focuses only on downside risk 'left tail' and extreme negative
outcomes. Including consideration of right-hand tail values will give a better understanding of the
risk-return trade off.
5. Sensitivity to Correlation Risk: It is well known that correlations increase or spike during periods of financial
stress. Increasing correlations mean that VaR measures based on normal levels of correlation will
overestimate diversification benefits and underestimate the magnitude of potential losses.
6. Quantification: While VaR is a single number that can be used to quantify risk, as with any summary
measure, many aspects of risk are not quantified or included.
7. Misunderstanding the Meaning of VaR: VaR is not a worst-case scenario. Losses can and will exceed VaR.
8. Vulnerability to different Volatility Regimes: A portfolio might remain under its VaR limit everyday but loose
an amount approaching the limit each day. If market volatility during the last year is lower than in the
lookback period, the portfolio could accumulate a substantial loss without technically breaching the VaR
constraint.
21
Fig 1: Applications of Risk Measures
Risk Measures
Banks
Sensitivity Measures (Duration of HTM and Foreign Exchange Risk Exposure),
Scenario Analysis, Stress Testing, Leverage Risk Measures, VaR, Liquidity Gap,
Economic Capital, Asset Liability Mismatching
Traditional Asset Managers
Sensitivity Measures (Interest Rate Risk and Market Risk), Scenario Analysis,
Beta Sensitivity, VaR, Position Limits, Active Share, Redemption Risk, Options
Risk, Ex-Ante Tracking Error
Hedge Funds
Sensitivity Measures, Scenario Analysis, Stress Testing, Leverage Risk
Measures, VaR, Gross Exposure, Maximum Drawdown
Pension Funds
Interest Rate and Curve Risk, Surplus at Risk, Glide Path, Liability Hedging
Exposure vs. Return Generating Exposure.
Insurance Companies
Sensitivity Measures, Scenario Analysis, Stress Testing, VaR, Liquidity Gap,
Economic Capital, Asset Liability Mismatching
Sensitivity Measure: Sensitivity analysis complements VaR in understanding portfolio risk, but unlike VaR, it doesn't involve any prediction of the
probability of losses of any specific amount. Sensitivities for Equities is Beta, Fixed Income is Duration (small changes) and Covexity (Large
changes), Options are Delta, Gamma and Vega. Both Convexity and Gamma are considered 'Second-Order Effects', while Duration and Delta are
'First-Order Effects' of risk factor changes. Sensitivity measures address some of the shortcomings of position size measures. Sensitivity measures
based on standard deviation or beta may be misleading for large changes in risk factors when returns are non-normal.
Scenario Risk Measure: The three types of scenario risk measures are (a) Historical Scenarios: Instead of assuming the shock is a single instant
event, historical scenario approach assumes it takes place over a no. of days and on each day the portfolio manager can take such actions as
selling assets or rebalancing hedges. Many risk managers are skeptical of this approach because they produce smaller potential loss measures (by
design) and don't answer important questions that have been relevant in real crises such as 'What if the actions?'. In this case, stress testing seems
helpful. (b) Hypothetical Scenarios: could have more extreme changes in risk factors than those that have occurred in the past but that have some
non-zero probability of occurring in the future. To design an effective hypothetical scenario, it is necessary to identify the portfolio's most
significant exposures. Hypothetical scenarios may incorrectly specify how assets will co-move, they may get the magnitude of movements wrong
and they may attempt to adjust for the effects of liquidity and concentration, but might do so incorrectly. Targeting these material exposures and
assessing their behavior in various environments is a process called 'Reverse Stress Testing'. Reverse Stress Testing is particularly helpful in
estimating potential losses if more than one important exposure is affected in a market crises, as often happens when participants 'crowd' into the
same exposures; sometimes, apparently unrelated markets experience stress at the same time. (c) Stress Test: intentionally focus on extreme
negative events to assess the impact of such an event on the portfolio. Scenario Risk Measures and Stress Tests are best used as the final screen.
Liquidity Gap: The extent of any liquidity and asset/liability mismatch.
Economic Capital: It is the amount of capital a firm needs to hold for it to survive severe losses due to the risks in the business.
Position Limits: are one way to limit risk because they ensure some minimum level of diversification by limiting risk exposures. Position limits
include restrictions on country, currency, sector and asset class.
Active Share: It is the difference between the weight of a security in the portfolio and its weight in the benchmark.
Redemption Risk: Open-end fund managers often assess wheat percentage of the portfolio could be redeemed at peak times and track this
behavior across the funds and asset classes they manage.
Ex-Post Tracking Error (Backward Looking) and Ex-Ante Tracking Error (Forward Looking): Ex-Ante provides an estimate of the degree to
which the current portfolio could underperform its benchmark.
Gross Exposures: sum of absolute value of long plus short positions and is important guide to the importance of correlation risk for portfolio.
Maximum Drawdown: Hedge Funds with significantly non-normal returns distribution use a risk measure referred to as Maximum Drawdown i.e.
the largest decrease in value over prior periods of a specific length i.e. defined as the worst-returning month or quarter for the portfolio or the
worst peak-to-trough decline in a portfolio returns.
Surplus at Risk: a VaR for plan assets minus liabilities. If ever pension fund's surplus at risk exceeds this limit, pension staff will change the fund's
asset allocation to make the assets in the fund better match the liabilities. The liability focused form of pension investing a commonly referred to
as 'Liability Driven Investing'.
Glide Path: as a tool for managing surplus at risk i.e. refers to a multi-year plan for adjusting pension fund contributions to reverse a significant
overfunded or underfunded status.
22
Example 3:
The corporate baking division's capital allocation is $2,800m. This amount considers market risk, credit risk
and operational risk or which the minimum required return is 11%. Capital Limits, Position Limits and StopLoss Limits are assigned to manage both overall exposure and exposure to single-name event risk.
Lending Group
Secured
Cashflow
Real Estate
Return
120
120
120
Capital Limit
800
1200
1000
Position Limit
32
24
10
Stop-Loss Limit
80
60
70
With respect to capital allocation, which lending group is least likely attractive?
A. Secured
120/800 = 15%
120/1200 = 10%; 10% return is below 11% minimum return (hurdle rate)
B. Cashflow
C. Real Estate 120/1000 = 12%
✓
NOTES
1. Constraints used in managing market risks are: (a) Risk Budgeting: refers to a risk management process
that first determines the acceptable total risk for an organization and then allocates that risk to different
activities, strategies or asset classes as appropriate. Risk Budgeting typically rests on a foundation of VaR or
Ex-Ante tracking error, (b) Position Limits: Position Limits don't take into account duration, volatility and
correlation as VaR does, but they are excellent controls or over-concentration, (c) Scenario Limits: are limits
on expected loss for a given scenario; if exceeded would require corrective action in the portfolio,
(d) Stop-Loss Limits: require that a risk exposure be reduced if losses exceed a specified amount over a
certain period of time. An example for a single stop-loss limit is a requirement to reduce the portfolio
allocation to a stock or asset class by a given amount, if it declines in value by more than a specified
percentage or currency amount. A slightly more complex type of stop-loss limit is a requirement that a risk
exposure be hedged as the value of a security or index falls. This is referred to as 'Portfolio Insurance or
Drawdown Control' when the value of a portfolio is hedged by index puts. Portfolio Insurance method is
more dynamic and spphisticated than the stop-loss limit.
23
CHAPTER 46
Economics and Investment Markets
The value of an asset depends on (i) Expected Future Cashflows and (ii) Discount Rate used to value those
cashflows. The uncertainty about future cashflows is reflected in the discount rate. Investors judge economic
data releases relative to their expectations for the data. Prices may fall (rise) despite good (bad) news, if the
expectation was for better (worse) news. Thus, for valuation, one more important distinction is information
that is 'news' or 'new information and information that has been anticipated. Therefore, news is a surprise
relative to fully anticipated information. Investor sentiment (i.e. enthusiasm or despair) affects asset values
through direct effects on discount rates via. higher or lower risk premiums (and possibly indirect effects on
future cashflows).
n
V0 = Σ E (CF t )
t=1
(1 + r) t
B
A
Real R f
+
C
Expected Inflation
D
Inflation Uncertainty
Risk Premium
E
+
Credit Rp
F
+
Equity Rp
G
+
Illiquidity Rp
Real Risk-Free Rate
Consider a single individual who has to choose between consuming today or invest in a default free
bonds i.e. investor would require no return on such a bond because there is no risk of loosing
money over the investment period in either nominal or real terms; But the choice to invest today
involves the opportunity cost of not consuming today; It is the aggregated opportunity cost of all
investors that will determine the price of this asset today and its return over the investment today:
If the return increases, the investor substitutes away from current consumption to future
consumption by purchasing an asset.
This trade-off is measured by the 'Marginal Utility of Consumption' 'n' periods in the future relative
to the 'Marginal Utility of Consumption' today 't'. The marginal utility of consumption is the
additional satisfaction or utility that a consumer derives from one additional unit of consumption.
The trade-off is known as the 'Inter-Temporal Rate of Substitution' i.e. trade-off between real
consumption now and real consumption in the future.
E (m t ) = Marginal Utility of Consuming 1 unit in the Future = m t
Inter-Temporal Rate of
Marginal Utility of Current Consumption of 1 unit m0
↶
lllll
+
Substitution
If E (m t ) = 0.8, then 1$ consumption in future is worth $0.8 today.
For a given quantity of consumption, investors always prefer current consumption over future
consumption (m 0 > m t ) and m t < 1 as a result.
To demonstrate the link between the bond price and these consumption/investment decisions,
imagine for the moment that the (i) market price of this bond is too 'low' for an individual investor.
In this case, the investor with a higher initial E (m t ) would buy more of the bond. As a result of this
purchase the investor will consume less today leading to an increase in today's marginal utility, but
he or she would expect to have more consumption and thus lower marginal utility in the future.
24
Consequently, the E (m t ) would fall. (ii) If the market price of the bond is too 'high' for a group of
investors, then the investors with a lower E (m t ) would buy less of the bond. They would have more
consumption and lower marginal utility today, but they would expect to have less consumption and
higher marginal utility in the future. As a result, the E (m t ) would rise and the demand and price of
the bond would fall. This process would again continue until P0 = E (m t ) is true for all individuals.
Real R f = P1 - P0 = P1 - E (m t ) = 1 - 1
P0
E (m t )
E (m t )
Diminishing marginal utility of wealth means that an investor's marginal utility of consumption
declines as wealth increases. This suggests that marginal utility of consumption is higher during
periods of scarcity, such as during economic contractions. Thus investors would receive a large
benefit (utility) from an asset that pays off more in bad economic times relative to one that pays off
in good economic times.
If investors expect higher incomes in the future, their expected marginal utility of future
consumption is decreased relative to current consumption. When investor expectations about the
economy change to better economic times ahead, the expectation of higher incomes in the future
will lead to an increase in current consumption and a reduction in savings. Investors will derive
greater utility from current consumption relative to future consumption and would therefore, save
less. Conversely, investors expecting worse times ahead would prefer to increase future
consumption by reducing current consumption and saving more. Investors increase their savings
rate when expected returns are high or when uncertainty about their future income increases.
↑
↓
E (m t )
P0
↓
↑
E (m t )
P0
Boom
m0
m1
Recession
m0
m1
↓
↑
↓
Rf
↑
Real GDP Growth
Save Less
↑
Rf
↓
Real GDP Growth
Save More
↑
↓
The higher rate of economic growth occurs for developing economies compared to developed
economies because a developing economy is typically below its steady state growth, so it grows
faster to catch up. During these periods, the marginal product of capital would be expected to be
higher, so the real default-free interest rate should also be expected to be higher. Ofcourse, this
advantage will dissipate as the economy matures.
The real interest rates (short-dated) are higher in an economy in which GDP growth is more volatile
compared with real interest rates in an economy in which growth is more stable.
The interest rates will still be positively related to the expected growth rate of GDP, but additionally
it will be positively related to the expected volatility of GDP growth due to a positive risk premium.
Example 1:
Which of the following financial assets is likely to offer the most effective hedge against bad consumption
outcomes?
A. Equities
B. Short-dated, default-free government bonds
C. Long-dated, default-free government bonds
✓
25
lllll
Expected Inflation and Inflation Uncertainty
Nominal R f interest rates include a premium for expected inflation 'π'. However, actual inflation is
uncertain. This additional risk gives risk to an additional risk premium for the uncertainty about
actual inflation 'θ'. This risk premium is higher for longer maturity bonds.
Nominal Rate
Short-Term : Nr = R f + π
Long-Term : Nr = R f + π + θ
Short-term nominal interest rates will be positively related to short-term real interest rates and to
short-term inflation expectation.
↶
Fig 1: Business Cycle and Shape of Yield Curve
LTR - STR
Term Spread
θ
Shape of Yield
Curve
Economic Scenario
Expected
Inflation
Positive
Upward
Rising
Negative
Downward
Late Recessions
(Boom)
Late Expansion
(Peak)
Falling
↑: Boom
↓: Recession
Long-Term : Rates ↑: Going to be Recession
↓: Will be Expansion (Boom)
Short-Term : Rates
Example 2:
An economy just getting out of recession would most likely have:
A. High short-term rates and an inverted yield curve
B. Low short-term rates and an inverted yield curve
C. Low short-term rates and an upward sloping yield curve
✓
One interpretation of an upward sloping yield curve is that
short-dated bonds are less positively (more negatively)
correlated with bad times than are long-dated bonds.
Central banks are usually charged with setting policy rates so as to (i) maintain price stability and
(ii) achieve the maximum sustainable level of employment. The Taylor Rule links the central bank's
policy rate to economic conditions (employment level and inflation).
Output Gap
Pr = Rn + π + 0.5 (π - π T ) + 0.5 (GDP - GDP T )
As Planned
↑ Inflation
↑ Employment
If Not Planned
If +ve : Economy is producing beyond its sustainable capacity
If -ve : Economy is producing below its sustainable capacity
Whereas,
Pr : Central bank policy rate implied by the Taylor Rule
Rn : Neutral real policy interest rate
π : Current inflation rate
π T : Central bank's target inflation rate
GDP : Log of current level of output
GDP T : Log of central bank's target sustainable output
26
If π above π T
If π below π T
:
:
Pr above Rn
Pr below Rn
:
:
Output is +ve
Output is -ve
Central banks can moderate the business cycle by making appropriate changes to the policy rate or
can magnify the cycle by not responding appropriately to changing economic conditions
(e.g. committing policy errors such as keeping rates too low). Finally, it is important to remember
that the neutral rate will also vary with the level of real economic growth and with the expected
volatility of that growth. In additional, the neutral rate might also change the level of inflation
targeted or preferred by the central bank changes. A responsible central bank or monetary
authority will usually set its policy rate with reference to the level of expected economic activity and
the expected rate of increase in price i.e. inflation.
Example 3:
Suppose that an analyst estimates that the real risk-free rate is 1.25% and that average inflation over the next
year will be 2.5%. If the analyst observes the price of a default-free bond with a face value of $100 and one
full year to maturity as being equal to $95.92, what would be the implied premium embedded in the bond's
price for inflation uncertainty?
95.92 =
100
(1 + 0.0125 + 0.025 + θ)
θ = 0.504%
The difference between the yield of a non-inflation-indexed risk-free bond and the yield of an
inflation-indexed risk-free bond of the same maturity is the Breakeven Inflation Rate (BEI). BEI
makes investors indifferent between nominal and inflation-indexed bonds.
BEI
=
YTM Nominal
-
Expected Inflation
Expected (π + θ)
YTM TIPS (Adjusted for Real Inflation)
Inflation-Indexed [Real Yield]
Real + Expected
Inflation
Real + Actual Inflation
Real (π + θ)
BEI is composed of two elements: Expected Inflation 'π' and a risk premium for uncertainty about
actual inflation 'θ'; BEI = π + θ.
Example 4:
The BEI rate is expected to be 2% over the next year, what is the credit spread for a 2% annual pay corporate
bond maturing in one year with a market price of $96.91 ($100 par), if the real risk-free over the next year is
1%?
YTM Corporate Bond
PMT = 2
n=1
Fv = 100
Pv = -96.91
Credit Spread = Yield - BEI - R f
= 5.25% - 2% - 1%
= 2.25%
I/Y = ? (5.25%)
27
lllll
Risk Premium
If the underlying cashflows are uncertain, investors demand a risk premium for bearing the risk that
comes with uncertainty. The loss of utility from a lower wealth is larger than the gain from an
equivalent increase in wealth. An individual who requires compensation for this uncertainty is
called 'Risk Averse'. An investor's absolute risk-aversion declines with their wealth; wealthier
investors are less risk-averse and more willing to take risk relative to their poorer counterparts i.e.
E (m t ) is lower for the wealthier investors. However, the marginal utility of holding risky assets
declines as an investor holds more risky assets in her portfolio. When the markets are in
equilibrium, wealthy and poorer investors would have the same willingness to hold risky assets. In
this case, the risk premium for a given risk is lower for wealthier individuals because the average
loss of marginal utility (slope of utility) from any risk taking is smaller.
Consider a risk-free inflation-indexed ZCB that an investor will sell prior to maturity. The uncertainty
about the sale price gives rise to a risk premium.
P0 = E (P1 )
(1 + R f )
①
Risk Neutral present value
because it represents a risky
asset's value, if investors didn't
require compensation for
bearing risk.
+
Cov. [E (P1 ), E (m 1)]
②
Risk Premium
The Covariance term is the discount for risk.
With one period default-free bond, the
covariance term is '0' as there is no uncertainty
about the terminal value i.e. there is no risk
premium. But with two-period default-free
bond, the future price of $1 two-periods in the
future is known with certainty, but the price
one period in the future is not. Consequently,
the covariance term is not '0'.
Rp = Cov. [E (P1 ), E (m t )]
With Risk-Averse investors, the covariance term for most risky assets is expected to be negative i.e.
Cov. [E (P1 ) , E (m t ) ].
↑
↓
↑ Labor Income 1 and ↑ Risky Asset Values are High (P1 )
(m t ↓)
Bad Times: ↓ Labor Income 1 and ↓ Risky Asset Values are Low (P1 )
(m t ↑)
Good Times:
The lower current price (P0 ) increases expected return. This higher expected return is due to a
positive risk premium i.e. the lower the current price, the larger the magnitude of the negative
covariance term. On the other hand, any asset that tended to have relatively high returns when the
marginal utility of consumption was high would provide a type of hedge against bad times and bear
a negative risk premium and have a relatively high price and low required rate of return.
Short-dated bonds have been more reliable hedges against bad economic times than long-dated
bonds, which in turn means that the bond risk premium should be higher for longer dated
government bonds than for their short-dated equivalents. Bond risk premium will tend to rise in
times when investors place less value on the consumption hedging properties of government
bonds.
Small-cap stocks tend to have higher volatility and command a higher risk premium.
For the required return < risk-free rate, the asset's risk premium would need to
be negative, because the asset supplies relatively high returns in economic
conditions in which the marginal utility of consumption is relatively high i.e.
Covariance term is positive and the asset thus bears negative risk premium.
28
↶
Example 5:
A risky asset offers high positive returns during business downturns. A colleague argues that the nominal
require rate of return on the asset may be less than the nominal risk-free rate. Is the colleague correct?
A. Yes
B. No, the return must be higher than the nominal risk-free rate
C. No, the relationship between the asset's nominal return and the nominal risk-free rate is indeterminate.
✓
lllll
Credit Risk Premium
Credit Risk Premium or Credit Spread ' γ ' is the difference in yield between a credit risky bond and a
default free bond of the same maturity. It is the credit risk component of a corporate bond and the
evolution of bond spreads that will cause corporate and comparable government bond returns to
diverge over time.
Required Rate of Return for Credit Risky Bonds = R f + π + θ + γ
PD
↓
Boom
Recovery
When Credit Spread
Recession
PD
Recovery
↑
↑
Narrows
When Credit Spread
Risky Bonds > Default Free Bonds
(Lower Rated)
(Higher Rated)
Widens
Risky Bonds < Default Free Bonds
(Lower Rated)
(Higher Rated)
Leverage
Cyclical
Non-Cyclical
↓
High
Low
Credit Spread
↑
↓
Spreads for issuers in the consumer cyclical sector increase significantly during economic
downturns compared to spreads for issuers in the consumer non-cyclical sector. Cyclical industries
(e.g. durable goods manufacturers and consumer discretionary) tend to be relatively more sensitive
to the phase of the business cycle. Defensive or non-cycle industries (e.g. consumer
non-discretionary) tend to be relatively immune to fluctuations in economic activity; their earnings
tend to be relatively stable throughout the business cycle.
Credit risky bond shares the same risk as default-free bonds, which market participants often refer
to as 'Interest Rate Risk'. Interest rate component of a corporate bond will be driven by the same
factors that drive government bond yields and returns. In other words, they are both subject to
interest rate risk.
Example 6:
With regard to the credit risk of the sovereign debt issued by country governments, which of the following
statements is correct? The credit risk premium on such debt is?
A. Zero because governments can print money to settle their debt.
B. Negligibly small because no country has defaulted on sovereign debt.
C. A non-zero and positive quantity which varies depending on a country's creditworthiness.
✓
lllll
Equity Risk Premium
The discount rate used to value equity securities includes an additional risk premium, the equity
risk premium.
↷
Discount Rate for Equity = R f + π + θ + γ + K
Additional risk premium
relative to risky debt for
an investment in equities
29
'λ' Equity Risk Premium
Usually assets that provide a higher payoff during economic downturn are more highly valued
because of the consumption hedging property of the asset. This property reduces the risk premium
on as asset. Equity prices are generally cyclical, with higher values during economic expansions,
when the marginal utility of consumption is lower. Equity investments, therefore, are not the most
effective hedge against bad consumption outcomes. Because of this poor consumption hedging
ability, equity risk premium is positive. It is impossible to quantify equity risk premium Ex-Ante. But
we can atleast look at its Ex-Post value using very long term runs of data. Ex-Post risk premiums on
equity sectors can be computed as the difference between the average return on a sector and the
short-term risk-free rate.
Example 7:
Risk averse investors demanding a large equity risk premium are most likely expecting their future
consumption outcomes and equity returns to be:
A. Uncorrelated
B. Positively Correlated
C. Negatively Correlated
✓
lllll
Illiquidity Risk Premium
Commercial Real Estate is a special asset class. It can be viewed as being past equity (e.g. capital
appreciation), part bond (e.g. rental-coupon) and it is usually very illiquid 'Φ'
Discount Rate for Commercial Real Estate = R f + π + θ + γ + K + Φ
Illiquidity acts to reduce an asset class's usefulness as a large against bad consumption outcomes,
because of this, investors will demand a liquidity risk premium 'Φ'. But what should the discount
rate look like? Consider the following tenants and associated rental/leasing agreements:
(i) Developed economy government
tenant that agrees to pay rental
income that is indexed to:
1 + Rf + K + Φ
(ii) Developed economy government
tenant that agrees to pay fixed
nominal rental income:
1 + Rf + π + θ + K + Φ
(iii) Corporate tenant that agrees to
pay a fixed nominal rental income
1 + Rf + π + θ +γ + K + Φ
Boom: Commercial Property Value
Recession: Commercial Property Value
↓
↑
The pro-cyclical nature of commercial property prices means that investors will generally demand a
relatively high risk premium in return for investing in this asset class. The reason is because
commercial property doesn't appear to be a very good hedge against bad economic outcomes.
Commercial property values tend to be very cyclical; because of this, the correlation of commercial
property values with those of other asset classes (e.g. equities) tends to be positive. Therefore, the
risk premium required by investors for investment in commercial properties will be relatively high
and often close to the risk premium required for equity investments.
30
Earnings
Boom
↑
Recession
↓
P/E
↑
(g↑, r↓)
↓
(g↓, r↑)
Dividend Yield
↓
Style
Growth Stocks
Found-in
Immature Markets
↑
Value Stocks
Mature Markets
ERP
↓
↑
Price multiples are positively correlated with expected earnings growth rates and negatively correlated to
required returns. Therefore, price multiples rise with increases in expected future earnings growth and with a
decrease in any of the components of the required rate of return (the real rate, falling volatility in real GDP
growth, expected inflation, risk premium for inflation uncertainty or equity risk premium).
NOTES
1. Shiller's CAPE called Real Cyclically Adjusted P/E Ratio reduces the volatility of unadjusted P/E ratios by
using: 'P' represents the real or inflation-adjusted price of equity market and 'E' is a 10-year moving
average of the market's real or inflation adjusted earnings.
31
CHAPTER 47
Analysis of Active Portfolio Management
Active Management seeks to add value by outperforming a passively managed benchmark portfolio. We are
going to assume that the systematic risk of the active portfolio is the same as the systematic risk of the
benchmark portfolio i.e. the beta of the active portfolio relative to the benchmark, active return is computed
as the difference in risk-adjusted returns and is known as 'Alpha' 'L ' i.e. L p = R p- β p. R p .
Ex-Post: Ex-Post Active Return is the difference between the realized return of the actively managed portfolio
and its benchmark portfolio.
Σ
△
E (R A ) = E (R P ) - E (R B )
w i . E (R i ) = Σ w P i . E (R i ) - Σ w B i . E (R i )
Whereas,
w i : Active Weight = w P i - wB i (i.e. Active Weights must sum to '0').
△
Ex-Ante: Ex-Ante Active Return is the expected return on the active portfolio minus the expected return on
the benchmark.
E (R A ) = E (R P ) - E (R B )
E (R A ) = Σ w P j . E (R P j ) - Σ w B j . E (R B j)
Alternatively, Active Return can be decomposed into two parts:
E (R A ) = Σ
△ w i . E (R B i ) + Σ w P i . E (RA i )
①
②
Return from Asset Allocation Return from Security Selection
From deviations of asset class
From active returns within
portfolio weights from
asset classes
benchmark weights
Example 1:
Optima Fund invests in three asset classes: US Equities, US Bonds and International Equities. The asset
allocation weights of Optima and the expected performance of each asset class and the benchmark are
shown in the following table:
US Equities
US Bonds
International Equities
wP i
wBi
E (R P i )
E (R B i )
45%
30%
25%
40%
30%
30%
11%
6%
4%
12%
5%
12%
Calculate the expected active return?
△
E (R A ) = Σ
w j . E (R B j ) + Σ w P j . E (R A j )
US Equities = (0.45 - 0.4) (0.12) + (0.45) (0.11 - 0.12)
US Bonds = (0.3 - 0.3) (0.05) + (0.3) (0.06 - 0.05)
International Equities = (0.25 - 0.3) (0.12) + (0.25) (0.14 - 0.12)
TOTAL =
0
0.35%
It can be seen that all of the expected active return is attributable to security selection.
32
Example 2:
Forecasted portfolio statistics for funds X, Y and Z and the benchmark.
Fund X
Fund Y
Fund Z
Benchmark
60
40
65
35
68
32
60
40
Portfolio Weights
Global Equities (%)
Global Bonds (%)
The expected active return from asset allocation for Fund X is?
A. Negative
B. Zero
Σ
w j . E (R B j) = (60 - 60) . R Be+ (40 - 40) . R Bb= 0
C. Positive
✓
△
Example 3:
When measuring value added by active management, it is more accurate to state that the active weights in an
actively managed portfolio:
A. Must add to 100%.
B. Are the differences between an individual asset's weight in the actively managed portfolio vs. the
corresponding weight in an equally-weighted portfolio.
C. Must be positively correlated with realized asset returns for value added to be positive.
✓
↷
Fig 1: The Sharpe Ratio and the information ratio are two different methods of measuring a portfolio's
risk-adjusted rate of return:
Ex-Ante Information Ratio is generally +ve.
Information Ratio
Def: The Sharpe Ratio (SR) is calculated as excess
return per unit of risk.
I
SR = R P- R f
o P↷
= Average Portfolio Return
Portfolio Risk
An important attribute of the Sharpe Ratio is that it
is unaffected by the addition of cash or leverage in
the portfolio.
Investors should form portfolios using two funds
i.e. Rf asset and risky asset portfolio with the
highest Sharpe Ratio known as Two-Fund
Separation.
↑
If Volatility is
- Hold more Cash (R f )
& less Risky Asset
↓
If Volatility is
- Hold less Cash (R f )
& more Risky Asset
Revised Weights: (Due to change in Volatility)
I
w C = o P = Desired
oC
Actual
1 - wC = w R
I
Risky Asset
Risk Free
f
IR = RP - RB = RA
o (RP - RB ) o RA
I
Total Risk
Def: The Information Ratio (IR) tells the investor
how much active return has been earned for
incurring the level of active risk.
I
Sharpe Ratio
Ex-Post Information Ratio often -ve.
= Active Return
Active Risk
Benchmark
Tracking Risk
Unlike Sharpe Ratio, Information Ratio is affected
by the addition of cash or leverage in portfolio.
The Information Ratio of an unconstrained portfolio
is unaffected by the aggressiveness of the active
weights. If the active weights of a portfolio are
tripled, the active return and active risk both triple,
leaving the Information Ratio unchanged.
If we combine an active managed portfolio with an
allocation to the benchmark portfolio, the resulting
blended portfolio will have the same Information
Ratio as the original actively managed portfolio. As
we increase the weight of the benchmark portfolio,
the active return and active risk decrease
proportionally, leaving the Information Ratio
unchanged.
33
A Closet Index Fund is a fund that is purported to be actively managed but in reality closely tracks the
underlying benchmark index. These funds will have a sharpe ratio similar to that of the benchmark index, a
very low information ratio and little active risk. After fees, the the information ratio of a closet index fund is
often negative i.e. while there may be little active risk, the information ratio of a closet fund will likely be close
to '0' or slightly negative if value added cannot overcome the management fees. Ofcourse, if one has the
actual holdings of the fund, closet indexing is easy to detect on the basis of a measurement called 'Active
Share'.
A fund with '0' systematic risk (e.g. a market-neutral long-short equity fund) that uses the risk-free as its
benchmark would have an information ratio that is equal to its sharpe ratio. This is because active return will
be equal to the portfolio's return minus risk-free rate, and active risk will be equal to total risk. In fact, under
the zero-sum property of active management, the average realized information ratio across investment funds
with the same benchmark should be about '0'. If Systematic risk is '0' then IR = SR.
If you want to increase the sharpe ratio, you need to keep the information ratio positive at all times. If
positive: SR P > SR B . The optimal risky portfolio is the portfolio with the highest sharpe ratio. The SR P > SR B
based on the information ratio of the actively managed portfolio. The portfolio with the highest IR will also
be the portfolio with the highest SR, so investors will choose the active manager with the highest IR; the
actively managed portfolio with the highest IR is the optimal (active) portfolio for all investors regardless of
their risk tolerance. As active risk increases,
SR also increases.
P
Example 4:
Consider an investor choosing between two risky portfolios: a large-cap stock portfolio and a small-cap stock
portfolio. Although forecasts about the future are usually subjectively determined. The current R f rate is 2.8%.
The forecasted 0.5 sharpe ratio of the small-cap portfolio is higher than the 0.47 ratio of the large-cap
portfolio, but suppose the investor doesn't want the high 21.1% volatility associated with the small-cap
stocks.
Expected Return
Expected Volatility
Sharpe Ratio
Large-Cap
10%
15.2%
0.47
Small-Cap
13.4%
21.1%
0.5
a. How much would an investor need to hold in cash (% terms) to reduce the risk of a portfolio invested in the
small-cap portfolio and cash to the same risk level as that of the large-cap portfolio?
I
I
Revised Weights = w C = o P = 0.152 = 0.72
o C 0.211
wR = 1 - wC = 1 - 0.72 = 0.28
f
With that amount of cash, the volatility of the combined portfolio will be 0.72 (0.211) = 15.2%, the same as
the large-cap portfolio.
b. Compare the expected return of the small-cap plus cash portfolio with the expected return of the large-cap
portfolio.
Expected Return of the Combined Portfolio = (0.134) (0.72) + (0.28) (0.028) = 10.4%
It is 40 bps higher than the 10% expected return on the large-cap portfolio but with the same risk as the
large-cap portfolio. To reconfirm, the sharpe ratio of the combined portfolio is (10.4% - 2.8%) / 15.2% = 0.5,
the same as the original 0.5 value.
34
Example 5:
Assume active risk is 10% and active return 5%. If you combine benchmark with an active portfolio:
b. Benchmark: An investor invests 70% in active
portfolio and 30% in S&P500.
R A = 5%
o A = 10%
I
I
RA = 0
oA = 0
IR = 5% = 0.5
10%
R A = (0.7) (0.05) + (0.3) (0) = 3.5%
o A = (0.7) (0.1) + (0.3) (0) = 7%
I
a. What is the impact on Information Ratio?
IR = 3.5% = 0.5
7%
The information ratio is unaffected by the aggressiveness of the active weights (i.e. deviations from
benchmark weights) in the managed portfolio, because both the active return and active risk increase
proportionally.
For an unconstrained active portfolio, the optimal amount of active risk is the level of active risk that
maximizes the portfolio's sharpe ratio. The optimal amount of active risk can be calculated as:
I
I
o A* = IR . o B
SR B
Investing with highest information ratio manager, will produce highest sharpe ratio for investor's portfolio:
2
SRP = SR B + IR
2
①
I
SR B = RB - R f
oB
Equation 1 is not practical for comparisons of investment skill investment skill involving negative information
ratio because the sign is lost in squaring. As a consequence, according to mean-variance theory, the expected
information ratio is the single best criterion for assessing active performance among various actively
managed funds with the same benchmark.
2
2
2
I
I
I
o P=oB+oA
Unconstrained active portfolios have optimal weights for each of the securities in the portfolio based on ExAnte expectations of active return and active risk. Sometimes constraints (e.g. long only positions) are
imposed on active portfolios, resulting in less than optimal weights.
Example 6:
Omega Fund has an information ratio of 0.3 and active risk 8%. The benchmark portfolio has a sharpe ratio of
0.4 and total risk of 16%. If a portfolio (Portfolio P) with an optimal level of active risk has been constructed by
combining Omega Fund and the benchmark portfolio, Calculate:
a. Optimal Risk
b. Sharpe Ratio
c. Expected Active Return
d. Portfolio P's excess Return
e. The proportion of Benchmark and Omega Fund in Portfolio P
35
a. 0.3 x 0.16 = 12%
0.4
2
2
b. (0.4) + (0.3) = 0.5
c. E (R A ) = 0.3 (0.12) = 3.6%
→ (RP - R B )
d. Portfolio P's Excess Return
(R P - R f ) = (R P - R B ) + (RB - R f )
10% = 3.6% + (R B - R f )
(R B - R f ) = 6.4%
2
2
2
I
I
o P = (0.16) + (0.12)
o P = 0.02
I
SR P = R P - R f
oP
(R P - R f ) = 0.5 (0.02) = 0.1 or 10%
e. The optimal level of active risk is 12% and Omega Fund has an active risk of 8%, so we calculate that
12%/8% = 1.5 of portfolio P's allocation will be to the Omega Fund and - 0.5 (1 - 1.5) to the benchmark
portfolio.
The Fundamental Law
There are three factors that determine the information ratio:
1. Information Coefficient: Information Coefficient (IC) is a measure of a manager's skill i.e. higher the
better. Ex-Ante IC must be positive or the investor would generally not pursue active management
and would simply invest in the passive benchmark. Ex-Post IC might be either positive or negative,
leading to positive or negative value added.
↶ IC = Corr. R A i , E (R A i )
oi
I
[o i
I
IC can take values
anywhere from -1 to +1
]
2. Transfer Coefficient: The optimal active weight for a security is positively related to its expected
active return and negatively related to its expected active risk. Transfer Coefficient (TC) can take on
values anywhere from -1 to +1, although TC values are typically positive and range from about 0.2 to
0.9. In fact, at TC = 0, there would be no correspondence between the active return forecasts and
active weights taken and thus no expectation of value added from active management. For an
unconstrained active portfolio, the active weights will be equal to the optimal weights and TC = 1 i.e.
allowing the full expected value added to be reflected in the portfolio structure. For a constrained
portfolio (e.g. constraints on short positions or active risk), TC < 1 i.e. the portfolio needs rebalancing.
TC can take values
anywhere from -1 to +1
oi
①
TC is the cross-sectional
correlation between the
forecasted active returns
and the actual weights
adjusted for risk
②
I
△ w i o i ]= Corr. (△ w*i o i , △ w i o i )
I
[
I
↶TC = Corr. E (R A i ) ,
I
lllll
TC can be thought of as
the correlation between
actual active weights and
optimal active weights
36
I
I
I
I
△
I
Whereas,
w *i : E (R A i ) .
oA
= E (R A i ) . o A
2
2
oi
Σ [ E (RA i ) ]
o i2
IC BR
2
oi
①
②
Mean-Variance
Optimal Active
Weights
I
The desired deviations (positive or
negative) from the benchmark weight for
security ' i ' are higher for larger values of
the forecasted active return E(R A i ), but
are reduced by forecasted volatility o i .
3. Breadth: Breadth (BR) is the number of independent active bets taken per year. For example, if a
manager takes active positions in 10 securities each month, then BR is 10 x 12 = 120. Alternatively, if
the investor makes quarterly or monthly forecasts about a security that are truly independent over
time, then BR can be as high as the no. of securities times the no. of rebalancing periods per year. BR
increases with no. of rebalancing periods only if the active returns are uncorrelated over times.
BR < No. of Securities: If active returns are positively correlated
BR > No. of Securities: If active returns are negatively correlated
BR
E (IR)
↓
↑
↓
↑
IC
↑
↓
The Grinold Rule allows us to compute the expected active return based on the IC, active risk and a
standardized score 'S i ' (Score of Security i , standardized with an assumed variance of 1):
I
E (R A i ) = IC . o i . S i
△ w i . E (R A i)
RA = Σ △ wi . RA i
= Cov. (△ w i . R A i ) . N
= Corr. (△ w i . RA i ) . o (△ w i ) . o R i . N
I
I
E (R A i ) = Σ
A
Realized value added from active management is the Ex-Post Alpha that the manager achieves:
I
↶E (R A / IC R ) = IC R . TC . BR . o A
R A = E (R A / IC R ) + Noise
↶
Represents the expected value
added, given the realized skill of
the investor that period
Ex-Post Information
Ex-Post i.e. realized decomposition of the portfolio's active return variance into two parts: Variation due to
the realized information coefficient and variation due to constraint-induced noise. Clarke de Silva and Thorley
2
2
(2005) showed that the two parts of the realized variance are proportional to TC and 1 - TC . For example,
with a TC value of, say 0.6, only TC 2 = 36% of the realized variation in performance is attributed to variation in
the realized information coefficient and 1 - TC 2= 64% comes from constraint-induced noise. Low TC investors
will frequently experience periods when the forecasting process succeeds but actual performance is poor or
when actual performance is good even though the return forecasting process fails.
37
Example 7:
Caramel Associates uses the fundamental law to estimate its expected active returns. Two things have
changed. First, Caramel will lower its estimate of the IC because they felt their prior estimates reflected over
confidence. Second, their major clients have relaxed several constraints on their portfolios, including social
screens, prohibition on short selling and constraints on turnover. Which of these changes will increase the
expected active return?
A. Only the lower IC
B. Only the relaxation of several portfolio constraints
C. Both the lower IC and the relaxation of portfolio constraints
✓
U: Unconstrained
C: Constrained
*: Optimal
Fig 2: Constrained Vs. Unconstrained Portfolios
Constrained
1
Information Ratio (IR)
(
△
Unconstrained
IR* = IC . BR
as TC = 1
IR = IC . TC . BR
as TC < 1
w i and
w*i will differ)
△
E (R A ) = IC . TC . BR . o A
E (R A )* = IC . BR . o A
3
Optimal o A or
Optimal Level of
Aggressiveness
o A* = IR*U . o B . TC
SR B
o A* = IR U . o B
SR B
4
Highest Sharpe Ratio
SR P = SR B + IR U + TC
5
oP
o P = o B + o A + TC
I
I
2
2
2
SR P = SR B + IR U
2
2
2
2
I
I
oP =o B+oA
I
2
I
2
I
I
I
1
I
I
I
2
2
I
Expected Active Return
E (R A )
I
2
If we assume that the individual securities all have the same residual volatility, then the correlation formulas for IC and TC
don't need to be risk weighted. As IR gets close to '0' either because of constraints or because the manager is judged to be
less skilled, the optimal amount of active risk goes to '0'; i.e. the optimal portfolio becomes passive benchmark portfolio.
I
I
Because TC < 1, the IR < IR* and E (R A ) < E (R A )* . This implies that the o A* of a constrained portfolio will be
less than the o A* of an unconstrained portfolio. Similarly, the SR of a constrained portfolio is lower than the
SR of an unconstrained portfolio.
Example 8:
Consider the simple case of four individual securities whose active returns are uncorrelated with each other
and forecasts are independent from year to year. The active return forecasts, active risks and the active
weights for each security are shown in the exhibit below:
E (R A )
oA
1
2
3
4
5%
10%
- 5%
- 10%
25%
50%
25%
50%
I
Security
wA
18%
9%
- 18%
- 9%
38
a. Suppose that the benchmark portfolio for these four securities is equally weighted (i.e. w B = 25% for each
security) and that the forecasted return on the benchmark portfolio is 10%. What are the portfolio weights
and the total expected returns for each of the four securities?
Security
1
2
3
4
Total Weight
25% + 18% = 43%
25% + 9% = 34%
25% - 18% = 7%
25% - 9% = 16%
100%
Total Return Forecast
10% + 5% = 15%
10% + 10% = 20%
10% - 5% = 5%
10% - 10% = 0%
b. Calculate the forecasted total return and active return of the managed portfolio.
Forecasted Total Return of the Portfolio = 0.43 (0.15) + 0.34 (0.2) + 0.07 (0.05) + 0.16 (0) = 13.6%
E (R A ) = R P - R B = 13.6% - 10% = 3.6%
or Σ wA x E (R A ) = 0.18 (0.05) + 0.09 (0.1) - 0.18 (- 0.05) - 0.09 (- 0.1) = 3.6%
c. Calculate the active risk of the managed portfolio.
2
2
2
2
2
2
2 1/2
2
[(0.18) (0.25) + (0.9) (0.5) + (- 0.18) (0.25) + (- 0.09) (0.5) ]
= 9%
I
I
d. Verify the basic fundamental law of active management using the E (R A ) and o A of the managed portfolio.
The individual security active return forecasts and active weights were sized using an IC = 0.2, BR = 4 and
o A = 9%.
I
The basic fundamental law states that the expected active portfolio return is IC . BR . o A = (0.2) ( 4) (0.09)
= 3.6%, which is consistent with the calculation in the solution 'b'. Alternatively, the IR = 3.6% / 9% = 0.4,
confirms the basic fundamental law that IR* = IC . BR = (0.2) ( 4) = 0.4.
Strengths and Limitations of the Fundamental Law of Active Management
The fundamental law can be used to evaluate a range of active strategies, include security selection, market
timing and sector rotation. The practical limitations of the fundamental law of active management can be
summarized as 'Garbage In, Garbage Out', poor inputs estimates lead to incorrect evaluations. The limitations
are generally derived from inaccurate estimates of the two inputs:
I
I
I
1. Ex-Ante Measurement of Skill: Problem being managers tend to overestimate their ability to outperform
the market and hence overestimate their IC. For example, Qian and Hur (2004) expanded the basic form of
the fundamental law by including the uncertainty about the level of skill or the reality that the realized IC
can vary over time. Specifically, they showed that realized active portfolio risk o A is a product of both the
benchmark tracking risk predicted by the risk model, denoted o TR and the additional risk induced by the
uncertainty of the information coefficient, denoted o IC .
I
I
I
o A = o TR . o IC . N
Their insight about 'Strategy Risk' is derived under the simplifying assumptions that portfolio positions are
unconstrained, TC = 1 and that breadth (BR) is the no. of securities (N) BR = N, but can be expanded to
include both refinements.
I
I
E (R A ) = IC . o A
o IC
39
2. Independence of Investment Decisions: BR measures the no. of independent decisions made by the
investor each year and is equal to the no. of securities only if the active returns are cross-sectionally
uncorrelated. Similarly, BR increases with the no. of rebalancing periods only if the active returns are
uncorrelated over time. If two or more decisions rely on same (or similar) information, then they are not
independent i.e. correlated.
Independent or Uncorrelated : BR = N
Dependent or Correlated : BR =
(BR < N)
N
Correlations between the
1 + (N - 1) r↷ active security returns
Decision independence may be compromised by systemic influences within a strategy, the cross-sectional
dependency. For example, a value strategy applied to different stocks within an industry may not be truly
independent (most stocks will have similar fundamentals, such as P/E ratio). Similarly, decision
independence can be compromised by time-series dependency. Monthly rebalancing decisions may not be
truly independent from period to period.
Market timing is simply a bet on the direction of the market (or a segment of the market). For a market timer,
the IC is based on the proportion of correct calls.
IC = 2 (%Correct ) - 1
or
IC = %Correct - %Incorrect
Market timing can also be used to make sector rotation decisions. For example, an active manager may
allocate assets into sectors that are expected to outperform. Consider a two-sector market made up of
Sectors X and Y.
Correct: 55%
Incorrect: 45%
I
I
I
I
E (R A ) = IC . BR . o A
Investment Grade
o Q = 2.84%
High Yield
o Q = 4.64%
I
rXY
o (R X- R Y ) = 0.038 or 3.8%
o (RX - R Y) = 0.038 4 = 0.076 or 7.6%
I
Quarterly
Annual
I
o A = o (RX - R Y ) . BR
I
For Example,
2
I
Annualized Active Return
2
I
Annualized Active Risk
2
o (R X- R ) = o X - 2 o X o Y r XY + o Y
Y
I
Standard Deviation
rXY
I
Correlation
I
E (R Y )
oY
I
E (R X )
oX
Time Series IC = 0.55 - 0.45 = 0.1 or 10%
40
Type 1: Without a limit on active risk
Quarterly
or
E (R A ) = 0.55 (3.8) + (0.45) (- 3.8) = 38 Bps
I
E (R A ) = IC . o A . Score = (0.1) (3.8) (1) = 38 Bps
Type 2: Limit on active risk to 2%
w A = 2% = 26.3%
7.6%
Let's assume benchmark weights are 70% for investment grade bonds and 30% for high yield bonds.
(a) When investor believe credit risk will pay off
IG : 70% - 26.3% = 43.7%
HY : 30% + 26.3% = 56.3%
(b) When investor believes credit risk will not pay off
IG : 70% + 26.3% = 96.3%
HY : 30% - 26.3% = 3.7%
or
Quarterly
BR . o A = 0.1 .
I
E (R A ) = IC .
4 . 2 = 40 Bps
E (R A ) = w A . o A = (0.263) (0.38) = 10 Bps
I
Annual
Therefore, market timer will have lower BR compared to security sector.
NOTES
1. Float-adjusted market capitalization weighted indexes represent an incremental improvement over
non-float-adjusted indexes by accounting for the percentage of a security or asset that is not privately held
and thus available to the general investing public. One important consequence of using a float-adjusted
capitalization weighted market index as benchmark is that when all relevant assets are included in the
market, the value added from active management becomes a zero-sum game with respect to the market.
41
CHAPTER 48
Trading Costs and Electronic Markets
Fixed Costs 1
Explicit Costs 2
(Direct Costs)
Costs of Trading
Variable Costs
Implicit Costs 3
(Indirect Costs)
1. Fixed Cost: For buy-side institutions, fixed trading costs include the cost of employing buy-side traders, the
costs of equipping them with proper trading tools (electronic systems and data), and the costs of office space
(trading rooms or corners). Small buy-side institutions often avoid these costs by not employing buy-side
traders. Their portfolio managers submit their orders directly to their brokers.
2. Explicit Cost: are the direct costs of trading such as broker commission costs, transaction taxes, stamp duties
and fees paid to exchanges. They are costs for which a trader could receive a receipt.
3. Implicit Cost: By contrast, are indirect costs caused by the market impact of trading. Although no receipt can
be given for implicit costs, they are real nonetheless. Buyers often must raise prices to encourage sellers to
trade with them and sellers often must lower prices to encourage buyers. Small market orders generally
have small market impact because these orders often are immediately filled by traders willing to trade at
quoted bid and offer prices, or even better prices. Larger orders have greater market impact when traders
must move the market to fill their orders. In these cases, traders must accept larger price concessions (less
attractive prices) to execute their orders in entirely. Implicit costs result from the following issues: (a) Bid-Ask
Spread (Ask Price minus Bid Price), (b) Market Impact (or Price Impact), (c) Delay Costs (or also called
Slippage): Traders fail to profit when they fill their orders after prices move as they expect and
(d) Opportunity Costs (or Unrealized Profit/Loss): Traders fail to profit when their orders fail to trade and
price move as they expect.
The Best Bid i.e. Inside Bid, is the offer to buy with the highest bid price.
The Best Ask i.e. Inside Ask, is the offer to sell with the lowest ask price.
The spread between the Best Bid price and the Best Ask price in a market Bid-Ask spread, which is also known
as the Inside Spread. It will be smaller (tighter or narrower) than the individual dealer spreads if the dealer
with the highest bid price is not also the dealer with the lowest ask price.
Example 1:
Suppose that a portfolio manager gives the firm's trading desk an order to buy 1000 shares of Economical
Chemical Systems Inc. (ECSI). Three dealers A, B and C make a market in those shares. When the trader views
the market in ECSI at 10:22 am on his computer screen, the three dealers have put in the following limit
orders to trade in an exchange market.
Dealer A
Dealer B
Dealer C
Bid: 98.85 for 600 Shares
Bid: 98.84 for 500 Shares
Bid: 98.82 for 700 Shares
The Bid-Ask Spreads of Dealers A, B and C are respectively,
A: 100.51 - 98.85 = 1.66
B: 100.55 - 98.84 = 1.71
C: 100.49 - 98.82 = 1.67
Ask: 100.51 for 1000 Shares
Ask: 100.55 for 500 Shares
Ask: 100.49 for 200 Shares
42
The Best Bid price, 98.85 by Dealer A is lower than the Best Ask price 100.49 by Dealer C. The market spread
is thus 100.49 - 98.85 = 1.64, which is lower than any of the dealer's spreads. The trader might see the quote
information organized on his screen in a display called 'Limit Order Book', the bids and asks are separately
ordered from best to worst with the best at the top. The trader also notes that the mid-quote price (halfway
between the market bid and ask prices) is (100.49 + 98.85) / 2 = 99.67.
Example 2:
Three dealers make market for the shares of Light Systems. Based on these prices, SAMN's trading desk
executes a market sell order for 1,100 shares of Light Systems.
Dealer B
Dealer C
Dealer A
Bid: $17.15 for 900 Shares (10:10 am)
Bid: $17.14 for 1500 Shares (10:11 am)
Bid: $17.12 for 1100 Shares (10:11 am)
Ask: $17.19 for 1200 Shares (10:11am)
Ask: $17.2 for 800 Shares (10:10 am)
Ask: $17.22 for 1100 Shares (10:12 am)
a. Based on the above exhibit, the Inside Bid-Ask Spread for the limit order book for Light Systems is closest
to: $ 17.19 - $ 17.15 = $ 0.04. The highest bid price for Light Systems is $ 17.15 and the lowest ask price is
$ 17.19.
b. The total amount that SAMN will receive, on a per share basis for executing the market sell order is closest
to:
SAMN's trading desk executes a market sell order for 1,100 shares. Based on the limit order book, the
trader would first sell 900 shares at $ 17.15 (highest bid, Dealer B) and then sell the remaining 200 shares
at $ 17.14 (second highest bid, Dealer C). Therefore, the approximate price per share received by SAMN for
selling the 1,100 shares is equal to:
(900 x 17.15) + (200 x 17.14) = $ 17.1482 per Share
1100
lllll
Implicit Transaction Cost Estimates
To estimate transaction costs, analysts compares trade prices to a benchmark price. Commonly used
price benchmarks include the mid-quote price at the time of the trade, the mid-quote price at the time
of the order submission and a volume weighted average price around the time of the trade. These
three benchmarks respectively, correspond to the effective spread, implementation shortfall and VWAP
methods of transaction costs estimation.
1. Effective Spreads: The market spread is a measure of trade execution costs. The loss is the cost of trading,
because this strategy otherwise accomplishes nothing. Given that two trades generated the cost, the cost
per trade is one half of the quoted spread. It uses the mid-quote price (the average or midpoint) as the
benchmark price.
(Orders)
Effective Spread = Trade Size x Trade Price - (Bid + Ask) / 2
BUY
(Bid + Ask) / 2 - Trade Price
SELL
{
For a buy order filled at the ask, the estimated implicit cost of trading is half the bid-ask spread, because
Ask - [(Bid + Ask) / 2] = (Ask - Bid) / 2. Multiplying this mid-quote price benchmark transaction cost estimate
by 2 produces a statistic called the 'Effective Spread'. It is the spread that traders would have observed if
the quoted ask (for a purchase) or bid (for a sale) were equal to the trade price.
Smaller Orders often fill at better prices, Larger Orders often fill worse prices and Standing Orders offering
liquidity fill at same-side prices i.e. Buy at Bid, Sell at Ask, if they fill at all.
43
The effective spread is a sensible estimate of transaction costs when orders are filled in single trades. If an
order fills at a price better than the quoted price (e.g. a buy order fills at a price below the ask price), the
order is said to receive 'Price Improvement' and the spread is effectively lower. If an order that fills at a
price outside the quoted spread, then it has an effective spread that is larger than the quoted spread. Such
results occur when trade execution prices are worse than quoted prices. The effective spread is a poor
estimate of transaction costs when traders split large orders into many parts to fill over time. Such orders
often move the market and cause bid and ask prices to rise or fall. The impact of the order on market
prices called 'Market Impact' makes trading expensive, especially for the last parts to fill.
Effective spreads also don't measure 'Delay Costs' i.e. Slippage that arises from the inability to complete
the desired trade immediately because of its size in relation to the available market liquidity. Delay is costly
when price moves away from an order (up for a buy order, down for a sell order), often because
information leaks into the market before or during the execution of the order. When delays in execution
causes a portion of the order to go unfilled, the associated cost is called 'Opportunity Cost'. For example,
suppose a futures trader places an order to buy 10 contracts with a limit price of $99, good for one day,
when the market quote is $99.01 to $99.04. The order doesn't execute and the contract closes at $99.8. If
the order could have been filled at $99.04, the difference ($99.8 - $99.04 = $0.76) reflects the opportunity
cost per contract. By trading more aggressively, the trader could avoid these costs. Opportunity costs are
difficult to measure.
2. Implementation Shortfall: This method of measuring trading costs addresses the problems associated with
the effective spread method. It measures the total cost of implementing an investment decision by
capturing all explicit and implicit costs including market impact costs, delay costs and opportunity costs,
which are often significant for large orders. Implementation Shortfall compares the values of the actual
portfolio with that of a paper portfolio constructed on the assumption that trades could be arranged at the
prices that prevailed when the decision to trade is made i.e. the prevailing price is decision price, the actual
price or strike price. The excess of the paper value over the actual value is the implementation shortfall.
3. VWAP Transaction Cost Estimates: Analysts typically compute the Volume-Weighted Average Price (VWAP)
using all trades that occurred from the start of the order until the order was completed, a measure that is
often referred to as 'Internal VWAP'. The VWAP is the sum of the total order value of the benchmark trades
divided by the total quantity of the trades.
(Orders)
VWAP = Trade Size x Trade VWAP - VWAP Benchmark
BUY
VWAP Benchmark - Trade VWAP
SELL
{
Interpreting VWAP transaction cost estimates is problematic when the trades being evaluated are a
substantial fraction of all trades in the VWAP Benchmark. In both cases, the Trade VWAP and the VWAP
Benchmark will be nearly equal, which would suggest that the evaluated trades were not costly. But this
conclusion would be misleading if the trade has substantial price impact. For example, if a large trader
were the only buyer for a given trading period, the VWAP transaction cost estimate would be '0' regardless
of the market impact.
Large orders have price impact when they move down the book as they full. The price impact of an order
depends on its size and the available liquidity.
Example 3:
Arapahoe Tanager a portfolio manager of a Canadian small-cap equity mutual fund, and his firm's chief
trader Lief Schrader, are reviewing the execution of a ticket to sell 12,000 shares of Alpha Company, limit
C$9.95. The order was traded over the day. Schrader split the ticket into 3 orders that executed that day as
follows:
44
A. A market order to sell 2000 shares executed a price of C$10.15. Upon order submission, the market was
C$10.12 bid for 3000 shares, 2000 shares offered at C$10.24.
B. A market order to sell 3000 shares executed at a price of C$10.11. Upon order submission, the market was
C$10.11 bid for 3000 shares, 2000 shares offered at C$10.22.
C. Towards the end of the trading day, Schrader submitted an order to sell the remaining 7000 shares, limit
C$9.95. The order submitted in part, with 5000 shares trading at an average price of C$10.01. Upon order
submission, the market was C$10.05 bid for 3000 shares, 2000 shares offered at C$10.19. This order
exceeded the quoted bid size and 'walked down' the limit order book i.e. after the market bid was filled, the
order continued to buy at lower prices. After the market closed, Schrader allowed the order to cancel.
Tanager did want to buy the 2000 unfilled shares on the next trading day.
Only two other trades in Alpha Company occurred on this day: 2000 shares at C$10.2 and 1000 shares at
C$10.15. The last trade price of the day was C$9.95; it was C$9.5 on the following day.
1. For each of the three fund trades, compute the quoted spread. Also compute the average quoted spreads
prevailing at the time of each trade.
A. Sell 2000 @ 10.15
(bid) 10.12 for 3000 Sh.
(ask) 10.24 for 2000 Sh.
B. Sell 3000 @ 10.11
C. Sell 3000 @ 10.11
10.01 for 5000 Sh.
(bid) 10.11 for 3000 Sh.
(bid) 10.05 for 3000 Sh.
(ask) 10.22 for 2000 Sh.
(ask) 10.19 for 2000 Sh.
The quoted spread is the difference between the ask and bid prices:
A. 10.24 - 10.12 = C$ 0.12
B. 10.22 - 10.11 = C$ 0.11
C. 10.19 - 10.05 = C$ 0.14
Average Quoted Spread = 0.12 + 0.11 + 0.14 = C$ 0.1233
3
2. For each of the three fund trades, compute the effective spread (use the average fill price for the third
trade). Also, compute the average effective spread.
Effective Spread
A : 2 x [(10.12 + 10.24 / 2) - 10.15] = C$ 0.06
B : 2 x [(10.11 + 10.22 / 2) - 10.11] = C$ 0.11
C : 2 x [(10.05 + 10.19 / 2) - 10.01] = C$ 0.22
Average Effective Spread = 0.06 + 0.11 + 0.22 = C$ 0.13
3
3. Explain the relative magnitudes of quoted and effective spreads for each of three fund trades.
A
B
C
Quoted Spread
0.12
0.11
0.14
Effective Spread
0.06
0.11
0.22
Selling Price (10.15) > Bid (10.12) ; ES < QS
Selling Price (10.11) = Bid (10.11) ; ES = QS
Selling Price (10.01) < Bid (10.05) ; ES > QS
4. Calculate the VWAP for all 13,000 Alpha Company shares that traded that day and for the 10,000 shares
sold by the mutual fund. Compute the VWAP transaction cost estimate for the 10,000 shares sold.
45
VWAP Benchmark = (2000 x 10.15) + (3000 x 10.11) + (5000 x 10.01) + (2000 x 10.2) + (1000 x 10.15)
13,000
= C$ 10.0946
VWAP Trade = (2000 x 10.15) + (3000 x 10.11) + (5000 x 10.01) = C$ 10.068
10,000
VWAP Transaction Cost Estimate (SELL) = Trade Size x (VWAP Benchmark - VWAP Trade)
= 10,000 x (10.0946 - 10.068)
= C$ 266.15
Fig 1: Types of Electronic Traders
Proprietary Traders
Proprietary Traders who are
registered as broker/dealers
usually send their orders
directly to exchanges. Those
who are not broker/dealers
must send their orders to
brokers, who then forward
them to exchanges. These
brokers are said to provide
sponsored access to their
proprietary electronic trader
clients. E.g. Dealers,
Arbitraguers and Front
Runners. All of whom are
profit motivated traders, (a)
HFTs: High-Frequency Traders,
trade in and out of an actively
traded security or contract
more than a thousand times
but usually only in small sizes
i.e. as quickly as a few
milliseconds and (b) LLTs: LowLatency Traders include news
traders who trade on
electronic news feeds and
certain parasite traders* that
include front runners, who
trade in front of traders who
demand liquidity and quote
matchers, who trade in front
of traders who supply liquidity.
In contrast to HFTs, LLTs may
hold their positions for as long
as a day and sometimes
longer.
Buy-Side Traders
Buy-Side Traders trade to fill
orders for investment and risk
managers who use the
markets to establish positions
from which they derive various
utilitarian and profit motivated
benefits. New electronic buyside order management
systems (OMS) also decreased
buy-side trading costs by
allowing a smaller no. of buyside traders to process more
orders and to process them
more efficiently than manual
traders, E.g. OMS. Buy-side
OMSs generally allow the buyside trader to route orders to
brokers for further handling,
along with instructions for how
the orders should be handled.
Buy-side traders often use
electronic brokers and their
systems for advanced orders
(generally are limit orders with
limit prices that changes as
market conditions change),
trading tactics (executing a
simple function that involves
submission of multiple orders)
and algorithms (may use
combinations or sequences of
simple orders, advanced
orders or multiple orders to
achieve their objectives)
provided by their electronic
brokers to search for liquidity.
Electronic Brokers
Electronic Brokers serve both
types of traders. Electronic
trading strategies are most
profitable or effective when
they can act on new
information quickly. In
addition to supporting
standard order instructions,
such as limit or market orders,
these brokers often provide a
full suite of advanced orders,
trading tactics and algorithms.
*Parasitic Traders are speculators who base their predictions about future prices on information they obtain about orders
that other traders intend or will soon intend to fill.
46
The most important electronic traders are Dealers, Arbitrageurs and Buy-Side Institutional Traders. Both
Dealers and Arbitrageurs offer liquidity i.e. liquidity benefits passed to Buy-Side in form of narrower spreads
quoted for larger sizes. Evidence suggests that any profits obtained by parasitic traders from front running
orders are smaller than the costs savings obtained by buy-side traders from trading in electronic markets
using algorithms. Accordingly, proprietary traders and electronic brokers build automated trading systems
that are extremely fast.
High levels of Fragmentation and Electronification now characterize most global trading markets. Market
fragmentation, trading the same instrument in multiple venues increases the potential for price and liquidity
disparities across venues because buyers and sellers often are not in the same venues at the same time.
Alternative Trading Systems (ATS) a.k.a. Electronic Communication Networks (ECNs) or Multilateral Trading
Facilities (MTFs), are increasingly important trading venues. They function like exchanges but don't exercise
regulatory authority over their subscribers except concerning the conduct of their trading in their trading
systems. Electronic algorithmic trading techniques such as 'Liquidity Aggregation' and 'Smart Order Routing',
help traders manage the challenges and opportunities presented by fragmentation. Liquidity Aggregators
create 'Super Books' that present liquidity across markets for a given instrument. Small Order Routing
algorithms send orders to the markets that display the best quoted prices and sizes. Recent empirical
research suggests that public investors would greatly benefit if their brokers provided them with direct access
to these systems as they presently do in the equity markets (Bond market vs. Equity market). Instead, most
broker/dealers commonly interpose themselves.
1. Electronic News Traders: Subscribe to high-speed electronic news feeds that report news releases made by
corporations, governments and other aggregators of information. Then they quickly analyze these releases
to determine whether the information they contain will move the markets and if so, in which direction.
News traders profit when they can execute against stale orders i.e. orders that don't yet reflect the new
information. Some news traders also process news releases that don't contain quantitative data. Using
natural language processing techniques, they try to identify the importance of the information for market
valuations. For example, a report stating that "Our main pesticide plant shut down because of the
accidental release of poisonous chemicals" might be marked as having strong negative implications for
values. Electronic news traders would sell on this information. If they are correct the market will drop as
other, slower traders read, interpret and act on the information. If they are wrong, the market will not react
to the information. In that case, news traders will reverse their position and lose the transaction costs
associated with their round-trip trades. Note that these transaction costs could be high if many news
traders made the same wrong influence. Because round-trip transaction costs usually are lower than the
profits that electronic news traders can occasionally make when significant news arrives, news traders
often may trade with the expectation of being right only occasionally.
2. Electronic Dealers: Dealers provide liquidity to other traders when they allow traders to buy and sell when
those traders want to trade. Dealers help markets function well by being continuously available to take the
other side of a trade when other traders want to trade. Dealers thus make markets more continuous.
Practitioners say that dealers make market when they offer to trade. On the first indication that prices may
move against their inventory positions (i.e. price decreases if they are long or own the asset; price
increases if they are short or sold an asset they don't own), they immediately take liquidity by executing on
the opposite side to reduce their exposure. They generally will not hold large inventory positions in actively
traded stocks. As soon as they reach their inventory limit on one side of the market or the other, they cease
bidding or offering on that side. Electronic dealers like all other dealers, also keep track of scheduled news
releases. They cancel their orders just before releases to avoid offering liquidity to traders who can act
faster than they can. They also may try to reduce their inventories before a scheduled release to avoid
holding a risky position. Dealers profit by selling at ask prices that are higher than the bid prices at which
they buy.
47
Buying Interest > Selling Interest : Dealers raise their ask prices to
discourage buyers and raise their
bid prices to encourage sellers.
Buying Interest < Selling Interest : Dealers lower their ask prices to
encourage buyers and lower their
bid prices to discourage sellers.
3. Electronic Arbitrageurs: Look across markets for arbitrage opportunities in which they can buy an
undervalued instrument and sell a similar overvalued one. Electronic arbitrageurs try to construct their
arbitrage portfolios at minimum cost and risk.
4. Electronic Front-Runners: Are low-latency traders who use artificial intelligence methods to identify when
large traders or many small traders are trying to fill orders on the same side of the market. They will
purchase when they believe that an imbalance of buy orders over sell orders will push the market up and
sell when they believe the opposite. Their order anticipation strategies try to identify predictable patterns
in order submission. They may search for patterns in order submissions, trades or the relation between
trades and other events. For example, suppose that a trader sees that trades of a given size have been
occurring at the offer every 10 minutes for an hour. If the trader has seen this pattern of trading before,
the trader may suspect that the activity will continue. If so, the trader may buy on the assumption that a
trader is in the market filling a large buy order by breaking it into smaller pieces. Electronic front-runners
look for these patterns, often using very advanced, automated data-mining tools. When the time between
the stimulus and the response is short, electronic traders have a clear advantage.
5. Electronic Quote Matchers: Try to exploit the option values of standing orders. Standing orders are limit
orders waiting to be filled. Options to trade are valuable to quote matchers because they allow them to
take positions with potentially limit losses. For example, a fast quote matcher may buy when a slow trader
is bidding at 20. If the price subsequently rises, the quote matcher will profit. If the quote matcher believes
that the price will fall, the quote matcher will sell the position to the buyer at 20 and thereby limit his
losses. The main risk of the quote-matching strategy is that the standing order may be unavailable when
the quote matcher needs it. Standing orders disappear when filled by another trader or when cancelled.
Electronic Trading Strategies
1. Hidden Orders: Hidden Orders are orders that are exposed or shown only to the brokers or exchanges who
receive them. Traders especially large traders submit them when they don't want to reveal the existence of
the trading options that their standing orders provide to the markets. Some electronic traders try to
discover hidden orders by pinging the market i.e. they submit a small IOC (Immediate or Cancel Order) limit
order for only a few shares at the price at which they are looking for hidden orders. If the pinging order
trades, they know that a hidden order is present at the price; however, they don't know the full size of the
order (which they can discover only trading with it). Traders then may use this information to adjust their
trading strategies. However, only the pinger will know on which side of the market the hidden liquidity lies.
2. Leapfrog: means to go ahead of each other in turn. When bid-ask spreads are wide, dealers often are
willing to trade at better price than they quote. They quote wide spreads because they hope to trade at
more favorable prices. When another trader quotes a better price, dealers often immediately quote an
even better price. This behavior frustrates buy-side traders, who then must quote a better price to
maintain order precedence. If the spread is sufficiently wide, a game of leapfrog may ensue as the dealer
jumps ahead again.
48
3. Flickering Quotes: Quotes that change in less than 1 second are called flickering quotes i.e. electronic
markets often have flickering quotes, which are exposed limit orders that electronic traders submit and
then cancel shortly thereafter, often within a second. Traders who wish to trade with a flickering quote can
place a hidden limit order at the price where the quote is flickering.
4. Electronic Arbitrage: [(a) and (c) must use fast trading systems]
(a) Take liquidity on both sides: The costliest and least risky arbitrage trading strategy.
(b) Offer liquidity on one side: This strategy produces lower-cost executions, but is is bit riskier.
(c) Offer liquidity on both sides: Riskiest strategy because arbitrageurs are exposed to substantial price risk
when one leg is filled and the other is not. Moreover, if prices are moving because well-informed traders
are on the same side in both markets, as they might be if the well-informed traders possess information
about common risk factors - the leg providing liquidity to the informed traders will fill quickly, whereas
the other leg probably will not fil..
5. Machine Learning: i.e. Data Mining, methods produce models based on observed empirical regularities
rather than on theoretical principles identified by analysts. However, these systems are often useless when
trading becomes extraordinary e.g. when volatilities shoot up.
Abusive Trading Practices
1. Front Running: Front Running involves buying in front of anticipated purchases and selling in front of
anticipated sales. In most jurisdictions, front running is illegal if the front runners acquire their information
about orders improperly, for example, by a tip from a broker handling a large order. This front running
strategy is legal if the information on which it is based is properly obtained, for example by matching a
market data feed. Front running increases transaction costs for the traders whose orders are front-run
because the front runners take liquidity that the front-run traders otherwise would have taken for
themselves.
2. Trading for Market Impact: involves trading to raise or lower prices deliberately to influence other traders'
perceptions of value.
3. Rumormongering: is the dissemination of false information about fundamental values or about other
traders' trading intentions to alter investors' value assessments.
4. Wash Trading: The purpose of wash trading is to fool investors into believing that a market is more liquid
than it truly is and to thereby increase investor's confidence both in their ability to exit positions without
substantial cost and in their assessments of security values. Manipulators also can achieve these purposes
by falsely reporting trades that news occurred, which is essentially what happens when they arrange trades
among commonly controlled accounts.
5. Spoofing: a.k.a. layering, is a trading practice in which traders place exposed standing limit orders to
convey an impression to other traders that the market is more liquid than it is or to suggest to other
traders that the security is under or overvalued. Example, spoofers place both buy and sell order to make it
look like the market is liquid enough to trade. Spoofing is risky because the spoofing orders that spoofers
submit might execute before their intended orders execute. To effectively manage these processes,
spoofers use electronic systems to monitor trading and to ensure that they can quickly cancel their orders
as soon as they no longer want to them to stand.
6. Bluffing: involves submitting orders and arranging trades to influence other traders' perceptions of value.
Bluffers often prey on momentum traders, who buy when prices are rising and sell when prices are falling.
49
Note also that bluffers may time their purchases to immediately follow the release of valid positive
information about the security and thereby fool traders into overvaluing the material significance of the
new information. Consider typical 'Pump and Dump' schemes in which bluffers buy stock to raise its price
and thereby encourage momentum traders to buy. In a pump and dump manipulation, the bluffer tries to
raise prices. Similar manipulations can occur on the short side, though they are common. In such
manipulations, manipulators like short positions and then try to repurchase shares at lower prices. These
manipulations are often called 'Short and Distorts'.
7. Gunning: Gunning the market is a strategy used by market manipulators to force traders to do
disadvantageous trades. A manipulator generally guns the market by selling quickly to push prices down
with the hope of triggering stop-loss sell orders.
8. Squeezing and Cornering: In a squeeze or corner, the manipulator obtains control over resources
necessary to settle trading contracts. The manipulator then unexpectedly withdraws those resources from
the market, which causes traders to default on their contracts, some of which the manipulators may hold.
The manipulator profits by providing the resources at high prices or by closing the contracts at
exceptionally high prices. A short squeeze is a situation in which a stock's price increase triggers a rush of
buying activity among short sellers. Short squeezes result when short sellers of a stock move to cover their
positions, purchasing large volumes of stock relative to the market volume. Since covering their positions
involves buying shares, the short squeeze causes a further rise in the stock's price. This scramble to buy
only adds to the upward pressure on the stock's price. To corner a market means to acquire enough shares
of a particular security type, such as those of a firm in a niche industry or to hold a significant commodity
position to be able to manipulate its price. The term implies that the market has been backed into a corner
and there is nowhere for the market to move to find other sellers and buyers.
Fig 2: Systematic Risks
Runaway Algorithms: produce streams of unintended orders that result from programming
mistakes. The problems sometime occur when programmers don't anticipate some contingency,
Fat Finger Errors: occur when a manual trader submits a large order than intended. They are
called fat finger errors because they sometimes occur when a trader hits the wrong key or hits a
key more often than intended.
Overlarge Orders: demand more liquidity then the market can provide. In these events, a trader
often inexperienced, will try to execute a marketable order that is too large for the market to
handle without severely disrupting prices in the time given to fill the order. The increase in trading
volumes causes the algorithm to increase the rate of its order submissions, which exacerbates
the problems.
Malevolent Order Streams: are created deliberately to disrupt the markets. The perpetrators
may be market manipulators i.e. aggrieved employees, such as traders or software engineers or
terrorists. Traders conducting denial-of-service attacks designed to overwhelm their competitors'
electronic trading systems with excessive quotes also may create malevolent order streams.
NOTES
1. Many exchanges allow electronic traders to place their servers in the rooms where the exchange servers
operate, a practice called 'Collocation'. Exchanges charge substantial fees for collocation space and related
services, such as air conditioning and power.
50
2. Some electronic trading problems change so frequently that speed of coding is more important than speed
of execution. For such problems, traders use high-level languages e.g. Python, because they can code faster
and more accurately in these languages than in lower-level languages such as C++.
3. Some electronic traders also reduce latency by creating contingency tables that contain prearranged action
plans. For example, traders presumably would also have precomputed response for a decrease in the bid,
among many other contingencies.
4. Points for traders to consider:
- Traders must test software thoroughly before using it in live trading.
- Must ensure that only those orders coming from approved sources enter electronic order-matching
systems.
- Must ensure that only authorized software developers can change software. Best practice mandates that
these controls also include the requirement that all software be real, understood and vouched for by
atleast one developer besides its author.
- Orders received must conform to preset parameters that characterize its expected volume, size and other
characteristics. When the order flow is different than expected, automatic controls must shut it off
immediately.
- Brokers must monitor all client orders. Brokers must not allow their clients to enter orders directly into
exchange trading systems - a process called sponsored naked access, because it would allow clients to
avoid broker oversight.
- Some exchanges have adopted price limits and trade halts to stop trading when prices move too quickly.
1
Derivatives
PAGE NOS. 33
VOL. 8
CHAPTERS. 2
F 0 = S 0 (1 + R f ) T
CHAPTER 37
Pricing and Valuation of Forward Commitments
S0 =
F0
(1 + R f ) T
No Arbitrage
A Forward Commitment is a derivative instrument in the form of a contract that provides the ability to lock in
a price or rate at which one can buy or sell the underlying instrument at some future date or exchange an
agreed upon amount of money at a series of dates. Consider a contract under which Party A agrees to buy a
$1000 face value 90-day Treasury Bill from Party B 30 days from now at a price of $990. Party A is the long
and Part B is the short. If 30 days from now T-bills are trading at $992, the short must delivery the T-bill to the
long in exchange for a $990 payment. If T-bills are trading at $998 on the future date, the long must purchase
the T-bill from the short for $990, the contract price.
From Long's
Perspective
0
t
T
0
St
F0
Method 1:
∴
Method 2:
Vt = F t - F0
(1 + R f )T-t
V T = ST - F0
T-t
Ft = S t (1 + R f )
Vt = S t -
F0
(1 + R f )T-t
= S t - Pv (F0 )
If any Benefits 'γ' i.e. dividends, foreign interest and bond coupon payments; and Costs 'θ' i.e. waste, storage
and insurance. Financial instruments have '0' carry costs. Then:
F0 = Fv (S 0 + θ - γ)
Compounding
F0 = S0 . e
(rc + θ - γ) (T)
Carry Arbitrage Model: a no-arbitrage approach in which the underlying instrument is either bought or sold
along with a forward position, hence the term 'Carry'. Carry arbitrage models are also known as 'Cost of Carry
Arbitrage' models or 'Cash and Carry Arbitrage' models. Based on no-arbitrage approach, a portfolio offering
'0' cashflow in the future is expected to be valued at '0' at time 0. F 0 = Fv (S 0 )
Breakeven
F0 = S T
Continuous Compounding
F 0 = S 0 . e rc (T)
Annual Coupon
F 0 = S 0 (1 + r) T
2
I : Cash and Carry Arbitrage when forward contract is Overpriced
Suppose the forward contract is actually trading at $510. A short position in the forward contract requires
the delivery of this bond 3 months from now. The arbitrage that we examine in this case amounts to
borrowing $500 at the risk free rate of 6%.
Buy and Invest:
Loan Repayment:
Arbitrage Profit (Long):
$ 500
$ 500 (1.06) 3/12 = $ 507.34
$ 510 - $ 507.34 = $ 2.66
II : Reverse Cash and Carry Arbitrage when the forward contract is Underpriced
Suppose the forward contract is actually trading at $502 instead. We invest the $500 proceeds from the
short sale at the 6% annual rate for 3 months.
Short Sell and Invest:
Investment Proceeds:
Arbitrage Loss (Long):
$ 500
$ 500 (1.06) 3/12 = $ 507.34
$ 502 - $ 507.34 = - $ 5.34
Therefore we can see that the no-arbitrage forward price that yields a 'zero' value for both the long and short
positions in the forward contract at inception is the no-arbitrage price of $507.34.
Forward Overpriced: Borrow money,
Buy (go long) the spot asset,
Go short the asset in the forward market
Forward Underpriced: Borrow Asset,
Sell (go short) spot asset,
Lend money,
Long (buy) forward
Underlying is purchased and
forward contract is sold
Underlying is sold short and
forward contract is purchased
Like forward contracts, futures contracts have no value at contract initiation. Unlike forward contracts, future
contracts don't accumulate value changes over the term of the contract. Since futures accounts are marked
to market daily, the value after the margin deposit has been adjusted for the day's gains and losses in
contract value is always zero. The value of a futures contract strays from zero only during the trading periods
between the times at which the account is marked to market. If the future price increases, the value of the
long position increases. The value is set back to zero by the mark-to-market at the end of the mark-to-market
period.
Value of Futures Contract = Current Futures Price - Previous Mark to Market Price
lllll
Equity Pricing and Valuation
A stock, a stock portfolio or an equity index may have expected dividend payments over the life of
the contract.
* With Discrete Dividends
In order to price such a contract, we must either adjust the spot price for the present value of the
expected dividends (PvD) over the life of the contract or adjust the forward price for the future
value of the dividends (FvD) over the life of the contract.
or
F 0 = (S 0 - PvD) (1 + R f ) T
T
F 0 = S 0 (1 + R f ) - FvD
Pricing
3
Vt =
]
or
]
Vt = (S t - PvD t ) -
From Long's
Perspective
F0
(1 + R f )T-t
Valuation
]
Ft - F 0
(1 + R f )T-t
]
Example 1:
Calculate the no-arbitrage forward price for a 100-day forward on a stock that is currently priced at $30 and is
expected to pay a dividend of $0.4 in 15 days, $0.4 in 85 days and $0.5 in 175 days. The annual risk free rate
is 5% and the yield curve is flat.
0
15 Days
S0 : 30
D1 : 0.4
85 Days 100 Days
D2 : 0.4 Contract Maturity
Usually two dividends will be given, just be concerned about the dividend during the life, not the one after the
life of the contract.
FvD 1 = 0.4 (1.05) 85/365 = 0.4169
FvD 2 = 0.4 (1.05) 15/365 = 0.4029
F 0 = S 0 (1 + R f )T - FvD
= 30 (1.05)100/365 - (0.4169 + 0.4029)
= $ 29.5838
After 60 days, the value of the stock in the previous example is $36. Calculate the value of the equity forward
contract on the stock to the long position, assuming the risk free rate is still 5% and the yield curve is flat.
There's only one dividend remaining (25 days) before the contract matures in 40 days, as shown below:
0
60 Days
85 Days 100 Days
S60 : 36
D2 : 0.4
Contract Maturity
PvD2 = 0.4
= 0.3987
25/365
(1.05)
]
= 36 - 0.3987 -
F0
(1 + R f )T-t
]
V60= (S t - PvDt ) -
]
29.5838
(1.05) 40/365
]
= $ 6.16
Example 2:
An analyst who mistakenly ignores the dividends when valuing a short position in a forward contract on a
stock that pays dividends will most likely:
A. Overvalue the position by the PvD
If dividends are ignored, the long position will be
B. Undervalue the position by the PvD
overvalued by the PvD and the short position will
be undervalued by the same amount.
C. Overvalue the position by the PvD
✓
4
* With Continuous Dividends
The relationship between the periodically compounded risk free rate R f and the continuously
c
c
compounded rate R f is R f = In (1+ R f ). For example, 5% compounded annually is equal to
In (1.05) = 0.04879 or 4.879% compounded continuously.
(rc - DYc ) (T)
F0 = S 0 . e
Pricing
Dividend Yield
V t = F t - F 0 ↶ Compounded
e (rc - DYc ) (T-t)
From Long's
Perspective
Valuation
∴ F t = S t . e (r - DY ) (T-t)
c
lllll
c
Fixed Income Pricing and Valuation
* Fixed Income Forwards
In order to calculate the no-arbitrage forward price on a coupon paying bond, we can use the same
formula as we need for a dividend paying stock or portfolio, simply substituting the present value of
the expected coupon payments (PvC) over the life of the contract for PvD or the future value of the
coupon payments (FvC) for FvD, to get the following formulas:
or
F 0 = (S 0 - PvC) (1 + R f ) T
Pricing
T
F 0 = S 0 (1 + R f ) - FvC
Vt =
]
or
]
V t = (S t - PvC t ) -
From Long's
Perspective
Ft - F0
(1 + R f ) T-t
F0
(1 + R f ) T-t
]
Valuation
]
Example 3:
Louise Michelle entered into a 250-day forward contract on a 7% US Treasury bond at a forward price of
$1057.37. 100 days later, the forward price has changed to $1081.04. The risk free rate is 6%. Calculate the
value to Michelle of the forward contract.
Vt =
]
F t - F0
T-t
(1 + R f )
=
]
1081.04 - 1057.37
(1.06)150/365
]
]
= $ 23.11
Example 4:
1 month ago, Trambadour purchased € / ¥ forward contracts with 3 months to expiration, at a quoted price
of 100.2. The contract's notional principle amount is ¥100,000. The current forward price is 100.05 and the
current annualized R f is 0.3%.
]
Vt =
Ft - F0
(1 + R f ) (T-t)
]
5
]
=
100.05 - 100.2
(1.0003) 2/12
]
= - ¥ 0.149925
* Fixed Income Futures
Bond futures contract often allow the short an option to deliver any of several bonds, which will
satisfy the delivery terms of the contract. This is called a delivery option and is valuable to the short.
When multiple bonds can be delivered for a particular maturity of a futures contract, a
cheapest-to-deliver bond typically emerges after adjusting for the conversion factor. Quoted Price is
called 'Clean Price' and Full Price is called 'Dirty Price' which includes Accrued Interest (AI 0 ).
Days since last coupon payment
Last Coupon
Purchase Bond
Next Coupon
Days between coupon payment
(
)
Accrued Interest = Days since last coupon payment
(AI 0 )
Days between coupon payment
x Coupon Amount
(Semi Annual)
Full Price = Clean Price + AI 0
T
↶ F 0 = Full Price (1 + R f ) - AI T - FvC
Futures
QFP = F 0 = Full Price (1 + R f ) T- AI T - FvC
Adjusted Price
CF
CF
↶
due to CF
Whereas,
CF : Since bonds trade at different prices based on maturity and stated coupon, an
adjustment known as the 'Conversion Factor' is used in an effort to make all
deliverable bonds roughly equal in price. The CF adjustment, however, is not
precise. Thus, the seller will deliver the bond that is least expensive. For example,
Underlying Asset
Yield: 6%
QFP
Maturity: 30 years
(Hypothetical Bond)
Yield 4%
6%
8%
CF
Less than 1
1
More than 1
T
For Arbitrage, Full Price (1 + R f ) - FvC = QFP x CF + AI T
Investment
Receipts
Example 5:
Suppose you needed to calculate the quoted futures price of a 1.2 year Treasury bond futures contract. The
cheapest-to-deliver bond is 7% T-bond will exactly 10 years remaining to maturity and a quoted price of
$1040 with a conversion factor of 1.13. There is currently no accrued interest because the bond has just paid
6
a coupon. The annual risk free rate is 5%. The accrued interest on the bond at maturity of the futures
contract will be $14.
0.7
0
FvC1 = 35 (1.05) = 36.216
0.2
FvC2 = 35 (1.05) = 35.343
0.5
1
1.2 years
C1 : 35
C2 : 35
QFP = Full Price (1 + R f )T- AI T - FvC
CF
= 1040 (1.05)1.2 - 14 - 36.216 - 35.343
1.13
= $ 900.13
Example 6:
The following information is available for a futures contract:
Market Quoted Price = $ 112
Time to Maturity = 60 Days
CF = 0.92
AI T = 0
Underlying Bond (CTD)
Quoted Price = $ 102.16
AI 0 = 0.22
AI T = 0.49
Calculate any arbitrage profits if the R f is 0.05%.
T
F 0 = Full Price (1 + R f ) - AI T - FvC
CF
F 0 = (102.16 + 0.22) (1.0005) 60/365 - 0 - 0.49
0.92
= $ 110.84
Since futures price is different from quoted market price ($112), there is a probability of arbitrage.
(a) Investment = Full Price (1 + R f )T- FvC = (102.16 + 0.22) (1.0005) 60/365 - 0 = $ 102.46
(b) Receipts = QFP x CF + AI T = 112 x 0.92 + 0.49 = $ 103.53
Arbitrage Profit at Maturity = | 103.53 - 102.46 | = $ 1.07
Arbitrage Profit Today =
lllll
1.07
= $ 1.06
60/365
(1.0005)
FRA Pricing and Valuation
A Forward Rate Agreement (FRA) is an over-the-counter (OTC) forward contract in which the
underlying is an interest rate on a deposit. LIBOR stands for London Interbank Offered Rate, is a
widely used interest rate that serves as the underlying for many derivative instruments. It
represents the rate at which London banks can borrow from other London banks. Average LIBOR
rates are derived and posted each day at 11:30 am London time. FRAs are typically settled based on
'Advanced Set and Advanced Settled' whereas swaps and interest rate options are normally based
on 'Advanced Set and Settled in Arrears'. The term advanced set is used because the reference
7
interest rate is set at the time the money is deposited. LIBOR is quoted as an annualized rate based
on a 360-day year. LIBOR is an add-on-rate, like a yield quote on a short term certificate of deposit.
LIBOR is used as a reference rate for floating rate US dollar denominated loans worldwide. An FRA
involves two counterparties: Fixed Receiver (Short) and Floating Receiver (Long).
LIBOR
(Borrower: Long receives payment)
LIBOR
(Lender: Short receives payment)
(At Maturity) FRA
If the floating rate at contract expiration (LIBOR for US dollar deposits and Euribor for Euro
deposits) is above the rate specified in the forward agreement, the long position in the contract can
be viewed as the right to borrow at below market rates and the long will receive a payment. If the
floating rate at the contract expiration is below the rate specified in the forward agreement, the
short position in the contract can be viewed as the right to lend at above market rates and the short
will receive a payment.
Fixed 4%
Borrower Gains
LIBOR > Fixed
Receives $ 400
+
Pays $ 600
LIBOR 6%
Fixed 4%
+
Lender Gains
LIBOR < Fixed
Receives $ 400
Pays $ 200
LIBOR 2%
For example, the notation for FRAs is unique. A 2 x 3 FRA is a contract that expires in two months
(60 days) and the underlying loan is settled in 3 months (90 days). The underlying rate is 1 month
(30 days) LIBOR on a 30-day loan in 60 days. Because there is no initial exchange of cashflows to
eliminate arbitrage opportunities, the FRA price is the fixed interest rate such that the FRA value is
'zero' on the initiation date.
1 month
2 months
3 months
LOAN PERIOD
2 months
FRA
Initiation
1 month
FRA Expiration /
Loan Initiation
Loan
Maturity
3 months
Example 7:
Compute the amount that must be repaid on a $1000,000 loan for 30 days if 30-day LIBOR is quoted at 6%.
8
LIBOR = 0.06 x 30/360 = 0.005 or 5%
Add-on-Interest = 1000,000 x 0.06 x 30/360 = $ 5000
The borrower would repay $ 1000,000 + $ 5000 = $ 1005,000 at the end of 30 days.
Example 8:
Calculate the price of a 1 x 4 FRA (i.e. a 90-day loan, 30 days from now). The current 30-day LIBOR is 4% and
120-day LIBOR is 5%.
5%
30
60
90
4%
120
x
Pricing requires you to be indifferent between 30 days or 30 days and roll over 90 days.
(1 + L 120 . 120/360)
(1 + 0.05 . 120/360)
1.01666
1.0033
1.0132 - 1
x
= (1 + L 30 . 30/360) (1 + L 90 . 90/360)
= (1 + 0.04 . 30/360) (1 + x . 90/360)
= 1 + 0.25 x
= 0.25 x
= 0.053 or 5.3%
5.3% is the arbitrage forward rate, the forward rate that will make the values of the long and the short
position in the FRA both 0 at the initiation of the contract.
Example 9:
It is 1 x 4 FRA, assume a notional principle of $1000,000. Suppose that at contract expiration, the 90-day rate
has increased at 6% which is above the contract rate of 5.32%. Calculate the value of the FRA at maturity,
which is equal to cash payment at settlement.
FRA Expiration
30
FRA Value
$ 1674.88
60
Before: 5.32%
Now: 6%
90
120
Interest Savings
$ 1700
(?)
(?)
The interest savings at the end of the loan term (compared to the market rate of 6%) will be:
[ (0.06 x 90/360) - (0.0532 x 90/360) ] x $ 1000,000 = $ 1700
The present value of this amount at the FRA settlement date (90 days prior to the end of the loan term)
discounted at the current rate of 6% is:
1700
= $ 1674.88
(1 + 0.06 x 90/360)
9
Example 10:
Value a 5.32% 1 x 4 FRA with a principle amount of $1000,000 10 days after initiation of 110-day LIBOR is 5.9%
and 20-day LIBOR is 5.7%.
FRA before
Settlement
$ 1487.39
(?)
5.9%
10
30
Interest Savings
$ 1514.2
60
5.7%
90
120
(?)
Before: 5.32%
Now: 5.925% ( x )
(1 + L 110 . 110/360) = (1 + L 20 . 20/360) (1 + L 90 . 90/360)
(1 + 0.05 . 110/360) = (1 + 0.04 . 20/360) (1 + x . 90/360)
1.01802 = 1 + 0.25 x
1.00316
1.014813 - 1 = 0.25 x
x = 0.05925 or 5.925%
Interest Savings = [ (0.05925 x 90/360) - (0.0532 x 90/360) ] x $ 1000,000 = $ 1514.2
Discounting =
1514.2
= $ 1487.39
(1 + 0.059 x 110/360)
Example 11:
Current 6-month LIBOR
6-month Forward Rate in 6 months
1%
1.15%
Using a 2-step binomial model to value the current at-the-money interest rate call option, the value of the
underlying instrument at Node 0, would most likely be:
A. 1.25%
The value of the underlying instrument at Node 0 is
B. 1.15%
the spot rate. The spot rate and the at-the-money
C. 1%
strike price is the current LIBOR rate of 1%.
✓
Currency Contracts Pricing and Valuation
The calculation of the currency forward rate is just an application of 'Covered Interest Parity'.
Interest Pate Parity is based on a no-arbitrage condition: a higher interest rate currency will trade at
a forward discount to offset the extra interest income.
↷Price Currency Interest Rate
F 0 = S 0 x (1 + R P ) T
(1 + R B ) T
Pricing
↶
F t or
F & S are quoted in
price currency per
unit of base currency
F t or
From Long's
Perspective
Base Currency Interest Rate
)T
F 0 = (1 + R P
S 0 (1 + R B ) T
c
; F 0 > S 0 then R P > R B = - ve
F 0 < S 0 then R P < R B = +ve
c
F 0 = S 0 . e (RP - R B) (T)
↶
lllll
Difference is going to be in Price Currency
Vt = (F t - F 0 ) x Contract Size
(1 + R P ) T-t
Valuation
10
Example 12:
David Hastings entered into a 4-month forward contract to buy €10,000,000 at a price of $1.112 per Euro.
1-month later, the 3-month forward price is $1.109 per Euro. The USD interest rate is 0.3% and the Euro
interest rate is 0.4%. Calculate the value of Hastings forward position.
Vt = (F t - F 0 ) x Contract Size
(1 + R P ) T-t
= (1.109 - 1.112) x 10,000,000 = - $ 29,978
(1.003) 90/365
The value of the forward position is negative to Hastings as the forward price of Euros has dropped since the
contract initiation.
Example 13:
Trambadour holds a short position in a ¥ / $ forward contract with a notional value of $1000,000. At contract
initiation, the forward rate was 112.10 ¥ / $. The forward contract expires in 3 months. The current spot
exchange rate is 112 ¥ / $ and the annually compounded risk free rates are - 0.2% for the Yen and 0.3% for
the US dollar. The current quoted price of the forward contract is equal to the no-arbitrage price.
F t = S t x (1 + R P ) T = 112 x (1 - 0.002) 0.25 = ¥ 111.8602
(1 + R B ) T
(1 + 0.003) 0.25
V t = (F t - F 0 ) x Contract Size = (111.8602 - 112) x 1000,000 = - ¥ 239,963 (Long)
(1 + R P )T-t
(1 - 0.002) 0.25
¥ 239,963 (Short)
Interest Rate Swaps Pricing and Valuation
Interest Rate Swaps are OTC, agreed upon by the counterparties. Swaps can be designed with an
infinite no. of variations. Under Interest Rate Swaps, one party agrees to pay floating (borrow at the
floating rate) and receive fixed (lend at a fixed rate). At initiation of the swap the fixed rate is
selected so that the present value of the floating rate payments is equal to the present value of the
fixed rate payments, which means the swap value is 0 to both parties.
Initially,
Pv Fixed Payments = Pv Floating Payments
Floating
↷ Rate
Floating = CF F = AP x F r =
No. of Accrued Days
x Fr
(F)
Total no. of Days during the year applicable to CF i
Fixed = CF FS = AP x FS r =
No. of Accrued Days
x FS r
(FS)
Total no. of Days during the year applicable to CF i
Fixed
↶
lllll
Rate
Floating interest rate is assumed to be Advanced Set and Settled in Arrears.
F - FS = AP (F r - FS r )
As per long: Receive floating & Pay Fixed
FS - F = AP (FS r - F r )
As per short: Receive Fixed & Pay Floating
This fixed rate is called the Swap Rate or Swap Fixed Rate (SFR). The SFR is derived from the LIBOR
curve corresponding to the swap tenor.
11
Discount
Factor
Z =
1
1 + LIBOR (Days/360)
Pricing
SFR (Period) = 1 - Last Discount Factor
Sum of Discount Factor
SFR (Annual) = SFR (Period) x No. of Settlement Periods per year
From Long's
Perspective
Value to the Payer = Σ Z (SFR New - SFR Old ) x Days x Notional Principle
360
Valuation
We are only responsible for valuation of an interest rate swap on settlement dates (after settlement
has occurred) and not between settlement dates. After the initiation of an interest rate swap, the
swap will take on a positive or negative value as interest rate changes. The party that is the fixed
rate payer (long) benefits if rates increase because they are paying the older (and lower) fixed rate
and receiving the newer (and higher) floating rate. Similarly, the fixed rate receiver (short) benefits if
rates decrease because they are receiving the older (and higher) fixed rate while paying a newer
(and lower) floating rate. The value is calculated as present value of the difference in payments.
Example 14:
Annualized LIBOR spot rates today are:
R 90 : 0.03
R 180 : 0.035
R 270 : 0.04
R
: 0.045
360
You're analyzing a 1-year swap with quarterly payments and a notional principle amount of $5000,000.
Calculate fixed rate in percentage terms and quarterly fixed payments in $.
Z 90 =
Z
1
= 0.99256
1 + 0.03 (90/360)
=
1
= 0.98280
1 + 0.035 (180/360)
Z 270 =
1
= 0.97087
1 + 0.04 (270/360)
Z 360 =
1
= 0.95694
1 + 0.045 (360/360)
180
SFR (Period) = 1 - Last Discount Factor =
1 - 0.95694
= 0.011 or 1.1%
Sum of Discount Factor
0.99256 + 0.98280 + 0.97087 + 0.95694
SFR (Annual) = 0.011 x 4 = 0.044 or 4.4%
Quarterly Fixed Payments = $ 5000,000 x 0.011
= $ 55000
12
Example 15:
Let's continue our previous example of a 1-year quarterly settlement, $5000,000 notional swap with a SFR at
initiation of 4.4%. Suppose that after 180 days, the LIBOR term structure is flat at 3.5% over the next year.
Calculate the value of the swap to the fixed rate receiver.
Z 90 =
Z 180 =
1
= 0.9913
1 + 0.035 (90/360)
Σ Z = 0.9913 + 0.9828 = 1.9741
1
= 0.9828
1 + 0.035 (180/360)
Value of the Swap to the Payer = 1.9741 (0.035 - 0.044) x 90 x 5000,000
360
= - $ 22209
Value to the Receiver = $ 22209
Example 16:
Two parties enter into a 2-year fixed-for-floating interest rate swap with semi-annual payments. After 180
days the swap is marked-to-market when the 180, 360 and 540 day annualized LIBOR rates are 4.5%, 5% and
6% respectively. The present value factors respectively are 0.9780, 0.9524 and 0.9174. What is the market
value of the swap per $1 notional principle and which of the two counterparties (the fixed rate payer or the
fixed rate receiver) would make the payment to mark the swap to market? Old SFR being 6.62%.
SFR New =
1 - 0.9174
= 0.0290
0.9780 + 0.9524 + 0.9174
SFR New (Annual) = 0.0290 x 2 = 0.0580
Payer makes the payment = 2.8478 (0.058 - 0.0662) x 180 = - 0.01166
360
lllll
Currency Swaps Pricing and Valuation
The interest rates in a Currency Swap are simply the swap rates calculated from each country's yield
curve in the relevant country's currency. Don't forget, with Currency Swaps, there are two yield
curves and two swap fixed rates, one for each currency. Principle amounts are exchanged at
initiation based on the exchange rate at initiation. Periodic payments are based on each currency's
fixed rate. The value to any party in a Currency Swap is the present value of the CFs they expect to
receive minus the present value of the CFs they are obliged to pay.
Example 17:
90 day $
180 day $
270 day $
360 day $
Rate
0.03
0.035
0.04
0.045
Z
0.99256
0.98280
0.97087
0.95694
SFR
= 1.1%
Quarterly
90 day £
180 day £
270 day £
360 day £
Rate
0.04
0.05
0.06
0.07
Z
0.99010
0.97561
0.95694
0.93458
SFR
= 1.7%
Quarterly
13
The current exchange rate is £0.5 per $. Determine the fixed rate on a 1-year quarterly $5000,000 interest
rate swap. Then determine £ principle amount and the quarterly cashflow on a Pay $ fixed, receive £ fixed
currency swap.
$ Principle Amount = $ 5000,000
£ Principle Amount = $ 5000,000 x 0.5 = £ 2500,000
At the initiation of the swap, we would exchange £2500,000 for $5000,000. We would pay interest, pay 1.1% of
$5000,000 = $55,000 and receive 1.7% of £2500,000 = £425,000. At the end of 1 year we would exchange the
original principle amounts.
The main advantage of the approach is that all foreign exchange considerations are moved to the
initial exchange rate. We don't need to address future foreign currency translation. There are two
main sources of risk related to Currency Swap valuation (i) Interest Rates and (ii) Exchange Rates.
From Long's
Perspective
Va = NPa [ (FR a ) (Σ Z a ) + Last Discount Factora ] - S 0 a .
NP b [ (FR b ) (Σ Z b ) + Last Discount Factor b ]
Valuation
Whereas,
NPa : Notional Principle of Currency A
NPb : Notional Principle of Currency B
FR : Fixed Rate
Example 18:
Annual LIBOR rates for $ is 4.4% and £ is 6.8%. After 300 days, the 60-day $ interest rate is 5.4%, the 60-day £
interest rate is 6.6% and the exchange rate is £0.52 per $. Calculate the value of a $5000,000 swap to the
counterparty that receives $ fixed and pays £ fixed. Z $60 = 0.99108 and Z £60 = 0.98912.
Method 1:
$ Principle + Interest = 5000,000 + 55,000 = 5,055,000
£ Principle + Interest = 2500,000 + 42,500 = 2,542,500
Value of $ Fixed Side = 0.99108 x 5,055,000 = $ 5,009,909
value of £ Fixed Side = 0.98912 x 2,542,500 = £ 2,514,838
Convert '£' to '$' using 0.52 £ / $ = £ 2,514,838 = $ 4,836,227
0.52
Finally the value of the received $ fixed, pay £ fixed side of the swap is equal to the value of
the $ fixed side minus the £ fixed side = $ 5,009,909 - $ 4,836,227 = $ 173,682
Method 2:
V a = NPa [ (FR a ) (Σ Z a ) + Last Discount Factor a ] - S 0 a .
NP b [ (FR b ) (Σ Z b ) + Last Discount Factor b ]
V $ = $ 5000,000 [ (0.011) (0.99108) + 0.99108 ] - 1.92307692 .
£ 2500,000 [ (0.017) (0.98912) + 0.98912)
= $ 5,009,909.4 - $ 4,836,032.70
= $ 173,876.69
14
Example 19:
A bank entered into a 1-year currency swap with quarterly payments. 200 days ago, by agreeing to swap
$1000,000 for €800,000. The bank agreed to pay an annual fixed rate of 5% on the €800,000 and receive a
fixed rate of 4.2% on the $1000,000. Current LIBOR and Euribor rates are as follows:
70 day $
90 day $
160 day $
180 day $
Rate
0.04
0.044
0.048
0.052
Z
0.9923
0.9891
0.9791
0.9747
Rate
0.052
0.056
0.061
0.063
70 day €
90 day €
160 day €
180 day €
Z
0.99
0.9862
0.9736
0.9695
The current spot exchange rate is 0.75 € / $. The value of the swap to the bank today is?
270
x
$ 10,500
€ 10,000
70
160
①
360
↷
200
$ 1000,000 x 1.05%
$ 1000,000 + $ 10,500
€ 800,000 + € 10,000
↶
Method 1:
②
①
$: 0.9923 x 10,500 = $ 10,419.15
€: 0.99 x 10,000 = € 9,900, we convert '€' to '$', 9,900 / 0.75 = $ 13,200
②
$: 0.9791 x 1,010,500 = $ 989,380.55
€: 0.9695 x 10,000 = € 785,295, we convert '€' to '$'. 785,295 / 0.75 = $ 1,047,060
€ 800,000 x 1.25%
(10,419.15 + 989,380.55) - (13,200 + 1,047,060) = $ 60,460.45
V a = NP a [ (FR a ) (Σ Z a ) + Last Discount Factor a ] - S 0a .
NP b [ (FR b ) (Σ Z b ) + Last Discount Factor b ]
↷
Method 2:
1 $ = 0.75 €
or 1 € = 0.333 $
(0.9923 + 0.9791)
↶
V $ = $ 1000,000 [ (0.0105) (1.9714) + 0.9791 ] - 1.333 .
€ 800,000 [ (0.0125) (1.9636) + 0.9736 ]
(0.99 + 0.9736)
= $ 1,010,185.25 - $ 1,077,837.33
= $ 67,652.08
Example 20:
The current $ / C$ exchange rate is 0.7. In a $1000,000 currency swap, the party that is entering the swap to
hedge existing exposure to C$ denominated fixed rate liability will:
A. Pay Floating in C$
They will pay 1000,000 / 0.7 = C$ 1,428,571 (principle) at swap inception
B. Receive Floating in C$
(in exchange for $ 1000,000) and get the same amount (C$ 1,428,571)
C. Pay C$ 1,428,571 at the beginning of the swap back at termination (in exchange for paying back the $ 1000,000)
✓
lllll
Equity Swaps Pricing and Valuation
An Equity Swap is an OTC derivative contract in which two parties agree to exchange series of
cashflows, whereby one party pays a variable series that will be determined by an equity and the
other party pays either (i) a variable series determined by a different equity or rate or (ii) a fixed
series. An Equity Swap is used to convert the returns from an equity investment into another series
of return, which as noted either can be derived from another equity series or can be a fixed rate.
Equity Swaps are widely used in equity portfolio investment management to modify returns and
risks.
15
VFS =
]
From Long's
Perspective
FS r
x Σ Z + Last Discount Factor x NP
No. of Settlements per year
]
Valuation
Fig 1: Types of Equity Swaps
1. Receive Equity, Pay Fixed
= NP (Equity Return - Fixed Rate)
Fixed CFs = NP x AP x FS r
Equity CFs = NP x E r
2. Receive Equity, Pay Floating
= NP (Equity Return - Floating Rate)
Equity CFs = NP x E r
3. Receive Equity, Pay Equity *
= NP (Equity Return a - Equity Return b )
Equity CFs = NP x E r
* For an equity, for equity swap can be viewed simply as a receive-equity 'a', pay fixed swap combined with a pay-equity 'b',
receive fixed swap. The fixed payments cancel out and we have synthetically created an equity for equity swap.
Example 21:
A $10,000,000 principle, quarterly settlement equity swap has a fixed rate of 6.05%. The index at inception is
985. After 30 days have passed, the index stands at 996 and the term structure of LIBOR is 6%, 6.5%, 7% and
7.5% for terms of 60, 150, 240 and 330 days respectively. The respective discount factors are 0.9901, 0.97363,
0.95541 and 0.93567. Calculate the value of the swap to the fixed rate payer on day 30.
VFS =
]
Value of the
Fixed Side
]
FS r
x Σ Z + Last Discount Factor x NP
No. of Settlements per year
= [ 0.0605 (0.9901 + 0.97363 + 0.95541 + 0.93567) + 0.93567 ] x $ 10,000,000
4
= $ 9,940,000
Value of Index
after 30 days.
V I = $ 10,000,000 x 996 = $ 10,111,675
985
From the standpoint of the fixed rate payer, the value of the swap after 30 days is,
$10,111,675 - $ 9,940,000 = $ 171,675
↷
Example 22:
The current value of the equity swap would be 0 if the equity index was currently trading at?
Usually at par $ 100 or $ 1000
Vt = C 0 - S t x NP
S0
0 = 9,940,000 - S t x 100; S t = 9.7911
10,000,000 985
NOTES
1. LIBOR based contracts such as FRA, Swaps, Caps, Floor = 360 days
Equity, Equity Index, Bond, Currency, Stock Options = 365 days
2. "If you are long a futures or forward contract and the price of the underlying has risen, the value of a
futures contract is is most likely lower than that of the equivalent forward contract", because futures
contracts are marked-to-market daily, profits are paid out and the value is reset to 0. Therefore, the
forward contract will likely have a higher value than the futures contract.
16
CHAPTER 38
Valuation of Contingent Claims
A Contingent Claim is a derivative instrument that provides its owner a right but not an obligation to a payoff
determined by an underlying asset, rate or other derivative e.g. options, option valuation models, like their
counterparts are based on the principle of no-arbitrage meaning that the appropriate price of an option is
the one that makes it impossible for any party to earn an arbitrage profit at the expense of any other party.
The price that precludes arbitrage profits is the value of the option. The first approach is the Binomial Model
based on discrete time and the second is the Black Scholes Merton Model based on continuous time. The
no-arbitrage approach can be applied to either European Style or American Style options because it provides
the intuition for the fair value of options.
lllll
The Binomial Model
This model is extensively used to value path dependent options, which are options whose values
depend not only on the value of the underlying at expiration but also how it got there. To construct a
Binomial Model, we need to know the beginning asset value, the size of the two possible changes and
the probability of each of these changes occurring.
lllll
One-Period Binomial Model
Expectations Approach results in an identical value as the No-Arbitrage Approach. The Expectations
Approach to option valuation differs in two significant ways from the DCF approach to securities
valuation. Firstly, the expectation is not based on the investor's beliefs regarding the future course
of the underlying i.e. probability is not objectively determined and not based on the investor's
personal view. The Expectations Approach is a result of this arbitrage process, not an assumption
regarding risk preferences. Hence, they are called Risk-Neutral Probabilities, though, they are not
true probabilities if up and down moves. Secondly, the discount rate is not risk-adjusted. The
Expectations Approach here is often viewed as superior to the DCF approach because both the
subjective future expectation as well as subjective risk-adjusted discount rate have been replaced
with more objective measures.
S 0 : Current Stock Price
U : Up move Factor (1 + % Up)
D : Down move Factor (1 - % Down) = 1 / U
S + : Stock Price if an Up move occurs = S 0 x U
S - : Stock Price if a Down move occurs = S 0 x D
π U : Probability of an Up move = 1 + R f - D / U - D
π D : Probability of a Down move = 1 - π U
Example 1:
Calculate the value today of a 1-year option on a stock that has an exercise price or strike price of $30.
Assume that the periodically compounded as opposed to continuously compounded risk free rate is 7%, the
current value of the stock is $30, the up move factor is 1.33 and the down move factor is 0.75. Calculate both
call and put option.
π U = 1 + R f - D = 1 + 0.07 - 0.75 = 0.55 or 55%
U-D
1.33 - 0.75
π D = 1 - π U = 1 - 0.55 = 0.45 or 45%
17
Call Option Value
+
S = 30 x 1.33 = 40
C + = 40 - 30 = 10
0.
55
E (Call option value in 1 year)
= (10 x 0.55) + (0 x 0.45) = 5.5
S 0 = 30
0.
Value of the option today = C 0 = 5.5 = 5.14
1.07
45
S - = 30 x 0.75 = 22.5
C - = 22.5 - 30 = 0
Yr 0
Yr 1
Put Option Value
+
S = 30 x 1.33 = 40
P + = 30 - 40 = 0
0.
55
E (Put option value in 1 year)
= (0 x 0.55) + (7.5 x 0.45) = 3.375
S 0 = 30
45
0.
Value of the option today = P0 = 3.375 = 3.154
1.07
S - = 30 x 0.75 = 22.5
P - = 30 - 22.5 = 7.5
Yr 0
Yr 1
Fig 1: Strategies for Arbitrage
Call
Valuation
Put
Position
. Call is Overvalued
Valuation
. Put is Overvalued
(i) Borrow
(ii) Buy HR Stock
(iii) Sell the Call
. Call is Undervalued
. Put is Undervalued
(i) Invest
(ii) Sell HR Stock
(iii) Buy the Call
Position
(i) Invest
(ii) Sell the Stock
(iii) Sell the Put
(i) Borrow
(ii) Buy the Stock
(iii) Buy the Put
We determine the Hedge Ratio or Delta (HR) such that we are indifferent to the underlying going up or down.
Thus, we are hedged against moves in the underlying.
HR = C + - C - ≥ 0
S+ - S -
or
(+ve)
Value of Call Option: HR x S 0 - C 0 = HR x S + - C +
(1 + R f )
-
= HR x S - C
(1 + R f )
-
-
HR = P + - P ≤ 0
S+ - S
(-ve)
C 0 = HR x S 0 + Pv (- HR x S + + C + )
or
-
-
C 0 = HR x S 0 + Pv (- HR x S + C )
18
Value of Put Option: HR x S 0 - P0 = HR x S + - P +
(1 + R f )
-
= HR x S - P
(1 + R f )
-
C = HR x S 0 + Pv (- HR x S + + P + )
or
-
-
C = HR x S 0 + Pv (- HR x S + P )
Fig 2: Arbitrage Opportunities
+
S
+
HR x S - C
+
or
+
+
-
-
HR x S - P
Portfolio = HR x S 0 - C 0
Long
Stock
Short
Call
S0
(Should be equal)
= HR x S 0 - P 0
Long
Stock
Short
Put
S
-
-
HR x S - C
-
or
HR x S - P
Example 2:
In the previous example, a 1-year call option on a $30 stock (exercise price of strike price of $30) was valued
at $5.14. We were given that the risk free rate is 7% and the up move factor and down move factor are 1.33
and 0.75 respectively. If the market price of the call option is $6.5, illustrate how this opportunity can be
exploited to earn an arbitrage profit. Assume we trade 100 call options.
HR = C + - C - = 10 - 0 = 0.5714 shares per option
S+ - S - 40 - 22.5
To check if there's any arbitrage opportunity,
HR x S 0 - C 0 = HR x S + - C +
(1 + R f )
(0.5714) (30) - 6.5 = (0.5714) (40) - 10
1.07
10.64 ≠ 12.014
Arbitrage Profit = 12.014 - 10.64
= 1.37295 and
= 1.37295 x 100 = $ 137.29
Example 3:
Identify the trading strategy that will generate the payoffs of taking a long position in a call option within a
single period binomial framework?
A. Buy HR = C + + C - / S+ + S - units of underlying and financing of - Pv (-HR x S - + C - )
B. Buy HR = C +- C - / S + - S - units of underlying and financing of - Pv (-HR x S - + C - )
C. Short Sell HR = C + - C - / S+ - S - units of underlying and financing of Pv (-HR x S - + C - )
✓
Therefore, a call option can be replicated with the underlying and financing. Specifically, the call option is
equivalent to a leveraged position in the underlying.
19
Example 4:
Identify the trading strategy that will generate the payoffs of taking a long position in a put option within a
single period binomial framework?
A. Short Sell - HR = - (P + - P - ) / (S+ + S - ) units of underlying and financing of - Pv (-HR x S - + P - )
B. Buy - HR = (P+ - P - ) / (S +- S - ) units of underlying and financing of - Pv (-HR x S - + P - )
C. Short Sell HR = (P+ - P - ) / (S + - S - ) units of underlying and financing of Pv (-HR x S - + P - )
✓
Therefore, a put option can be replicated with the underlying and financing. Specifically, the put option is
equivalent to a short position in the underlying with financing in the form of lending.
lllll
Two-Period Binomial Model
The Two Period Binomial Option Valuation Model illustrates two important concepts i.e.
Self- Financing and Dynamic Replication. Self Financing implies that the replicating portfolio will not
require any additional funds from the arbitrageur during the life of this dynamically rebalanced
portfolio. If additional funds are needed, then they are financed externally. Dynamic Replication
means that the payoffs from the option can be exactly replicated through a planned trading
strategy.
Example 5:
Suppose you own a stock currently priced at $50 and that a two period European call and put option on the
stock is available with a strike price of $45. The up move factor is 1.25 and the down move factor is 0.8. The
risk free rate per period is 7%. Compute the value of the call and put option using a two period binomial
model.
π U = 1 + R - D = 1 + 0.07 - 0.8 = 0.6 or 60%
U-D
1.25 - 0.8
π D = 1 - π U = 1- 0.6 = 0.4 or 40%
Call Option Value
1.
25
++
S = 62.5 x 1.25 = 78.13
++
C = 78.13 - 45 = 33.13
+
8
1.
0.
25
S = 50 x 1.25 = 62.5
S+- = 62.5 x 0.8 = 50
C+- = 50 - 45 = 5
8
0.
1.
25
S 0 = 50
-
S = 50 x 0.8 = 40
8
0.
Yr 0
Yr 1
--
S = 40 x 0.8 = 32
-C = 32 - 45 = 0
Yr 2
C + = (0.6 x 33.13) + (0.4 x 5) = 20.45
1.07
20
-
C = (0.6 x 5) + (0.4 x 0) = 2.8
1.07
C 0 = (0.6 x 20.45) + (0.4 x 2.8) = 12.51
1.07
Put Option Value
++
1.
2
5
S = 62.5 x 1.25 = 78.13
P++= 78.13 - 45 = 0
+
8
0.
1.
2
5
S = 50 x 1.25 = 62.5
8
0.
1.
25
S 0 = 50
+-
S = 62.5 x 0.8 = 50
+P = 50 - 45 = 0
-
S = 50 x 0.8 = 40
8
0.
Yr 0
Yr 1
S -- = 40 x 0.8 = 32
-P = 32 - 45 = 13
Yr 2
P + = (0.6 x 0) + (0.4 x 0) = 0
1.07
P - = (0.6 x 0) + (0.4 x 13) = 4.85
1.07
P 0 = (0.6 x 0) + (0.4 x 4.85) = 1.813
1.07
Fig 3: Put-Call Parity
Protective Put = Fiduciary Call
S 0 + P0 = C 0 + X
T
Stock Put
Call (1 + R f )
Investment
in ZCB with a
Face value =
Strike Price
Note: Both options are on
the same underlying
stock have the same
exercise price and
same maturity.
American Style Options allow for exercise any point until maturity of the option
European Style Options can only be exercised at maturity
Bermuda Style Options allow for early exercise but only on specific dates
21
Early exercise feature is not value for call options on dividend paying stocks because the minimum
price of the option exceeds its exercise value but deep-in-the money put options could benefit from
early exercise. When an investor exercises an option early, he/she captures only the intrinsic value
of the option and forges the time value (at expiration time value is 0). While the intrinsic value can
be invested at the risk free rate, the interest so earned is less than the time value in most cases. For
a deep-in-the money put option, the upside is limited because the stock price cannot fall below 0. In
such cases, the interest or intrinsic value can exceed the option's time value. When early exercise
has value, the no-arbitrage approach is the only way to value American style options.
Example 6: [Without Dividends]
Suppose the periodically compounded interest rate is 3%, the non-dividend paying underlying stock is
currently trading at $72, exercise price is $75, U = 1.356, D = 0.541 and the put option expires in two years.
Calculate European Put Option and American Put Option.
π U = 1 + R - D = 1 + 0.03 - 0.541 = 0.6 or 60%
U-D
1.356 - 0.541
π D = 1 - π U = 1 - 0.6 = 0.4 or 40%
European Put Option
1.
35
S + = 72 x 1.356 = 97.632
+
P = 8.61401
HR = - 0.27876
41
5
0.
S - = 72 x 0.541 = 38.952
P - = 33.86353
HR = - 1
35
6
1.
+-
35
6
S = 97.632 x 0.541 = 52.818
+P = 52.818 - 75 = 22.18109
1.
1.
S = 97.632 x 1.356 = 132.388
++
P = 132.388 - 75 = 0
1
54
0.
Yr 1
++
1
54
Yr 0
S - = 72 x 0.541 = 38.952
PE- = 33.86353
✓ PA- = 38.952 - 75 = 36.048
HR = - 1
S -- = 38.952 x 0.541 = 21.073
P-- = 21.073 - 75 = 53.92697
0.
1
54
Put Option
0.
S 0 = 72
↶ AP0 = 19.0171
American HR = - 0.46752
35
6
American Put Option (Higher or Lower)
S+ = 72 x 1.356 = 97.632
✓ P+E = 8.61401
+
PA
= 97.632 - 75 = 0
HR = - 0.27876
S+- = 97.632 x 0.541 = 52.818
+P = 52.818 - 75 = 22.18109
1.
35
6
1
54
0.
1
54
Put Option
0.
S 0 = 72
P = 18.1687
↶E 0
European HR = - 0.43029
1.
35
6
6
++
S = 97.632 x 1.356 = 132.388
++
P = 132.388 - 75 = 0
--
S = 38.952 x 0.541 = 21.073
-P = 21.073 - 75 = 53.92697
Yr 2
22
Thus, the early exercise premium at Time 0 is 0.85 = 19.02 - 18.17. American Style Options can be valued as
the present value of the expected future option payout in a single period setting.
Example 7: [With Dividends]
Consider a call on a $100 stock with exercise price of $95. The periodically compounded interest rate is 1%,
the stock will pay a $3 dividend at Time 1, U = 1.224, D = 0.796 and the American Call option expires in two
years.
π U = 1 + R - D = 1 + 0.01 - 0.796 = 0.5 or 50%
U-D
1.224 - 0.796
π D = 1 - π U = 1 - 0.5 = 0.5 or 50%
1.
22
1.
22
6
79
0.
Yr 0
Yr 1
+-
S = 118.764 x 0.796 = 94.536
+C = 94.536 - 95 = 0
1.
22
-
S = 97.029 x 0.796 = 77.2356
CE = 0
C -A = 0
HR = 0
4
6
79
0.
6
79
0.
S 0 = 100
- 2.97
97.0297
A C 0 = 13.2497
HR = 0.6445
4
+
S = 97.029 x 1.224 = 118.764
+
CE = 24.9344
✓ C +A = 118.76 - 95 + 3 = 26.764
HR = 0.9909
4
++
S = 118.764 x 1.224 = 145.3675
++
C = 145.3675 - 95 = 50.3676
S -- = 77.2356 x 0.796 = 61.4795
-C = 61.479 - 95 = 0
Yr 2
Our approach here is known as the Escrow Method. Because dividends lower the value of the stock,
a call option holder is hurt. Thus, by exercising early, the call buyer acquires the stock just before it
goes ex-dividend and thus is able to capture the dividend. The American Style Call Option is worth
more than the European Style Call Option because at Time 1, when an up move occurs, the call is
exercised early, capturing additional value. For dividend paying stocks, the stock price falls when
the stock goes ex-dividend and it may make sense to exercise the call option before such a decline
in price.
lllll
Interest Rate Options
We can use an estimate of the volatility of an interest rate to create a set of possible rate paths for
interest rates in the future called a Binomial Interest Rate Tree. The interest rates are selected so
that the risk- neutral (RN) probabilities of up and down moves are both equal to 0.5.
[R > X]
Positive Payoff
Call Payoff = NP x [Max (0, Reference Rate - Exercise Rate)] x Days
360
[R < X]
Positive Payoff
Put Payoff = NP x [Max (0, Exercise Rate - Reference Rate)] x Days
360
23
↶
Example 8:
Suppose we seek to value two year European Style Call and Put Options an the periodically compounded
1-year spot interest rate (the underlying). Assume the notional amount of the options is $1000,000 and the
call and put exercise rate is 3.25% of par. Assume the risk-neutral probability is 50% and these option cash
settle at Time 2 based on the observed rates.
Reference
Rates
3.9706
3.9084
3.0454
3.2542
2.6034
2.2593
Yr 0
Call Option
[R > X]
Yr 1
Yr 2
++
C = $ 1000,000 (3.9706% - 3.25%) = $ 7206
+C = $ 1000,000 (3.2542% - 3.25%) = $ 42
C -- = $ 1000,000 (2.2593% - 3.25%) = $ 0
C + = (0.5 x 7206) + (0.5 x 42) = $ 3487.68
1.039084
C - = (0.5 X 42) + (0.5 X 4) = $ 20.4671
1.026034
C 0 = (0.5 x 3487.68) + (0.5 x 20.4671) = $ 1702.333
1.030454
Put Option
[R < X]
P++= $ 1000,000 (3.25% - 3.9706%) = $ 0
P+- = $ 1000,000 (3.25% - 3.2542%) = $ 0
P -- = $ 1000,000 (3.25% - 2.2593%) = $ 9907
P + = (0.5 x 0) + (0.5 x 0) = $ 0
1.039084
P - = (0.5 x 0) + (0.5 x 9907) = $ 4827.81
1.026034
P 0 = (0.5 x 0) + (0.5 x 4827.81) = $ 2342.56
1.0304454
24
lllll
Black Scholes Merton Model
The innovation of the Black Scholes Merton Model (BSM) is essentially the no-arbitrage approach
introduced in the previous section but applied with a continuous time process, which is equivalent to a
binomial model in which the length of the time step essentially approaches zero. It is also consistent
with the basic statistical fact that the binomial process with a 'large' no. of steps converges to the
standard normal distribution. This characteristic of how an asset evolves randomly is called a
Stochastic Process. The specific assumptions of the BSM model are as follows:
Assumption 1: The underlying asset price follows a 'Geometric Brownian Motion Process' implies continuous
prices, meaning that the price of underlying instrument doesn't jump from one value to another, rather it
moves smoothly from value to value. The return on the underlying asset follows a lognormal distribution. In
other words, the logarithmic continuously compounded return is normally distributed.
Assumption 2: The continuously compounded risk free rate is constant and known. Borrowing and lending
are both as the risk free rate. The volatility of the returns on the underlying asset is constant and known. The
price of the underlying changes smoothly i.e. doesn't jump abruptly. The continuously compounded yield on
the underlying asset is constant.
Assumption 3: Markets are 'Frictionless'. There are no taxes, no transaction costs and no restrictions on
short sales or the use of short sale proceeds. Continuous trading is possible and there are no arbitrage
opportunities in the marketplace. The underlying instrument is liquid i.e. it can be easily bought and sold.
Assumption 4: The options are European Options i.e. they can only be exercised at expiration. American
Options use binomial model not BSM.
Investing
Financing
I
C 0 = S 0 . N(d 1 ) - X .
Stock
II
-Rf (T)
e
. N(d 2 )
Bond
C 0 = e-R f (T) [ S 0 . e Rf (T) . N(d1 ) - X . N(d2 ) ]
Present Value
Expected Payoff at Maturity
Call option value is the present value of expected payoff at maturity. For call options, - N(d 2 ) implies
borrowing or buying N(d1 ) units of stock by short selling N(d 2 ) shares of a ZCB trading at X . e R f (T) . If the
value of the underlying S 0 increases, then the value of N(d 1 ) also increases because S 0 has a positive effect
on d 1 . Thus, the replicating strategy for calls requires continually buying shares in a rising market and selling
shares in a falling market.
Financing
I
-R f (T)
P0 = X . e
Investing
II
. N(-d 2 ) - S 0 . N(-d 1 )
Bond
Stock
P0 = e-R f (T) [ X . N(-d 2 ) - S 0 . e R f (T) . N(-d 1) ]
Present Value
Expected Payoff at Maturity
Put option value is the present value of expected payoff at maturity. For put options, N(-d2 ) implies lending
money or buying N(-d 2 ) units of bond and selling N(-d1 ) units of stocks.
25
Whereas,
d1 : In (S 0 / x) + (R f + o 2 / 2)T
o T
d2 : d1 - o T
N(d1 ) : Hedge Ratio or Delta
N(d2 ) : Probability that call option expires in the money. P (S T > X)
N(-d 2 ) : Probability that put option expires in the money. P (X > S T )
R f : Continuously compounded risk free rate.
o : Annual Volatility of Asset Returns.
N(x) : Cumulative Standard Normal Probability i.e. which is the probability of obtaining a
value of less than 'x' based on standard normal distribution.
N(-x) : 1 - N(x)
X . e-R f (T) : Price of a ZCB
Within the BSM model theory, the aggregate losses from this 'Buy High or Sell Low' strategy over
the life of the option adds up exactly to the BSM model option premium received for the option at
inception. This result must be the case, otherwise there would be arbitrage profits available
because transaction costs are not in fact 0, the frequent rebalancing by buying and selling the
underlying adds significant costs for the hedger. Also, markets can often move discontinuously,
contrary to the BSM model's assumption that prices move continuously, thus allowing for
continuous hedging adjustments. Hence, in reality hedges are imperfect. For example, if a company
announces a merger, then the company's stock price may jump substantially higher contrary to the
BSM model's assumption. In addition, volatility cannot be known in advance. For these reasons,
options are typically more expensive than they would be as predicted by the BSM model theory.
Example 9:
Stock of Swipe Wire Inc. is currently trading at $50. Suppose that the return volatility is 25% and the
continuously compounded risk free rate is 3%. Calls and puts with a strike price of $45 and expiring in six
months (T = 0.5) are trading at $7 and $1 respectively. If N(d1 ) = 0.779 and N(d 2) = 0.723, calculate the value
of the replicating portfolios and any arbitrage profits on both options.
C0 = S 0 . N(d 1) - X . e- R f (T) . N(d 2 )
-0.03 (0.5)
= 50 (0.779) - 45 . e
(0.723)
= 38.95 - 32.05
= $ 6.9
P0 = X . e- R f (T) . N(-d 2 ) - S 0 . N(-d 1 )
= 45 . e-0.03 (0.5) (1 - 0.723) - 50 (1 - 0.779)
= 12.28 - 11.05
= $ 1.23
Call: It entails writing a call at $ 7 and buying the replicating portfolio for $ 6.9 to yield an arbitrage profit of
$ 7 - $ 6.9 = $ 0.1 per call.
Put: Selling the replicating portfolio of $ 1.23 and buying puts for an arbitrage profit of $ 0.23 per put.
26
lllll
Options on Dividend Paying Stocks
C 0 = S 0 . e- DY (T). N(d1 ) - X . e- R f (T) . N(d 2 )
Stock
Bond
P0 = X . e- R f (T) . N(-d 2) - S 0 . e- DY (T). N(-d 1 )
Bond
Stock
Whereas,
d1 : In (S 0 / x) + (R f - DY + o 2/ 2) T
o T
d2 : d1 - T
DY : Continuously compounded dividend yield.
Both 'd 1 ' and 'd 2' are reduced by carry benefits. An increase in carry benefits will lower the value of
the call option and raise the value of the put option. Also, higher dividends will lower the no. of
bonds to short sell for calls and lower the no. of bonds to buy for puts. Also, higher dividends will
lower the no. of shares to buy for calls and lower the no. of shares to short sell for puts. The
dividend yield BSM model can again be interpreted as a dynamically managed portfolio of the stock
and ZCBs.
Example 10:
Suppose now that the stock doesn't pay dividend i.e. DY = 0%. Identify the correct statement:
A. The BSM model option value is the same as the previous because options are not dividend adjusted.
B. The BSM model option values will be different because there is an adjustment term applied to the
exercise price i.e. e - DY (T) , which will influence the option values.
C. The BSM model option value will be different because 'd 1 ', 'd2 ' and the stock component are all adjusted
for dividends.
✓
lllll
Options on Currencies
C 0 = S 0 . e- Rb (T). N(d1 ) - X . e- Rp (T). N(d 2 )
Foreign Exchange
N(d1 ) units of base
currency invested
as the base
currencies 'Rf '.
N(d 1 ) units of
Investing
Bond
Investment in base
currency funded by
borrowing in the
priced currency.
[Short N(d 2) units
ZCB]
N(d2 ) units of
Borrowing
- R (T)
- R (T)
P0 = X . e p . N(-d2 ) - S 0 . e b . N(-d 1 )
Bond
Foreign Exchange
Whereas,
R p : Continuously compounded price currency interest rate i.e. R f .
R b : Continuously compounded base currency interest rate i.e. Carry Rate.
27
Fig 4: BSM Put-Call Parity
lllll
Basic BSM
[Without Dividends]
P0 + S 0 = C 0 + X . e - Rf (T)
Basic BSM
[With Dividends]
P0 + S 0 . e- DY (T) = C 0 + X . e - Rf (T)
Black Model
In 1976, Fischer Black introduced a modified version of the BSM model approach that is applicable to
options on underlying instruments that are costless to carry, such as options on futures contracts for
example, equity index futures and options on forward contracts. The latter includes interest rate based
options such as caps, floors and swaptions. Black model is just the BSM model with substituted
- R (T)
F 0 . e f for S 0. Black Model is simply the BSM model in which the futures contract is assumed to
reflect the carry arbitrage model. (Settlement: 1)
C 0 = e- R f (T) [ F 0 . N(d1 ) - X . N(d 2 ) ]
C 0 = F 0 . e- R f (T) . N(d1 ) - X . e- R f (T) . N(d 2 )
Futures
Bond
P0 = e- R f (T) [ X . N(-d 2) - F 0 . N(-d1 ) ]
P0 = X . e - R f (T) . N(-d2 ) - F 0 . e- R f (T) . N(-d1 )
Bond
Futures
Whereas,
d 1 : In (F 0 / x) + (o 2 / 2) T
o T
d 2 : d1 - o T
F0 : Futures Price
Fig 5: Black's Future Option Put-Call Parity
P0 + F 0 . e- R f (T) = C 0 + X . e- R f (T)
lllll
Interest Rate Options
Interest Rate Options are options on forward rates or options on FRAs. A call option on an FRA gains
when rates rise and a put option on an FRA gains when rates fall. Interest rates are fixed in advance
(i.e. beginning of the loan term) and settled in arrears (i.e. paid at maturity of the loan). While FRAs
generally use a 30/360 convention, options on FRAs use an Actual/365 convention. Consider an
interest rate call option on an (MxN) FRA expiring in 'M' months that has a strike rate of X. The
underlying is an (N-M) month forward rate. The model presented here is known as the Standard
Market Model and is a variation of Black's future options valuation model. Interest rate options
28
give option buyers the right to certain cash payments based on observed interest rates. For
example, an interest rate call option gives the call buyer the right to a certain cash payment when
the underlying interest rate exceeds the exercise rate. An interest rate put option gives the put
buyer the right to a certain cash payment when the underlying interest rate is below the exercise
rate. (Settlement: 1)
C 0 = AP . e- Rf (Actual/365) [ FRA (MxN) . N(d1 ) - R x . N(d 2 ) ]
C 0 = AP [ FRA (MxN) . N(d 1 ) - R x . N(d 2) ]
After dividing by e
- Rf (Actual/365)
After dividing by e
- Rf (Actual/365)
- R (Actual/365)
P0 = AP . e f
[ R x . N(-d 2 ) - FRA (MxN) . N(-d 1) ]
P0 = AP [ R x . N(-d 2 ) - FRA (MxN) . N(-d 1 ) ]
Whereas,
↶ Time to option expiration not maturity
d 1 : In (FRA (MxN) / R x ) + (o 2/ 2) Tt
o Tt
d 2 : d1 - o T t
R x : Exercise Rate expressed on an annual basis.
AP : Accrued Period i.e. (Actual/365)
NP : Notional Principle on FRA
If Exercise Rate = Current FRA Rate, then long an interest rate call option and short an interest rate
put option is equivalent to a receive floating, pay fixed FRA. If Exercise Rate = Current FRA Rate, then
long an interest rate put option and short an interest rate call option is equivalent to a received
fixed, pay floating FRA.
The underlying interest rate call options are termed Caplets. Thus, a set of floating rate loan
payments can be hedged with a long position in an interest rate cap encompassing a series of
interest rate call options.
The underlying interest rate put options are termed Floorlets. Thus, a set of floating rate loan
payments can be hedged with a long position in an interest rate floor encompassing a series of
interest rate put options.
Long Cap, Short Floor: Payer Swap (i.e. Pay Fixed, Received Floating)
Short Cap, Long Floor: Receiver Swap (i.e. Pay Floating, Received Fixed)
If the exercise rate on a floor and a cap are set equal to a market swap fixed rate, the value of the
cap will be equal to the value of the floor. When an exercise rate is selected such that the
Cap = Floor value, then the initial cost of being long a cap and short the floor is also 0. This occurs
when the cap and floor strike are equal to the swap rate.
lllll
Swaptions
A Swap Option or Swaption is simply an option on a swap. It gives the holder the right, but not the
obligation to enter a swap at the pre-agreed swap rate - the exercise rate. A Payer Swaption is the
right to enter into a specific swap at some date in the future at a predetermined rate as the fixed
rate payer. If interest rates increase, payer swaption becomes more valuable. The holder of a payer
swaption would exercise if the market rate > exercise rate at expiration. A Receiver Swaption is the
right to enter into a specific swap at some date in the future as the fixed rate receiver. If interest
29
rates decrease, receiver swaption becomes more valuable. The holder of a receiver swaption would
exercise if the market rate < exercise rate at expiration. (Settlement: Many)
PAYs = AP (PVA) [ SFR . N(d1 ) - R x . N(d 2) ]
Payer Swaption
PAYs = AP (PVA) SFR . N(d1 ) - AP (PVA) R x . N(d 2)
Swap
Bond
E (PAYs) = e- R f (T) . PAYs
RECs = AP (PVA) [ R x . N(-d2 ) - SFR . N(-d 1) ]
Receiver Swaption
RECs = AP (PVA) R x . N(-d 2 ) - AP (PVA) SFR . N(-d1 )
Bond
Swap
E (RECs) = e- R f (T) . RECs
Whereas,
d1 : In (SFR / R x ) + (o 2/ 2) T
o T
d 2 : d1 - o
T
R x : Exercise Rate.
AP : Accrual Period i.e. (Actual/365)
SFR : Current Market Swap Fixed Rate.
PVA : Present Value of Annuity i.e. Annuity Discount Factor.
Long Receiver Swaption and Short Payer Swaption with same exercise rate is equivalent to entering
a received fixed, pay floating forward swap. Long Payer Swaption and Short Receiver Swaption with
same exercise rate is equivalent to entering a receive floating, pay fixed forward swap. Note that if
the exercise rate is selected such that the receiver and payer swaptions have the same value, then
the exercise rate is equal to the at-market forward swap rate (When an interest rate swap is
initiated, its current value is 0 and is known as an at-market swap). A long callable bond can be
replicated using a long option-free bond + short receiver swaption.
Option Greeks
There are 5 inputs to the BSM model: asset price, exercise price, asset price volatility, time to expiration and
the risk free rate. The measures examined here are known as Greeks and includes Delta, Gamma, Theta, Vega
and Rho. We seek to address how much a particular portfolio will change for a given small change in the
appropriate parameter. These measures are sometimes referred to as static risk measures.
1. Delta: describes the relationship between changes in asset prices and changes in option prices. Delta is
also the hedge ratio (change in the price of an option for a 1 unit change in the price of the underlying
stock). Call option deltas are positive because as the underlying asset price increases, call option value also
increases. Conversely, the delta of a put option is negative because the put value falls as the asset price
increases.
Dividends
Delta =
C
e - DY (T)
. N(d 1 )
Delta = - e - DY (T) . N(-d 1 )
P
No Dividends
Delta = C 1 - C 0 / S1 - S 0 =
C
Delta C = N(d 1) Continuous
△C/△S
Discrete
30
Call Value
Call
60
Call Delta
50
40
30
Delta (The slope of prior-to-expiration curve)
↷
Illustrates call value
based on the BSM model
and call value based on
delta approximation
20
10
0
10
20
30
40
50
60
70
80
90
100 110 120 130
Stock Value
-10
-20
Approximation are close for small changes in stock price, but the approximation becomes less accurate as
the
S becomes larger i.e. the estimation error also increases. For example, if the stock value rises from
100 to 101, notice that both the call line and call delta estimate line are almost the same value. If however,
the stock value rises from 100 to 150, the call line is now significantly above the call delta estimated line.
The delta approximation is biased low for both a down move and an up move.
△
△ C ≅ e- DY (T). N(d1) x △ S
△ P ≅ - e- DY (T) . N(-d1) x △ S
Based on the graph, the BSM model assumption of continuous trading is essential to avoid hedging risk.
This hedging risk is related to the difference between these two lines and the degree to which the
underlying price experiences large changes. Delta Hedging refers to manipulating the underlying portfolio
delta by appropriately changing the positions in the portfolio. A delta neutral portfolio refers to setting the
portfolio Delta all the way to 0. In theory, the Delta neutral portfolio will not change in value for small
changes in the stock instrument.
↷
Long position in Stock with a Short position in a Call
No. of Short Call Options needed to Delta Hedge = No. of Shares Hedged
Delta C
↷
Long position in Stock with a Long position in a Put
No. of Long Put Options needed to Delta Hedge = - No. of Shares Hedged
Delta P
A key consideration in delta neutral hedging is that the delta hedged asset position is only Rf for a very
small change in the value of the underlying stock. The delta neutral portfolio must be continually
rebalanced to maintain the hedge, for this reason it is called a Dynamic Hedge. Hence, continuously
maintaining a delta neutral position involves significant transaction costs.
Call Option
0
Put Option
0
Dividends: 0 to e- DY (T)
No Dividends: 0 to 1
Dividends: - e- DY (T) to 0
No Dividends: -1 to 0
Out of Money
Out of Money
Delta
Delta
In the Money
e- DY (T) or 1
- DY (T)
In the Money
-e
or - 1
31
Delta neutral portfolio can be created with any of the following combinations: Long Stock and Short Calls,
Short Stock and Long Calls, Long Stock and Long Puts, Short Stocks and Short Put.
Example 11:
Suppose you own 60,000 shares of ABC Company common stock that is currently selling for $50. A call option
on ABC with a strike price of $50 is selling for $4 and has a delta of 0.6. Determine the no. of call options
necessary to create a delta neutral hedge.
No. of Options needed to Delta Hedge = 60,000 = 100,000 Options
0.6
Because we are
long the stock,
we need to short
the options,
Now calculate the effect on portfolio value of a $1 increase in the price of ABC stock.
△ S = $1 +
Gain = 60,000 x 1 = 60,000
Loss = 100,000 x 0.6 x 1 = - 60,000
Total change in portfolio value = 60,000 - 60,000 = $ 0
2. Gamma: measures the rate of change in delta as the underlying stock price changes. Gamma captures the
curvature of the option value vs. stock price relationship. Option gamma is defined as the change in a given
option data for a given small change in the stock's value, holding everything else constant. Thus, the
gamma of a long or short position in one share of stock is 0, because the data of a share of stock never
changes.
Gamma C = Gamma P = e- DY (T). N(d1 )
S0 . o T
Call and put options on the same underlying asset with the same exercise price and time to expiration will
have equal gammas. Long positions in calls and puts have positive gamma. Gamma approximates the
estimation error in data for options because the option price with respect to the stock is non-linear and
delta is a linear approximation. Thus, Gamma is a risk measure; specifically, gamma measures the
non-linearity risk or the risk that remains over the portfolio is delta neutral. A delta hedged portfolio with a
long position in stocks and a short position in calls will have negative net gamma exposure.
Gamma
Out of
Money
At the
Money
In the
Money
Gamma is highest for at the money options. Deep in the money or deep out of money options have low
gamma. Also, as the stock price changes and as time to expiration changes, the gamma is also changing.
The lower (increase) the overall gamma of a portfolio, one should short (go long) options. Lower gamma is
better for delta hedging.
32
For example, a gamma of 0.04 implies that a $1 increase in the price of the underlying stock will cause a
call option's delta to increase by 0.04, making the call option more sensitive to changes in the stock price.
Importantly, the call delta plus gamma estimated line is significantly closer to the BSM model call values.
Thus, we see that even for fairly large changes in the stock, the delta plus gamma approximation is
accurate. Delta plus gamma approximation is biased low for a down move but biased high for an up move.
Delta plus gamma approximation is an improvement when compared with using the delta approximation
on its own.
△ C ≅ DeltaC x △ S + 1/2 Gamma x △ S2
△ P ≅ Delta P x △ S + 1/2 Gamma x △ S2
Gamma risk is the risk that the stock price might abruptly 'jump', leaving an otherwise delta hedged
portfolio unhedged or leaving us suddenly unhedged. Consider a delta hedge involving a long position in
stock and short position is calls. If the stock price falls abruptly, the loss in the long stock position will not
equal the gain in the short call position. This is the gamma risk of the hedge.
3. Vega: measures the sensitivity of the option price to changes in the volatility of returns on the underlying
asset. The higher the volatility, so vega is positive for both calls and puts. Vega gets larger when options are
at or near the money.
↶
4. Theta: measures the sensitivity of option price to the passage of time. Option theta is the rate at which the
option time value declines as the option approaches expiration. If calendar time or passage of time passes,
then time to expiration declines, because stock don't have an expiration date, the stock theta is 0. Like
gamma, theta cannot be adjusted with stock trades. The gain or loss of an option portfolio in response to
the mere passage of calendar time is known as time decay. Typically, theta is negative for options. That is,
as calendar time passes, expiration time declines and the option value also declines. As time passes and a
call option approaches maturity, its speculative value declines, all else equal. That behavior also applies for
most put options (though deep-in-the-money put options close to maturity may actually increase in value
as time passes). An exception being European put options thetas are negative.
Option = Intrinsic Value + Time Value
θ 'Theta'
Example 12:
Stock Price
Call and Put Option Strike Price
$ 77
$ 75
If the price of GI stock approaches $75 over the next 30 days, which of the following changes in option
parameter measures will most likely be observed?
The speed of the option value decline increases, however, as time
A. Decrease in Vega and the absolute value of Theta.
to expiration decreases. During the next 30 days, the options will
B. Increase in Vega and the absolute value of Theta.
approach expiration and approach being at the money.
C. Decrease in Vega and an increase in the absolute value of Theta.
✓
5. Rho: measures the sensitivity of the option price to changes in the R f rate. The impact on option prices
when interest rate change is relatively small when compared with that for volatility changes and that for
changes in the stock. Hence, rho, is not a very important sensitivity measure. Rho is positive for calls
and negative for puts. When interest rates are 0, the call and put options values are the same for
at-the-money options. As interest rates rise, the difference between call and put options increase.
33
Option Value
Call Value
25
20
15
10
5
0
5
10
15
20
25
30
35
40
Put Value
Interest Rate %
Fig 6: European Option Prices for a Change in the 5 BSM Model Inputs
Greeks
Input
Delta
Gamma
Vega
Rho
Theta
Asset Price (S)
Delta
Volatility (o )
Rf
* Time to Expiration (T)
Exercise Price (X)
↑
↑
↑
↑
↑
↑
CALL
PUT
Time Value : $ 0
Maturity :
Time Value : $ 0
Maturity :
↑
↑
↑
↑
↓
↓
↓
↑
↑
↓
↓
↑
* The relationship between option value and time to maturity is positive. All else equal,
shorter maturity options have lower values. The relationship between option value and
the passage of time is negative. All else equal, as time passes and the option
approaches maturity, the value of the option decays.
Implied Volatility is the standard deviation of continuously compounded asset returns that is 'implied' by the
market price of the option. The volatility parameter in the BSM model, however is the future volatility.
Volatility can be inferred from option prices, this inferred volatility is called the Implied Volatility. Implied
Volatility must be found by iteration i.e. trial and error. Implied Volatilities in options with different exercise
prices on the same underlying may reflect different implied volatilities (a violation of the BSM assumption of
constant volatility). It is common to construct a 3D plot of the implied volatility with respect to both expiration
time and exercise price, a visualization known as the Volatility Surface. If the BSM model assumptions were
true, then one would expect to find the volatility surface flat. If we plug these four inputs (stock price, exercise
price, R f rate and time to maturity) into the BSM model, we can solve for the value of volatility that makes the
BSM model value equal the market price. Volatility is the 'guess factor'.
Example 13:
Suppose an options dealer offers to sell a 3-month at-the-money call on the FTSE index option at 19% implied
volatility and a 1-month in-the-money put on Vodaphone (VOD) at 24%. An option trader believes that based
on the current outlook, FTSE volatility should be closest to 25% and VOD volatility should be closer to 20%.
What actions might the trader take to benefit from her views?
A. Buy FTSE Call and VOD Put
B. Buy FTSE Call and Sell VOD Put
C. Sell FTSE Call and VOD Put
✓
1
Fixed Income
PAGE NOS. 46
CHAPTER 32
VOL. 9
CHAPTERS. 5
The Term Structure and Interest Rate Dynamics
Spot Rate is the annualized market interest rates for a single payment to be received in the future.
Future Rate is an interest rate agreed today for a loan to be made at some future date.
Forward
Price
Forward
Pricing Model
PT =
1
(1 + S T )T
P(a+b) = Pa . F (a,b)
1
a
2
b
When the spot curve is upward sloping, the forward curve will lie above the spot curve: F0 > S 0
Undervalued
When the spot curve is downward sloping, the forward curve will lie below the spot curve: F 0 < S0 Overvalued
Example 1:
Compute the price and YTM of a 3-year 4% annual pay, $1000 face value bond given the following spot rate
curve, S1 = 5%, S 2 = 6% and S 3 = 7%.
Price = 40 + 40 + 1040 = $ 922.64
(1.05)1 (1.06)2 (1.07)3
n=3
Pv = - 922.64
Fv = 1000
PMT = 40
I/Y = 6.94%
Note that the yield on a 3-year bond is a weighted average of 3 spot rates, so in this case we would expect
S 1 < YTM 3 < S 3 . The YTM3 is closest to S3 , because the par value dominates the value of the bond and
therefore S 3 has the highest weight.
Example 2:
Calculate the forward price two years from now for a $1 par, zero coupon, 3-year bond given the following
spot rates: S 2 = 4% and S 5 = 6%.
5%
2
4%
5
x
(1.06) 5 = (1.04) 2 (1 + x)3
1.33822 = 1.0816 (1 + x)
(1.23725) = 1 + x
7.35% = x or f (2, 3)
2
Since 7.35% is a forward rate; the forward price is P T =
1
= 0.8084
(1.0735) 3
In other words, $0.8084 is the price agreed today, to pay in two years for 3-year bond that will pay $1 at
maturity.
Example 3:
The 1-year spot rate is 5% and the forward price for a 1-year ZCB beginning in one year is 0.9346. The spot
price of a 2-year zero coupon bond is closest to?
x
1
5%
2
0.9346
P = 1 = 0.9523
1.05
S2 or x = (0.9523) (0.9346) = 0.89
Example 4:
The model that equates the buying a long maturity ZCB to entering into a forward contract to buy a ZCB that
matures at the same time is known as:
A. The forward rate model
Forward rate f(a, b) should make investors indifferent between buying a long
B. The forward pricing model
maturity ZCB vs. buying a shorter maturity ZCB and reinvesting the principal.
C. The forward arbitrage model
✓
Example 5:
Is it possible for a coupon bond to earn less than the YTM, if held to maturity?
A. Yes
If reinvestment rates for the future coupons are lower than the initial YTM, a bondholder may experience realized return
(realized return on a bond refers to the actual return that the investor experiences over the investment's holding period.
B. No
Realized Return is based on actual reinvestment rates).
C. There is not enough information provided to determine this. For a coupon bond, if the spot rate curve is not flat, the
✓
YTM will not be the same as the spot rate. If the yield
curve is not flat, the coupon payments will not be
reinvested at the YTM and the expected return will differ
from the yield.
Bootstrapping or Forward Substitution i.e. spot rates or zero coupon rates can be derived from the par
curve. Bootstrapping involves using the output of one step as an input to the next step.
Example 6:
Given the following (annual-pay) par curve, compute the corresponding spot rate curve:
Maturity
Par Rate
1
2
3
1%
1.25%
1.5%
Spot Rate
x 1%
x 1.252%
x 1.51%
3
Consider 2-year bond
100 = 1.25 + 101.25
(1.01)1 (1 + x) 2
x = S 2 = 1.252%
Consider 3-year bond
100 = 1.5 +
1.5
+ 101.5
(1.01)1 (1.01252) 2 (1 + x) 3
x = S 3 = 1.51%
When spot rates turn out to be lower (higher) than implied by the forward curve, the forward price will
increase (decrease). If an investor believes that future spot rates will be lower than corresponding forward
rates, then he/she will purchase bonds at a presumably attractive price because the market appears to be
discounting future cashflows at 'too high' of a discount rate (a trader expecting lower future spot rates than
implied by the current forward rates would purchase the forward contract to profit from its appreciation) i.e.
the investor would perceive the bond to be undervalued. If an investor believes that future spot rates will be
higher than corresponding forward rates, then the trader or investor is likely to sell the forward contract i.e.
the investor would perceive the bond to be overvalued. Moreover, if a portfolio manager's projected spot
curve is below (above) the forward curve and his or her expectations turn out to be true, the return will be
more (less) than the one-period risk free interest rate. If the future spot rates actually evolve as forecasted by
the forward curve, the forward price will remain unchanged i.e. for a bond investor, the return on a bond
over a one year horizon is always equal to the 1-year risk free rate, if the spot rates evolve as predicted by
today's forward curve. If the spot curve one year from today is not the same as that predicted by today's
forward curve, the return over the 1-year period will differ, with the return depending on the bond's maturity.
Example 7:
Jane Dash CFA, has collected benchmark spot rates as shown below:
Maturity
Spot Rate
1
2
3
3%
4%
5%
The expected spot rates or implied forward rates 'f (a, b)' at the end of one year are as follows:
Year
Expected Spot Rate
1
2
5.01%
6.01%
Calculate the 1-year holding period return of a:
a. 1-year ZCB
100
1.03
$97.08
$100
0
1
3%
Return = 100 - 97.08 = 3%
97.08
100
(1.04) 2
$92.455
4%
↷
b. 2-year ZCB
0
$95.22
100
1.0501
$100
1
2
5.01%
4
Return = 95.22 - 92.45 = 3%
92.45
100
(1.05)3
$86.38
5%
↷
c. 3-year ZCB
0
$88.983
100
(1.0601)2
1
$100
2
3
6.01%
Return = 88.98 - 86.38 = 3%
86.38
Hence, regardless of the maturity of the bond, the holding period return will be the 1-year spot rate if the
spot rates evolve consistently with the forward curve as it existed when the trade was initiated.
Maturity Matching i.e. Investor's purchasing bonds that have a maturity equal to the investor's investment
horizon. However with an upward-sloping interest rate term structure, investors seeking superior returns
may pursue a strategy called 'Riding the Yield Curve' a.k.a. 'Rolling Down the Yield Curve'. Under this strategy,
an investor will purchase bonds with maturities longer than his investment horizon; if the yield curve remains
unchanged over the investment horizon, riding the yield curve strategy will produce higher returns than a
simple maturity matching strategy, increasing the total return of a bond portfolio. In an upward-sloping yield
curve, shorter maturity bonds have lower yields than longer maturity bonds. As the bond approaches
maturity i.e. rolls down the yield curve, It is valued using successively lower yields and therefore, at
successively higher prices. The greater the difference between the forward rate and the spot rate and the
longer the maturity of the bond, the higher total return i.e. the more sensitive its total return is to be spread.
A bond investor with an investment horizon of 5 years could purchase a bond maturing in 5 years and earn
the 3% coupon but no capital gain i.e. maturity matching vs. the investor could instead purchase a 30-year
bond for $63.67, hold it for 5 years and sell it for $71.81, earnings an additional return beyond the 3% coupon
over the same period. In the aftermath of the financial crisis of 2007-2008, central banks kept those
short-term rates low, giving yield curves a steep upward slope. Many active managers took advantage by
borrowing at short-term rates and buying long maturity bonds. The risk of such a leveraged strategy is the
possibility of an increase in spot rates.
The Swap Rate Curve i.e. In a plain vanilla interest rate swap, one party makes payments based on a fixed
rate (Swap Fixed Rate or SFR) while the counterparty makes payments based on a floating rate (LIBOR).
Market participants prefer the swap rate curve as a benchmark interest rate curve rather than a government
bond yield curve for the following reasons:
- Swap rates reflect the credit risk of commercial banks rather than the credit risk of governments.
- The swap market is not regulated by any government, which makes swap rates in different countries
more comparable. Government bond yield curves additionally reflect sovereign risk unique to each
country.
- The swap curve typically has yield quotes at many maturities, while the US government bond yield curve
has on-the-run issues trading at only a small no. of maturities.
- Wholesale banks that manage interest rates risk with swap contracts are more likely to use swap curves
to value their A&L. Retail banks, on the other hand, are more likely to use a government bond yield curve.
- In countries in which the private sector is much bigger than the public sector, the swap curve is a far
more relevant measure of the time value of money than is the government's cost of borrowing.
T
t=1
SFR T +
1
=1
t
(1 + S t ) (1 + S T ) T Value of Floating
↷
Σ
Value of Fixed Leg
Leg, always 1 at
the Initiation
5
Example 8:
Suppose a government spot curve implies the following discount factors:
P (1) = 0.9524
P (2) = 0.89
P (3) = 0.8163
P (4) = 0.735
? = 1 / (1 + x)1 = 0.9524
? = 1 / (1 + x) 2= 0.89
? = 1 / (1 + x)3 = 0.8163
? = 1 / (1 + x) 4 = 0.735
↷Spot Rates
; x = 5%
; x = 6%
; x = 7%
; x = 8%
Given the information, determine the swap rate curve.
Compute SFR 1 = SFR + 1 = 1
(1.05) 1 (1.05)1
; SFR 1 = 5%
Compute SFR 2 = SFR + SFR + 1 = 1
(1.05) 1 (1.06) 2 (1.06) 2
; SFR 2 = 5.97%
Compute SFR 3 = SFR + SFR + SFR + 1 = 1 ; SFR 3 = 6.91%
(1.05) 1 (1.06) 2 (1.07) 3 (1.07) 3
Compute SFR 4 = SFR + SFR + SFR + SFR + 1 = 1 ; SFR 4 = 7.81%
(1.05) 1 (1.06) 2 (1.07) 3 (1.08) 4 (1.08) 4
The swap rates, spot rates and discount factors are all mathematically linked together. Having access to data
for one of the series allows you to calculate the other two.
lllll
Spread Measures
The swap spread is a popular way to indicate credit spreads in a market. The swap spread is defined as
the spread paid by the fixed rate payer of an interest rate swap over the rate of the 'on-the-run' (most
recently issued) government security with the same maturity as the swap.
Swap Spread t = Swap Rate t - Treasury Yield t
(YTM)
(R f )
If the bond is default free, then the swap spread could provide an indication of the bond's liquidity or it
could provide evidence of market mispricing. The higher the swap spread, the higher the return that
investors require for credit and/or liquidity risk.
↶
↷
1. I-Spread: I-Spread for a credit-risky bond is the amount by which the yield on the risky bond exceeds the
swap rate for the same maturity. In a case where the swap rate for a specific maturity is not available, the
missing swap rate can be estimated from the swap rate curve using linear interpolation, hence I-Spread.
I-Spread = YTM - Swap Rate
Interpolated Rate
Only reflects
compensation for credit
and liquidity risks
Example 9:
6% Zinni Inc. bonds are currently yielding 2.35% and mature 1.6 years. From the provided swap curve,
compute I-Spread.
Tenor
Swap Curve
0.5
1%
1
1.25%
1.5
1.35%
2
1.5%
6
Since, 1.6 years falls in the 1.5 - 2 year interval,
Interpolated Rate = Rate for Lower Bond + (No. of Years for Interpolated Rate - No. of Years for Lower Bond)
(Higher Bond Rate - Lower Bond Rate)
(No. of Years for Upper Bond - No. of Years for Lower Bond)
1.6 Swap Rate = 1.5 Year Swap Rate + 0.1 (2-Year Swap Rate - 1.5 Year Swap Rate)
0.5
= 1.35% + 0.1 (1.5% - 1.35%)
0.5
= 1.38%
I-Spread = YTM - Swap Rate
= 2.35% - 1.38%
= 0.97%
2. Z-Spread: Z-Spread is the constant basis point spread that would need to be added to the implied spot
yield curve so that the discounted cashflows of a bond are equal to its current market price. This spread
will be more accurate than a linearly interpolated yield, particularly with steep interest rate swap curves.
The term 'Zero Volatility' in the Z-Spread refers to the assumption of zero interest rate volatility. Z-Spread is
not appropriate to use to value bonds with embedded options; without any interest rate volatility, options
are meaningless.
$8
+
$ 108
(1 + 0.04 + z) (1 + 0.05 + z)
(S1 )
(S 2 )
↶
$ 104.12 =
Compensation for option risk as well as
compensation for credit and liquidity risk
3. TED Spread: TED Spread is an acronym that combines the 'T' in 'T-Bill' with 'ED' (the ticker symbol for the
Eurodollar futures contract). Conceptually, the TED Spread is the amount by which the interest rate on
loans between banks (formally, 3-month LIBOR) exceeds the interest rate on short-term US government
debt (3-month T-bills).
TED Spread = 3-month LIBOR Rate - 3-month T-Bill Rate
Because T-bills are considered to be risk-free white LIBOR reflects the risk of lending to commercial banks,
the TED Spread is seen as an indication of the risk of interbank loans. An increase (decrease) in the TED
Spread is a sign that lenders believe because the risk of default on interbank loans is increasing
(decreasing) and that risk-free T-bills are becoming more valuable (non-valuable) in comparison. The TED
Spread can also be thought of as a measure of counterparty risk. Compared with the 10-year swap spread,
the TED Spread more accurately reflects risk in the banking system, whereas the 10-year swap spread is
more often a reflection of differing supply and demand conditions.
4. LIBOR-OIS Spread: OIS stands for overnight indexed swap i.e. rate for overnight unsecured lending
between banks. An OIS is an interest rate swap in which the periodic floating rate of the swap is equal to
the geometric average of an overnight rate (or overnight index rate) over every day of the payment period.
The LIBOR-OIS Spread is the amount by which the LIBOR rate (which includes credit risk) exceeds the OIS
rate (which includes only minimal credit risk).
LIBOR-OIS Spread = LIBOR Rate - OIS Rate ↷
Roughly reflects the federal funds
rate and minimal counterparty risk
7
LIBOR-OIS Spread is a useful measure of credit risk and an indication of the overall wellbeing of the
banking system i.e. the spread is considered an indicator of the risk and liquidity of money market
securities. A low LIBOR-OIS Spread is a sign of high market liquidity while a high LIBOR-OIS Spread is a sign
that banks are unwilling to lend due to concerns about creditworthiness.
lllll
Term Structure of Interest Rates
lllll
Unbiased Expectations Theory
Under the Unbiased Expectations Theory or the Pure Expectations Theory, we hypothesize that it is
investor's expectations that determine the shape of the interest rate term structure. It says that the
forward rate is an unbiased predictor of the future spot rate and that every maturity strategy has
the same expected return over a given investment horizon. In other words, long-term interest rates
equal the mean of future expected short-term rates. This implies that an investor should earn the
same return by investing in a 5-year bond or by investing in a 3-year bond and then a 2-year bond
after the 3-year bond matures. Similarly, an investor with a 3-year investment horizon would be
indifferent between investing in a 3-year bond or in a 5-year bond that will be sold 2 years prior to
maturity. The underlying principal behind the Pure Expectations Theory is 'Risk Neutrality' i.e.
investors don't demand a risk premium for maturity strategies that differ from their investment
horizon. In a risk-neutral world, investors are unaffected by uncertainty and risk premiums don't
exist. If the yield curve is upward (downward) sloping, short-term rates are expected to rise (fall). A
flat yield curve implies that the market expects short-term rates to remain constant.
lllll
Local Expectations Theory
In Local Expectations Theory, it preserves the risk-neutrality assumption only for short holding
periods. In other words, over longer periods, risk premiums should exist. This implies, that over
short-time periods, every bond every long-maturity risky bonds should earn the risk-free rate. The
Local Expectations Theory can be shown not to hold because the short-holding period returns of
long-maturity bonds are higher than short-holding period returns on short-maturity bonds due to
liquidity premiums and hedging concerns. Thus, both the yields and actual returns for short-dated
securities are typically lower than those for long-dated securities.
lllll
Liquidity Preference Theory
Liquidity Preference Theory of the term structure addresses the shortcomings of the Pure
Expectations Theory by proposing that forward rates reflect investor's expectations of future spot
rates plus a liquidity premium to compensate investors for exposure to interest rate risk. Liquidity
premium is positively related to maturity i.e. a 25-year bond should have a larger liquidity premium
that a 5-year bond. Due to liquidity premium, forward rates derived from the current yield curve
provide an upwardly biased estimate of expected future spot rates. Although downward sloping
may sometimes occur (deflation), the existence of liquidity premium implies yield curve will typically
be upward sloping
lllll
Segmented Markets Theory
Under Segmented Markets Theory, the shape of the yield curve is determined by the preferences of
borrowers and lenders, which drives the balance between supply and demand for loans of different
maturities. Each maturity sector can be thought of as a segmented market in which yield is
determined independently from the yields that prevail in other maturity segments. Bond market
participants are limited to purchase of maturities that match the timing of their liabilities.
lllll
Preferred Habitat Theory
The Preferred Habitat Theory also proposes that forward rates represent expected future spot rates
8
plus a premium, but it doesn't support the view that this premium is directly related to maturity.
However, the theory contends that if the expected additional returns to be gained become large
enough, institutions will be willing to deviate from their preferred maturities or habitats. For
example, if the expected returns on longer-term or (shorter-term) securities exceed those on
short-term or (long-term) securities by large enough margin, money market funds will lengthen or
(shorten) the maturities of their assets as per requirements. The preferred Habitat Theory is based
on the realistic notion that agents and institutions will accept additional risk in return for additional
expected returns. Borrowers require cost savings (i.e. lower yields) and lenders require a yield
premium (i.e. higher yields) to move out of their preferred habitats.
Modern Term Structure Models
Equilibrium Term Structure Models
↷Fundamental Economic Variables
1. Cox Ingersoll Ross Model: is based on the idea that interest rate movements are driven by
individuals choosing between consumption today vs. investing and consuming at a later time. The
CIR model is an instance of a so-called continuous-time finance model. We have assumed that the
term premium of the CIR model is equal to zero. Thus, the CIR model assumes that the company
has a constant long-run interest rate that the short-term interest rate converges to over time.
I
dr = a (b - r) dt + o
This term forces the interest rate
to mean-revert toward the
long-run value 'b' at a speed
determined by the mean reversion
parameter 'a'. If 'a' is high (low)
mean reversion to the long-run
rate 'b' would occur quickly (slowly)
Stochastic or
Random Walk
Volatility increases (o ) with the
interest rate 'r'.
I
Deterministic
or Drift Term
r dz
It also avoids the possibility of
negative interest rates.
Whereas,
dr : Change in the short-term interest rate or
r
dt : Small increase in time or
t
dz : Small random walk movement
a : Speed of mean reversion parameter i.e.
a,
Speed
b : Long-run mean reverting value of the short-term interest rate
r : Short-term interest rate
o : Volatility
△
△
↑ ↑
2. Vasicek Model: is viewed as an equilibrium term structure model. Like the CIR model, the Vasicek
model suggests that interest rates are mean-reverting to some long-run value.
dr = a (b - r) dt + o dz
I
lllll
I
lllll
Deterministic
or Drift Term
Stochastic or
Random Walk
Volatility in this model doesn't increase
as the level of interest rates increase.
It doesn't force interest rates to be
positive.
9
lllll
Arbitrage Free Models
The term structure of interest rates begin with the assumption that bonds trading in the market
↷ are correctly priced and the model is calibrated to match current market prices. The market yield
curve can be modeled accurately for valuing derivatives and bonds with embedded options.
1. Ho-Lee Model: can be calibrated to market data by inferring the form of the time dependent drift
term 'θ t ' from market prices, which means the model can precisely generate the current term
structure. This calibration is typically performed via binomial lattice model; this probability is
called the 'Implied Risk Neutral Probability' which is somewhat misleading because arbitrage-free
models don't assume market professionals are risk neutral as does the Local Expectations
Theory. Since, the model usually generates a symmetrical bell-shaped or normal distribution of
future rates, negative interest rates are possible. Arbitrage-free models don't try to justify the
current yield curve, rather they take this curve as given, for this reason they are sometimes
labeled as 'Partial Equilibrium Model'.
I
dr t = θ t dt + o dz
NOTES
1. The spot rate for a long-maturity security will equal the geometric mean of the one period spot rate and a
series of 1-year forward rate.
2. Short-term interest rates are generally more volatile than are long-term rates. Volatility at the
long-maturity and is thought to be associated with uncertainty regarding the real economy and inflation,
while volatility at the short-maturity and reflects risks regarding monetary policy.
I
3. Interest rate volatility at time 't' for a security with maturity of 'T' is denoted as o (t, T). This variable
measures the annualized standard deviation of the change in bond yield.
I
I
o (t, T) = o [
△ r (t, T) / r (t, T)]
△t
10
CHAPTER 33
The Arbitrage-Free Valuation Framework
The Arbitrage-Free Valuation framework is used extensively in the pricing of securities. The basic principal of
the 'Law of One Price' in freely functioning markets derives this analytical framework. There are two types of
arbitrage opportunities:
Value Additivity: when the value of whole differs from the sum of the values of parts and
Dominance: when one asset trades at a lower price than another asset with identical characteristics.
If the principal of value additivity doesn't hold, arbitrage profits can be earned by stripping or reconstitution.
Portfolio of Strips < Intact Bond : Purchase strips, combine them and
sell them as an intact bond.
(Reconstitution)
Portfolio of Strips > Intact Bond : Purchase intact bond, split them to
strips and sell the strips.
(Stripping)
Example 1:
The following information has been collected:
Security
A
B
C
D
Current Price
$ 99
$ 990
$ 100
$ 100
Payoff in 1 Year
$ 100
$ 1010
$ 102
$ 103
Securities A & B are identical in every respect other than as noted. Similarly, securities C & D are identical in
every other respect. Demonstrate the exploitation of any arbitrage opportunities.
a. Arbitrage due to violation of the value additivity principal.
Short 10 Units of Security A
Long 1 Unit of Security B
Net Cashflow
t=0
990
- 990
0
t=1
- 1000
1010
10
t=0
100
- 100
0
t=1
- 102
103
1
b. Arbitrage due to the occurrence of Dominance.
Short 1 Unit of Security C
Long 1 Unit of Security D
Net Cashflow
lllll
Types of Valuations
lllll
Binomial Model
The Binomial Interest Rate Tree framework assumes that interest rates have equal probability of
taking one of two possible values in the next period.
11
i 2UU
i1U
i 2LU or i 2UL
i0
i 1L
i 2LL
Year 1
Year 3
i 2LU = i 2LL . e 2 o
i 3UUL = i 3LLL . e 4 o
I
I
i 3UUU = i 3LLL . e 6 o
i 2UU = i 2LL . e
I
4o
I
I
i 1U = i 1L . e
Year 2
2o
I
i 3LLU = i 3LLL . e 2 o
e ≈ 2.7183
The binomial interest rate tree framework is a lognormal random walk model with two desirable
properties: (i) Higher volatility at higher rates and (ii) non-negative interest rates.
Example 2:
Ngnyen has asked a colleague Alok Nath, to generate a binomial interest rate tree consistent with this data
and an assumed volatility of 20%. Nath completed a partial interest rate tree shown below:
Maturity
1
2
3
Par Rate
3%
4%
5%
Spot Rate
3%
4.020%
5.069%
I
Binomial tree with o = 20%, 1-year forward rates.
Time 0
3%
Time 1
5.7883%
A
Time 2
B
C
D
Calculate forward rates A, B, C and D.
i 1L = i 1U . e -2 o = (0.057883) . e -2 (0.2) = 0.0388 or 3.88%
C=?
Forward 'C' rate is the middle rate for period 3 and hence the best estimate for that rate is the
1-year forward rate in two years i.e. f (2, 1) or 1 f 2 .
I
A=?
5.069%
0
1
4.020%
2
3
x
12
3
2
(1 + S 3 ) = (1 + S 2 ) (1 + 1 f 2 )
(1.05069)3 = (1.0402) 2 (1 + x)
x = 7.199%
i 2LL = i 2LU . e -2 o = (0.07198) . e -0.4 = 0.0483 or 4.83%
B=?
i 2UU = i 2LU . e 2 o = (0.07198) . e 0.4 = 0.1074 or 10.74%
I
I
D=?
Example 3:
Samuel Faure is interested in valuing the same 3-year 3% annual pay treasury bond. The spot rate curve is as
before, but this time Faure wants to use a binomial interest rate tree with the following rates. Below are the
one-period forward rates in year:
Year 0
3%
Year 1
5.7883%
3.88%
Year 2
10.7383%
7.1981%
4.8250%
93.01 + 3
I UU = 10.733%
103
96.08 + 3
I UL = 7.1981%
103
98.26 + 3
I LL = 4.825%
103
Yr 2
Yr 3
0.
5
Compute the value of the $100 par option-free bond?
5
0.
0.
5
92.21 + 3
IU = 5.7883%
5
0.
0.
5
94.49
I 0 = 53%
96.43 + 3
I L = 3.88%
5
0.
Yr 0
Yr 1
V 2UU = (0.5) (103) + (0.5) (103) = 93.01
1.107383
V 2UL = (0.5) (103) + (0.5) (103) = 96.08
1.071981
V 2LL = (0.5) (103) + (0.5) (103) = 98.26
1.048250
V1U = (0.5) (93.01 + 3) + (0.5) (96.08 + 3) = 92.21
1.057883
V 1L = (0.5) (96.08 + 3) + (0.5) (98.26 + 3) = 96.43
1.0388
13
V0 = (0.5) (92.21 + 3) + (0.5) (96.43 + 3) = 94.485
1.03
lllll
Pathwise Valuation
An alternative approach to backward induction in a binomial tree is called pathwise valuation. For a
binomial interest rate tree with 'n' periods, there will be 2(n-1) unique paths, e.g. for a 3-year period,
there will be 2 (3-1) , 2 2 = 4 Paths.
Example 4:
Paul Smith wants to value the same 3-year, 3% annual-pay treasury bond. The interest rate tree is the same
as before but this time, Smith wants to use a pathwise valuation approach, using one period forward rate in a
year.
Year 0
3%
Year 1
5.7883%
3.88%
Year 2
10.7383%
7.1981%
4.8250%
Compute the value of the $100 par option-free bond.
Path
1
2
3
4
Year 1
3%
3%
3%
3%
Year 2
5.7883%
5.7883%
3.88%
3.88%
Year 3
10.7383%
7.1981%
7.1981%
4.825%
Average
Value
$ 91.03
$ 93.85
$ 95.52
$ 97.55
$ 94.49
Path 1: V1 = 3 +
3
+
103
= $ 91.03
(1.03) (1.03) (1.057883) (1.03) (1.057883) (1.107383)
Path 2: V2 = 3 +
3
+
103
= $ 93.85
(1.03) (1.03) (1.057883) (1.03) (1.057883) (1.071981)
Path 3: V3 = 3 +
3
+
103
= $ 95.52
(1.03) (1.03) (1.0388) (1.03) (1.0388) (1.071981)
Path 4: V4 = 3 +
3
+
103
= $97.55
(1.03) (1.03) (1.0388) (1.03) (1.0388) (1.04825)
lllll
Monte Carlo Simulation
It involves randomly generating a large number of interest rate paths, using a model that
incorporates a volatility assumption and an assumed probability distribution. The simulated paths
should be calibrated, so the benchmark interest rate paths value benchmark securities at their
market price i.e. arbitrage-free valuation. The calibration process entails adding (substracting) a
constant to all rates when the value obtained from the simulated paths is too high (too low) relative
to market prices. This calibration process results in a 'Drift Adjusted Model'. Monte Carlo Simulation
may impose upper and lower bonds on interest rates i.e. based on mean reversion. Increasing the
no. of paths using Monte Carlo method does increase the estimate's statistical accuracy. It doesn't
however provide a value that is closer to the bond's true fundamental value.
Binomial Valuation process is that the value of the cashflows at a given point in time is independent of the
path i.e. cashflows are not path dependent, whereas, under Monte Carlo, cashflows can be path dependent.
14
NOTES
1. After calibrating a binomial interest rate tree and once its accuracy is confirmed, the interest rate tree can
be used to value bonds with embedded options.
2. Potential interest rate volatility in a binomial interest rate tree can be estimated using historical interest
rate volatility or observed market prices from interest rate derivatives.
15
CHAPTER 34
Valuation and Analysis: Bonds with Embedded Options
V CALL = V STRAIGHT - V CALLABLE
V PUT = V PUTABLE - V STRAIGHT
Call Option is valuable when yield curve flattens
Put Option is valuable when yield curve steepens
Call Option's value increases as the yield curve flattens and increases further if the yield curve inverts.
Put Option's value decreases upward sloping to flat to downward sloping i.e. inverted.
Callable Bonds give the issuer the option to call back the bond. Most callable bonds have a lockout period
during which the bond cannot be called. A Make-Whole Call provision is a call provision attached to a bond,
whereby the borrower must make a payment to the lender in an amount equal to the NPV of the coupon
payments that the lender will forgo if the borrower pays the bonds off early i.e. the investor is short the call
option.
Putable Bonds allow the investor to put the bond back to the issuer prior to maturity. A related bond is an
extendible bond, which allows the investor to extend the maturity of the bond i.e. bondholder has the right to
keep the bond for a no. of years after maturity. An Estate Put which includes a provision that allows the heirs
of an investor to put the bond back to the issuer upon the death of the investor. The value of this contingent
put option is inversely related to the investor's life expectancy; the shorter the life expectancy, the higher the
value i.e. the investor is long the underlying put option.
Example 1:
Consider a 3-year 4.25% annual pay $100 per bond option-free, callable or putable at $100 each year.
0.
5
Option-Free Bond
98.791 + 4.25
I UU = 5.5258%
104.25
99.7376 + 4.25
I UL = 4.5242%
104.25
100.526 + 4.25
I LL = 3.7041%
104.25
5
0.
0.
5
99.658 + 4.25
I U = 3.8695%
5
0.
0.
5
102.114
I 0 = 2.5%
100.922 + 4.25
I L = 3.1681%
5
0.
Yr 0
Yr 1
Yr 2
Yr 3
16
98.791 + 4.25
I UU = 5.5258%
104.25
99.7376 + 4.25
I UL = 4.5242%
104.25
100.526 + 4.25
100
I LL = 3.7041%
104.25
99.658 + 4.25
I U = 3.8695%
0.
5
0.
5
Callable Bond
0.
5
0.
100.921 + 4.25
100
I L = 3.1681%
0.
5
5
101.540
I 0 = 2.5%
5
0.
Yr 0
Yr 1
Yr 2
Yr 3
V CALL = V STRAIGHT - VCALLABLE = 102.114 - 101.54 = $ 0.574
Assume that nothing changes relative to the initial setting except that the bond is now callable at 102 instead
of 100. The new value of the callable bond is? Call price is too high for the call option to be exercised in any
scenario. Thus, the value of the call option is '0' and the value of the callable bond is equal to the value of the
straight bond that is 102.114.
0.
5
Putable Bond
98.791 + 4.25
100
I UU = 5.5258%
104.25
99.738 + 4.25
100
I UL = 4.5242%
104.25
100.526 + 4.25
I LL = 3.7041%
104.25
5
0.
0.
5
100.366 + 4.25
I U = 4.25%
5
0.
0.
5
102.522
I 0 = 2.5%
100.3036 + 4.25
I L = 3.1681%
5
0.
Yr 0
Yr 1
Yr 2
Yr 3
V PUT = V PUTABLE - V STRAIGHT = 102.522 - 102.114 = $ 0.408
Assume that nothing changes relative to the initial setting except that the bond is now putable at 95 instead
of 100. The new value of the putable bond is? The put price is too low for the put option to be exercised in
any scenario. Thus, the value of the put option is '0' and the value of the putable bond is equal to the value of
the straight bond that is 102.114.
17
Option Adjusted Spread (OAS) is the constant spread, when added to all one-period forward rates on the
interest rate tree which makes the arbitrage-free value of the bond (calculated value) equal to its market
price. The Z-Spread for an option-free bond is simply its OAS at zero volatility. The OAS is added to the tree
after the adjustment for the embedded option i.e. the mode values are adjusted according to the call/put
rule. Hence the OAS is calculated after the option risk has been removed.
Z-Spread - OAS = Option Cost
↑ Volatility ↓ Volatility
Callable Bond = Z-Spread ≥ OAS =
Putable Bond = Z-Spread ≤ OAS =
Positive
Negative
Negative
Positive
Option Cost
Option Cost
OAS is used by analysts in relative valuation, bonds with similar credit risk should have the same OAS. If the
OAS for a bond is higher than the OAS of its peers, it is considered to be undervalued i.e. attractive
investment meaning it offers a higher compensation for a given level of risk (cheap). Conversely, bonds with
low OAS relative to peers are considered to be overvalued (rich) and should be avoided.
Example 2:
A $100 par, 3-year 6% annual pay ABC Inc. callable bond trades at $99.95. The underlying call option is a
Bermudan-Style option that can be exercised in one or two years at par. The benchmark interest rate tree
assuming volatility of 20% is provided below.
One-Period Forward Rates
Year 0
Year 1
Year 2
3%
5.7883%
10.7383%
3.88%
7.1981%
4.825%
Compute the OAS on the bond.
101.5 + 6
100
I L = 3.88%
0.
5
5
5
0.
106
98.88 + 6
I UL = 7.1981%
106
101.12 + 6
100
I LL = 4.825%
106
97.65 + 6
I U = 5.7883%
101.77
I 0 = 3%
5
0.
Yr 0
95.72 + 6
I UU = 10.7383%
0.
0.
5
0.
5
Callable Bond
Yr 1
Yr 2
Yr 3
18
94.86 + 6
I UU = 10.7383%+ 1%
106
97.97 + 6
I UL = 7.1981% + 1%
106
100.17 + 6
100
I LL = 4.825% + 1%
106
Yr 2
Yr 3
95.91 + 6
I U = 5.7883% +1%
0.
5
0.
5
To force the computed value to be equal to the current market price of $ 99.95, a constant spread (OAS) of
100 Bps or 1% (OAS computed here i.e. 1% is largely an iterative process and beyond the scope) is added to
each interest rate in the tree below.
5
0.
0.
5
99.95
I 0 = 3% + 1%
5
0.
100.1 + 6
100
I L = 3.88% + 1%
5
0.
Yr 0
Yr 1
A Floating-Rate Bond (floater) pays a coupon that adjusts every period based on an underlying reference rate.
The coupon is typically paid in arrears, meaning the coupon rate is determined at the beginning of a period
but is paid at the end of that period.
Capped Floater: The cap provision in a floater prevents the coupon rate from increasing above a specified
maximum rate. (This option favors the issuer)
Floored Floater: The floor provision in a floater prevents the coupon rate from decreasing below a specified
minimum rate. (This option favors the bondholder or lender)
VCF = Value of a Straight Floater - Value of the Embedded Cap
V FF = Value of a Straight Floater + Value of the Embedded Floor
As with the valuation of a bond with embedded options, we must adjust the value of the floater at each node
to reflect the exercise of an in-the-money option, in this case, a cap or floor. Variable-Rate Bond (floater) are
not much sensitive to interest rate change. At each coupon reset date, value of floater = par value.
Example 3:
Susane Albright works as a fixed income analyst with Zedone Banks. She has been asked to value a $100 par.
2-year floating rate note that pays LIBOR (set in arrears). The underlying bond has the same credit quality as
reflected in the LIBOR swap curve.
One-Period Forward Rate
Year 0
Year 1
4.5749%
7.1826%
5.3210%
19
98.9 + 4.57
I U = 7.1826%
0.
5
a. Compute the V CF and Value of the Embedded Cap, assuming it is capped at a rate of 6%.
100 + 7.18
6
99.47
I 0 = 4.5749%
5
0.
Yr 0
100 + 4.57
I L = 5.321%
Yr 1
100 + 5.32
Yr 2
Value of the Embedded Cap = Par - V CF
= 100 - 99.47
= 0.53
0.
5
b. Compute the V FF and Value of the Embedded Floor, assuming it is floored at a rate of 5%.
100 + 4.57
5
I U = 7.1826%
100 + 7.18
100 + 4.57
5
I L = 5.321%
100 + 5.32
100.41
I 0 = 4.5749%
5
0.
Yr 0
Yr 1
Yr 2
Value of the Embedded Cap = V FF - Par
= 100.41 - 100
= 0.41
Example 4:
Peter has been asked to value a $100 par, 3-year floating rate note that pays LIBOR (set in arrears).
One-Period Forward Rates
Year 0
Year 1
Year 2
2.5%
3.8695%
5.5258%
3.1681%
4.5242%
3.7041%
a. Compute the V CF and Value of the Embedded Cap, assuming it is capped at a rate of 4.5%.
99.028 + 3.8695
I UU = 5.5258%
100 + 5.5258
4.5
99.521 + 2.5
I U = 3.8695%
0.
5
0.
5
20
0.
5
0.
0.
5
5
99.761
I 0 = 2.5%
99.977 + 3.8695
99.977 + 3.1681
I UL = 4.5242%
100 + 4.5242
4.5
99.989 + 2.5
I L = 3.1681%
5
0.
Yr 0
Yr 1
100 + 3.1681
I LL = 3.7041%
Yr 2
100 + 3.7041
Yr 3
Value of Embedded Cap = Par - V CF
= 100 - 99.761
= 0.239
100 + 3.8695
I UU = 5.5258%
100 + 5.5258
100 + 3.8695
100 + 3.1681
3.5
I UL = 4.5242%
100 + 4.5242
100 + 3.1681
3.5
I LL = 3.7041%
100 + 3.7041
0.
5
0.
5
b. Compute the V FF and Value of the Embedded Floor, assuming it is floored at a rate of 5%.
5
0.
100 + 2.5
3.5
I U = 3.8695%
0.
5
101.133
I 0 = 2.5%
5
0.
100.322 + 2.5
3.5
I L = 3.1681%
5
0.
Yr 0
Yr 1
Yr 2
Yr 3
Value of Embedded Floor = V FF - Par
= 101.133 - 100
= 1.133
lllll
Duration
The duration of a bond measures the sensitivity of the bond's full price including accrued interest to
changes in the bond's yield to maturity (YTM) (in case of yield duration) or to changes in benchmark
interest rates (in case of yield curve duration).
21
lllll
Modified Duration
Modified Duration is a linear estimate of the relation between a bond's price and YTM, whereas the
actual return is convex, not linear i.e. modified duration measure provides good estimate of bond
prices for small changes in yield, but increasingly poor estimates for larger changes due to the
curvature for option-free bonds. Modified Duration and Convexity are not useful for bonds with
embedded options.
lllll
Effective Duration
Effective Duration measures price sensitivity to small 100 Bps parallel shifts in the yield curve i.e.
benchmark yield curve, assuming no change in the bond's credit spread, but it is not an accurate
measure of interest rate sensitivity to non-parallel shifts in the yield curve like those described by
'Shaping Risk'. Shaping Risk refers to changes in portfolio value due to changes in the shape of the
benchmark yield curve. However, parallel shifts explain more than 75% of the variation in bond
portfolio returns.
Effective Duration =
V - - V+
2 x V0 x ( Curve)
△
Whereas,
V - : Estimated price if yield decreases by Y
V + : Estimated price if yield increases by Y
V 0 : Initial observed bond price
Curve : Change in required yield or Y
△
△
△
△
In-the-Money
When the embedded option (call or put) is deep in the money, the effective duration of the bond
with an embedded option resembles that of the straight bond maturing on the first exercise date,
reflecting the fact that the bond is highly likely to be called or put on that date.
Out-of-Money
Both call and put option have the potential to reduce the life of a bond:
Effective Duration Callable ≤ Effective Duration Straight
Effective Duration Putable ≤ Effective Duration Straight
Effective Duration ZCB ≈ Maturity of the Bond
Effective Duration Fixed Rate Coupon < Maturity of the Bond
Effective Duration Floater ≈ Time in Years to Next Reset
At-the-Money
The effective duration of the callable bond shortens when interest rate falls, which is when the call
option moves into the money, limiting the price appreciation of the callable bond. The effective
duration of the putable bond shortens when interest rates rise, which is when the put option moves
into the money, limiting the price depreciation of the putable bond. While effective duration of
straight bonds is relatively unaffected by changes in interest rates.
Duration
Longer Maturity
Coupon
YTM
Callable/Putable
↑
↑
↑
↓
↓
↓
22
Fig 1: Convexity
Callable
Putable
Price
Price
Call Option
Value
Option-Free Bond
(Non-Callable Bond)
(Over-Positivity Convexity)
Putable Bond
Callable Bond
0
Negative
Convexity
Option-Free Bond
(Non-Putable Bond)
y1
Positive
Convexity
[Option-Free Bond]
Yield
Call Option value is inversely related to the level of
interest rates.
YTM
↓
↑
Bond
(Smaller)
0
Put Option value varies directly with the level of
interest rates.
YTM
YTM
Bond
(Larger)
Bond
(Larger)
YTM
↓
Effective convexity of the callable bond turns
negative when the call option is near the money
which indicates that the upside for a callable bond
is much smaller than the downside. When rates are
high, callable bonds are unlikely to be called and
will exhibit positive convexity.
Yield
Positive
Convexity
↑
↓
Put Option
Value
↑
↑
↓
Bond
(Smaller)
Putable bonds always have positive convexity.
When the option is near the money, the upside for
a putable bond is much larger than the downside
because the price of a putable bond is floored by
the price of the put option, if it is near the exercise
date.
When interest rates decline, Putable Bonds have more upside potential than otherwise identical
Callable Bonds. When interest rates rise, Putable Bonds also have less downside risk than otherwise
identical Callable Bonds. Straight Bonds have positive convexity i.e. the increase in the value of an
Option-Free Bond is higher when rates fall than the decrease in value when rates increase by an
equal amount.
Convexity of Callable Bond < Convexity of Option-Free Bond
(In-the-Money) Convexity of Callable Bond < Convexity of Putable Bond (Out-of-Money)
Duration is an approximation of the expected bond price responses to changes in interest rates
because actual changes in bond prices are not linear, particularly for bonds with embedded
options. Thus, it is useful to measure 'Effective Convexity'.
Effective Convexity = V - + V + - 2. V 0
V 0 x ( Curve) 2
△
A longer maturity, a lower coupon rate or a lower YTM, will all increase convexity and vice versa. For
two bonds with equal duration, the one with cashflows that are more dispersed over time will have
a greater convexity.
23
Example 5:
Bond X, 3%, 15-Year Option-Free Bond
Bond Y, 3%, 15-Year Callable Bond
Bond Z, 3%, 15-Year Putable Bond
Which bond is most likely to experience an increase in effective duration due to an increase in interest rates?
A. Bond X
Duration
When interest rates , Option-free bond
B. Bond Y
When interest rates , Callable bond becomes less likely to be called
C. Bond Z
✓
↑
↑
When interest rates ↓, Putable bond becomes likely to be exercised
lllll
↑
↓
↓
One-Sided Duration
The One-Sided Durations are better at capturing the interest rate sensitivity of a callable or putable
bond than the 'two-sided' effective duration, particularly when the embedded option is near the
money. This metric captures interest rate risk reasonably well for small changes in the yield curve
and for option-free bonds.
It is more sensitive to interest rate
declines than to interest rate rises.
Putable Bond = One-Sided Down Duration > One-Sided Up Duration
Option-Free Bond =
lllll
At-the-Money or
Near-the-Money
↷
Callable Bond = One-Sided Down Duration < One-Sided Up Duration
↶
It is more sensitive to interest rate
rises than to interest rate declines.
↑ One-Sided Down Duration and ↑ One-Sided Up Duration
Key Rate Duration
The impact of non-parallel shifts can be measured using Key-Rate Duration i.e. is defined as the
sensitivity of the value of a bond or portfolio to small changes in the spot rate for a specific
maturity, holding other spot rates constant. Numerically, Key Rate Duration is defined as the
approximate percentage change in the value of a bond portfolio in response to a 100 Bps change in
the corresponding key rate, holding all other rates constant.
An alternative to decomposing yield curve risk into sensitivity to changes at various maturities (key
rate duration) is to decompose the risk into sensitivity to the following 3 categories of yield curve
movements:
Level ( x L ): A parallel increase or decrease of interest rates
Steepness ( x S ): A non-parallel shift in the yield curve when either short-term rates change
more than long-term rates or long-term rates change more than short-term rates.
Curvature ( x C ): It is a reference to movement in three segments of the yield curve, the
short-term and long-term segments rise while the middle term segment falls or vice versa.
It has been found that all yield curve movements can be described using a combination of one or
more of these movements.
△
△
△
Consider a portfolio of 1-Year, 5-Year and 10-Year ZCB with $100 value in each position; total
portfolio value is therefore $300. Also consider the hypothetical set of factor movements shown in
the following table: The rates change by 100 Bps.
Parallel
Steepness
Curvature
Year 1
1
-1
1
Year 5
0
0
0
Year 10
1
1
1
Effective Duration = 1 ( 1 + 5 + 10) = 5.333
Weighted Sum of
3
↶
Effective Duration of
each bond position
24
Key Rate Duration =
△ P = - D 1 . △ r1 - D 5 . △ r 5 - D 10 . △ r 10
P
D1 =
1
= 0.333
300 (0.01)
D5 =
5
= 1.667
300 (0.01)
D 10 =
10
= 3.333
300 (0.01)
∴ Sum of these Durations = 0.333 + 1.667 + 3.333 = 5.333
△ P = - 0.333 . △ r1 - 1.667 . △ r5 - 3.333 . △ r 10
P
In this case, Key Rate Durations = Effective Duration of the portfolio; Key Rate Duration measures
the portfolio risk exposure to each key rate. If all the key rates move by the same amount, then the
yield curve has made a parallel shift and as a result, the proportional change in value has to be
consistent with Effective Duration.
Here's the Decomposed Version
△ P = - D L . △ x L - D S . △ x S - DC . △ x C
P
Whereas,
D L : D1 + D 5 + D 10
Portfolio's sensitivities to
changes in yield curve's level,
D S : - D1 + D 10
steepness and curvature.
D C : D1 + D 10
= 0.333 + 1.667 + 3.333
= - 0.333 + 3.333
= 0.333 + 3.333
△ P = - 5.333 . △ x L - 3 . △ x S - 3.667 . △ x C
P
16
- (1 - 10)
1 + 10
300 (0.01) 300 (0.01) 300 (0.01)
△ x L = - 0.005, △ x S = 0.002 and △ x C = 0.001
Therefore, △ P = - 5.333 (- 0.005) - 3 (0.002) - 3.666 (0.001) = 0.01699 or 1.7%
If
P
The following generalizations can be made about key rates:
- If an option-free bond is trading at par, the bond's maturity-matched rate is the only rate that
affects the bond's value. Its maturity key rate duration is the same as its effective duration and all
other rate durations are zero.
- For an option-free bond not trading at par, the maturity-matched rate is still the most important
rate, because the largest cashflow of a fixed-rate bond occur at maturity with the payment of both
the final coupon and the principal.
- A bond with a low or zero coupon rate may have negative key rate durations for horizons other
than the bond's maturity.
- Higher coupon bonds are more likely to be called and lower coupon bonds are more likely to be
put and therefore the time-to-exercise rate will tend to dominate the time-to-maturity.
- Shifting a par rate up or down at a particular maturity point, however, respectively increases or
decreases the discount rate at that maturity point.
Key Rate Durations and a measure based on sensitivities to level, slope and curvature movements can
address shaping risk but Effective Duration cannot.
25
Example 6:
Factor
Level
Steepness
Curvature
Time to Maturity
5-Year
10-Year
- 0.4352%
- 0.5128%
- 0.0515%
- 0.3015%
0.3963%
0.5227%
a. Calculate the expected change in yield on the 20-year bond resulting from a 2 standard deviation increase
in the steepness factor. The yield on the 20-year bond is decreasing by:
A. 0.3015%
B. 0.6030%
Change in 20-year bond yield = - 0.3015% x 2 = - 0.6030%
C. 0.8946%
✓
b. Calculate the expected change in yield on the 5-year bond resulting from a 1 standard deviation decrease
in the level factor and 1 standard deviation decrease in the curvature factor. The yield on the 5-year bond
is?
A. Decreasing by 0.8315%
Change in 5-year bond yield (Level) = - 0.4352% x - 1 = 0.4352%
B. Decreasing by 0.0389%
Change in 5-year bond yield (Curvature) = 0.3963% x - 1 = - 0.3963%
C. Increasing by 0.0389%
Total = 0.4352% - 0.3963% = 0.0389%
✓
lllll
Convertible Bonds
The owner of a convertible bond has the right to convert the bond into a fixed no. of common shares of
the issuer during a specified timeframe (conversion period) and at a fixed amount of money
(conversion price). Convertible bond includes a conversion option, which is a call option on the issuer's
common stock. Investors usually accept a lower coupon for convertible bonds than for otherwise
identical non-convertible bonds because they can participate in the potential upside on the issuer's
stock, although this comes at a cost of lower yield. The issuer of a convertible bond benefits from a
lower borrowing cost but existing shareholders may face dilution if the conversion option is exercised.
In case of conversion, an added benefit for the issuer is that it no longer has to repay the debt that was
converted into equity. If not converted, the issuer must repay the debt that was converted into equity.
If not converted, the issuer must repay the debt or refinance it potentially at a higher cost and the
bondholders will have lost interest income relative to an otherwise identical non-convertible bond.
Conversion Price = Issue Price or Par Value
Conversion Ratio
Change-of-Control events are defined in the prospectus or offering circular and if such an event occurs,
convertible bondholders usually have the choice between (a lower conversion price may be specified in
the event of control):
- A put option that can be exercised during a specified period following the change-of-control event
and that provides full redemption of the nominal value of the bond; or
- An adjusted conversion price that is lower than the initial conversion price. This downward
adjustment gives the convertible bondholders the opportunity to convert their bonds into shares
earlier and at more advantageous terms, and thus allow them to participate in the announced
merger or acquisition as common shareholders.
Options embedded in a convertible bond could be: (a) Put Options can be hard puts (redeemable for
cash) or soft puts (the issuer decides whether to redeem the bond for cash, stock, subordinated
debentures or a combination of the three) and (b) The issuer has an incentive to Call the bond when
the underlying share price increases above the conversion price in order to avoid paying further
26
coupons. Such an event is called 'Forced Conversion' because it forces bondholders to convert their
bonds into shares. The forced conversion strengthens the issuer's capital structure and eliminates the
risk that a subsequent correction in equity prices prevents conversion and require redeeming the
convertible bonds at maturity.
Conversion Value = Market Price of Stock x Conversion Ratio
Minimum Value of Convertible Bond = Max (Straight Value, Conversion Value)
or Floor Value
Non-Convertible or Option-Free Underlying
↑ Interest Rate, ↓ Straight Value = Floor Fall
This must be the case, or arbitrage opportunities would be possible. If a convertible bond were to sell
for less than its conversion value, investors will find the convertible bond attractive, buy it and push its
price until the convertible bond price returns to the straight value and the arbitrage opportunity
disappears i.e. the convertible bond price will increase until it reaches the conversion value and the
arbitrage opportunity disappears.
Market Conversion Premium or Discount = Market Conversion Price - Stock's Market Price
Per Share
Market Conversion Price = Convertible Bond Price
(Conversion Parity Price)
Conversion Ratio
Once the Underlying Market Price > Market Conversion Price, any further rise in the underlying market
price is certain to increase the value of the convertible bond by atleast the same percentage.
Market Conversion Premium Ratio = Market Conversion Premium Per Share
Stock's Market Price
As the underlying share price falls, the convertible bond price will not fall below the straight value.
There is a fundamental difference, however between the buyers of a call option and the buyers of a
convertible bond. The former (before) know exactly the amount of the downside risk, whereas the
letter (after) know only that the most they can loose is the difference between the convertible bond
price and the straight value because the straight value is not fixed. This downside risk is measured by
the 'Premium over Straight Value', which is calculates as:
Premium over Straight Value = Convertible Bond Price - 1
(PSV)
Straight Value
The greater the Premium over Straight Value (PSV), the less attractive the convertible bond, PSV is a
flawed measure of downside risk because straight value is not fixed but rather, fluctuates with changes
in interest rates and credit spreads.
Example 7:
At the end of trading on 2nd September 2015, the market conversion price for Soyang bonds was $5025 and
its common shares closed at $2985. The next day, Soyang, which had never previously paid a dividend to
common shareholders, paid a dividend of $400 per share. The dividend paid to Soyang's common
shareholders will most likely affect the bond's:
A. Issuer Call Price
The cash dividend is higher than the threshold dividend, which will typically result in a
reduction in the conversion price, which results in an increase in the conversion ratio.
B. Conversion Ratio
C. Change of Control Conversion Price
✓
27
Example 8:
Structured Data for DE Convertible Bond:
Issue Price
Initial Conversion Price
Threshold Dividend
Change of Control Conversion Price
Common Stock Share Price on Issue Date
Share Price on 17th September 2015
Convertible Bond Price on 17th September 2015
$
1000
10
0.5
8
8.7
9.1
1123
a. 'I expect that Delille Enterprises will soon announce a common stock dividend of $0.7 per share'. If Delille
Enterprises pays the dividend expected by Gilette, the conversion price of the DE Bond will? Be adjusted
downward.
b. What is the Market Conversion Premium Per Share for the DE Bond on 17th September 2015?
Market Conversion Price = Convertible Bond Price = 1123 = $ 11.23
Conversion Ratio
1000 / 10
Market Conversion Premium Per Share = Market Conversion Price - Underlying Share Price
= 11.23 - 9.1
= $ 2.13
Fig 2: Convertible Bond Vs. Straight Bond
Conversion Premium
Unlimited Upside
Potential
Reduced Upside
Potential
Price Floor
x
x
x
Downside Protection
(Limits Downside Risk)
Buying Convertible bonds instead of stocks, limits downside risk; the price floor set by the
straight bond value provides this downside protection. The cost of the downside protection
is reduced upside potential due to the conversion premium. Convertible bond investors
must be concerned with credit risk, interest rate risk and liquidity risk.
Contingent Convertible Bonds (CoCos) pay a higher coupon than otherwise identical non-convertible
bonds, whereas, Convertible Contingent Convertible Bonds (CoCoCos) combine a traditional convertible
bond and a CoCo. They are convertible at the discretion of the investor, thus offering upside potential if
the share price increases, but they are also converted into equity or face principal write-downs in the
event of a regulatory capital breach.
28
Value of Convertible Bond = Value of Straight Bond
+ Value of Call Option on Stock
Callable Convertible Bond Value = Value of Straight Bond
+ Value of Call Option on Stock
- Value of Call Option on Bond
Putable Convertible Bond Value = Value of Straight Bond
+ Value of Call Option on Stock
+ Value of Put Option on Bond
Callable and Putable Convertible Bond Value = Value of Straight Bond
+ Value of Call Option on Stock
- Value of Call Option on Bond
+ Value of Put Option on Bond
Fig 3: Price Behavior of a Convertible Bond and the Underlying Common Stock
Price
Conversion
Price
[CP]
0
Convertible
Bond Price
[CBP]
The call option component
increases significantly in
value as the SP approaches
the CP. Any further increase
in SP is certain to increase
the value of the convertible
bond by atleast the same
percentage.
Share Price
[SP]
A
B
C
D
Bond Equivalent or
Busted Convertible or
Fixed Income Equivalent
CBP behavior is similar to
straight bond price
behavior. When underlying
SP is well below CP, the
convertible bond is called
'Busted Convertible'.
Hybrid I
As SP increases towards CP,
CBP increases but at a
lower rate than SP. Returns
converge when CP is
exceeded. When the SP
rises, the convertible bond
will underperform because
of the conversion premium.
Hybrid II
As SP decreases towards
CP, CBP decreases but at a
lower rate than SP because
it has a floor in the value of
the straight bond. When the
SP falls, the returns on
convertible bonds exceed
those of the stock because
the CBP has a floor equal to
its straight bond.
Stock Equivalent
CBP behavior is similar to
underlying SP behavior.
When SP is above the CP,
the convertible bond
behaves as though it is an
equity security.
1. Call option is
out-of-money.
2. SP movements don't
significantly affect the
price of the call option
and CBP.
3. Interest rate movements
and credit spreads
significantly affect the
CBP.
1. Call option is
in-the-money.
2. SP movements
significantly affect the
price of the call option
and CBP.
3. CBP is mostly unaffected
by interest rate
movements and credit
spreads.
If the SP remains stable like the CP, the return on a convertible bond may exceed the stock return due to the coupon
payments received from the bond, assuming no change in interest rates or credit risk of the issuer.
Time
29
NOTES
1. (a) Check whether Put-Call Parity holds:
C - P = Pv (Forward Price of Bond on Exercise Date) - Pv (Exercise Price)
(b) Check and make sure that the valuation of option-free bonds should be independent of the assumed
level of volatility used to generate the interest rate tree.
(c) Check that the volatility term structure slopes downward. In order for the interest rate process to be
stable, the implied volatilities should decline as the term lengthens.
2. Call Vs. Put
When interest rate volatility
increases, the values of both call
and put options increase. The
value of a straight bond is
affected by changes in the level
of interest rates but is unaffected
by changes in the volatility of
interest rates. The greater the
volatility, the more opportunities
exist for the embedded option to
be exercised and wider the
dispersion in interest rates.
Price of Underlying Asset
Exercise Price
Risk-Free Rate
Volatility of the Underlying
Time to Expiration
Costs of Holding Asset
Benefits of Holding Asset
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
CALL
↑
↓
↓
↑
↑
↓
↑
↓
↑
↓
↑
↓
↓
↑
PUT
↓
↑
↑
↓
↓
↑
↑
↓
↑
↓
↓
↑
↑
↓
30
CHAPTER 35
Credit Analysis Models
The terms 'Default Risk' and 'Credit Risk' may be used interchangeably. Default Risk is the narrower term
because it addresses the likelihood of an event of default. Credit Risk is the broader term because it
considers default risk and how much is expected to be lost if default occurs.
Expected Exposure is the amount of money a bond investor in a credit risky bond stands to loose at a point
in time before any recovery is factored in. A bond's expected exposure changes over time.
Recovery Rate is the percentage recovered in the event of a default,
Recovery Rate % = 1 - Loss Severity %
Recovery $ = Recovery Rate % x Exposure
Loss Severity or Loss Given Default (LGD) is the amount of loss if a default occurs.
Loss Severity % = 1 - Recovery Rate %
Loss Severity $ = Loss Severity % x Exposure
Probability of Default (PD) is the likelihood of default occurring in a given year. One reason for the
difference between actual or historical PD and risk-neutral PD is that actual PD don't include the default risk
premium associated with uncertainty over the timing of possible default loss. Another reason being the
observed spread over the yield on a risk-free bond in practice also includes liquidity and tax considerations in
addition to credit risk. Under the risk-neutral option pricing methodology, the expected value for the payoffs
is discounted using the risk-free interest rate. The initial probability of default is also known as the 'Hazard
Rate'. The probability of default in each subsequent year is calculated as the conditional probability of default
given that default has previously not occurred.
PD t = Hazard Rate x PS (t-1)
Probability of Survival (PS) is 1 - Cumulative Conditional Probability of Default.
PS t = (1 - Hazard Rate) t
Expected Loss is Conditional Probability of Default (PD) x Loss Given Default (LGD).
Credit Valuation Adjustment (CVA) is the sum of the Pv of the expected loss for each period. CVA is the
monetary value of the credit risk in present value terms i.e. is the difference in value between a risk-free bond
and an otherwise identical risky bond.
CVA = Price of R f Bond - Price of Risky Bond
The estimated risk-neutral probabilities of default and recovery rates are positively correlated
(PD , Recovery Rate ). The higher the expected loss, the higher the credit risk. Doubling the default
probability has a greater impact on the credit spread than halving the recovery rates.
↑
↑
Example 1: [No Coupon]
Consider a 1-Year ZCB, $100 par bond trading at $95. The benchmark 1-Year is 3% and recovery rate is
31
assumed to be 60%. Calculate the risk-neutral probability of default, which is the probability of default
implied in the current market price?
95
Default
'p'
Let's assume 'p' as probability of default,
No Default
'(1 - p)'
60
95 = 60 p + 100 (1 - p)
1.03
100
97.85 = 100 - 40 p
p = 5.38%
If we assumed a probability of default greater than 5.38%, then the implied recovery rate would be higher.
Example 2: [Coupon]
Consider a 1-Year 4% annual payment corporate bond priced at par value $100 and the assumed recovery
rate is 40%. Calculate the implied risk neutral probability, risk-free rate being 3%?
100
Default
'p'
41.6
Let's assume 'p' as probability of default,
No Default
'(1 - p)'
100 = 41.6 p + 104 (1 - p)
1.03
104
100 = 104 - 62.4 p
p = 1.6%
Example 3:
If the annual hazard rate for a bond is 1.25%, the probability of default (PD) in year 1 is?
A. Less than 1.25%
The hazard rate is the PD in the 1st year.
B. Equal to 1.25%
PD < Hazard rate in all years after the 1st year.
C. Greater than 1.25%
✓
Example 4:
A fixed income analyst is considering the credit risk over the next year for three corporate bonds currently
held in her bond portfolio.
$100 Par
Bond
Exposure
Recovery
PD
A
B
C
104
98
92
40
35
30
0.75 %
0.90 %
0.80 %
Although all three bonds have every similar YTM, the differences in the exposures arise because of
differences in their coupon rates. On a relative basis, which bond has the highest risk?
Expected Loss = LGD x PD
A
0.48 = 64 X 0.75%
B
0.567 = 63 x 0.9%
C
0.496 = 62 x 0.8%
Based on expected losses, Bond B has highest credit risk and Bond A the lowest. The ranking is B, C and A.
32
Example 5: [No Coupon]
A 3-Year, $100 par ZCB has a hazard rate of 2% per year. Its recovery rate is 60% and the benchmark rate
curve is flat at 3%. Calculate the CVA and IRRs (when bond defaults and when it doesn't).
A
B
C
D
Year
Exposure
LGD
PS
PD
1
2
3
94.26
97.087
100
37.704
38.835
40
98%
96.04%
94.119%
2%
1.96%
1.9208%
Exposure
Year 1: 100 / (1.03) 2 = 94.26
Year 2: 100 / (1.03) 1 = 97.087
Year 3: 100
E
F
Expected Loss x
0.754077
0.761165
0.76832
DP
0.970874
0.942596
0.915142
CVA
DF
1 / (1.03) = 0.970874
1 / (1.03) 2 = 0.942596
1/ (1.03) 3 = 0.915142
= Pv of Expected Loss
0.732113
0.717471
0.703122
2.152706
Recovery
94.26 x 0.6 = 56.556
97.087 x 0.6 = 58.2522
100 x 0.6 = 60
1
CVA = Price of R f Bond - Price of Risky Bond
2.15 = (100 / 1.03 3 ) - Price of Risky Bond
2.15 = 91.5 - Price of Risky Bond
Price of Risky Bond = 91.5 - 2.15
= $ 89.36
We not calculate the credit spread on the corporate bond,
100 = 89.36
(1 + YTM) 3
YTM = 3.82%
Credit Spread = YTM - R f = 3.82% - 3% = 0.82%
or
Approximation Credit Spread = Hazard Rate (1 - Recovery Rate) = 0.02 (1 - 0.6) = 0.8%
A key point is that the compensation for credit risk received by the investor can be expressed in two ways:
(a) as the CVA of 2.15 in terms of a present value per 100 of per value on date '0' and (b) as a credit spread of
82 Bps in terms of an annual percentage rate for 3 years.
Yr 0
89.36
Default
Yr 1
56.556
89.36 = 56.556
(1 + IRR) 1
Yr 2
Yr 3
No Default
IRR 1 = - 36.7099%
0
Default
No Default
58.2522
0
89.36 =
0
+ 58.25
Default
(1 + IRR)1 (1 + IRR) 2
IRR 2 = - 19.26%
60
No Default
100
89.36 =
0
+
0
+ 60
(1 + IRR)1 (1 + IRR) 2 (1 + IRR) 3
89.36 =
0
+
0
+ 100
(1 + IRR)1 (1 + IRR) 2 (1 + IRR) 3
IRR 3 = - 12.43%
IRR 3 = 3.816%
↷
If the bond doesn't default over its life,
the investor would earn 3.816% IRR.
33
Example 6: [Coupon]
A fixed income trader at a hedge fund observes a 3-Year, 5% annual payment corporate bond trading at 104
per 100 of par value. The research team at the hedge fund determines that the risk-neutral probability of
default used calculate the conditional PD for each data for the bond 1.5% given a recovery rate 40%. The
government bond yield curve is flat at 2.5%. Based on these assumptions, does the trader deem the
corporate bond to be overvalued or undervalued?
A
B
C
D
E
F
Year
Exposure
LGD
PS
PD
Expected Loss
DP
Pv of Expected Loss
1
2
3
109.8186
107.439
105
0.9884
0.9524
0.9169
0.975610
0.951814
0.928599
CVA
0.9643
0.9066
0.8514
2.7222
65.8911
98.5%
1.5%
64.4634 97.0225% 1.4775%
63
95.5672% 1.4553%
Exposure
Year 1: 5 + 5 / (1.025) 1 + 105 / (1.025) 2 = 109.8186
Year 2: 5 + 105 / (1.025)1 = 107.439
Year 3: 105
DF
1/
= 0.970874
1 / (1.025) 2 = 0.951814
1 / (1.025) 3 = 0.928599
(1.025) 1
Recovery
109.8186 x 0.4 = 43.9274
107.439 x 0.4 = 42.9756
105 x 0.4 = 42
CVA = Price of R f Bond - Price of Risky Bond
2.7222 = [5 / (1.025)1 + 5 / (1.025) 2 + 5 / (1.025) 3 ] - Price of Risky Bond
2.7222 = 107.1401 - Price of Risky Bond
Price of Risky Bond = 107.1401 - 2.7222
= $ 104.4178
We not calculate the credit spread on the corporate bond,
5
+
5
+
105
= 104.4178
(1 + YTM) 1 (1 + YTM) 2 (1 + YTM) 3
YTM = 3.57%
Credit Spread = YTM - R f = 3.57% - 2.5% = 1.07%
The fixed income trader at the hedge fund would deem this corporate bond to be undervalued by 0.4178 per
100 of par value, if it is trading at a price of 104.
Yr 0
104
Default
Yr 1
43.9274
104 = 43.9274
(1 + IRR) 1
Yr 2
IRR 1 = - 57.93%
104 =
Yr 3
No Default
5
Default
No Default
42.9756
5
+ 42.9756
(1 + IRR) 1 (1 + IRR) 2
Default
IRR 2 = - 33.27%
104 =
5
42
5
+
5
+
42
(1 + IRR)1 (1 + IRR) 2 (1 + IRR) 3
IRR 3 = - 22.23%
No Default
105
104 =
5
+
5
+ 105
(1 + IRR)1 (1 + IRR) 2 (1 + IRR) 3
IRR 3 = 3.57%
↷
If the bond doesn't default over its life,
the investor would earn 3.57% IRR.
34
lllll
Structural Model
Structural Models a.k.a. Company Value Model, of corporate credit risk are based on the structure
of a company's balance sheet and rely on insights provided by option pricing theory.
↶
A (T) = D (T) + E (T)
or K (Face Value of Debt)
Call: E (T) = Max [A (T) - K, 0]
D (T) = A (T) - Max [A (T) - K, 0]
Equity is essentially a purchase call option on the assets of the company,
whereby the strike price is the face value of the debt.
If A (T) > K: Call Option In-the-Money
Then, E (T) = A (T) - K
D (T) = A (T) - [A (T) - K] = K
If A (T) < K: Call Option Out-of-Money
Then, E (T) = 0
D (T) = A (T) - 0 = A (T) i.e. Debtholders own the
remaining assets
Put: E (T) = A (T) - K + Max [K - A (T), 0]
D (T) = K - Max [K - A (T), 0]
Equity investors as long the net assets of the company [A (T) - K] and long
a put option allowing them to sell the assets at an exercise price of K.
If A (T) > K: Put Option Out-of-Money
Then, E (T) = A (T) - K + 0 = A (T) - K
D (T) = K - 0 = K
If A (T) < K: Put Option In-the-Money
Then, E (T) = A (T) - K + [K - A (T)] = 0
D (T) = K - [K - A (T)] = A (T) i.e. Debtholders own the
remaining assets
Value of Call Option = Max [A (T) - K, 0]
Value of Put Option = Max [K - A (T), 0]
Value of Risky Debt = Value of R f Debt - Value of Put Option
or
Value of Risky Debt = Value of R f Debt - CVA
∴ Value of Put Option = CVA
Default
Barrier
Probability of
Default
K
A (T)
If the value of assets falls below the Default Barrier 'K',
the company defaults. This default probability increases
with the variance of the future asset value, with greater
time-to-maturity and with greater financial leverage. Less
debt in the capital structure lowers the 'K-Line' and
reduces the probability of default. These factors indicate
that credit risk is linked to option pricing theory.
Disadvantages
1. Because structural models assume a simple balance sheet structure, complex balance sheets
cannot be modeled. Additionally, when companies have off-balance sheet debt, the default
barrier under structure models 'K' would be inaccurate and hence the estimated outputs of the
model will be inaccurate.
35
2. One of key assumptions of the structural models is that the assets of the company are traded in
the market. This restrictive assumption makes the structural model impractical.
lllll
Reduced Form Model
Unlike the structural model, Reduced Form Models don't explain why default occurs. Instead, they
statistically model when default occurs. This is known as the 'Default Intensity' or 'Default Time' and
can be modeled using a Poisson Stochastic Process. The key parameter in this process is the
'Default Intensity', which is the probability of default over the next small time period. Reduced Form
credit risk models are thus also called 'Intensity-Based' and 'Stochastic Default Rate' models.
Default Intensity can be estimated using regression models that employ several independent
variables including company specific variables as well as macro-economic variables. Reduced-Form
Models have the advantage that the inputs are observable variables including historical data.
Default Intensity is allowed to vary as company fundamentals change, as well as when the state of
the economy changes.
Disadvantages
1. Reduced Form Models don't explain why the economic reasons for default.
2. Here, default is treated as a random event i.e. surprise, but in reality, default is rarely a surprise
(it is often preceded by several downgrades).
Fig 2: Difference between Structured Vs. Reduced Form Model
Structured
1. Default
→ Endogenous (Internal Event)
Reduced Form
1. Default
→ Exogenous (External Event)
2. Explains 'why' default occurs.
[A (T) < K]
2. Doesn't explain 'why', but only explains
'when'.
3. Defaults are explained.
3. Defaults are random surprises.
4. Assumes assets are traded.
4. Doesn't assume.
5. Probability Default based on options pricing.
5. Calculate 'Default Intensity' using regression
models.
Credit Spread = YTM of Risky Bond - YTM of Benchmark
The value of a risky bond, assuming it doesn't default, is its 'Value given No Default' (VND). VND is calculated
using the risk-free rate to value the risky bond.
Fair Value of the Risky Bond = VND - CVA
Example 7:
Jack Gordon, a fixed income analyst for Omega Bank Plc. is evaluating a AA corporate bond for inclusion in
the bank's portfolio. The $100 par 3.5% annual-pay, 5-Year bond is currently priced with a credit spread of
135 Bps over the benchmark per rate of 2%. Calculate the bond's CVA implied in its market price?
VND:
n=5
PMT = 3.5
I/Y = 2%
Fv = 100
Pv = - 107.07
36
Value of Risky Bond:
n=5
PMT = 3.5
I/Y = 2% + 1.35%
Fv = 100
Pv = - 100.68
CVA = VND - Value of Risky Bond
= 107.07 - 100.68
= $ 6.39
Example 8:
For a 3-Year annual pay 4% coupon, $100 par corporate bond using interest rate tree. Calculate VND for the
bond and the expected exposure for each year.
93.92
IUU = 10.7383%
104
97.02
I UL = 7.1981%
104
99.21
I LL = 4.825%
104
94.02
I U = 5.7883%
VND = 97.24
I 0 = 3%
98.3
I L = 3.88%
Exposure: (Here the exposure will be the average)
Year 1: (94.02 + 4) + (98.3 + 4) = $ 100.16
2
Year 2: (93.92 + 4) + (97.02 + 4) + (97.02 + 4) + (99.21 + 4) = $ 100.79
4
Year 3: 104
Changes in the fair value of a corporate bond arising from a change in the assumed rate volatility occur only
when there are embedded options.
A benchmark yield should be equal to the risk-free rate plus expected inflation as well as risk premium for
uncertainty in future inflation. Credit spreads include compensation for default, liquidity and taxation risks
relative to the benchmark. Adjustment to the price for all these risk factors together is known as the XVA.
Example 9:
Joan De Silva, a junior analyst works for a regional bank. Currently, De Silva is evaluating a 3-Year annual pay
3% XYZ corporate bond priced at $102. The benchmark yield curve is flat at 1.75%. De Silva assumes hazard
rate of 1.25% and a recovery rate of 70% and calculates CVA of 1.12. Now, Collins states that based on the
expectations of a slowdown in the economy, a 1.5% hazard rate and a recovery rate of 60% would be more
37
appropriate for XYZ and calculate CVA of 1.79.
a. Using De Silva's estimates of hazard rate and recovery rate XYZ bond is currently most likely?
b. Using the market price of the bond, the credit spread on XYZ bond is closest to?
c. Assuming the market price changes to reflect Collin's expectations of PD and recovery rate, the new credit
spread would be closest to?
a. n = 3
PMT = 3
I/Y = 1.75%
Fv = 100
Pv = - 103.62
Value of Risky Bond = VND - CVA = 103.62 - 1.12 = 102.5
The market price of $ 102 for the bond implies that the bond is undervalued.
b. n = 3
PMT = 3
Pv = - 102
Fv = 100
I/Y = 2.3%
Credit Spread = YTM - R f = 2.3% - 1.75% = 0.55%
c. Based on the revised PD and recovery rate, CVA = 1.79
Value of Risky Bond = VND - CVA = 103.62 - 1.79 = 101.83
n=3
PMT = 3
Pv = - 101.83
Fv = 100
I/Y = 2.36%
Credit Spread = YTM - R f = 2.36% - 1.75% = 0.61%
Credit Scoring is used for small businesses and individuals. A higher credit score indicates better credit
quality. FICO i.e. Fair Isaac Corporation is a well known example of a credit scoring model used in the US.
FICO scores are computed using data from consumer credit files collected by 3 national credit bureaus:
Experian, Equifax and Transunion. Five primary factors are included in the proprietary algorithm used to get
the score, 35% for the payment history, 30% for the debt burden, 15% for the length of credit history, 10% for
the types of credit used and 10% for recent searches for credit. FICO scores are higher for those with
(a) Longer credit histories i.e. age of oldest account, (b) Absence of delinquencies i.e. outright defaults,
(c) Lower Utilization i.e. outstanding balances divided by available line, (d) Fewer credit inquiries and (e) Wider
variety of types of credit used.
Credit Rating are issued for corporate debt, asset-backed securities and government and quasi-government
debt. Similar to credit score, credit ratings are ordinal ratings i.e. higher the better. Three major global credit
rating agencies are Moody's Investor Service, S&Ps and Fitch Ratings. The issuer rating for a company is
typically for its senior unsecured debt. 'Notching' is the practice of lowering the rating by one or more levels
for more subordinate debt of the issuer. Notching accounts for LGD differences between different classes of
debt by the same issuer i.e. higher LGD for issues with lower seniority.
Credit Migration, bond portfolio managers often want to evaluate the performance of a bond in the event of
credit migration i.e. change in rating. A change in a credit rating generally reflects a change in the bond's
credit risk.
△ % P = - Modified Duration x △ Spread
Example 10:
Suppose a bond with a modified duration of 6.32 gets downgraded from AAA to AA. The typical AAA credit
38
spread is 60 Bps, while the typical AA credit spread is 87 Bps. Calculate the percentage change in price of the
bond assuming that the bond is priced at typical spreads.
△ % P = - 6.32 x (87 Bps - 60 Bps) = - 0.0171 or - 1.71%
Example 11:
Charles uses a 10-Year A-rated corporate bond that has a modified duration of 7.2 at the end of the year.
Calculate the expected percentage price change as the product of the modified duration and the change in
the spread?
From/To (%)
AAA
AA
A
BBB
BB
B
CCC
AAA
AA
A
BBB
BB
B
CCC
Credit Spread
90
1.5
0.05
0.02
0.01
0.6%
9
88
2.5
0.3
0.06
0.05
0.01
0.9%
0.6
9.5
87.5
4.8
0.3
0.15
0.12
1.1%
0.15
0.75
8.4
85.5
7.75
1.4
0.87
1.5%
0.1
0.15
0.75
6.95
79.5
9.15
1.65
3.4%
0.1
0.05
0.6
1.75
8.75
76.6
18.5
6.5%
0.05
0.03
0.12
0.45
2.38
8.45
49.25
9.5%
From A to AAA: - 7.2 (0.6% - 1.1%) = 3.6%
From A to AA: - 7.2 (0.9% - 1.1%) = 1.44%
From A to A: - 7.2 (1.1% - 1.1%) = 0%
From A to BBB: - 7.2 (1.5% - 1.1%) = - 2.88%
From A to BB: - 7.2 (3.4% - 1.1%) = - 16.56%
From A to B: - 7.2 (6.5% - 1.1%) = - 38.88%
From A to CCC: - 7.2 (9.5% - 1.1%) = - 60.48%
(0.0005 x 3.6%) + (0.025 x 1.44%) + (0.875 x 0%) + (0.084 x - 2.88%) +
(0.0075 x - 16.56%) + (0.006 x - 38.88%) + (0.0012 x - 60.48%) = - 0.6342%
Therefore, the expected return on the bond over the next year is its yield to maturity minus 0.6342% i.e.
Expected Return 1 = YTM - 0.6342%, assuming no default. Credit spread migration typically reduces the
expected return for two reasons: First, the probabilities for change are not symmetrically distributed around
the current rating. They are skewed towards a downgrade rather than an upgrade. Second, the increase in
the credit spread is much larger for downgrades than the decrease in the spread for upgrades.
Example 12:
Crownwell selects a 10-Year maturity 5% coupon bond currently trading at par and rated A+. He asks Thames
to calculate the expected total return over a 1-Year horizon, assuming the bond is downgraded by 2 notches
and to determine why the year-end duration for the bond is 6.9.
S&P Rating
AAA+
A
A-
G-Spread
0.7
0.85
1
1.1
39
Expected Return = - Duration (New Spread - Initial Spread) + Coupon
= - 6.9 (1.1% - 0.85%) + 5%
= 3.275%
The credit spread is inversely related to the recovery rate and positively related to the probability of default. If
default probabilities are expected to be higher or recovery rates lower in the future, the credit curve would be
expected to be positively sloped or upward sloping. Empirical studies also tend to support the view that the
credit spread term structure is upward sloping for investment grade bond portfolio i.e. corporates. Flat credit
curves indicate stable expenditures over time. When an issuer refinances a near-dated bond with a longerterm bond, the spread may appear to narrow for the longer maturity, possibly leading to an inverted credit
spread curve. Issuer or industry-specific factors such as the chance of a future leverage decreasing event, can
cause the credit spread curve to flatten or invert.
When a bond is very likely to default, it often trades close to its recovery value at various maturities. Now
assuming recovery in a bankruptcy scenario and cross default provisions across maturities, the CVA
representing the sum of expected losses is simply the difference between the VND and the recovery rate; we
now end up with a steep and inverted credit spread curve.
Key determinants of the slope of the credit curve spread include expectations about (a) Future Recovery
Rates and Default Probabilities, (b) Credit Quality, (c) Financial Conditions, (d) Market Demand and Supply and
(e) Equity Market Volatility.
Example 13:
Mossimo Gulzar has complied data on spreads of AAA corporate bonds. Using a typical 4% coupon AAA bond
for each maturity category, Mossimo would conclude the credit spread curve to be upward sloping,
downward sloping or flat?
Benchmark Par Rate
AAA CVA
VND
Year 1:
Year 3:
Year 5:
Year 10:
Year 1
0.75%
$ 1.79
Year 3
1.25%
$ 2.12
Year 5
1.75%
$ 3.24
Year 10
2.25%
$ 5.84
Year 1
Year 3
Year 5
Year 10
Benchmark Par Rate
0.75%
1.25%
1.75%
2.25%
VND
103.23
108.05
110.68
115.52
Value of Risky Bond
(VND - CVA)
101.44
105.93
107.44
109.68
YTM
2.53%
1.95%
2.4%
2.88%
Credit Spread
(YTM - Benchmark Rate)
1.78%
0.7%
0.65%
0.63%
n=1
n=3
n=5
n = 10
PMT = 4
PMT = 4
PMT = 4
PMT = 4
I/Y = 0.75%
I/Y = 1.25%
I/Y = 1.75%
I/Y = 2.25%
Fv = 100
Fv = 100
Fv = 100
Fv = 100
Pv = - 103.23
Pv = - 108.05
Pv = - 110.68
Pv = - 115.52
40
YTM
Year 1:
Year 3:
Year 5:
Year 10:
n=1
n=3
n=5
n = 10
PMT = 4
PMT = 4
PMT = 4
PMT = 4
Pv = - 101.44
Pv = - 105.93
Pv = - 107.44
Pv = - 109.68
Fv = 100
Fv = 100
Fv = 100
Fv = 100
I/Y = 2.53%
I/Y = 1.95%
I/Y = 2.4%
I/Y = 2.88%
Components of Credit Analysis of Secured Debt
1. Collateral Pool: Homogeneity of a pool refers to the similarity of the assets within the collateral pool.
Granularity refers to the transparency of assets within the pool. A more discrete pool of a few loans would
warrant examination of each obligation separately. Short-term granular and homogeneous structured
financed vehicles are evaluated using a 'Statistical-based Approach'. Medium-term granular and
homogenous obligations are evaluated using a 'Portfolio-based Approach' because the portfolio
composition varies over time. Discrete and non-granular heterogenous portfolios, a 'Loan-by-Loan
Approach' or at the individual loan level to credit analysis is more appropriate.
2. Servicer Quality: is important to evaluate the ability of the servicer to manage the origination and servicing
of the collateral pool. After origination, investors in secured debt face the operational and counterparty risk
of the servicer.
3. Structure: examples of internal credit enhancements include tranching of credit risk among classes with
differing seniority i.e. distribution waterfall, overcollateralization and excess servicing spread. Third party
guarantees like bank, insurance companies or loan originators are an example of external credit
enhancements. A special structure is the case of a 'Covered Bond'; issued by a financial institution or bank,
covered bonds are senior secured bonds backed by a collateral pool as well as by the issuer i.e. covered
bonds have recourse rights (a recourse is a legal agreement which gives the lender the right to pledge
collateral, if the borrower is unable to satisfy the debt obligation i.e. also known as 'Legal Right to Collect').
Under the European Union Bank Recovery and Resolution Directive (BRRD), covered bonds enjoy unique
protection among bank liabilities in the event of restructuring or regulatory intervention. As a result, rating
agencies often assign a credit rating to covered bonds that are several notches above that of the issuing
financial institution.
41
CHAPTER 36
Credit Default Swaps
Credit Default Swap (CDS) is a contract between two parties in which one party purchases protection from
another party against losses from the default of a borrower. The protection seller is assuming (i.e. long) credit
risk, while the protection buyer is (i.e. short) credit risk. CDS doesn't provide protection against market-wide
interest rate risk, only against credit risk. For a protection buyer, a CDS has some of the characteristics of a
put option - when the underlying performs poorly, the holder of the put option has the right to exercise the
option. Consider that any bondholder is a buyer of credit and interest rate risk. If the bondholder wants only
credit risk, it can obtain it by selling a CDS, which would require far less capital and incur potentially lower
overall transaction costs than buying the bond. Moreover, the CDS can easily be more liquid than the bond.
1. Single-Name CDS
A CDS on one specific borrower is called a 'Single-Name CDS'. The borrower is called the 'Reference Entity'
and the contract specifies a 'Reference Obligation', a particular debt instrument issued by the borrower
that is the designated instrument (i.e. senior unsecured obligation) being covered. Reference obligation
(assuming third party) is not the only instrument covered by the CDS, any debt obligation issued by the
borrower that is 'Pari Passu' (ranked equivalently in priority of claims) or higher relative to the reference
obligation is covered. The CDS payoff is based on the market value of the 'Cheapest-to-Deliver' (CTD) bond
that has the same seniority as the reference obligation, that can be purchased and delivered at the lowest
cost.
2. Index CDS
Index CDS involves a combination of borrowers, allowing market participants to take on an exposure to the
credit risk of several companies simultaneously in the same way that stock indexes allow investors to take
on equity exposure to several companies at once. In this case, the protection for each issuer is equal
(i.e. equally weighted) and the total notional principal is the sum of the protection on all the issuers. The
pricing of an index CDS is dependent on the correlation of default called 'Credit Correlation'. The more
correlated the defaults, the more costly it is to purchase protection for a combination of the companies
i.e. the higher the correlation of default among index constituents, the higher the spread on the index CDS.
In contrast, for a diverse combination of companies, whose defaults have low correlations, it will be much
less expensive to purchase protection.
3. Tranche CDS
Tranche CDS covers a combination of borrowers but only up to pre-specified levels of losses like the ABS.
Common credit events include bankruptcy, failure to pay and restructuring.
42
Example 1: [Single-Name CDS]
Party X is a protection buyer in a $10,000,000 notional principal senior CDS of Alpha Inc. There is a credit
event i.e. Alpha defaults and the market prices of Alpha's bond after the credit event are as follows:
Cheapest among the two
is 'Q'. Criteria:
- Better quality bond
- Select low% value
Bond P, a subordinated unsecured debenture is trading at 15% of par.
Bond Q, a 5-Year senior unsecured debenture is trading at 25% of par.
Bond R, a 3-Year senior unsecured debenture is trading at 30% of par.
What will be the payoff on the CDS?
Payoff = Notional Principal - Market Value
= 10,000,000 - (0.25) (10,000,000)
= $ 7,500,000
Example 2: [Index CDS]
Party X is a protection buyer in a 5-Year $100,000,000 notional principal CDS for CDX-IG, which contains 125
entities. One of the index constituents, company A, defaults and its bond trade at 30% of par after default.
What will be the payoff on the CDS?
Notional Principal attributable to Entity A = 100,000,000 = $ 800,000
125
Payoff to Party X = 800,000 - (0.3) (800,000)
= $ 560,000
Post the default event, the remainder of the CDS continues with a notional principal of $ 99,200,000.
The International Swaps and Derivatives Association (ISDA), the unofficial governing body of the industry,
publishes standardized contract terms and conventions is facilitate smooth functioning of the CDS market. A
15-member group of the ISDA is called the 'Determinations Committee' (DC) declares when a credit event has
occurred. Each region of the world has a DC, which consists of 10 CDS dealer banks and 5 non-bank end
users. A supermajority vote of atleast 12 members is required for a credit event to be declared. Parties to
CDS contracts generally agree that their contracts will conform to ISDA specifications i.e. parties sign a ISDA
master agreement before any transactions are made. When there is a credit event, the swap will be settled in
cash or by physical delivery (after the credit event).
Physical Settlement
Cash Settlement
Reference Obligation
[Bond or Loan]
Par Value
Par Value - Market Value
Payout Amount = Payout Ratio x Notional Principal
↶
Payout Ratio = 1 - Recovery Rate
Expected Credit Loss
43
Example 3:
A French company files for bankruptcy, triggering various CDS contracts. It has two series of senior bonds
outstanding: Bond A trades at 30% of par and Bond B trades at 40% of par. Investor X owns $10,000,000 of
Bond A and owns $10,000,000 of CDS protection. Investor Y owns $10,000,000 of Bond B and owns
$10,000,000 of CDS protection.
a. Determine the recovery rate for both CDS contracts.
Bond A is the Cheapest-to-Deliver obligation, trading at 30% of par, so the recovery rate for both CDS
contrasts is 30%.
b. Explain whether Investor X would prefer to cash settle or physically settle.
Investor X Cash Settlement: 7000,000 [10,000,000 - (0.3) (10,000,000)] and sell her bond for 3000,000, for
total proceeds of $10,000,000.
Physical Settlement: She can physically deliver her entire $10,000,000 face amount of bonds to the
counterparty in exchange for $10,000,000 in cash.
c. Explain whether Investor Y would prefer to cash settle or physically settle.
Investor Y Cash Settlement: 7000,000 [10,000,000 - (0.3) (10,000,000)] and sell her bond for 4000,000, for
total proceeds of $11,000,000.
Physical Settlement: He can physically deliver his entire $10,000,000 face amount of bonds to the
counterparty in exchange for $10,000,000 in cash.
Therefore, Investor X has no preference between cash settlement and physical settlement, but Investor Y
would prefer a cash settlement because he owns Bond B, which is worth more than the Cheapest-to-Delivery
obligation e.g. 4000,000 greater than 3000,000 and 11,000,000 greater than 10,000,000.
The factors that influence the pricing of CDS i.e. CDS spread include the Probability of Default (PD), Loss Given
Default (LGD) and coupon rate on the swap.
Example 4:
Assume that a company's hazard rate is a constant 8% per year or 2% per quarter. An investor sells 5-Year
CDS protection on the company with the premiums paid quarterly over the next 5 years.
a. What is the probability of survival for the first quarter? 100% - 2% = 98%
b. What is the conditional probability of survival for the second quarter?
We call the probability of default given that it has not already occurred the conditional probability of
default or hazard rate. The conditional probability of survival for the second quarter is also 98% because
the hazard rate is constant at 2%.
c. What is the probability of survival through the second quarter? (0.98) (0.98) = 96.04%
Standardization in the market has led to a fixed coupon on CDS products: 1% for investment grade securities
and 5% for high yield securities. Clearly, not all investment grade companies have equivalent credit risk and
44
not all high yield companies have equivalent credit risk. In effect, standard rate may be too high or too low.
This discrepancy is accounted for by an upfront payment, commonly called the Upfront Premium.
Protection Leg: which is the contingent payment that the credit protection
seller may have to make to the credit protection buyer.
Premium Leg: which is the series of payments the credit protection buyer
promises to make to the credit protection seller.
Upfront Payment = Pv (Protection Leg) - Pv (Premium Leg)
If +ve: Protection buyer pays the Protection seller
If -ve: Protection seller pays the Protection buyer
Valuation at
Inception
Upfront Premium (%) ≈ (CDS Spread - CDS Coupon) x CDS Duration
It is the compensation for
bearing the credit risk of
the reference obligation.
If CDS Coupon < CDS Spread : +ve
If CDS Coupon > CDS Spread : - ve
Protection buyer pays Protection seller
Protection seller pays Protection buyer
Price of CDS (per $100 notional) ≈ $100 - Upfront Premium (%)
Example 5:
Imagine an investor sold 5-Year protection on an investment grade company and has to pay a 2% upfront
premium to the buyer of protection. Assume the duration of the CDS to be 4 years. What are the company's
credit spreads and the price of the CDS per 100 par?
Upfront Premium (%) ≈ (CDS Spread - CDS Coupon) x CDS Duration
- 50 Bps = - 0.02 ≈ (CDS Spread - CDS Coupon)
4
Price of CDS (per $100 notional) ≈ 100 - (- 2) = 102
Example 6:
Which of the following statements about hazard rate is most accurate? Hazard rate:
Hazard Rate,
Expected Value of Payoffs made
A. is the PD given that default has already occurred in a previous period.
by the protection seller upon default: Protection
B. affects both the premium leg as well as the protection leg in a CDS.
Leg. Hazard rate also effects the premium leg
C. is higher for higher LGD.
because once default occurs, the CDS ceases to
↑
✓
↑
exist and premium income would also cease.
Valuation after
Inception
Profit for Protection Buyer ≈ Change in Spread x CDS Duration x NP
Profit for Protection Buyer (%) ≈ Change in Spread (%) x CDS Duration
or Change in CDS Price (%)
If Credit Quality
If Credit Quality
↑ : Protection seller benefits
↓ : Protection buyer benefits
45
The terminology related to CDS is counterintuitive: the protection buyer is the short party (short the credit
risk of the reference asset and short the CDS), while the protection seller is the long party (long the CDS and
long the credit risk of the reference asset).
As the company's credit quality changes through time, the market value of the CDS changes, giving rise to
gains and losses for the CDS counterparties. The counterparties can realize those gains and losses by
entering into new offsetting contracts, effectively selling their CDS positions to other parties. The protection
buyer (or seller) can unwind an existing CDS exposure (prior to expiration or default) by entering to an
offsetting transaction and might either capture a gain or realize a loss. This process of capturing value from
an in-the-money CDS exposure is called Monetizing a gain or loss. For example, suppose that the buyer of
the original CDS wants to unwind his position; he would then enter into the new CDS as a protection seller
and receive the newly calculated upfront premium. As we noted, this value is less than what he paid
originally. Likewise, the seller could offset his original position by entering into this new CDS as a protection
buyer. He would pay an upfront premium that is less than what he originally received. The original protection
buyer monetizes a loss and the seller monetizes a gain. (Strategy 1)
The credit spread can be expressed roughly as the probability of default multiplied by the loss given default,
with the latter in terms of percentage. The credit spreads for a range of maturities of a company's debt make
up its Credit Curve. If the longer maturity bonds have a higher credit spread compared to shorter maturity
bonds i.e. implies a greater likelihood of default in later years will be upward sloping credit curve. Whereas, if
the longer maturity bonds have a lower credit spread compared to shorter maturity bonds i.e. implies near
term stress in the financial markets will be downward sloping credit curve. A constant hazard rate will tend to
flatten the credit curve.
In a Naked CDS, an investor with no underlying exposure purchases protection in the CDS market. Some say
naked CDS bring liquidity to the credit market, potentially providing more stability. Naked CDS is banned in
Europe for sovereign debt. (Strategy 2)
In a Long/Short Trade, an investor purchases protection on one reference entity, while simultaneously
selling protection on another (often related) reference entity. The investor is betting that the difference in
credit spreads between the two reference entities will change to the investor's advantage. This is similar to
the Monetizing. (Strategy 3)
Curve Trade involves buying a CDS of one maturity and selling a CDS on the same reference entity with a
different maturity. For example, if the investor expects that an upwards sloping credit curve on a specific
corporate issuer will flatten, she may take the position of protection buyer in a short maturity CDS and the
position of protection seller in a long maturity CDS. (Strategy 4)
Short-Term
Long-Term
IF
Strategy
High CR
Buy Protection
Low CR
Sell Protection
IF
Strategy
Low CR
Sell Protection
High CR
Buy Protection
CR: Credit Risk
A Basis Trade is an attempt to exploit the difference in credit spreads between bond markets and the CDS
market. For example, if a specific bond is trading at a credit spread of 4% over LIBOR in the bond market but
the CDS spread on the same bond is 3%, a trader can profit by buying the bond and taking the protection
buyer position in the CDS market. If the expected convergence occurs, the trader will make a profit of 1%
differential in the two markets.
46
Example 7:
Dean Advisor's chief credit analyst recently reported that Tollunt Corporation's 5-Year bond is currently
yielding 7% and a comparable CDS contract has a credit spread of 4.25%. Since, LIBOR is 2.5%, Watt has
recommended executing a basis trade to take advantage of the pricing of the Tollunt's Bonds and CDS. The
basis trade would consist of purchasing both the bond and the CDS contract. If convergence occurs in the
bond and CDS markets, the trade will capture a profit closest to?
A. 0.25%
If the spread is higher in the bond market than the CDS market, it is
said to be a negative basis. The credit risk is cheap in the CDS market.
B. 1.75%
Bond Market - CDS Market = (7% - 2.5%) - 4.25% = 4.5% - 4.25% = 0.25%
C. 2.75%
✓
NOTES
1. In a Leveraged Buyout (LBO), the firm issues a great amount of debt in order to repurchase all of the
company's traded equity. This additional debt will increase the CDS spread because the default is now
more likely. An investor who anticipates an LBO, might purchase both the stock and CDS protection, both
of which will increase in value when the LBO eventually occurs. On the other hand, a synthetic
collateralized Debt Obligation or Synthetic CDO is created by combining a portfolio of default free
securities with a combination of CDS undertaken as protection sellers, because they effectively contain
securities subject to default. If a synthetic CDO can be created at a cost lower than that of the cash CDO,
investors can buy the synthetic CDO and sell the cash CDO, engaging in a profitable arbitrage.
Alternative
Investments
PAGE NOS. 37
CHAPTER 39
VOL. 10
1
CHAPTERS. 4
Private Real Estate Investments
Debt
Equity
Private
(Direct)
Mortgage
(Lending money against
real estate)
Direct Investments such as
sole ownership, joint
ventures, partnerships &
other forms of commingled
funds. E.g. Commingled Real
Estate Fund (CREF)
Public
(Indirect)
Mortgage-Backed Securities
(Residential & Commercial)
Shares of Real Estate
Investment Trust (REIT) and
Real Estate Operating
Company (REOC)
A debt investor is a lender that owns a mortgage or mortgage securities. Usually, the mortgage is
collateralized (secured) by the underlying real estate. An equity investor has an ownership interest in real
estate or securities of an entity that owns real estate. Equity investors control decisions such as borrowing
money, property management and the exit strategy. In this case, the lender has a superior claim over an
equity investor in the event of default. Since the lender must be repaid first, the value of an equity investor's
interest is equal to the value of the property less the outstanding debt.
Private real estate investments are usually larger than public investments because real estate is indivisible
and illiquid. Public real estate investment allow the property to remain undivided while allowing investors
divided ownership. As a result, public real estate investments are more liquid and enable investors to
diversify by participating in more properties.
- Long-term horizon & lower liquidity: Debt & Equity, Endowments, Pension Funds
- Short-term horizon & higher liquidity: REITs
Private real estate investment requires property management expertise on the part of the owner or a
property management company. In the case of a REIT or REOC, the real estate is professionally managed,
thus, investors need no property management expertise.
Equity investors usually require a higher rate of return than mortgage lenders because of higher risk. As
previously discussed, lenders have a superior claim in the event of default. As financial leverage increases,
return requirements of both lenders and equity investors increasing as a result of higher risk.
Lenders expect to receive returns from promised cashflows and don't participate in the appreciation of the
underlying property. Equity investors expect to receive an income stream as a result of renting the property
and the appreciation of value over time.
2
Real estate, especially private equity investment is less than perfectly correlated with the returns of stocks
and bonds. Thus, adding private real estate investment to a portfolio can reduce risk relative to the expected
return.
Fig 1: Real Estate Characteristics
Stock/Bond
Real Estate
1
Heterogeneity
Similar
Not Similar
2
High Unit Value
Lower
Higher
3
Active Management
4
Transaction Costs
5
Depreciation &
Desirability
6
Cost & Availability of
Debt Capital
7
8
Not Necessary
Necessary
(Specially for Private RE)
Low
High
Low &
High
High &
Low
Not so Sensitive
Sensitive
Liquid
Illiquid
Easy
Difficult
Lack of Liquidity
Difficulty in
Determining Price
Buildings wear
out over time
Debt
(Scarce)
↓
Interest
Rate
↑
(a) Residential: Single-Family
(b) Commercial: Warehouse, Multi-Family, Office Spaces
Capital Appreciation (c) Farmland & Timberland
Rental Income
Market Value: the most probable sales price a typical investor is willing to pay.
Investment Value: the value or worth that considers a particular investor's
motivations, and how well the property fits into the investors portfolio,
investor's risk tolerance, the investor's tax circumstances and so on.
Value In Use: the value to a particular user such as a manufacturer that is
using the property as a part of its business.
Assessed Value: that is used by a taxing authority.
Mortgage Lending Value: It is for purposes of valuing collateral; lenders may
ask for more conservative value i.e. mortgage lending value.
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Real Estate Valuation Approaches
lllll
Income Approach
The value is based on the expected rate of return required by a buyer to invest in the subject
property. With the income approach, value is equal to the present value of the expected future
income from the property, including proceeds from resale at the end of a typical investment
holding period. The income approach is most useful in commercial real estate transactions.
Real
Estate
↓
3
Highest and Best Use
The highest and best use of a vacant site is the use that would result in the highest value for the
land. The implied land value is equal to the value of the property once construction is completed
less the cost of construction and lease out. Note that the highest and best use is not based on the
highest value after construction, but rather, the highest implied land value.
Example 1:
An investor is considering a site to build either an apartment building or a shopping center. Once
construction is complete, the apartment building would have an estimated value of $50,000,000 and the
shopping center would have an estimated value of $40,000,000. Construction costs including developer profit
are estimated at $45,000,000 for the apartment building and $34,000,000 for the shopping center. Calculate
the highest and best use of the site.
Value when Completed
- Construction Costs
Implied Land Value
Apartment
$50,000,000
$45,000,000
$5000,000
Shopping Center
$40,000,000
$34,000,000
$6000,000
Therefore, the shopping center is the highest and best use for the site because 6000,000 > 5000,000.
Direct Capitalization Method
The direct capitalization method estimates the value of an income - producing property based on
the level and quality of its net operating income.
Rental Income at Full Occupancy
+ Other Income (e.g. parking)
= Potential Gross Income
- Vacancy and Collection Loss
= Effective Gross Income
- Operating Expense*
Net Operating Income
x
x
x
(x)
x
(x)
x
*Operating expenses include property taxes, insurance, maintenance, utilities,
repairs and insurance but before deducting any costs associated with financing
and before deducting federal income taxes. It simply means that NOI is a
before-tax unleveraged measure of income.
Example 2:
A 50-unit apartment building rents for $1000 per unit per month. It currently has 45 units rented. Operating
expenses, including property taxes, insurance, maintenance and advertising are typically 40% of effective
gross income. The property manager is paid 10% of effective gross income. Other income from parking and
laundry is expected to average $500 per rented unit per year. Calculate the NOI.
Rental Income at Full Occupancy (1000 x 50 x 12)
+ Other Income (500 x 50)
= Potential Gross Income
- Vacancy Loss (62,500 x 10% x 5/50)
= Effective Gross Income
- Property Management (562,500 x 10%)
- Other Operating Expense (562,500 x 40%)
Net Operating Income
600,000
25,000
625,000
(62,500)
562,500
(56,250)
(225,000)
281,250
4
* Direct Capitalization Method capitalizes the current NOI at a rate known as the capitalization rate.
The cap and discount rates are closely linked but are not the same. Briefly, the discount rate is the
return required from an investment i.e. the risk-free rate plus a risk premium. The cap rate is lower
the discount rate because it is calculated using the current NOI. The cap rate is applied to first year
NOI, and the discount rate is applied to first year and future NOI. Properties with greater income
growth potential have higher ratios of price to current NOI and thus lower cap rates.
Cap Rate = NOI 1 or
NOI 1
Value
Comparable Sales Price
ARY = Rent 1 or
Rent 1
Value
Comparable Sales Price
Whereas,
Cap Rate : Discount Rate - Growth Rate = r - g; Since cap rate is
based on first year NOI, it is sometimes called the
going-in cap rate.
ARY : All Risks Yield.
1. If rent will be level in the foreseeable future (like a perpetuity)
Cap Rate = Discount Rate
ARY = IRR or YTM
2. If investor expects growth in both NOI and Value
Cap Rate ≠ Discount Rate
3. If rents are expected to increase after every rent review
Cap Rate < Discount Rate
4. If rents are expected to increase at a constant compound rate
Cap Rate + Growth = Discount Rate
Example 3:
The Royal Oaks office building has annual net operating income of $130,000. A similar office building with net
operating income of $200,000 recently sold for $2,500,000. Using the direct capitalization method, the market
value of Royal Oaks is closest to?
Cap Rate =
NOI
= 200,000 = 0.08
Comparable Sales Price 2,500,000
Value =
NOI = 130,000 = 1,625,000
Cap Rate
0.08
Example 4:
Suppose you have collected the information in the table below for 4 comparable companies:
Property
A
B
C
D
NOI
$200,000
$220,000
$250,000
$230,000
Mv (Selling Price)
$2,250,000
$2000,000
$2,500,000
x
5
Using the market extraction method in conjunction with an average capitalization rate, the market value for
Property D is estimated to be closest to?
C A = 200,000 = 8.89%
2,250,000
C B = 220,000 = 11%
2000,000
C C = 250,000 = 10%
2,500,000
Estimated Cap Rate = 8.89% + 11% + 10% = 9.96%
3
Mv D = 230,000 = $ 2,309,236.95
0.0996
* If NOI is not representative of the NOI of similar properties because of a temporary issue, the
subject property's NOI should be stabilized. For example, suppose a property is temporarily
experiencing high vacancy during a major renovation. In this case, the first year NOI should be
stabilized; NOI should be calculated as if the renovation is complete. A cap rate will be used from
properties that are not being purchased before the renovation is complete, a slightly lower price
will be paid because the purchaser has to wait for the renovation to be complete to get the higher
NOI. Applying the cap rate to the lower NOI that is occurring during the renovation will understate
the value of the property because it implicitly assumes that the lower NOI is expected to continue.
Example 5:
On January 1 of this year, renovation began on a shopping center. This year, NOI is forecasted at $6000,000.
Absent renovations, NOI would have been $10,000,000. After this year, NOI is expected to increase 4%
annually. Assuming all renovations are completed by the seller at their expense, estimate the value of the
shopping center as of the beginning of this year assuming investors require a 12% rate of return.
Value = Stabilized NOI = 10,000,000 = $ 125,000,000
Cap Rate
12% - 4%
The present value of the temporary decline in NOI during renovation is,
n=1
I/Y = 12%
PMT = 0
Fv = 4000,000
Total Value of the Shopping Center
Value after Renovations
Loss in Value during Renovations
Total Value
Pv = 3,571,429
$
125,000,000
(3,571,429)
121,428,571
Gross Income Multiplier
A shortfall of the gross income multiplier is that it ignores vacancy rates and operating expenses.
Thus, if the subject property's vacancy rate and operating expenses are higher than those of the
comparable transactions, an investor will pay more for the same rent. Gross Income is also
considered a form of direct capitalization but is generally not considered as reliable as using a
capitalization rate.
Gross Income Multiplier from = Sales Price or Mv
Comparable Companies
Gross Income
Discounted Cashflow Method
DCF method sometimes referred to as a yield capitalization method involves projecting income
6
beyond the first year and discounting that income at a discount rate. The cap rate used to estimate
the resale price or terminal value is referred to as a 'Terminal Cap Rate' or 'Residual Cap Rate'. The
value that can be obtained by selling the property at some point in the future is often referred to as
the 'Reversion'.
Example 6:
NOI is expected to be level at $100,000 per year for the next 5 years because of existing leases. Starting in
year 6, the NOI is expected to increase to $120,000 because of lease rollovers and increase at 2% year
thereafter. The property value is also expected to increase at 2% year after year 5. Investors require a 12%
return and expect to hold the property for 5 years. What is the current value of the property?
0
1
2
3
4
5
100,000
100,000
100,000
100,000
100,000
120,000
Terminal Value = 120,000 = 1,200,000
(12% - 2%)
Discount the level NOI or the first 5 years and the terminal value price,
n=5
I/Y = 12%
PMT = 100,000
Fv = 1,200,000
Pv = 1,041,390
Implied Going-in Cap Rate = NOI = 100,000 = 9.6%
Value 1,041,390
The Going-in Cap Rate < Terminal Cap Rate. An investor is willing to pay a higher price for the current NOI
because he or she knows that it will increase when the lease is renewed at market rents in 5 years. The
expected rent jump on lease renewal is implicit in cap rate.
Example 7:
Suppose the NOI from a property is expected to be level at $600,000 per year for a long period of time such
that, for all practical purposes, it can be assumed to be a perpetuity. What is the value of the property
assuming investors want a 12% rate of return?
Value = NOI = 600,000 = 5000,000
Cap Rate 0.12 - 0
The Going-in Cap Rate = Terminal Cap Rate. In this case the cap rate will be the same as the discount rate.
This is true when there is no expected change in income and value overtime.
* If the current market rent is greater than the contract rent, then the rent is likely to be adjusted
upward at the time of the rent review and the property; which is referred to in the UK as a
'Reversionary Potential' because of the higher rent at the next rent review. 'Term and Reversion'
approach in UK simply splits the income into two components. The term rent is the fixed passing
(current contract) rent from the date of appraisal to the next rent review, and the reversion is the
Estimated Rental Value (ERV). The values of the two components of the income stream are
appraised separately by the application of different capitalization rates. The reversion cap rate is
derived from comparable, fully let properties. The discount rate applied to the contract rent will
likely be lower than the reversion rate because the contract rent is less risky (the existing tenants
are not likely to default on a below-market lease).
7
Example 8:
A property was let for a 5-Year term 3 years ago at $400,000 per year. Rent review occurs every 5 years, the
estimated rental value (ERV) in the current market is $450,000 and all risks yield (cap rate) on comparable
fully let properties is 5%. A lower rate of 4% is considered appropriate to discount the term rent because it is
less risky than market rent (ERV). Estimate the value of the property.
1
2
400,000
400,000
450,000
↶
0
450,000 = 9000,000
0.05
Pv @ 4% : 754,437.86
Pv @ 5% : 8,163,265.30
8,917,703
* With the layer method, one source (layer) of income is the contract (term) rent that is assumed to
continue in perpetuity. The second layer is the increase in rent that occurs when the lease expires
and the rent is reviewed. A cap rate similar to the ARY is applied to the term rent because the term
rent is less risky. A higher cap rate is applied to the incremental income that occurs as a result of
the rent review.
Example 9:
Consider the same property as in Example 8. The current contract (term) rent is to be discounted at 5% and
the incremental rent is to be discounted at 6%. Estimate the value of the property using the layer method.
1
400,000
2
400,000 = 8000,000
0.05
↶
0
400,000
8000,000
450,000 ↷
Pv @ 6% : 741,664
50,000
=
833,333
50,000
8,741,664
0.06
A concept proposed by Investment Property Databank (IPD) in the UK is to show the effective yield
for a property being valued, where the effective yield is an IRR calculation based on reasonable
assumptions for future rent reviews beyond the first one.
Example 10:
Total operating expenses for a multi-tenant office building are 30% fixed and 70% variable. If the 100,000 sq.
feet building was fully occupied, operating expenses would total $6 per sq. feet. The building is currently 90%
occupied. If the total operating expenses are allocated to the occupied space, calculate the operating expense
per occupied sq. foot.
If Building is 100% Occupied
Fixed
$180,000
Variable
$420,000
Total
$600,000
(100,000 x 6)
(30% x 600,000)
(70% x 600,000)
Variable Expense = 420,000 = $ 4.2
100,000
If Building is 90% Occupied
Fixed
$180,000
Variable
$378,000
Total
$558,000
(90% x 100,000 x 4.2)
8
Operating Expenses = Total Operating Expenses = 558,000 = $ 6.2
Occupied Sq. Foot
90,000
Example 11:
A property is being valued using an 8% discount rate. A terminal cap rate if 5.5% was used to estimate the
resale price. After solving for value, the appraiser calculates the implied going-in cap rate to be 6%. Market
rents and property values have been increasing about 1% per year, and that is expected to continue in the
foreseeable future. Current mortgages rates for a loan on the property would be 7.5%. Do the assumed
discount and terminal cap rates seem reasonable?
There are several red flags or warnings signs. First, the discount rate is only 50 Bps above the mortgage rate.
Whether this is a sufficient risk premium for an equity investor is questionable. Second, the terminal cap rate
< going-in cap rate, which suggests either interest rates will fall in the future or NOI and property values will
increase at an even faster rate in the future. Usually, terminal cap rates are the same as or slightly higher
than going-in cap rates to reflect the fact that the property will be older when sold, and older properties
usually have less NOI growth. Finally, the difference between the discount rate of 8% and going-in cap rate of
6% implies 2% per year growth. Yet NOI and property values are expected to increase only about 1% per year.
Overall, it appears that the appraiser may be overvaluing the property.
DCF Common Errors
1. The discount rate doesn't reflect the risk.
2. Income growth is greater than expense growth.
3. The terminal cap rate is not logical compared with the implied going-in cap rate.
4. The terminal cap rate is applied to an income that is not typical.
5. The cyclical nature of real estate markets is not recognized.
Fig 2: Direct Capitalization Vs. DCF
Direct Capitalization
DCF
A cap rate or income
multiplier is applied to
first year NOI.
The future cashflows are
projected each year until
sale of property.
Thus, expected growth
increase in NOI in the
future must be implicit in
the multiplier or cap rate.
Thus, the future income
pattern, including the
effect of growth, is explicit
in a DCF.
Direct Capitalization
relies on comparable
transaction.
DCF doesn't rely on
comparable transaction as
long as an appropriate
discount rate is chosen.
Simple Calculations
Complex Calculations
Direct Capitalization
doesn't consider other
cashflows.
DCF considers other
cashflows e.g. capital
expenditures.
9
lllll
Cost Approach
The premise behind the cost approach is that a buyer is unlikely to pay more for a property than it
would cost to purchase land and build a comparable building. Because of the difficulty in measuring
depreciation and obsolescence, the cost approach is most useful when the subject property is
relatively new. The cost approach is often used for unusual properties or properties where
comparable transactions are limited. The cost approach is sometimes considered the upper limit of
value since an investor would never pay more than the cost to build a comparable building. That
said, market values that exceed the implied value of the cost approach are questionable.
Step 1
+
Step 2
-
Land Cost
Construction Cost
Mv of Land
+ Replacement Cost
+ Builder's Profit
- Curable Physical
Deterioration
The depreciation being
estimated for the cost approach
may have little relationship to
the amount of depreciation that
would be used on financial
statements using a historical
cost approach to accounting.
Step 3
Economic Depreciation
+ Incurable Physical
Deterioration
[(Effective/Total Life)
x Step 2]
+ Curable Functional
Obsolescence
+ Incurable Functional
Obsolescence
[x/Cap Rate]
+ Locational
Obsolescence
[x/Cap Rate]
+ Economic
Obsolescence Loss [x]
Effective Age = Total Economic Life - Remaining Life
Depreciation Cost = Replacement Cost - Total Depreciation
Includes Physical Deterioration
(Curable & Incurable), Functional
Obsolescence (Curable & Incurable)
and Locational Obsolescence
Whereas,
Physical Deterioration : It is related to the building's age and occurs as a result of
normal wear and tear over time. There are two types of
physical deterioration i.e. curable and incurable. Curable
means that fixing the problem will add value that is at least as
great as the cost of the curve. An item is incurable if the
problem is not economically feasible to remedy. Incurable
items increase the effective age of the property.
Functional Obsolescence : Is the loss in value resulting from defaults in design that
impairs a building's utility e.g. Bad floor plan. NOI is usually
lower than it otherwise would be because of lower rent or
higher operating expenses. (Internal Obsolescence)
Locational Obsolescence : Occurs when the location is no longer optimal e.g. a prison is
built near a luxury apartment. Care must be taken in
deducting the loss in value because part of the loss is likely
already reflecting in the market value of the land. (External
Obsolescence)
10
Economic Obsolescence : Occurs when new construction is not feasible under current
economic conditions. This can occur when rental rates are not
sufficient to support the property. Consequently, the
replacement cost of the subject property exceeds the value of
a new building if it was developed. (External Obsolescence)
lllll
Sales Comparison Approach
The premise is that a buyer would pay no more for a property than others are paying for similar
properties. With the sales comparison approach, the sale prices of similar (comparable) properties
are adjusted for differences with the subject property. The sales comparison approach is most
useful when there are a no. of properties similar to the subject that have recently sold, as is usually
the case with single-family homes. When the market is active, the sales comparison approach can
be quite reliable. But when the market is weak, there tends to be fewer transactions, which makes it
difficult to find comparable properties. This approach assumes purchasers are active rationally;
However, there are times when purchasers become overly exuberant and market bubbles occur.
Example 12:
Details for Property 1, 2 and 3 mentioned below. Calculate estimated value for the subject property:
Reference
2% Dep
5%
10%
0.5%
Unit of Comparison
Size (Sq. Feet)
Age (Years)
Physical Condition
Location
Sale Date (Months)
Sales Price ($)
Subject Property
30,000
5
Average
Prime
Subject Property
Sales Price
Age (9000,000 x 2% x 4)*
Physical Condition (9000,000 x 5%)*
Location (Same: Prime)*
Sale Date (9000,000 x 0.5% x 6)*
Adjusted Sales Price
Size
Adjusted Sales Price (per Sq. Feet)
Comparable Transactions
1
2
3
40,000
20,000
35,000
9
4
5
Good
Average
Poor
Prime
Secondary
Prime
6
18
12
9000,000 4,500,000 8000,000
Comparable Transactions
1*
2
3
9000,000
4,500,000
8000,000
+720,000
- 90,000
-450,000
+400,000
+450,000
+270,000
+405,000
+480,000
9,540,000
5,265,000
8,880,000
40,000
20,000
35,000
238.5
263.25
253.71
Average Sales Price
per Sq. Feet
$251.82
Estimated Value = 30,000 x $ 251.82 = $ 7,554,600
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Private Equity Real Estate Investment Indexes
Investors should also be aware that the apparent low correlation of real estate with other asset classes
may be due to limitations in real estate index construction.
11
1. Appraisal-Based Indexes
Appraisal-Based Indexes combine valuation information from individual properties and provide a
measure of market movements. A popular index in the US is the NCREIF Property Index (NPI).
Members of NCREIF, mainly investment managers and pension fund sponsors, submit appraisal date
quarterly and NCREIF calculates the return as follows:
Return = NOI - Capital Expenditures + (Ending Market Value - Beginning Market Value)
Beginning Market Value
The index is then value-weighted based on the returns of the separate properties. The return is
known as a holding-period return and is equivalent to a single-period IRR. Appraisal-based indices
tend to lag actual transactions because actual transactions occur before appraisals are performed.
Thus, a change in price may not be reflected in appraised values until the next quarter or longer if a
property is not appraised every quarter. Appraisal lag tends to smooth the index, meaning that it has
less volatility. It behaves somewhat like a moving average. Thus, appraisal-based indexes may
underestimate the volatility of real estate returns. Due to lag in appraisal-based real estate indexes,
they will also tend to have a lower correlation with other asset classes. Appraisal lag can be adjusted
by unsmoothing the index or by using a transaction-based index.
2. Transaction-Based Indexes
Indexes have been created that are based on actual transactions rather than appraised values.
Transaction-based indices can be constructed using a repeat-sales index and a hedonic index. The
challenge for those creating these indexes is to try to keep the noise (include random elements in
the observations) to a minimum through use of appropriate statistical techniques and collecting as
much data as possible.
Repeat-Sales Index: relies on repeat sales of the same property. A change in market conditions can
be measured once a property is sold twice. Ofcourse, the more sales, the more reliable is the index.
The property size and location are not required. Accordingly, a regression is developed to allocate
the change in value to each quarter.
Hedonic Index: requires only one sale. A regression is developed to control for differences in
property characteristics such as size, age, location and quality of construction etc.
Real Estate Financial Ratios
1. Debt Service Coverage Ratio (DSCR): Debt service (loan payment) includes interest and principal, if
required. Principal payments reduce the outstanding balance of the loan. An interest-only loan doesn't
reduce the outstanding balance. Interest-only loans typically either revert to amortizing loans at some
point or have a specified maturity date. Lenders typically require a DSCR of 1.2 or greater to provide a
margin of safety that the NOI from the property can cover the debt service.
DSCR = First Year NOI
Debt Service
2. Loan-to-Value Ratio (LTV): Higher LTV results in higher leverage and thus, higher risk because lenders have
a superior claim in the event of default.
LTV = Loan Amount
Appraisal Value
12
3. Equity Dividend Rate: as a measure of how much cashflow they are getting as a percentage of their equity
investment. This is sometimes referred to as a 'cash-on-cash' return because it measures the cash return
on the amount of cash invested. Equity dividend rate only covers one period. It is not the same as the IRR
that measures the return over the entire holding period.
Equity Dividend Rate = First Year Cashflow or NOI - Debt Service
Equity
Equity
Example 13:
A real estate lender agreed to make a 10% interest-only loan on a property that was recently appraised at
$1,200,000 as long as the debt service coverage ratio is atleast 1.5 and loan-to-value ratio doesn't exceed
80%. Calculate the maximum loan amount assuming the property's NOI is $135,000.
DSCR = First Year NOI
Debt Service
1.5 = 135,000
x
x = 90,000
LTV = Loan Amount
Appraisal Value
80% =
x
1,200,000
x = $960,000
Loan Amount = 90,000 = $900,000
0.1
In this case, the maximum loan amount is the $900,000 which is the lower of the two amounts. At $900,000,
the LTV is 75% (900,000/1,200,000) and the DSCR is 1.5 (135,000/90,000).
Example 14:
Property was recently appraised at $1,200,000 and the property's NOI is $135,000. The debt service
amounted to $90,000. Calculate the equity dividend rate assuming the property is purchased for the
appraised value.
1,200,000
Debt
900,000
Equity
300,000
Equity Dividend Rate = NOI - Debt Service = 135,000 - 90,000 = 15%
Equity
300,000
Example 15: [Leveraged IRR]
Returning to the last example, calculate the IRR if the property is sold at the end of 6 years for 1,500,000.
Assume that NOI growth is zero.
n=6
PMT = 45,000
(First Year Cashflow)
Pv = - 300,000
Fv = 600,000
(Equity) (1,500,000 - 900,000)
I/Y = 24.1%
Since the property was financed with debt, the cashflows that are received at the end of the holding period
i.e. net sales proceeds - outstanding mortgage balance. 24.1% is the IRR based on the equity invested in the
property.
13
Example 16: [Unleveraged IRR]
Refer to previous examples, what would the IRR be if the property were purchased on an all-cash basis (no
loan)?
n=6
PMT = 135,000
(NOI)
Pv = - 1,200,000
(Appraised Value)
Fv = 1,500,000
(Sales Proceed)
I/Y = 14.2%
The annual cashflows are higher because there is no debt service and the terminal cashflow is higher because
no mortgage balance is repaid at the end of the holding period. 14.2% is an IRR based on an unleveraged
(all-cash) investment in the property.
As per Example 15 & 16
Leveraged IRR > Unlevered IRR
24.1% > 14.2%
Therefore as a result, the equity investor benefits by financing the property with debt because of positive
leverage. Remember however, that leverage will also magnify negative returns.
14
CHAPTER 40
Publicly Traded Real Estate Securities
Publicly traded real estate securities can take several forms: Real Estate Investment Trusts (REITs), Real Estate
Operating Companies (REOCs), RMBS and CMBS.
Equity REIT: are tax-advantaged companies (trusts) that are for the most part exempt from corporate income
tax (tax advantage is a result of being allowed to deduct dividends paid). Equity REITs are actively managed,
own income-producing real estate and seek to profit by growing cashflows, improving existing properties and
purchasing additional properties. REITs offer greater liquidity and opportunities for broader diversification.
REOC: REOCs are ordinary taxable real estate ownership companies. Compared with REIT, REOCs can retain
more of their income for re-investment when they believe attractive opportunities exist to create value for
investors REOCs are free to use wider range of capital structure and degrees of financial leverage. A business
will form as a REOC if it is ineligible to organize as REIT. For example, the firm may intend to develop and sell
real estate rather than generating cash from rental payments or the firm may be based in a country that
doesn't allow tax-advantaged REITs. Consequently, in many markets there is a tendency for REOCs to
experience less access to equity capital and lower market valuations (and higher cost to equity) than REITs.
REOCs are usually able to elect to covert to REIT status, if they meet the general requirements of REIT but,
depending on their countries of domicile.
MBS: RMBS or CMBS are publicly traded asset-based securitized debt obligations that receive cashflows from
an underlying pool of mortgage loans. Real estate debt securities represent a far larger aggregate market
value than do publicly traded real estate equity securities.
Mortgage REIT: invest primarily in mortgages, mortgage securities or loans that are secured by real estate.
Hybrid REIT: that own and operate income-producing real estate, as do equity REITs but invest in mortgages
as well. There are relatively few hybrid REITs.
Advantages [REIT & REOC]
1. Superior Liquidity: The ability to trade shares (REIT & REOC) on stock exchanges provides greater liquidity
compared to buying and selling real estate in property markets. The low liquidity of a direct real estate
investment stems from the relatively high value of an individual real estate property and the unique nature
of each property.
2. Lower Minimum Investment: REIT & REOC shares trade for much smaller dollar amounts, when compared
to multi-million dollar direct investment.
3. Limited Liability: The financial liability of a REIT investor is limited to the amount invested.
4. Access to Premium Properties: Some prestigious properties such as high-profile shopping malls or other
prominent or landmark buildings are difficult to invest in directly. Investors can gain access to such
properties by purchasing the shares of REITs that own them.
5. Active Professional Management: Investors in publicly traded real estate equity securities don't require
such expertise or skills. REITs and REOCs employ professional management to control expenses, maximize
rents and occupancy rates and sometimes to acquire additional properties unlike property management.
6. Protections accorded to Publicly Traded Securities: REITs and REOCs must meet the same requirements
applicable to other publicly traded companies, including rules related to financial reporting, disclosure and
governance.
REIT Specific Advantages
7. Greater Potential for Diversification
↓
8. Exemption from Taxation: REITs are typically exempt from the double taxation of income that comes from
taxes being due at the corporate level and again when dividends or distributions are made to shareholders.
15
In most jurisdictions, there are no taxes payable by a REIT if it (a) meets certain requirements for types of
assets held, typically rental property (75% of total assets in real estate for US REITs), (b) derives the bulk of
its income from rents or mortgage interest on real estate (75% for US REITs), (c) has limited non-rental
property assets and (d) pays out in dividends/distributions nearly all of its taxable income (atleast 90% in
US). The dividends/distributions that REIT shareholders receive are typically divided into ordinary taxable
income, capital gains (qualifying for lower capital gains tax rates) and return of capital (the portion of
distributions in excess of a REITs earnings, treated as a return of capital and deducted from the investor's
share cost basis for tax purposes).
9. Predictable Earnings: The earnings of REITs tend to be relatively consistent overtime because REIT's rental
income is fixed by contracts, unlike the income of companies in other industries.
10. High Income Payout Ratios and Yields: To maintain their tax-advantaged status, REITs are obligated to
payout most of their taxable income as dividends. Because of this high income payout ratios, the yields of
REITs are higher than the yields on most other publicly traded equities.
Disadvantages [REIT & REOC]
1. Taxes Vs. Direct Ownership: Depending on local laws, investors that make direct investments in properties
may be able to deduct losses on real estate from taxable income or replace one property for a similar
property (like-kind exchange in the US) without taxation on the gains. For investors in REITs or REOCs,
these specific tax benefits are not available.
2. Lack of Control: Minority shareholders in a publicly traded REIT have less control over property-level
investment decisions than do direct property owners.
3. Costs: The maintenance of a publicly traded REIT structure is costly. However, there are also related costs,
which may not be worthwhile for smaller REITs.
4. Stock Market determined Pricing: While the appraisal-based value of a REIT may be relatively stable, the
market determined price of a REIT share is likely to be much more volatile. Appraisals tend to be infrequent
and backward looking, while the stock market is continuous and reflects forward-looking values. Appraised
values tend to underestimate volatility.
5. Structural Conflicts of Interest: When a REIT is structured as an UPREIT or a DOWNREIT, there is a potential
for conflict of interest. When the opportunity arises to sell properties or take on additional borrowing, a
particular action may have different tax implications for REIT shareholders and for the general partners,
which may tempt the general partners to act in their own interest rather than in the interest of all
stakeholders.
(UPREITs and DOWNREITs structures have the purpose of avoiding recognition of taxable income if
appreciated property is transformed to the REIT. UPREIT is an 'Umbrella' partnership REIT structure, where
the REIT is the general partner and holds a controlling interest in a partnership that owns and operates the
properties. DOWNREIT has an ownership interest in more than one partnership and can own properties
both at the partnership level and at the REIT level. A DOWNREIT can form partnerships for each property
acquisition it undertakes).
6. Limited Potential for Income Growth: REITs high rates of income payout limit REITs ability to generate
future growth through reinvestment. This limits future income growth and may dampen the share price of
REITs (the stock market's tendency to focus on earnings growth can cause REIT shares to underperform in
periods during which the market highly values fast-growing companies).
7. Forced Equity Issuance: REITs typically use financial leverage and are regularly in the debt markets to
refinance their maturing debt. If a REIT's management of its overall financial leverage and the timing and
type of its debt maturities is flawed, a REIT may be forced to issue equity at a disadvantageous price.
lllll
REIT Valuations
lllll
Net Asset Value Approach
Net Asset Value Per Share (NAVPS) is the pre share amount by which assets exceed liabilities, using
16
current market values rather than accounting book values. NAVPS is a superior measure of the net
worth of a REIT and REOC. NAV is the largest component of the intrinsic value of a REIT or REOC; the
balance being the value of non-asset based income streams, the value added by management, and
the value of any contingent liabilities. If the market price of a REIT varies from NAVPS, this is seen as
a sign of over or undervaluation. Investors must understand the implications of using a private
market valuating tool on a publicly traded security. Reason being, the stock market tends to focus
more on the outlook for short-term future changes in income and asset value than the property
market, which is more focused on long term valuation. If in general, the market is trading at a
premium to NAVPS, a value investor would select the investments with lowest premium.
Fig 1: Disclosures
Fair Value
1. Calculation of Fair Value
2. Reconciliation between the
beginning and ending
carrying amounts of
investment property.
Cost Model
1. Depreciation Method
2. Useful Lives
3. Fair Value of investment
property.
NOI 1
÷ Assumed Cap Rate
Property Value
+ Tangible Assets 2
- Tangible Debt and Other Liabilities 3
NAV
÷ Shares Outstanding
NAVPS
1. Non-cash rent is the difference between the average rent over the
term of a lease contract (i.e. straight-line rent) vs. the amount of
cash rent actually received in a period.
2. Goodwill, deferred financing expenses and DTA will be excluded.
3. DTL will be removed.
Example 1:
Vinny Cestone CFA is undertaking a valuation of the Anyco Shopping Center REIT Inc. Given the following
financial data for Anyco, estimate NAVPS based on forecasted cash NOI.
Last 12-months NOI
Cash and Equivalents
Accounts Receivable
Total Debt
Other Liabilities
Non-Cash Rents
Full-year Adjustment for Acquisitions
Land held for Future Development
Prepaid/Other Assets (excluding Intangibles)
Shares Outstanding
Estimate of next 12 months growth NOI
Cap Rate based on recent comparable transactions
$
80
20
15
250
50
2
1
10
5
15
1.25%
8%
17
Last 12-months NOI
- Non-Cash Rents
+ Full-year Adjustment for Acquisitions
Proforma Cash NOI for Last 12 months
+ Next 12 months Growth in NOI (@ 1.25%)
Estimated next 12 months Cash NOI
÷ Cap Rate
Estimated Value of Operating Real Estate
+ Cash and Equivalents
+ Land held for Future Development
+ Accounts Receivable
+ Prepaid/Other Assets
- Total Debt
- Other Liabilities
Net Asset Value (NAV)
$
80
2
1
79
1
80
8%
1000
20
10
15
5
250
50
750
NAVPS = Assets - Liabilities =
NAV
= 750 = $ 50
Shares O/S
Shares O/S 15
lllll
Funds From Operations
Funds From Operations (FFO) adjusts earnings and is a popular measure of the continuing
operating income of a REIT or REOC. FFO has some shortcoming but because it is the most
standardized measure of a REIT or REOC's earning power, P/FFO is the most frequently used
multiple in analyzing the sector since P/AFFO relies more on estimated and is considered more
subjective.
Accounting Net Earnings
+ Depreciation Expense
+ Deferred Tax Expenses (i.e. DTL)
- Gains from Sales of Property and Debt Restructuring
+ Loss from Sales of Property and Debt Restructuring
FFO
lllll
Adjusted Funds From Operations
Adjusted Funds From Operations (AFFO) also known as Funds Available for Distribution (FAD) or
Cash Available for Distribution (CAD) is a refinement of FFO that is designed to be a more accurate
measure of current economic income. AFFO is superior to FFO as a measure of economic income;
however, it is open to more variation and error in estimation than FFO. Although many REITs and
REOCs compute and refer to AFFO is their disclosures, their methods of computation and their
assumptions vary. Firms that compile statistics and estimates of publicly traded enterprises for
publications, such as Bloomberg and Thomson Reuters, tend not to gather AFFO estimates because
of the absence of a universally accepted methodology for computing AFFO and inconsistent
corporate reporting of actual AFFO figures, which hinders corroboration of analysts' estimates.
FFO
- Non-Cash (Straight-Line) Rent Adjustment
- Recurring Maintenance type Capital Expenditures & Leasing Commissions
AFFO
18
Example 2:
An increase in the capitalization rate will most likely decrease a REITs:
A. Cost of Debt
Incorrect, because estimated NOI is
B. Estimated NOI
based on income growth not cap rate.
C. Estimated NAV
✓
lllll
Discounted Cashflow Approach
REITs and REOCs generally return a significant portion of their income to their investors and tend to
be high-dividend paying shares. Thus, dividend discount models for valuation are applicable. DDM
and DCF are used in private real estate in the same way that they are used to value stocks in
general. The other key component in DCF and DDM models is required returns. The conventional
arguments is that the risk premium and thus the discount rate associated with REIT and REOC
should be lower than the average stock in the broader market because the underlying business of
owning income producing real estate should be less volatile because of contractual revenue
streams from leases. A long-term look at the betas of REIT shares suggests values tend to be less
than 1, which supports this view. The key drawbacks to using DCF and DDM for valuing REITs and
REOCs is the high sensitivity of these valuation models to the key inputs of growth and discount
rates.
Key Notes [FFO & AFFO]
- The higher the expected growth, the higher the multiple or relative valuation growth can be driven
by business model e.g. REIT and REOC successful in real estate development often generate
above-average FFO/AFFO growth over time.
- As financial leverage increases, equities' FFO and AFFO multiples decrease because required
return increases as risk increases.
- The purpose of the adjustments to net earnings made in computing FFO and AFFO is to obtain a
more tangible, cash-focused measure of sustainable economic income that reduces reliance on
non-cash accounting estimates and excludes non-economic, non-cash charges.
- Applying a multiple to FFO & AFFO may not capture the intrinsic value of all real estate assets held
by the REIT & REOC e.g. land parcels and empty buildings may not currently produce income and
hence don't contribute to FFO but have value.
- An increased level of such one-line items as gains and accounting charges as well as new revenue
recognition rules, have affected the income statement, thus making P/FFO and P/AFFO more
difficult to compute complicating comparisons between companies.
- Neither FFO nor AFFO take into account differences in leverage: leverage ratios can be used to
adjust for differences in leverage among REITs when comparing valuations based on FFO and AFFO
multiples. REIT investors tend to be averse to high leverage.
Fig 2: Sub-Types of Equity REITs
REIT Type
Retail or
Shopping Center
Office
Economic Value
Investment Characteristics
- Retail Sales Growth
- Job Creation
Lease: 3-10 Years
Revenue: Fixed Rental price (minimum lease)
plus a Percentage of Sales over a certain level
i.e. Anchor Tenants. [Net Lease]
- New Space Supply vs.
Demand
- Job Creation
Lease: 5-25 Years
Revenue: Rent, Property Taxes, Operating
Expenses and other common costs. [Net Lease]
19
Multi-Family or
Residential
Health Care
Industrial
Hotel
Storage
- Population Growth
- Job Creation
Lease: 1 Year
Revenue: Rent, Operating Expenses, Tax and
Maintenance Costs. [Gross Lease]
- Population Growth
- New Space Supply vs.
Demand
Lease: To maintain the tax-free status, REITs
rent properties to health care providers.
Revenue: Rent, Litigation Costs and Insurance
Costs. [Net Lease]
- Population Growth
- Retail Sales Growth
Lease: 5-25 Years
Revenue: Rent. [Net Lease]
- New Space Supply vs.
Demand
- Job Creation
Lease: Sector is cyclical because it is not
protected by long-term leases.
Revenue: Rent, Operating Expenses and
Maintenance Costs.
Analysts compare a number of statistics
against industry averages (Operating Profit
Margins, Occupancy Rates, Average Room
Rates, RevPAR = Average Occupancy Rate x
Average Room Rate and Food & Beverage
Sales).
- Population Growth
- Job Creation
Lease: Monthly
Revenue: Rent. [Gross Lease]
National GDP growth is the largest driver of economic value for all REIT types.
Net Lease entails that the tenant (not owner) incurs the operating expense for the property.
Office: Employment Growth
Industrial: Economy & Economic Growth and Import & Export Activity
Retail: Consumer Spending
Multi-Family: Population Growth
20
CHAPTER 41
Private Equity Valuation
Private Equity Firms make investments ranging from investments in early stage companies called a venture
capital investment to invests in mature companies generally in a buyout transactions.
(Outsider)
Partnership Interests
Equity Position
Investment
Investment
Limited Partners (LP): Don't have active
role in the management and liability is
limited to the amount invested.
General Partners (GP): Has management
control & unlimited liability. GP is the
manager of the private equity fund.
It is commonly believed that Private Equity (PE) firms have the ability to add greater value to their portfolio
companies than do publicly governed firms, since private companies don't face the scrutiny of analysts,
shareholders and the broader market. In private equity firms, debt is more heavily utilized and is quoted as a
multiple of EBITDA as opposed to a multiple of equity as for public firms. The sources of this increased value
are:
a. The ability to obtain debt financing on more advantageous terms. In buyout transactions, management
often has a substantial stake in the company's equity.
b. The ability to re-engineer the portfolio company and operate it more effectively. In many venture capital
investments, the private equity offers advice and management expertise.
c. Superior alignment of interests between management and private equity ownership.
Public companies are bought and sold on regulated exchanges daily. Private companies however, are bought
by buyers with specific interests at specific points in time with each potential buyer possibly having a different
valuation for the company. Furthermore, valuing a private company is more difficult than valuing public
companies.
Fig 1: Private Equity Valuation Methodologies
Valuation Technique
Application
Income Approach:
Discounted Cashflow
(DCF)
Value is obtained by discounting expected future cashflows at an
appropriate cost of capital. The cost of capital is generally estimated using
the WACC formula used for public companies. There's a serious challenge in
assessing cost of equity in private equity settings is the lack of public
historical data on share prices and returns. Therefore β must be estimated
by means of proxy. This is performed typically by estimating the beta for
comparable companies, and then adjusting for financial and operating
leverage. Typically a terminal value is calculated using a price multiple of the
company's EBITDA.
Relative Value or
Market Approach:
Earnings Multiples
The earnings multiple is frequently obtained from the average of a group of
public companies operating in a similar business and of comparable size e.g.
P/E, EV/EBITDA and EV/Sales. Generally applies to companies with significant
operating history and predictable stream of cashflows. May also apply with
caution to companies operating at the expansion stage.
21
Real Option
The right to undertake a business decision (call or put). Requires judgmental
assumptions about key option parameters. It is applicable for immature
companies with flexibility in their future strategies. Therefore, generally
applies to some companies operating at the seed or start-up phase.
Replacement Cost
Estimated cost to recreate the business as it stands as of the valuation date.
Generally applies to early (seed and start-up) stage companies or companies
operating at the development stage and generating negative cashflows.
Rarely applies to mature companies as it is difficult to estimate the cost to
recreate a company with a long operating history e.g. Coca-Cola.
Venture Capital
Capital invested in a project in which there is a substantial element of risk,
typically a new or expanding business like business in seed stage (financing
provided to research business ideas, develop prototype products or conduct
market research), start-up stage (financing to recently created companies
with well articulated business and marketing plans), expansion stage and
replacement capital (financing provided to purchase shares from other
existing venture capital investors or to reduce financial leverage).
Leverage Buyout
The buyer acquires a controlling equity position in a target company. For
example, acquisition capital (financing in the form of debt, equity or
quasi-equity provided to a company to acquire another company), leverage
buyout (typically involves senior debt, junk bonds, equity and mezzanine
finance) and management buyout (financing provided to the management to
acquire a company, specific product line or division carve out).
Other considerations for valuing private equity portfolio companies are control premiums, country risk and marketability and liquidity
discounts. The purpose of valuation is about assessing a company's ability to generate superior cashflows from a distinctive competitive
advantage and serving as a benchmark for negotiations with the seller.
Types of Exit Routes
The means and timing of the exit strongly influence the exit value e.g. if a portfolio company cannot be sold
due to weak capital markets, the private equity firm may want to consider buying another portfolio company
at depressed prices, merging the two companies and waiting until capital market conditions improve to sell
both portfolio companies as one.
1. Initial Public Offering: In an IPO, a company's equity is offered for public sale. An IPO usually results in the
highest exit value due to increased liquidity, greater access to capital and the potential to hire better
quality managers. However, an IPO is less flexible, more costly and more cumbersome process than the
other alternatives. IPOs are most appropriate for companies with strong growth prospects and a significant
operating history and size.
2. Secondary Market Sale: Sale of stake held by a financial investor to other financial investors or to strategic
investors (companies operating or willing to establish in the same sector or market of the portfolio
company). Two main advantages of secondary market transactions are (i) the possibility to achieve the
highest valuation multiples in the absence of an IPO and (ii) with the segmentation of private equity firms,
specialized firms have the skill to bring their portfolio companies to the next level (restructuring, merger,
new market) and sell either to a strategic investor seeking to exploit synergies or to another private equity
to further add value to the portfolio company.
3. Management Buyout: In an MBO, the company is sold to management who utilize a large amount of
leverage; the resulting high leverage may limit management's flexibility.
22
4. Liquidation: The outright sale of the company's assets is pursued when the company is deemed no longer
viable and usually results in a low value.
There are two important differences between investing in public equity and in a private equity fund: First,
funds are committed in the private investments and later drawn down as capital is invested in portfolio
companies. In a public firm, the committed capital is usually immediately deployed. Second, the returns on a
private equity investment typically follow a J-curve pattern through time. Initially, returns are negative but
then positive as portfolio companies are sold at exit. Private equity investments are usually regulated such
that they are only available to 'qualified' investors because of the high risks associated with it.
Fig 2: Private Equity Fund Structure
Commitments by Investors
Multiple 'Closings'
Cashflows back to Investors
Indications of Fund Performance
1 Yr
Marketing
10 Yrs
Drawdown/
Investment
2 Yrs
Realization of
Returns and Exit
Extension
[Follow-on-Fund]
The terms in a fund prospectus are a result of negotiation between GP and the LPs. If the fund is
oversubscribed i.e. has more prospective investors than needed, the GP has greater negotiating power. In a
prospectus, the valuation is related to the fund's NAV which is the value of fund assets minus liabilities.
Undrawn LP capital commitments are not included in the NAV calculation but are essentially liabilities for the
LP. The value of the commitments depends on the cashflows generated from them, but these are quite
uncertain. When a GP has trouble raising funds, this implies that the value of these commitments is low.
Private equity firms have no obligation to disclose publicly their fund performance information. Before
investing, outside investors should conduct a thorough due diligence of a private equity fund due to the
following characteristics:
- Private equity funds have returns that tend to persist i.e. outperformers tend to keep outperforming and
underperformers tend to keep underperforming or go out of business.
- The return discrepancy between outperformers and underperformers is very large and can be as much
as 20%.
- Private equity is typically illiquid and thus LPs are locked for the long-term. On the other hand, when
private equity funds exit an investment, they return the cash to the investors immediately. Therefore, the
'duration' of an investment in private equity is typically shorter than the maximum life of the fund.
The most common form of ownership structure for private equity funds is the limited partnership. Most fund
structures are 'Closed End' i.e. which restricts existing investors to entering the fund only at predefined time
periods, at the discretion of the GP.
23
Key Terms:
1. Management Fees: These are fees paid to the GP on an annual basis as a percent of committed capital and
are commonly 2%.
2. Transaction Fees: These are paid by third parties to the GP in their advisory capacity e.g. investment
banking services. These fees are usually split evenly with the LPs and when received, are deducted from
management fees.
3. Carried Interest or Performance Fees or Incentive Fees: This is the GP's share of the fund profits and is
usually 20% of profits, after management fees.
4. Ratchet: This specifies the allocation of equity between stockholders and management of the portfolio
company and allows management to increase their allocation depending on company performance.
5. Hurdle Rate: This is the IRR that the fund must meet before the GP can receive carried interest. It usually
varies from 7% to 10% and incentivizes the GP.
6. Vintage: This is the year the fund was started and facilitates performance comparisons with other funds.
7. Committed Capital: Is the amount of funds promised by investors to private equity funds.
8. Paid-in-Capital or Invested Capital: Is the amount of funds actually received from investors.
9. Trailer Fee: Is the compensation paid by the fund manager to the person selling the fund to investors.
10. Key Man Clause: If a key named executive leaves the fund or doesn't spend a sufficient amount of time at
the fund, the GP may be prohibited from making additional investments until another key executive is
selected.
11. Clawback: Under a clawback provision, if the fund subsequently underperforms, the GP is required to
payback a portion of the early profits to the LPs. The clawback provision is usually settled at termination
of the fund but can also be settled annually a.k.a. true-up.
12. Distribution Waterfall: This provision specifies the method in which profits will flow to the LPs and when
the GP receives carried interest:
(a) Deal by Deal method, carried interest can be distributed after each individual deal. The disadvantage
of this method from the LPs perspective is that one deal could earn $10,000,000 and another could
loose $10,000,000, but the GP will receive carried interest, even though the LPs have not earned an
overall positive return.
(b) Total Return method, carried interest is calculated on the entire portfolio. These are two variants of the
total return method (i) carried interest can be only after the entire committed capital is returned to LPs
or (ii) carried interest can be paid when the value of the portfolio exceeds invested capital by some
minimum amount (typically 20%). Notice that the former uses committed capital whereas the latter
uses only the capital actually invested.
13. Tag-Along, Drag-Along Clause: Anytime an acquirer acquires control of the company, they must extend
the acquisition offer to all shareholders, including firm management.
14. No-Fault Divorce: A GP may be removed without cause, provided that a super majority (generally above
75%) of LPs approve that removal.
24
15. Co-Investment: This provision allows the LPs to invest in other funds of the GP at low or no management
fees. This provides GP another source of funds. The provision also prevents the GP from using capital
from different funds to invest in the same portfolio company. A conflict of interest would arise if the GP
takes capital from one fund to invest in a troubled company that had received capital earlier from another
fund.
Example 1:
Suppose a fund has committed capital of $100,000,000, carried interest of 20% and a hurdle rate of 9%. The
firm called 80% of its commitments in the beginning of year 1 of this, $50,000,000 was invested in Company A
and $30,000,000 in Company B. At the end of year 2, a $7000,000 profit is realized on the exit from Company
A. The investment in Company B is unchanged. The carried interest is calculated on a deal-by-deal basis i.e.
the IRR for determining carried interest is calculated for each deal upon exit. Determine the theoretical
carried interest and the actual carried interest.
Carried Interest = 20% x 7000,000 = $ 1,400,000
Pv = - 50
Fv = 57
n=2
I/Y = 6.8% (IRR)
Because 6.8% < 9% i.e. IRR < Hurdle Rate, no carried interest is actually paid to the GP.
Example 2:
Suppose a fund has committed capital of $100,000,000 and carried interest of 20%. An investment of
$40,000,000 is made. Later in the year, the fund exits the investment and earns a profit of $22,000,000.
Determine whether the GP receives any carried interest under the three distribution waterfall methods.
1. Deal by Deal Method (Carried interest can be distributed after each individual deal)
Carried Interest = 20% x $22,000,000 = $4,400,000 is paid to the GP, the rest $17,600,000 to LP.
2. a. Total Return Method (Carried interest can be paid only after portfolio value exceeds committed capital)
Committed Capital $100,000,000 > Portfolio Value $62,000,000; Hence, no carried interest to the GP, but
the entire proceeds of sale of $62,000,000 are entitled to the LPs.
b. Total Return Method (Carried interest can be paid when the value of the portfolio exceeds invested
capital by some minimum amount typically 20%)
Invested Capital ($40,000,000 x 1.2) $48,000,000 < Portfolio Value $62,000,000; So carried interest of
$4,400,000 is paid to the GP.
Example 3:
Continuing with the previous example, suppose that in the second year, another investment of $25,000,000 is
exited and results in a loss of $4000,000. Assume the deal by deal method and a clawback with annual
true-up apply. Determine whether the GP must return any former profits to the LPs.
Carried Loss = 20% x 4000,000 = $800,000
In practice, an escrow account is used to regulate these fluctuations until termination of the fund.
Example 4:
The GP for private equity fund A charges a management fee of 2% and paid-in-capital and carried interest of
25
20%, using the first total return method. The total committed capital for the fund was $150,000,000. The
statistics for years 2011-2016 are shown below:
Capital Paid-in Management Operating
NAV
Carried Distributions*
NAV
Called Capital
Fees
Results*
before
Interest
after
Down*
Distribution
Distribution
2011
2012
2013
2014
2015
2016
50
20
30
20
10
10
50
70
100
120
130
140
1
1.4
2
2.4
2.6
2.8
-10
-25
25
50
60
110
39
32.6
85.6
153.2
200
267.8
0.6
9.4
13.6
20
40
60
39
32.6
85.6
132.6
150.6
174.2
*Capital Called Down, Operating Results and Distributions are given, the rest are calculated.
Paid-in Capital = Cumulative Capital Called Down
Management Fees = 2% x Paid-in Capital
NAV before Distributions = NAV after Distributions + Capital Called Down - Management Fees + Operating Results
NAV after Distributions = NAV before Distributions - Carried Interest - Distributions
Carried Interest: Carried Interest is paid if (i) NAV before Distributions > Committed Capital
2014: 20% (153.2 - 150) = 0.6
(ii) NAV before Distributions (Yr 2) > NAV before Distributions (Yr 1)
2015: 20% (200 - 153.2) = 9.4
IRR, a cashflow weighted rate of return (cash weighted or money weighted) is deemed the most appropriate
measure of private equity performance by the GIPS, Venture Capital and Private Equity Valuation Principals,
and by other Venture Capital and Private Equity standards. The interpretation of IRR in private equity should
however be subject to caution because an implicit assumption behind the IRR calculation is that the fund is
fully liquid. Whereas a significant portion of the NAV is illiquid during a substantial part of a private equity
fund's life. Therefore, IRR calculation should be interpreted cautiously.
Note also that the private equity IRR is cashflow weighted whereas most other asset class index returns are
time weighted. One solution to this problem has been to covert publicly traded equity benchmark returns to
cash weighted returns using the cashflow patterns of private equity funds. This method, however, has some
significant limitations.
The distinction between gross and net IRR is also important. Gross IRRs are estimated from capital called
down and the previous year's operating result. Whereas, Net IRRs are estimated from capital called down,
previous year's operating results, net of management fees and carried interest. Gross IRR related cashflows
between the private equity fund and its portfolio companies and Net IRR related cashflows between the
private equity fund and LPs; and so captures the returns enjoyed by investors. Fees and profit shares create
significant deviations between gross and net IRRs, because there are cyclical trends in IRR returns, the net IRR
should be benchmarked against a peer group of comparable private equity funds of the same vintage and
strategy.
Paid-in-Capital (PIC) =
DPI, RVPI, TVPI
are net of
management
fees and carried
interest
Paid in Capital
Committed Capital
Distributed to Paid-in-Capital (DPI) = Cumulative Distributions paid to LPs
or Cash on Cash Return
Cumulative Invested Capital
Residual Value to Paid-in-Capital (RVPI) = Value of LPs holdings in Fund
Cumulative Invested Capital
Total Value to Paid-in-Capital (TVPI) = DPI + RVPI
Capital utilized by
the GP
DPI measures the
LP's realized return
RVPI measures the
LP's unrealized
return
TVPI measures the
LP's realized and
unrealized return
26
If RVPI > DPI, this indicates that the firm has not successfully harvested many of its investments and that the
fund may have an extended J-curve (it is taking longer than realized to earn a positive return on its
investments). The investors should carefully examine the GP's valuations of the remaining portfolio
companies, potential write-offs and whether the routes for future exit have dried up.
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Buyout Investment
In a buyout transaction, the buyer acquires a controlling equity position in a target company.
Buyouts include takeovers, management buyouts (MBO) and leveraged buyouts (LBO). MBO is a
transaction of acquisition of stocks or organized part of a company from its owners. LBO is
characterized by a high level of external financing. The LBO has three main input parameters:
(i) cashflow forecasts of the target company, (ii) expected return from the providers of financing
(equity, senior debt, high yield bonds, mezzanine) and (ii) the amount of financing available for the
transaction.
Exit Value = Investment Cost + Earnings Growth + Increase in Price Multiple + Debt Reduction
One purpose of calculating the exit value is to determine the investment's IRR sensitivity in the exit
year. Each of the three variables should be examined using scenario analysis to determine the
plausibility of their forecasted values and the forecasted exit values. The LBO model provides the
maximum price that can be paid to the seller while satisfying the target returns for the providers of
financing.
Example 5:
An LBO transaction is valued at $1000 and has the following characteristics: Exit occurs in 5 years at a
projected multiple of 1.8 of the company's initial cost. It is financed with 60% debt and 40% equity. The $400
equity investment is composed of: $310 preference shares held by the private equity firm, $80 in equity held
by private equity firm and $10 in equity held by management equity participation (MEP). Preference shares
are guaranteed a 14% compound annual return payable at exit. The equity of the private equity firm is
promised 90% of the company's residual value at exit after creditors and preference shares are paid.
Management equity receives the other 10% residual value. By exit, the company will have paid off $350 of the
initial $600 in debt using operation cashflow. Calculate the payoff for the company's claimants and IRR and
payoff multiple for the equity claimants.
Exit Value = $1000 x 1.8 = $1800
Debt = $600 - $350 = $250
Preference Shares (14%) = $310 (1.14) 5 = $596.88
Private Equity Firm (90%) = 0.9 ($1800 - $250 - $596.88) = $857.81
Management Equity Participation (10%) = 0.1 ($1800 - $250 - $596.88) = $95.31
Preference Shares
Private Equity
Management
Debt
Before
After
310
80
10
600
596.88
857.81
95.31
250
Multiple
(After/Before)
1.92
10.722
9.53
0.4167
IRR
(Pv = - Before, Fv = After, n = 6, I/Y = ?)
14%
60.7%
56.97%
- 16.06%
Both the equity sold to managers a.k.a. MEP and the equity held by the private equity firm are most sensitive
to the level of the exit. The larger the exit multiple, the larger the upside potential for both the MEP and
equity held by the private equity firm.
27
Example 6:
Tsang and Collin analyze a potential investment in the leveraged buyout of Stone Industries. They specifically
assess the expected gain if they elect to purchase all the preference shares and 90% of the common equity in
the LBO. The buyout requires an initial investment of $10. Financing for the deal includes $6 in debt, $3.6 in
preference shares that promise a 15% annual return paid at exit and $0.4 in common equity. The expected
exit value in 6 years is $15, with an estimated reduction in debt of $2.8 over the 6 years prior to exit. What is
the multiple of expected proceeds at exit to invested funds for JRR's Stone LBO investment?
Debt = ($6 - $2.8) = $3.2
Preference Shares = $3.6 (1.15) 6 = $8.33
Common Equity = ($15 - $3.2 - $8.33) = $3.47
Initial Investment = $3.6 + 0.9 ($0.4) = $3.96
Proceeds at Exit = $8.33 + 0.9 ($3.47) = $11.45
Multiple = $11.45 = 2.89x
$3.96
Venture Capital
A private equity firm makes an investment (INV) in an early-stage start-up company. The two
fundamental concepts in venture capital investments are pre-money valuation (PRE) and
post-money valuation (POST). The pre-money valuation and investment will be negotiated between
the investee company and VC investor. Additionally, VC investor should keep in mind that his
ownership could be diluted in the future due to future financing, conversion of convertible debt into
equity and the issuance of stock options to management.
Single-Round Financing
POST = Exit Value or Pv (Exit Value)
(1 + r) n
PRE = POST - INV
f = INV
POST
or
f = Fv (INV)
Exit Value
(NPV Method)
Shares VC = Shares Founders
Price =
(IRR Method)
( )
lllll
f
1-f
INV
Shares VC
As long as the same compound rate is used to calculate the Pv of exit value and to calculate the Fv
of VC investment, the fractional ownership 'f' required is the same under either method
(NPV method and IRR method).
Multi-Round Financing
For multiple rounds of VC financing, we work backwards (from last round to first), using the NPV
method. The second round of financing is considered less risky than the first round, if lower
discount is used in the second round of financing.
28
POST 2 = Exit Value
n
(1 + r2 ) 2
POST 1 = PRE 2
n
(1 + r1 ) 1
PRE 2 = POST2 - INV 2
f 1 = INV1
POST1
( )
Shares VC = (Shares VC + Shares Founders )
2
1
Price 2 =
INV 2
Shares VC
f2
1 - f2
Shares VC = Shares Founders )
1
Price 1 =
2
INV 1
Shares VC
( )
f 2 = INV 2
POST 2
f1
1 - f1
1
After the second round, the first round investor's share dilutes from 'f 1 ' to 'f 1 (1 - f 2 )'.