CFA® Level II
Formula Sheet – 2022 Syllabus
QUANTITATIVE METHODS
QUANTITATIVE METHODS
INTRODUCTIONTO
TOLINEAR
LINEARREGRESSION
REGRESSION
INTRODUCTION
Simple Linear Regression
Y: Dependent variable/explained variable
X: Independent variable/explanatory variable
Y = b! + b" X + ϵ,
where b! is the intercept, b" is the slope coefficient,
and ϵ is the error term
The parameters can be estimated by:
Cov[Y, X] ∑$#%" (Y# − 4
Y)(X # − 4
X)
b'" =
=
4 )'
Var[X]
∑$#%" (X # − X
Y − b'" 4
X
b'! = 4
r=
Cov[Y, X]
6Var[Y]6Var[X]
To test a hypothesis about the intercept:
b'! − b!
t=
s)*"
4 )'
(X
1
s)*" = PMSE R + $
S
n ∑#%" (X # − 4
X) '
Estimated variance of the prediction error for Y:
1 (X − 4
X) '
U
s+' = s(' T1 + +
n (n − 1)s,'
General Form
Y# = b! + b" X",# + b' X ',# + ⋯ + b. X .,# + ε#
Predicting the Dependent Variable
:"# + b'' X
: '# + ⋯ + b'. :
:
X .#
Y# = b'! + b'" X
Serial Correlation
Errors are correlated across observations
Hypothesis Testing
b'/ − b/
; df = n − k − 1
t=
S)*#
Sources of Uncertainty
- Random error term
- Parameter estimates
:# ;'
SSE = ∑$#%" 9Y# − Y
F=
Sum of squares regression (SSR): Explained
variation in Y
:# − 4
Y;
SSR = ∑$#%" 9Y
'
Sum of squares total (SST): Total variation in Y
Y)'
SST = SSE + SSR = ∑$#%" (Y# − 4
Coefficient of determination:
SSR
R' =
SST
= r ' (if there is only one independent variable)
F-statistic:
MSR
SSR/k
F=
=
MSE SSE/(n − [k + 1])
Standard error of regression:
s( = √MSE =
:# ;'
−Y
n−2
$
P∑#%" 9Y#
Hypothesis Testing of Linear Regression
Coefficients
To test a hypothesis about the slope:
b'" − b"
t=
s)*!
s(
s)*! =
4)'
6∑$#%" (X # − X
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Consequences of Heteroskedasticity
- Mistakes in inference
- Unreliable F-test and t-test
- Incorrect standard error, test statistics
Testing Conditional Heteroskedasticity
- Breusch-Pagan test (one-tailed)
- Reject null hypothesis of no conditional
heteroskedasticity if nR' > critical χ' value
MULTIPLE REGRESSION
REGRESSION
Assumptions of Simple Linear Regression
Model
- Linear relationship between X and Y
- Homoscedasticity (i.e., constant variance of
residuals)
- Independence between X and Y
- Normality of the residuals
Analysis of Variance
Sum of squares error (SSE): Unexplained variation
in Y
Heteroskedasticity
Variance of error term differs across observations
- Heteroskedasticity is unconditional if error term
is uncorrelated with independent variables and
conditional if variance is correlated
- Conditional heteroskedasticity is more
problematic than the unconditional version
Significance of Multiple Regression Model
F-test
Null hypothesis is that all coefficients are equal to
zero in the population; Alternative hypothesis is
that at least one coefficient is non-zero
Mean regression sum of squares (MSR)
Mean squared error (MSE)
RSS⁄k
=
SSE⁄(n − k − 1)
- F = t ')! if only one independent variable (k = 1)
- Numerator df = k
- Denominator df = n– (k + 1)
Reject H! if F > F0
Adjusted 𝐑𝐑𝟐𝟐
Adjusted R' = 1 − d
n−1
e (1 − R' )
n−k−1
- Adjusted R' will always be less than R' because k
is greater than 0.
- It is possible for adjusted R' to be negative.
- A high adjusted R' does not necessarily mean the
regression is well specified.
Dummy Variables in a Multiple Linear
Regression
Dummy variable = 1, if true; 0, if false
To distinguish among n categories, use n – 1
dummy variable.
Intercept Dummies
Y = b! + d!2 + b"3 + ϵ
Slope Dummies
Y = b! + b" X + d" (D × X) + ϵ
Intercept and Slope Dummies
Y = b! + d! D + b" X + d" (D × X) + ϵ
Correcting for Heteroskedasticity
- Compute robust standard error
- Generalized least squares method
Consequences of Serial Correlation
- Incorrect estimate of regression coefficient
standard errors
- Inflated F-statistic, t-statistic
- Ordinary least squares (OLS) standard error
underestimates true standard error
Testing for Serial Correlation
∑64%'(εj4 − εj45" )'
DW =
≈ 2 (1 − r)
∑64%" εj'4
DW = Durbin-Watson test statistic
H! : No positive serial correlation
Outcome
DW < d7
d7 < DW < d8
DW > d8
Conclusion
Reject H!
Inconclusive
Fail to reject H!
Correcting for Serial Correlation
- Adjust coefficients’ standard errors (typically by
using the Hansen or Newey-West methods)
- Modify the regression equation
Multicollinearity
2+ independent variables are highly correlated
Consequences of Multicollinearity
- Unreliable regression coefficient estimates
- Inflated standard errors
- Low t-statistics
Detecting Multicollinearity
High R' , significant F-statistic coupled with
insignificant t-statistic for slope coefficients
Correcting for Multicollinearity
Exclude one or more of the independent variables
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Model Specification and Errors in Specification
Common misspecified functional form errors:
- Omitting key independent variables
- Failing to transform variables to account for nonlinear relationships
- Using data from different samples that should not
be combined
Common time series misspecification errors:
- Including lagged dependent variables as
independent variables with serially correlated
errors
- Including a function of a dependent variable as an
independent variable
- Independent variables that are measured with
error
- Nonstationarity, the most common time-series
misspecification error, resulting from using data
with parameters (i.e., mean and variance) that
have changed over time
Models with Qualitative Dependent Variables
For models with qualitative dependent variables, it
is often preferable to use logit model.
- p is the probability that an event happens
9
- The logistic transformation is ln n o
-
9
"59
- ln n
"59
is known as the odds of an event happening
9
"59
o is the natural logarithm of the odds of an
event happening, which is known as log odds or
logits
ANALYSIS
TIME SERIES
SERIES ANALYSIS
Time Series Challenges
- Linear regression assumptions violated
- Correlated residual errors
- Mean/variance of time series changes over time
Linear Trend Model
y4 = b! + b" t + ε4 ,
t = 1, 2, … , T
Log-Linear Trend Model
y4 = e)":)!4 ,
t = 1, 2, … , T
t = 1, 2, … , T
ln y4 = b! + b" t + ε4 ,
Autoregressive Time-Series Model
A first-order autoregression, AR(1), predicts a
variable (x4 ) based on its most recent value (x45" ):
x4 = b! + b" x45" + ε4
A model using values for p periods, AR(p), is:
x4 = b! + b" x45" + b' x45' + ⋯ + b9 x459 + ε4
Covariance Stationary Assumption
For inferences from AR models to be valid, it is
assumed that the time series’ mean and variance
are constant over time
Detecting Serial Correlation of Error
Residual autocorrelation
t=
1⁄√T
T = number of observations in time series
Mean Reversion for an AR(1) Model
Mean reverting level =
- xj4:" = x4 when x4 =
- xj4:" > x4 when x4 <
- xj4:" < x4 when x4 >
!!
"#!"
)"
"5)!
)"
"5)!
)"
"5)!
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Root Mean Squared Error (RMSE)
- In-sample forecast errors are the residuals from
the time period used to estimate the parameters
of the model.
- Out-of-sample forecast errors are the residuals
from a time period not used to fit the data.
- The root mean squared error (RMSE) is the square
root of the average squared error. A relatively low
RMSE for out-of-sample data indicates a good fit.
Random Walk and Unit Root
x4 = x45" + ε4
E(ε4 ) = 0, E(ε'4 ) = σ' , E(ε4 ε; ) = 0, if t ≠ s
Random Walk with Drift
E(ε4 ) = 0
x4 = b! + x45" + ε4 ,
Take first difference before analyzing
y4 = x4 − x45"
b! ≠ 0
y4 = b ! + ε4 ,
Unit Root Test of Nonstationarity
For an AR(1) time series to be covariance
stationary, the absolute value of b1 must be < 1.
Dickey-Fuller Test
x4 = b! + b" x45" + ε4
x4 − x45" = b! + (b" − 1)x45" + ε4
x4 − x45" = b! + g" x45" + ε4
H! : g" = 0 (has unit root)
H< : g" < 0 (does not have unit root)
If the time series has a unit root, we can model the
first-differenced series using an autoregressive
time series.
Moving Average Time-Series Model
MA(q) model
x4 = ε4 + θ" ε45" + θ' ε45' + ⋯ + θ9 x45=
E(ε4 ) = 0, E(ε'4 ) = σ' , E(ε4 ε; ) = 0, if t ≠ s
Seasonality in Time Series
- Regular pattern within a year
- Significant seasonal autocorrelation of error term
Autoregressive Moving Average Model
An ARMA(p, q) model includes p autoregressive
parameters and q moving-average parameters
- An AR(1) model has a “one-period memory” and
all autocorrelations other than the first will be 0
Limitations of AR Models
- Highly unstable parameters
- Imperfect criteria for deciding p and q
- Should not be used for <80 observations
Autoregressive Conditional
Heteroskedasticity Models
ARCH(1) model
ε4 ~ N(0, a! + a" ε'45" )
If a" = 0, variance of error in every period is a! .
The variance is constant over time and does not
depend upon past errors
If a" > 0, variance of error in one period depends
on how large the squared error was in the previous
period. If a large error occurs in one period,
variance of error in the next period will be larger:
εj'4 = a! + a" εj'45" + u4
If a time-series model has ARCH(1) errors, the
variance of error in period t+1 can be predicted in
period t:
z'4:" = aj! + aj" εj'4
σ
Cointegration of Time Series
Two time series are co-integrated when they
have a financial or economic relationship that
prevents them from diverging without bound in
the long run.
Cointegration Detection
Engle-Granger or Dickey-Fuller test
Other Issues in Time Series
- Large forecast uncertainty
- Need to consider uncertainty of error term and
estimated parameters
MACHINE
MACHINE LEARNING
LEARNING
Used for client profiling, asset allocation, stock
selection, portfolio construction, trading, etc.
Supervised ML: Uses labeled data to infer patterns
between inputs and outputs
- Dependent variable (Y) is the target and
independent variables (X) are features
- Can be used for regression (linear and non-linear)
and classification problems
Unsupervised ML: Finds patterns within unlabeled
data; there is no dependent variable
- Can be used for dimension reduction and
clustering problems
Deep Learning: Sophisticated algorithms for tasks
such as image classification, face recognition, and
natural language processing
Reinforcement Learning (RL): An algorithm learns
from the data that it generates
Neural networks: Highly flexible ML algorithms
used for classification, regression, deep learning,
and reinforcement learning
ML Methods for Different Types of Variables
Supervised Unsupervised
Variables
ML
ML
Regression
Dimensionality
- Linear,
Reduction
LASSO
- PCA
Continuous - Logistic
Clustering
- CART
- K-Means
- Random
- Hierarchical
Forest
Classification Dimensionality
- Logit
Reduction
- SVM
- PCA
Categorical
- KNN
Clustering
- CART
- K-Means
- Hierarchical
Neural
Neural
Continuous networks,
networks,
Deep
Deep Learning,
or
RL
Categorical Learning,
RL
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Training an ML Model: Sampling
Training an ML model requires a dataset to be
divided into three non-overlapping samples:
1. Training sample: In-sample data used to find
relationships
2. Validation sample: Out-of-sample data used to
validate relationships found in training sample
3. Test sample: Out-of-sample data used to test the
model’s predictive powers
Overfit models explain the training data well but
do not generalize to the out-of-sample data.
K-fold cross-validation can be used to prevent
overfitting.
Training an ML Model: Errors
Bias error: Does not explain training data well
(underfit); more likely for linear functions
Variance error: Model performs differently with
out-of-sample data because it has incorporated
noise from training data (overfit); more likely for
non-linear functions
Base error: Unavoidable errors due to randomness
The trade-off between bias error and variance
error can be shown on a fitting curve
Model
complexity
Low
High
Bias error
Variance
error
Higher
Lower
Lower
Higher
Supervised ML Methods
Penalized regression/Regularization: Seeks to
reduce the risk of overfitting by imposing a penalty
on additional features; Least Absolute Shrinkage
and Selection Operator (LASSO) uses a
hyperparameter, λ, as a penalty
Support Vector Machine (SVM): Linear classifier
model used for binary classification, regression,
and outlier detection; Soft margin classification is a
non-linear alternative to SVM
K-Nearest Neighbor (KNN): Non-parametric
method typically used for classification (e.g., credit
rating prediction), but also used for regression
Classification and Regression Tree (CART):
Produces a tree with a root node, decision nodes,
and terminal nodes; Iterative structure is used to
find relationships in non-linear data, but it is a
black box method
Ensemble learning: Use multiple models to reduce
error rate relative to relying on one model.
Examples include majority-vote classifier,
bootstrap aggregating, and random forest.
Unsupervised ML Methods
Principal Components Analysis (PCA): Features are
grouped together to reduce the number of
independent variables in a model. It significantly
reduces model complexity but is a black box
method.
Clustering: K-means clustering, hierarchical
clustering, and dendrograms are used to organize
observations into groups.
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TP
TP + FP
TP
Recall (R) =
TP + FN
TP + TN
Accuracy (A) =
TP + FP + TN + FN
PROJECTS
BIG DATA
DATA PROJECTS
Precision (P) =
4Vs of Big Data
1. Volume
2. Variety
3. Velocity
4. Veracity
ML Model Building Steps
1. Conceptualization
2. Data Collection/Curation
3. Data Preparation and Wrangling
- Data Cleansing
- Data Preprocessing
4. Data Exploration
- Exploratory Data Analysis
- Feature Selection
- Feature Engineering
5. Model Training
- Method Selection
- Performance Evaluation
- Tuning
F1 score =
FP
TN + FP
TP
True Positive Rate (TPR) =
TP + FN
False Positive Rate (FPR) =
Receiver Operator Characteristic (ROC) technique
plots FPR on x-axis and TPR on y-axis.
Root Mean Squared Error (RMSE) is calculated as
the square root of the sum of mean squared errors.
$
RMSE = }~
Text ML Model Building
1. Text problem formulation
2. Text curation
3. Text preparation and wrangling
4. Text exploration
#%"
Errors Addressed with Data Cleansing
Incompleteness error: Data not present
Invalidity error: Outside meaningful range
Inaccuracy error: Not a measure of true value
Inconsistency error: Conflicts between data points
Non-uniformity error: Multiple formats used
Duplication error: Duplicate observations present
Data Wrangling Methods
Feature extraction: Creating a new variable from an
existing variable to improve analysis
Aggregation: Combining similar variables
Filtration: Removing irrelevant rows
Selection: Removing irrelevant columns
Conversion: Making adjustments to increase
relevance
Addressing Outlier Data
Trimming: Removing the top/bottom X%
Winsorization: Replacing extreme high/low
observations with maximum/minimum values
Normalization:
X # − X @#$
X $>?@<7#A(B =
X @<, − X @#$
Standardization:
X# − µ
X ;4<$B<?B#A(B =
σ
Text Wrangling Methods
1. Lowercasing
2. Stop words
3. Stemming
4. Lemmatization
Model Performance Evaluation
Actual
Actual
training
training
label - 1
label - 0
Predicted
result - 1
Predicted
result - 0
True
positive
(TP)
False
negative
(FN) (Type
II Error)
2 ×P×R
P+R
False
positive
(FP)
True
negative
(TN)
(Predicted# − Actual#)'
n
A lower RMSE indicates potentially better model
performance if historical relationships hold.
ECONOMICS
ECONOMICS
CURRENCY
EXCHANGERATES
RATES
CURRENCY EXCHANGE
Factors Influencing Bid/Ask Spreads
Currency pair: Wider for less liquid currencies
Time of day: Wider when NY/London closed
Market conditions: Wider when more volatile
Contract term: Wider for longer-term forward
contracts due to lower liquidity and increased
exposure to credit risk and interest rate risk
Covered Interest Rate Parity
Days
1 + i+ n
o
360 Ç
F+⁄B = S+⁄B 
Days
o
1 + iB n
360
Uncovered Interest Rate Parity
%∆S+(⁄B = i+ − iB
Estimated Future Spot Rate
F+⁄B − S+⁄B
= %∆S+(⁄B = i+ − iB
S+⁄B
Absolute Version of PPP
P+ = S+⁄B × PB
S+⁄B = P+ ⁄ PB
Relative Version of PPP
%∆S+⁄B ≅ π+ − πB
Ex-ante Version of PPP
%∆S+(⁄B ≅ π(+ − π(B
The Fisher Effect and Real Interest Rate Parity
i = r + π(
(i+ − iB ) = (r+ − rB ) + (π(+ − π(B )
(r+ − rB ) = (i+ − iB ) − (π(+ − π(B )
Real Interest Rate Parity
(r+ − rB ) = 0
International Fisher Effect
(i+ − iB ) = (π(+ − π(B )
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3
Mundell-Fleming Model
High capital mobility
Fiscal Policy
Expansionary
Restrictive
Expansionary
Restrictive
DC ↓
Ambiguous
Ambiguous
Low capital mobility
Fiscal Policy
Expansionary
Restrictive
Monetary Policy
DC ↑
Monetary Policy
Expansionary
DC ↓
Ambiguous
Restrictive
Ambiguous
DC ↑
Currency Crisis Warning Signs
- Recent liberalization of capital markets
- Large foreign capital inflows, especially s/t funds
- Banking crises, either just before or concurrent
- Fixed or partially fixed exchange rates
- Sudden, sharp decline in FX reserves
- Recent spike in domestic currency value
- Deteriorating terms of trade
- Money supply growing faster than bank reserves
- Recent high inflation
ECONOMICGROWTH
GROWTHAND
ANDINVESTMENT
ECONOMIC
DECISIONS
INVESTMENT DECISIONS
Factors Affecting Growth: Developing Countries
- Low rates of savings and investment
- Poorly developed financial markets
- Weak legal systems and failure to enforce laws
- Lack of property rights and political stability
- Poor public education and health services
- Excessive taxes and regulations
- Restrictions on international trade/capital flows
Potential Growth and Stock Market Returns
𝑃𝑃
𝐸𝐸
+ %∆
%∆𝑃𝑃 = %∆𝐺𝐺𝐺𝐺𝐺𝐺 + %∆
𝐸𝐸
𝐺𝐺𝐺𝐺𝐺𝐺
Cobb-Douglas Production Function
F (K, L) = K D L"5D
Capital Deepening vs. Technological Progress
Growth Accounting
∆K
∆L
∆Y ∆A
=
+ α d e + (1 − α) d e
A
K
L
Y
Growth rate in potential GDP
= L/T growth rate of labor force
+ L/T growth rate in labor productivity
Labor Supply Inputs
- Population growth
- Labor force participation
- Net migration
- Average hours worked
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Classical Model
- Labor productivity increases population growth
- Population growth accelerates as per capita
incomes increase
- Diminishing marginal returns to labor input will
lead to decline in per capita income
Classical model failed because:
1. Population growth slowed as incomes rose
2. Diminishing marginal returns to labor input
were more than offset by technological progress
Neoclassical Model
- Based on Cobb-Douglas production function
- Both capital (K) and labor (L) are subject to
diminishing marginal productivity
- In the steady state, the output-to-capital ratio is
constant because they grow at the same rate
- In the long run, output per capita is driven by:
1. Savings/investment rate
2. Rate of technological change
3. Population growth
∆y
∆A
∆k
=
+ αd e
y
A
k
∆k
Y
= sd e− δ − n
k
K
∆y
θ
Growth rate of output per capita =
=
y
1−α
θ
∆Y
=
+n
Growth rate of output =
1−α
Y
Implications:
- Capital accumulation, capital deepening, and an
increase in savings rate can only temporarily
increase growth, but technological improvements
can have a permanent impact
- Per capita income growth will converge across
countries
Endogenous Growth Theory
- Output per worker is proportional to stock of
capital per worker (k ( )
- c is constant marginal product of capital
y( = f(k ( ) = ck (
∆y(⁄y( = ∆k ( ⁄k ( = sc − δ − n
- A higher savings rate can permanently increase
the potential GDP growth rate
ECONOMICS
OFREGULATION
REGULATION
ECONOMICS OF
Types of Regulation
- Statutes enacted by legislatures
- Administrative regulations for agencies
- Judicial laws established by legal rulings
Classification of Regulators
- Independent regulators: Granted the ability to
make regulations by government
- Self-regulatory bodies: Private organizations that
regulate members, typically industry peers
- Self-regulatory organizations: Independent
industry bodies that have been granted law
enforcement powers
- Standard-setting bodies: Establish rules but lack
any enforcement powers (e.g., IFRS)
Regulatory Interdependencies
- Regulatory capture: Businesses use relationship
with regulators to serve their interest
- Regulatory competition: Regulators from different
jurisdictions compete to attract certain entities
- Regulatory arbitrage: Businesses exploit
differences between economic substance and
regulatory interpretation
Regulatory Tools
- Price mechanisms (e.g., taxes, subsidies)
- Mandates and restrictions
- Provision of public goods
- Public financing of private projects
Cost-Benefit Analysis of Regulation
Regulatory burden: Private costs of regulation, both
direct and indirect
Net burden: Private costs less private benefits
FINANCIAL REPORTING
FINANCIAL REPORTING
INTERCORPORATE INVESTMENTS
INVESTMENTS
Classification
- Financial assets (<20%): Buyer has no significant
control over the investee
- Associates (20% - 50%): Buyer has significant
influence but not control
- Joint venture: Entity is operated by companies
that share control
- Business combinations (>50%): Buyer has control
Investments in Financial Assets
- Under IFRS 9, all financial assets are initially
measured at fair value (cost basis at acquisition)
- In subsequent periods, financial assets may be
measured at either amortized cost or fair value
- To be measured at amortized cost, financial
assets must meet the following criteria:
o Held to collect contractual cash flows
o Cash flows may only be principal and interest
- Financial assets that fail to meet both criteria
must be measured at either fair value through
profit or loss (FVPL) or fair value through other
comprehensive income (FVOCI)
- Securities are classified as follows:
o Debt securities: Carried at amortized cost if
held to maturity, but must be carried at fair
value if it is possible that they may be sold
o Equities: Can be held at FVPL or FVOCI
o Derivatives: Must be carried at FVPL unless
they are being used as hedging instruments
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4
Investments in Associates and Joint Ventures
- Equity method (one-line consolidation):
o Initial investment is recorded at cost on the
balance sheet as a non-current asset
o Carrying amount is adjusted upward to
reflect a proportionate share of earnings
o Dividends received are treated as a return of
capital and reduce the carrying amount
o If carrying value falls to 0, equity method is
discontinued and no further losses recorded
- Profits from transactions with associates cannot
be realized until the products are sold/used
Classifying Business Combination
- Merger: A + B = A
- Acquisition: A+ B = (A + B)
- Consolidation: A + B = C
Acquisition Method
- The fair value of the consideration given by the
acquiring company is used
- Direct costs of the business combination are
expensed as incurred
- IFRS: Full or partial goodwill
US GAAP: Full goodwill only
Partial goodwill
= Fair value of consideration given
− Acquirer E s shares of the fair value of A/L
Full goodwill
= Fair value of acquiredE s entity
− Fair value of the entity E s A/L
- Non-controlling interest (NCI) is the portion of
subsidiary’s equity held by third parties
o Full goodwill: NCI is measured at fair value
o Partial goodwill: NCI is based on the
proportionate share of net identifiable assets
EMPLOYEE COMPENSATION:
COMPENSATION:POSTPOSTEMPLOYEE
SHARE BASED
EMPLOYMENT AND SHARE-BASED
Types of Pension Plan
Defined benefit (DB): Firm makes periodic
payments to employee after retirement;
Employer’s contributions may vary depending on
assumptions and the performance of plan assets
Defined contribution (DC): Employer’s obligation is
limited to periodic contribution; Future benefits
depend on investment performance
Pension Obligation
Present value of estimated future payments to
employees for benefits earned to date
Net Pension Asset/Liability
Overfunded
- Plan assets exceed the pension obligation
- Sponsor reports net pension asset
Underfunded
- Pension obligation exceeds plan assets
- Sponsor reports net pension liability
Components of Pension Expense (IFRS)
- Current service cost (P&L)
- Past service cost (P&L)
- Net interest expense (P&L)
o Net liability × Interest rate
- Net return on plan assets (OCI, not amortized)
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o Actual return − (Assets × Interest rate)
- Actuarial losses (OCI, not amortized)
Components of Pension Expense (US GAAP)
- Current service cost (P&L)
- Past service cost (OCI, then amortized)
- Interest expense (P&L)
o Liability × Interest rate
- Expected return (P&L)
o Assets × Expected return
- Net return on plan assets (OCI, then amortized)
o Actual return − (Assets × Expected return)
- Actuarial losses (OCI, then amortized)
DB Plan Assumptions
- Assuming a higher discount rate reduces the
pension liability and service costs
- Assuming a higher rate of compensation growth
increases liability and service costs
- Assuming a higher expected return on assets
reduces the pension expense under US GAAP but
not IFRS and has no impact on the value of assets
Benefit Calculations
- Annual benefit = (Estimated final salary × Benefit
formula) × Years of service
- Value at retirement date of estimated future
benefits = PV of annual benefit during retirement
- Annual unit credit = Value at retirement
date/Years of service
Cash flows for DB Plan Sponsor
The difference between periodic contributions to a
plan and total pension costs of the period can be
viewed as financing activity
- If contributions to a plan are more (less) than the
total pension costs of the period, the excess is
similar to repaying debt (additional borrowing)
Share-Based Compensation
- Aims to align employees’ interest with owners’
- Requires no current-period cash outlays
- Treated as an expense, which reduces earnings
- Potentially dilutes EPS by increasing shares
Stock Grants
- May be outright, restricted, or contingent
- Recipients are fully exposed to downside risk,
may become risk-averse
Stock grants’ compensation expense:
- Amount based on stock’s fair value on grant date
- Allocated over the employee’s service period
Stock Options
Must be valued using a pricing model based on key
assumptions such as:
- Exercise price
- Stock price volatility
- Estimated life of each award
- Estimated number of options to be forfeited
- Dividend yield
- Risk-free interest rate
MULTINATIONAL OPERATIONS
OPERATIONS
MULTINATIONAL
Presentation and Functional Currencies
- Presentation currency: Currency in which the
company presents its financial statements
- Functional currency: Currency in which the
company conducts its primary activity
- Local currency: Used within the country in which
the company operates
- Often, functional currency of subsidiary ≠
functional and presentation currency of parent
Remeasurement/Translation Methods
Current rate method
- Use when the subsidiary’s functional currency is
different from the parent’s functional currency
- FX gain/loss reported in shareholders’ equity as
part of cumulative translation adjustment (CTA)
- Exposure is net assets (assets minus liabilities)
Temporal method
- Use when the subsidiary’s functional currency is
the same as the parent’s functional currency
- Remeasurement gain/loss reported in
income statement
- Exposure is net monetary assets (monetary assets
minus monetary liabilities)
Exchange Rate for Each Line Item
Monetary assets
Monetary
liabilities
Nonmonetary
assets
Nonmonetary
liabilities
Common stock
Revenues/
Expenses
COGS
Depreciation
Current
Rate
Current
Temporal
Current
Current
Historical
Current
Current
Historical
Average
Average
Average
Current
Historical
Historical
Average
Historical
Historical
Translation Adjustments
Balance
Foreign Currency
Sheet
Strengthens
Weakens
Exposure
Positive
Negative
Net asset
translation
translation
adjustment
adjustment
Negative
Positive
Net
translation
translation
liability
adjustment
adjustment
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5
Effect of Translation Method on Ratios
- Pure income statement and pure balance sheet
ratios are unaffected by the current rate method
- Under both the current rate and temporal
methods, ratios using both income statement and
balance sheets figures will be different from the
same ratios calculated using local currency
statements before translation
Examples:
1. Receivables turnover (sales/receivables) is the
same under both current and temporal methods
- Sales are translated at the average exchange
rate under both
- Receivables are translated at the current
exchange rate under both
2. Current ratio (current assets/current liabilities)
will be different under the two methods
- Inventory is translated at the current
exchange rate under the current method, but
at the historical exchange rate under the
temporal method
- If the subsidiary’s currency appreciates
relative to the parent, the current ratio will
be higher under the current rate method than
the temporal method
Subsidiaries in Hyperinflationary Economies
Under IFRS
- Restate subsidiary’s local currency financial
statements for local inflation
- Translate inflation-restated foreign currency
financial statements into the parent’s
presentation currency using the current FX rate
Under U.S. GAAP
- Use the temporal method to translate the
subsidiary’s local currency financial statements
- Include the resulting translation adjustment as a
gain or loss in determining net income
ANALYSIS OF
OF FINANCIAL
FINANCIALINSTITUTIONS
INSTITUTIONS
ANALYSIS
Key Components of Basel III:
- Minimum capital requirements: Based on riskweighted assets
- Minimum liquidity: Enough to handle a 30-day
liquidity stress scenario
- Stable funding: To cover needs over a one-year
horizon; based on the length of the deposits and
the type of depositor
Minimum Capital Requirements
under Basel III:
- Common Equity Tier 1 capital ≥ 4.5%
- Total Tier 1 capital ≥ 6.0%
- Tier 1 plus Tier 2 capital ≥ 8.0%
The CAMELS Approach
1. Capital adequacy
2. Asset quality
3. Management capabilities
4. Earnings sufficiency
5. Liquidity position
6. Sensitivity to market risk
Relevant Factors Not Covered by CAMELS
Banking-Specific Analytical Considerations
- Government support
- Government ownership
- Mission of banking entity
- Corporate culture
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Considerations relevant to any company
- Competitive environment
- Off-balance-sheet items
- Segment information
- Currency exposure
- Risk factors
- Basel III disclosures
Analyzing Property and Casualty Insurers
Loss and loss adjustment expense ratio:
Loss expense + Loss adjustment expense
Net premiums earned
Underwriting expense ratio:
Underwriting expense
Net premiums written
Combined ratio:
Loss and loss adjustment expense ratio
+ Underwriting expense ratio
Dividends to policyholders ratio:
Dividends to policyholders
Net premiums earned
Combined ratio after dividends:
Combined ratio + Dividends to policyholders ratio
EVALUATINGQUALITY
QUALITYOF
OFFINANCIAL
FINANCIAL
REPORTS
EVALUATING
REPORTS
Quality Spectrum of Financial Reports
1. GAAP-compliant; decision-useful, high-quality
earnings (adequate return/sustainable)
2. GAAP-compliant; decision-useful, low-quality
earnings (inadequate return/not sustainable)
3. GAAP-compliant; not decision-useful
4. GAAP-compliant; earnings management
5. Non-compliant accounting
6. Fictitious transactions
Beneish Model
M-scores: Higher values indicate a greater
likelihood of earnings manipulation
- Threshold M-scores:
o −1.78 (3.8% likelihood of manipulation)
o −1.49 (6.8% likelihood of manipulation)
- Days Sales in Receivables Index (DSRI): Large
increase could mean revenue inflation
- Gross Margin Index (GMI): >1 means gross margin
has decreased, possible pressure to manipulate
- Asset Quality Index (AQI): Increase could indicate
excessive capitalization of expenses
- Sales Growth Index (SGI): Rapid sales growth can
create pressure to manipulate earnings
- Depreciation Index (DEPI): >1 means depreciation
rate has decreased, indicating manipulation
- SG&A Expenses Index (SGAI): >1 means increasing
SG&A expense, might encourage manipulation
- Accruals: More accruals indicate manipulation
- Leverage index (LEVI): >1 shows increasing debtto-asset ratio, greater pressure to manipulate
In the Beneish model, SGAI and LEVI are negatively
correlated with the M-score; other variables are
positively correlated
Measures of Earnings Persistence and Accruals
- Earnings forecast should not include nonrecurring items
- Cash components are more persistent than
accrual components
- Earnings with significant accruals will experience
a faster mean reversion
- A company that consistently performs slightly
better than benchmark should be scrutinized
- Regulatory enforcement actions and restatements
of previous financial statements are red flags
Altman model: A lower Z-score indicates a higher
probability of bankruptcy
Cash Flow Quality
- A startup company might be expected to have
negative operating and investing cash flows,
funded by positive financing cash flows from
equity issues or borrowing
- For established companies, high-quality
operating cash flow (OCF) should be:
- Positive
- Derived from sustainable sources
- Enough for capex, dividends, and debt service
- Stable relative to peers
Balance Sheet Quality Indicators
High financial reporting quality requires:
- completeness
- unbiased measurement
- clear presentation
A high-quality balance sheet requires:
- optimal amount of leverage
- adequate liquidity
- economically successful asset allocation
Sources of Information about Risk
- Financial statements, including notes
- Audit opinion or discretionary change in auditor
- Management commentary
- Regulatory disclosures
- Financial press
INTEGRATION
FINANCIAL STATEMENT
STATEMENT
INTEGRATION OF
OF FINANCIAL
ANALYSIS TECHNIQUES
TECHNIQUES
Framework for Analysis
1. Define purpose for analysis
2. Collect input data
3. Process data
4. Analyze/interpret processed data
5. Develop and communicate conclusions
6. Follow up
Sources of Earnings and ROE
Net Income
Sales
Assets
ROE =
×
×
Sales
Assets
Equity
ROE =
ROE =
Net Profit
Asset
×
× Leverage
Margin
Turnover
NI
EBT EBIT Sales Assets
×
×
×
×
EBT EBIT Sales Assets Equity
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6
Accruals and Earnings Quality
Balance sheet approach
AccrualsFG = NOA($B − NOA)(H
Accruals RatioFG =
NOA($B − NOA)(H
9NOA($B + NOA)(H ;⁄2
Cash flow statement approach
AccrualsIJ = NI − CFO − CFI
Accruals RatioIJ =
NI − CFO − CFI
9NOA($B + NOA)(H ;⁄2
CORPORATECORPORATE
FINANCE
FINANCE
CAPITAL STRUCTURE
STRUCTURE
Modigliani and Miller Propositions
MM Proposition I: A firm's capital structure would
have no effect on its value, assuming:
1. Investors have homogeneous expectations
2. No market frictions (e.g., transaction costs, taxes,
or costs of financial distress)
3. No agency costs
4. Investors can borrow and lend at risk-free rate
5. Investing/financing decisions are independent
MM Proposition II: Cost of equity increases with the
debt-to-equity ratio.
Without Taxes
Firm value
WACC
Cost of
Equity
With Taxes
Firm value
WACC
Cost of
Equity
VK = VL
D
E
d e rB + d e r(
V
V
D
r( = r! + (r! − rB ) d e
E
VK = VL + tD
D
E
d e rB (1 − t) + d e r(
V
V
D
r! = r" + (r" − r# )(1 − t) * E
r" = cost of capital for a firm financed only by equity
Costs of Financial Distress
Companies with negative earnings can experience
financial distress, which leads to explicit and
implicit costs:
- The explicit, or direct, costs of financial distress
include the legal and administrative cash
expenses associated with bankruptcy.
- Implicit, or indirect, costs include foregone
investment opportunities and the loss of
trust from customers, creditors, suppliers,
and employees.
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The expected cost of financial distress is a function
of two factors:
1. the total cost (direct and indirect) incurred in
the event of financial distress or bankruptcy
2. the likelihood that financial distress or
bankruptcy will actually occur
The probability that a company will
experience financial distress increases with its
use of leverage.
Agency Costs
Agency costs arise from conflicts between
managers and owners. The interests of
managers, shareholders, and bondholders
are not always aligned.
The components of net agency costs of equity are:
- Monitoring costs
- Bonding costs
- Residual loss
Free Cash Flow Hypothesis
Including more debt in a company’s capital
structure will impose discipline on managers.
It reduces the management’s opportunity to
waste cash.
Pecking Order Theory
Because they have an asymmetric information
advantage, managers prefer capital sources that
reveal the least amount of information:
1. Internally generated earnings (best option)
2. New debt
3. New equity (least attractive for managers)
Debt can be viewed as a positive signal because it
implies management is confident the company can
support the interest and principal payments.
Static Trade-Off Theory
Balances costs of financial distress with tax shield
benefits from using debt
VK = VL + tD − PV(costs of financial distress)
Country-Specific Factors Impacting Capital
Structures of Companies
Factors
D/E ratio
Maturity
Auditors and analysts
Lower
Longer
Efficient legal system
Common law system
Taxes favoring equity
Active capital market
Bank-based lending
Institutional investors
High inflation
High GDP growth rate
Lower
Lower
Lower
Higher
Lower
Lower
Lower
Longer
Longer
Longer
Longer
Shorter
Longer
ANALYSIS OF DIVIDENDS AND SHARE
REPURCHASES
SHARE
REPURCHASES
Stock Dividends
- In lieu of paying a cash dividend, companies may
distribute additional shares.
- Such a distribution increases the number of
shares outstanding without affecting the
company's total market value.
- The stock dividend does not affect the company’s
balance sheet or income statement, so liquidity
and financial leverage ratios are unchanged.
Stock Splits
- They have no economic effect on the company
and should not impact shareholders’ wealth.
- Reverse stock splits reduce the number of shares
outstanding, but they still have no economic
impact on the company or shareholder.
Dividend Theories
- Dividends are irrelevant: MM Propositions
- Bird-in-the-hand argument: Investors prefer cash
dividends over unrealized capital gains
- Signalling: Managers want to increase dividends
as a signal of strength to investors
- Agency cost: Paying dividends to owners limits
managers’ ability to fund negative NPV projects
Factors Affecting Dividend Policy
- Investment opportunities: Company with more
(less) investment opportunities will pay out less
(more) in dividends
- Expected volatility of future earnings: Companies
with greater earnings volatility are less likely to
increase dividends; Increasing dividends
increases chances of not maintaining the dividend
- Financial flexibility: Companies seeking more
flexibility are less likely to initiate or increase
dividends
- Tax considerations: Investors consider dividends
on an after-tax basis
- Flotation costs: Make using newly issued stock
more expensive than internally generated funds;
Smaller companies face higher flotation costs
- Contractual and legal restrictions: Bond
indentures, obligations to preferred shareholders,
rules against capital impairment, etc.
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7
Tax System and Dividend Policy
- Double Taxation: Corporate earnings are first
taxed at the corporate level and then again at the
shareholder level when distributed as dividends.
ETR = Corp. tax rate
+(1 − Corp. tax rate)(Ind. tax rate)
- Split-rate: Corporate earnings distributed as
dividends are taxed at a lower rate than the
earnings retained. Dividends are taxed as
ordinary income for investors. The effective tax
rate on dividends is calculated with the same
formula used under a double taxation system, but
the lower corporate tax rate is used for dividend
distributions
ETR = Corp. tax rate2
+(1 − Corp. tax rate2 )(Ind. tax rate)
- Imputation: Corporate earnings distributed as
dividends are taxed at the shareholder’s tax rate.
Individual shareholders receive a tax credit if
their marginal tax rate is less than the corporate
tax rate. If their marginal tax rate is higher, they
only pay tax based on the difference between
their marginal tax rate and the corporate rate.
Payout Policies
Stable Dividend Policy
D" = D! + (E" × PR 6 − D! ) × A
where
E" is expected earnings
PR 6 is target payout ratio
A is an adjustment factor (1/Years to target ratio)
Constant Dividend Payout
Dividend as a constant share of earnings
Residual Dividend Payout
Pay out earnings remaining after funding all
positive NPV capital projects consistent with its
target capital structure.
D = max 0, Earnings −
Capital Equity %
Ç
×
Budget in capital
structure
Share Repurchase Methods
- Buy in the open market
- Buy back a fixed number of shares at a fixed price
- Dutch auction
- Repurchase by direct negotiation
Financial Statement Effects of Repurchases
Changes in Earnings per Share
- If the net income stays the same, the EPS will
increase after a share repurchase because there
are fewer shares outstanding.
- Share repurchases made with borrowed funds: If
the earnings yield is greater (lower) than the
after-tax cost of the funds, the EPS will increase
(decrease).
- An increase in EPS does not necessarily imply an
increase in shareholders’ wealth.
Changes in Book Value per Share
- If the market price per share is greater (less) than
its book value per share, a stock repurchase will
decrease (increase) the book value per share.
Valuation Equivalence
- Ignoring the tax and information effects, cash
dividends and share repurchases should have the
same effect on shareholder wealth.
- However, if shares are repurchased at a premium,
wealth will be transferred from remaining
shareholders to the seller.
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Advantages of Share Repurchases vs. Dividends
- Potential tax advantages
- Signaling
- Managerial flexibility
- Offsets dilution from employee stock options
- Adjusting capital structure
- Increasing EPS
Analysis of Dividend Safety
All else being equal, the risk of a dividend cut or
omission increases when we have:
- A higher dividend payout ratio (dividends/net
income)
- A lower dividend coverage ratio (net
income/dividends)
FCFE
FCFE
coverage =
Dividends + Share repurchases
ratio
- If the ratio is equal to 1, the company is
distributing all available cash to shareholders.
- If the ratio is significantly greater than 1, the
company is keeping some earnings to enhance
liquidity.
- If the ratio is significantly less than 1, the
company is borrowing cash to pay dividends,
thereby decreasing liquidity. This is
unsustainable because the company is paying out
more than it can afford.
CORPORATE GOVERNANCE
CORPORATE
GOVERNANCEAND
ANDESG
ESG
Ownership Structure
- Dispersed: Many small minority shareholders
- Concentrated: One majority shareholder or a few
shareholders with large minority positions
- Horizontal: Cross-holding share arrangements
- Vertical: Controlling interest in two or more
holding companies with their own subsidiaries
- Dual-class shares: Disproportionate voting rights
for certain classes of shares
Conflicts with Different Ownership Structures
- Dispersed ownership, dispersed voting power:
Leads to principal-agent problem where strong
managers take advantage of relatively weak
shareholders
- Concentrated ownership, concentrated voting
power: Leads to the principal-principal problem
where a captive board makes decisions to the
detriment of minority owners
- Dispersed ownership, concentrated voting power:
Also leads to the principal-principal problem, but
for the benefit of large minority owners holding
dual-class shares
- Concentrated ownership, dispersed voting power:
Can occur if voting caps are used
Board Composition
One-tier: Executive and non-executive directors
Two-tier: Independent supervisory board oversees
management board
CEO duality: The same individual serves as both
CEO and chair of a one-tier board, raising concerns
about a lack of independent oversight
Special voting arrangements: May require directors
to secure dual majorities among both controlling
and minority shareholders
Stewardship codes: Encourage asset managers to
increase engagement in corporate governance
Board Policies and Practices
- Board of Directors Structure
- Board Independence
- Board Committees
- Board Skill and Experience
- Board Composition
- Other Considerations in Board Evaluation
Executive Remuneration
- Companies may have a say on pay provision that
allows shareholders to vote or at least provide
feedback on compensation.
- A claw-back policy allows companies to regain
previously paid compensation if mismanagement
or misconduct is subsequently uncovered.
Shareholder Voting Rights
- Under a straight voting share structure, each
share has the same voting rights.
- In a dual-class structure, certain classes of shares
have enhanced voting rights, which can create a
conflict of interest between minority
shareholders and founders/management.
Approaches to Identify ESG Factors
- Proprietary methods
- Third-party vendors
- Non-profit industry organizations and initiatives
Evaluating ESG-related Risks and Opportunities
- Equity analysts consider both the opportunities
and risks related to ESG factors, while fixedincome analysts primarily consider the downside
risks associated with ESG issues.
- Companies may issue green bonds to raise capital
for projects intended to benefit the environment.
- Misrepresenting a project’s benefits when issuing
green bonds is known as greenwashing.
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8
MERGERS
MERGERS AND
ANDACQUISITIONS
ACQUISITIONS
Forms of Integration
Statutory
Target is entirely absorbed
merger
Subsidiary
merger
Consolidation
Types of Mergers
Type
Horizontal
Vertical
Target becomes a subsidiary
Two companies consolidate
to form new company
Target is…
In the same industry
In the same production chain
Conglomerate In an unrelated core business
Motives for Mergers
- Synergy
- Growth
- Increased market power
- Acquiring unique capabilities and resources
- Diversification
- Bootstrapping earnings
- Managers’ personal incentives
- Tax considerations
- Unlocking hidden value
- Cross-border motivations
Bootstrap Effect
- It is observed when earnings per share (EPS)
increase as a result of a merger. This occurs if the
acquirer trades at a higher price-to-earnings ratio
(P/E) than the target.
- If the acquirer's P/E ratio remains the same after
the merger, the market share price will benefit
because of the EPS increase.
Merger and Industry Life Cycles
Pioneer or Development
- Large development costs, uncertain product
acceptance, slowly increasing sales growth
- Motives: Acquire resources, Pool management
- Type: Conglomerate, Horizontal
Rapid Growth
- Rapid sales growth, high profit margins
- Motives: Large capital to fund growth or expand
- Type: Conglomerate, Horizontal
Mature Growth
- Reduced profit margins, growth potential
remains
- Motives: Economies of scale, Cost savings,
Operational efficiencies
- Type: Vertical, Horizontal
Stabilization
- Competition has eliminated most of the growth
potential
- Motives: Economies of scale, Consolidate market
share
- Type: Horizontal
Decline
- Industry overcapacity, declining profit margins,
lower demand
- Motives: Survival, Increase efficiency and profit
margins (vertical), New growth opportunities
(conglomerate)
- Type: Horizontal, Vertical, Conglomerate
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Transaction Characteristics
Form of Acquisition
- Stock purchase: The acquirer pays for the stock of
the target with a combination of its own stock
and cash. Payment is made to the target’s
shareholders.
- Asset purchase: The acquirer pays the target
company directly for the assets.
Method of Payment
The acquirer can pay for the target company with
cash, securities, or a combination of the two.
- If the acquirer's managers are very confident
about the merged entity's prospects, then they
would tend to prefer a cash offer in order to avoid
sharing the deal's upside with the target
company's shareholders.
- If the acquirer's managers believe that their
shares are overvalued relative to the target
company's shares, then the acquirer would prefer
to pay in stock.
Mindset of Target Management
- Friendly Mergers: Those that are endorsed by the
target company's managers
- Hostile Mergers: Those that are opposed by the
target company's managers
- Bear hug: The tactic of presenting a hostile
merger offer directly to the target company’s
board of directors
- Proxy fight: An alternative method available to a
would-be acquirer to take control of the target
company through a shareholder vote
Defense Mechanisms
Pre-offer
- Poison pill: Current shareholders have option to
buy shares at a discount
- Poison put: Gives bondholders option to demand
immediate repayment
- Restrictive takeover laws: Reincorporate in state
with stricter anti-takeover laws
- Staggered board: Staggered elections ensure that
at least 2 yrs are needed to gain majority control
- Restricted voting rights: Loss of voting rights
above a certain equity ownership percentage
- Supermajority voting provision: Higher percentage
of approval required for mergers
- Fair price amendments: Restrict a merger unless
price to shareholders is fair
- Golden parachutes: Offer target’s management
lucrative payouts for leaving after merger
Post-offer
- Just say no: Simply say no
- Litigation: File lawsuit against acquirer
- Greenmail: Pay off acquirer to terminate
takeover attempt
- Share repurchase: Target submits tender offer for
own shares and forces acquirer to raise bid
- Leveraged recapitalization: Target assumes large
amounts of debt to finance share repurchases
- Crown jewel defense: Target sells subsidiary or
major asset after takeover offer
- Pac-man defense: Target makes counteroffer to
acquire acquirer
- White knight defense: Friendly third party
acquires target
- White squire defense: Friendly third party
acquires minority stake in target
Valuing Target Company
Discounted cash flow (DCF) analysis
Terminal value is calculated based on the free cash
flows in the final year of the first stage, the longterm growth rate, and the weighted average cost of
capital.
FCF6 × (1 + g)
Terminal value6 =
WACC − g
Comparable company analysis
1. Select comparable companies
- Public companies similar to target
- Same or similar industry
2. Calculate relative value measures
- Multiples based on price or enterprise value
3. Apply metrics to target
- Judgment needed to select appropriate metric
4. Estimate takeover price
- The takeover premium is:
Deal Price − Stock Price
T. Premium =
Stock Price
- The takeover price is:
Estimated Stock Value (1 + T. Premium)
Comparable Transaction Analysis
1. Collect data on recent acquisitions of
comparable companies
2. Calculate multiples for comparisons
3. Estimate value based on multiples
Evaluating Merger Bids
Gain4<?H(4 = TP = P4<?H(4 − V4<?H(4
Gain<0=8#?(? = S − TP = S − 9P4<?H(4 − V4<?H(4 ;
V0>@)#$(B = V<0=8#?(? + V4<?H(4 + S − C
Effect of Price and Payment Method
Cash offer
- Acquirer assumes risk, receives potential reward
- More likely if acquirer is more confident in
realizing synergies or in target’s value
Stock offer
- Some risks and potential rewards shift to target
- More likely if acquirer is less confident of
unlocking the target’s hidden value
Mergers that Create Value
- Strong buyer
- Low transaction premium
- Few bidders
- Favorable market reaction to news
Reasons for Restructuring
- Change in strategic focus
- Poor fit
- Reverse synergy
- Financial or cash flow needs
Forms of Restructuring
- Sale: A sale to another company could involve the
sale of assets or an equity carve-out, which is the
creation of a new legal entity that issues its own
equity.
- Spin-off: In a spin-off, shareholders of the parent
company receive shares in a new entity. No cash
is received by the parent company. A split-off is
similar, but the shareholders of the parent
company exchange some of their shares for new
entity shares.
- Liquidation: This involves breaking up a company
and selling its assets, often due to bankruptcy.
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9
CAPITAL
CAPITAL BUDGETING
BUDGETING
Cash Flow Equations (Expansion)
Outlay = FCInv + NWCInv
CF = (S − C − D)(1 − t) + D
CF = (S − C)(1 − t) + Dt
TNOCF = NWCInv + Sal6 − T(Sal6 − B6 )
Cash Flow Equations (Replacement)
Reduce outlay by salvage value of old asset:
Outlay = FCInv + NWCInv − Sal" + t(Sal" − B" )
Use incremental operating cash flows:
CF = (ΔS − ΔC − ΔD)(1 − t) + ΔD
CF = (ΔS − ΔC)(1 − t) + ΔDt
Use incremental terminal year cash flows:
TNOCF = NWCInv + (Sal6M(N − Sal6O7B)
− t[(Sal6M(N − Sal6O7B )
− (B6M(N − B6O7B)]
Depreciation Method
Compared to the straight-line method, the
accelerated depreciation method results in greater
tax savings in the early years, which subsequently
results in a greater NPV.
Effects of Inflation on Capital Budgeting
- Projected cash flows type must match the
discount rate (nominal or real).
- Higher-than-expected inflation decreases the
value of future cash flows.
- Higher-than-expected inflation decreases the real
value of payments to bondholders.
- Inflation does not affect all revenues and costs
uniformly.
Mutually Exclusive Projects with Unequal Lives
Two logically equivalent approaches:
Least common multiple of lives approach
Extend the time horizon to the least common
multiple of all projects. The NPV is computed over
the entire time horizon, with a new outlay required
each time a project is renewed. Choose the project
with the highest NPV.
Equivalent annual annuity (EAA) approach
Compute the NPV for each project. Convert the
NPV to an annuity using the financial calculator's
TVM function to solve for PMT. The result
represents an EAA. The best project is the one with
the highest EAA.
Capital Rationing
- This occurs if a company's capital budget is
insufficient to accept all positive NPV projects
that have been identified as investment
opportunities.
- The objective of a company that is constrained by
capital rationing rules is to choose the group of
projects that offers the highest combined NPV.
- Where capital rationing rules are used, they may
be "hard" (i.e., fixed limits that cannot be
violated) or "soft" (i.e., violations of fixed limits
are permitted under certain circumstances).
Risk Analysis: Stand-Alone Methods
- Sensitivity analysis: Examines effect on NPV of
changing one input variable at a time
- Scenario analysis: Examines effect on NPV of a set
of changes that reflect a scenario (e.g., recession,
normal, or economic boom)
- Simulation analysis (Monte Carlo): Examines
effect on NPV when all uncertain inputs follow
respective probability distributions
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Risk Analysis: Market Risk Methods
The discount rate should reflect the level of risk
that investors are being asked to take on. The
capital asset pricing model (CAPM) and arbitrage
pricing theory (APT) are equilibrium models that
are often used to determine an appropriate risk
premium.
Real Options
- Timing option: Option to delay the investment
- Sizing option: Option to expand, grow, or abandon
- Flexibility option: Option to alter operations,
such as changing prices or substituting inputs
- Fundamental option: Option to alter decisions
based on future events (e.g., drill based on price
of oil, continue R&D depending on initial results)
Analyzing Projects with Real Options
- Use the discounted cash flow (DCF) analysis
without considering real options.
- Adjust the stand-alone DCF analysis by including
the present value of the expected costs and
benefits options.
- Use option pricing models.
- Use decision trees.
Common Capital Budgeting Pitfalls
- Failure to consider economic responses
- Misusing templates
- Undertaking pet projects
- Basing decisions on earnings metrics (instead of
incremental cash flows)
- Basing decisions on IRR (instead of NPV)
- Poor accounting for cash flows
- Over/Underestimating overhead costs
- Errors in estimating a discount rate
- Over/Underspending capital budget
- Failure to consider investment alternatives
- Improper handling of sunk costs and
opportunity costs
EQUITY
EQUITY
EQUITY VALUATION
Equity Value vs. Market Price
VP – P = (V – P) + (VP – V)
VP : estimate of intrinsic value
P: market price
V: unobservable intrinsic value
(V – P): “true” mispricing
Definitions of Value
- Going concern value: Assumes company will
continue operating for the foreseeable future
- Liquidation value: Assumes assets must be sold
immediately, likely at a discount
- Orderly liquidation value: Higher than if assets
must be sold immediately
- Fair market value: Price negotiated between a
willing buyer and seller
- Investment value: Subjective valuation for a
specific investor
RETURN CONCEPTS
RETURN
CONCEPTS
Holding Period Returns (HPR)
DQ DQ − P!
+
r=
P!
P!
= Dividend yield + Price appreciation
Required Return
D"
r?(=8#?(B =
+ g
P
Alpha
Expected (ex ante) alpha = r(,9(04(B − r?(=8#?(B
Realized (ex post) alpha = r<048<7 − r?(=8#?(B
Expected Return for Mispriced Equity
Composed of required return and return
from convergence to intrinsic value over τ years
V! − P!
E(R R) = rR +
P!
Geometric vs. Arithmetic Mean Return
'
Geometric mean = Arithmetic mean − 𝜎𝜎 ü2
Equity Risk Premium
Gordon growth model risk premium
= Dividend yield
+ Long-term earnings growth rate
− Long-term govt bond yield
Ibbotson-Chen equity risk premium model
ERP = [(1 + EINFL)(1 + EGREPS)(1 + EGPE) – 1]
+ Expected income – Expected risk-free rate
EINFL: expected inflation
EGREPS: expected growth in real EPS
EGPE: expected growth in P/E
Beta Estimation
Blume adjustment for public companies
1
2 Unadjusted
Adjusted
o + d e (1)
= d en
3
3
beta
beta
For private companies
Step 1: Calculate unlevered beta of public company
1
¢β
βL ≈ °
1 + (D⁄E) P
Step 2: Calculate levered beta of private company
βEP ≈ [1 + (D′⁄E′)]βL
Multifactor Models
Fama-French model
r# = R J + βST6
RMRF + βG#A(
SMB + βU<78(
HML
#
#
#
RMRF = R S − R J
SMB = Small-cap return − Large-cap return
HML = High vs. low book-to-market return
Pastor-Stambaugh model
Adds a liquidity factor to Fama-French model
5-factor BIRR Model
- Confidence risk
- Time horizon risk
- Inflation risk
- Business cycle risk
- Market timing risk
Build-Up Method for Private Companies
Risk-free rate
+ Equity risk premium
+ Size premium
+ Company-specific premium
Bond Yield plus Risk Premium Method
𝑟𝑟V = YTM on long-term debt + Risk premium
Weighted Average Cost of Capital
D
E
r (1 − t) +
r
WACC =
D+E B
D+E (
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10
DISCOUNTED
DISCOUNTEDDIVIDEND
DIVIDENDVALUATION
VALUATION
Single Period
D" + P"
V! =
(1 + r)"
Multiple Period
$
V! = ~
4%"
D4
P$
+
(1 + r)4 (1 + r)$
Gordon Growth Model (GGM)
D"
D! (1 + g)
V! =
=
r−g
r−g
Value of Fixed-rate Perpetual Preferred Stock
D
r
Present Value of Growth Opportunities
E"
V! =
+ PVGO
r
V! =
Required Rate of Return Using GGM
D"
r=
+g
P!
Two-Stage DDM
$
D! (1 + g G )4 D! (1 + g G )$ (1 + g K )
V! = ~
+
(1 + r)$ (r − g K )
(1 + r)4
4%"
H-Model
D! (1 + g " ) + D! H(g # − g " )
V! =
r − g"
Required Return from H-Model
D!
r = d e [(1 + g K ) + H(g G − g K )] + g K
P!
Sustainable Growth Rate
g = b × ROE
b = retention rate = 1 − dividend payout ratio
Dividend Growth Rate and ROE Analysis
Net Income
ROE =
Shareholder′ equity
=d
Net income Sales Assets
ed
ed
e
Sales
Assets Equity
NI − Dividends
NI
Sales Assets
g=d
ed
ed
ed
e
NI
Sales Assets Equity
CASHFLOW
FLOWVALUATION
VALUATION
FREE CASH
FCFF Valuation Model
X
FCFF4
V+#?@ ~
(1 + WACC)4
4%"
Constant Growth Model
FCFF"
FCFF! (1 + g)
V+#?@ =
=
WACC − g
WACC − g
FCFE Valuation Model
X
FCFE4
= V+#?@ − MVB()4
V(=8#4Y = ~
(1 + r)4
4%"
Constant Growth Model
V(=8#4Y =
FCFE" FCFE! (1 + g)
=
r−g
r−g
Calculating FCFF from Net Income
FCFF = Net income to common shareholders
+ Net non-cash charges
+ Interest expense (1 – tax rate)
- Investment in fixed capital
- Investment in working capital
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FCFF from NI: Adjustments for Noncash Items
Adjustment to
Noncash Item
NI
Depreciation/Amortization Added back
Impairment of intangibles
Added back
Restructuring charges
Added back
Expense reversals
Subtracted
Losses
Added back
Gains
Subtracted
Amortization of long-term
Added back
bond discounts
Amortization of long-term
Subtracted
bond premiums
Added back
Deferred taxes
but special
attention
Calculating FCFF from Statement of Cash Flows
FCFF = CFO + Int(1 − tax) − FCInv
CFO = NI + NCC − WCInv
FCInv = CapEx − Proceeds from sale of l/t assets
Calculating FCFE from FCFF
FCFE = FCFF − Int(I − tax) + Net borrowing
Finding FCFF/FCFE from EBIT or EBITDA
FCFF = EBIT(1 − tax) + Dep − FCInv − WCInv
NI + Int(1 − tax)
EBIT =
1 − tax
FCFF = EBITDA(1 − tax) + Dep(tax) − FCInv
− WCInv
Uses of FCFF and FCFE
Uses of FCFF =
± Δ Cash balance
+ Interest expense (1-t)
+ Debt repayment
+ Cash dividends
+ Share repurchases
Uses of FCFE =
± Δ Cash balance
+ Cash dividends
+ Share repurchases
Forecasting of FCFF and FCFE
Assuming fixed debt ratio (DR) in capital structure
Net borrowing = DR(FCInv – Dep) + DR(WCInv)
FCFE = NI − (1 − DR)(FCInv − Dep)
− (1 − DR)(WCInv)
International Application of Single-Stage Model
Required rate of return (real)
= Country return(real)
± Industry adjustment
± Size adjustment
± Leverage adjustment
Free Cash Flow Model Variations
Sensitivity Analysis of FCFF and FCFE Valuations
Sensitivity analysis can be performed on key
variables such as beta, risk-free rate, equity risk
premium, FCFE growth rates, and the initial FCFF
or FCFE.
Two-Stage Free Cash Flow Models
$
V+#?@ = ~
4%"
FCFF4
(1 + WACC)4
FCFF$:"
1
°
¢
WACC − g (1 + WACC)$
Three-Stage Growth Models
Three-stage models usually assume one of the
following:
- Constant growth rate in all three stages
- Constant growth rate in the first and third stages,
with a declining growth rate in stage two
+
Non-operating Assets and Firm Value
Firm value
= Value of operating assets
+ Value of non-operating assets
MARKET-BASED VALUATION:
VALUATION: PRICE
PRICE AND
ENTERPRISE
VALUE
MULTIPLES
AND
ENTERPRISE
VALUE
MULTIPLES
Trailing (current) P/E
Current stock price
Most recent four quarters’ EPS
Forward (leading) P/E
Current stock price
Next year’s expected EPS
Justified P/E
P! D"⁄E" 1 − b
Leading
=
=
E"
r−g
r−g
P! D! (1 + g)⁄E! (1 − b)(1 + g)
Trailing
=
=
E!
r−g
r−g
P/E to Growth (PEG) Ratio
- P/E divided by expected earnings growth
- Stocks with lower PEG are more attractive than
stocks with higher PEGs, all else equal
Fed Model
- Compares the S&P 500 earnings yield (inverse of
P/E) to the 10-year Treasury yield
- Market is overvalued if the earnings yield, based
on next 12 months’ expected earnings, is less than
the 10-year Treasury yield
Yardeni Model
CEY = CBY − b × LTEG + residual
- CEY = current earnings yield on the market index
- CBY = current yield on A-rated corporate bonds
- LTEG = consensus 5-year market EPS growth rate
- b = weight the market gives to earnings estimates
1
P
=
E (CBY − b × LTEG)
Justified Price Multiples
P! ROE − g
Price/Book
=
B!
r−g
Price/Sales
E!
P! n üS! o (1 − b)(1 + g)
=
S!
r−g
Price to Cash Flow
Advantages vs. P/E
- Cash flows are more stable than earnings
- Cash flows are less subject to manipulation
- Differences between CFs and earnings should be
eliminated over time
Disadvantages vs. P/E
- Cash flows may still be manipulated
- FCFE is theoretically better but more volatile than
FCFF
- Cash flows are different under IFRS vs. US GAAP
Justified Dividend Yield
D! r − g
=
P! 1 + g
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11
Enterprise Value to EBITDA
Enterprise Value
= Market Value of Common Equity
+ Market Value of Preferred Stock
+ Market Value of Debt
+ Minority Interest
− Cash and Investment
EBITDA = NI + Int + Taxes + Dep & Amort
EV/EBITDA vs. P/E
Advantages:
- Controls for differences in leverage
- Controls for differences in depreciation
- EBITDA can be used even if earnings are negative
Disadvantages:
- EBITDA overestimates cash flows if working
capital is growing
- EBITDA ignores the impact of revenue
recognition policies on CFO
Momentum Indicators
Earnings Surprise / Unexpected Earnings
UE4 = EPS4 − E(EPS4 )
Percent earnings surprise = UE/Expected EPS
Standardized Unexpected Earnings
EPS4 − E(EPS4 )
SUE4 =
σ[EPS4 − E(EPS4)]
RESIDUAL INCOME
INCOMEVALUATION
VALUATION
RESIDUAL
Residual Income (RI)
RI = NI – Equity charge
Economic Value Added (EVA)
EVA = NOPAT − (C% × TC)
NOPAT = EBIT(1 − tax)
C% = Cost of capital
TC = Total capital
Market Value Added (MVA)
= Market value of the company
− Accounting book value of total capital
Residual Income Model
RI4 = E4 − rB45" = (ROE − r)B45"
X
RI4
V! = B! + ~
(1 + r)4
4%"
Clean Surplus Relationship
B4 = B45" + E4 − D4
Constant Growth
ROE − r
V! = B! +
B
r−g !
Multistage RI Valuation Model
6
V! = B! + ~
4%"
E4 − rB45" P6 − B6
+
(1 + r)4
(1 + r)6
If RI fades over time:
65"
V! = B! + ~
4%"
E4 − rB45"
E6 − rB65"
+
(1 + r)4
(1 + r − ω)(1 + r)65"
- ω = persistence factor; 0 < ω < 1
- ω = 0: RI will fall to zero after forecast horizon
- ω = 1: RI will persist at current level indefinitely
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Strengths
- Terminal value does not make up a large portion
of the total present value as in other models
- RI models use readily available accounting data
- Can be applied to companies that do not pay
dividends or positive expected near-term FCF
- Can be used when cash flows are unpredictable
- Appealing focus on economic profitability
Weaknesses
- The models are based on accounting data that can
be subject to manipulation by management
- Accounting data used as inputs may require
significant adjustments
- Additional adjustments are needed when clean
surplus accounting relationships do not hold
- The RI model’s use of accounting income assumes
that the cost of debt is reflected appropriately by
interest expense
PRIVATE COMPANY
PRIVATE
COMPANY VALUATION
VALUATION
Approaches to Valuation
Income approach
Market approach
Asset-based approach
Capitalized Cash Flow Method
V+#?@ = FCFF" ⁄(WACC − g + )
V(=8#4Y = FCFE" ⁄(r − g)
Excess Earnings Method
Residual income × (1 + g)
V#$4<$H#)7(; =
r−g
V)8;#$(;; = WC + FA + Intangibles
Valuation Discounts and Premiums
Discounts for lack of control
1
DLOC = 1 −
1 + Control premium
FIXED INCOME FIXED INCOME
TERM STRUCTURE
ANDINTEREST RATE
STRUCTURE AND
DYNAMICSRATE DYNAMICS
INTEREST
Spot Rates
DFM =
1
(1 + ZN )M
Forward Pricing Model
B-A
(1 + zB )B = (1 + zA )A 91 + fA,B-A ;
1
F^,F5^ =
F5^
´1 + f^,F5^ ¨
DFF
F^,F5^ =
DF^
Relationship between Spot and Forward Rates
For an upward-sloping (downward-sloping)
spot curve, the forward curve will be above
(below) the spot curve
Swap Rate
6
~
4%"
s6
1
+
=1
(1 + z4 )4 (1 + z6 )6
6
~ s6 DF4 + DF6 = 1
4%"
Spreads
- Swap spread = Swap rate − Treasury yield
- I-spread = Bond yield − Swap rate
- Z-spread: Spread added to each spot rate so
that PV of bond’s cash flows equals bond’s
market price
- TED spread = LIBOR rate − T-bill rate
- LIBOR-OIS spread = LIBOR rate − OIS rate
Term Structure of Interest Rates: Theories
Pure expectations: Forward rates are unbiased
predictors of future spot rates
Local expectations: All bonds are expected to
generate the risk-free rate over short time periods
Liquidity preference: Forward rates reflect
expectations of future spot rates plus liquidity
premiums for longer maturities
Segmented markets: Yield curve shape is
determined by preferences of borrowers and
lenders
Preferred habitat: Investors have preferred
maturity ranges, but they will shift if yields are
attractive
Yield Curve Risk Based on Key Rate Durations
Decomposes changes in yield curve into changes in
level (L), steepness (S), and curvature (C):
∆P
d e ≈ −DK ∆xK − DG ∆xG − DI ∆xI
P
Flattening and Steepening of Yield Curve
Bullish steepening: Short-term rates falling faster
than long-term rates; observed when central bank
loosens monetary policy to stimulate economy
Bearish flattening: Short-term rates increasing
more than long-term rates; occurs when central
bank raises rates by restricting money supply
Bullish flattening: Long-term rates falling more
than short-term rates; observed in the aftermath of
market turmoil as investors flock to government
bonds in a flight to quality
ARBITRAGE-FREE VALUATION
VALUATION FRAMEWORK
FRAMEWORK
Arbitrage Concepts
Value additivity: Exists if the whole does not equal
sum of parts
Dominance: Exists if the present value of a risk-free
payoff is non-zero
Stripping: Turning coupon bonds into multiple
zero-coupon bonds
Reconstitution: Combining zero-coupon bonds to
replicate a coupon-paying bond
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12
Binomial Interest Rate Tree
Arbitrage-Free Models
- Based on the assumption that bond prices and the
term structure implied by those prices are correct
- Allow parameters to vary deterministically
over time
Convexity Improves Duration-based Estimates
Ho-Lee Model:
dr4 = θ4 dt + σdZ
- Drift term: Time-dependent
- Volatility: Constant
- Negative rates: Possible
i",Q = i",K e'_
Kalotay-Williams-Fabozzi (KWF) Model:
d ln(r4) = θ4 dt + σdZ
- Drift term: Time-dependent
- Volatility: Constant
- Negative rates: Not possible
BONDS
WITH EMBEDDED
EMBEDDED OPTIONS
OPTIONS
BONDS WITH
Binomial Valuation Framework
Bond value plus coupon
payment if higher rate
is realized.
Bond value
at any node
Bond value plus coupon
payment if lower rate is
realized.
VH + C VL + C
+
e
1+i
1+i
C + 0.5(VH + VL)
=
1+i
Bond value at node = 0.5 d
Term Structure Models
Equilibrium Models
- Use fundamental economic variables to describe
the term structure
- Require parameters to be specified
- Preferable for dynamic applications
- Allow for possibility of multiple different future
interest rate paths
Valuation of Callable and Putable Bonds
V0<77<)7( = V;4?<#H`4 − V0<77
V984<)7( = V;4?<#H`4 + V984
Effective Convexity =
(PV5 ) + (𝑃𝑃𝑉𝑉: ) − [2 × (𝑃𝑃𝑉𝑉! )]
(∆Curve)' (PV!)
Convexity for Interest Rates
and Bond Structure
Interest
Straight and
putable bonds
rates
High
Low
Positive
Positive
Callable
bonds
Positive
Negative
Effects of Interest Rates and Yield Curve
- Callable (putable) bond value is inversely
(directly) related to interest rate volatility
- Value of option-embedded bonds declines as
upward-sloping yield curve flattens
Option Adjusted Spread (OAS)
Constant spread is added to each forward rate in a
benchmark binomial interest rate tree to equate PV
of credit risky bond’s cash flow to its market price
- OAS0<77<)7( is inversely related to assumed
volatility of benchmark binomial tree rates
- OAS984<)7( is directly related to assumed volatility
of benchmark binomial tree rates
Effective Duration of Callable & Putable Bonds
(PV5) − (𝑃𝑃𝑉𝑉: )
Effective duration =
2 × (∆Curve)(PV! )
- EffDur0<77<)7( ≤ EffDur;4?<#H`4
- EffDur984<)7( ≤ EffDur;4?<#H`4
Capped or Floored Floating-rate Bonds
Capped floater
Bond with option that prevents coupon rate from
rising above specified maximum
V0<99(B = V;4?<#H`4 − V(@)(BB(B
+7><4(?
+7><4(?
0<9
+7><4(?
+7><4(?
+7>>?
Floored floater
Embedded option sets minimum coupon rate:
V+7>>?(B = V;4?<#H`4 + V(@)(BB(B
Cox-Ingersoll-Ross (CIR) Model:
dr4 = k(θ − r4 )dt + σ6r4 dZ
- Drift term: Mean reverting
- Volatility: Proportional to the square root of the
short-term rate
- Negative rates: Not possible
Vasicek Model:
dr4 = k(θ − r4 )dt + σdZ
- Drift term: Mean reverting
- Volatility: Constant
- Negative rates: Possible
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13
Convertible Bonds
Bondholder has right to convert bond into
common shares at conversion ratio:
Market price
Conversion
Conversion
×
=
value
ratio
of stock
Market
Market price
Conversion
conversion = of convertible∞
ratio
price
bond
Market
Market
Market conversion
= conversion − price of
premium per share
price
stock
Minimum value
Straight Conversion
,
o
of convertible = max n
value
value
bond
Callable and putable convertible bond value
= V;4?<#H`4 + V0<77 >$ − V0<77 >$ + V984 >$
)>$B
;4>0.
)>$B
)>$B
ANALYSIS MODELS
MODELS
CREDIT ANALYSIS
Measures of Credit Risk
- Recovery rate = 100% − Loss severity
- Loss given default = Exposure × (100% − RR)
- The conditional probability of default in a given
year is the probability of default assuming no
prior defaults. If this probability of default is a
constant rate, then the probability of survival to
year t can be calculated as:
(100% − Hazard rate)4
Measures of Credit Risk (continued)
PV of Expected Loss
= Discount factor × Expected loss
= Discount factor × Loss given default
× Prob. of default
= Discount factor × (Exposure × Loss severity)
× Prob. of default
where Prob. of default = POSFa × Hazard rate4
Credit Valuation Adjustment (CVA)
$
CVA = ~ PV of expected loss4
4%"
Structural and Reduced Form Credit Models
Structural credit models can interpret debt and
equity values with option terminology. Debt and
equity are also random values because they are
functions of the asset random values. At time T, the
balance sheet can be represented with:
A(T) = D(T) + E(T)
Assuming the maturity value at time T of the debt
is K, a default will occur if the asset value at time T
is less than K. The value of the equity and debt at
time T are:
- E(T) = max[A(T) − K, 0]
- D(T) = A(T) − max[A(T) − K, 0]
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This implies that:
- Equity is a call option on the assets with a strike
price equal to the face value of the debt
- Debtholders have written the call option to the
equity holders
The value of the equity and debt at time T can also
be expressed as follows using put-call parity:
- E(T) = A(T) − K + max[K − A(T),0]
- D(T) = K − max[K − A(T),0]
This implies that:
- Equity can also be viewed as a long position in
the assets, long put option, and short bond
- Debt can be interpreted as a long bond and a
short put option
Factors for the Term Structure of Credit
Spreads
- Credit quality
- Financial conditions
- Market supply and demand
- Company-value model
Factors for the Credit Analysis on Securitized
Debt
- Homogeneity
- Granularity
- Underlying collateral
- Structure of the secured debt transaction
CREDIT DEFAULT
DEFAULT SWAPS
SWAPS
Settlement Protocol
- Loss given default (LGD) = 1 − Recovery rate (%)
- Payout amount = LGD × Notional amount
CDS Pricing Conventions
PV of credit spread = Upfront premium
+ PV of fixed coupon
Upfront
Credit
Fixed
≈d
−
e × CDS Duration
premium
spread coupon
CDS price per 100 par = 100 − Upfront premium %
Valuation Changes during CDS Term
% Change in CDS price
= Change in spread (bps) × Duration
Applications of CDS
- Naked credit default swap: Purchase of credit
protection without holding the reference
obligation
- Long/short trade: Long position in one CDS
and short position in another
- Curve trade: Buy a CDS of one maturity and
sell a CDS with a different maturity for the
same reference entity
- Basis trade: Exploit differences in credit
spreads offered by the CDS market and
the bond market
DERIVATIVES
DERIVATIVES
PRICING AND VALUATION OF
FORWARD COMMITMENTS
COMMITMENTS
FORWARD
Principles of Arbitrage-Free Pricing
To implement the no-arbitrage argument, the
following simplifying assumptions are made:
- Replicating instruments are available.
- Market frictions are absent.
- Short selling is allowed.
- Investors can borrow and lend at the
risk-free rate.
Pricing and Valuation of Forwards and Futures
Forward Price
F! (T) = FV!,6 (S! + CC! − CB! )
- Carry benefits could include dividends and bond
coupon payments. They reduce the forward price.
- Carry costs could include waste, storage, and
insurance. They increase the forward price.
- If interest rates, carry benefits, and carry costs
are continuously compounded, then:
F! (T) = S! e(?$:II5IF)6
- If there is no carry benefit, then set CB = 0.
- If there is no carry cost, then set CC = 0.
Valuing Forward Contracts
V4 (T) = PV4,6[F4 (T) − F! (t)] for long
V4 (T) = PV4,6[F! (t) − F4 (T)] for short
V6(T) = S6 − F!(t) for long
V6(T) = F! (t) − S6 for short
Futures Contract
- Right before marking to market:
v4 (T) = f4 (T) − f45(T) for long
v4 (T) = f45 (T) − f4(T) for short
- The value after the daily marking to market is 0.
Interest Rate Forward and Futures
Forward Rate Agreement
- The two FRA counterparties are the fixed-rate
receiver (short party) and the floating-rate
receiver (long party).
- FRAs are identified in an “X × Y” format. A 3 × 9
FRA will expire in three months with a payoff
based on the six-month Libor when the FRA
expires.
- FRAs typically use the advanced set, advanced
settled approach. This means interest rate is set
and the FRA settlement is made at time h, when
the FRA expires.
- The settlement amount for the floating receiver
(long position) is:
NA[L@ − FRA! ]t @
1 + D@ t @
- The FRA fixed rate is the forward m-day rate in h
days:
1
1 + L6 t 6
FRA! = °
− 1¢ d e
t@
1 + L` t `
- The value of an existing long FRA (floating
receiver) can be calculated using an offsetting
transaction at the new rate applicable when the
FRA expires:
´FRAH − FRA! ¨t @
VH =
1 + D65H t 65H
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14
Fixed-Income Forward and Futures
Accrued interest
Accrued interest = Accrual period
× Periodic coupon amount
C
NAD
e×d e
AI = d
n
NTD
where:
- NAD is the number of days since the last coupon
payment
- NTD is the total days during a coupon payment
period
- n is the number of coupon payments per year
- C is the annual coupon amount
If accrued interest is included in the bond price
quote, then F! = FV!,6 (S! + CC! − CB! ).
- S! is the full bond price (including accrued
interest).
- There are no carry costs, so CC! = 0.
- The carry benefits are the coupon interest
payments, so CB! = PVCI!,6 .
If accrued interest is not included in the bond
price quote, then:
F! = FV!,6 ´B! + AI! − PVCI!,6 ¨
= FV!,6 [B! + AI! ] − AI6 − FVCI!,6
Often, the delivery of more than one bond is
permitted. Then, the actual futures price is:
F! = Q ! × CF
Valuing Fixed-Income Forwards and Futures
- The value of a bond futures contract is the price
change since the previous day’s settlement.
- The value of a bond forward contract at a later
date is the present value of the current forward
price less the original forward price.
Pricing and Valuing Swap Contracts
Swap pricing equation
1.0 − PV!,4% (1)
rJd3 = $
∑/%" PV!,4& (1)
VALUATION OF CONTINGENT
CONTINGENTCLAIMS
CLAIMS
Binomial Option Valuation Model
Exercise values for calls and puts
c6 = Max(0, S6 − X)
p6 = Max(0, X − S6 )
Hedge ratio
c: − c5
h= :
≥ 0 for call
S − S5
:
5
p −p
≤ 0 for put
h= :
S − S5
- A long call is equivalent to buying stocks with
borrowed funds.
- A long put is equivalent to selling stocks short and
lending the proceeds.
Expectations approach
c = PV[πc : + (1 − π)c 5 ]
p = PV[πp: + (1 − π)p5 ]
":?5B
π=
85B
American-Style Options
Solve for the values from right to left. At each node,
the value is the greater of the calculated value or
the immediate exercise value.
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The Black-Scholes-Merton Formula
Stock
For nondividend-paying stock:
c = SN(d" ) − e5?6 XN(d' )
p = e5?6 XN(−d' ) − SN(−d")
ln(S⁄X) + [r + (σ' ⁄2)]T
d" =
σ√T
d' = d" − σ√T
- N(d' ) represents the probability the call option
expires in the money.
- 1 − N(d' ) represents the probability the put
option expires in the money.
Futures
c = F! (T)e5?6 N(d" ) − e5?6 XN(d' )
p = e5?6 XN(−d' ) − F! (T)e5?6 N(−d" )
d" =
ln(F! (T)e5?6 ⁄X) + [r + (σ' ⁄2)]T
d' = d" − σ√T
σ√T
Interest Rate Options
- An interest rate cap is a portfolio of call options
on interest rates (a.k.a. caplets), each with the
same exercise rate and with sequential
maturities.
- An interest rate floor is a portfolio of put
options on interest rates (a.k.a. floorlets),
each with the same exercise rate and with
sequential maturities.
Swaptions
A swaption is an option on a swap. It gives the
holder the right to enter into a swap with
specified terms, including the fixed rate.
- A payer swaption is an option on a swap to
pay a fixed rate and receive a floating rate.
- A receiver swaption is an option on a swap
to pay a floating rate and receive a fixed rate.
Option Greeks
Delta
- Delta is the change in an instrument's value
for a given change in the underlying value.
- The delta of a long stock is 1.
- Using the BSM model:
Delta0 = e5e6 N(d" ); 0 ≤ Delta0 ≤ e5e6
Delta9 = −e5e6 N(−d" ); −e5e6 ≤ Delta9 ≤ 0
- As the stock price increases:
Delta0 → 1
Delta9 → 0
- Delta hedging is used to offset the exposure of
the portfolio to changes in the underlying.
Portfolio Delta
NQ = −
DeltaQ
If NQ < 0, short the hedging instrument.
If NQ > 0, long the hedging instrument.
- Delta approximation:
¥prıce ≈ option price + Delta9S' − S;
optıon
Gamma
- Gamma measures the change in an instrument’s
delta for a small change in the underlying stock.
- The gamma for a stock position is 0.
- Gamma0 = Gamma9
- Gamma is always non-negative.
- The largest value occurs when the option
is near the money.
- Delta-plus-gamma approximation:
¥prıce ≈ option price + Delta9S' − S;
optıon
Gamma
'
9S' − S;
+
2
Theta
- Theta is the change in an instrument's value for a
small change in calendar time.
- Stock theta is zero.
- Option theta is negative and more pronounced
the closer the option is to expiration.
Vega
- Vega is the change in an instrument's value for a
small change in volatility.
- Vega0 = Vega9
- Positive for both call and put
Rho
- Rho is the change in an instrument's value for a
small change in the risk-free interest rate.
- Positive for calls, negative for put
Implied Volatility
- The implied volatility measure provides
information about the market consensus on price
uncertainty and demand for options.
- The volatility smile is a plot of the implied
volatility with respect to the exercise price.
- The volatility surface is a plot of the implied
volatility with respect to various combinations of
time to expiration and exercise prices.
ALTERNATIVE
INVESTMENTS
ALTERNATIVE
INVESTMENTS
PRIVATE
REALINVESTMENTS
ESTATE INVESTMENTS
REAL ESTATE
Public
Equity
Debt
REOC shares
REIT shares
ETFs/Index funds
Mortgage REITs
MBSs
Unsecured REIT debt
Real Estate Characteristics
- Heterogeneity and fixed location
- High unit value
- Management intensive
- High transaction costs
- Depreciation
- Need for debt capital
- Illiquidity
- Price determination
Private
Direct
Indirect
Mortgages
Private debt
Bank debt
Risk Factors of Real Estate Investment
Property demand and supply:
- Business conditions
- Demographics
- Excess supply
Valuation:
- Cost and availability of capital
- Availability of information
- Lack of liquidity
- Rising interest rates
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15
Property operations:
- Management
- Lease provisions
- Leverage
- Environmental
- Obsolescence
- Recent and ongoing market disruption
Benefits of Real Estate Investment
- Current income
- Price appreciation
- Inflation hedge
- Diversification
- Tax benefits
Types of Leases
Gross lease: The owner pays all operating expenses
Net lease: The tenant covers certain operating
expenses
Triple-net lease: The tenant pays their share of
common area maintenance costs, property taxes,
and building insurance
Sale-leaseback: The owner sells its property and
leases it back from the new owner
Appraisal-Based Index
Relevant appraisal data in the U.S. are provided by
the NCREIF Property Index (NPI).
Holding period return:
HPR =
NOI − Capex + (End. MV − Beg. MV)
Beginning MV
Disadvantages:
- Appraisal lag occurs
- Volatility and correlations with other assets are
understated
- Index requires complicated unsmoothing
techniques
Transaction-Based Index
Repeat sales index uses repeat sales of the same
property.
Hedonic index performs regression based on
properties that are sold.
Disadvantages:
- Market trends contain random noises
Valuation Approaches
- Income approach
- Cost approach
- Sales comparison approach
Net Operating Income (NOI)
Rental income at full occupancy
+ Other income
= Potential gross income
− Vacancy and collection loss
= Effective gross income
− Operating expenses
= Net operating income
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Direct Capitalization Method
Cap rate = Discount rate − Growth rate
NOI
=
Value
Rent
ARY =
Sale price of comparables
DCF Method
NOI
V=
r−g
Debt Service Coverage Ratio (DSCR)
DSCR = NOI⁄Debt service
Publicly Traded Real Estate Securities
Real estate investment trusts (REITs)
- Tax-advantaged entities
Real estate operating companies (REOCs):
- Ordinary taxable entities
Mortgage-backed securities (MBS):
- Asset-backed securitized debt obligations
Advantages and Disadvantages of REITs
Advantages:
- Liquidity
- Transparency
- Diversification
- High-quality portfolios
- Active professional management
- High, stable income
- Tax efficiency
Disadvantage:
- Frequent need to raise new capital in order to
expand
Net Asset Value Per Share (NAVPS)
NAV
NAVPS =
Shares outstanding
MV of assets − MV of liabilities
=
Shares outstanding
Funds from Operations (FFO)
= Accounting net earnings
+ Depreciation charges on real estate
+ Deferred tax charges
+ Loss from property sales & debt restructuring
− Gain from property sales & debt restructuring
Adjusted Funds from Operations (AFFO)
= FFO − Non-cash rent − Maintenance-type capital
expenditures and leasing costs
Advantages of Using P/FFO and P/AFFO
- Widely accepted in global stock markets
- Valuations of REITs and REOCs are easily
compared to other investment alternatives
- FFO estimates are readily available
- Multiples can be used with expected growth and
leverage levels to deepen understanding
Disadvantages of Using P/FFO and P/AFFO
- FFO or AFFO may not capture all intrinsic value
(like land parcels)
- Does not adjust for the right recurring capex
- Income statement rules have changed, so P/FFO
and P/AFFO are difficult to calculate
PRIVATEEQUITY
EQUITYINVESTMENT
INVESTMENT
PRIVATE
Contrasting Valuation in Venture
Capital and Buyout Settings
Buyout investments
- Predictable cash flow
- Strong, experienced management team
- Extensive leverage
- Low need for working capital
- Established products
- Predictable exit
Venture capital investments
- Unpredictable cash flow
- New management team with strong track records
- Primarily equity funded and weak asset base
- Significant cash burn rate
- Unpredictable exit
VC Methods for Valuation of Venture Capital
Transactions
Post-money valuation:
POST = PRE + Investment
Exit Value
=
ROI
Required rate of return:
Exit Value
(1 + IRR)4 = ROI =
POST
New Share Creation
S$
=F
S( + S$
Exit Routes
Initial public offering (IPO): Often a higher
valuation and more liquidity, but expensive,
cumbersome, and less flexible option
Secondary market: Sale to another financial or
strategic investor
Management buyout: Significant leverage
Liquidation: Applies if company is no longer viable
Corporate Governance Terms
Key man clause: Future investment is limited if key
people leave
Clawback provision: GP returns some capital to LP
if early exits are very profitable and later ones are
not as profitable
Distribution waterfall: Provides an order of
distributions between LPs and GPs
- Deal-by-deal allows earlier distribution of
carried interest to GP after the deal
- In total return, GP may not get any carried
interest until LPs get back all committed capital
Tag-along, drag along: Future acquirer can’t get
control without making an offer to all shareholders
No-fault divorce: LPs can remove GP with super
majority vote (~75%)
Removal for cause: If GP commits gross negligence
Investment restrictions: Set a minimum level of
diversification or limits on borrowing
Co-investment: LPs can invest alongside the GP
Performance Evaluation Multiples
PIC (paid in capital)
Paid in capital / committed capital
DPI (distributed to paid in)
Cumulative distributions/cumulative invested
capital
RVPI (residual value to paid in)
LP’s shareholding / cumulative invested capital
TVPI (total value to paid in)
Distributed and undistributed value / cumulative
invested capital
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16
INTRODUCTION
INTRODUCTIONTO
TOCOMMODITIES
COMMODITIESAND
AND
COMMODITY
COMMODITY DERIVATIVES
DERIVATIVES
Commodities Overview
- Energy (crude oil, natural gas, and refined
products)
- Grains
- Industrial (base) metals
- Livestock
- Precious metals
- Softs
Spot and Futures Pricing
Basis: Spot price − Futures price
- Spot price > Futures price: Backwardation
- Spot price < Futures price: Contango
Calendar spread
= Futures contract with an earlier expiration
−Futures contract with a later expiration
Components of Futures Returns
- Price return
- Roll return
- Collateral return
- Rebalance return
PORTFOLIO
MANAGEMENT
PORTFOLIO
MANAGEMENT
EXCHANGE-TRADEDFUNDS:
FUNDS:MECHANICS AND
EXCHANGE-TRADED
MECHANICS AND APPLICATIONS
APPLICATIONS
The Creation/Redemption Process of ETFs
- ETF shares are created or redeemed continuously
to match supply and demand.
- The primary market is over-the-counter (OTC)
with trades between authorized participants
(APs), which are large broker/dealers, and the
ETF manager (a.k.a. ETF sponsor or ETF issuer).
APs are the only investors who can create or
redeem new shares of an ETF.
Tracking Error
- Tracking error is the standard deviation of return
differentials between an ETF and its index.
- Sources of ETF tracking error include:
- Fees and expenses
- Representative sampling/optimization
- Depository receipts and ETFs
- Index changes
- Fund accounting practices
- Regulatory and tax requirements
- Asset manager operations
ETF Bid-Ask Spreads
ETF spreads are typically less than or equal to the
sum of the factors listed below:
- Direct trading costs, such as brokerage and
exchange fees, as well as creation/redemption
fees paid to the sponsor
- Bid-ask spreads for the underlying securities
- Compensation for the market maker’s risk of
carrying positions
- The market maker’s desired profit spread
Premiums and Discounts
Sources of ETF premiums and discounts
to NAV include:
- Timing differences
- Stale pricing
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Total Costs of ETF Ownership
The main components of ETF cost are:
- Management fees
- Trading costs (e.g., commissions, bid-ask spreads,
premiums/discounts)
- Taxes
- Tracking error
- Portfolio turnover
- Security lending
Trading costs are more significant for active
shorter-term investors.
Factor Models in Risk Attribution
Active risk = Tracking error = s(R h − R F )
Active risk square
= s ' (R h − R F )
= Active factor risk + Active specific risk
$
Active specific risk = ~(w#<)' σ'i&
Information Ratio
4h − R
4F
R
IR =
s(R h − R F )
#%"
Ongoing costs have an increasing impact on
returns for longer-term investors.
MEASURING AND
MEASURING
AND MANAGING
MANAGINGMARKET
MARKETRISK
RISK
Portfolio Uses of ETF
- Efficient Portfolio Management
- Asset Class Exposure Management
- Active and Factor Investing
Estimating VaR
Parametric Method
- This method (a.k.a. the analytical method or the
variance-covariance method) typically estimates
VaR by assuming the returns are normally
distributed.
- Advantage: Simplicity; Works best when the
normal distribution assumption is reasonable and
the parameter estimates are reliable
- Disadvantage: Does not work well for portfolios
that contain options because the returns on
options are not normally distributed
Risks
- Exchange-traded notes (ETNs) carry
counterparty risks of default.
- Any fund that uses OTC derivatives will carry
settlement risk because mark-to-market gains are
subject to default.
- ETFs often lend securities to short-sellers for
additional income to investors, creating the risk
of counterparty default.
- The closing of an ETF fund can trigger unexpected
tax liabilities and force investors to find another
fund.
USING MULTIFACTOR
USING
MULTIFACTOR MODELS
MODELS
Arbitrage Pricing Theory (APT)
The APT is an equilibrium pricing model, but it
makes these weaker assumptions:
- A factor model describes the asset returns.
- A well-diversified portfolio can be created to
eliminate asset-specific risk.
- No arbitrage opportunities exist in welldiversified portfolios.
E9R 9 ; = R J + λ" β9," + ⋯ + λ. β9,.
Carhart Model
E´R 9 ¨ = R J + β9," RMRF + β9,' SMB + β9,f HML
+ β9,g WML
Fundamental Factor Models
R # = a# + b#" F" + b#' F' + ⋯ + b#T FT + ε#
Factors are stated as returns rather than return
surprises in relation to predicted values
Macroeconomic Factor Models
R # = a# + b#" F" + b#' F' + ⋯ + b#T FT + ε#
Asset returns are correlated with the surprises in
certain factors
Surprise = Predicted value − Expected value
Factor Models in Return Attribution
Active return = R h − R F
T
Benchmark
Portfolio
Active
= ~ °d
e ¢
e −d
sensitivity .
sensitivity .
return
T%"
Factor
× n
o
return .
+ Security selection
Understanding Value at Risk
- Value at Risk (VaR) is the minimum loss expected
a certain percentage of time over a certain period
of time.
- It can be measured in currency units or as a
percentage of the portfolio value.
Historical Simulation Method
- This method analyzes the return of the current
portfolio composition over a historical period.
The VaR is then calculated by sorting the returns
from the largest loss to the largest gain and
choosing the one based on the desired confidence
interval.
- Advantage: Can incorporate events that actually
happened and it does not require specification of
a distribution or the estimation of parameters;
can also handle options
- Disadvantage: Only useful if the historical period
is representative of the future
Monte Carlo Simulation Method
- This method allows the analyst to develop
assumptions about the statistical
characteristics of the distribution, then
use them to generate random outcomes.
- Advantage: Very flexible; Works well
with a portfolio that has many assets
and not constrained by the normal
distribution assumption
- Disadvantage: Can be complex and
time-consuming to use
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17
Advantages and Limitations of VaR
Advantages of VaR include:
- Simple concept
- Easily communicated concept
- Basis for risk comparison
- Facilitates capital allocation decisions
- Can be used for performance evaluation
- Reliability can be verified
- Widely accepted by regulators
Limitations of VaR include:
- Subjectivity
- Underestimating the frequency of extreme events
- Failure to consider liquidity
- Sensitivity to correlation risk
- Vulnerability to volatility regimes
- Misunderstanding of its meaning
- Oversimplification
- Disregard of right-tail events
Extensions of VaR
- Conditional VaR (CVaR), a.k.a. expected shortfall
or expected tail loss, measures the expected loss if
VaR is exceeded.
- Incremental VaR (IVaR) measures the impact of
changing a position's size within a portfolio.
- Marginal VaR (MVaR) is similar to incremental
VaR in that it measures the change in VaR for a
small change in a position, but it uses formulas
derived from calculus.
- Relative VaR, a.k.a. ex ante tracking error,
indicates the amount that a portfolio may deviate
from its benchmark.
Sensitivity Risk Measures
Equity Exposure Measures - Beta
E(R # ) = R J + β#[E(R S ) − R J ]
Fixed-Income Exposure Measures – Duration,
Convexity
∆F
F
≈ −D
∆Y
":Y
"
+ C
'
∆Y'
(":Y)'
Option Risk Measures – Delta, Gamma, Vega
I`<$H( #$ k<78( >+ >94#>$
∆(delta) ≈
I`<$H( #$ k<78( >+ 8$B(?7Y#$H
Г(gamma) ≈
I`<$H( #$ B(74<
I`<$H( #$ k<78( >+ 8$B(?7Y#$H
"
c + ∆c ≈ c + ∆0 ∆S + Г0 (∆S)
Vega ≈
'
'
I`<$H( #$ k<78( >+ >94#>$
I`<$H( #$ k>7<4#7#4Y >+ 8$B(?7Y#$H
Scenario Risk Measures
- Historical scenarios are scenarios that measure
the portfolio return if historical markets repeat
themselves.
- Hypothetical scenarios model the impact of
extreme movements and co-movements in
different markets that have not previously
happened.
- Reverse stress testing is the process of targeting
and stressing the portfolio’s material exposures.
- Sensitivity and scenario risk measures
complement VaR in the following ways:
o Sensitivity measures address some of the
shortcomings of position size measures.
o They do not have to rely on historical
volatility and correlation data, so their
utility is not limited by the parameters of
the lookback period.
o Normal distributions do not have
to be assumed.
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o Scenarios can target key exposures of the
portfolio, such as concentrated positions.
- One limitation of sensitivity risk measures is they
often do not distinguish by volatility.
- The limitations of scenario measures include:
o Hypothetical scenarios are necessary to fill in
the gaps of historical scenarios.
o Hypothetical scenarios may incorrectly specify
the correlation between assets or fail to adjust
correctly for factors such as liquidity or
concentrated positions.
o Hypothetical scenarios are difficult to
maintain. There is no certainty the range of
scenarios tested is adequate.
o It is difficult to establish limits based on
scenario analysis.
o Extreme hypothetical scenarios are not likely
to be taken seriously by management.
However, using only plausible scenarios could
be too limiting.
Using Constraints in Market Risk Management
- Risk budgeting first sets limits for the entire firm,
then allocates the firm's overall risk budget
among sub-activities. The risk will generally be
based on ex ante tracking error or VaR.
- Position limits are effective controls against
overconcentration. They may be specified in
terms of the market value of securities or the
notional principal of derivatives.
- Scenario limits place limits on the loss in a given
scenario. This can be used to address
shortcomings in VaR. If results are not within the
limits, corrective action should be taken.
- Stop-loss limits require changes if losses over a
given magnitude occur in a specified period. This
can catch trending losses that are staying just
below the VaR daily limits.
- Capital allocation aligns risks and rewards by
placing limits on capital assigned to each of the
firm's activities.
BACKTESTING
BACKTESTING AND SIMULATION
Backtesting Process
1. Strategy design
2. Historical investment simulation
3. Analysis of backtesting output
Backtesting Multifactor Models
- Give equal weight to each parameter
- Use a risk-based weighting scheme
Common Problems in Backtesting
- Survivorship bias: This can be combated using a
point-in-time approach
- Look-ahead bias: This includes reporting lags,
data revisions, and addition of new companies to
indexes/databases
- Data snooping: To combat this practice, analysts
may use a higher statistical t-value, perform
cross-validation, or test theories on different data
Simulation Analysis
Historical simulation: A non-deterministic form of
rolling window backtesting
Monte Carlo simulation: A model that specifies a
statistical distribution for the underlying data
ECONOMICS AND
ECONOMICS
AND INVESTMENT
INVESTMENT MARKETS
MARKETS
Pricing a default-free nominal coupon
paying bond
M
P4# = ~
#
CF4:;
;
91 + l4,; + θ4,; + π4,; ;
l4,; : yield to maturity on a real default-free
investment today
θ4,; : expected inflation rate
π4,; : compensation for uncertainty in inflation
G%"
Credit Premiums and Business Cycle
M
#
∑ 4:;
¨
E4 ´CF
P4# = ~
# ;
+
θ
+
π
91
+
l
4,;
4,;
4,; + γ4,; ;
G%"
#
γ4,; : credit premium
Equities and the Equity Risk Premium
M
#
∑ 4:;
¨
E4 ´CF
P4# = ~
;
91 + l4,; + θ4,; + π4,; + γ#4,; + κ#4,; ;
G%"
κ#4,; : equity premium relative to credit risky bonds
Commercial Real Estate
M
#
∑ 4:;
¨
E4´CF
P4# = ~
;
91 + l4,; + θ4,; + π4,; + γ#4,; + κ#4,; + ϕ#4,; ;
ϕ#4,; : liquidity risk premium
G%"
ANALYSISOF
OFACTIVE
ACTIVE PORTFOLIO
ANALYSIS
MANAGEMENT
PORTFOLIO
MANAGEMENT
Measuring Value Added
R^ = Rh − RF
where:
M
R F = ~ wF,# R #
#%"
M
R h = ~ wh,# R #
#%"
M
R ^ = ~ Δw# R # ; Δw# = wh.# − wF,#
#%"
M
R ^ = ~ Δw# R ^# ; R ^# = R # − R F
#%"
Decomposition of Value Added
S
S
/%"
/%"
R ^ = ~ Δw/ R F,/ + ~ Δwh,/ R ^,/
Sharpe Ratio
Rh − RJ
SR h =
STD(R h )
Information Ratio
Rh − RF
IR =
STD(R h − R F )
Constructing Optimal Portfolios
SR'h = SR'F + IR'
IR
STD(R F )
STD(R ^ ) =
SR F
Active Security Returns
µ# = ICσ# S#
Mean-variance active return weights
µ# σ^
Δw#∗ = '
σ# IC√BR
Basic Fundamental Law
Expected active portfolio return
E(R ^)∗ = IC√BRσ^
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18
Full Fundamental Law
E(R ^) = (TC)(IC)√BRσ^
Ex Post Performance Measurement
E(R ^ |ICn ) = (TC)(IC)√BRσ^
R ^ = E(R ^ |ICn ) + Noise
The portfolio’s active return variance can be
divided into two parts:
- Variation due to the realized information
coefficient, TC '
- Variation due to constraint-induced noise,
1 − TC '
Ex Ante Measurement of Skill
σ^ = σdI √NσnS
IC
E(R ^) =
σ
σdI ^
Practical Measure of Breadth
N
BR =
1 + (N − 1)ρ
TRADING COSTS
COSTSAND
ANDELECTRONIC
ELECTRONICMARKETS
MARKETS
TRADING
Costs of Trading
- Explicit costs are direct costs of trading, such as
brokerage commissions, transaction taxes,
stamp duties, and exchange fees.
- Implicit costs are indirect costs caused by the
market impact of trading. These include:
o Bid-ask spread = Ask price – Bid price
Traders seeking quick completion of orders
may accept less attractive prices
o Market impact (or price impact) is the
negative effect of large order sizes
o Delay (slippage) costs occur when prices
move before trades can be fully executed
o Opportunity costs (unrealized profit/loss)
result from partially filled orders
- Effective spread:
o For buy orders: (Trade price – Midquote) x 2
o For sell orders: (Midquote – Trade price) x 2
- Implementation shortfall: Difference between
the value of a hypothetical portfolio based on
the assumption that an order was executed at
the decision price and the actual portfolio value
- VWAP transaction cost:
o For buy orders: VWAP6?<B( − VWAPF($0`@<?.
o For sell orders: VWAPF($0`@<?. − VWAP6?<B(
Types of Electronic Traders
- Proprietary traders include dealers, arbitrageurs,
and front runners. Broker/dealers send orders
directly to exchanges; other proprietary traders
send orders to outside brokers to be forwarded to
exchanges. Brokers may provide sponsored
access to clients.
- High-frequency traders (HFT) make round-trip
trades (buy/sell) within milliseconds.
- Low-latency traders speculate based on
predictions about future prices and order activity
using analysis of market data. They are willing to
hold positions for a day or more, unlike highfrequency traders. They include news traders and
some parasitic traders.
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- News traders need to quickly collect and process
information in order to take positions based on
predicted directional movement. They hope to
profit from stale orders that are not updated to
reflect the latest news.
- Parasitic traders include front runners
and quote matches.
- Front runners use artificial intelligence to detect
trading patterns and predict orders.
- Quote matchers seek to exploit the option value
of standing orders.
- Buy-side traders fill orders for investment and
risk managers who use the markets to establish
positions from which they derive profitmotivated benefits.
- Electronic brokers serve both proprietary and
buy-side traders.
- Electronic dealers hope to profit by trading at
positive net spreads. They take liquidity to reduce
exposure if prices move against inventory
positions, and they generally avoid holding large
inventories of actively traded stocks.
- Electronic arbitrageurs seek to buy undervalued
securities and sell overvalued securities. They try
to construct an arbitrage portfolio at minimum
cost and risk.
Systemic Risks of Electronic Trading
- Runaway algorithms result in unintended errors
due to programming mistakes.
- Fat finger errors occur when a manual trader
submits the wrong order size. This is not unique
to electronic trading, but the consequences can be
far more severe.
- Overlarge orders demand too much liquidity from
the market. This may result in severely disrupted
prices and even a temporary halt to trading.
- Malevolent order streams are created deliberately
to disrupt markets. This may be done by
disgruntled employees or even terrorists.
Solutions to Systemic Risks
- Test software thoroughly before using it in live
trading operations.
- Exchanges must ensure that order matching
systems only accept orders from approved
parties (market access controls).
- Adopt controls to ensure that algorithms can only
be changed by authorized parties.
- Shut off trading immediately if the order flow is
observed to be abnormal.
- Ensure broker oversight by not allowing broker’s
clients to enter orders directly (sponsored naked
access).
- Use price limits and trade halts to stop trading
when prices move too quickly or there is
excessive demand for liquidity.
Abusive Trading Practices
- Market manipulation strategies usually involve
one or more of these improper market activities:
o Trading for market impact deliberately raises
or lowers prices to influence others’
perceptions. Traders may pay significant
transaction costs to do this.
o Rumor mongering spreads false information
that affects a stock’s fundamental value.
o Wash trading is trading between accounts
controlled by the same entity to create the
impression that liquidity is greater than
actually exists.
o Spoofing, a.k.a. layering, is a practice of
placing exposed standing limit orders to
give the false impression of liquidity or a
misleading valuation.
- Market manipulation strategies include
the following:
o Bluffing involves submitting orders or making
trades to influence other traders’ perceptions
of value. Momentum traders are usually
targeted because they buy when prices are
rising and sell when prices are falling.
o “Pump-and-dump” is an example of a bluffing
tactic that tries to raise prices.
o Gunning the market attempts to force traders
into a disadvantageous trade.
o Squeezing and cornering is a practice of
manipulators to control the resources
necessary to settle trading contracts.
Then the manipulator takes the resources
off the market, forcing some to default on
their contract or buy the resources at very
high prices.
ETHICAL AND
ETHICAL
AND PROFESSIONAL
PROFESSIONALSTANDARDS
STANDARDS
I(A) Knowledge of the Law
Obey strictest applicable law. Disassociate
immediately from any illegal or unethical activity.
I(B) Independence and Objectivity
Do not offer or accept gifts that might impair
independence and objectivity. Gifts from clients
may be permissible.
I(C) Misrepresentation
Cite sources. Do not plagiarize or omit important
information. Act quickly to correct any errors.
I(D) Misconduct
Does not apply to personal behavior unless it
reflects poorly on the investment profession.
II(A) Material Nonpublic Information
Do not act or cause others to act on material
nonpublic information. Seek public dissemination.
II(B) Market Manipulation
Do not take any actions that distort prices or
trading volume. Market making and legitimate
trading strategies are allowed.
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III(A) Loyalty, Prudence, and Care
Place clients’ interest above yours. Disclose
policies on proxy voting and soft commissions.
III(B) Fair Dealing
Treat all clients fairly. Treat non-immediate family
like other clients. Communicate investment
recommendations and changes simultaneously.
III(C) Suitability
Use a regularly updated IPS during investment
decisions. Evaluate decisions in a portfolio context.
III(D) Performance Presentation
Performance data should be fair, accurate, and
complete. Do not promise returns for risky assets.
III(E) Preservation of Confidentiality
Keep all client information confidential unless:
client is involved in illegal activity, you are legally
required, or you have the client’s permission.
IV(A) Loyalty
Get permission before taking outside work (even
unpaid) that competes with employer. Abide by
non-compete agreement (if applicable) and do not
take employer’s property.
IV(B) Additional Compensation Arrangements
Obtain written permission from all parties before
receiving any compensation for outside work.
IV(C) Responsibilities of Supervisors
Supervisors must adequately train and monitor
subordinates. Responsibilities may be delegated.
V(A) Diligence and Reasonable Basis
Exercise diligence and thoroughness. Support
actions with research and investigation.
V(B) Communication with Clients and
Prospective Clients
Make appropriate disclosures. Distinguish between
fact and opinion in analysis and recommendations.
V(C) Record Retention
Maintain records to support recommendations and
decisions. 7-year retention period recommended.
VI(A) Disclosure of Conflicts
Disclose any matters that may impair
independence and objectivity, prominently and in
plain language
VI(B) Priority of Transactions
Execute clients’ transactions before accounts in
which you have a beneficial interest.
VI(C) Referral Fees
Disclose referral fees to clients and employer,
including non-monetary arrangements.
VII(A) Conduct as Participants in
CFA Institute Program
Do not share confidential exam details. Expressing
opinions about CFAI policies is permissible.
VII(B) Reference to CFA Institute, the CFA
Designation, and the CFA Program
Do not misrepresent the meaning of CFA Institute
membership, designation, or candidacy.
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BA II PLUS CALCULATOR TIPS
BA II PLUS CALCULATOR TIPS
Basic Operations
2ND : Access secondary functions (in yellow)
ENTER : Send value to a variable
2ND + ENTER : Toggle between options
↑ ↓ : Navigate between variables/options
STO + 0 - 9 : Store current value into memory
RCL + 0 - 9 : Recall value from memory
Time Value of Money (TVM)
For annuity, loan, and bond calculations
N : Number of periods
I/Y : Effective interest rate per period (in %)
PV : Present value
PMT : Payment/coupon amount
FV : Future value/redemption value
CPT + one of the above : Solve for unknown
2ND + BGN : Toggle between ordinary annuity
and annuity due
2ND + CLR TVM : Clear TVM worksheet
Note:
- Always clear the TVM worksheet before starting
a new calculation.
- For bonds, PMT and FV should have the same
sign, and opposite signs for PV.
Cash Flow Worksheet ( CF , NPV , IRR )
For non-level payments
Input ( CF )
CF0: Initial cash flow
C01: 1st distinct cash flow after initial cash flow
F01: Frequency of CO1
C0n: nth distinct cash flow
F0n: Frequency of C0n
Note:
- Always clear the CF worksheet before starting a
new calculation.
- The use of F0n is optional. You can leave them as
1 and input repeating cash flows multiple times. If
you do so, C01 will be the cash flow at time 1, C02
will be the cash flow at time 2, and so on.
Output ( NPV , IRR )
I: Effective interest rate per period (in %)
NPV + CPT : Solve for net present value
IRR + CPT : Solve for internal rate of return
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