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Quantitative Methods
The Time Value of Money
The Time Value of Money
Effective Annual Rates
EAR = (1 + Periodic interest rate) N − 1
The Future Value of a Single Cash Flow
FVN = PV (1 + r) N
The Present Value of a Single Cash Flow
PV =
FV
(1 + r) N
The Present and Future Value of an Ordinary Annuity
PVAnnuity: # periods N; % interest per period I/Y; amount FV or amount PMT → PV
FVAnnuity: # periods N; % interest per period I/Y; amount FV or amount PMT → FV
The Present and Future Value of an Annuity Due
PVAnnuity Due = PVOrdinary Annuity × (1 + r)
FVAnnuity Due = FVOrdinary Annuity × (1 + r)
Present Value of a Perpetuity
PMT
I/Y
PVPerpetuity =
Continuous Compounding and Future Values
FVN = PVe r ⋅N
s
2
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Discounted Cash Flow Applications
Discounted Cash Flow Applications
Net Present Value
N
CFt
t
t=0 (1 + r )
NPV = ∑
where:
CFt = the expected net cash flow at time t
N = the investment’s projected life
r = the discount rate or appropriate cost of capital
Internal Rate of Return
n
CFt
t
t=0 (1 + IRR )
NPV= 0 = ∑
Solved as an iterative process using calculator TVM functions.
Bank Discount Yield
rBD =
D 360
×
F
t
where:
rBD = the annualized yield on a bank discount basis
D = the dollar discount (face value – purchase price)
F = the face value of the bill
t = number of days remaining until maturity
Holding Period Yield
HPY =
P1 − P0 + D1 P1 + D1
=
−1
P0
P0
where:
P0 = initial price of the investment.
P1 = price received from the instrument at maturity/sale.
D1 = interest or dividend received from the investment.
Money-weighted rate of return
rmw = IRRCF
Time-weighted rate of return
rtw = [ (1 + HPY1 )(1 + HPY2 )...(1 + HPYn ) ]
 n

=  ∏ (1 + HPYt ) 
 t =1

1
n
1
n
−1
−1
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3
Discounted Cash Flow Applications
Effective Annual Yield
EAY = (1 + HPY)365/ t − 1
where:
HPY = holding period yield
t = numbers of days remaining till maturity
HPY = (1 + EAY) t /365 − 1
Money Market Yield
R MM =
360 × rBD
360 − (t × rBD )
R MM = HPY × (360/t)
Bond Equivalent Yield
BEY = [(1 + EAY)0.5 − 1] × 2
4
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Statistical Concepts
Statistical Concepts
Population Mean
N
µ=
∑ xi
i =1
N
where:
xi = is the ith observation.
Sample Mean
n
X=
∑ xi
i =1
n
Geometric Mean
1 + R G = T (1 + R1 ) × (1 + R 2 ) ×…× (1 + R T )
OR
G = n X1X 2 X 3 … X n
with X i > 0 for i = 1, 2,…, n.
1
T
 T
R G =  ∏ (1 + R t )  − 1
 t =1

Harmonic Mean
Harmonic mean: X H =
N
with X i > 0 for i = 1,2,…,N.
1
∑x
i =1 i
N
Percentiles
Ly =
( n + 1) y
100
where:
y = percentage point at which we are dividing the distribution
Ly = location (L) of the percentile (Py) in the data set sorted in ascending order
Range
Range = Maximum value − Minimum value
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5
Statistical Concepts
Mean Absolute Deviation
n
MAD =
∑ Xi − X
i =1
n
where:
n = number of items in the data set
X = the arithmetic mean of the sample
Population Variance
N
σ2 =
∑ (X i − µ)2
i =1
N
where:
Xi = observation i
μ = population mean
N = size of the population
Population Standard Deviation
N
σ=
∑ (X i −
µ)2
i =1
N
Sample Variance
n
Sample variance = s2 =
∑ (X i −
i =1
X)2
n −1
where:
n = sample size.
Sample Standard Deviation
n
s=
6
∑ (X i − X)2
i =1
n −1
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Statistical Concepts
Coefficient of Variation
Coefficient of variation =
s
X
where:
s = sample standard deviation
X = the sample mean.
Sharpe Ratio
Sharpe ratio =
rp − rf
sp
where:
rp = mean portfolio return
rf = risk‐free return
sp = standard deviation of portfolio returns
Sample skewness, also known as sample relative skewness, is calculated as:
n
(X i − X)3
∑


n
i =1
SK = 

 ( n − 1)( n − 2 ) 
s3
As n becomes large, the expression reduces to the mean cubed deviation.
n
1
SK ≈
n
∑ (X i − X)3
i =1
s3
where:
s = sample standard deviation
Sample Kurtosis uses standard deviations to the fourth power. Sample excess kurtosis is
calculated as:
n


(X i − X)4 
∑

n(n + 1)
3(n − 1)2
i =1
−
KE = 
4
s
 (n − 1)(n − 2)(n − 3)
 (n − 2)(n − 3)




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7
Statistical Concepts
As n becomes large the equation simplifies to:
n
KE ≈
(X i − X)4
∑
1
i=1
n
s4
−3
where:
s = sample standard deviation
For a sample size greater than 100, a sample excess kurtosis of greater than 1.0 would be
considered unusually high. Most equity return series have been found to be leptokurtic.
8
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Probability Concepts
Probability Concepts
Odds for an Event
P (E) =
a
(a + b)
Where the odds for are given as “a to b”, then:
Odds for an Event
P (E) =
b
(a + b)
Where the odds against are given as “a to b”, then:
Conditional Probabilities
P(A B) =
P(AB)
given that P(B) ≠ 0
P(B)
Multiplication Rule for Probabilities
P(AB) = P(A B) × P(B)
Addition Rule for Probabilities
P(A or B) = P(A) + P(B) − P(AB)
For Independent Events
P(A B) = P(A), or equivalently, P(B A) = P(B)
P(A or B) = P(A) + P(B) − P(AB)
P(A and B) = P(A) × P(B)
The Total Probability Rule
P(A) = P(AS) + P(ASc )
P(A) = P(A S) × P(S) + P(A Sc ) × P(Sc )
The Total Probability Rule for n Possible Scenarios
P(A) = P(A S1 ) × P(S1 ) + P(A S2 ) × P(S2 ) + + P(A Sn ) × P(Sn )
where the set of events {S1 , S2 ,…, Sn } is mutually exclusive and exhaustive.
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9
Probability Concepts
Expected Value
E(X) = P(X1 )X1 + P(X 2 )X 2 + … P(X n )X n
n
E(X) = ∑ P(X i )X i
i =1
where:
Xi = one of n possible outcomes.
Variance and Standard Deviation
σ 2 (X) = E{[X − E(X)]2}
n
σ 2 (X) = ∑ P(X i ) [X i − E(X)]2
i =1
The Total Probability Rule for Expected Value
1.
2.
E(X) = E(X | S)P(S) + E(X | Sc)P(Sc)
E(X) = E(X | S1) × P(S1) + E(X | S2) × P(S2) + . . . + E(X | Sn) × P(Sn)
where:
E(X) = the unconditional expected value of X
E(X | S1) = the expected value of X given Scenario 1
P(S1) = the probability of Scenario 1 occurring
The set of events {S1, S2, . . . , Sn} is mutually exclusive and exhaustive.
Covariance
Cov(XY) = E{[X − E(X)][Y − E(Y)]}
Cov(R A ,R B ) = E{[R A − E(R A )][R B − E(R B )]}
Correlation Coefficient
Corr(R A ,R B ) = ρ(R A ,R B ) =
10
Cov(R A ,R B )
(σ A )(σ B )
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Probability Concepts
Expected Return on a Portfolio
N
E(R p ) = ∑ wi E(R i ) = w1E(R1 ) + w2 E(R 2 ) + + w N E(R N )
i =1
where:
Weight of asset i =
Market value of investment i
Market value of portfolio
Portfolio Variance
N N
Var(R p ) = ∑ ∑ wi w jCov(R i ,R j )
i =1 j=1
Variance of a 2 Asset Portfolio
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w B Cov(R A ,R B )
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + 2w A w Bρ(R A ,R B )σ (R A )σ (R B )
Variance of a 3 Asset Portfolio
Var(R p ) = w2A σ 2 (R A ) + w2B σ 2 (R B ) + w2C σ 2 (R C )
+ 2w A w B Cov(R A ,R B ) + 2w B wC Cov(R B ,R C ) + 2wC w A Cov(R C ,R A )
Bayes’ Formula
P(Event Information) =
P (Information Event) × P (Event)
P (Information)
Counting Rules
The number of different ways that the k tasks can be done equals n1 × n2 × n3 × … nk .
Combinations
n Cr
n
n!
=  =
 r  ( n − r )!( r!)
Remember: The combination formula is used when the order in which the items are
assigned the labels is NOT important.
Permutations
n Pr
=
n!
( n − r )!
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11
Common Probability Distributions
Common Probability Distributions
Discrete Uniform Distribution
F(x) = n × p(x) for the nth observation.
Binomial Distribution
P(X=x) = n Cx (p)x (1 − p)n-x
where:
p = probability of success
1 − p = probability of failure
nCx = number of possible combinations of having x successes in n trials. Stated differently,
it is the number of ways to choose x from n when the order does not matter.
Mean of a Binomial Random Variable
B(n, p) = np
Variance of a Binomial Random Variable
σ 2x = n × p × (1 − p)
The Continuous Uniform Distribution
P(X < a), P (X > b) = 0
P (x1 ≤ X ≤ x 2 ) =
x 2 − x1
b−a
Confidence Intervals
For a random variable X that follows the normal distribution:
The 90% confidence interval is x − 1.65s to x + 1.65s
The 95% confidence interval is x − 1.96s to x + 1.96s
The 99% confidence interval is x − 2.58s to x + 2.58s
The following probability statements can be made about normal distributions
•
•
•
•
12
Approximately 50% of all observations lie in the interval
Approximately 68% of all observations lie in the interval
Approximately 95% of all observations lie in the interval
Approximately 99% of all observations lie in the interval
μ ± (2/3)σ
μ ± 1σ
μ ± 2σ
μ ± 3σ
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Common Probability Distributions
z‐Score
z = (observed value − population mean)/standard deviation = (x − µ)/σ
Roy’s Safety‐First Criterion
Minimize P(RP< RT)
where:
RP = portfolio return
RT = target return
Shortfall Ratio
Shortfall ratio (SF Ratio) =
E (RP ) − RT
σP
Continuously Compounded Returns
EAR = e r − 1
cc
HPR t = e r
cc
×t
rcc = continuously compounded annual rate
−l
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13
Sampling and Estimation
Sampling and Estimation
Sampling Error
Sampling error of the mean = Sample mean − Population mean = x − µ
Standard Error of Sample Mean when Population Variance is known
σx = σ
n
where:
σ x = the standard error of the sample mean
σ = the population standard deviation
n = the sample size
Standard Error of Sample Mean when Population Variance is not known
sx =
s
n
where:
s x = standard error of sample mean
s = sample standard deviation.
Confidence Intervals
Point estimate ± (reliability factor × standard error)
where:
Point estimate = value of the sample statistic that is used to estimate the population
parameter
Reliability factor = a number based on the assumed distribution of the point estimate and
the level of confidence for the interval (1 − α).
Standard error = the standard error of the sample statistic (point estimate)
14
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Sampling and Estimation
x ± z α /2
σ
n
where:
x = The sample mean (point estimate of population mean)
zα/2 = The standard normal random variable for which the probability of an observation
lying in either tail is σ / 2 (reliability factor).
σ
= The standard error of the sample mean.
n
x ± tα
2
s
n
where:
x = sample mean (the point estimate of the population mean)
tα
= the t‐reliability factor
2
s
= standard error of the sample mean
n
s = sample standard deviation
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15
Hypothesis Testing
Hypothesis Testing
Test Statistic
Sample statistic − Hypothesized value
Standard error of sample statistic
Test statistic =
Power of a Test
Power of a test = 1 − P(Type II error)
Decision Rules for Hypothesis Tests
Decision
Do not reject H0
H0 is True
Correct decision
Incorrect decision
Type I error
Significance level =
P(Type I error)
Reject H0
H0 is False
Incorrect decision
Type II error
Correct decision
Power of the test
= 1 − P(Type II error)
Confidence Interval
 sample   critical   standard    population   sample   critical   standard  
 statistic −  value   error   ≤  parameter  ≤  statistic +  value   error  




x
(s n)
x
(s n)
− (z α /2 )
≤
µ0
≤
+ (z α /2 )
Summary
H0 : μ ≤ μ0
Alternate
hypothesis
Ha : μ > μ0
One tailed
(lower tail)
test
H0 : μ ≥ μ0
Ha : μ < μ0
Test statistic <
critical value
Test statistic ≥
critical value
Probability that lies
below the computed test
statistic.
Two‐tailed
H0 : μ = μ0
Ha : μ ≠ μ0
Test statistic <
lower critical
value
Test statistic >
upper critical
value
Lower critical
value ≤ test
statistic ≤
upper critical
value
Probability that lies
above the positive
value of the computed
test statistic plus the
probability that lies
below the negative
value of the computed
test statistic.
Type of test
One tailed
(upper tail)
test
16
Null
hypothesis
Fail to reject
null if
Reject null if
Test statistic >
critical value
Test statistic ≤
critical value
P‐value represents
Probability that lies
above the computed test
statistic.
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Hypothesis Testing
t‐Statistic
t-stat =
x − µ0
s n
where:
x = sample mean
μ0 = hypothesized population mean
s = standard deviation of the sample
n = sample size
z‐Statistic
z-stat =
x − µ0
σ n
z-stat =
where:
x = sample mean
μ = hypothesized population mean
σ = standard deviation of the population
n = sample size
x − µ0
s n
where:
x = sample mean
μ = hypothesized population mean
s = standard deviation of the sample
n = sample size
Tests for Means when Population Variances are Assumed Equal
t=
(x1 − x2 ) − (µ1 − µ 2 )
 s2p s2p 
n +n 
 1
2
1/2
where:
s2p =
(n1 − 1)s12 + (n 2 − 1)s22
n1 + n 2 − 2
s12 = variance of the first sample
s22 = variance of the second sample
n1 = number of observations in first sample
n2 = number of observations in second sample
degrees of freedom = n1 + n2 −2
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17
Hypothesis Testing
Tests for Means when Population Variances are Assumed Unequal
t-stat =
df =
(x1 − x2 ) − (µ1 − µ 2 )
 s12 s22 
 n + n 
1
2
 s12 s22 
 n + n 
1
2
1/2
2
(s12 n1 )2 + (s22 n2 )2
n1
n2
where:
s12 = variance of the first sample
s22 = variance of the second sample
n1 = number of observations in first sample
n2 = number of observations in second sample
Paired Comparisons Test
t=
d − µ dz
sd
where:
d = sample mean difference
sd
s d = standard error of the mean difference =
n
sd = sample standard deviation
n = the number of paired observations
Hypothesis Tests Concerning the Mean of Two Populations ‐ Appropriate Tests
18
Population
distribution
Normal
Relationship
between
samples
Independent
Assumption
regarding
variance
Equal
Normal
Independent
Unequal
t‐test with
variance not
pooled
Normal
Dependent
N/A
t‐test with
paired
comparisons
Type of test
t‐test pooled
variance
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Hypothesis Testing
Chi Squared Test‐Statistic
χ2 =
( n − 1) s2
σ 20
where:
n = sample size
s2 = sample variance
σ 20 = hypothesized value for population variance
Test‐Statistic for the F‐Test
F=
s12
s22
where:
s12 = Variance of sample drawn from Population 1
s22 = Variance of sample drawn from Population 2
Hypothesis tests concerning the variance
Hypothesis Test Concerning
Variance of a single, normally distributed
population
Appropriate Test Statistic
Chi‐square stat
Equality of variance of two independent,
normally distributed populations
F‐stat
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19
Technical Analysis
Technical Analysis
Setting Price Targets with Head and Shoulders Patterns
Price target = Neckline - (Head − Neckline)
Setting Price Targets for Inverse Head and Shoulders Patterns
Price target = Neckline + (Neckline − Head)
Momentum or Rate of Change Oscillator
M = (V − Vx ) × 100
where:
M = momentum oscillator value
V = last closing price
Vx = closing price x days ago, typically 10 days
Relative Strength Index
RSI = 100 −
100
1 + RS
where:
RS =
Σ (Up changes for the period under consideration)
Σ(| Down changes for the period under consideration|)
Stochastic Oscillator
C − L14 
%K = 100 
 H14 − L14 
where:
C = last closing price
L14 = lowest price in last 14 days
H14 = highest price in last 14 days
%D (signal line) = Average of the last three %K values calculated daily.
Short Interest ratio
Short interest ratio =
Short interest
Average daily trading volume
Arms Index
Arms index =
20
Number of advancing issues / Number of declining issues
Volume of advancing issues / Volume of declining issues
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Economics
Topics in Demand and Supply Analysis
Topics in Demand and Supply Analysis
The demand function captures the effect of all these factors on demand for a good.
Demand function: QDx = f(Px, I, Py , …) … (Equation 1)
Equation 1 is read as “the quantity demanded of Good X (QDX) depends on the price of
Good X (PX), consumers’ incomes (I) and the price of Good Y (PY), etc.”
The own‐price elasticity of demand is calculated as:
EDPx =
%∆QDx
… (Equation 6)
%∆Px
If we express the percentage change in X as the change in X divided by the value of X,
Equation 6 can be expanded to the following form:
Slope of demand
function.
Coefficient on
own‐price in
market demand
function
EDPx
%∆QDx
=
=
%∆Px
∆QDx
∆Px
QDx
Px
 ∆QDx   Px 
=
… (Equation 7)
 ∆Px   QDx 
Arc elasticity is calculated as:
(Q 0 - Q1 )
× 100
% change in quantity demanded % ∆ Q d (Q 0 + Q1 )/2
=
=
EP =
(P0 - P1 )
% change in price
%∆P
× 100
(P0 + P1 )/2
22
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Topics in Demand and Supply Analysis
Income Elasticity of Demand
Income elasticity of demand measures the responsiveness of demand for a particular good
to a change in income, holding all other things constant.
%∆QDx
ED I =
=
%∆I
EI =
∆QDx
∆I
QDx
I
= 

∆QDx   I 
… (Equation 8)

∆I   QDx 
Same as coefficient
on I in market
demand function
(Equation 11)
% change in quantity demanded
% change in income
Cross‐Price Elasticity of Demand
Cross elasticity of demand measures the responsiveness of demand for a particular good to
a change in price of another good, holding all other things constant.
EDPy
EC =
%∆QDx
=
=
%∆Py
∆QDx
∆Py
QDx
Py
 ∆QDx   Py 
=

 … (Equation 9)
 ∆Py   QDx 
Same as coefficient
on PY in market
demand function
(Equation 11)
% change in quantity demanded
% change in price of substitute or complement
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23
Topics in Demand and Supply Analysis
Accounting Profit
Accounting profit (loss) = Total revenue − Total explicit (accounting) costs.
Economic Profit
Also known as abnormal profit or supernormal profit:
Economic profit = Total revenue − Total economic costs
= Total revenue − (Explicit costs + Implicit costs)
= Accounting profit − Implicit costs
Under perfect competition, economic profit = 0
Normal Profit
Normal profit = Accounting profit − Economic profit
= Implicit costs
Total, Average, and Marginal Revenue
Table: Summary of Revenue Terms
24
Revenue
Calculation
Total revenue (TR)
Price times quantity (P × Q), or the sum of individual
units sold times their respective prices; Σ(Pi × Qi)
Average revenue (AR)
Total revenue divided by quantity; (TR / Q)
Marginal revenue (MR)
Change in total revenue divided by change in quantity;
(ΔTR / ΔQ)
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Topics in Demand and Supply Analysis
Total, Average, Marginal, Fixed, and Variable Costs
Table: Summary of Cost Terms
Costs
Calculation
Total fixed cost (TFC)
Sum of all fixed expenses; here defined to include all
opportunity costs
Total variable cost (TVC)
Sum of all variable expenses, or per unit variable cost
times quantity; (per unit VC × Q)
Total costs (TC)
Total fixed cost plus total variable cost; (TFC + TVC)
Average fixed cost (AFC)
Total fixed cost divided by quantity; (TFC / Q)
Average variable cost (AVC)
Total variable cost divided by quantity; (TVC / Q)
Average total cost (ATC)
Total cost divided by quantity; (TC / Q) or (AFC + AVC)
Marginal cost (MC)
Change in total cost divided by change in quantity;
(ΔTC / ΔQ)
Breakeven, Shutdown, and Exit Points
Revenue/ Cost Relationship
Short-run Decision
Long-run Decision
TR = TC
Continue operating
Continue operating
TR > TVC, but < TC
Continue operating
Exit market
TR < TVC
Shut down production
Exit market
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25
The Firm And Market Structures
The Firm And Market Structures
Relationship of Marginal Revenue to Price Elasticity
The relationship between MR and price elasticity can be expressed as:
MR = P[1 − (1/E p )]
In a monopoly, MC = MR so:
P[1 − (1/E p )] = MC
QSMax where MC = MR in any market, and MC = MR = ATC in perfectly competitive
markets.
Income Elasticity
EY =
%∆QD
%∆Y
Inferior goods: EY < 0
Normal goods: 0 < EY < 1
Luxury goods: EY > 1
Changes shift entire demand curve rather than cause movement along it.
Cross-Price Elasticity
EP =
Y
%∆QDX
%∆PY
Substitutes: EPY > 0 (i.e., direct relationship; PY increases, QDX increases)
Complements: EPY < 0 (i.e., inverse relationship; PY increases, QDX decreases)
Changes shift entire demand curve rather than cause movement along it.
Concentration Measures
N‐firm concentration ratio: Aggregate market share of N largest firms in the industry. The
ratio will equal 0 for perfect competition and 100 for a monopoly.
Herfindahl‐Hirschman Index (HHI): Adds up the squares of the market shares of each of
the largest N companies in the market. The HHI equals 1 for a monopoly. If there are M
firms in the industry with equal market shares, the HHI will equal 1/M.
26
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Aggregate Output, Price, And Economic Growth
Aggregate Output, Price, And Economic Growth
Nominal GDP refers to the value of goods and services included in GDP measured at
current prices.
Nominal GDP = Quantity produced in Year t × Prices in Year t
Real GDP refers to the value of goods and services included in GDP measured at
base‐year prices.
Real GDP = Quantity produced in Year t × Base-year prices
GDP Deflator
GDP deflator =
Value of current year output at current year prices
× 100
Value of current year output at base year prices
GDP deflator =
Nominal GDP
× 100
Real GDP
The Components of GDP
Based on the expenditure approach, GDP may be calculated as:
GDP = C + I + G + (X − M)
C = Consumer spending on final goods and services
I = Gross private domestic investment, which includes business investment in capital goods
(e.g. plant and equipment) and changes in inventory (inventory investment)
G = Government spending on final goods and services
X = Exports
M = Imports
Expenditure Approach
Under the expenditure approach, GDP at market prices may be calculated as:
GDP = Consumer spending on goods and services
+ Business gross fixed investment
+ Change in inventories
+ Government spending on goods and services
+ Government gross fixed investment
+ Exports − Imports
+ Statistical discrepancy
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This equation is just
a breakdown of the
expression for GDP
we stated in the
previous LOS, i.e.
GDP = C + I + G +
(X − M).
27
Aggregate Output, Price, And Economic Growth
Income Approach
Under the income approach, GDP at market prices may be calculated as:
GDP = National income + Capital consumption allowance
+ Statistical discrepancy
… (Equation 1)
National income equals the sum of incomes received by all factors of production used to
generate final output. It includes:
•
•
•
•
•
Employee compensation
Corporate and government enterprise profits before taxes, which includes:
○○ Dividends paid to households
○○ Corporate profits retained by businesses
○○ Corporate taxes paid to the government
Interest income
Rent and unincorporated business net income (proprietor’s income): Amounts
earned by unincorporated proprietors and farm operators, who run their own
businesses.
Indirect business taxes less subsidies: This amount reflects taxes and subsidies that
are included in the final price of a good or service, and therefore represents the
portion of national income that is directly paid to the government.
The capital consumption allowance (CCA) accounts for the wear and tear or depreciation
that occurs in capital stock during the production process. It represents the amount that
must be reinvested by the company in the business to maintain current productivity levels.
You should think of profits + CCA as the amount earned by capital.
Personal income = National income
− Indirect business taxes
− Corporate income taxes
− Undistributed corporate profits
+ Transfer payments
… (Equation 2)
Personal disposable income = Personal income − Personal taxes … (Equation 3)
Personal disposable income = Household consumption + Household saving
28
… (Equation 4)
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Aggregate Output, Price, And Economic Growth
Household saving = Personal disposable income
− Consumption expenditures
− Interest paid by consumers to businesses
− Personal transfer payments to foreigners … (Equation 5)
Business sector saving = Undistributed corporate profits
+ Capital consumption allowance … (Equation 6)
GDP = Household consumption + Total private sector saving + Net taxes
The equality of expenditure and income
S = I + (G − T) + ( X − M) … (Equation 7)
The IS Curve (Relationship between Income and the Real Interest Rate)
Disposable income = GDP − Business saving − Net taxes
S − I = (G − T) + ( X − M) … (Equation 7)
The LM Curve
Quantity theory of money: MV = PY
The quantity theory equation can also be written as:
M/P and MD/P = kY
where:
k = I/V
M = Nominal money supply
MD = Nominal money demand
MD/P is referred to as real money demand and M/P is real money supply.
Equilibrium in the money market requires that money supply and money demand be equal.
Money market equilibrium: M/P = RMD
Solow (neoclassical) growth model
Y = AF(L,K)
where:
Y = Aggregate output
L = Quantity of labor
K = Quantity of capital
A = Technological knowledge or total factor productivity (TFP)
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29
Aggregate Output, Price, And Economic Growth
Growth Accounting Equation
Growth in potential GDP = Growth in technology + WL (Growth in labor)
+ WK (Growth in capital)
Growth in per capita potential GDP = Growth in technology
+ WK (Growth in capital-labor ratio)
Measures of Sustainable Growth
Labor productivity = Real GDP/Aggregate hours
Potential GDP = Aggregate hours × Labor productivity
This equation can be expressed in terms of growth rates as:
Potential GDP growth rate = Long‐term growth rate of labor force + Long‐term labor
productivity growth rate
30
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Understanding Business Cycles
Understanding Business Cycles
Unit labor cost (ULC) is calculated as:
ULC = W/O
where:
O = Output per hour per worker
W = Total labor compensation per hour per worker
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31
Monetary And Fiscal Policy
Monetary And Fiscal Policy
Required reserve ratio = Required reserves / Total deposits
Money multiplier = 1/ (Reserve requirement)
Quantity Theory of Money
M = PY / V
Where:
M = Money supply
V = velocity of transactions
P = price level
Y = real output
Quantity Equation of Exchange
MV = PY
The Fischer effect states that the nominal interest rate (RN) reflects the real interest rate
(RR) and the expected rate of inflation (IIe).
R N = R R + Πe
The Fiscal Multiplier
Ignoring taxes, the multiplier can also be calculated as:
○○
1
1
=
= 10
(1 − MPC) (1 − 0.9)
Assuming taxes, the multiplier can also be calculated as:
1
[1 − MPC(1 − t)]
32
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International Trade And Capital Flows
International Trade And Capital Flows
Balance of Payment Components
A country’s balance of payments is composed of three main accounts:
• The current account balance largely reflects trade in goods and services.
• The capital account balance mainly consists of capital transfers and net sales of
non‐produced, non‐financial assets.
• The financial account measures net capital flows based on sales and purchases of
domestic and foreign financial assets.
Current Account
CA = X – M = Y – (C + I + G)
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33
Currency Exchange Rates
Currency Exchange Rates
Real exchange rate
Real exchange rate DC/FC = SDC/FC × (PFC /PDC )
where:
SDC/FC = Nominal spot exchange rate
PFC = Foreign price level quoted in terms of the foreign currency
PDC = Domestic price level quoted in terms of the domestic currency
Relative Currency Movement
Where P is the price currency (or quote currency) and B is the base currency:
E (%∆SP B ) =
E ( SP B )
SP B
−1
Premium when E(S) > S; expect depreciation of price currency.
Discount when E(S) < S; expect appreciation of price currency.
Currency Cross Rates
P1 P2 P1 B P1
=
×
=
÷
B B
B P2 P2
Arbitrage Relationship
Forward pricing is based on a sum of money invested domestically in the base currency
at the domestic interest rate, rB, as equivalent to the same sum of money converted at the
spot rate for P units of price (foreign) currency, invested at the foreign rate rP for the same
time, and converted back to domestic currency at a forward price set at the beginning of
the term.
 1 
(1 + rB ) = SP B (1 + rP ) 

 FP B 
(1 + rP )
FP B = SP B
(1 + rB )
The last term 1/FP/B in the first equation can also be written FB/P or Fd/f.
34
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Currency Exchange Rates
Forward Premium/(Discount)
Using the previous relationship between forward and spot rates, the forward premium/
(discount) approximately equates to the price currency (foreign) interest rate versus base
currency (domestic) interest rate.
FP B
SP B
FP B
SP B
=
−1 =
=
(1 + rP )
(1 + rB )
(1 + rP )
−1
(1 + rB )
(1 + rP ) (1 + rB ) rP − rB
−
=
(1 + rB ) (1 + rB ) (1 + rB )
A higher base currency (domestic) interest rate results in a forward discount of
approximately the interest differential percentage, leading to base currency (domestic)
appreciation of that percentage.
This relationship must hold or arbitrage will take place to realign spot and forward prices
with the ratedifferential. However, the expected spot exchange rate may differ from the
forward exchange rate.
The forward rate may be calculated as:
FDC/FC =
1
SFC/DC
×
(1 + rDC )
(1 + rDC )
or FDC/FC = SDC/FC ×
(1 + rFC )
(1 + rFC )
This version of the
formula is perhaps
easiest to remember
because it contains
the DC term in
numerator for all
three components:
FDC/FC, SDC/FC and
(1 + rDC)
Forward rates are sometimes interpreted as expected future spot rates.
Ft = St +1
(St +1 )
(r − r )
− 1 = ∆%S(DC/FC)t +1 = DC FC
S
(1 + rFC )
Exchange Rates and the Trade Balance
Marshall-Lerner condition: ω x ε x + ω M (ε M − 1) > 0
where:
ωx = Share of exports in total trade
ωM = Share of imports in total trade
εx = Price elasticity of demand for exports
εM = Price elasticity of demand for imports
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35
Financial Reporting and Analysis
Financial Reporting Mechanics
Financial Reporting Mechanics
Income and Comprehensive Income
Net Income = Revenue + Other income − Expenses
= Income − Expenses
Net income + Other comprehensive income = Comprehensive income
Comprehensive income = Profit after tax
+ Exchange differences
+ Actuarial gains/losses
+ Cash flow hedges
+ Available for sale financial assets (marketable
securities)
+ Deferred taxes
+ Share of profits/losses in equity-accounted
investments
2 Exhibit
10, Vol 3, CFA Program Curriculum 2018
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37
Financial Reporting Mechanics
Cash Flows
Change in cash and equivalents = Cash from operating activities
+ Cash from investing activities
+ Cash from financing activities
Balance Sheet
Assets (basic equation) = Liabilities + Owner’s equity
Owner's equity = Assets − Liabilities
= Contributed capital + Retained earnings
Ending retained earnings = Beginning retained earnings
+ Revenues − Expenses − Dividends
Assets (expanded equation) = Liabilities + Contributed capital
+ Beginning retained earnings + Revenue −
Expenses − Dividends
38
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Understanding Income Statements
Understanding Income Statements
Basic EPS
Basic EPS =
Net income − Preferred dividends
Weighted average number of shares outstanding
Diluted EPS
Diluted EPS =

Preferred 
 Net income − dividends  +


Weighted
average +
shares
Shares from
conversion of
convertible
preferred shares
 Convertible

Convertible

preferred +
× (1 − t ) 
debt


dividends
 interest

Shares from
conversion of
+
convertible
debt
Shares
issuable
from
+
stock options1
Comprehensive Income
Net income + Other comprehensive income = Comprehensive income
Ending Shareholders’ Equity
Ending shareholders’ equity = Beginning shareholders’ equity + Net income +
Other comprehensive income − Dividends declared
1Option warrant conversion uses treasury stock method; i.e., as if company used option value to repurchase
shares at average market price during the period. Convertible preferred and debt shares are ‘if-converted’ at
beginning of period.
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39
Understanding the Balance Sheets
Understanding the Balance Sheets
Gains and Losses on Marketable Securities
Balance Sheet
Items recognized
on the income
statement
Held‐to‐Maturity
Securities
Reported at cost or
amortized cost.
Interest income.
Realized gains and
losses.
Available‐for‐Sale Securities
Reported at fair value.
Trading Securities
Reported at fair value.
Unrealized gains or losses due
to changes in market values are
reported in other comprehensive
income within owners’ equity.
Dividend income.
Dividend income.
Interest income.
Interest income.
Realized gains and losses.
Realized gains and losses.
Unrealized gains and losses
due to changes in market
values.
Liquidity Ratios
Liquidity ratios indicate a company’s ability to meet current obligations.
Current ratio =
Current assets
Current liabilities
Quick ratio
Cash + Marketable securities + Receivables
=
(acid test)
Current liabilities
Cash ratio =
Cash + Marketable securities
Current liabilities
Solvency Ratios
Solvency ratios indicate a company’s financial leverage and financial risk.
L-T debt-to-equity =
40
Total L-T debt
Total equity
Debt-to-equity =
Total debt
Total equity
Total debt =
Total debt
Total assets
Financial leverage =
Total assets
Total equity
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Understanding Cash Flow Statements
Understanding Cash Flow Statements
Cash Flow Classification under U.S. GAAP
CFO
Inflows
Cash collected from customers.
Interest and dividends received.
Proceeds from sale of securities held for trading.
CFI
Inflows
Sale proceeds from fixed assets.
Sale proceeds from long‐term investments.
Outflows
Cash paid to employees.
Cash paid to suppliers.
Cash paid for other expenses.
Cash used to purchase trading
securities.
Interest paid.
Taxes paid.
Outflows
Purchase of fixed assets.
Cash used to acquire LT investment
securities.
CFF
Inflows
Proceeds from debt issuance.
Proceeds from issuance of equity instruments.
Outflows
Repayment of LT debt.
Payments made to repurchase stock.
Dividends payments.
Cash Flow Statements under IFRS and U.S. GAAP
IFRS
U.S. GAAP
Classification of Cash Flows
Interest and dividends received
Interest paid
CFO or CFI
CFO or CFF
CFO
CFO
Dividend paid
Dividends received
Taxes paid
CFO or CFF
CFO or CFI
CFO, but part of the tax can be categorized
as CFI or CFF if it is clear that the tax arose
from investing or financing activities.
CFF
CFO
CFO
Bank overdraft
Included as a part of cash equivalents.
Not considered a part of cash equivalents and
included in CFF.
Direct or indirect method. The former is
preferred.
Direct or indirect method. The former is
preferred. However, if the direct method is
used, a reconciliation of net income and CFO
must be included.
Taxes paid should be presented separately on
the cash flow statement.
If taxes and interest paid are not explicitly
stated on the cash flow statement, details can be
provided in footnotes.
Presentation Format
CFO
(No difference in CFI and CFF
presentation)
Disclosures
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41
Understanding Cash Flow Statements
Free Cash Flow to the Firm
FCFF = NI + NCC + [Int * (1 − tax rate)] − FCInv − WCInv
FCFF = CFO + [Int * (1 − tax rate)] − FCInv
Free Cash Flow to Equity
FCFE = CFO − FCInv + Net borrowing
Performance Ratios
CFO
Net Revenue
CFO
Cash return on assets =
Average total assets
CF to revenue =
Cash return on equity =
Cash to income =
CF per share =
CFO
÷
Average shareholders' equity
CFO
Operating income
( CFO − Preferred dividiends )
Number of common shares outstanding
Coverage Ratios
CFO
Total debt
CFO + Interest paid + Taxes paid
Debt coverage =
Interest paid
Debt coverage =
42
Reinvestment =
CFO
Cash paid for L-T assets
Debt payment =
CFO
Cash paid for L-T debt repayment
Dividend payment =
CFO
Cash paid for dividends
Investing/financing =
CFO
Cash outflows for investing/financing
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Understanding Cash Flow Statements
Financial Analysis Techniques
Activity Ratios
Inventory turnover =
Cost of goods sold
Average inventory
Days of inventory on hand (DOH) =
Receivables turnover =
Revenue
Average receivables
Days of sales outstanding (DSO) =
Payables turnover =
365
Inventory turnover
365
Receivables turnover
Purchases
Average trade payables
Number of days of payables =
Working capital turnover =
365
Payables turnover
Revenue
Average working capital
Fixed asset turnover =
Revenue
Average fixed assets
Total asset turnover =
Revenue
Average total assets
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43
Understanding Cash Flow Statements
Liquidity Ratios
Current ratio =
Quick ratio =
Cash ratio =
Current assets
Current liabilities
Cash + Short-term marketable investments + Receivables
Current liabilities
Cash + Short-term marketable investments
Current liabilities
Defensive interval ratio =
Cash + Short-term marketable investments + Receivables
Daily cash expenditures
Cash conversion cycle = DSO + DOH − Number of days of payables
Solvency Ratios
Debt -to-assets ratio =
Total debt
Total assets
Debt -to-capital ratio =
Total debt
Total debt + Shareholders’ equity
Debt -to-equity ratio =
Total debt
Shareholders’ equity
Financial leverage ratio =
Interest coverage ratio =
44
Average total assets
Average total equity
EBIT
Interest payments
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Understanding Cash Flow Statements
Fixed charge coverage ratio =
EBIT + Lease payments
Interest payments + Lease payments
Profitability Ratios
Gross profit margin =
Gross profit
Revenue
Operating profit margin =
Pretax margin =
Operating profit
Revenue
EBT (earnings before tax, but after interest)
Revenue
Net profit margin =
Net profit
Revenue
Return on Investment Ratios
ROA =
Net income
Average total assets
Adjusted ROA =
Net income + Interest expense (1 − Tax rate)
Average total assets
Operating ROA =
Operating income or EBIT
Average total assets
Return on total capital =
Return on equity =
EBIT
Short-term debt + Long-term debt + Equity
Net income
Average total equity
Return on common equity =
Net income − Preferred dividends
Average common equity
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45
Understanding Cash Flow Statements
DuPont Decomposition of ROE
Net income
Average shareholders’ equity
ROE =
2‐Way Dupont Decomposition
Net income
Average total assets
×
Average total assets Average shareholders’ equity
ROE =
↓
↓
ROA
Leverage
3‐Way Dupont Decomposition
Net income
Average total assets
Revenue
×
×
Revenue
Average total assets Average shareholders’ equity
ROE =
↓
↓
Net profit margin
↓
Asset turnover
Leverage
5‐Way Dupont Decomposition
ROE =
Interest burden
Asset turnover
↓
↓
Net income EBT
Average total assets
EBIT
Revenue
×
×
×
×
EBT
EBIT Revenue Average total assets Avg. shareholders’ equity
↓
↓
↓
Tax burden
EBIT margin
Leverage
Valuation Ratios
P /E =
Price per share
Earnings per share
P /CE =
P /S =
Price per share
Sales per share
P /BV =
46
Price per share
Cash flow per share
Price per share
Book value per share
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Understanding Cash Flow Statements
Per Share Ratios
Cash flow from operations
Average number of shares outstanding
Cash flow per share =
EBITDA per share =
EBITDA
Average number of shares outstanding
Dividends per share =
Common dividends declared
Weighted average number of ordinary shares
Dividend-Related Measures
Dividend payout ratio =
Retention Rate =
Common share dividends
Net income attributable to common shares
Net income attributable to common shares − Common share dividends
Net income attributable to common shares
Sustainable growth rate = Retention rate × ROE
Credit Analysis Ratios
EBIT
Gross interest
EBIT
EBITDA interest coverage =
Gross interest
FFO + interest paid − operating lease adj
FFO1 interest coverage =
Gross interest
EBIT
Return on capital =
Avg (equity + non-current deferred taxes + debt)
EBIT interest coverage =
FFO
Total debt
Adj CFO − capex
Free Operating CF to debt =
Total debt
CFO − capex − dividends paid
Discretionary CF to debt =
Total debt
FFO − dividends
Net CF to capex =
capex
FFO1 to debt =
Debt to EBITDA =
1FFO
Total debt
EBITDA
(free funds to operations) = net income + non-cash charges
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47
Inventories
Inventories
LIFO versus FIFO (with rising prices and stable inventory levels.)
LIFO versus FIFO when Prices are Rising
LIFO
COGS
Higher
Income before taxes
Lower
Income taxes
Lower
Net income
Lower
Cash flow
Higher
EI
Lower
Working capital
Lower
FIFO
Lower
Higher
Higher
Higher
Lower
Higher
Higher
Type of Ratio
Profitability ratios
NP and GP margins
Effect on
Numerator
Income is lower
under LIFO because
COGS is higher
Effect on
Denominator
Sales are the same
under both
Debt-to-equity
Same debt levels
Lower equity under
LIFO
Higher under LIFO
Current ratio
Current assets are
lower under LIFO
because EI is lower
Current liabilities
are the same
Lower under LIFO
Quick ratio
Assets are higher
as a result of lower
taxes paid
Current liabilities
are the same
Higher under LIFO
Inventory turnover
COGS is higher
under LIFO
Average inventory is Higher under LIFO
lower under LIFO
Total asset turnover
Sales are the same
Lower total assets
under LIFO
Effect on Ratio
Lower under LIFO
Higher under LIFO
The LIFO Method and the LIFO Reserve:
EI FIFO = EI LIFO + LR
where
LR = LIFO Reserve
COGSfifo = COGSlifo − (Change in LR during the year)
Net income after tax under FIFO will be greater than LIFO net income after tax by:
Change in LIFO Reserve × (1 − Tax rate)
48
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Inventories
When converting from LIFO to FIFO assuming rising prices:
Equity (retained earnings) increase by:
LIFO Reserve × (1 − Tax rate)
Liabilities (deferred taxes) increase by:
LIFO Reserve × (Tax rate)
Current assets (inventory) increase by:
LIFO Reserve
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49
Long-Lived Assets
Long-Lived Assets
Financial Statement Effects of Capitalizing versus Expensing
Initially when the cost is
capitalized
In future periods when the asset is
depreciated or amortized
When the cost is expensed
Effect on Financial Statements
• Noncurrent assets increase.
• Cash flow from investing activities decreases.
•
•
•
•
•
•
•
•
Net income (first year)
Net income (future years)
Total assets
Shareholders’ equity
Cash flow from operations
Cash flow from investing
Income variability
Debt-to-equity
50
Noncurrent assets decrease.
Net income decreases.
Retained earnings decrease.
Equity decreases.
Net income decreases by the entire after‐tax
amount of the cost.
No related asset is recorded on the balance
sheet and therefore, no depreciation or
amortization expense is charged in future
periods.
Operating cash flow decreases.
Expensed costs have no financial statement
impact in future years.
Capitalizing
Higher
Lower
Higher
Higher
Higher
Lower
Lower
Lower
Expensing
Lower
Higher
Lower
Lower
Lower
Higher
Higher
Higher
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Long-Lived Assets
Original cost − Salvage value
Depreciable life
Depreciation expense =
DDB depreciation in Year X =
2
× Book value at the beginning of Year X
Depreciable life
Depreciation Components
Estimated useful life =
Gross investment in fixed assets
Annual depreciation expense
Average age of asset =
Accumulated depreciation
Annual depreciation expense
Remaining useful life =
Net investment in fixed assets
Annual depreciation expense
Gross investment in fixed assets
Accumulated depreciation
Net investment in fixed assets
=
+
Annual depreciation expense
Annual depreciation expense
Annual depreciation expense
Estimated useful or depreciable
life
The historical cost of an asset
divided by its useful life equals
annual depreciation expense under
the straight line method. Therefore,
the historical cost divided by
annual depreciation expense
equals the estimated useful life.
Average age of asset
Remaining useful life
Annual depreciation expense
times the number of years that
the asset has been in use equals
accumulated depreciation.
Therefore, accumulated
depreciation divided by annual
depreciation equals the average
age of the asset.
The book value of the asset divided
by annual depreciation expense
equals the number of years the asset
has remaining in its useful life.
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51
Income Taxes
Income Taxes
Tax Base
Originally booked asset value
less =
previous tax depreciation/amortization
Amount deductible in future periods (Tax base)
Carrying amount – Tax base = Temporary difference
Effective Tax rate
Effective tax rate =
Income tax expense
Pretax income
Income Tax Expense
Income tax expense = Taxes Payable + Change in DTL − Change in DTA
Treatment of Temporary Differences
Balance Sheet Item
Asset
Asset
Liability
Liability
Carrying Value versus Tax Base
Carrying amount is greater.
Tax base is greater.
Carrying amount is greater.
Tax base is greater.
Results in…
DTL
DTA
DTA
DTL
Income Tax Accounting under IFRS versus U.S. GAAP
IFRS
ISSUE SPECIFIC TREATMENTS
Revaluation of fixed assets Recognized in equity as
and intangible assets.
deferred taxes.
52
U.S. GAAP
Revaluation is prohibited.
Treatment of undistributed
profit from investment in
subsidiaries.
Recognized as deferred
taxes except when the
parent company is able to
control the distribution of
profits and it is probable that
temporary differences will
not reverse in future.
No recognition of
deferred taxes for foreign
subsidiaries that fulfill
indefinite reversal criteria.
No recognition of deferred
taxes for domestic
subsidiaries when amounts
are tax‐free.
Treatment of undistributed
profit from investments in
joint ventures.
Recognized as deferred
taxes except when the
investor controls the sharing
of profits and it is probable
that there will be no reversal
of temporary differences in
future.
No recognition of deferred
taxes for foreign corporate
joint ventures that fulfill
indefinite reversal criteria.
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Income Taxes
IFRS
Treatment of undistributed
profit from investments in
associates.
Recognized as deferred
taxes except when the
investor controls the sharing
of profits and it is probable
that there will be no reversal
of temporary differences in
future.
DEFERRED TAX MEASUREMENT
Tax rates.
Tax rates and tax laws
enacted or substantively
enacted.
Deferred tax asset
recognition.
Recognized if it is probable
that sufficient taxable profit
will be available in the
future.
U.S. GAAP
Deferred taxes are
recognized from temporary
differences.
Only enacted tax rates and
tax laws are used.
Deferred tax assets are
recognized in full and then
reduced by a valuation
allowance if it is likely that
they will not be realized.
DEFERRED TAX PRESENTATION
Offsetting of deferred tax
Offsetting allowed only if
Same as in IFRS.
assets and liabilities.
the entity has right to legally
enforce it and the balance is
related to a tax levied by the
same authority.
Balance sheet
classification.
Classified on balance sheet
as net noncurrent with
supplementary disclosures.
Classified as either current
or noncurrent based on
classification of underlying
asset and liability.
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53
Non-Current (Long-Term) Liabilities
Non-Current (Long-Term) Liabilities
Income Statement Effects of Lease Classification
Income Statement Item
Operating expenses
Nonoperating expenses
EBIT (operating income)
Total expenses‐ early years
Total expenses‐ later years
Net income‐ early years
Net income‐ later years
Finance Lease
Lower
Higher
Higher
Higher
Lower
Lower
Higher
Operating Lease
Higher
Lower
Lower
Lower
Higher
Higher
Lower
Balance Sheet Effects of Lease Classification
Balance Sheet Item
Assets
Current liabilities
Long term liabilities
Total cash
Capital Lease
Higher
Higher
Higher
Same
Operating Lease
Lower
Lower
Lower
Same
Cash Flow Effects of Lease Classification
CF Item
CFO
CFF
Total cash flow
Capital Lease
Higher
Lower
Same
Operating Lease
Lower
Higher
Same
Impact of Lease Classification on Financial Ratios
Ratio
Asset turnover
Numerator
under Finance
Lease
Sales‐ same
Denominator
Ratio Better or
under Finance
Worse under
Lease
Effect on Ratio Finance Lease
Assets‐ higher
Lower
Worse
Return on
assets*
Net income
lower in early
years
Assets‐ higher
Lower
Worse
Current ratio
Current assetssame
Current
liabilitieshigher
Lower
Worse
Leverage ratios
(D/E and D/A)
Debt‐ higher
Equity same
Assets higher
Higher
Worse
Return on
equity*
Net income
lower in early
years
Equity same
Lower
Worse
* In early years of the lease agreement.
54
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Non-Current (Long-Term) Liabilities
Financial Statement Effects of Lease Classification from Lessor’s Perspective
Total net income
Net income (early years)
Taxes (early years)
Total CFO
Total CFI
Total cash flow
Financing Lease
Same
Higher
Higher
Lower
Higher
Same
Operating Lease
Same
Lower
Lower
Higher
Lower
Same
Definitions of Commonly Used Solvency Ratios
Solvency Ratios
Description
Numerator
Denominator
Debt‐to‐assets ratio
Expresses the percentage of
total assets financed by debt
Total debt
Total assets
Debt‐to‐capital ratio
Measures the percentage of a
company’s total capital (debt +
equity) financed by debt.
Total debt
Total debt + Total
shareholders’ equity
Debt‐to‐equity ratio
Measures the amount of debt
financing relative to equity
financing
Total debt
Total shareholders’
equity
Financial leverage ratio
Measures the amount of total
assets supported by one money
unit of equity
Average total assets Average shareholders’
equity
Leverage Ratios
Coverage Ratios
Interest coverage ratio
Measures the number of times a EBIT
company’s EBIT could cover its
interest payments.
Interest payments
Fixed charge coverage ratio
Measures the number of
times a company’s earnings
(before interest, taxes and
lease payments) can cover the
company’s interest and lease
payments.
Interest payments +
Lease payments
EBIT + Lease
payments
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55
Financial Reporting Quality
Financial Reporting Quality
Relationship between Financial Reporting Quality and Earnings Quality
Financial Reporting Quality
Low
Earnings High
(Results)
Quality
LOW financial reporting
quality impedes assessment
of earnings quality and
Low impedes valuation.
56
High
HIGH financial reporting quality
enables assessment.
HIGH earnings quality increases
company value.
HIGH financial reporting quality
enables assessment.
LOW earnings quality decreases
company value.
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Corporate Finance
Capital Budgeting
Capital Budgeting
Net Present Value (NPV)
n
CFt
− Outlay
(1
+ r) t
t =1
NPV = ∑
where:
CFt = after‐tax cash flow at time, t.
r = required rate of return for the investment. This is the firm’s cost of capital adjusted for
the risk inherent in the project.
Outlay = investment cash outflow at t = 0.
Internal Rate of Return (IRR)
n
n
CFt
∑ (1 + IRR)
t = Outlay
t =1
CF
t
∑ (1 + IRR)
t − Outlay = 0
t =1
Payback Period
Payback period and discounted payback period have the same formula, but discounted
payback uses cash flows discounted by the appropriate rate:
n
n
∑ CFn+1 −∑ CFn
Payback
= n + t =0
period
n
t =0
∑ CFn+1
t =0
n
n
where n = # of periods that ∑ CFn < 0 and
t =0
n
∑ CFn+1 −∑ CFn
t =0
n
t =0
∑ CFn+1
is the partial year (in year n+1)
t =0
CF required to recover the remaining investment amount.
Profitability Index
PI =
58
PV of future cash flows
NPV
= 1+
Initial investment
Initial investment
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Cost of Capital
Cost of Capital
Weighted Average Cost of Capital
WACC = (wd )(rd )(1 − t) + (wp )(rp ) + (we )(re )
where:
wd = Proportion of debt that the company uses when it raises new funds
rd = Before‐tax marginal cost of debt
t = Company’s marginal tax rate
wp = Proportion of preferred stock that the company uses when it raises new funds
rp = Marginal cost of preferred stock
we = Proportion of equity that the company uses when it raises new funds
re = Marginal cost of equity
To Transform Debt‐to‐equity Ratio into a Component’s Weight
D
E = D =w
d
D
1+ E D + E
wd + we = 1
Valuation of Bonds


 n PMT 
M
t

P0 =  ∑
t +
n
 t =1  rd    rd 
1
+
1
+
 
2
2   
where:
P0 = current market price of the bond.
PMTt = interest payment in period t.
rd = yield to maturity on BEY basis.
n = number of periods remaining to maturity.
M = Par or maturity value of the bond.
Valuation of Preferred Stock
Vp =
Dp
rp
where:
Vp = current value (price) of preferred stock.
Dp = preferred stock dividend per share.
rp = cost of preferred stock.
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59
Cost of Capital
Required Return on a Stock
Capital Asset Pricing Model
re = R F + β i [E(R M ) − R F ]
where:
[E(RM) − Rf] = Equity risk premium.
RM = Expected return on the market.
βi = Beta of stock. Beta measures the sensitivity of the stock’s returns to changes in market
returns.
RF = Risk‐free rate.
re = Expected return on stock (cost of equity).
Dividend Discount Model
P0 =
D1
re − g
where:
P0 = current market value of the security.
D1 = next year’s dividend.
re = required rate of return on common equity.
g = the firm’s expected constant growth rate of dividends.
Rearranging the above equation gives us a formula to calculate the required return on
equity:
re =
D1
+g
P0
Sustainable Growth Rate
D 
g = 1 −
× ( ROE )
 EPS 
Where (1 − (D/EPS)) = Earnings retention rate
Bond Yield plus Risk Premium Approach
re = rd + risk premium, where rd is the required return on debt.
60
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Cost of Capital
Unlevered beta for a comparable asset
β ASSET




1
= β EQUITY 

 1 +  (1 − t ) D  
 
E  
Beta for a project using a comparable asset re-levered for target company
D 

β PROJECT = β ASSET 1 +  (1 − t )  

E

Country Risk Premium
re = R F + β [E(R M ) − R F + CRP]
Country equity risk
=
premium
Break point =
Sovereign yield
×
spread
Annualized standard deviation of equity index
Annualized standard deviation of sovereign
bond market in terms of the developed market
currency
Amount of capital at which a component’s cost of capital changes
Proportion of new capital raised from the component
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61
Measures of Leverage
Measures of Leverage
Degree of Operating Leverage
DOL =
Percentage change in operating income
Percentage change in units sold
DOL =
Q × (P − V)
Q × (P − V) − F
where:
Q = Number of units sold
P = Price per unit
V = Variable operating cost per unit
F = Fixed operating cost
Q × (P − V) = Contribution margin (the amount that units sold contribute to covering fixed
costs)
(P − V) = Contribution margin per unit
Degree of Financial Leverage
DFL =
Percentage change in net income
Percentage change in operating income
DFL =
[Q(P − V) − F](1 − t)
[Q(P − V) − F]
=
[Q(P − V) − F − C](1 − t) [Q(P − V) − F − C]
where:
Q = Number of units sold
P = Price per unit
V = Variable operating cost per unit
F = Fixed operating cost
C = Fixed financial cost
t = Tax rate
62
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Measures of Leverage
Degree of Total Leverage
Percentage change in net income
Percentage change in the number of units sold
DTL =
DTL = DOL × DFL
DTL =
Q × (P − V)
[Q(P − V) − F − C]
where:
Q = Number of units produced and sold
P = Price per unit
V = Variable operating cost per unit
F = Fixed operating cost
C = Fixed financial cost
Breakeven point
PQ BE = VQ BE + F + C
where:
P = Price per unit
Q = Number of units produced and sold
V = Variable cost per unit
F = Fixed operating costs
C = Fixed financial cost
The breakeven number of units can be calculated as:
Q BE =
F+C
P−V
Note that taxes are not considered in breakeven analysis because there is no taxable income.
Operating breakeven point
PQ OBE = VQ OBE + F
Q OBE =
F
P−V
Net income at various levels of sales
Net income = [Q ( P − V ) − F − C ](1 − t )
Note that this formula considers taxes, unlike the variant formula for breakeven point
(which has no taxable income).
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63
Working Capital Management
Working Capital Management
Liquidity Measures
Current Ratio =
Quick Ratio =
Current assets
Current liabilities
Cash + Short term marketable investments + Receivables
Current liabilities
Accounts receivable turnover =
Credit sales
Average receivables
Number of days of receivables =
=
Inventory turnover =
Accounts receivable
Sales on credit / 365
Cost of goods sold
Average inventory
Number of days of inventory =
=
Payables turnover =
Accounts receivable
Average days sales on credit
Inventory
Average day’s cost of goods sold
Inventory
Cost of goods sold / 365
Purchases
Average trade payables
Number of days of payables =
=
Accounts payables
Average day’s purchases
Accounts payables
Purchases / 365
Purchases = Ending inventory + COGS − Beginning inventory
Operating cycle = Number of days of inventory + Number of days of receivables
Net operating cycle = Number of days of inventory + Number of days of receivables
− Number of days of payables
64
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Working Capital Management
Short-term Investment Returns
 360 
Face value − price   360 
Money market yield = 
= Holding period yield × 
 × 


 Days 
 Days 
Price
 365 
Face value − price   365 
Bond equivalent yield = 
= Holding period yield × 
 × 

 Days 
 Days 
Price
 360 
Face value − price   360 
Discount basis yield = 
= % discount × 
×



  Days 
 Days 
Face value
% Discount =
Face value − Price
Price
Working Capital Effectiveness Measure
Inventory turnover =
Cost of goods sold
Average inventory
Number of days of inventory =
Inventory
Average days cost of goods sold
=
Inventory
Cost of goods sold / 365
=
365
Inventory turnover
 365
Discount  

Implicit rate = Cost of trade credit =  1 +
 1 − Discount 

Number of days

beyond discount period 
−1
Accounts payable
Average day’s purchases
Accounts payable
365
=
Purchases / 365
Payables turnover
Number of days of payables =
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65
Working Capital Management
Short-term Funding Cost Measures
Line of credit cost =
Interest + Commitment fee
Loan amount
Banker’s acceptance cost =
Interest
Interest
=
Net proceeds Loan amount − Interest
Interest + Dealer’s commission + Backup costs
Loan amount − Interest
Portfolio Management: An Overview
The Portfolio Management Process
Planning
Understanding client needs
Preparing an investment policy statement (IPS)
Execution
Asset allocation
Security analysis
Portfolio construction
Feedback
Portfolio monitoring and rebalancing
Performance measurement and reporting
66
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Portfolio Management
Risk Management: An Introduction
Risk Management: An Introduction
Figure 1-1: The Risk Management Framework in an Enterprise
Risk Governance
Risk Drivers
Board
Management
Goals
Strategies
Risk
Tolerance
Establish Risk
Management
Infrastructure
MODIFY Risk Mitigation
(Allocate to)
Risky Activities
& Management
Identify Risks
Measure Risks
NO
Risk
Budgeting
Risk Exposures
Policies & Processes
Risks
in
Line?
Montitor Risks
YES
Reports
(Communications)
Strategic
Analysis
68
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Portfolio Risk and Return: Part I
Portfolio Risk and Return: Part I
Holding Period Return
Pt − Pt −1 + D t Pt − Pt −1 D t
=
+
= Capital gain + Dividend yield
Pt −1
Pt −1
Pt −1
P + DT
= T
−1
P0
R=
where:
Pt = Price at the end of the period
Pt-1 = Price at the beginning of the period
Dt = Dividend for the period
Holding Period Returns for more than One Period
R = [(1 + R1 ) × (1 + R 2 ) × . . . × (1 + R n )] − 1
where:
R1, R2, . . . , Rn are sub‐period returns
Geometric Mean Return
R = {[(1 + R1 ) × (1 + R 2 ) × . . . × (1 + R n )]1/n} − 1
Annualized Return
rannual = (1 + rperiod )n − 1
where:
r = Return on investment
n = Number of periods in a year
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69
Portfolio Risk and Return: Part I
Portfolio Return
R p = w1R1 + w2 R 2
where:
Rp = Portfolio return
w1 = Weight of Asset 1
w2 = Weight of Asset 2
R1 = Return of Asset 1
R2 = Return of Asset 2
Variance of a Single Asset
T
∑ (R t − µ)2
t =1
σ2 =
T
where:
Rt = Return for the period t
T = Total number of periods
μ = Mean of T returns
Variance of a Representative Sample of the Population
T
s2 =
∑ (R t − R)2
t =1
T −1
where:
R = mean return of the sample observations
s2 = sample variance
Standard Deviation of an Asset
T
σ=
T
∑ (R t − µ)2
t =1
s=
T
∑ (R t − R)2
t =1
T −1
Variance of a Portfolio of Assets
σ 2P =
N
∑ wi w jCov(R i ,R j )
i, j=1
N
σ 2P = ∑ w2i Var(R i ) +
i =1
70
N
∑
i, j=1, i ≠ j
wi w jCov(R i ,R j )
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Portfolio Risk and Return: Part I
Standard Deviation of a Portfolio of Two Risky Assets
σ p = w12 σ12 + w22 σ 22 + 2w1w2 σ1σ 2ρ1,2 or
w12 σ12 + w22 σ 22 + 2w1w2 Cov1,2
Utility Function
1
U = E(R) − Aσ 2
2
where:
U = Utility of an investment
E(R) = Expected return
σ2 = Variance of returns
A = Additional return required by the investor to accept an additional unit of risk.
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71
Portfolio Risk and Return: Part II
Portfolio Risk and Return: Part II
Capital Allocation Line
The CAL has an intercept of RFR and a constant slope that equals:
[E(R i ) − RFR]
σi
Expected Return on portfolios that lie on CML
E(R p ) = w1R f + (1 − w1 ) E(R m )
Variance of portfolios that lie on CML
σ 2 = w12 σ 2f + (1 − w1 )2 σ 2m + 2w1 (1 − w1 )Cov(R f , R m )
Equation of CML
E(R p ) = R f +
E(R m ) − R f
× σp
σm
where:
y‐intercept = Rf = risk‐free rate
slope =
E(R m ) − R f
= market price of risk.
σm
Systematic and Nonsystematic Risk
Total Risk = Systematic risk + Unsystematic risk
Return‐Generating Models
k
k
j=1
j= 2
E(R i ) − R f = ∑ β ij E(Fj ) = β i1[E(R m ) − R f ] + ∑ β ij E(Fj )
The Market Model
R i = α i + βi R m + ei
Calculation of Beta
βi =
72
Cov(R i ,R m ) ρi,m σ i σ m ρi,m σ i
=
=
σm
σ 2m
σ 2m
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Portfolio Risk and Return: Part II
The Capital Asset Pricing Model
E(R i ) = R f + β i [E(R m ) − R f ]
Sharpe ratio
Sharpe ratio =
Rp − Rf
σp
Treynor ratio
Treynor ratio =
Rp − Rf
βp
M‐squared (M2)
M2 = (R p − R f )
σm
− (R m − R f )
σp
Jensen’s alpha
α p = R p − [R f + β p (R m − R f )]
Security Characteristic Line
R i − R f = α i + β i (R m − R f )
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73
Equity
Market Organization and Structure
Market Organization and Structure
Leveraged Position Metrics
PQ
Position size
1
Leverage
=
= 0 0 =
ratio
Equity size P0 Q 0 M0 M0
Equity per share
% Equity
=M=
margin
Price per share
Equity after disposal
Return using
=
margin
Initial Equity
=
Pt + D − C − P0 (1 − M 0 )(1 + r CM )
P0 M 0 + C
Where:
P0 = initial share price
Pt = share disposal price
D = dividend per share during the period
C = commission per share
rCM = call money rate
M0 = initial margin
Security Price at which the Investor Would Receive a Margin Call
P0 ×
(1 − Initial margin)
(1 − Maintenance margin)
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75
Security Market Indices
Security Market Indices
The value of a price return index is calculated as follows:
N
VPRI =
∑ ni Pi
i =1
D
where:
VPRI = Value of the price return index
ni = Number of units of constituent security i held in the index portfolio
N = Number of constituent securities in the index
Pi = Unit price of constituent security i
D = Value of the divisor
Price Return
The price return of an index can be calculated as:
PR I =
VPRI1 − VPRI0
VPRI0
where:
PRI = Price return of the index portfolio (as a decimal number)
VPRI1 = Value of the price return index at the end of the period
VPRI0 = Value of the price return index at the beginning of the period
The price return of each constituent security is calculated as:
PR i =
Pi1 − Pi0
Pi0
where:
PRi = Price return of constituent security i (as a decimal number)
Pi1 = Price of the constituent security i at the end of the period
Pi0 = Price of the constituent security i at the beginning of the period
The price return of the index equals the weighted average price return of the constituent
securities. It is calculated as:
PRI = w1PR1 + w2PR2 + . . . + wNPRN
where:
PRI = Price return of the index portfolio (as a decimal number)
PRi = Price return of constituent security i (as a decimal number)
wi = Weight of security i in the index portfolio
N = Number of securities in the index
76
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Security Market Indices
Total Return
The total return of an index can be calculated as:
TR I =
VPRI1 − VPRI0 + Inc I
VPRI0
where:
TRI = Total return of the index portfolio (as a decimal number)
VPRI1 = Value of the total return index at the end of the period
VPRI0 = Value of the total return index at the beginning of the period
IncI = Total income from all securities in the index held over the period
The total return of each constituent security is calculated as:
TR i =
P1i − P0i + Inc i
P0i
where:
TRi = Total return of constituent security i (as a decimal number)
P1i = Price of constituent security i at the end of the period
P0i = Price of constituent security i at the beginning of the period
Inci = Total income from security i over the period
The total return of the index equals the weighted average total return of the constituent
securities. It is calculated as:
TRI = w1TR1 + w2TR2 + . . . + wNTRN
where:
TRI = Total return of the index portfolio (as a decimal number)
TRi = Total return of constituent security i (as a decimal number)
wi = Weight of security i in the index portfolio
N = Number of securities in the index
Calculation of Index Returns over Multiple Time Periods
Given a series of price returns for an index, the value of a price return index can be
calculated as:
VPRIT = VPRI0 (1 + PR I1 ) (1 + PR I2 ) . . . (1 + PR IT )
where:
VPRI0 = Value of the price return index at inception
VPRIT = Value of the price return index at time t
PRIT = Price return (as a decimal number) on the index over the period
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77
Security Market Indices
Similarly, the value of a total return index may be calculated as:
VTRIT = VTRI0 (1 + TR I1 ) (1 + TR I2 ) . . . (1 + TR IT )
where:
VTRI0 = Value of the index at inception
VTRIT = Value of the index at time t
TRIT = Total return (as a decimal number) on the index over the period
Price Weighting
wPi =
Pi
N
∑ Pi
i =1
Equal Weighting
wEi =
1
N
where:
wi = Fraction of the portfolio that is allocated to security i or weight of security i
N = Number of securities in the index
Market‐Capitalization Weighting
wiM =
Q i Pi
N
∑ Q jPj
j=1
where:
wi = Fraction of the portfolio that is allocated to security i or weight of security i
Qi = Number of shares outstanding of security i
Pi = Share price of security i
N = Number of securities in the index
The float‐adjusted market‐capitalization weight of each constituent security is calculated as:
wiM =
fi Q i Pi
N
∑ f jQ jPj
j=1
where:
fi = Fraction of shares outstanding in the market float
wi = Fraction of the portfolio that is allocated to security i or weight of security i
Qi = Number of shares outstanding of security i
Pi = Share price of security i
N = Number of securities in the index
78
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Security Market Indices
Fundamental Weighting
wFi =
Fi
N
∑ Fj
j=1
where:
Fi = A given fundamental size measure of company i
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79
Overview of Equity Securities
Overview of Equity Securities
Return Characteristics of Equity Securities
Total Return, Rt = (Pt – Pt−1 + Dt) / Pt−1
where:
Pt‐1 = Purchase price at time t − 1
Pt = Selling price at time t
Dt = Dividends paid by the company during the period
Accounting Return on Equity
ROE t =
80
NI t
NI t
=
Average BVE t (BVE t + BVE t −1 )/2
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Equity Valuation: Concepts and Basic Tools
Equity Valuation: Concepts and Basic Tools
Dividend Discount Model (DDM)
V0 =
D1
D2
D∞
1 +
2 + +
(1 + k e )∞
(1 + k e ) (1 + k e )
n
Dt
t
t =1 (1 + k e )
V0 = ∑
One year holding period:
V0 =
dividend to be received year-end price
+
(1 + k e )1
(1 + k e )1
Multiple‐Year Holding Period DDM
V0 =
D1
D2
Pn
1 +
2 + +
(1 + k e )n
(1 + k e ) (1 + k e )
where:
Pn = Price at the end of n years.
Infinite Period DDM (Gordon Growth Model)
V0 =
D0 (1 + gc )1 D0 (1 + gc )2 D 0 (1 + gc )3
D0 (1 + gc )∞
+
+
+
+
(1 + k e )∞
(1 + k e )1
(1 + k e )2
(1 + k e )3
This equation simplifies to:
V0 =
D0 (1 + gc )1
D1
1 =
k e − gc
(k e − gc )
The long‐term (constant) growth rate is usually calculated as:
gc = RR × ROE
Where RR is the reinvestment rate, or 1 – dividend payout rate.
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81
Equity Valuation: Concepts and Basic Tools
Multi‐Stage Dividend Discount Model
V0 =
D1
D2
Dn
Pn
n +
1 +
2 + +
(1 + k e )
(1 + k e )n
(1 + k e ) (1 + k e )
where:
Dn +1
Pn =
k e − gc
Dn = Last dividend of the supernormal growth period
Dn+1 = First dividend of the constant growth period
The Free‐Cash‐Flow‐to‐Equity (FCFE) Model
FCFE = CFO − FC Inv + Net borrowing
Analysts may calculate the intrinsic value of the company’s stock by discounting their
projections of future FCFE at the required rate of return on equity.
∞
FCFE t
t
t =1 (1 + k e )
V0 = ∑
Value of a Preferred Stock
When preferred stock is non‐callable, non‐convertible, has no maturity date and pays
dividends at a fixed rate, the value of the preferred stock can be calculated using the
perpetuity formula:
V0 =
D0
r
For a non‐callable, non‐convertible preferred stock with maturity at time, n, the value of
the stock can be calculated using the following formula:
n
Dt
F
t +
(1
+
r)
(1
+
r)n
t =1
V0 = ∑
where:
V0 = value of preferred stock today (t = 0)
Dt = expected dividend in year t, assumed to be paid at the end of the year
r = required rate of return on the stock
F = par value of preferred stock
82
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Equity Valuation: Concepts and Basic Tools
Price Multiples
P0 D1 /E1
=
E1
r−g
Price to cash flow ratio =
Market price of share
Cash flow per share
Price to sales ratio =
Market price per share
Net sales per share
Price to sales ratio =
Market value of equity
Total net sales
P/BV =
Current market price of share
Book value per share
P/BV =
Market value of common shareholders’ equity
Book value of common shareholders’ equity
where:
Book value of common shareholders’ equity =
(Total assets – Total liabilities) – Preferred stock
Enterprise Value Multiples
EV/EBITDA
where:
EV = Enterprise value and is calculated as the market value of the company’s common
stock plus the market value of outstanding preferred stock if any, plus the market value of
debt, less cash and short term investments (cash equivalents).
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83
Fixed Income
Fixed-Income Securities: Defining Elements
Fixed-Income Securities: Defining Elements
Bond Coupon
Coupon = Coupon rate × Par value
Coupon Rate (Floating)
Coupon Rate = Reference rate + Quoted margin
Coupon Rate (Inverse Floaters)
Coupon rate = K − L × (Reference rate)
Callable Bonds
Value of callable bond = Value of non‐callable bond − Value of embedded call option
Value of embedded call option = Value of non‐callable bond − Value of callable bond
Putable Bonds
Value of putable bond = Value of non‐putable bond + Value of embedded put option
Value of embedded put option = Value of putable bond − Value of non‐putable bond
Traditional Analysis of Convertible Securities
Conversion value = Market price of common stock × Conversion ratio
Market conversion price =
Market price of convertible security
Conversion ratio
Market conversion premium per share = Market conversion price − Current market price
Market conversion premium ratio =
Premium payback period =
Market conversion premium per share
Market price of common stock
Market conversion premium per share
Favorable income differential per share
Favorable income differential per share =
Premium over straight value =
Coupon interest − (Conversion ratio × Common stock dividend per share)
Conversion ratio
Market price of convertible bond
−1
Straight value
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85
Fixed-Income Securities: Defining Elements
Fixed Income Markets: Issuance, Trading, and Funding
Discount Interest
The borrower receives less than the full amount that must eventually be repaid:
YD =
(VM − VB )  360  =
VM


t 
D  360 


VM  t 
t 
D = VM YD 
 360 
t 

VB = VM 1 − YD 
 360  

D  360 
id =

 , and VB < VM ∴ id < YD
VB  t 
Where
YD = Discount rate (or discount yield)
VM = Value repaid at maturity (i.e., amount of the loan)
VB = Borrower’s proceeds
D = Discount from value repaid at maturity (i.e., VM – VB)
Id = borrower’s interest rate, which is greater than the discount yield.
Add-On Interest
The borrower borrows one amount and repays another amount that includes both the
borrowed amount and interest on the borrowed amount:
t 

VM = VB 1 + id 
 360  

86
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Introduction to Fixed-Income Valuation
Introduction to Fixed-Income Valuation
Pricing Bonds with Spot Rates
PV =
PMT
PMT
PMT + FV
1 +
2 +…+
(1 + z N ) N
(1 + z1 ) (1 + z 2 )
z1 = Spot rate for Period 1
z2 = Spot rate for Period 2
zN = Spot rate for Period N
Flat Price, Accrued Interest and the Full Price
Figure: Valuing a Bond between Coupon‐Payment Dates
PV Full = PV Flat + AI
AI = t/T × PMT
PV Full = PV × (1 + r) t/T
Semiannual bond basis yield or semiannual bond equivalent yield
 1 + SAR M 


M 
M
 SAR N 
= 1 +


N 
N
Important: What we refer to as stated annual rate (SAR) is referred to in the curriculum as APR or annual percentage rate. We
stick to SAR to keep your focus on a stated annual rate versus the effective annual rate. Just remember that if you see an annual
percentage rate on the exam, it refers to the stated annual rate.
Current yield
Current yield =
Annual cash coupon payment
Bond price
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87
Introduction to Fixed-Income Valuation
Option‐adjusted price
Value of non‐callable bond (option‐adjusted price) = F
lat price of callable bond
+ Value of embedded call option
Money Market Pricing, Discount-rate basis
 Days

PV = FV ×  1 −
× DR 


year
 Year   FV − PV 
DR = 
×
 Days   FV 
Money Market Pricing, Add-on Rate Basis
PV=
FV
 1 + Days × AOR 


Year
 Year   FV − PV 
AOR = 
×
 Days   PV 
Yield Spreads over the Benchmark Yield Curve
PV =
•
•
PMT
PMT
PMT + FV
1 +
2 + ...+
(1 + z N + Z) N
(1 + z1 + Z) (1 + z 2 + Z)
The benchmark spot rates z1, z2, zN are derived from the government yield curve
(or from fixed rates on interest rate swaps).
Z refers to the z‐spread per period. It is constant for all time periods.
Option‐adjusted Spread (OAS)
OAS = z‐spread − Option value (bps per year)
Implied Forward Rates
(1 + z A ) A (1 + IFRA, B− A ) B− A = (1 + z B ) B

(1 + z B ) B 
IFRA, B− A = ( B− A)
 −1
(1 + z A ) A 

88
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Introduction to Asset-Backed Securities
Introduction to Asset-Backed Securities
Parties to the Securitization
Party
Seller
Issuer/Trust
Servicer
SMMt =
Description
Originates the loans and sells loans to the SPV
The SPV that buys the loans from the seller
and issues the asset-backed securities
Services the loans
Party in
Illustration
ABC Company
SPV
Servicer
Prepayment in month t
Beginning mortgage balance for month t − Scheduled principal payment in month t
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89
Understanding Fixed-Income Risk and Return
Understanding Fixed-Income Risk and Return
Macaulay Duration
1 + r 1 + r + [N × (c − r)] 
MacDur = 
−
 − (t/T)
c × [(1 + r) N − 1] + r 
 r
c = Coupon rate per period (PMT/FV)
Modified Duration
ModDur =
MacDur
1+ r
Modified duration has a very important application in risk management. It can be used to
estimate the percentage price change for a bond in response to a change in its yield‐to‐
maturity.
%∆PV Full ≈ − AnnModDur × ∆Yield
If Macaulay duration is not already known, annual modified duration can be estimated
using the following formula:
ApproxModDur =
(PV− ) − (PV+ )
2 × ( ∆Yield) × (PV0 )
We can also use the approximate modified duration (ApproxModDur) to estimate
Macaulay duration (ApproxMacDur) by applying the following formula:
ApproxMacDur = ApproxModDur × (1 + r)
Effective Duration
EffDur =
(PV− ) − (PV+ )
2 × ( ∆Curve) × (PV0 )
Duration of a Bond Portfolio
Portfolio duration = w1D1 + w2 D2 +…+ w N D N
Annual ModDur =
90
Annual MacDur
1+ r
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Understanding Fixed-Income Risk and Return
Money Duration
MoneyDur = AnnModDur × PVFull
The estimated (dollar) change in the price of the bond is calculated as:
ΔPVFull = – MoneyDur × ΔYield
Price Value of a Basis Point
PVBP =
(PV− ) − (PV+ )
2
Basis Point Value (BPV)
BPV = MoneyDur × 0.0001 (1 bps expressed as a decimal)
Annual Convexity
ApproxCon =
(PV− ) + (PV+ ) − [2 × (PV0 )]
( ∆Yield)2 × (PV0 )
Price Effects Based on Duration and Convexity
1
%∆PV Full ≈ (− AnnModDur × ∆Yield) +  × AnnConvexity × ( ∆Yield)2 

2
Money convexity
1
∆PV Full ≈ (− MoneyDur × ∆Yield) +  × MoneyCon × ( ∆Yield)2 

2
Effective convexity
EffCon =
[(PV− ) + (PV+ )] − [2 × (PV0 )]
( ∆Curve)2 × (PV0 )
Yield Volatility
1
%∆PV Full ≈ (− AnnModDur × ∆Yield) +  × AnnConvexity × ( ∆Yield)2 

2
Duration Gap
Duration gap = Macaulay duration − Investment horizon
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91
Fundamentals of Credit Analysis
Fundamentals of Credit Analysis
Expected Loss
Expected loss = Default probability × Loss severity given default
Yield on a corporate bond:
Yield on a corporate bond = Real risk-free interest rate + Expected inflation rate
+ Maturity premium + Liquidity premium + Credit spread
Yield Spread:
Yield spread = Liquidity premium + Credit spread
For small, instantaneous changes in the yield spread, the return impact (i.e. the percentage
change in price, including accrued interest) can be estimated using the following formula:
Return impact ≈ − Modified duration × ∆Spread
For larger changes in the yield spread, we must also incorporate the (positive) impact of
convexity into our estimate of the return impact:
Return impact ≈ −(MDur × ∆Spread) + (1/2 × Convexity × ∆Spread 2 )
92
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Derivatives
Basics of Derivative Pricing and Valuation
Basics of Derivative Pricing and Valuation
Fundamental Value of an Asset
 E(ST ) 
S0 = 
T −θ+ γ
 (1 + r + λ) 
Where:
r = risk free rate
λ = asset risk premium
θ = Storage, insurance, and other costs at present value
γ = Convenience and other benefits at present value
Arbitrage and Replication:
Asset + Derivative = Risk-free asset
Asset − Risk-free asset = − Derivative
Derivative − Risk-free asset = − Asset
Forward Contract Payoffs:
Long position
Short position
ST > F(0,T)
ST – F(0,T)
(Positive payoff)
ST < F(0,T)
ST – F(0,T)
(Negative payoff)
–[ST – F(0,T)]
(Negative payoff)
–[ST – F(0,T)]
(Positive payoff)
Forward price:
F(0,T) = S0 (1 + r)T
F(0,T) = (S0 − γ + θ)(1 + r)T or F(0,T) = S0 (1 + r)T − ( γ − θ)(1 + r)T
*Note
that benefits (γ) and costs (θ) are expressed in terms of present value.
Value of a forward contract:
Vt (0,T) = St − [F(0,T) / (1 + r)T− t ]
Vt (0,T) = St − ( γ − θ)(1 + r) t − [F(0,T) / (1 + r)T− t ]
94
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Basics of Derivative Pricing and Valuation
Time
At initiation
Long Position Value
Zero, as the contract is
priced to prevent arbitrage
Short Position Value
Zero, as the contract is
priced to prevent arbitrage
During
life of the
contract
At expiration
 F(0,T) 
St − 
T− t 
 (1+r ) 
 F(0,T) 

T − t  − St
 (1+r ) 
ST – F(0,T)
F(0,T) – ST
Net payment made (received) by the fixed‐rate payer on a swap:
Net fixed rate payment t = [Swap fixed rate − (LIBOR t −1 + spread)]* (No. of days/360)* NP
Call Option Payoffs
Option Position
Call option holder
Call option writer
Descriptions
Choice to buy the
underlying asset for X
Obligation to sell the
underlying asset for X if
the option holder chooses
to exercise the option
Payoff
ST > X
Option holder
exercises the
option
ST – X
– (ST – X)
ST < X
Option holder
does not exercise
the option
0
0
Moneyness and Exercise Value of a Call Option
Moneyness
In‐the‐money
At‐the‐money
Out‐of‐the‐money
Current Market Price (St)
versus Exercise Price (X)
St is greater than X
St equals X
St is less than X
Intrinsic Value Max
[0, (St – X)]
St – X
0
0
Put Option Payoffs
Option Position
Descriptions
Put option holder
Choice to sell the
underlying asset for X
Obligation to buy the
underlying asset for X if
the option holder chooses
to exercise the option
Put option writer
Payoff
ST > X
ST < X
Option holder
Option holder
exercises the
does not exercise
option
the option
0
X – ST
– (X – ST)
0
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95
Basics of Derivative Pricing and Valuation
Moneyness and Exercise Value of a Put Option
Current Market Price (St)
versus Exercise Price (X)
St is less than X
St equals X
St is greater than X
Moneyness
In‐the‐money
At‐the‐money
Out‐of‐the‐money
Intrinsic Value Max
[0, (X – St)]
X – St
0
0
Fiduciary Call and Protective Put Payoffs
Security
Call option
Zero coupon bond
Fiduciary call payoff
Value if ST > X
ST – X
X
ST
Value if ST < X
Zero
X
X
Put option
Stock
Protective put payoff
Zero
ST
ST
X – ST
ST
X
Put‐Call Parity
c0 +
X
= p 0 + S0
(1 + R F )T
Combining Portfolios to Make Synthetic Securities
Strategy
fiduciary
call
Consisting
of
Value
long call +
X
long bond c 0 +
long call
long call
c0
=
long put
long put
p0
=
long
underlying
asset
long bond
long
underlying
asset
long bond
S0
=
X
(1 + R F )T
=
96
(1 + R F )T
Equals Strategy
Consisting of
=
Protective put long put + long
underlying asset
Synthetic call long put + long
underlying asset
+ short bond
Synthetic put long call + short
underlying asset
+ long bond
Synthetic
long call + long
underlying
bond + short put
asset
Synthetic
long put + long
bond
underlying asset
+ short call
Value
p0 + S0
p 0 + S0 −
X
(1 + R F )T
c 0 − S0 +
X
(1 + R F )T
c0 +
X
− p0
(1 + R F )T
p0 + S0 – c0
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Basics of Derivative Pricing and Valuation
Lowest Prices of European Calls and Puts
c 0 ≥ Max[0,S0 −
p0 ≥ Max[0,
X
]
(1 + R F )T
X
− S0 ]
(1 + R F )T
Put‐Call Forward Parity
p0 − c 0 =
[X − F(0,T)]
(1 + R F )T
Binomial Option Pricing
c=
πc + + (1 − π)c −
(1 + r)
π=
(1 + r − d)
(u − d)
Where u =
S1+
S−
and d = 1
S0
S0
Hedge ratio
n=
c+ − c −
S+ − S−
Lowest Prices of American Calls and Puts
C0 ≥ Max[0, S0 − X/(1 + RFR)T ]
P0 ≥ Max[0, (X − S0 )]
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97
Basics of Derivative Pricing and Valuation
Summary of Options Strategies
Holder
Writer
Call
CT = max(0,ST − X)
Value at expiration = CT
Profit: Π = CT − C0
Maximum profit = ∞
Maximum loss = C0
Breakeven: ST* = X + C0
CT = max(0,ST − X)
Value at expiration = –CT
Profit: Π = –CT − C0
Maximum profit = C0
Maximum loss = ∞
Breakeven: ST* = X + C0
Put
PT = max(0,X − ST)
Value at expiration = PT
Profit: Π = PT − P0
Maximum profit = X − P0
Maximum loss = P0
Breakeven: ST* = X − P0
PT = max(0,X − ST)
Value at expiration = –PT
Profit: Π = –PT − P0
Maximum profit = P0
Maximum loss = X − P0
Breakeven: ST* = X − P0
Where:
C0, CT = price of the call option at time 0 and time T
P0, PT = price of the put option at time 0 and time T
X = exercise price
S0, ST = price of the underlying at time 0 and time T
V0, VT = value of the position at time 0 and time T
Π = profit from the transaction: VT − V0
r = risk‐free rate
Covered Call
Value at expiration: VT = ST − max(0,ST − X)
Profit: Π = VT − S0 + C0
Maximum profit = X − S0 + C0
Maximum loss = S0 − C0
Breakeven: ST* = S0 − C0
Protective Put
Value at expiration: VT = ST + max(0,X − ST)
Profit: Π = VT − S0 − P0
Maximum profit = ∞
Maximum loss = S0 + P0 − X
Breakeven: ST* = S0 + P0
98
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Alternative Investments
Introduction to Alternative Investments
Introduction to Alternative Investments
Hedge Fund Fee Structure
When quoted as 2 and 20, 1 and 10, etc., the first number is the percent management
fee and the second number is the percent performance fee. Where management fees and
performance fees are calculated independently, it means that management fees are charged
against AUM gross of fees and performance fees are charged against profit calculated net
of fees.
Hedge Fund Management Fee
Management fee = FM × AUM t −1
AUMt-1 = beginning assets under management (end of last period)
FM = percent management fee
Note that hedge funds may apply a management fee to ending AUM, but this is not as
common because performance fees already realize a share of investor profits during the
period.
Hedge Fund Performance Fee
In general, performance fees equal a percentage fee multiplied by investor profit. Profit
may be described as gross of fees (i.e., independent of management fees) or net of fees.
A performance fee may be subject to a high-water mark based on the highest end-ofperiod AUM value (to avoid additional fees against a profit that only improves an investor
position to a previous high water mark):
IF AUM t −1 < HWM < AUM t , then
Performance fee = FP × ( AUM t − HWM ) , ELSE
Performance fee = FP × ( AUM t − AUM t −1 )
Where
AUMt = ending assets under management
HWM = high-water mark
FP = percent performance fee
Note that an investor usually must make a profit before performance fees are applied.
100
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Introduction to Alternative Investments
Hurdle Rate
The percentage performance fee applied may be subject to a hurdle rate (e.g., risk-free
rate, S&P500 index return, etc.):


AUM t
FP = 
− 1 − rH
 AUM t −1 or HWM 
= rA − rH
Where
rA = percent return of the asset (i.e., fund)
rH = percent hurdle rate (i.e., benchmark rate)
Hedge Fund Net-of-Fee Returns
net-of-fees return = rI =
AUM t
−1
AUM t −1
Where
rI = investor return
Note that AUM will be actual, net-of-fee returns for this measure.
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101