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Basit graduated Magna Cum Laude from the world-renowned Wharton School of Business at the
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his own private wealth management firm. He started teaching CFA courses more than five years
ago, and upon discovering how much he enjoyed teaching, he founded Elan Guides with a view to
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in the CPA and CFA prep industries. Previously he worked as an auditor for Deloitte & Touche, was
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financial statement analysis, and tax at both the graduate and undergraduate levels at Fordham
University’s business school. He graduated Magna Cum laude from Pace University and went on to
earn his JD degree from Fordham University School of Law.
© 2013 ELAN GUIDES
QUANTITATIVE METHODS
CORRELATION AND REGRESSION
n
Sample covariance = Cov (X,Y) =
 (X  X)(Y  Y)/(n  1)
i
i
i=1
where:
n = sample size
Xi = ith observation of Variable X
X = mean observation of Variable X
Yi = ith observation of Variable Y
Y = mean observation of Variable Y
Cov (X,Y)
Sample correlation coefficient = r =
s Xs Y
n
Sample variance =
2
sX
=
 (X  X) /(n  1)
i
2
i=1
Sample standard deviation = sX =
2
sX
Test statistic
r n2
Test-stat = t =
1  r2
Where:
n = Number of observations
r = Sample correlation
Linear Regression with One Independent Variable
Regression model equation = Yi = b0 + b1Xi + i, i = 1,...., n




b1 and b0 are the regression coefficients.
b1 is the slope coefficient.
b0 is the intercept term.
 is the error term that represents the variation in the dependent variable that
is not explained by the independent variable.
© 2013 ELAN GUIDES
QUANTITATIVE METHODS
Regression line equation = Yˆi = bˆ 0 + bˆ1Xi , i = 1,...., n
Regression Residuals
n
 [Y  (bˆ  bˆ X )]
i
1 i
0
2
i=1
where:
Yi = Actual value of the dependent variable
bˆ 0 + bˆ 1Xi = Predicted value of dependent variable
The Standard Error of Estimate
(
SEE =
n
n

(Yi  bˆ 0  bˆ 1Xi)2
i=1
) ( )
 (ˆ )
1/2
i
=
n2
2
1/2
i=1
n2
=
(
SSE
n2
)
1/2
The Coefficient of Determination
Total variation = Unexplained variation + Explained variation
R2 =
Explained variation
=
Total variation  Unexplained variation
Total variation
Total variation
Unexplained variation
=1
Total variation
Hypothesis Tests on Regression Coefficients
CAPM: RABC = RF + ABC(RM – RF)
RABC – RF =  + ABC(RM – RF) + 


The intercept term for the regression, b0, is .
The slope coefficient for the regression, b1, is ABC
The regression sum of squares (RSS)
n
RSS =
 (Y^  Y )
2
i
 Explained variation
i=1
The sum of squared errors or residuals (SSE)
n
SSE =
 (Y  Y^ )
i
i
2
 Unexplained variation
i=1
© 2013 ELAN GUIDES
QUANTITATIVE METHODS
ANOVA Table
Source of Variation
Degrees of Freedom
Sum of Squares
k
RSS
n k + 1)
SSE
n 1
SST
Regression (explained)
Error (unexplained)
Total
k = the number of slope coefficients in the regression.
Prediction Intervals
2
sf = s
2
Y^  tc sf
© 2013 ELAN GUIDES
[
1
1
n
(X  X)2

2
(n  1) sx
]
Mean Sum of Squares
MSR =
RSS
k
=
MSE =
RSS
1
SSE
n 2
= RSS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
Multiple regression equation
Multiple regression equation = Yi = b0 + b1X1i + b2X2i + . . .+ bk Xki + i, i = 1,2, . . . , n
Yi
Xji
b0
b1, . . . , bk
i
n
= the ith observation of the dependent variable Y
= the ith observation of the independent variable Xj , j = 1,2, . . . , k
= the intercept of the equation
= the slope coefficients for each of the independent variables
= the error term for the ith observation
= the number of observations
Residual Term
ˆi = Yi  Yˆi = Yi  (bˆ0 + bˆ 1X1i + bˆ 2X2i + . . .+ bˆ k Xki)
Confidence Intervals
bˆj ± (tc  sbˆj)
estimated regression coefficient ± (critical t-value)(coefficient standard error)
F-statistic
F-stat =
MSR
RSS/k
=
MSE
SSE/[n k + 1)]
R2 and Adjusted R2
R2 =
Total variation  Unexplained variation
Total variation
Adjusted R2 = R2 = 1 
(
n 1
n k1
)
=
SST  SSE
SST
=
RSS
SST
(1 R2)
Testing for Heteroskedasticity- The Breusch-Pagan (BP) Test
2 = nR2 with k degrees of freedom
n = Number of observations
R2 = Coefficient of determination of the second regression (the regression when the squared
residuals of the original regression are regressed on the independent variables).
k = Number of independent variables
© 2013 ELAN GUIDES
MULTIPLE REGRESSION AND ISSUES IN REGRESSION
Testing for Serial Correlation- The Durban-Watson (DW) Test
DW  2(1 – r); where r is the sample correlation between squared residuals from one period and
those from the previous period.
Value of Durbin-Watson Statistic
(H0: No serial correlation)
Reject H0,
conclude
Positive Serial
Correlation
dl
0
Do not Reject
H0
Inconclusive
du
Inconclusive
4  du
Reject H0,
conclude
Negative Serial
Correlation
4  dl
4
Problems in Linear Regression and Solutions
Problem
Effect
Solution
Heteroskedasticity
Incorrect standard errors
Use robust standard errors
(corrected for conditional
heteroskedasticity)
Serial correlation
Incorrect standard errors (additional
problems if a lagged value of the
dependent variable is used as an
independent variable)
Use robust standard errors
(corrected for serial correlation)
Multicollinearity
High R2 and low t-statistics
Remove one or more independent
variables; often no solution based
in theory
Model Specification Errors
Yi = b0 + b1lnX1i + b2X2i + 
Linear Trend Models
yt = b0 + b1t + t,
t = 1, 2, . . . , T
where:
yt = the value of the time series at time t (value of the dependent variable)
b0 = the y-intercept term
b1 = the slope coefficient/ trend coefficient
t = time, the independent or explanatory variable
t = a random-error term
© 2013 ELAN GUIDES
TIME SERIES ANALYSIS
TIME-SERIES ANALYSIS
Linear Trend Models
yt = b0 + b1t + t,
t = 1, 2, . . . , T
where:
yt = the value of the time series at time t (value of the dependent variable)
b0 = the y-intercept term
b1 = the slope coefficient/ trend coefficient
t = time, the independent or explanatory variable
t = a random-error term
Log-Linear Trend Models
A series that grows exponentially can be described using the following equation:
yt = eb0 + b1t
where:
yt = the value of the time series at time t (value of the dependent variable)
b0 = the y-intercept term
b1 = the slope coefficient
t = time = 1, 2, 3 ... T
We take the natural logarithm of both sides of the equation to arrive at the equation for the loglinear model:
ln yt = b0 + b1t + t,
t = 1,2, . . . , T
AUTOREGRESSIVE (AR) TIME-SERIES MODELS
xt = b0 + b1xt  1 + t
A pth order autoregressive model is represented as:
xt = b0 + b1xt  1 + b2xt  2+ . . . + bpxt  p + t
Detecting Serially Correlated Errors in an AR Model
t-stat =
Residual autocorrelation for lag
Standard error of residual autocorrelation
where:
Standard error of residual autocorrelation = 1/ T
T = Number of observations in the time series
© 2013 ELAN GUIDES
TIME SERIES ANALYSIS
Mean Reversion
xt =
b0
1  b1
Multiperiod Forecasts and the Chain Rule of Forecasting
^x = ^b + ^b x
t+1
0
1 t
Random Walks
xt = xt  1 + t , E(t) = 0, E(t2) = 2, E(ts) = 0 if t s
The first difference of the random walk equation is given as:
yt = xt  xt  1 = xt  1 + t  xt  1= t , E(t) = 0, E(t2) = 2, E(ts) = 0 for t s
Random Walk with a Drift
xt = b0 + b1xt  1 + t
b1 = 1, b0 0, or
xt = b0 + xt  1 + t , E(t) = 0
The first-difference of the random walk with a drift equation is given as:
yt = xt  xt  1 , yt = b0 + t , b0 0
The Unit Root Test of Nonstationarity
xt  b0 + b1xt  1 + t
xt  xt  1  b0 + b1xt  1  xt  1 + t
xt  xt  1  b0 + (b1  1)xt  1 + t
xt  xt  1  b0 + g1xt  1 + t
© 2013 ELAN GUIDES
TIME SERIES ANALYSIS
Autoregressive Moving Average (ARMA) Models
xt = b0 + b1xt  1 + . . . + bpxt  p + t + 1t  1 +. . . + qt  q
E(t) = 0, E(t2) = 2, E(ts) = 0 for t s
Autoregressive Conditional Heteroskedasticity Models (ARCH Models)
^ 2 = a + a ^ 2 + u
t
0
1 t 1
t
The error in period t+1 can then be predicted using the following formula:
^ 2 = a^ + a^ ^ 2
t+1
0
1 t
© 2013 ELAN GUIDES
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
Currency Cross Rates
For example, given the USD/EUR and JPY/USD exchange rates, we can calculate the cross rate
between the JPY and the EUR, JPY/EUR as follows:
JPY
JPY USD
=

EUR
USD EUR
Cross Rate Calculations with Bid-Ask Spreads
USD/EURask = 1.3806
USD/EURbid = 1.3802
 Represents the price of EUR
 An investor can buy EUR with
USD at this price.
 Represents the price of EUR (base
currency).
 An investor can sell EUR for USD
at this price (as it is the bid price
quoted by the dealer).
Determining the EUR/USDbid cross rate:
EUR/USDbid = 1/(USD/EURask)
Determining the EUR/USDask cross rate:
EUR/USDask = 1 / (USD/EURbid)
Forward exchange rates (F) - One year Horizom
FFC/DC = SFC/DC 
(1 + iFC)
(1 + iDC)
FPC/BC = SPC/BC 
Forward exchange rates (F) - Any Investment Horizom
FFC/DC = SFC/DC 
1 + (iFC Actual 360)
1 + (iDC Actual 360)
FPC/BC = SPC/BC 
1 + (iPC Actual 360)
1 + (iBC Actual 360)
Currencies Trading at a Forward Premium/Discount
FFC/DC  SFC/DC = SFC/DC
FPC/BC  SPC/BC = SPC/BC
© 2013 ELAN GUIDES
(
(
)
)
(iFC  iDC) Actual 360
1 + (iDC Actual 360)
(iPC  iBC) Actual 360
1 + (iBC Actual 360)
(1 + iPC)
(1 + iBC)
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
Covered Interest Rate Parity
1 + (iPC Actual 360)
1 + (iBC Actual 360)
FPC/BC = SPC/BC 
The forward premium (discount) on the base currency can be expressed as a percentage as:
FPC/BC  SPC/BC
Forward premium (discount) as a % =
SPC/BC
The forward premium (discount) on the base currency can be estimated as:
Forward premium (discount) as a %  FPC/BC  SPC/BC  iPC  iBC
Uncovered Interest Rate Parity
Expected future spot exchange rate:
SeFC/DC = SFC/DC 
(1 + iFC)
(1 + iDC)
The expected percentage change in the spot exchange rate can be calculated as:
Expected % change in spot exchange rate = SePC/BC =
SePC/BC – SPC/BC
SPC/BC
The expected percentage change in the spot exchange rate can be estimated as:
Expected % change in spot exchange rate  SePC/BC iPCiBC
Purchasing Power Parity (PPP)
Law of one price: PXFC = PXDC  SFC/DC
Law of one price: PXPC = PXBC  SPC/BC
Absolute Purchasing Power Parity (Absolute PPP)
SFC/DC = GPLFC / GPLDC
SPC/BC = GPLPC / GPLBC
Relative Purchasing Power Parity (Relative PPP)
Relative PPP: E(S
T
FC/DC)
= S
0
FC/DC
(
T
1  FC
1 + DC
)
Ex Ante Version of PPP
Ex ante PPP: %SeFC/DC  eFC eDC
Ex ante PPP: %SePC/BC ePCeBC
© 2013 ELAN GUIDES
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
Real Exchange Rates
The real exchange rate (qFC/DC) equals the ratio of the domestic price level expressed in the foreign
currency to the foreign price level.
qFC/DC =
PDC in terms of FC
PFC
=
PDC  SFC/DC
PFC
= SFC/DC
( )
PDC
PFC
The Fisher Effect
Fischer Effect: i = r + e
International Fisher effect: (iFC – iDC) = (eFC – eDC)
Figure 1: Spot Exchange Rates, Forward Exchange Rates, and Interest Rates
Ex Ante PPP
Foreign-Domestic
Expected Inflation
Differential
eFC  eDC
International Fisher
Effect
Expected change
in
Spot Exchange Rate
%SeFC/DC
Forward Rate as
an Unbiased
Predictor
Uncovered Interest
Rate Parity
Foreign-Domestic
Interest rate
Differential
iFC  iDC
Forward Discount
FFC/DC SFC/DC
SFC/DC
Covered
Interest Rate
Parity
Balance of Payment
Current account + Capital account + Financial account = 0
Real Interest Rate Differentials, Capital Flows and the Exchange Rate
qL/H – qL/H = (iH – iL) – (eH – eL) – (H – L)
© 2013 ELAN GUIDES
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
The Taylor rule
i = rn +  + y y*)
where
i = the Taylor rule prescribed central bank policy rate
rn = the neutral real policy rate
 = the current inflation rate
* = the central bank’s target inflation rate
y = the log of the current level of output
y* = the log of the economy’s potential/sustainable level of output
qPC/BC = qPC/BC + ( rnBC rnPC) + BCBCPCPC
yBC y*BC) yPC y*PC)] BC PC)
© 2013 ELAN GUIDES
ECONOMIC GROWTH AND THE INVESTMENT DECISION
ECONOMIC GROWTH AND THE INVESTMENT DECISION
Relationship between economic growth and stock prices
P = GDP
( )( )
E
GDP
P
E
P = Aggregate price or value of earnings.
E = Aggregate earnings
This equation can also be expressed in terms of growth rates:
P = (GDP) + (E/GDP) + (P/E)
Production Function
Y = AKL1-
Y = Level of aggregate output in the economy
L = Quantity of labor
K = Quantity of capital
A = Total factor productivity. Total factor productivity (TFP) reflects the general level
of productivity or technology in the economy. TFP is a scale factor i.e., an increase
in TFP implies a proportionate increase in output for any combination of inputs.
 = Share of GDP paid out to capital
1   = Share of GDP paid out to labor
y = Y/L = A(K/L)(L/L)1- = Ak 
y = Y/L = Output per worker or labor productivity.
k = K/L = Capital per worker or capital-labor ratio
Cobb-Douglas production function
Y/Y =A/A + K/K + (1  )L/L
Potential GDP
Growth rate in potential GDP = Long-term growth rate of labor force
+ Long-term growth rate in labor productivity
Labor Supply
Total number of hours available for work = Labor force Average hours worked per worker
© 2013 ELAN GUIDES
ECONOMIC GROWTH AND THE INVESTMENT DECISION
Neoclassical Model (Solow’s Model)
( )[( )
Y
1
=
K
s
]

+ + n 
(1-)
s = Fraction of income that is saved
 = Growth rate of TFP
 = Elasticity of output with respect to capital
y = Y/L or income per worker
k = K/L or capital-labor ratio
 = Constant rate of depreciation on physical stock
n = Labor supply growth rate.
Savings/Investment Equation:
sy =
[(
)
]

+ + n k
(1 )
Growth rates of Output Per Capita and the Capital-Labor Ratio
y

Y
=
+ s

y
(1)
K
( ) (
k

Y
=
+s

k
(1)
K
)
Production Function in the Endogenous Growth Model
ye = f(ke) = cke
© 2013 ELAN GUIDES
INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS
Ending Inventory = Opening Inventory + Purchases - Cost of goods sold
LIFO and FIFO Comparison with Rising Prices and Stable Inventory Levels
LIFO
FIFO
COGS
Higher
Lower
Income before taxes
Lower
Higher
Income taxes
Lower
Higher
Net income
Lower
Higher
Total cash flow
Higher
Lower
EI
Lower
Higher
Working capital
Lower
Higher
LIFO versus FIFO with Rising Prices and Stable Inventory Levels
Type of Ratio
Profitability ratios
NP and GP margins
Solvency ratios
Debt-to-equity and
debt ratio
Liquidity ratios
Current ratio
Quick ratio
Activity ratios
Inventory turnover
Total asset turnover
© 2013 ELAN GUIDES
Effect on
Numerator
Effect on
Denominator
Income is lower
under LIFO because
COGS is higher
Sales are the same
under both
Lower under LIFO
Same debt levels
Lower equity and
assets under LIFO
Higher under LIFO
Current assets are
lower under LIFO
because EI is lower
Current liabilities are
the same.
Lower under LIFO
Quick assets are
higher under LIFO as
a result of lower taxes
paid
Current liabilities are
the same
Higher under LIFO
COGS is higher
under LIFO
Average inventory is
lower under LIFO
Higher under LIFO
Sales are the same
Lower total assets
under LIFO
Higher under LIFO
Effect on Ratio
INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
LIFO reserve (LR)
EIFIFO = EILIFO + LR
where LR = LIFO Reserve
COGSFIFO is lower than COGSLIFO during periods of rising prices:
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)
When converting from LIFO to FIFO assuming rising prices:
Equity (retained earnings) increases by:
LIFO Reserve  (1 Tax rate)
Liabilities (deferred taxes) increase by:
LIFO Reserve  (Tax rate)
Current assets (inventory) increase by:
LIFO Reserve
© 2013 ELAN GUIDES
INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
Impact of an Inventory Write-Down on Various Financial Ratios
Type of Ratio
Profitability ratios
NP and GP margins
Solvency ratios
Debt-to-equity and
debt ratio
Liquidity ratios
Current ratio
Activity ratios
Inventory turnover
Total asset turnover
© 2013 ELAN GUIDES
Effect on
Numerator
Effect on
Denominator
COGS increases so
profits fall
Sales remain the
same
Lower (worsens)
Debt levels remain
the same
Equity decreases
(due to lower profits)
and current assets
decrease (due to
lower inventory)
Higher (worsens)
Current assets
decrease (due to
lower inventory)
Current liabilities
remain the same.
Lower (worsens)
COGS increases
Average inventory
decreases
Higher (improves)
Sales remain the
same
Total assets decrease
Higher (improves)
Effect on Ratio
LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS
LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND
RATIOS
Effects of Capitalization
Initially when the cost is capitalized
In future periods when the asset is
depreciated or amortized
Effects on Financial Statements
Noncurrent assets increase.
Cash flow from investing activities decreases.
Noncurrent assets decrease.
Net income decreases.
Retained earnings decrease.
Equity decreases.
Effects of Expensing
When the item is expensed
Effects on Financial Statements
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.
Financial Statement Effects of Capitalizing versus Expensing
Capitalizing
Net income (first year)
Higher
Net income (future years)
Lower
Total assets
Higher
Shareholders’ equity
Higher
Cash flow from operations activities
Higher
Cash flow from investing activities
Lower
Income variability
Lower
Debt to equity ratio
Lower
Expensing
Lower
Higher
Lower
Lower
Lower
Higher
Higher
Higher
Straight Line Depreciation
Depreciation expense =
Original cost  Salvage value
Depreciable life
Accelerated Depreciation
DDB depreciation in Year X =
2
× Book value at the beginning of Year X
Depreciable life
© 2013 ELAN GUIDES
LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS
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.
Income Statement Effects of Lease Classification
Income Statement Item
Finance Lease
Operating expenses
Nonoperating expenses
EBIT (operating income)
Total expenses- early years
Total expenses- later years
Net income- early years
Net income- later years
Lower (Depreciation)
Higher (Interest expense)
Higher
Higher
Lower
Lower
Higher
Cash Flow Effects of Lease Classification
CF Item
Finance Lease
CFO
Higher
CFF
Lower
Total cash flow
Same
© 2013 ELAN GUIDES
Operating Lease
Lower
Higher
Same
Operating Lease
Higher (Lease payment)
Lower (None)
Lower
Lower
Higher
Higher
Lower
LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS
Table 9: Impact of Lease Classification on Financial Ratios
Ratio
Asset turnover
Denominator
Ratio Better or
Numerator under under Finance
Worse under
Finance Lease
Lease
Effect on Ratio Finance Lease
Sales- same
Return on assets* Net income- lower
Assets- higher
Lower
Worse
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
Equity- same
Lower
Worse
Return on equity* Net income- lower
**Notice that both the numerator and the denominator for the D/A ratio are higher when classifying
the lease as a finance lease. Beware of such exam questions. When the numerator and the denominator
of any ratio are heading in the same direction (either increasing or decreasing), determine which of the
two is changing more in percentage terms. If the percentage change in the numerator is greater than the
percentage change in the denominator, the numerator effect will dominate.
Firms usually have lower levels of total debt compared to total assets. The increase in both debt and
assets by classifying the lease as a finance lease will lead to an increase in the debt to asset ratio because
the percentage increase in the numerator is greater.
Financial Statement Effects of Lease Classification from Lessor’s Perspective
Financing Lease Operating Lease
Total net income
Same
Same
Net income (early years)
Higher
Lower
Taxes (early years)
Higher
Lower
Total CFO
Lower
Higher
Total CFI
Higher
Lower
Total cash flow
Same
Same
© 2013 ELAN GUIDES
INTERCORPORATE INVESTMENTS
INTERCORPORATE INVESTMENTS
Summary of Accounting Treatment for Investments
In Financial Assets
In Associates
Business
Combinations In Joint Ventures
Influence
Not significant
Significant
Controlling
Typical
percentage
interest
Usually < 20%
Usually 20%  50% Usually > 50% Varies
Accounting Classified into one of
Treatment four categories based on
management intent and
type of security.
Equity method
Consolidation
Debt only:
 Held-to-maturity
(amortized cost,
changes in value
ignored unless
deemed as impaired)
Debt and Equity:
 Held for trading
(fair value, changes
in value recognized
in profit or loss)
 Available-for-sale
(fair value, changes
in value recognized
in equity)
 Designated at fair
value (fair value,
changes in value
recognized in profit
or loss)
Combination
Merger
Acquisition
Consolidation
© 2013 ELAN GUIDES
Description
Company A + Company B = Company A
Company A + Company B = (Company A + Company B)
Company A + Company B = Company C
Shared Control
IFRS: Equity method
or proportionate
consolidation
U.S. GAAP: Equity
method
(except for
unincorporated
ventures in
specialized industries)
INTERCORPORATE INVESTMENTS
Adjusted Values Upon Reclassification of Sale of Receivables:
CFO
Lower
CFF
Higher
Total cash flow
Same
Current assets
Higher
Current liabilities
Higher
Current ratio
Lower
(Assuming it was greater than 1)
Difference between QSPE and SPE
Securitized Transaction: Qualified Special
Purpose Entity
Securitized Transaction: Special
Purpose Entity
 Originator of receivables sells financial
assets to an SPE.
 The originator does not own or hold
or expect to receive beneficial interest.
 SFAS 140 (before 2008 revision)
allowed seller to derecognize the sold
assets if transferred assets have been
isolated from the transferor and are
beyond the reach of bankruptcy, and are
financial assets.
 Originator of receivables sells financial
assets to an SPE.
 Seller is primary beneficiary; absorbs
risks and rewards.
 Seller maintains some level of control.
 Seller is required to consolidate.
 Seller’s balance sheet would still
show receivables as an asset.
 Debt of SPE would appear on seller’s
balance sheet.
Impact of Different Accounting Methods on Financial Ratios
Equity Method
Proportionate
Consolidation
Acquisition Method
Leverage
Better (lower) as liabilities are
lower and equity is the same
In-between
Worse (higher) as liabilities are
higher and equity is the same
Net Profit
Margin
Better (higher) as sales are lower
and net income is the same
In-between
Worse (lower) as sales are higher
and net income is the same
ROE
Better (higher) as equity is lower Same as under the
and net income is the same
equity method
Worse (lower) as equity is higher
and net income is the same
ROA
Better (higher) as net income is
the same and assets are lower
Worse (lower) as net income is
the same and assets are higher
In-between
© 2013 ELAN GUIDES
EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
Final year’s salary = Current salary × [(1 + Annual compensation increase)years until retirement]
Estimated annual payment = (Estimated final salary × Benefit formula) × Years of service
Annual unit credit = Value at retirement / Years of service
Types of Post-Employment Benefits
Amount of PostEmployment Benefit to
Employee
Obligation of Sponsoring
Company
Defined contribution
pension plan
Amount of future benefit is
not defined. Actual future
benefit will depend on
investment performance of
plan assets.
Investment risk is borne by
employee.
Amount of the company’s Not applicable.
obligation (contribution)
is defined in each period.
The contribution, if any, is
typically made on a periodic
basis with no additional
future obligation.
Defined benefit
pension plan
Amount of future benefit is
defined, based on the plan’s
formula (often a function of
length of service and final
year’s compensation).
Investment risk is borne by
company.
Amount of the future
obligation, based on the
plan’s formula, must be
estimated in the current
period.
Companies typically prefund the DB plans by
contributing funds to a
pension trust. Regulatory
requirements to pre-fund
vary by country.
Other postemployment benefits
(e.g., retirees’ health
care)
Amount of future benefit
depends on plan
specifications and type of
benefit.
Eventual benefits are
specified. The amount of the
future obligation must be
estimated in the current
period.
Companies typically do not
pre-fund other postemployment benefit
obligations.
Type of Benefit
A company’s pension obligation will increase as a result of:
 Current service costs.
 Interest costs.
 Past service costs.
 Actuarial losses.
A company’s pension obligation will decrease as a result of:
 Actuarial gains.
 Benefits paid.
© 2013 ELAN GUIDES
Sponsoring Company’s
Pre-funding of Its Future
Obligation
EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
Reconciliation of the Pension Obligation:
Pension obligation at the beginning of the period
+ Current service costs
+ Interest costs
+ Past service costs
+ Actuarial losses
– Actuarial gains
– Benefits paid
Pension obligation at the end of the period
The fair value of assets held in the pension trust (plan) will increase as a result of:
 A positive actual dollar return earned on plan assets; and
 Contributions made by the employer to the plan.
The fair value of plan assets will decrease as a result of:
 Benefits paid to employees.
Reconciliation of the Fair Value of Plan Assets:
Fair value of plan assets at the beginning of the period
+ Actual return on plan assets
+ Contributions made by the employer to the plan
 Benefits paid to employees
Fair value of plan assets at the end of the period
Balance Sheet Presentation of Defined Benefit Pension Plans
Funded status = Pension obligation  Fair value of plan assets


If Pension obligation > Fair value of plan assets:
Plan is underfunded  Positive funded status  Net pension liability.
If Pension obligation < Fair value of plan assets:
Plan is overfunded Negative funded status  Net pension asset.
Calculating Periodic Pension Cost
Priodic pension cost = Ending funded status  Beginning funded status + Employer contributions
Periodic pension cost = Current service costs + Interest costs + Past service costs
+ Actuarial losses  Actuarial gains  Actual return on plan assets
Under the corridor method, if the net cumulative amount of unrecognized actuarial gains and losses at the
beginning of the reporting period exceeds 10% of the greater of (1) the defined benefit obligation or (2) the
fair value of plan assets, then the excess is amortized over the expected average remaining working lives of
the employees participating in the plan and included as a component of periodic pension expense on the P&L.
© 2013 ELAN GUIDES
EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
Components of a Company’s Defined Benefit Pension Periodic Costs
IFRS Component IFRS Recognition
U.S. GAAP Component
U.S. GAAP Recognition
Service costs
Recognized in P&L.
Current service costs
Past service costs
Recognized in P&L.
Recognized in OCI and
subsequently amortized to
P&L over the service life of
employees.
Net interest
income/ expense
Recognized in P&L as
the following amount:
Net pension liability or
asset × interest rate(a)
Interest expense on
pension obligation
Expected return on
plan assets
Recognized in P&L.
Actuarial gains and losses
including differences
between the actual and
expected returns on plan
assets
Recognized immediately in
P&L or, more commonly,
recognized in OCI and
subsequently amortized to
P&L using the corridor or
faster recognition method.(b)
 Difference between
expected and actual return
on assets = Actual return 
(Plan assets × Expected
return).
 Actuarial gains and losses
= Changes in a company’s
pension obligation arising
from changes in actuarial
assumptions.
Remeasurements: Recognized in OCI and
Net return on plan not subsequently
assets and actuarial amortized to P&L.
gains and losses
 Net return on plan
assets = Actual return
 (Plan assets ×
Interest rate).
 Actuarial gains and
losses = Changes in a
company’s pension
obligation arising from
changes in actuarial
assumptions.
Recognized in P&L as the
following amount: Plan assets
× expected return.
(a) The interest rate used is equal to the discount rate used to measure the pension liability (the yield on highquality corporate bonds.)
(b) If the cumulative amount of unrecognized actuarial gains and losses exceeds 10 percent of the greater of the
value of the plan assets or of the present value of the DB obligation (under U.S. GAAP, the projected benefit
obligation), the difference must be amortized over the service lives of the employees.
© 2013 ELAN GUIDES
EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
Impact of Key Assumptions on Net Pension Liability and Periodic Pension Cost
Impact of Assumption on Net Impact of Assumption on Periodic
Assumption
Pension Liability (Asset)
Pension Cost and Pension Expense
Higher discount
rate
Lower obligation
Pension cost and pension expense will
both typically be lower because of lower
opening obligation and lower service
costs
Higher rate of
compensation
increase
Higher obligation
Higher service and interest costs will
increase periodic pension cost and
pension expense.
Higher expected
return on plan
assets
No effect, because fair value
of plan assets are used on
balance sheet
Not applicable for IFRS
No effect on periodic pension cost under
U.S. GAAP
Lower periodic pension expense under
U.S. GAAP
© 2013 ELAN GUIDES
MULTINATIONAL OPERATIONS
MULTINATIONAL OPERATIONS

The presentation currency (PC) is the currency in which the parent company reports
its financial statements. It is typically the currency of the country where the parent is
located. For example, U.S. companies are required to present their financial results in
USD, German companies in EUR, Japanese companies in JPY, and so on.

The functional currency (FC) is the currency of the primary business environment in
which an entity operates. It is usually the currency in which the entity primarily generates
and expends cash.

The local currency (LC) is the currency of the country where the subsidiary operates.
Table 1
Transaction
Export sale
Import purchase
Type of Exposure
Asset (account receivable)
Liability (account payable)
Foreign Currency
Strengthens Weakens
Gain
Loss
Loss
Gain
Methods for Translating Foreign Currency Financial Statements of Subsidiaries
Current Rate/
Temporal Method
Local
Currency
T
Functional
Currency
CR
Presentation
Currency
Temporal
Method
Local
Currency
T
Functional
Currency
=
Presentation
Currency
Current Rate
Method
Local
Currency
=
Functional
Currency
CR
Presentation
Currency



The current rate is the exchange rate that exists on the balance sheet date.
The average rate is the average exchange rate over the reporting period.
The historical rate is the actual exchange rate that existed on the original transaction date.
© 2013 ELAN GUIDES
MULTINATIONAL OPERATIONS
Rules for Foreign Currency Translation
Current Rate Method
FC = LC
Income Statement Component
Sales
Cost of goods sold
Selling expenses
Depreciation expense
Amortization expense
Interest expense
Income tax
Net income before translation
gain (loss)
Translation gain (loss)
Net income
Less: Dividends
Change in retained earnings
Balance Sheet Component
Temporal Method
FC = PC
Exchange Rate Used
Average rate
Average rate
Average rate
Average rate
Average rate
Average rate
Average rate
Average rate
Historical rate
Average rate
Historical rate
Historical rate
Average rate
Average rate
Computed as Rev – Exp
N/A
Computed as Rev – Exp
Historical rate
Computed as NI – Dividends
Used as input for translated
B/S
Plug in Number
Computed as RE +
Dividends
Historical rate
From B/S
Exchange Rate Used
Cash
Accounts receivable
Monetary assets
Inventory
Nonmonetary assets measured at
current value
Property, plant and equipment
Less: Accumulated depreciation
Nonmonetary assets measured at
historical cost
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Historical rate
Current rate
Current rate
Current rate
Current rate
Historical rate
Historical rate
Historical rate
Accounts payable
Long-term notes payable
Monetary liabilities
Nonmonetary liabilities:
Measured at current value
Measured at historical cost
Capital stock
Retained earnings
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Historical rate
From I/S
Current rate
Historical rate
Historical rate
To balance Used as input for
translated I/S
N/A
Cumulative translation adjustment
Plug in Number
© 2013 ELAN GUIDES
MULTINATIONAL OPERATIONS
Balance Sheet Exposure
Foreign Currency (FC)
Strengthens
Weakens
Positive translation adjustment Negative translation adjustment
Negative translation adjustment Positive translation adjustment
Balance Sheet Exposure
Net asset
Net liability
Effects of Exchange Rate Movements on Financial Statements
Temporal Method,
Net Monetary
Liability Exposure
Temporal Method,
Net Monetary Asset
Exposure
Foreign currency
strengthens
relative to
parent’s
presentation
currency
Revenues
Assets
Liabilities
 Net income
 Shareholders’ equity
Translation loss
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Translation gain
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Positive translation
adjustment
Foreign currency
weakens relative
to parent’s
presentation
currency
 Revenues
 Assets
 Liabilities
Net income
Shareholders’ equity
Translation gain
 Revenues
 Assets
 Liabilities
 Net income
 Shareholders’ equity
Translation loss
 Revenues
 Assets
 Liabilities
 Net income
 Shareholders’ equity
Negative translation
adjustment
Current Rate Method
Measuring Earnings Quality
Aggregate accruals = Accrual-basis earnings – Cash earnings
Balance Sheet Approach
Net Operating Assets (NOA)
NOAt = [(Total assetst Casht) (Total liabilitiest Total debtt)]
Aggregate Accruals
Aggregate accrualstb/s = NOAtNOAt1
Aggregate Ratio
Accruals ratiotb/s =
© 2013 ELAN GUIDES
(NOAt NOAt1)
(NOAt + NOAt1)/2
INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES
INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES
A Financial Statement Analysis Framework:
Phase
Sources of Information
Examples of Output
1. Define the purpose
and context of the
analysis.
 The nature of the analyst’s
function, such as evaluating an
equity or debt investment or
issuing a credit rating.
 Communication with client or
supervisor on needs and
concerns.
 Institutional guidelines related
to developing specific work
product.
 Statement of the purpose or
objective of analysis.
 A list (written or unwritten) of
specific questions to be
answered by the analysis.
 Nature and content of report
to be provided.
 Timetable and budgeted
resources for completion.
2. Collect input data.
 Financial statements, other
financial data, questionnaires,
and industry/economic data.
 Discussions with management,
suppliers, customers, and
competitors.
 Company site visits (e.g., to
production facilities or retail
stores)
 Organized financial statements.
 Financial data tables.
 Completed questionnaires, if
applicable.
3. Process input data,
as required, into
analytically useful
data.
 Data from the previous phase.
 Adjusted financial statements.
 Common-size statements.
 Forecasts.
4. Analyze/interpret
the data.
 Input data and processed data
 Analytical results
5. Develop and
communicate
conclusions and
recommendations
(e.g., with an
analysis report).
 Analytical results and previous
reports
Institutional guidelines for
 published reports
 Analytical report answering
questions posed in Phase 1
 Recommendations regarding
the purpose of the analysis,
such as whether to make an
investment or grant credit.
6. Follow-up.
 Information gathered by
periodically repeating above
steps as necessary to determine
whether changes to holdings
or recommendations are
necessary
 Update reports and
recommendations
DuPont Analysis
ROE = Tax Burden × Interest burden × EBIT margin × Total asset turnover × Financial leverage
ROE =
NI
EBT
×
EBT
EBIT
×
EBIT
Revenue
×
Revenue
Average Asset
×
Average Asset
Average Equity
© 2013 ELAN GUIDES
CAPITAL BUDGETING
CAPITAL BUDGETING
Expansion Project
Initial investment outlay for a new investment = FCInv + NWCInv
NWCInv = Non-cash current assets – Non-debt current liabilities
Annual after-tax operating cash flows (CF)
CF = (S – C – D) (1 – t) + D
or
CF = (S – C) (1 – t) + tD
Terminal year after-tax non-operating cash flow (TNOCF):
TNOCF = SalT + NWCInv – t(SalT – BVT)
Replacement Project
Investment outlays:
Initial investment for a replacement project = FCInv + NWCInv – Sal0 + t(Sal0 – BV0)
Annual after-tax operating cash flow:
CF = (S – C) (1 – t) + tD
Terminal year after-tax non-operating cash flow:
TNOCF = SalT + NWCInv – t(SalT – BT)
Mutually Exclusive Projects with Unequal Lives

Least Common Multiple of Lives Approach
In this approach, both projects are repeated until their ‘chains’ extend over the same time
horizon. Given equal time horizons, the NPVs of the two project chains are compared
and the project with the higher chain NPV is chosen.

Equivalent Annual Annuity Approach (EAA)
This approach calculates the annuity payment (equal annual payment) over the project’s
life that is equivalent in present value (PV) to the project’s NPV. The project with the
higher EAA is chosen.
SML
Ri = RF + ßi[E(RM) – RF]
Ri = Required return for project or asset i
RF = Risk-free rate of return
ßi = Beta of project or asset i
[E(RM) – RF] = Market risk premium
© 2013 ELAN GUIDES
CAPITAL BUDGETING
Economic Income
Economic income = After-tax operating cash flow + Increase in market value
Economic income = After-tax operating cash flow + (Ending market value – Beginning market value)
Economic income = After-tax operating cash flow – (Beginning market value – Ending market value)
Economic income = After-tax cash flows – Economic depreciation
Economic Profit
Economic profit = [EBIT (1 – Tax rate)] – $WACC
Economic profit = NOPAT – $WACC
NOPAT = Net operating profit after tax
$WACC = Dollar cost of capital = Cost of capital (%) × Invested capital
Under this approach, a project’s NPV is calculated as the sum of the present values of economic profit earned
over its life discounted at the cost of capital.

NPV = MVA =
EPt
 (1 + WACC)
t
Residual Income
Residual income = Net income for the period – Equity charge for the period
Equity charge for the period = Required return on equity × Beginning-of-period book value of equity
The RI approach calculates value from the perspective of equity holders only. Therefore, future residual income
is discounted at the required rate of return on equity to calculate NPV.

NPV =
RIt
 (1 + r )
E
t
Claims Valuation



First, we separate the cash flows available to debt and equity holders
Then we discount them at their respective required rates of return.
o Cash flows available to debt holders are discounted at the cost of debt,
o Cash flows available to equity holders are discounted at the cost of equity.
The present values of the two cash flow streams are added to calculate the total value of the company/asset.
© 2013 ELAN GUIDES
CAPITAL STRUCTURE
CAPITAL STRUCTURE
The Capital Structure Decision
rWACC =
rD(1  t) +
rE
rD = Marginal cost of debt
rE = Marginal cost of equity
t = Marginal tax rate
D = Market value of the company’s outstanding debt
E = Market value of shareholders’ equity
V = D + E = Value of the company
MM Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity
rWACC =
() ( )
rD +
rE = r0
Company’s cost of equity (rE) under MM Proposition II without taxs is calculated as:
Intercept
Independent variable
rE = r0 + (r0  rD)
Dependent variable
Slope
The total value of the company is calculated as:
V=
Interest
EBIT  Interest
+
rD
rE
The systematic risk (ß) of the company’s assets can be expressed as the weighted average of the systematic
risk of the company’s debt and equity.
A =
() ()
+
This formula can also be expressed as:
E =
+ (A  D)
© 2013 ELAN GUIDES
()
CAPITAL STRUCTURE
Relaxing the Assumption of no Taxes
=
+ tD
The WACC is then calculated as:
rWACC =
rD(1  t) +
rE
And the cost of equity is calculated as:
rE = r0 + (r0  rD) (1  t)
Modigilani and Miller Propositions
Without Taxes
Proposition I
Proposition II
With Taxes
=
rE = r0 + (r0  rD)
=
+ tD
rE = r0 + (r0  rD) (1  t)
The Optimal Capital Structure: The Static Trade-Off Theory
VL = VU + tD – PV(Costs of financial distress)
© 2013 ELAN GUIDES
DIVIDENDS AND SHARE REPURCHASE
DIVIDENDS AND SHARE REPURCHASE
The expected decrease in share price when it goes ex-dividend can be calculated using the following equation:
PW
PX =
D
Pw = Share price with the right to receive the dividend
PX = Share price without the right to receive the dividend
D = Amount of dividend
TD = Tax rate on dividends
TCG = Tax rate on capital gains
Double Taxation System
ETR = CTR + [(1 – CTR) × MTRD]
ETR = Effective tax rate
CTR = Corporate tax rate
MTRD = Investor’s marginal tax rate on dividends
Split-Rate Tax System
ETR = CTRD + [(1 – CTRD) × MTRD]
CTRD = Corporate tax rate on earnings distributed as dividends.
Stable Dividend Policy
The expected increase in dividends is calculated as:
Expected dividend increase = Increase in earnings × Target payout ratio × Adjustment factor
Adjustment factor = 1/N
N = Number of years over which the adjustment is expected to occur
Analysis of Dividend Safety
Dividend payout ratio = (dividends / net income)
Dividend coverage ratio = (net income / dividends)
FCFE coverage ratio = FCFE / [Dividends + Share repurchases]
© 2013 ELAN GUIDES
MERGERS AND ACQUISITION
MERGERS AND ACQUISITION
Mergers and the Industry Life Cycle
Industry Life
Cycle Stage
Industry
Description
Motives for Merger
Types of
Merger
Pioneering
development
 Low but slowly
increasing sales
growth.
 Substantial
development costs.
 Younger, smaller companies may sell
themselves to larger firms in mature
or declining industries to enter into a
new growth industry.
 Young companies may merge with
firms that allow them to pool
management and capital resources.
 Conglomerate
 Horizontal
Rapid
accelerating
growth
 High profit
margins.
 Low competition.
 To meet substantial capital
requirements for expansion.
 Conglomerate
 Horizontal
Mature
growth
 Decrease in the
entry of new
competitors.
 Growth potential
remains.
 To achieve economies of scale,
savings, and operational efficiencies.
 Horizontal
 Vertical
Stabilization
and market
maturity
 Increasing capacity
constraints
 Increasing
competition.
 To achieve economies of scale in
research, production, and marketing
to match low costs and prices of
competitors.
 Large companies may buy smaller
companies to improve management
and provide a broader financial base.
 Horizontal
Deceleration
of growth
and decline
 Overcapacity.
 Eroding profit
margins.
 Horizontal mergers to ensure survival.
 Vertical mergers to increase efficiency
and profit margins.
 Conglomerate mergers to exploit
synergy.
 Companies in the industry may
acquire companies in young
industries.
 Horizontal
 Vertical
 Conglomerate
Source: Adapted from J. Fred Weston, Kwang S. Chung, and Susan E. Hoag, Mergers, Restructuring, and
Corporate Control (New York: Prentice Hall, 1990, p.102) and Bruno Solnik and Dennis McLeavy, International
Investments, 5th edition (Boston: Addison Wesley, 2004, p. 264 – 265).
© 2013 ELAN GUIDES
MERGERS AND ACQUISITION
Major Differences of Stock versus Asset Purchases
Approval
Stock Purchase
Target shareholders receive
compensation in exchange for
their shares.
Shareholder approval required.
Tax: Corporate
No corporate-level taxes.
Payment
Tax: Shareholder Target company’s shareholders
are taxed on their capital gain.
Liabilities
Acquirer assumes the target’s
liabilities.
Asset Purchase
Payment is made to the selling
company rather than directly to
shareholders.
Shareholder approval might not be
required.
Target company pays taxes on any
capital gains.
No direct tax consequence for target
company’s shareholders.
Acquirer generally avoids the
assumption of liabilities.
Herfindahl-Hirschman Index (HHI)
n

i
(
Sales or output of firm i
Total sales or output of market
 100
)
2
HHI Concentration Levels and Possible Government Response
Post-Merger HHI
Less than 1,000
Between 1,000 and 1,800
More than 1,800
Concentration
Not concentrated
Moderately concentrated
Highly concentrated
Change in HHI
Any amount
100 or more
50 or more
Government Action
No action
Possible challenge
Challenge
FCFF is estimated by:
Net income
+
Net interest after tax
=
Unlevered income
+
Changes in deferred taxes
=
NOPLAT (net operating profit less adjusted taxes)
+
Net noncash charges
–
Change in net working capital
–
Capital expenditures (capex)
Free cash flow to the firm (FCFF)
Net interest after tax = (Interest expense – Interest income) (1 – tax rate)
Working capital = Current assets (excl. cash and equivalents) – Current liabilities (excl. short-term debt)
© 2013 ELAN GUIDES
MERGERS AND ACQUISITION
Comparable Company Analysis
TP =
(DP  SP)
SP
TP = Takeover premium
DP = Deal price per share
SP = Target’s stock price per share
Bid Evaluation
Target shareholders’ gain = Takeover premium = PT – VT
Acquirer’s gain = Synergies – Premium
= S – (PT – VT)
S = Synergies created by the merger transaction
The post-merger value of the combined company is composed of the pre-merger value of the
acquirer, the pre-merger value of the target, and the synergies created by the merger. These
sources of value are adjusted for the cash paid to target shareholders to determine the value of
the combined post-merger company.
VA* = VA + VT + S – C
VA* = Value of combined company
C = Cash paid to target shareholders
© 2013 ELAN GUIDES
EQUITY VALUATION: APPLICATIONS AND PROCESSES
EQUITY VALUATION: APPLICATIONS AND PROCESSES
Perceived mispricing:
Perceived mispricing = True mispricing + Error in the estimate of intrinsic value.
VE – P = (V – P) + (VE – V)
VE = Estimate of intrinsic value
P = Market price
V = True (unobservable) intrinsic value
© 2013 ELAN GUIDES
RETURN CONCEPTS
RETURN CONCEPTS
Holding Period Return
Holding period return =
PH – P0 + DH
P0
PH = Price at the end of the holding period
P0 = Price at the beginning of the period
DH = Dividend
Required Return


The difference between an asset’s expected return and its required return is known as
expected alpha, ex ante alpha or expected abnormal return.
o Expected alpha = Expected return – Required return
The difference between the actual (realized) return on an asset and its required return
is known as realized alpha or ex post alpha.
o Realized alpha = Actual HPR – Required return for the period
When the investor’s estimate of intrinsic value (V0) is different from the current market price
(P0), the investor’s expected return has two components:
1.
2.
The required return (rT) earned on the asset’s current market price; and
The return from convergence of price to value [(V0 – P0)/P0].
Internal Rate of Return
Next year’s expected dividend
Intrinsic Value =
Required return – Expected dividend growth rate
V0 =
D1
ke – g
If the asset is assumed to be efficiently-priced (i.e. the market price equals its intrinsic value), the IRR would
equal the required return on equity. Therefore, the IRR can be estimated as:
Required return (IRR) =
ke (IRR) =
D1
P0
Next year’s dividend
Market price
+ Expected dividend growth rate
+g
© 2013 ELAN GUIDES
RETURN CONCEPTS
Equity Risk Premium
The required rate of return on a particular stock can be computed using either of the following two approaches.
Both these approaches require the equity risk premium to be estimated first.
1.
Required return on share i = Current expected risk-free return + ßi(Equity risk premium)

2.

A beta greater (lower) than 1 indicates that the security has greater-than-average (lower-thanaverage) systematic risk.
Required return on share i = Current expected risk-free return + Equity risk premium
± Other risk premia/discounts appropriate for i

This method of estimating the required return is known as the build-up method. It is discussed
later in the reading and is primarily used for valuations of private businesses.
Gordon Growth Model (GGM) Estimates
GCM equity risk premium estimate =
D1
P0
+ g – rLTGD
Macroeconomic Model Estimates
Equity risk premium = {[(1 + EINFL) (1 + EGREPS) (1 + EGPE) – 1] + EINC} – Expected RF
Expected inflation =
1 + YTM of 20-year maturity T-bonds
1 + YTM of 20-year maturity TIPS
– 1
The Captial Asset Pricing Model (CAPM)
Required return on i = Expected risk-free rate + Betai (Equity risk premium)
The Fama-French Model
ri = RF + imktRMRF + isizeSMB + ivalueHML
ßmkt = Market beta
ßsize = Size beta
ßvalue = Value beta
The Pastor-Stambaugh model (PSM)
ri = RF + imktRMRF + isizeSMB + ivalueHML + iliqLIQ
ßliq = Liquidity beta
© 2013 ELAN GUIDES
RETURN CONCEPTS
BIRR model
ri = T-bill rate + (Sensitivity to confidence risk × Confidence risk)
+ (Sensitivity to time horizon risk × Time horizon risk)
+ (Sensitivity to inflation risk × Inflation risk)
+ (Sensitivity to business cycle risk × Business cycle risk)
+ (Sensitivity to market timing risk × Market timing risk)
Build-up method
ri = Risk-free rate + Equity risk premium + Size premium + Specific-company premium
For companies with publicly-traded debt, the bond-yield plus risk premium approach can be
used to calculate the cost of equity:
BYPRP cost of equity = YTM on the company’s long-term debt + Risk premium
Adjusting Beta for Beta Drift
Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0)
Estimating the Asset Beta for the Comparable Publicly Traded Firm:
BASSET reflects only
business risk of the
comparable
company. Therefore
it is used as a proxy
for business risk of
the project being
studied.
1
ßASSET = ßEQUITY
(
1 + (1 - t)
D
E
)
BEQUITY reflects
business and
financial risk of
comparable
company.
where:
D/E = debt-to-equity ratio of the comparable company.
t = marginal tax rate of the comparable company.
To adjust the asset beta of the comparable for the capital structure (financial risk) of the project
or company being evaluated, we use the following formula:
BPROJECT reflects
business and
financial risk of the
project.
ßPROJECT = ßASSET 1 + (1 - t)
D
E
BASSET reflects
business risk of
project.
where:
D/E = debt-to-equity ratio of the subject company.
t = marginal tax rate of the subject company.
Country Spread Model
ERP estimate = ERP for a developed market + Country premium
© 2013 ELAN GUIDES
RETURN CONCEPTS
Weighted Average Cost of Capital (WACC)
WACC =
MVD
MVD + MVCE
rd (1 – Tax rate ) +
MVCE
MVD + MVCE
MVD = Market value of the company’s debt
rd = Required rate of return on debt
MVCE = Market value of the company’s common equity
r = Required rate of return on equity
© 2013 ELAN GUIDES
r
DISCOUNTED DIVIDEND VALUATION
DISCOUNTED DIVIDEND VALUATION
One-Period DDM
V0 =
D1
(1 + r)1
+
P1
(1 + r)1
=
D1 + P1
(1 + r)1
V0 = The value of the stock today (t = 0)
P1 = Expected price of the stock after one year (t = 1)
D1 = Expected dividend for Year 1, assuming it will be paid at the end of Year 1 (t = 1)
r = Required return on the stock
Multiple-Period DDM
V0 =
D1
Dn
Pn
1 + ... +
n+
(1 + r)
(1 + r) (1 + r)n
n
V0 =
Dt
 (1 + r)
t
+
t=1
Pn
(1 + r)n
Expression for calculating Value of a share of stock

V0 =
Dt
 (1 + r)
t
t=1
Gordon Growth Model
V0 =
D0 (1 + g)
D1
, or V0 =
(r – g)
(r – g)
Present value of Growth Opportunities
V0 =
E1
+ PVGO
r
P/E ratio
Justified leading P/E ratio =
P0
=
D1/E1
E1
Justified trailing P/E =
P0
E0
=
D1/E0
rg
rg
=
=
(1 b)
rg
D0 (1 g) / E0
rg
=
(1 b)(1 g)
rg
© 2013 ELAN GUIDES
DISCOUNTED DIVIDEND VALUATION
Value of Fixed-Rate Perpetual Preferred Stock
D
r
V0 =
Two-Stage Dividend Discount Model
n
V0 =

t=1
D0 (1 + gS)t D0 (1 + gS)n(1 + gL)
+
(1 + r)t
(1 + r)n(r – gL)
gS = Short term supernormal growth rate
gL = Long-term sustainable growth rate
r = required return
n = Length of the supernormal growth period
The H-Model
V0 =
D0 (1 + gL) D0H (gs – gL)
+
r – gL
r – gL
gS = Short term high growth rate
gL = Long-term sustainable growth rate
r = required return
H = Half-life = 0.5 times the length of the high growth period
The H-model equation can be rearranged to calculate the required rate of return as follows:
r=
( )
D0
[(1 + gL) + H(gs – gL)] + gL
P0
The Gordon growth formula can be rearranged to calculate the required rate of return given the other variables.
r=
D1
+g
P0
Sustainable growth rate (SGR)
g = b × ROE
b = Earnings retention rate, calculated as 1 – Dividend payout ratio
© 2013 ELAN GUIDES
DISCOUNTED DIVIDEND VALUATION
ROE can be calculated as:
ROE =
Net income
Sales
Total assets
×
×
Sales
Total assets
Shareholders’ equity
PRAT model
g = Profit margin × Retention rate × Asset turnover × Financial leverage
g=
Net income - Dividends
Net income
Sales
Total assets
×
×
×
Net income
Sales
Total assets
Shareholders’ equity
© 2013 ELAN GUIDES
FREE CASH FLOW VALUATION
FREE CASH FLOW VALUATION
FCFF/FCFE

Firm Value =
FCFFt
 (1+WACC)
t
t=1
WACC =
MV(Equity)
MV(Debt)
r
rd (1  Tax Rate) +
MV(Debt) + MV(Equity)
MV(Debt) + MV(Equity)
Equity Value = Firm Value  Market value of debt

Equity Value =
FCFEt
 (1 + r)
t
t=1
Computing FCFF from Net Income
FCFF = NI + NCC + Int(1  Tax Rate)  FCInv  WCInv
Investment in fixed capital (FCInv)
FCInv = Capital expenditures  Proceeds from sale of long-term assets
Investment in working capital (WCInv)
WCInv = Change in working capital over the year
Working capital = Current assets (exc. cash)  Current liabilities (exc. short-term debt)
Table: Noncash Items and FCFF
Noncash Item
Depreciation
Amortization and impairment of intangibles
Restructuring charges (expense)
Restructuring charges (income resulting from reversal)
Losses
Gains
Amortization of long-term bond discounts
Amortization of long-term bond premiums
Deferred taxes
© 2013 ELAN GUIDES
Adjustment to NI to
Arrive at FCFF
Added back
Added back
Added back
Subtracted
Added back
Subtracted
Added back
Subtracted
Added back but requires
special attention
FREE CASH FLOW VALUATION
Computing FCFF from CFO
Table: IFRS versus U.S. GAAP Treatment of Interest and Dividends
IFRS
U.S. GAAP
Interest received
CFO or CFI
CFO
Interest paid
CFO or CFF
CFO
Dividend received
Dividends paid
CFO or CFI
CFO or CFF
CFO
CFF
FCFF = CFO + Int(1  Tax rate)  FCInv
Computing FCFF from EBIT
FCFF = EBIT(1 – Tax rate) + Dep – FCInv – WCInv
Computing FCFF from EBITDA
FCFF = EBITDA(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv
Computing FCFE from FCFF
FCFE = FCFF – Int(1– Tax rate) + Net borrowing
Computing FCFE from Net Income
FCFE = NI + NCC – FCInv – WCInv + Net Borrowing
Computing FCFE from CFO
FCFE = CFO + FCInv – Net borrowing
Computing FCFE from EBIT
FCFE = EBIT(1 – Tax rate) – Int(1 – Tax rate) + Dep – FCInv – WCInv + Net borrowing
Computing FCFE from EBITDA
FCFE = EBITDA(1 – Tax rate) – Int(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv + Net
borrowing
© 2013 ELAN GUIDES
FREE CASH FLOW VALUATION
Uses of FCFF
Increases in cash balances
Plus: Net payments to providers of debt capital
+ Interest expense (1 – tax rate)
+ Repayment of principal
 New borrowings
Plus: Net payments to providers of equity capital
+ Cash dividends
+ Share repurchases
 New equity issues
= Uses of FCFF
Uses of FCFE
Increases in cash balances
Plus: Net payments to providers of equity capital
+ Cash dividends
+ Share repurchases
 New equity issues
= Uses of FCFE
Constant Growth FCFF Valuation Model
Value of the firm =
FCFF1
FCFF0 (1 + g)
=
WACC - g
WACC - g
WACC = Weighted average cost of capital
g = Long-term constant growth rate in FCFF
Constant Growth FCFE Valuation Model
Value of equity =
FCFE1 FCFE0 (1 + g)
=
r-g
r-g
r = Required rate of return on equity
g = Long-term constant growth rate in FCFE
An International Application of the Single-Stage Model
Value of equity =
© 2013 ELAN GUIDES
FCFE0 (1 + greal)
rreal  greal
FREE CASH FLOW VALUATION
General expression for the two-stage FCFF model:
n
Firm value =
FCFFn+1
FCFFt
1
 (1 + WACC) + (WACC  g) (1 + WACC)
t
n
t=1
Firm value = PV of FCFF in Stage 1 + Terminal value × Discount Factor
General expression for the two-stage FCFE model:
n
Equity value =
FCFEt
 (1 + r) +
FCFFn+1
1
rg
(1 + r)n
t
t=1
Equity value = PV of FCFE in Stage 1 + Terminal value × Discount Factor
Determining Terminal Value
Terminal value in year n = Justified Trailing P/E × Forecasted Earnings in Year n
Terminal value in year n = Justified Leading P/E × Forecasted Earnings in Year n + 1
Non-operating Assets and Firm Value
Value of the firm = Value of operating assets + Value of non-operating assets
© 2013 ELAN GUIDES
MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE
MULTIPLES
Price to Earnings Ratio
Trailing P/E ratio =
Current Stock Price
Last year’s EPS
Forward P/E ratio =
Current Stock Price
Expected EPS
Price to Book Ratio
P/B ratio =
Market price per share
Book value per share
P/B ratio =
Market value of common shareholders’ equity
Book value of common shareholders’ equity
Book value of equity = Common shareholders’ equity
= Shareholders’ equity – Total value of equity claims that are senior to common stock
Book value of equity = Total assets – Total liabilities – Preferred stock
Price to Sales Ratio
P/S ratio =
Market price per share
Sales per share
Relationship between the P/E ratio and the P/S ratio
P/E × Net profit margin = (P / E) × (E / S) = P/S
Price to Cash Ratio
P/CF ratio =
Market price per share
Free cash flow per share
Dividend Yield
Justified trailing dividend yield
Trailing dividend yield = Last year’s dividend / Current price per share
Justified leading dividend yield
Leading dividend yield = Next year’s dividend / Current price per share
© 2013 ELAN GUIDES
MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
Justified P/E Multiple Based on Fundamentals
D1
V0 =
(r  g)
Justified leading P/E multiple
Justified leading P/E =
P0
=
E1
D1/E1
rg
=
(1 b)
rg
(1 – b) is the payout ratio.
Justified trailing P/E multiple
Justified trailing P/E =
P0
E0
=
D1/E0
rg
=
D0 (1 g) / E0
rg
=
(1 b)(1 g)
rg
Justified P/B Multiple Based on Fundamentals
P0
B0
=
ROE g
rg
ROE = Return on equity
r = required return on equity
g = Sustainable growth rate
Justified P/S Multiple Based on Fundamentals
P0
S0
=
(E0/S0)(1 b)(1 g)
rg
E0/S0 = Net profit margin
1 – b = Payout ratio
Justified P/CF Multiple Based on Fundamentals
FCFE0 (1  g)
V0 =
(r  g)
Justified Dividend Yield
D0
P0
=
rg
1g
© 2013 ELAN GUIDES
MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
P/E-to-growth (PEG) ratio
PEG =
P/E
Growth (%)
Terminal price based on fundamentals
TVn = Justified leading P/E  Forecasted earningsn +1
TVn = Justified trailing P/E  Forecasted earningsn
Terminal price based on comparables
TVn = Benchmark leading P/E  Forecasted earningsn +1
TVn = Benchmark trailing P/E  Forecasted earningsn
EV/EBITDA Multiple
Enterprise value = Market value of common equity + Market value of preferred stock
+ Market value of debt – Value of cash and short-term investments
EBITDA = Net income + Interest + Taxes + Depreciation and amortization
Alternative Denominators in Enterprise Value Multiples
Free Cash Net
plus
minus
plus
plus
less
less
Flow to the Income Interest Tax Savings Depreciation Amortization Investment in
Investment in
Firm =
Expense on Interest
Working Capital Fixed Capital
EBITDA=
Net
plus
plus
Income Interest Taxes
Expense
plus
plus
Depreciation Amortization
EBITA =
Net
plus
plus
Income Interest Taxes
Expense
plus
Amortization
EBIT =
Net
plus
plus
Income Interest Taxes
Expense
Justified forward P/E after accounting for Inflation
P0
E1
=
1
(1  ) I
= The percentage of inflation in costs that the company can pass through to revenue.
= Real rate of return
I = Rate of inflation
© 2013 ELAN GUIDES
MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
Unexpected earnings (UE)
UEt = EPSt – E(EPSt)
Standardized unexpected earnings (SUE)
SUEt =
EPSt  E(EPSt)
[EPSt  E(EPSt)]
EPSt = Actual EPS for time t
E(EPSt) = Expected EPS for time t
[EPSt  E(EPSt)] = Standard deviation of [EPSt  E(EPSt)]
© 2013 ELAN GUIDES
RESIDUAL INCOME VALUATION
RESIDUAL INCOME VALUATION
The Residual Income
Residual income = Net income – Equity charge
Equity charge = Cost of equity capital × Equity capital
Residual income = After-tax operating profit  Capital charge
Capital charge = Equity charge + Debt charge
Debt charge = Cost of debt × (1 – Tax rate) × Debt capital
Economic Value Added
EVA = NOPAT – (C% × TC)
NOPAT = Net operating profit after tax = EBIT (1 – Tax rate)
C% = Cost of capital (WACC)
TC = Total capital
Market Value Added
MVA = Market value of the company – Accounting book value of total capital
Market value of company = Market value of debt + Market value of equity.
The Residual Income Model
RIt = Et – (r × Bt-1)
RIt = Residual income at time t
Et = Earnings at time t
r = Required rate of return on equity
Bt-1 = Book value at time t-1
Intrinsic value of a stock:

V0 = B0 +

RIt
 (1 + r)
t
i=1
Et  rBt-1
 (1 + r)
= B0 +
t
i=1
V0 = Intrinsic value of the stock today
B0 = Current book value per share of equity
Bt = Expected book value per share of equity at any time t
r = Required rate of return on equity
Et = Expected EPS for period t
RIt = Expected residual income per share
© 2013 ELAN GUIDES
RESIDUAL INCOME VALUATION
Residual Income Model (Alternative Approach)
RIt = EPSt - (R × Bt-1)
RIt = (ROE - r)Bt-1

V0 = B0 +

(ROEt  r)Bt-1
t=1
V0 = B0 +
(1 + r)t
ROE  r
B0
rg
Tobin’s q
Tobin’s q =
Market value of debt and equity
Replacement cost of total assets
Multi-Stage Residual Income Valuation
T
V0 = B0 +

t=1
(Et  rBt 1)
t
(1 + r)
+
PT  BT
(1 + r)T
When residual income fades over time as ROE declines towards the required return on equity, the intrinsic
value of a stock is calculated using the following formula:
T-1
V0 = B0 +

t=1
(Et  rBt 1)
t
(1 + r)
+
ET  rBT-1
(1 + r  )(1 + r)T1
= Persistence factor.
Implied Growth Rate
g=r
[
(ROE  r) × B0
V0  B0
]
© 2013 ELAN GUIDES
PRIVATE COMPANY VALUATION
PRIVATE COMPANY VALUATION
The Capitalized Cash Flow Method
Vf =
FCFF1
WACC  gf
Vf = Value of the firm
FCFF1 = Free cash flow to the firm for next twelve months
WACC = Weighted average cost of capital
gf = Sustainable growth rate of free cash flow to the firm
V=
FCFE1
rg
V = Value of the equity
FCFE1 = Free cash flow to the equity for next twelve months
r = Required return on equity
g = Sustainable growth rate of free cash flow to the equity
Methods Used to Estimate the Required Rate of Return for a Private Company
Capital Asset Pricing Model
Required return on equity = Risk-free rate + (Beta × Market risk premium)
Expanded CAPM
Required return on equity = Risk-free rate + (Beta × Market risk premium)
+ Small stock premium + Company-specific risk premium
Build-Up Approach
Required return on equity = Risk-free rate + Equity risk premium + Small stock premium
+ Company-specific risk premium + Industry risk premium
Discount for Lack of Control (DLOC)
DLOC = 1 -
© 2013 ELAN GUIDES
1
1 + Control Premium
PRIVATE REAL ESTATE INVESTMENTS
PRIVATE REAL ESTATE INVESTMENTS
Net Operating Income
Rental income at full occupancy
+ Other income (such as parking)
= Potential gross income (PGI)
 Vacancy and collection loss
= Effective gross income (EGI)
 Operating expenses (OE)
= Net operating income (NOI)
The Direct Capitalization Method
Cap rate = Discount rate – Growth rate
The cap rate can be defined as the current yield on an investment:
Capitalization rate =
NOI1
Value
Rearranging the above equation, we can estimate the value of a property by dividing its firstyear NOI by the cap rate.
Value =
NOI1
Cap rate
An estimate of the appropriate cap rate for a property can be obtained from the selling price of
similar or comparable properties.
Cap rate =
NOI
Sale price of comparable property
The cap rate derived by dividing rent by recent sales prices of comparables is known as the all
risks yield (ARY). The value of a property is then calculated as:
Market value =
Rent1
ARY
Other Forms of the Income Approach
Gross income multiplier =
Selling price
Gross income
Value of subject property = Gross income multiplier  Gross income of subject property
© 2013 ELAN GUIDES
PRIVATE REAL ESTATE INVESTMENTS
The Discounted Cash Flow Method (DCF)
Value =
NOI1
(r – g)
The Terminal Capitalization Rate
Terminal value =
NOI for the first year of ownership for the next investor
Terminal cap rate
Appraisal-Based Indices
Return =
NOI  Capital expenditures + (Ending market value  Beginning market value)
Beginning market value
Loan to Value ratio
LTV ratio =
Loan amount
Appraised value
Debt Service Coverage ratio
DSCR =
NOI
Debt service
Equity dividend rate/Cash-on-cash return
Equity dividend rate =
First year cash flow
Equity investment
© 2013 ELAN GUIDES
PUBLICLY TRADED REAL ESTATE SECURITIES
PUBLICLY TRADED REAL ESTATE SECURITIES
VALUATION: NET ASSET VALUE APPROACH
Capitalization rate
Capitalization rate =
NOI of a comparable property
Total value of comparable property
Net Asset Value per Share
NAVPS =
Net Asset Value
Shares outstanding
VALUATION: RELATIVE VALUATION (PRICE MULTIPLE) APPROACH
Funds from operations (FFO)
Accounting net earnings
Add: Depreciation charges on real estate
Add: Deferred tax charges
Add (Less): Losses (gains) from sales of property and debt restructuring
Funds from operations
Adjusted funds from operations (AFFO)
Funds from operations
Less: Non-cash rent
Less: Maintenance-type capital expenditures and leasing costs
Adjusted funds from operations
AFFO is preferred over FFO as it takes into account the capital expenditures necessary to maintain
the economic income of a property portfolio.
© 2013 ELAN GUIDES
PRIVATE EQUITY VALUATION
PRIVATE EQUITY VALUATION
Quantitative Measures of Return

PIC (paid in capital): Ratio of paid in capital to date to committed capital.

DPI (distributed to paid-in) or cash-on-cash return: Value of cumulative distributions
paid to LPs as a proportion of cumulative invested capital.
o (DPI = Cumulative distributions / PIC)

RVPI (residual value to paid-in): Value of LPs’ shareholdings held with the fund as a
proportion of cumulative invested capital.
o RVPI = NAV after distributions / PIC

TVPI (total value to paid-in): Value of portfolio companies’ distributed (realized) and
undistributed (unrealized) value as a proportion of cumulative invested capital.
o TVPI = DPI + RVPI
NAV before distributions = Prior year’s NAV after distributions + Capital called
down – Management Fees + Operating results
NAV after distributions = NAV before distributions – Carried interest – Distributions
Total Exit Value
Exit value = Initial cost + Earnings growth + Multiple expansion + Debt reduction
Post-money valuation (POST)
POST = PRE + I
Proportionate ownership of the VC investor
= I / POST
Post-money value
Post-money value =
Exit value
(1 + Required rate of return )Number of years to exists
Required wealth
Required wealth = Investment  (1 + IRR) Number of years to exit
Ownership propotion
Ownership proportion = Required wealth / Exit value
© 2013 ELAN GUIDES
PRIVATE EQUITY VALUATION
Shares to be issued
Shares to be issued =
Proportion of venture capitalist investment Shares held by
company founders
Proportion of investment of company founders
Price per share
Price per share =
Amount of venture capital investment
Number of shares issued to venture capital investment
Adjusted discount rate
Adjusted discount rate =
1+r
–1
1–q
r = Discount rate unadjusted for probability of failure.
q = Probability of failure.
© 2013 ELAN GUIDES
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 × Spread2)
© 2013 ELAN GUIDES
TERM STRUCTURE AND VOLATILITY OF INTEREST RATES
TERM STRUCTURE AND VOLATILITY OF INTEREST RATES
Measuring Historical Yield Volatility
Xt = 100  ln
yt
( )
yt1
where
yt = yield on day t
Annualizing the Standard Deviation
Annualized standard deviation = Daily standard deviation of days in a year
Calculating Variance of Daily Yield Changes
T
Xt2
Variance =
t = 1 T 1

... assigns an equal weight to all observations
T
Wt Xt2
Variance =
t = 1 T 1

... attaches a greater weight to more recent information
where:
Wt = the weight assigned to each daily yield change observation such that the sum of the weights
equals 1.
© 2013 ELAN GUIDES
VALUING BONDS WITH EMBEDDED OPTIONS
VALUING BONDS WITH EMBEDDED OPTIONS
Treasury Market Benchmark
Spread Measure
Nominal
Zero-volatility
Option-adjusted
Benchmark
Treasury yield curve
Treasury spot rate curve
Treasury spot rate curve
Reflects Compensation For
Credit risk, liquidity risk and option risk
Credit risk, liquidity risk and option risk
Credit risk and liquidity risk
Specific Bond Sector with a Given Credit Rating Benchmark
Spread Measure
Nominal
Zero-volatility
Option-adjusted
Benchmark
Sector yield curve
Sector spot rate curve
Sector spot rate curve
Reflects Compensation For
Credit risk, liquidity risk and option risk
Credit risk, liquidity risk and option risk
Credit risk and liquidity risk
Issuer-Specific Benchmark
Spread Measure
Nominal
Zero-volatility
Option-adjusted
Benchmark
Issuer yield curve
Issuer spot rate curve
Issuer spot rate curve
Reflects Compensation For
Liquidity risk and option risk
Liquidity risk and option risk
Liquidity risk
Summary of Relationships between Benchmark, OAS and Relative Value
Benchmark
Treasury market
Negative OAS
Overpriced (rich) security
Bond sector with a
Overpriced (rich) security
given credit rating
(assumes credit rating higher
(assumes credit rating than security being analyzed)
higher than security
being analyzed)
© 2013 ELAN GUIDES
Zero OAS
Overpriced (rich)
security
Positive OAS
Comparison must be made
between security OAS and OAS
of comparable securities
(required OAS):
If security OAS > required OAS,
security is cheap
If security OAS < required OAS,
security is rich
If security OAS = required OAS,
security is fairly priced
Overpriced (rich)
security
(assumes credit rating
higher than security
being analyzed)
Comparison must be made
between security OAS and OAS
of comparable securities
(required OAS):
If security OAS > required OAS,
security is cheap
If security OAS < required OAS,
security is rich
If security OAS = required OAS,
security is fairly priced
VALUING BONDS WITH EMBEDDED OPTIONS
Summary of Relationships between Benchmark, OAS and Relative Value (Contd.)
Benchmark
Positive OAS
Zero OAS
Negative OAS
Bond sector with a
given credit rating
(assumes credit rating
lower than security
being analyzed)
Comparison must be made
between security OAS and OAS
of comparable securities
(required OAS):
If security OAS > required OAS,
security is cheap
If security OAS < required OAS,
security is rich
If security OAS = required OAS,
security is fairly priced
Issuer’s own securities Overpriced (rich) security
Underpriced (cheap) Underpriced (cheap) security
security
(assumes credit rating lower than
(assumes credit rating security being analyzed)
lower than security
being analyzed)
Fairly valued
Under priced (cheap) security
Determining Bond Value at a Node Applying Backward Induction
Bond's value in higher-rate
state 1-year forward
1-year rate at the
node at which we
are calculating the
bond's value, VHHL

VHHHL  C
r4,HHHL
NHHHL
Cash flow in higher
rate state
NHHL
r3,HHL
VHHL
NHHLL
r4,HHLL
VHHLL C
Cash flow in lower
rate state
Bond's value in lower-rate
state 1-year forward
The present values of the these two cash flows discounted at the 1-year rate (r3,HHL) at Node NHHL
are:
1.
2.
(
(
VHHHL + C
1 + r3,HHL
VHHLL + C
1 + r3,HHL
)
)
 Present value in the higher one-year rate scenario
 Present value in the lower one-year rate scenario
© 2013 ELAN GUIDES
VALUING BONDS WITH EMBEDDED OPTIONS
Finally, the expected value of the bond, VHHL at Node NHHLis calculated as:
1
2
(
VHHHL + C
1 + r3,HHL
)(
+
VHHLL + C
1 + r3,HHL
)
Determining Call Option Value
Value of call option = Value of option-free bond – Value of callable bond.
Determining Put Option Value
Value of put option = Value of putable bond  Value of option-free bond
Effective Duration and Effective Convexity
Duration =
VV+
2V0 (y)
Convexity =
VV+  2V0
2V0 (y)2
Traditional Analysis of a Convertible Security
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
Favorable income differential per share
Favorable income differential per share =
Premium over straight value =
© 2013 ELAN GUIDES
Market conversion premium per share
Market price of common stock
Coupon interest  (Conversion ratio Common stock dividend per share)
Conversion ratio
Market price of convertible bond
Straight value

VALUING BONDS WITH EMBEDDED OPTIONS
An Option-Based Valuation Approach
Covertible security value = Straight value Value of the call option on the stock
Covertible callable bond value = Straight value Value of the call option on the
stock Value of the call option on the bond
Covertible callable and putable bond value = Straight value
Value of the call option on the stock
Value of the call option on the bond
Value of the put option on the bond
© 2013 ELAN GUIDES
MORTGAGE-BACKED SECTOR OF THE BOND MARKET
MORTGAGE-BACKED SECTOR OF THE BOND MARKET
Single Monthly Mortality Rate (SMM)
SMMt =
Prepayment in month t
Beginning mortgage balance for month t  Scheduled principal payment in month t
Prepayment in month t = SMM × (Beginning mortgage balance for month t
 Scheduled principal payment in month t)
Conditional Prepayment Rate (CPR)
CPR = 1 (1  SMM)12
Given the CPR, the SMM can be computed as:
SMM = 1 (1  CPR)1/12
Average Life
Average life =
T
t Projectedprincipal recieved at time t
t=1
12Totalprincipal

t = Number of months
Distribution of Prepayment Risk in a Sequential-Pay CMO
Tranche
Contraction Risk Extension Risk
A (sequential pay)
HIGH
LOW
B (sequential pay)
C (sequential pay)
Z (accrual pay)
LOW
HIGH


Prepayment Risk in Different PAC Tranches
Tranche
Prepayment Risk
PAC I - Senior
LOW
PAC I - Junior
PAC II
Support
HIGH

© 2013 ELAN GUIDES
ASSET-BACKED SECTOR OF THE BOND MARKET
ASSET-BACKED SECTOR OF THE BOND MARKET
Parties to the Securitization
Party
Description
Party in Illustration
Seller
Originates the loans and
sells loans to the SPV
ABC Company
Issuer/Trust
The SPV that buys the
loans from the seller
and issues the assetbacked securities
SPV
Servicer
Services the loans
Servicer
Manufactured Housing-Backed Securities
SMM =
ABS
1 – [ABS × (M – 1)]
ABS =
SMM
1 + [SMM × (M – 1)]
VALUING MORTGAGE-BACKED AND ASSET-BACKED SECURITIES
Cash Flow Yield
ABS and MBS typically have monthly cash flows, so the cash flow yield on these securities is compared to
the yield on Treasury coupon securities based on their bond equivalent yields. The bond equivalent yield for
MBS/ABS is calculated as:
Bond equivalent yield = 2 [(1 + monthly cash flow yield)6 – 1]
Option Cost
Option cost = Zero-volatility spread – Option-adjusted spread
Duration
Duration =
V V+
2V0 ( y)
© 2013 ELAN GUIDES
DERIVATES
DERIVATIVES
FORWARD MARKETS AND CONTRACTS
Value of a Forward Contract
Time
Long Position Value
Zero, as the contract is priced to
prevent arbitrage
At initiation
During life of the
contract
F(0,T)
St 
At expiration
Short Position Value
Zero, as the contract is priced to
prevent arbitrage
F(0,T)
T-t
(1 + r)T-t
(1 + r)
ST  F(0,T)
St
F(0,T)  ST
Price of an Equity Forward with Discrete Dividends
n
PV(D,0,T) =
Di
 (1 + r)
i=1
ti
... Approach I
T
F(0,T) = [S0 – PV(D,0,T)] (1 + r)
n
FV(D,0,T) =
 D (1 + r)
Tti
i
i=1
... Approach II
T
F(0,T) = S0 (1 + r) – FV(D,0,T)
Price of an Equity Forward with Continuous Dividends
c
c
F(0,T) = (S0e T)er T
c
c
F(0,T) = S0 e(r  )T
rc = Continuously compounded risk-free rate
c = Continuously compounded dividend yield
Value of an Equity Forward
Vt(0,T) = [St – PV(D,t,T)] – [F(0,T) / (1 + r)T – t]
PV(D,t,T) = PV of dividends expected to be received over the remainder of the contract
term (between t and T).
Assuming continuous compounding, the value of a forward contract on a stock index or portfolio
can be calculated as:
Vt(0,T) = Ste–c(T – t) – F(0,T)e–rc(T – t)
St
Vt(0,T) =
c(T – t)
e
© 2013 ELAN GUIDES
–
F(0,T)
erc(T – t)
DERIVATES
Calculating the No-Arbitrage Forward Price for a Forward Contract on a Coupon Bond
F(0,T) = [B0C(T+Y) – PV(CI,0,T)] × (1 + r)T
Or
F(0,T) = [B0C(T+Y)] (1 + r)T – FV(CI,0,T)
BC = Price of coupon bond
T = Time of forward contract expiration
Y = Remaining maturity of bond upon forward contract expiration
T+Y = Time to maturity of the bond at forward contract initiation.
PV(CI,0,T) = Present value of coupon interest expected to be received between time 0
(contract initiation) and time T (contract expiration).
FV(CI,0,T) = Future value of coupon interest expected to be received between time 0
(contract initiation) and time T (contract expiration).
Valuing a Forward Contract on a Coupon Bond
The value of the long position in a forward contract on a fixed income security prior to expiration
can be calculated as:
Vt(0,T) = BtC(T+Y) – PV(CI,t,T) – F(0,T) / (1 + r)T – t
PV(CI,t,T) = Present value of coupon payments that are expected to be received between time
t and time T.
C
Bt (T+Y) = Current value of coupon bond with time T+Y remaining until maturity
Pricing a Forward Rate Agreement
1 + L0(h + m)
FRA(0,h,m) =
( )
( )
( )
h+m
360
1
1 + L0( h )
h
360
360
m
FRA(0,h,m) = The annualized rate on an FRA initiated at Day 0, expiring on Day h, and based
on m-day LIBOR.
h = Number of days until FRA expiration
m = Number of days in underlying hypothetical loan
h+m = Number of days from FRA initiation until end of term of underlying hypothetical loan.
L0 = (Unannualized) LIBOR rate today
© 2013 ELAN GUIDES
DERIVATES
FRA Payoff
NP × [(Market LIBOR – FRA rate) × No. of days in the loan term / 360]
FRA payoff =
1 + [Market LIBOR × (No. of days in the loan term / 360)]
Valuing FRA prior to expiration
NP × [(Current forward rate – FRA rate) × No. of days in the loan term / 360]
1 + {Current LIBOR × [(No. of days in loan term + No. of days till contract expiration) / 360]}
Or:
1 + FRA(0,h,m)
1
Vg (0,h,m) =

hg
1 + Lg (h  g)
360
( )
1 + Lg (h + m  g)
(
m
360
( )
h+mg
360
)
g = Number of days since FRA initiation.
Pricing a Currency Forward Contracts
F(0,T) = S0 ×
(1 + RDC)T
(1 + RFC)T
F and S are quoted in terms of DC per unit of FC
RDC = Domestic risk-free rate
RFC = Foreign risk-free rate
T = Length of the contract in years. Remember to use a 365-day basis to calculate T if
the term is given in days.
Valuing a Currency Forward Contract
The value of the long position in a currency forward contract at any time prior to maturity can
be calculated as follows:
Vt (0,T) =
St

(1 + RFC)(Tt)
F (0,T)
(1 + RDC)(Tt)
Assuming continuous compounding, the price and value of a currency forward contract can be
calculated by applying the formulas below:
F(0,T) = (S0e– r
cFC × T
Vt(0,T) = [St / er
© 2013 ELAN GUIDES
) × er
cFC × (T – t)
cDC × T
or F(0,T) = S0 × e(r
cDC × (T – t)
] – [F(0,T) / er
]
cDC – rcFC) × T
rc here represents a
continuously
compounded riskfree rate in these
formulas.
FUTURES MARKETS AND CONTRACTS
FUTURES MARKETS AND CONTRACTS
If we ignore the effects of the mark-to-market adjustment on futures contracts, we can make the
simplifying assumption that the futures price and forward price are the same.
f0(T) = F(0,T) = S0 × (1 + r)T
f0(T) = Futures price today of a futures contract that expires at time T.
F(0,T) = Forward price of a forward contract that expires at time T.
S0 = Spot price of underlying asset today
r = Annual risk-free rate
The Effect of Storage or Carrying Costs on the Futures Price
f0(T) = S0 (1 + r)T + FV(SC,0,T)
The Effect of Monetary Benefits on the Futures Price
f0(T) = S0 (1 + r)T  FV(CF,0,T)
The Effect of Non-Monetary Benefits on the Futures Price
FV(CB,0,T) = Costs of storage – Nonmonetary benefits (Convenience yield)
If costs exceed benefits, FV(CB,0,T) is a positive number and is known as cost of carry. In this
case, the general futures pricing formula is given as:
f0(T) = S0 (1 + r)T + FV(CB,0,T)
Pricing Treasury Bond Futures
f0(T) = B0C(T+Y) [(1 + r0(T)]T – FV(CI,0,T)
BC = Price of coupon bond
T = Time of futures contract expiration
Y = Remaining maturity of bond upon futures contract expiration
T+Y = Time to maturity of the bond at futures contract initiation.
r0(T) = Interest rate at time 0 for period until time T.
FV(CI,0,T) = Future value of coupon interest expected to be received between time
0 (contract initiation) and time T (contract expiration).
The adjusted futures price of a t-bond futures contract is calculated as:
f0(T) =
B0C (T + Y) [1 + r0 (T)]T  FV (CI,0,T)
CF(T)
CF(T) = Conversion factor on CTD bond
© 2013 ELAN GUIDES
FUTURES MARKETS AND CONTRACTS
Pricing Stock Index Futures
f0(T) = S0 (1 + r)T – FV(D,0,T)
f0(T) = S0  e(r
c–
c)T
Pricing Currency Futures
f0(T) = S0 
(1 + rDC)T
(1 + rFC)T
F and S are quoted in terms of DC/FC
rDC = Domestic currency interest rate
rFC = Foreign currency interest rate
T = Length of the contract in years. Remember to use a 365-day year if maturity is given in days.
If interest rates are assumed to be continuously compounded, then the no-arbitrage futures price
is calculated as:
f0(T) = S0 × e(r
cDC – rcFC)×T
rc represents the continuously compounded risk-free rate.
© 2013 ELAN GUIDES
OPTION MARKETS AND CONTRACTS
OPTION MARKETS AND CONTRACTS
Put-Call Parity
C0 +
X
= P0 + S0
(1 + RF)T
Synthetic Securities
Strategy
Consisting of
fiduciary call
long call +
long bond
long call
long call
long put
long put
long
underlying
asset
long bond
Value
Equals
Strategy
Consisting of
Value
X
(1 + RF)T
=
Protective
put
long put + long
underlying asset
P0 + S0
C0
=
long put + long
Synthetic call underlying asset
+ short bond
P0 + S0 
X
(1 + RF)T
P0
=
Synthetic put
long call + short
underlying asset
+ long bond
C0  S0 +
X
(1 + RF)T
long
underlying
asset
S0
=
Synthetic
underlying
asset
long call + long
bond + short put
C0 +
long bond
X
(1 + RF)T
=
Synthetic
bond
long put + long
underlying asset
+ short call
C0 +
X
 P0
(1 + RF)T
P 0 + S 0  C0
© 2013 ELAN GUIDES
OPTION MARKETS AND CONTRACTS
One-Period Binomial Model
Computing the two possible values of the stock:
S+ = Su
S- = Sd
Binomia Call Option Pricing
Call payoff = Max(0, S+ – X)
Binomial Put Option Pricing
Put payoff = Max (0, X – ST)
Compute the risk-neutral probabilities:
=
(1 + r  d
(u  d)
Calculating the value of the call option:
 c+ + (1 – ) c-
c=
1+r
Calculating the value of the put option:
p=
 p+ + (1 – ) p1+r
Hedge Ratio
n=
c+  cS+  S-
Intrinsic value of caplet at expiration:
Caplet value =
Max {0, [(One-year rate – Cap rate)  Notional principal]}
1 + One-year rate
Intrinsic value of floorlet at expiration:
Floorlet value =
© 2013 ELAN GUIDES
max {0, [(Floor rate – One-year rate)  Notional principal]}
1 + One-year rate
OPTION MARKETS AND CONTRACTS
The Black-Scholes-Merton Formula
c
c = S0N(d1)  Xer TN(d2)
c
p = Xe-r T[1  N(d2)]  S0[1  N(d1)]
Where:
d1 =
ln(S0 X) + [rc + (

d2 = d1  
= the annualized standard deviation of the continuously compounded return on the stock
rc = the continuously compounded risk-free rate of return
N(d1) = Cumulative normal probability of d1.
Delta
Delta =
Change in option price
Change in underlying price
Change in option price = Delta  Change in underlying price
An approximate measure for option delta can be obtained from the BSM model:
 N(d1) from the BSM model approximately equals call option delta.
 N(d1) – 1 approximately equals put option delta.
Therefore:
 c  N(d1)   S
 p  N(d1) – 1]   S
Put-Call Parity for Forward Contracts
Value at Expiration
ST  X
Transaction
Current Value
Call and Bond
Buy call
Buy bond
Total
c0
[X – F(0,T)]/(1 + r)T
c0 + [X – F(0,T)]/(1 + r)T
0
X – F(0,T)
X – F(0,T)
ST – X
X – F(0,T)
ST – F(0,T)
p
X – ST
ST – F(0,T)
X – F(0,T)
0
ST – F(0,T)
ST – F(0,T)
Put and Forward
Buy put
Buy forward contract
Total
0
0
p0
ST > X
© 2013 ELAN GUIDES
OPTION MARKETS AND CONTRACTS
Put-call-forward parity
c0 +
X – F(0,T)
= p0
(1 + r)T
Forward Contract and Synthetic Forward Contract
Value at Expiration
Transaction
ST  X
Current Value
Forward Contract
Long forward contract
Synthetic Forward Contract
Buy call
Sell put
Buy (or sell) bond
Total
ST > X
0
ST – F(0,T)
ST – F(0,T)
c0
– p0
0
– ( X – S T)
X – F(0,T)
ST – F(0,T)
ST – X
0
X – F(0,T)
ST – F(0,T)
T
[X – F(0,T)]/(1 + r)
c0 – p0 + [X – F(0,T)]/(1 + r)T
The Black Model
The Black model is used to price European options on futures.
c
c = er T [f0(T)N(d1)  XN(d2)]
c
p = er T (X[1  N(d2)]  f0(T)[1  N(d1)])
Where:
d1 =
ln(f0(T) X) + (

d2 = d1  
f0(T) = the futures price
c
Notice that the Black model is similar to the BSM model except that er T f(T) is substituted for
S0. In fact, the price of a European option on a forward or futures would be the same as the price
of a European option on the underlying asset if the options and the forward/futures contract
expire at the same point in time.
© 2013 ELAN GUIDES
SWAP MARKETS AND CONTRACTS
SWAP MARKETS AND CONTRACTS
The Swap Fixed Rate
Swap fixed rate =
(
1 B0(N)
B0(1) + B0(2) + B0(3) + ... + B0(N)
)
100
Valuing a Swap
Value of pay-fixed side of plain-vanilla interest rate swap:
Present value of floating-rate payments  Present value of fixed-rate payments
Value of pay-floating side of plain-vanilla interest rate swap:
Present value of fixed-rate payments  Present value of floating-rate payments
Valuing Equity Swaps
‘Pay a fixed rate and receive the return on equity’ swap
[(1 + Return on equity)  Notional principal]  PV of the remaining fixed-rate payments
‘Pay a floating rate and receive the return on equity’ swap
[(1 + Return on equity)  Notional principal]  PV (Next coupon payment + Par value)
The value of a ‘pay the return on one equity instrument and receive the return on another equity
instrument’ swap is calculated as the difference between the values of the two (hypothetical)
equity portfolios:
[(1 + Return on Index 2)  NP] – [(1 + Return on Index 1)  NP]
Payer swaption
(Market fixed-rate – Exercise rate) 
No. of days in the payment period
360
Notional principal
Receiver swaption
(Exercise rate – Market fixed-rate) 
No. of days in the payment period
360
Notional principal
© 2013 ELAN GUIDES
INTEREST RATE DERIVATIVE INSTRUMENTS
INTEREST RATE DERIVATIVE INSTRUMENTS
Table: Caps, Floors, Interest Rate Options, Bond Options and Interest Rates
Security
Benefits when…
Long cap (floor)
Interest rates rise (fall)
Long call (put) option on interest rates
Interest rates rise (fall)
Long call (put) option on a fixed income instrument
Interest rates fall (rise)
Payoff to the buyer of an interest rate cap
Payoff = Max [0,(Market interest-rate – Cap rate) 
No. of days
Notional principal]
360
Payoff to the buyer of an interest rate floor
Payoff = Max [0,(Floor rate – Market interest-rate) 
© 2013 ELAN GUIDES
No. of days
360
Notional principal]
PORTFOLIO CONCEPTS
PORTFOLIO CONCEPTS
Expected return on Two-Asset Portfolio
E(RP) = w1E(R1) + w2E(R2)
E(R1) = expected return on Asset 1
E(R2) = expected return on Asset 2
w1 = weight of Asset 1 in the portfolio
w2 = weight of Asset 2 in the portfolio
Variance of 2-asset portfolio:
2P = w1221 + w2222 + 2w1w212
1= the standard deviation of return on Asset 1
2= the standard deviation of return on Asset 2
= the correlation between the two assets’ returns
Variance of 2-asset portfolio:
P2 = w1221 + w2222 + 2w1w2Cov1,2
Cov1,2 = 12
Expected Return and Standard Deviation for a Three-Asset Portfolio
Expected return on 3-asset portfolio:
E(RP) = w1E(R1) + w2E(R2) + w3E(R3)
Variance of 3-asset portfolio:
P2 = w1212 + w2222 + w3223 + 2w1w212 + 2w1w313 + 2w2w323
Variance of 3-asset portfolio:
P2 = w1212 + w2222 + w3223 + 2w1w2Cov + 2w1w3Cov + 2w2w3Cov
© 2013 ELAN GUIDES
PORTFOLIO CONCEPTS
Expected Return and Variance of the Portfolio
For a portfolio of n assets, the expected return on the portfolio is calculated as:
n
E(RP) =
 w E(R )
j
j
j=1
The variance of the portfolio is calculated as:
n
2P =
n
  w w Cov(R ,R )
i
i=1
j
i
j
j=1
Variance of an Equally-weighted Portfolio
P2 =
1 2 n1
 +
Cov
n
n
2P = 2
(
1 
n
+
)
Expected Return for a Portfolio Containing a Risky Asset and the Risk-Free Asset
E(RP) = RFR + P
[E(Ri) RFR]
i
Standard Deviation of a Portfolio Containing a Risky Asset and the Risk-Free Asset
P = wii
© 2013 ELAN GUIDES
PORTFOLIO CONCEPTS
CML
Expected return on portfolios that lie on CML:
E(RP) = w1Rf + (1  w1)E(Rm)
Variance of portfolios that lie on CML:
2 = w12f2 + (1  w1)2m2 + 2w1(1  w1)Cov(Rf , Rm)
Equation of CML:
E(RP) = Rf +
E(Rm)  Rf
 P
m
Calculation and Interpretation of Beta
i
Cov(Ri,Rm)
m2

i,mi,m
m2

i,mi
m
The Capital Asset Pricing Model
E(Ri) Rf +i[E(Rm) – Rf ]
The Decision to Add an Investment to an Existing Portfolio


E(Rnew)  RF
E(Rp)  RF

Corr(Rnew,Rp)
p
new
Market Model Estimates
Ri = i + i RM + i
Ri = Return on asset i
RM = Return on the market portfolio
i = Average return on asset i unrelated to the market return
i = Sensitivity of the return on asset i to the return on the market portfolio
i = An error term


i is the slope in the market model. It represents the increase in the return on asset i if
the market return increases by one percentage point.
i is the intercept term. It represents the predicted return on asset i if the return on the
market equals 0.
Expected return on asset i
E(Ri) = i + iE(RM)
© 2013 ELAN GUIDES
PORTFOLIO CONCEPTS
Variance of the return on asset i
Var(Ri) = 2i 2M+ 2i
Covariance of the returns on asset i and asset j
Cov(Ri,Rj) = ij2
M
Correlation of returns between assets i and j
Corr(Ri,Rj) =
2
ijM
2
(2i M2 + 2i )1/2 (2j M
+ 2j )1/2
Market Model Estimates: Adjusted Beta
Adjusted beta = 0.333 + 0.667 (Historical beta)
Macroeconomic Factor Models
Ri = ai + bi1FINT + bi2FGDP + i
Ri = the return to stock i
ai = the expected return to stock i
FINT = the surprise in interest rates
FGDP = the surprise in GDP growth
bi1 = the sensitivity of the return on stock i to surprises in interest rates.
bi2 = the sensitivity of the return on stock i to surprises in GDP growth.
i = an error term with a zero mean that represents the portion of the return to stock i
that is not explained by the factor model.
Fundamental Factor Models
Ri = ai + bi1FDY + bi2FPE + i
Ri = the return to stock i
ai = intercept
FDY = return associated with the dividend yield factor
FPE = return associated with the P-E factor
bi1 = the sensitivity of the return on stock i to the dividend yield factor.
bi2 = the sensitivity of the return on stock i to the P-E factor.
i = an error term
Standardized sensitivities are computed as follows:
bij =
Assets i’s attribute value  Average attribute value
Attribute values)
© 2013 ELAN GUIDES
PORTFOLIO CONCEPTS
Arbitrage Pricing Theory and the Factor Model
E(RP) = RF + 1p, p,
E(Rp) = Expected return on the portfolio p
RF = Risk-free rate
 j = Risk premium for factor j
p,j = Sensitivity of the portfolio to factor j
K = Number of factors
Active Risk
TE = s(Rp  RB)
Active risk squared = s2(Rp  RB)
Active risk squared = Active factor risk + Active specific risk
n
Active specific risk =
w 
i=1
a 
i i
Where:
wai = The ith asset’s active weight in the portfolio (i.e., the difference between the asset’s weight
in the portfolio and its weight in the benchmark).
 = The residual risk of the ith asset (i.e., the variance of the ith asset’s returns that is not explained
i
by the factors).
Active factor risk = Active risk squared – Active specific risk.
Active Return
Active return = Rp – RB
Active return = Return from fctor tilts + Return from asset selection
K
Active return =
[(Portfolio sensitivity)  (Benchmark sensitivity) ]  (Factor return) + Asset selection
j
j
j
j=1
© 2013 ELAN GUIDES
PORTFOLIO CONCEPTS
Factor’s Marginal Contribution to Active Risk Squared (FMCAR)
K
baj
FMCARj =
FMCARj =
 b Cov(F ,F )
a
i
j
i
i=1
Active risk squared
Active factor risk
Active risk squared
where:
baj = The portfolio’s active exposure to factor j
K
baj
 b Cov(F ,F ) = The active factor risk for factor j
a
i
j
i
i=1
The Information Ratio
IR =
Rp  RB
s(Rp  RB)
© 2013 ELAN GUIDES
THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
Weight of security k in the active portfolio (Portfolio A)
wk =
k / ek
n
 i / ei
i=1
The expression for the optimal weight, w*, of the active portfolio (Portfolio A) in the optimal
risky portfolio (Portfolio P) is given as:
A
w* =
2(eA)
A(1A) + RM
2 M
Assuming (for simplicity) that the beta of Portfolio A equals 1, the optimal weight, w0, of Portfolio
A in Portfolio P is calculated as:
A
w0 =
RM
=
2(eA)
A /2(eA)
RM /2M
2M
If the beta of Portfolio A does not equal 1, we can use the following equation to determine the
optimal weight, w*, of Portfolio A in Portfolio P.
w* =
w0
(1A)w0
Evaluation of Performance
Sharpe Ratio
The Sharpe ratio of the optimal risky portfolio (Portfolio P) can be separated into contributions
from the market and active portfolio as follows:
2
2
SP = SM
2A
+
2(eA)
2
=
  
RM
M
+
2
A
(eA)
Information Ratio
2
   
A
(eA)
n
=
i=1
2
i
(ei)
© 2013 ELAN GUIDES
THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
Imperfect Forecasts of Alpha Values
Actual (realized) alpha: RRM
To measure the forecasting accuracy of the analyst, we can regress alpha forecasts (f) on realized
alpha ().
f  aa
For simplicity, we assume that a and a equal 0 and 1 respectively. Given that forecast errors () are uncorrelated
with true alpha () i.e., Cov, equals 0, the variance of the forecast is given as:
f   + 
We can evaluate the quality of the analyst’s forecasts by calculating the coefficient of determination
of the regression described above.

R 

f


 + 
This estimate of R2 is used as a shrinking factor to adjust the analyst’s forecasts of alpha.
© 2013 ELAN GUIDES
THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY STATEMENT
THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY
STATEMENT
Risk Tolerance
Willingness to Take Risk
Below Average
Above Average
Ability to Take Risk
Below Average
Above Average
Below-average risk tolerance
Resolution needed
Resolution needed
Above-average risk tolerance
Return Requirements and Risk Tolerances of Various Investors
Type of Investor
Return Requirement
Risk Tolerance
Individual
Depends on stage of life,
circumstances, and obligations
Varies
Pension Plans (Defined
Benefit)
The return that will adequately
fund liabilities on an inflationadjusted basis
Depends on plan and
sponsor characteristics,
plan features, funding status,
and workforce characteristics
Pension Plans (Defined
Contribution)
Depends on stage of life of
individual participants
Varies with the risk
tolerance of individual
participants
Foundations and
Endowments
The return that will cover
annual spending, investment
expenses, and expected inflation
Determined by amount of
assets relative to needs, but
generally above- average
or average
Life Insurance
Companies
Determined by rates used to
determine policyholder reserves
Below average due to factors
such as regulatory constraints
Non-Life- Insurance
Companies
Determined by the need to price
policies competitively and by
financial needs
Below average due to factors
such as regulatory constraints
Banks
Determined by cost of funds
Varies
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