elan guides formula sheet cfa 2013 level 2
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7/18/2019 Elan Guides Formula Sheet CFA 2013 Level 2
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Peter has taught CPA and CFA Exam Review courses for the past ten years and is a real ‘celebrity’in the CPA and CFA prep industries. Previously he worked as an auditor for Deloitte & Touche, wasa tax attorney for Ernst and Young, and later spent nearly ten years teaching law, accounting,financial statement analysis, and tax at both the graduate and undergraduate levels at FordhamUniversity’s business school. He graduated Magna Cum laude from Pace University and went on to
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Basit graduated Magna Cum Laude from the world-renowned Wharton School of Business at theUniversity of Pennsylvania with majors in Finance and Legal Studies. After graduating, Basit ranhis own private wealth management firm. He started teaching CFA courses more than five yearsago, and upon discovering how much he enjoyed teaching, he founded Elan Guides with a view toproviding CFA candidates all around the globe access to efficient and effective CFA study materials
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OUR INSTRUCTORS
7/18/2019 Elan Guides Formula Sheet CFA 2013 Level 2
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Sample covariance = Cov (X,Y) = n
i = 1
(Xi X)(Yi Y)/(n 1)
where:n = sample size
Xi = ith observation of Variable XX = mean observation of Variable XYi = ith observation of Variable YY = mean observation of Variable Y
Sample correlation coefficient = r =
Cov (X,Y)
sXsY
Sample variance = s X
2=
n
i = 1
(Xi X)2/(n 1)
Sample standard deviation = s X = s X
2
Test-stat = t =r n 2
1 r 2
Where:
n = Number of observationsr = Sample correlation
Test statistic
Regression model equation = Y i = b0 + b1 X i + 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.
Linear Regression with One Independent Variable
CORRELATION AND R EGRESSION
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QUANTITATIVE METHODS
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Regression line equation = Y i = b0 + b1 X i , i = 1,...., nˆ ˆ ˆ
n
i = 1
[Y i (b0 b1 X i)]2ˆˆ
Regression Residuals
where:Y i = Actual value of the dependent variableb0 + b1 X i = Predicted value of dependent variableˆ ˆ
SEE =
1/2
( )n 2
n
i = 1
(Y i b0 b1 X i)2ˆ ˆ 1/2
( )n
i = 1
(i)2ˆ
n 2= = ( )SSE
n 2
1/2
The Standard Error of Estimate
Hypothesis Tests on Regression Coefficients
CAPM: R ABC = R F + ABC(R M – R F)
R ABC – R F = + ABC(R M – R F) +
The intercept term for the regression, b0, is . The slope coefficient for the regression, b1, is ABC
The Coefficient of Determination
R 2 = =Explained variation
Total variation
Total variation Unexplained variation
Total variation
= 1 Total variation
Unexplained variation
Total variation = Unexplained variation + Explained variation
n
i = 1
(Y i Y )2^RSS = Explained variation
The regression sum of squares (RSS)
n
i = 1
(Y i Y i )2^
SSE = Unexplained variation
The sum of squared errors or residuals (SSE)
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QUANTITATIVE METHODS
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s f 2
s2
= 1 1
n
(X X)2
(n 1) s x2[ ]
Y t c s f ^
Prediction Intervals
Source of Variation
Regression (explained)
Error (unexplained)
Total
Degrees of Freedom
k
n k + 1)
n 1
Sum of Squares
RSS
SSE
SST
Mean Sum of Squares
MSR = RSS
k
RSS
1= RSS=
MSE =SSE
n 2
k = the number of slope coefficients in the regression.
ANOVA Table
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QUANTITATIVE METHODS
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Multiple regression equation = Y i = b0 + b1 X 1i + b2 X 2i + . . .+ bk X ki + i, i = 1,2, . . . , n
Y i X jib0
b1, . . . , bk
i
n
= the ith observation of the dependent variable Y
= the ith observation of the independent variable X j , 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
Multiple regression equation
i = Y i Y i = Y i (b0 + b1 X 1i + b2 X 2i + . . .+ bk X k i)ˆ ˆ ˆ ˆ ˆ ˆ
Residual Term
MULTIPLE R EGRESSION AND ISSUES IN R EGRESSION ANALYSIS
b j ± (t c sb j)ˆ
ˆ
estimated regression coefficient ± (critical t -value)(coefficient standard error)
Confidence Intervals
F-stat =RSS/k
SSE/[n k + 1)]
MSR
MSE=
F -statistic
(1 R2)
n 1
n k 1( )Adjusted R2 = R2 = 1
R 2 =Total variation Unexplained variation
Total variation
SST SSE
SST=
RSS
SST=
R 2 and Adjusted R 2
2 = nR 2 with k degrees of freedom
n = Number of observationsR 2 = 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
Testing for Heteroskedasticity- The Breusch-Pagan (BP) Test
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MULTIPLE REGRESSION AND ISSUES IN REGRESSION
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Y i = b0 + b1lnX 1i + b2 X 2i +
Model Specification Errors
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 variablet = a random-error term
Linear Trend Models
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 andthose from the previous period.
Value of Durbin-Watson Statistic
Inconclusive
d l
d u
(H0: No serial correlation)
Reject H0,
conclude
Positive Serial
Correlation Inconclusive
Do not Reject
H0
Reject H0,
conclude
Negative Serial
Correlation
0 4 d l
44 d u
Problems in Linear Regression and Solutions
Problem
Heteroskedasticity
Serial correlation
Multicollinearity
Effect
Incorrect standard errors
Incorrect standard errors (additional
problems if a lagged value of the
dependent variable is used as anindependent variable)
High R 2 and low t-statistics
Solution
Use robust standard errors
(corrected for conditionalheteroskedasticity)
Use robust standard errors
(corrected for serial correlation)
Remove one or more independentvariables; often no solution based
in theory
© 2013 ELAN GUIDES
MULTIPLE REGRESSION AND ISSUES IN REGRESSION
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yt = eb0 + b1t
ln 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 termb1 = the slope coefficientt = time = 1, 2, 3 ... T
We take the natural logarithm of both sides of the equation to arrive at the equation for the log-linear model:
Log-Linear Trend Models
xt = b0 + b1 xt 1 + t
xt
= b0
+ b1
xt 1
+ b2
xt 2
+ . . . + b p
xt p
+ t
AUTOREGRESSIVE (AR) TIME-SERIES MODELS
Detecting Serially Correlated Errors in an AR Model
t-stat =Residual autocorrelation for lag
Standard error of residual autocorrelation
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 termb1 = the slope coefficient/ trend coefficient
t = time, the independent or explanatory variable
t = a random-error term
A series that grows exponentially can be described using the following equation:
A pth order autoregressive model is represented as:
where:Standard error of residual autocorrelation = 1/ T
T = Number of observations in the time series
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TIME SERIES ANALYSIS
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xt =b0
1 b1
Mean Reversion
xt+1 = b0 + b1 xt ^ ^^
Multiperiod Forecasts and the Chain Rule of Forecasting
xt = xt 1 + t , E(t ) = 0, E(t 2) = 2, E(t s) = 0 if t s
yt = xt xt 1 = xt 1 + t xt 1= t , E(t ) = 0, E(t
2) = 2, E(t s) = 0 for t s
Random Walks
The first difference of the random walk equation is given as:
xt = b0 + b1 xt 1 + t
b1 = 1, b0 0, or
xt = b0 + xt 1 + t , E(t ) = 0
yt = xt xt 1 , yt = b0 + t , b0 0
Random Walk with a Drift
The first-difference of the random walk with a drift equation is given as:
The Unit Root Test of Nonstationarity
xt b0 + b1 xt 1 + t
xt xt 1 b0 + b1 xt 1 xt 1 + t
xt xt 1 b0 + (b1 1) xt 1 + t
xt xt 1 b0 + g 1 xt 1 + t
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TIME SERIES ANALYSIS
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t
2 = a0 + a1t
2
1 + ut ^ ^
t + 1 = a0 + a1t 2^ ^2 ^^
Autoregressive Conditional Heteroskedasticity Models (ARCH Models)
xt = b0 + b1 xt 1 + . . . + b p xt p + t + 1t 1 +. . . + qt q
E(t ) = 0, E(t 2
) = 2
, E(t s) = 0 for t s
Autoregressive Moving Average (ARMA) Models
The error in period t +1 can then be predicted using the following formula:
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TIME SERIES ANALYSIS
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CURRENCY EXCHANGE R ATES: DETERMINATION AND FORECASTING
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:
=USD
EUR
JPY
EUR USD
JPY
Currency Cross Rates
Cross Rate Calculations with Bid-Ask Spreads
USD/EUR bid = 1.3802
Represents the price of EUR (base
currency). An investor can sell EUR for USD
at this price (as it is the bid pricequoted by the dealer).
USD/EUR ask = 1.3806
Represents the price of EUR
An investor can buy EUR withUSD at this price.
Determining the EUR/USD bid cross rate:
EUR/USD bid = 1/(USD/EUR ask )
Determining the EUR/USDask cross rate:
EUR/USDask = 1 / (USD/EUR bid)
Forward exchange rates (F) - One year Horizom
FFC/DC = SFC/DC (1 + iFC)
(1 + iDC)FPC/BC = SPC/BC
(1 + iPC)
(1 + iBC)
FPC/BC SPC/BC=1 + (iBC Actual 360)
1 + (iPC Actual 360)
FFC/DC SFC/DC= 1 + (iDC Actual 360)
1 + (iFC Actual 360)
Forward exchange rates (F) - Any Investment Horizom
Currencies Trading at a Forward Premium/Discount
FFC/DC SFC/DC = SFC/DC(iFC iDC) Actual 360
1 + (iDC Actual 360)( )
( )FPC/BC SPC/BC = SPC/BC(iPC iBC) Actual 360
1 + (iBC Actual 360)
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CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
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The expected percentage change in the spot exchange rate can becalculated as:
Covered Interest Rate Parity
FPC/BC 1 + (i
BC
Actual 360
)
1 + (iPC Actual 360)SPC/BC=
Forward premium (discount) as a % =FPC/BC SPC/BC
SPC/BC
Forward premium (discount) as a % FPC/BC SPC/BC iPC iBC
Uncovered Interest Rate Parity
Expected future spot exchange rate:
(1 + iFC)
(1 + iDC)Se
FC/DC = SFC/DC
Expected % change in spot exchange rate =SePC/BC =
The expected percentage change in the spot exchange rate can beestimated as:
Expected % change in spot exchange rate SePC/BC iPC iBC
SPC/BC
SePC/BC – SPC/BC
The forward premium (discount) on the base currency can be expressed as a percentage as:
The forward premium (discount) on the base currency can be estimated as:
Purchasing Power Parity (PPP)
Law of one price: PXFC = PX
DC SFC/DC
Law of one price: PXPC = PX
BC 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(STFC/DC) = S0
FC/DC
1 FC
1 + DC( )
T
Ex Ante Version of PPP
Ex ante PPP: %SeFC/DC e
FC eDC
Ex ante PPP: %SePC/BC e
PC eBC
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CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
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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 SFC/DCPFC
PDC in terms of FC PDC SFC/DC
PFC
= ==PDC
PFC( )
The Fisher Effect
Fischer Effect: i = r + e
International Fisher effect: (iFC – iDC) = (eFC – e
DC)
Figure 1: Spot Exchange Rates, Forward Exchange Rates, and Interest Rates
Ex Ante PPP
Forward Rate as
an Unbiased
Predictor
International Fisher
Effect
Covered
Interest Rate
Parity
Uncovered Interest
Rate Parity
Expected change
in
Spot Exchange Rate%Se
FC/DC
Foreign-DomesticExpected Inflation
Differentiale
FC eDC
Foreign-Domestic
Interest rate
DifferentialiFC iDC
Forward Discount
FFC/DC SFC/DC
SFC/DC
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 – e
L) – (H – L)
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CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
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The Taylor rule
i = r n + + y y*)
wherei = the Taylor rule prescribed central bank policy rater n = the neutral real policy rate = the current inflation rate* = the central bank’s target inflation rate
y = the log of the current level of outputy* = the log of the economy’s potential/sustainable level of output
© 2013 ELAN GUIDES
CURRENCY EXCHANGE RATES: DETERMINATION AND FORECASTING
qPC/BC = qPC/BC + ( r nBC
r nPC) + BC BC PC PC
yBC y*BC) yPC y*PC)] BC PC)
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ECONOMIC GROWTH AND THE INVESTMENT DECISION
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 economyL = Quantity of labor
K = Quantity of capital
A = Total factor productivity. Total factor productivity (TFP) reflects the general levelof productivity or technology in the economy. TFP is a scale factor i.e., an increasein 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
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
Relationship between economic growth and stock prices
Potential GDP
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ECONOMIC GROWTH AND THE INVESTMENT DECISION
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Neoclassical Model (Solow’s Model)
= + + nY
K
1
s( )
(1-)( )[ ]s = Fraction of income that is saved = Growth rate of TFP
= Elasticity of output with respect to capitaly = 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.
= + + n
(1 )( )[ ]sy k
Savings/Investment Equation:
Growth rates of Output Per Capita and the Capital-Labor Ratio
ye = f(k e) = ck e
=y
y
(1)
Y
K + s
=k k
(1)( ) Y
K ( )+ s
Production Function in the Endogenous Growth Model
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ECONOMIC GROWTH AND THE INVESTMENT DECISION
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INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENTS AND R ATIOS
Ending Inventory = Opening Inventory + Purchases - Cost of goods sold
LIFO and FIFO Comparison with Rising Prices and Stable Inventory Levels
COGS
Income before taxes
Income taxes
Net income
Total cash flow
EI
Working capital
LIFO
Higher
Lower
Lower
Lower
Higher
Lower
Lower
FIFO
Lower
Higher
Higher
Higher
Lower
Higher
Higher
LIFO versus FIFO with Rising Prices and Stable Inventory Levels
Income is lower under LIFO becauseCOGS is higher
Same debt levels
Profitability ratios
NP and GP margins
Solvency ratios
Debt-to-equity and
debt ratio
Sales are the sameunder both
Lower equity and
assets under LIFO
Lower under LIFO
Higher under LIFO
Type of Ratio
Effect on
Numerator
Effect on
Denominator Effect on Ratio
Liquidity ratios
Current ratio
Quick ratio
Activity ratios
Inventory turnover
Total asset turnover
Current assets are
lower under LIFO because EI is lower
Quick assets arehigher under LIFO asa result of lower taxes
paid
COGS is higher under LIFO
Sales are the same
Current liabilities are
the same.
Current liabilities arethe same
Average inventory islower under LIFO
Lower total assetsunder LIFO
Lower under LIFO
Higher under LIFO
Higher under LIFO
Higher under LIFO
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INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
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LIFO reserve (LR)
EIFIFO = EILIFO + LR
COGSFIFO = COGSLIFO (Change in LR during the year)
Change in LIFO Reserve (1 Tax rate)
When converting from LIFO to FIFO assuming rising prices:
Equity (retained earnings) increases by:
Liabilities (deferred taxes) increase by:
LIFO Reserve (Tax rate)
LIFO Reserve (1 Tax rate)
Current assets (inventory) increase by:
LIFO Reserve
where LR = LIFO Reserve
Net Income after tax under FIFO will be greater than LIFO net income after tax by:
COGSFIFO is lower than COGSLIFO during periods of rising prices:
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INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
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Impact of an Inventory Write-Down on Various Financial Ratios
COGS increases so profits fall
Debt levels remain
the same
Current assetsdecrease (due tolower inventory)
COGS increases
Sales remain thesame
Profitability ratios NP and GP margins
Solvency ratios
Debt-to-equity and
debt ratio
Liquidity ratiosCurrent ratio
Activity ratios
Inventory turnover
Total asset turnover
Sales remain thesame
Equity decreases
(due to lower profits)and current assetsdecrease (due tolower inventory)
Current liabilitiesremain the same.
Average inventorydecreases
Total assets decrease
Lower (worsens)
Higher (worsens)
Lower (worsens)
Higher (improves)
Higher (improves)
Type of Ratio
Effect on
Numerator
Effect on
Denominator Effect on Ratio
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INVENTORIES: IMPLICATIONS FOR FINANCIAL STATEMENT AND RATIOS
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LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND
R ATIOS
Straight Line Depreciation
Depreciation expense =Depreciable life
Original cost Salvage value
Accelerated Depreciation
DDB depreciation in Year X =2
Depreciable lifeBook value at the beginning of Year X×
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.
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 Capitalization
Financial Statement Effects of Capitalizing versus Expensing
Net income (first year)
Net income (future years)
Total assets
Shareholders’ equity
Cash flow from operations activities
Cash flow from investing activities
Income variability
Debt to equity ratio
Capitalizing
Higher
Lower
Higher
Higher
Higher
Lower
Lower
Lower
Expensing
Lower
Higher
Lower
Lower
Lower
Higher
Higher
Higher
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LONG-LIVED ASSETS: IMPLICATIONS FOR FINANCIAL STATEMENTS AND RATIOS
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Remaining useful life
The book value of the asset divided
by annual depreciation expense
equals the number of years the asset
has remaining in its useful life.
= +Gross investment in fixed assets
Annual depreciation expense
Accumulated depreciation
Annual depreciation expense
Net investment in fixed assets
Annual depreciation expense
Average age of asset
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.
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.
Income Statement Item
Operating expenses
Nonoperating expenses
EBIT (operating income)
Total expenses- early years
Total expenses- later years
Net income- early years Net income- later years
Finance Lease
Lower (Depreciation)
Higher (Interest expense)
Higher
Higher
Lower
Lower Higher
Operating Lease
Higher (Lease payment)
Lower (None)
Lower
Lower
Higher
Higher Lower
Income Statement Effects of Lease Classification
CF Item
CFO
CFF
Total cash flow
Finance Lease
Higher
Lower
Same
Operating Lease
Lower
Higher
Same
Cash Flow Effects of Lease Classification
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Effect on Ratio
Lower
Lower
Lower
Higher
Lower
Denominator
under Finance
Lease
Assets- higher
Assets- higher
Currentliabilities-
higher
Equity- same
Assets- higher
Equity- same
Numerator under
Finance Lease
Sales- same
Net income- lower
Current assets-same
Debt- higher
Net income- lower
Ratio Better or
Worse under
Finance Lease
Worse
Worse
Worse
Worse
Worse
Ratio
Asset turnover
Return on assets*
Current ratio
Leverage ratios
(D/E and D/A**)
Return on equity*
Table 9: Impact of Lease Classification on Financial Ratios
**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.
Operating Lease
Same
Lower
Lower
Higher
Lower
Same
Financing Lease
Same
Higher
Higher
Lower
Higher
Same
Total net income
Net income (early years)
Taxes (early years)
Total CFO
Total CFI
Total cash flow
Financial Statement Effects of Lease Classification from Lessor’s Perspective
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Summary of Accounting Treatment for Investments
Influence
Typical percentageinterest
AccountingTreatment
In Financial Assets
Not significant
Usually < 20%
Classified into one of four categories based on
management intent andtype of security.
Debt only: Held-to-maturity
(amortized cost,
changes in value
ignored unlessdeemed as impaired)
Debt and Equity:
Held for trading(fair value, changesin value recognizedin profit or loss)
Available-for-sale
(fair value, changesin value recognized
in equity) Designated at fair
value (fair value,changes in value
recognized in profitor loss)
In Associates
Significant
Usually 20% 50%
Equity method
BusinessCombinations
Controlling
Usually > 50%
Consolidation
In Joint Ventures
Shared Control
Varies
IFRS: Equity methodor proportionate
consolidation
U.S. GAAP: Equitymethod(except for unincorporated
ventures in
specialized industries)
INTERCORPORATE INVESTMENTS
Combination
Merger
Acquisition
Consolidation
Description
Company A + Company B = Company A
Company A + Company B = (Company A + Company B)
Company A + Company B = Company C
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CFO
CFF
Total cash flow
Current assets
Current liabilities
Current ratio
Adjusted Values Upon Reclassification of Sale of Receivables:
Lower
Higher
Same
Higher
Higher
Lower (Assuming it was greater than 1)
Difference between QSPE and SPE
Securitized Transaction: Qualified 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.
Securitized Transaction: Special
Purpose Entity
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
Leverage
Net Profit
Margin
ROE
ROA
Better (lower) as liabilities are
lower and equity is the same
Better (higher) as sales are lower
and net income is the same
Better (higher) as equity is lower and net income is the same
Better (higher) as net income isthe same and assets are lower
Equity Method
In-between
In-between
Same as under theequity method
In-between
Proportionate
Consolidation
Worse (higher) as liabilities are
higher and equity is the same
Worse (lower) as sales are higher
and net income is the same
Worse (lower) as equity is higher and net income is the same
Worse (lower) as net income isthe same and assets are higher
Acquisition Method
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EMPLOYEE COMPENSATION: POST-EMPLOYMENT AND SHARE-BASED
Types of Post-Employment Benefits
Type of Benefit
Defined contribution pension plan
Defined benefit
pension plan
Other post-employment benefits
(e.g., retirees’ healthcare)
Amount of Post-
Employment Benefit to
Employee
Amount of future benefit isnot defined. Actual future
benefit will depend oninvestment performance of plan assets.Investment risk is borne byemployee.
Amount of future benefit is
defined, based on the plan’sformula (often a function of length of service and finalyear’s compensation).
Investment risk is borne bycompany.
Amount of future benefitdepends on plan
specifications and type of benefit.
Obligation of Sponsoring
Company
Amount of the company’sobligation (contribution)
is defined in each period.The contribution, if any, istypically made on a periodic basis with no additionalfuture obligation.
Amount of the future
obligation, based on the plan’s formula, must beestimated in the current period.
Eventual benefits arespecified. The amount of the
future obligation must beestimated in the current period.
Sponsoring Company’s
Pre-funding of Its Future
Obligation
Not applicable.
Companies typically pre-
fund the DB plans bycontributing funds to a pension trust. Regulatoryrequirements to pre-fund
vary by country.
Companies typically do not pre-fund other post-
employment benefitobligations.
Estimated annual payment = (Estimated final salary × Benefit formula) × Years of service
Final year’s salary = Current salary × [(1 + Annual compensation increase)years until retirement]
Annual unit credit = Value at retirement / Years of service
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.
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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
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 paidPension 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 employeesFair value of plan assets at the end of the period
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.
Balance Sheet Presentation of Defined Benefit Pension Plans
Calculating Periodic Pension Cost
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) thefair 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.
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Components of a Company’s Defined Benefit Pension Periodic Costs
IFRS Component
Service costs
Net interest
income/ expense
Remeasurements:
Net return on plan
assets and actuarial
gains and losses
IFRS Recognition
Recognized in P&L.
Recognized in P&L as
the following amount:
Net pension liability or
asset × interest rate(a)
Recognized in OCI and
not subsequently
amortized to P&L.
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.
U.S. GAAP Component
Current service costsPast service costs
Interest expense on
pension obligation
Expected return on
plan assets
Actuarial gains and losses
including differences
between the actual and
expected returns on plan
assets
U.S. GAAP Recognition
Recognized in P&L.Recognized in OCI and
subsequently amortized to
P&L over the service life of
employees.
Recognized in P&L.
Recognized in P&L as the
following amount: Plan assets
× expected return.
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.
(a) The interest rate used is equal to the discount rate used to measure the pension liability (the yield on high-quality corporate bonds.)
(b) If the cumulative amount of unrecognized actuarial gains and losses exceeds 10 percent of the greater of thevalue of the plan assets or of the present value of the DB obligation (under U.S. GAAP, the projected benefitobligation), the difference must be amortized over the service lives of the employees.
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Impact of Key Assumptions on Net Pension Liability and Periodic Pension Cost
Assumption
Higher discountrate
Higher rate of
compensationincrease
Higher expectedreturn on plan
assets
Impact of Assumption on Net
Pension Liability (Asset)
Lower obligation
Higher obligation
No effect, because fair valueof plan assets are used on
balance sheet
Impact of Assumption on Periodic
Pension Cost and Pension Expense
Pension cost and pension expense will both typically be lower because of lower opening obligation and lower servicecosts
Higher service and interest costs willincrease periodic pension cost and
pension expense.
Not applicable for IFRS No effect on periodic pension cost under
U.S. GAAPLower periodic pension expense under U.S. GAAP
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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 inUSD, 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 generatesand expends cash.
The local currency (LC) is the currency of the country where the subsidiary operates.
Table 1
Export sale
Import purchase
Loss
Gain
Asset (account receivable)
Liability (account payable)
Foreign Currency
Gain
Loss
Transaction Type of Exposure Strengthens Weakens
Methods for Translating Foreign Currency Financial Statements of Subsidiaries
FunctionalCurrency
LocalCurrency
LocalCurrency
PresentationCurrency
PresentationCurrency
TemporalMethod
Current Rate
Method
Current Rate/
Temporal Method
FunctionalCurrency
FunctionalCurrency
PresentationCurrency
LocalCurrency
T
T
CR
CR
=
=
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.
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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
Current Rate Method
FC = LC
Temporal Method
FC = PC
Exchange Rate Used
Average rate
Average rate
Average rate
Average rate
Average rate
Average rate
Average rate
N/A
Computed as Rev – Exp
Historical rate
Computed as NI – Dividends
Used as input for translated
B/S
Average rate
Historical rate
Average rate
Historical rate
Historical rate
Average rate
Average rate
Computed as Rev – Exp
Plug in Number
Computed as RE +
Dividends
Historical rate
From B/S
Rules for Foreign Currency Translation
Balance Sheet Component Exchange Rate Used
Cash
Accounts receivableMonetary assets
Inventory
Nonmonetary assets measured at
current value
Property, plant and equipment
Less: Accumulated depreciation
Nonmonetary assets measured at
historical cost
Accounts payableLong-term notes payable
Monetary liabilities
Nonmonetary liabilities:
Measured at current value
Measured at historical cost
Capital stock
Retained earnings
Cumulative translation adjustment
Current rate
Current rate Current rate
Current rate
Current rate
Current rate
Current rate
Current rate
Current rate Current rate
Current rate
Current rate
Current rate
Historical rate
From I/S
Plug in Number
Current rate
Current rate Current rate
Historical rate
Current rate
Historical rate
Historical rate
Historical rate
Current rate Current rate
Current rate
Current rate
Historical rate
Historical rate
To balance Used as input for
translated I/S
N/A
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Balance Sheet Exposure
Net asset Net liability
Balance Sheet Exposure
Foreign Currency (FC)
Strengthens
Positive translation adjustment Negative translation adjustment
Weakens
Negative translation adjustmentPositive translation adjustment
Effects of Exchange Rate Movements on Financial Statements
Foreign currency
strengthens
relative to
parent’s
presentation
currency
Foreign currency
weakens relative
to parent’s
presentation
currency
Temporal Method,
Net Monetary
Liability Exposure
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Translation loss
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Translation gain
Temporal Method,
Net Monetary Asset
Exposure
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Translation gain
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Translation loss
Current Rate Method
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Positive translation
adjustment
Revenues
Assets
Liabilities
Net income
Shareholders’ equity
Negative translation
adjustment
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 accrualst b/s = NOAt NOAt1
Aggregate Ratio
Accruals ratiot
b/s
= (NOAt + NOAt1)/2
(NOAt NOAt1)
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A Financial Statement Analysis Framework :
Phase Sources of Information Examples of Output
Define the purposeand context of theanalysis.
Collect input data.
Process input data,as required, intoanalytically usefuldata.
Analyze/interpretthe data.
The nature of the analyst’sfunction, such as evaluating anequity or debt investment or issuing a credit rating.Communication with client or supervisor on needs and
concerns.
Institutional guidelines relatedto developing specific work product.
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)
Data from the previous phase.
Input data and processed data
Statement of the purpose or objective of analysis.A list (written or unwritten) of specific questions to beanswered by the analysis. Nature and content of report
to be provided.
Timetable and budgetedresources for completion.
Organized financial statements.Financial data tables.
Completed questionnaires, if applicable.
Adjusted financial statements.Common-size statements.Forecasts.
Analytical results
1.
2.
3.
4.
ROE = Tax Burden × Interest burden × EBIT margin × Total asset turnover × Financial leverage
ROE = NI
EBT EBIT
EBT
Revenue
EBIT
Average Asset
Revenue
Average Equity
Average Asset××××
DuPont Analysis
Develop andcommunicateconclusions and
recommendations
(e.g., with an
analysis report).
Follow-up.
5.
6.
Analytical results and previousreportsInstitutional guidelines for
published reports
Information gathered by
periodically repeating abovesteps as necessary to determine
whether changes to holdingsor recommendations arenecessary
Analytical report answeringquestions posed in Phase 1Recommendations regarding
the purpose of the analysis,
such as whether to make an
investment or grant credit.
Update reports and
recommendations
INTEGRATION OF FINANCIAL STATEMENT ANALYSIS TECHNIQUES
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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) + tD
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 comparedand 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 thehigher EAA is chosen.
SML
R i = R F + ßi[E(R M) – R F]
R i = Required return for project or asset i
R F = Risk-free rate of returnßi = Beta of project or asset i[E(R M) – R F] = Market risk premium
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Economic Income
Economic income = After-tax operating cash flow + Increase in market valueEconomic 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 earnedover its life discounted at the cost of capital.
(1 + WACC)t
EPt NPV = MVA =
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 incomeis discounted at the required rate of return on equity to calculate NPV.
NPV =
(1 + r E)t
RIt
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.
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CAPITAL STRUCTURE
r WACC = r D(1 t) + r E
The Capital Structure Decision
r WACC = r D + r E r 0=( ) ( )
MM Proposition II without Taxes: Higher Financial Leverage Raises the Cost of Equity
Independent variableIntercept
Dependent variable Slope
r E = +r 0 (r 0 r D)
The systematic risk (ß) of the company’s assets can be expressed as the weighted average of the systematicrisk of the company’s debt and equity.
=A +( ) ( )
This formula can also be expressed as:
=E + (A D)
( )
r D = Marginal cost of debtr E = Marginal cost of equity
t = Marginal tax rateD = Market value of the company’s outstanding debtE = Market value of shareholders’ equityV = D + E = Value of the company
Company’s cost of equity (r E) under MM Proposition II without taxs is calculated as:
The total value of the company is calculated as:
V = +r E
EBIT Interest
r D
Interest
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= +tD
Relaxing the Assumption of no Taxes
r WACC = r D(1 t) + r E
The WACC is then calculated as:
r E = +r 0 (r 0 r D) (1 t)
And the cost of equity is calculated as:
Modigilani and Miller Propositions
Without Taxes With Taxes
=
r E = +r 0 (r 0 r D)
+ tD=
r E = +r 0 (r 0 r D) (1 t)
Proposition I
Proposition II
The Optimal Capital Structure: The Static Trade-Off Theory
VL = VU + tD – PV(Costs of financial distress)
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DIVIDENDS AND SHARE R EPURCHASE
Pw = Share price with the right to receive the dividend
PX = Share price without the right to receive the dividendD = Amount of dividendTD = Tax rate on dividends
TCG = Tax rate on capital gains
Double Taxation System
ETR = CTR + [(1 – CTR) × MTR D]
ETR = Effective tax rateCTR = Corporate tax rateMTR D = Investor’s marginal tax rate on dividends
Split-Rate Tax System
ETR = CTR D + [(1 – CTR D) × MTR D]
CTR D = 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]
PW PX = D
The expected decrease in share price when it goes ex-dividend can be calculated using the following equation:
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Industry Life
Cycle Stage
Pioneering
development
Rapid
acceleratinggrowth
Mature
growth
Stabilization
and market
maturity
Deceleration
of growth
and decline
Industry
Description
Low but slowly
increasing salesgrowth.Substantialdevelopment costs.
High profit
margins.Low competition.
Decrease in theentry of newcompetitors.Growth potential
remains.
Increasing capacityconstraintsIncreasing
competition.
Overcapacity.Eroding profitmargins.
Types of
Merger
Conglomerate
Horizontal
Conglomerate
Horizontal
HorizontalVertical
Horizontal
HorizontalVerticalConglomerate
Motives for Merger
Younger, smaller companies may sell
themselves to larger firms in matureor declining industries to enter into anew growth industry.Young companies may merge with
firms that allow them to pool
management and capital resources.
To meet substantial capital
requirements for expansion.
To achieve economies of scale,savings, and operational efficiencies.
To achieve economies of scale inresearch, production, and marketingto match low costs and prices of
competitors.Large companies may buy smaller companies to improve management
and provide a broader financial base.
Horizontal mergers to ensure survival.Vertical mergers to increase efficiencyand profit margins.Conglomerate mergers to exploit
synergy.
Companies in the industry may
acquire companies in youngindustries.
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).
Mergers and the Industry Life Cycle
MERGERS AND ACQUISITION
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MERGERS AND ACQUISITION
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Major Differences of Stock versus Asset Purchases
Payment
Approval
Tax: Corporate
Tax: Shareholder
Liabilities
Stock Purchase
Target shareholders receive
compensation in exchange for their shares.
Shareholder approval required.
No corporate-level taxes.
Target company’s shareholders
are taxed on their capital gain.
Acquirer assumes the target’s
liabilities.
Asset Purchase
Payment is made to the selling
company rather than directly toshareholders.
Shareholder approval might not berequired.
Target company pays taxes on anycapital gains.
No direct tax consequence for target
company’s shareholders.
Acquirer generally avoids the
assumption of liabilities.
n
i ( Sales or output of firm i
Total sales or output of market100)
2
Herfindahl-Hirschman Index (HHI)
Post-Merger HHI
Less than 1,000
Between 1,000 and 1,800More than 1,800
Concentration
Not concentrated
Moderately concentratedHighly concentrated
Change in HHI
Any amount
100 or more50 or more
Government Action
No action
Possible challengeChallenge
HHI Concentration Levels and Possible Government Response
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)
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MERGERS AND ACQUISITION
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Comparable Company Analysis
TP = Takeover premium
DP = Deal price per shareSP = Target’s stock price per share
TP =SP
(DP SP)
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 theacquirer, 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 companyC = Cash paid to target shareholders
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MERGERS AND ACQUISITION
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Perceived mispricing:
Perceived mispricing = True mispricing + Error in the estimate of intrinsic value.
VE – P = (V – P) + (VE – V)
VE = Estimate of intrinsic valueP = Market price
V = True (unobservable) intrinsic value
EQUITY VALUATION: APPLICATIONS AND PROCESSES
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EQUITY VALUATION: APPLICATIONS AND PROCESSES
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R ETURN CONCEPTS
Holding period return = PH – P0 + DH
P0
Intrinsic Value = Next year’s expected dividend
Required return – Expected dividend growth rate
V0 =D1
k e – g
k e (IRR) = g+D1
P0
If the asset is assumed to be efficiently-priced (i.e. the market price equals its intrinsic value), the IRR wouldequal the required return on equity. Therefore, the IRR can be estimated as:
Required return (IRR) = + Expected dividend growth rateMarket price
Next year’s dividend
Holding Period Return
PH = Price at the end of the holding periodP0 = Price at the beginning of the periodDH = Dividend
Required Return
The difference between an asset’s expected return and its required return is known asexpected 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 returnis 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. The required return (r T) earned on the asset’s current market price; and
2. The return from convergence of price to value [(V0 – P0)/P0].
Internal Rate of Return
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RETURN CONCEPTS
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BIRR model
r i = 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
r i = 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 beused 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:
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 projector company being evaluated, we use the following formula:
where:D/E = debt-to-equity ratio of the subject company.t = marginal tax rate of the subject company.
BASSET reflects only
business risk of the
comparable
company. Thereforeit is used as a proxy
for business risk of
the project being
studied.
BEQUITY
reflects
business and
financial risk of
comparable
company.
ßASSET = ßEQUITY
)1
(1 + (1 - t)D
E
BPROJECT
reflects
business and
financial risk of the
project.
BASSET
reflects
business risk of
project. ßPROJECT = ßASSET 1 + (1 - t)
D
E
Country Spread Model
ERP estimate = ERP for a developed market + Country premium
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RETURN CONCEPTS
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Weighted Average Cost of Capital (WACC)
WACC = + r MVCE
MVD + MVCE
r d (1 – Tax rate )MVD
MVD + MVCE
MVD = Market value of the company’s debtr d = Required rate of return on debtMVCE = Market value of the company’s common equity
r = Required rate of return on equity
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RETURN CONCEPTS
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Justified leading P/E ratio =P0
E1
D1/E1
r g= =
(1 b)
r g
Justified trailing P/E = P0
E0
D1/E0
r g= = (1 b)(1 g)
r g
D0 (1 g) / E0
r g=
P/E ratio
Present value of Growth Opportunities
V0 = + PVGOE1
r
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
One-Period DDM
Multiple-Period DDM
DISCOUNTED DIVIDEND VALUATION
V0 += = D1
(1 + r)1
P1
(1 + r)1
D1 P1
(1 + r)1
+
V0 += D1
(1 + r)1
Pn
(1 + r)n
Dn
(1 + r)n+ +...
V0 += Dt
(1 + r)t
Pn
(1 + r)nn
t = 1
Expression for calculating Value of a share of stock
V0 = Dt
(1 + r)
t
t = 1
V0 = D0 (1 + g)
(r – g), or V0 =
D1
(r – g)
Gordon Growth Model
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DISCOUNTED DIVIDEND VALUATION
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gS = Short term supernormal growth rate
gL = Long-term sustainable growth rate
r = required returnn = Length of the supernormal growth period
Two-Stage Dividend Discount Model
gS = Short term high growth rategL = Long-term sustainable growth rater = 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:
The H-Model
r = [(1 + gL) + H(gs – gL)] + gL
D0
P0
)(
The Gordon growth formula can be rearranged to calculate the required rate of return given the other variables.
r = + gD1
P0
V0 =D
r
Value of Fixed-Rate Perpetual Preferred Stock
V0 = n
t = 1
D0 (1 + gS)t
(1 + r)t + D0 (1 + gS)
n(1 + gL)
(1 + r)n(r – gL)
V0 = D0 (1 + gL)
r – gL
+ D0H (gs – gL)
r – gL
Sustainable growth rate (SGR)
b = Earnings retention rate, calculated as 1 – Dividend payout ratio
g = b × ROE
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DISCOUNTED DIVIDEND VALUATION
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ROE can be calculated as:
ROE = × × Net income
Sales Total assets
Sales Total assets
Shareholders’ equity
g = × × Net income
Sales Total assets
Sales Total assets
Shareholders’ equity
Net income - Dividends
Net income×
PRAT model
g = Profit margin × Retention rate × Asset turnover × Financial leverage
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DISCOUNTED DIVIDEND VALUATION
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FCFF = NI + NCC + Int(1 Tax Rate) FCInv WCInv
Computing FCFF from Net Income
FCInv = Capital expenditures Proceeds from sale of long-term assets
Investment in fixed capital (FCInv)
WCInv = Change in working capital over the year
Working capital = Current assets (exc. cash) Current liabilities (exc. short-term debt)
Investment in working capital (WCInv)
WACC =MV(Debt)
MV(Debt) + MV(Equity)r d(1 Tax Rate)
MV(Equity)
MV(Debt) + MV(Equity)r +
=Firm Value
t =1
FCFFt
(1+WACC)t
Equity Value Firm Value Market value of debt=
Equity Value =
t =1
FCFEt
(1 + r )t
FCFF/FCFE
FREE CASH FLOW VALUATION
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
Adjustment to NI to
Arrive at FCFF
Added back
Added back
Added back
Subtracted
Added back
Subtracted
Added back
Subtracted
Added back but requiresspecial attention
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FREE CASH FLOW VALUATION
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FCFE = EBIT(1 – Tax rate) – Int(1 – Tax rate) + Dep – FCInv – WCInv + Net borrowing
Computing FCFE from EBIT
FCFE = EBITDA(1 – Tax rate) – Int(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv + Net
borrowing
Computing FCFE from EBITDA
FCFF = CFO + Int(1 Tax rate) FCInv
Computing FCFF from CFO
FCFF = EBIT(1 – Tax rate) + Dep – FCInv – WCInv
Computing FCFF from EBIT
FCFF = EBITDA(1 – Tax rate) + Dep(Tax rate) – FCInv – WCInv
Computing FCFF from EBITDA
FCFE = FCFF – Int(1– Tax rate) + Net borrowing
Computing FCFE from FCFF
FCFE = NI + NCC – FCInv – WCInv + Net Borrowing
Computing FCFE from Net Income
FCFE = CFO + FCInv – Net borrowing
Computing FCFE from CFO
Table: IFRS versus U.S. GAAP Treatment of Interest and Dividends
Interest received
Interest paid
Dividend received
Dividends paid
IFRS
CFO or CFI
CFO or CFF
CFO or CFI
CFO or CFF
U.S. GAAP
CFO
CFO
CFO
CFF
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FREE CASH FLOW VALUATION
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Value of the firm = FCFF1
WACC - gFCFF0 (1 + g)
WACC - g=
WACC = Weighted average cost of capitalg = Long-term constant growth rate in FCFF
Constant Growth FCFF Valuation Model
Value of equity =FCFE1
r - g
FCFE0 (1 + g)
r - g=
r = Required rate of return on equityg = Long-term constant growth rate in FCFE
Constant Growth FCFE Valuation Model
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
Increases in cash balances
Plus: Net payments to providers of equity capital
+ Cash dividends
+ Share repurchases
New equity issues
= Uses of FCFE
Uses of FCFF
Uses of FCFE
An International Application of the Single-Stage Model
Value of equity =r real greal
FCFE0 (1 + greal)
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FREE CASH FLOW VALUATION
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General expression for the two-stage FCFF model:
n
t = 1
Firm value =
(1 + WACC)
n
1FCFFt
(1 + WACC)
t
FCFFn+1
(WACC g)
+
Firm value = PV of FCFF in Stage 1 + Terminal value × Discount Factor
General expression for the two-stage FCFE model:
Equity value = n
t = 1
FCFEt
(1 + r)t
FCFFn+1
r g (1 + r)n
1+
Equity value = PV of FCFE in Stage 1 + Terminal value × Discount Factor
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
Determining Terminal Value
Non-operating Assets and Firm Value
Value of the firm = Value of operating assets + Value of non-operating assets
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FREE CASH FLOW VALUATION
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MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE
MULTIPLES
Trailing P/E ratio = Current Stock PriceLast year’s EPS
Forward P/E ratio =Current Stock Price
Expected EPS
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
P/S ratio =Market price per share
Sales per share
P/E × Net profit margin = (P / E) × (E / S) = P/S
P/CF ratio =Market price per share
Free cash flow per share
Leading dividend yield = Next year’s dividend / Current price per share
Trailing dividend yield = Last year’s dividend / Current price per share
Price to Book Ratio
Price to Sales Ratio
Relationship between the P/E ratio and the P/S ratio
Justified leading dividend yield
Justified trailing dividend yield
Price to Cash Ratio
Price to Earnings Ratio
Dividend Yield
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MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
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V0 =D1
(r g)
Justified P/E Multiple Based on Fundamentals
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= =
(1 b)(1 g)
r g
D0 (1 g) / E0
r g=
Justified P/B Multiple Based on Fundamentals
=P0
B0
ROE g
r g
ROE = Return on equityr = required return on equityg = Sustainable growth rate
Justified P/CF Multiple Based on Fundamentals
V0 =FCFE0 (1 g)
(r g)
P0
S0
=(E0/S0)(1 b)(1 g)
r g
Justified P/S Multiple Based on Fundamentals
E0/S0 = Net profit margin
1 – b = Payout ratio
Justified Dividend Yield
D0
P0
r g
1 g=
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MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
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P/E-to-growth (PEG) ratio
TVn = Justified leading P/E Forecasted earningsn +1
TVn = Justified trailing P/E Forecasted earningsn
TVn = Benchmark leading P/E Forecasted earningsn +1
TVn = Benchmark trailing P/E Forecasted earningsn
Terminal price based on fundamentals
Terminal price based on comparables
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 CashFlow to theFirm =
EBITDA=
EBITA =
EBIT =
NetIncome
Net
Income
NetIncome
Net
Income
plusInterestExpense
plus
InterestExpense
plusInterestExpense
plus
InterestExpense
minusTax Savingson Interest
plus
Taxes
plusTaxes
plus
Taxes
plusDepreciation
plus
Depreciation
plusAmortization
plus
Amortization
plusAmortization
lessInvestment inWorking Capital
lessInvestment inFixed Capital
Justified forward P/E after accounting for Inflation
1
(1 ) I
P0
E1
=
= The percentage of inflation in costs that the company can pass through to revenue. = Real rate of returnI = Rate of inflation
PEG =P/E
Growth (%)
EV/EBITDA Multiple
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MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
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Unexpected earnings (UE)
UEt = EPSt – E (EPSt)
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 )]
Standardized unexpected earnings (SUE)
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MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES
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R ESIDUAL INCOME VALUATION
The Residual Income
V0 = Intrinsic value of the stock today
B0 = Current book value per share of equityBt = Expected book value per share of equity at any time tr = Required rate of return on equity
Et = Expected EPS for period tRIt = Expected residual income per share
E t rBt-1
(1 + r )t V 0 = B0 +
i = 1
RIt
(1 + r )t
i = 1
= B0 +
Intrinsic value of a stock:
Residual income = Net income – Equity charge
Equity charge = Cost of equity capital × Equity capital
RIt = Et – (r × Bt-1)
The Residual Income Model
RIt = Residual income at time tEt = Earnings at time tr = Required rate of return on equity
Bt-1 = Book value at time t-1
Capital charge = Equity charge + Debt charge
Debt charge = Cost of debt × (1 – Tax rate) × Debt capital
Residual income = After-tax operating profit Capital charge
NOPAT = Net operating profit after tax = EBIT (1 – Tax rate)C% = Cost of capital (WACC)TC = Total capital
Economic Value Added
EVA = NOPAT – (C% × TC)
Market value of company = Market value of debt + Market value of equity.
MVA = Market value of the company – Accounting book value of total capital
Market Value Added
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RESIDUAL INCOME VALUATION
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RIt = EPSt - (R × Bt-1)
RIt = (ROE - r)Bt-1
V0 = B0 +
t = 1
(ROEt r )Bt-1
(1 + r )t
Residual Income Model (Alternative Approach)
Tobin’s q =Market value of debt and equity
Replacement cost of total assets
Tobin’s q
ROE r
r gB0 + B0V0 =
V0 = B0 +T - 1
t = 1
(Et rBt 1)
(1 + r)t+
ET rBT-1
(1 + r )(1 + r)T1
= Persistence factor.
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:
V0 = B0 +(Et rBt 1)
(1 + r)t
+PT BT
(1 + r)T
T
t = 1
Multi-Stage Residual Income Valuation
g = r (ROE r) × B0
V0 B0[ ]
Implied Growth Rate
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RESIDUAL INCOME VALUATION
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PRIVATE COMPANY VALUATION
DLOC = 1 -1
1 + Control Premium
V = Value of the equityFCFE1 = 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
V = FCFE1
r g
Vf =FCFF1
WACC gf
Vf = Value of the firm
FCFF1 = Free cash flow to the firm for next twelve monthsWACC = Weighted average cost of capitalgf = Sustainable growth rate of free cash flow to the firm
The Capitalized Cash Flow Method
Discount for Lack of Control (DLOC)
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
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PRIVATE COMPANY VALUATION
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PRIVATE R EAL 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 first-
year NOI by the cap rate.
Value = NOI
1Cap 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 = Sale price of comparable property
NOI
The cap rate derived by dividing rent by recent sales prices of comparables is known as theall
risks yield (ARY). The value of a property is then calculated as:
Market value =Rent1
ARY
Gross income multiplier =Selling price
Gross income
Value of subject property = Gross income multiplier Gross income of subject property
Other Forms of the Income Approach
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PRIVATE REAL ESTATE INVESTMENTS
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Value = NOI1
(r – g)
The Discounted Cash Flow Method (DCF)
Terminal value = NOI for the first year of ownership for the next investor
Terminal cap rate
The Terminal Capitalization Rate
Appraisal-Based Indices
Return = NOI Capital expenditures + (Ending market value Beginning market value)
Beginning market value
LTV ratio =Loan amount
Appraised value
Debt Service Coverage ratio
DSCR =Debt service
NOI
Loan to Value ratio
Equity dividend rate/Cash-on-cash return
Equity dividend rate =First year cash flow
Equity investment
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PRIVATE REAL ESTATE INVESTMENTS
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PUBLICLY TRADED R EAL ESTATE SECURITIES
Capitalization rate = NOI of a comparable property
Total value of comparable property
NAVPS = Net Asset Value
Shares outstanding
Net Asset Value per Share
Capitalization rate
VALUATION: RELATIVE VALUATION (PRICE MULTIPLE) APPROACH
Funds from operations (FFO)
Accounting net earningsAdd: Depreciation charges on real estate
Add: Deferred tax chargesAdd (Less): Losses (gains) from sales of property and debt restructuringFunds from operations
Adjusted funds from operations (AFFO)
Funds from operations
Less: Non-cash rentLess: Maintenance-type capital expenditures and leasing costsAdjusted 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.
VALUATION: NET ASSET VALUE APPROACH
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PUBLICLY TRADED REAL ESTATE SECURITIES
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Post-money value =Exit value
(1 + Required rate of return ) Number of years to exists
Required wealth = Investment (1 + IRR) Number of years to exit
Proportionate ownership of the VC investor
= I / POST
Post-money value
Required wealth
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) andundistributed (unrealized) value as a proportion of cumulative invested capital.
o TVPI = DPI + RVPI
NAV before distributions = Prior year’s NAV after distributions + Capital calleddown – 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
PRIVATE EQUITY VALUATION
Ownership proportion = Required wealth / Exit value
Ownership propotion
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PRIVATE EQUITY VALUATION
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Shares to be issued =
Proportion of venture capitalist investment Shares held bycompany founders
Proportion of investment of company founders
Price per share =Amount of venture capital investment
Number of shares issued to venture capital investment
Adjusted discount rate = – 11 + r 1 – q
r = Discount rate unadjusted for probability of failure.q = Probability of failure.
Adjusted discount rate
Shares to be issued
Price per share
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PRIVATE EQUITY VALUATION
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FUNDAMENTALS OF CREDIT ANALYSIS
Expected loss = Default probability Loss severity given default
Expected Loss
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 = Liquidity premium + Credit spread
For small, instantaneous changes in the yield spread, the return impact (i.e. the percentage changein price, including accrued interest) can be estimated using the following formula:
Spread – Modified duration ×Return impact
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)
Yield Spread:
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FUNDAMENTALS OF CREDIT ANALYSIS
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TERM STRUCTURE AND VOLATILITY OF INTEREST R ATES
X t = 100 ln yt
yt1( )where
yt = yield on day t
Measuring Historical Yield Volatility
Annualizing the Standard Deviation
Annualized standard deviation = Daily standard deviation of days in a year
Variance =T
t = 1
X t 2
T 1
Variance =T
t = 1
W t X t 2
T 1
where:W t = the weight assigned to each daily yield change observation such that the sum of the weights
equals 1.
Calculating Variance of Daily Yield Changes
... assigns an equal weight to all observations
... attaches a greater weight to more recent information
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TERM STRUCTURE AND VOLATILITY OF INTEREST RATES
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Specific Bond Sector with a Given Credit Rating Benchmark
Treasury Market Benchmark
Nominal
Zero-volatility
Option-adjusted
Treasury yield curve
Treasury spot rate curve
Treasury spot rate curve
Credit risk, liquidity risk and option risk
Credit risk, liquidity risk and option risk
Credit risk and liquidity risk
Spread Measure Benchmark Reflects Compensation For
Nominal
Zero-volatility
Option-adjusted
Sector yield curve
Sector spot rate curve
Sector spot rate curve
Credit risk, liquidity risk and option risk
Credit risk, liquidity risk and option risk
Credit risk and liquidity risk
Spread Measure Benchmark Reflects Compensation For
Issuer-Specific Benchmark
Nominal
Zero-volatility
Option-adjusted
Issuer yield curve
Issuer spot rate curve
Issuer spot rate curve
Liquidity risk and option risk
Liquidity risk and option risk
Liquidity risk
Spread Measure Benchmark Reflects Compensation For
Summary of Relationships between Benchmark, OAS and Relative Value
Benchmark
Treasury market
Bond sector with a
given credit rating
(assumes credit rating
higher than security
being analyzed )
Negative OAS
Overpriced (rich) security
Overpriced (rich) security
(assumes credit rating higher
than security being analyzed )
Zero OAS
Overpriced (rich)
security
Overpriced (rich)
security
(assumes credit rating
higher than security
being analyzed )
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
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
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VALUING BONDS WITH EMBEDDED OPTIONS
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Determining Bond Value at a Node Applying Backward Induction
1-year rate at the
node at which we
are calculating the
bond's value, VHHL
Bond's value in higher-rate
state 1-year forward
NHHL
r 3,HHL
VHHL
VHHHL
C
r 4,HHHL
NHHHL
NHHLL
r 4,HHLL
VHHLL C
Cash flow in higher
rate state
Cash flow in lower
rate state
Bond's value in lower-rate
state 1-year forward
Summary of Relationships between Benchmark, OAS and Relative Value (Contd.)
Underpriced (cheap) security
(assumes credit rating lower than
security being analyzed )
Under priced (cheap) security
Underpriced (cheap)
security
(assumes credit rating
lower than security
being analyzed )
Fairly valued
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
Bond sector with a
given credit rating
(assumes credit rating
lower than security
being analyzed )
Issuer’s own securities
Positive OASZero OASNegative OASBenchmark
The present values of the these two cash flows discounted at the 1-year rate (r 3,HHL) at Node NHHL
are:
VHHHL + C
1 + r 3,HHL( ) 1. Present value in the higher one-year rate scenario
VHHLL + C
1 + r 3,HHL( ) 2. Present value in the lower one-year rate scenario
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VALUING BONDS WITH EMBEDDED OPTIONS
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VHHHL + C
1 + r 3,HHL
( )
VHHLL + C
1 + r 3,HHL
( )+
1
2
Finally, the expected value of the bond, VHHL at Node NHHLis calculated as:
Effective Duration and Effective Convexity
V V+ 2V0
2V0 ( y)2Convexity =
V V+
2V0 ( y)Duration =
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 =
Market conversion premium per share
Market price of common stock
Premium payback period =Market conversion premium per share
Favorable income differential per share
Favorable income differential per share =Coupon interest (Conversion ratio Common stock dividend per share)
Conversion ratio
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
Traditional Analysis of a Convertible Security
Premium over straight value =Market price of convertible bond
Straight value
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VALUING BONDS WITH EMBEDDED OPTIONS
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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
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VALUING BONDS WITH EMBEDDED OPTIONS
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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
SMM = 1 (1 CPR )1/12
Given the CPR, the SMM can be computed as:
Average life = t = 1
T t Projected principal recieved at time t
12 Total principal
t = Number of months
Average Life
Prepayment Risk in Different PAC Tranches
TranchePAC I - Senior
PAC I - Junior
PAC II
Support
Prepayment Risk LOW
HIGH
Distribution of Prepayment Risk in a Sequential-Pay CMO
Tranche
A (sequential pay)
B (sequential pay)
C (sequential pay)
Z (accrual pay)
Contraction Risk
HIGH
LOW
Extension Risk
LOW
HIGH
MORTGAGE-BACKED SECTOR OF THE BOND MARKET
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MORTGAGE-BACKED SECTOR OF THE BOND MARKET
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ASSET-BACKED SECTOR OF THE BOND MARKET
Parties to the Securitization
Party
Seller
Issuer/Trust
Servicer
Description
Originates the loans andsells loans to the SPV
The SPV that buys theloans from the seller
and issues the asset- backed securities
Services the loans
Party in Illustration
ABC Company
SPV
Servicer
SMM =1 – [ABS × (M – 1)]
ABS
ABS =1 + [SMM × (M – 1)]
SMM
Manufactured Housing-Backed Securities
Cash Flow Yield
ABS and MBS typically have monthly cash flows, so the cash flow yield on these securities is compared tothe 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 = Zero-volatility spread – Option-adjusted spread
Option Cost
Duration =V V+
2V0 ( y)
Duration
VALUING MORTGAGE-BACKED AND ASSET-BACKED SECURITIES
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ASSET-BACKED SECTOR OF THE BOND MARKET
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DERIVATIVES
Value of a Forward Contract
St F(0,T)
(1 + r)T-t
Time
St
F(0,T)
(1 + r)T-t
ST F(0,T)
Long Position Value
Zero, as the contract is priced to prevent arbitrage
F(0,T) ST
Short Position Value
Zero, as the contract is priced to prevent arbitrage
At expiration
At initiation
During life of the
contract
Price of an Equity Forward with Discrete Dividends
PV(D,0,T) = n
i = 1
Di
(1 + r)ti
F(0,T) = [S0 – PV(D,0,T)] (1 + r)T
FV(D,0,T) =
n
i = 1
Di(1 + r)Tti
F(0,T) = S0 (1 + r)T – FV(D,0,T)
... Approach I
... Approach II
F(0,T) = (S0ecT)er cT
F(0,T) = S0 e(r cc)T
c = Continuously compounded dividend yield
r c = Continuously compounded risk-free rate
Price of an Equity Forward with Continuous Dividends
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 portfoliocan be calculated as:
Vt(0,T) = Ste – c(T – t) – F(0,T)e –rc(T – t)
Vt(0,T) = –
St
ec(T – t)
F(0,T)
erc(T – t)
FORWARD MARKETS AND CONTRACTS
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DERIVATES
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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 bondT = 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 expirationcan 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 timet and time T.
BtC
(T+Y) = Current value of coupon bond with time T+Y remaining until maturity
FRA(0,h,m) =360m( ) 1
1 + L0(h + m)360
h + m( )1 + L0( h )
h
360( )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
Pricing a Forward Rate Agreement
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DERIVATES
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Valuing FRA prior to expiration
FRA payoff =1 + [Market LIBOR × (No. of days in the loan term / 360)]
NP × [(Market LIBOR – FRA rate) × No. of days in the loan term / 360]
FRA Payoff
g = Number of days since FRA initiation.
Vg (0,h,m) =1
1 + Lg (h g) )( h g360
1 + Lg(h + m g)360
h + m g )(
1 + FRA(0,h,m)360
m )(
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]}
Pricing a Currency Forward Contracts
(1 + R DC)T
(1 + R FC)TF(0,T) = S0 ×
F and S are quoted in terms of DC per unit of FC
R DC = Domestic risk-free rateR FC = Foreign risk-free rateT = Length of the contract in years. Remember to use a 365-day basis to calculate T ifthe 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) =(1 + R FC)(Tt)
St (1 + R DC)(Tt)
F (0,T)
Assuming continuous compounding, the price and value of a currency forward contract can be
calculated by applying the formulas below:
Vt(0,T) = [St / er cFC × (T – t)] – [F(0,T) / er cDC × (T – t)]
r c here represents a
continuouslycompounded risk-free rate in theseformulas.
or F(0,T) = S0 × e(r cDC – r cFC) × TF(0,T) = (S0e – r cFC × T) × er cDC × T
Or:
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DERIVATES
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FUTURES MARKETS AND CONTRACTS
If we ignore the effects of the mark-to-market adjustment on futures contracts, we can make thesimplifying assumption that the futures price and forward price are the same.
f 0(T) = F(0,T) = S0 × (1 + r)T
f 0(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 todayr = Annual risk-free rate
The Effect of Storage or Carrying Costs on the Futures Price
f 0(T) = S0 (1 + r)T + FV(SC,0,T)
The Effect of Monetary Benefits on the Futures Price
f 0(T) = S0 (1 + r)T FV(CF,0,T)
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 ascost of carry. In thiscase, the general futures pricing formula is given as:
f 0(T) = S0 (1 + r)T
+ FV(CB,0,T)
The Effect of Non-Monetary Benefits on the Futures Price
Pricing Treasury Bond Futures
f 0(T) = B0C(T+Y) [(1 + r 0(T)]T – FV(CI,0,T)
BC = Price of coupon bondT = 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.r 0(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 time0 (contract initiation) and time T (contract expiration).
The adjusted futures price of a t-bond futures contract is calculated as:
f 0(T) =B0
C (T + Y) [1 + r 0 (T)]T FV (CI,0,T)
CF(T)
CF(T) = Conversion factor on CTD bond
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FUTURES MARKETS AND CONTRACTS
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Pricing Stock Index Futures
f 0(T) = S0 (1 + r)T – FV(D,0,T)
f 0(T) = S0 e(r
c
– c
)T
Pricing Currency Futures
f 0(T) = S0 (1 + r DC)T
(1 + r FC)T
F and S are quoted in terms of DC/FCr DC = Domestic currency interest rater FC = 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 priceis calculated as:
f 0(T) = S0 × e(r cDC – r cFC)×T
r c represents the continuously compounded risk-free rate.
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FUTURES MARKETS AND CONTRACTS
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Put-Call Parity
C0 + = P0 + S0
X
(1 + R F)T
Synthetic Securities
Value
C0 +X
(1 + R F)T
C0
P0
S0
X(1 + R F)
T
X
(1 + R F)TC0 S0 +
P0 + S0
X
(1 + R F)T
P0 + S0
P0 + S0 C0
C0 +X
(1 + R F)T P0
ValueStrategy
fiduciary call
long call
long put
long
underlying
asset
long bond
long call +
long bond
long call
long put
long
underlying
asset
long bond
Consisting of
=
=
=
=
=
Equals
Protective
put
Synthetic call
Synthetic put
Synthetic
underlying
asset
Synthetic
bond
Strategy
long put + long
underlying asset
long put + long
underlying asset
+ short bond
long call + short
underlying asset
+ long bond
long call + long
bond + short put
long put + long
underlying asset
+ short call
Consisting of
OPTION MARKETS AND CONTRACTS
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OPTION MARKETS AND CONTRACTS
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One-Period Binomial Model
Computing the two possible values of the stock:
S+
= SuS- = Sd
Binomia Call Option Pricing
Call payoff = Max(0, S+ – X)
=(1 + r d
(u d)
c = c+ + (1 – ) c-
1 + r
Calculating the value of the call option:
n =c+ c-
S+ S-
Hedge Ratio
Binomial Put Option Pricing
Put payoff = Max (0, X – ST)
Compute the risk-neutral probabilities:
p = p+ + (1 – ) p-
1 + r
Calculating the value of the put option:
Intrinsic value of caplet at expiration:
Caplet value =Max {0, [(One-year rate – Cap rate) Notional principal]}
1 + One-year rate
Floorlet value =max {0, [(Floor rate – One-year rate) Notional principal]}
1 + One-year rate
Intrinsic value of floorlet at expiration:
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OPTION MARKETS AND CONTRACTS
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Put-Call Parity for Forward Contracts
Call and Bond Buy call
Buy bond
Total
Put and Forward
Buy put
Buy forward contract
Total
Transaction
c0
[X – F(0,T)]/(1 + r)T
c0 + [X – F(0,T)]/(1 + r)T
p0
0
p0
Current Value
Value at Expiration
ST – X
X – F(0,T)
ST – F(0,T)
0
ST – F(0,T)
ST – F(0,T)
ST > X
0
X – F(0,T)
X – F(0,T)
X – ST
ST – F(0,T)
X – F(0,T)
ST X
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:
N(d1) – 1] p S
N(d1) Sc
The Black-Scholes-Merton Formula
c = S0 N(d1) Xer cT N(d2)
p = Xe
-r cT
[1 N(d2)] S0[1 N(d1)]Where:
d1 =
ln(S0 X) + [r c + (
d2 = d1
= the annualized standard deviation of the continuously compounded return on the stock
r c = the continuously compounded risk-free rate of return
N(d1) = Cumulative normal probability of d1.
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OPTION MARKETS AND CONTRACTS
Delta
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c0 + = p0
X – F(0,T)
(1 + r)T
Forward Contract and Synthetic Forward Contract
Forward Contract
Long forward contract
Synthetic Forward Contract
Buy call
Sell putBuy (or sell) bond
Total
Transaction
0
c0
– p0
[X – F(0,T)]/(1 + r)T
c0 – p0 + [X – F(0,T)]/(1 + r)T
Current Value
Value at Expiration
ST – F(0,T)
0
– ( X – ST)X – F(0,T)
ST – F(0,T)
ST X
ST – F(0,T)
ST – X
0X – F(0,T)
ST – F(0,T)
ST > X
Put-call-forward parity
The Black Model
The Black model is used to price European options on futures.
Where:
d1 =
ln(f 0(T) X) + (
d2 = d1
c = e
r cT
[f 0(T)N(d1) XN(d2)]
p = er cT (X[1 N(d2)] f 0(T)[1 N(d1)])
f 0(T) = the futures price
Notice that the Black model is similar to the BSM model except that er cT 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 priceof a European option on the underlying asset if the options and the forward/futures contractexpire at the same point in time.
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OPTION MARKETS AND CONTRACTS
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SWAP MARKETS AND CONTRACTS
Swap fixed rate = 1 B0(N)B0(1) + B0(2) + B0(3) + ... + B0(N) )( 100
The Swap Fixed Rate
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
[(1 + Return on equity) Notional principal] PV of the remaining fixed-rate payments
[(1 + Return on equity) Notional principal] PV (Next coupon payment + Par value)
[(1 + Return on Index 2) NP] – [(1 + Return on Index 1) NP]
Valuing Equity Swaps
‘Pay a fixed rate and receive the return on equity’ swap
‘Pay a floating rate and receive the return on equity’ swap
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:
Payer swaption
Notional principal(Market fixed-rate – Exercise rate) No. of days in the payment period
360
Receiver swaption
Notional principal(Exercise rate – Market fixed-rate) No. of days in the payment period
360
Valuing a Swap
Value of pay-fixed side of plain-vanilla interest rate swap:
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SWAP MARKETS AND CONTRACTS
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Table: Caps, Floors, Interest Rate Options, Bond Options and Interest Rates
Security
Long cap (floor)Long call (put) option on interest rates
Long call (put) option on a fixed income instrument
Benefits when…
Interest rates rise (fall)Interest rates rise (fall)
Interest rates fall (rise)
Payoff to the buyer of an interest rate cap
Payoff to the buyer of an interest rate floor
Payoff = Max [0,(Market interest-rate – Cap rate) Notional principal] No. of days
360
INTEREST R ATE DERIVATIVE INSTRUMENTS
Notional principal] No. of days
360Payoff = Max [0,(Floor rate – Market interest-rate)
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INTEREST RATE DERIVATIVE INSTRUMENTS
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Expected return on Two-Asset Portfolio
E(R P) = w1E(R 1) + w2E(R 2)
E(R 1) = expected return on Asset 1
E(R 2) = expected return on Asset 2w1 = weight of Asset 1 in the portfolio
w2 = weight of Asset 2 in the portfolio
Variance of 2-asset portfolio:
1 1P 222 = w22 + w22 + 2w1w2 12
1= the standard deviation of return on Asset 12= the standard deviation of return on Asset 2 = the correlation between the two assets’ returns
Variance of 2-asset portfolio:
Cov1,2 = 12
1 1P 222 = w22 + w22
+ 2w1w2Cov1,2
Expected Return and Standard Deviation for a Three-Asset Portfolio
Expected return on 3-asset portfolio:
Variance of 3-asset portfolio:
1 1P 22 332 = w22 + w22 + w22 + 2w1w2 12 + 2w1w3 13 + 2w2w3 23
Variance of 3-asset portfolio:
E(R P) = w1E(R 1) + w2E(R 2) + w3E(R 3)
1 1P 22 332 = w22 + w22 + w22 + 2w1w2Cov + 2w1w3Cov + 2w2w3Cov
PORTFOLIO CONCEPTS
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PORTFOLIO CONCEPTS
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2
P2 =
1
n+( )
P2
=1
n 2
+n 1
n Cov
Variance of an Equally-weighted Portfolio
E(R P) =n
j=1
w jE(R j)
The variance of the portfolio is calculated as:
P2 =
n
j=1
n
i=1
wiw jCov(R i ,R j)
For a portfolio of n assets, the expected return on the portfolio is calculated as:
Standard Deviation of a Portfolio Containing a Risky Asset and the Risk-Free Asset
E(R P) = RFR + P
[E(R i) RFR]
i
P = wii
Expected Return for a Portfolio Containing a Risky Asset and the Risk-Free Asset
Expected Return and Variance of the Portfolio
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PORTFOLIO CONCEPTS
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Equation of CML:
E(R P) = R f + P
E(R m) R f m
E(R P) = w1R f + (1 w1)E(R m)
Variance of portfolios that lie on CML:
2 = w22 + (1 w1)22 + 2w1(1 w1)Cov(R f , R m)1 f m
Expected return on portfolios that lie on CML:
CML
The Decision to Add an Investment to an Existing Portfolio
i Cov(R i,R m)
2 2
i,mi,m
i,mi
m m m
Calculation and Interpretation of Beta
E(R i) R f + i[E(R m) – R f ]
The Capital Asset Pricing Model
E(R new) R F
new
E(R p) R F p
Corr(R new,R p)
Market Model Estimates
R i = i + i R M + i
R i = Return on asset i
R M = Return on the market portfolioi = Average return on asset i unrelated to the market returni = 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.
E(R i) = i + iE(R M)
Expected return on asset i
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PORTFOLIO CONCEPTS
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Cov(R i,R j) = i j2
M
Covariance of the returns on asset i and asset j
Correlation of returns between assets i and j
Corr(R i,R j) =i j
2
M
j M(22 + 2 )1/2
jMi(22 + 2 )1/2
i
R i = ai + bi1FDY + bi2FPE + i
R i = the return to stock iai = 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
bij =Assets i’s attribute value Average attribute value
Attribute values)
Fundamental Factor Models
Standardized sensitivities are computed as follows:
Var(R i) = 22 + 2i M i
Variance of the return on asset i
R i = ai + bi1FINT + bi2
FGDP + i
Market Model Estimates: Adjusted Beta
Adjusted beta = 0.333 + 0.667 (Historical beta)
Macroeconomic Factor Models
R i = the return to stock i
ai = the expected return to stock i
FINT = the surprise in interest ratesFGDP = the surprise in GDP growthbi1 = 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.
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PORTFOLIO CONCEPTS
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Active specific risk =n
i=1i i
wa
Where:wa= 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 by the factors).
i
i
Active factor risk = Active risk squared – Active specific risk.
Active return = R p – R B
Active return = Return from fctor tilts + Return from asset selection
Active return =K
j=1
[(Portfolio sensitivity) j (Benchmark sensitivity) j] (Factor return) j + Asset selection
Active Return
E(R P ) = R F + 1 p, p,
Arbitrage Pricing Theory and the Factor Model
E(R p) = Expected return on the portfolio p
R F = 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(R p R B)
Active risk squared = s
2
(R p R B)
Active risk squared = Active factor risk + Active specific risk
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PORTFOLIO CONCEPTS
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FMCAR j =
K
i=1b
a j b
ai Cov(F j,Fi)
Active risk squared
FMCAR j = Active risk squared
Active factor risk
K
i=1b
a j b
ai Cov(F j,Fi) = The active factor risk for factor j
where:
ba = The portfolio’s active exposure to factor j j
Factor’s Marginal Contribution to Active Risk Squared (FMCAR)
IR =R p R B
s(R p R B)
The Information Ratio
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PORTFOLIO CONCEPTS
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THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
The expression for the optimal weight, w*, of the active portfolio (Portfolio A) in the optimalrisky portfolio (Portfolio P) is given as:
w* = A
A(1 A) + R M
2(e A)
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:
w0 =
A
R M
2(e A)
2
M
A /2(e A)
R M /2 M
=
If the beta of Portfolio A does not equal 1, we can use the following equation to determine theoptimal weight, w*, of Portfolio A in Portfolio P.
Information Ratio
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:
S P =2
S M
2+
2(e A)
2 A
=R M
M +
(e A)
A 22
Weight of security k in the active portfolio (Portfolio A)
wk =
ni=1
i / ei
k / ek
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THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
(e A)
A 2
(ei)
i 2
n
i=1=
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Imperfect Forecasts of Alpha Values
Actual (realized) alpha: R R M
R
f
+
f
+
To measure the forecasting accuracy of the analyst, we can regress alpha forecasts ( f
) on realizedalpha ().
We can evaluate the quality of the analyst’s forecasts by calculating the coefficient of determination
of the regression described above.
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THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
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:
This estimate of R 2 is used as a shrinking factor to adjust the analyst’s forecasts of alpha.
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Risk Tolerance
Willingness to Take Risk
Below Average
Above Average
Below AverageBelow-average risk tolerance
Resolution needed
Above AverageResolution needed
Above-average risk tolerance
Ability to Take Risk
Return Requirements and Risk Tolerances of Various Investors
Individual
Pension Plans (Defined
Benefit)
Pension Plans (Defined
Contribution)
Depends on stage of life,
circumstances, and obligations
The return that will adequately
fund liabilities on an inflation-
adjusted basis
Depends on stage of life of
individual participants
Varies
Depends on plan and
sponsor characteristics,
plan features, funding status,
and workforce characteristics
Varies with the risk
tolerance of individual
participants
Return RequirementType of Investor Risk Tolerance
Foundations and
Endowments
Life Insurance
Companies
Non-Life- Insurance
Companies
B k
The return that will cover
annual spending, investment
expenses, and expected inflation
Determined by rates used to
determine policyholder reserves
Determined by the need to price
policies competitively and by
financial needs
D t i d b t f f d
Determined by amount of
assets relative to needs, but
generally above- average
or average
Below average due to factors
such as regulatory constraints
Below average due to factors
such as regulatory constraints
V i
THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY
STATEMENT
THE PORTFOLIO MANAGEMENT PROCESS AND THE INVESTMENT POLICY STATEMENT