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Advanced Financial Management

MAF - 581

Abraham G. (Assistant Professor)

 Chapter One
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Risk, Return, and Asset Pricing

Models

 Required readings:
 Ehrhardt, M.C. Brigham, E. F. (2011), Financial Management: Theory
and Practice, 13th Ed., South-Western Cengage Learning. (Chapter 6, 24)
 Stephen A. Ross, Randolph W. Westerfield, Bradford D. Jordan (2013),
Fundamentals of Corporate Finance, 10 th ed. McGraw-Hill. (Chapter 13)
 Return on Investments
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 Investment is the current commitment of resources

for a period of time in expectation of receiving future
resources greater than the current outlay.
 In order to part with their money, investors require
compensation for:
 The time resources committed
 The expected rate of inflation
 The uncertainty of the future payments

 The gain (or loss) from the investment is return on

investment .
 Cont’d……….
4

 Return on investment usually have two components.

 Cash collection while owning the investment - the

income component of return.

 The value of the asset (or investment) will often

change. In this case, there is a capital gain or

capital loss on the investment.
 Thus, return is measured by taking the income plus

the price change.

Rate of return = (Income + Capital gains)/Purchase
price
 Cont’d………
5

The expected rate of return E(r) on investment is the

statistical measure of return; the sum of all possible
rates of returns for the same investment weighted by
probabilities:
n
E(r) =Σ pi × ri where; pi - probability of rate of return;
i=1 ri - rate of return.

 The decision of investment is based on the estimated

expected rate of return.
 Stand-Alone Risk
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 Risk is a chance that the actual outcome from an

investment will differ from the expected outcome.
 The more variable the possible outcomes that can
occur, the greater the risk.
 Stand-alone risk is the risk an investor would face if
an investor held only one asset.
 Risk of investments can be measured with the
common absolute measures used in statistics;
 Variance and Standard Deviation
 Cont’d………
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Variance and Standard Deviation

 Measure the “spread” in returns
 The potential average deviation of each possible
investment return from the expected rate of return:
 How far the actual return deviates from the
average in a typical year?
 The greater the volatility, the greater the
uncertainty/risk
 Cont’d……..
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 Variance of actual returns = sum of squared deviations

from the mean/(number of observations - 1)
V = (Σ(ri – E(r))2/(N – 1)
 Standard deviation of expected returns = positive
square root of the expected variance
δ2 = Σ pi × [ ri - E(r) ]2
i=1
 They are similar measures of risk and can be used for
the same purposes in investment analysis; however,
standard deviation is used more often in practice.
 Cont’d……..
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Year Actual Avera Deviation from Squared

Return ge the Mean Deviation
(Average)
Retur
n
1 -.20 .175 -.375 .140625
2 .50 .175 .325 .105625
3 .30 .175 .125 .015625
4 .10 .175 -.075 .005625
Totals .70 / 4 .000 .267500
= .175
Variance = sum of squared deviations from the mean / (number of observations –
1)
= .26750 / (4-1) = .0892
 Cont’d……….
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 The larger the variance, the more the actual returns

tend to differ from the average return.
 The larger the variance or standard deviation, the
more spread out the returns will be.
 Cont’d………..
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Historical average returns, standard

deviations, and frequency
Average Standard
distributions:
Series 1926 – 2011.
Return Deviation Distribution
Large-company
stocks 11.7% 20.3
Small-company *

stocks 18.7 39.2

Long-term
corporate bonds 6.6 7.0
Long-term
government Bonds 5.5 9.3
Intermediate-term
government Bonds 5.4 5.8
U.S. Treasury
bills 3.6 3.1
Inflation 3.2 4.5
-90% 0% 90%
 Cont’d………..
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NB:
 The 1933 small-company stock total return was

142.9 percent.
 The standard deviation for small-stock portfolio is

more than 10 times larger than the T-bill portfolio’s

standard deviation.
 Risk Premium
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 Risk premium is the excess return required from an

investment in a risky asset over that required from a
risk-free investment.
 The “extra” return earned for taking a risk
 Treasury bills are considered as a risk-free

investment.
 Cont’d………
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Coefficient of Variation (CV)

 If a choice has to be made between two investments

that have the same expected returns but different

standard deviations, choose the one with the lower
standard deviation.
 Given a choice between two investments with the

same risk but different expected returns, investors

generally prefer the investment with the higher
expected return.
 But, how do we choose between two investments if one
has a higher expected return and the other a lower risk?
 Cont’d………
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 Use the coefficient of variation (CV) to measure the

degree of risk,
CV = σ/r
 The coefficient of variation shows the risk per unit of
return, and it provides a more meaningful basis for
comparison than σ when the expected returns on two
alternatives are different.
 Return on Portfolio
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 Portfolio is a collection of investment vehicles

assembled to meet one or more investment goals.
 Growth-Oriented Portfolio: primary objective is

long-term price appreciation
 Income-Oriented Portfolio: primary objective is

current dividend and interest income

 Capital preserving Portfolio: primary objective is

getting marginal income without depleting capital

 Return on Portfolio is the weighted average of
returns on the individual assets in the portfolio.
 Cont’d……….
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Rp = w1r1 + w2r2 + w3r3 + w4r4

n
E R P    w
i1
i  E R i  

Where: E(RP) = expected portfolio return

wi = portfolio weight in portfolio asset i
E(Ri) = expected return for asset i

 Assume that a security analyst estimated the coming

year’s returns on the stocks of four large companies.
The investment is $1 million, divided among the
stocks.
 Cont’d……….
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 30% in Co. A (expected return of 15%), 10% in Co.

B (expected return of 12%), 20% in Co. C (expected
return of 10%), and 40% in Co. D (expected return
of 9%). The expected portfolio return is:
Rp = w1r1 + w2r2 + w3r3 + w4r4
= 0.3(15%) + 0.1(12%) + 0.2(10%) + 0.4(9%)

= 11.3%
 Portfolio Risk
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 Portfolio risk is the risk an investor would face if s/he

held many assets.
 Standard Deviation (σ) of a portfolio return is
calculated using all individual assets in the portfolio.
 Unlike returns, the risk of a portfolio, σp, is generally
not the weighted average of the standard deviations
of the individual assets in the portfolio.
 σp will be smaller than the assets’ weighted σ, and it is
theoretically possible to combine stocks that are
individually quite risky as measured by their σ and
form a portfolio that is completely riskless, with σp = 0.
 Cont’d……….
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 The reason assets can be combined to form a riskless

portfolio is that their returns move counter-cyclically
to each other when asset one returns fall, those of
other rise, and vice versa.
 The tendency of two variables to move together is
called correlation.
 Correlation coefficient measures this tendency and

symbolized by the Greek letter rho, ρ (pronounced

roe).
 Cont’d………
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 In statistical terms, the returns on Stocks A and B are;

 Perfectly negatively correlated, with ρ = −1.0.
 Perfectly positively correlated, with ρ = 1.0.
 Uncorrelated, with ρ = 0

 ρ can range from +1.0, denoting that the two variables

move in perfect synchronization to –1.0, denotes the
variables always move in exactly opposite directions.
n
p  
t 1
( r i , t  r i , Avg )( r j , t  r j , Avg )

 n

  r i , t  r i , Avg  2  n

   r j , t  r j , Avg  2 

 t 1   tt 
 Cont’d……….
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 Returns on two perfectly positively correlated stocks

move up and down together, and a portfolio
consisting of two such stocks would be exactly as
risky as each individual stock.
 Thus, diversification does nothing to reduce risk if

the portfolio consists of perfectly positively

correlated stocks.

 When stocks are perfectly negatively correlated, all

risks can be diversified away.
 Cont’d………..
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 Cont’d………
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 In reality, almost all stocks are positively correlated,

but not perfectly so.
 Studies have estimated on average, the ρ for the
monthly returns on two randomly selected stocks is
in the range of 0.28 to 0.35.
 So, combining stocks into portfolio reduces but

does not completely eliminate risk.

 To sum up, diversification can reduce risk but not

eliminate it.
 Cont’d………
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 The risk of a portfolio declines as the number of

stocks in the portfolio increases.
 A portfolio that provides the highest return for a
given level of risk is called efficient portfolio.
 In general, there are higher correlations
 Between the returns of two companies in the same

industry than for companies in different industries.

 Among similar “style” companies, such as large

versus small, and growth versus value.

 Thus, to minimize risk, portfolios should be
diversified across industries and styles.
 Cont’d…………
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 Cont’d……..
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 Financial data shows,

 σ of a one-stock portfolio (or an average stock), is
1
approximately 35%.
 However, a portfolio consisting of all stocks,

called the market portfolio, have a σM of 20%.

 So, almost half of the risk inherent in an average
individual stock can be eliminated if the stock is held
in a reasonably well-diversified portfolio.
 The risk that can be eliminated is called diversifiable
risk, while the part cannot be eliminated is called
market risk.
 Cont’d………
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 Diversifiable risk is caused by random events like

lawsuits, strikes, unsuccessful marketing programs,
losing a major contract, and others that are unique to
a particular firm.
 Their effects on a portfolio can be eliminated by

diversification; bad events in one firm offsets by

good events in another.
 Market risk stems from factors that are systematically
affects most firms and cannot be eliminated by
diversification: war, inflation, recessions, and high
interest rates.
 Cont’d………..
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 Diversifiable risk is also known as company-specific,

or unsystematic risk.
 Market risk is also known as non-diversifiable,
systematic; it is the risk that remains after
diversification.
 Cont’d………
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        2 A B A B P( R A RB )
2
p
2
A
2
A
2
B
2
B
Where: wA = portfolio weight of asset A
wB = portfolio weight of asset B
P = correlation coefficient
wA + wB = 1
 Cont’d……….
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 The other important measurement of relationships

between two assets in portfolio creation is
covariance.
 Covariance is the expected product of the deviations
of two returns from their means.
 The covariance between expected Ri and Rj is:
Cov (Ri, Rj) = E [(Ri - E [Ri])(Rj - E [Rj])]
 Covariance from Historical Data:
Cov (Ri, Rj) = ∑t (Ri,t - Ri)(Rj,t - Rj)
2 2 2 2 2 T – 1
        2  COV ( A, B )
p A A B B A B
 Cont’d……….
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Example
 The returns and risk of Johnson & Johnson (JNJ),

International Business Machines Corp. (IBM), and

Boeing Co. (BA), for April 2001 to April 2010 are;
 Cont’d……….
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 The weights are 20.87% in JNJ, 46.93% in IBM, and

32.2% in BA company.
 The return on this portfolio is:
R p   0.2087   0.0080   0.4693  0.0050    0.3220  0.0113
 0.76%
 The risk on portfolio:
 p2   0.2087  0.0025   0.4693  0.0071   0.3220  0.0083
2 2 2

 2 0.2087  0.4693  0.0007  2 0.2087  0.3220  0.0007  2 0.4693  0.3220  0.0006

 0.00302

σp = 5.495%
 Market Risk
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Market risk of a stock is measured by beta

coefficient.
 It measures a stock’s tendency to move up and

down with the market.

bi = (σi/σM)ρiM
Where; bi - beta coefficient of Stock i
ρiM - correlation between Stock i’s R and market R
σi - standard deviation of Stock i’s return
σM - standard deviation of the market’s return.

 The average beta for all stocks (market) is 1.0.

 Cont’d……….
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 If beta of a given stock equals 1.0, the stock is as

risky as the market, assuming it is held in a
diversified portfolio.
 If beta is less than 1.0 then the stock is less risky

than the market;

 If beta is greater than 1.0, the stock is more risky

than the market.

 Most stocks have betas in the range of 0.50 to 1.50.
 The beta of a portfolio is a weighted average of the
individual securities’ betas.
bp = w1b1 + w2b2 +…+ wnbn
 Risk and Return Relationships
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 Have a positive relationship between them.

 The higher the risk the higher the return.

 The market risk premium, RPM, is the premium that

investors require for bearing the risk of an average
stock, and it depends on the degree of risk aversion
that investors on average have.
 The reward for bearing risk depends only on the
systematic risk of an investment since unsystematic
risk can be diversified away.
RPM = RMkt − RRF
 Capital Asset Pricing Model
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 It describes the relationship between systematic risk

and expected return for assets, particularly stocks.
 The goal of the model is to evaluate whether a stock
is fairly valued when its risk and the time value of
money are compared to its expected return.
 CAPM was developed by Harry Markowitz in 1959.
 Cont’d………..
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 Assumptions underlying the CAPM’s development;

 All investors focus on a single holding period, and they
seek to maximize the expected utility.
 All investors can borrow or lend an unlimited amount at
a given risk-free rate of interest.
 All investors have identical estimates of the expected
returns, variances, and covariances among all assets
 All assets are perfectly divisible and perfectly liquid.
 There are no transaction costs.
 There are no taxes.
 All investors are price takers (assume their own buying
and selling activity will not affect stock prices).
 The quantities of all assets are given and fixed.
 Cont’d………
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 The CML specifies the relationship between risk and

return for an efficient portfolio.
 But, investors and managers are more concerned
about the relationship between risk and return of
individual assets.
 SML specifies the relationship between risk and

return of individual assets.

ri = rRF + (rM – rRF)bi
= rRF + (RPM)bi
 Cont’d……….
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 SML is the representation of the CAPM.

 It displays the expected rate of return of an
individual security as a function of systematic risk.
 Cont’d………
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 Cont’d………
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 Suppose the risk-free rate is 4%, the market risk

premium is 8.6%, and a stock has a beta of 1.3.
Based on the CAPM, what is the expected return of
the stock? What would the expected return be if the
beta were to double?
ri = rRF + bi(rM – rRF)
= 4% + 1.3*8.6%
= 15.18%
ri = rRF + bi(rM – rRF)
= 4% + 2.6*8.6%
= 26.36%
 Arbitrage Pricing Theory (APT)
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 CAPM is a single factor model that specifies risk as a

function of only one factor, the security’s beta.

 The risk–return relationship is more complex, with a

stock’s return is a function of more than one factor.
 Ross (1976) has developed APT that includes a
number of risk factors, so the required return be a
function of two, three, four, or more factors.
 Cont’d……..
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 APT model
ri = rRF + (r1 – rRF)bi1 + . . . + (rj – rRF)bij
ri = required rate of return on Stock i:
bi = sensitive only to economic Factor:
Example
 Assume that all stocks’ returns depends on inflation,

industrial production, and the aggregate degree of

risk aversion (the cost of bearing risk, which we
assume is reflected in the spread between the yields
on Treasury and low-grade bonds).
 Cont’d………
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 The risk-free rate is 8.0%;

 The required rate of return is 13% on a portfolio with
unit sensitivity to inflation;
 The required return is 10% on a portfolio with unit
sensitivity to industrial production;
 The required return is 6% on a portfolio with unit
sensitivity to the degree of risk aversion.
 Assume that Stock i has factor sensitivities (betas) of
0.9 to inflation portfolio, 1.2 to industrial production
portfolio, and −0.7 to the risk-bearing portfolio.
 Cont’d……….
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 Stock i’s required rate of return, according to APT

would be;
ri = 8% + (13% − 8%)0.9 + (10% − 8%)1.2 + (6% −
8%)(−0.7)
= 16.3%
 If the required rate of return on the market were

15.0% and if Stock i had a CAPM beta of 1.1, then

its expected return, according to the SML, would be;
ri = 8% + (15% − 8%)1.1
= 15.7%
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Questions?

Thank You!