SINGLE INDEX MODEL & THE

CAPITAL ASSET PRICING

MODEL

Financial Economics
Tute 04

1
Introductio
n

• Markowitz model require a large number of input data for calculations. An

investor must obtain estimates of return and variance of returns for all
securities and also co variance of returns for each pair of securities
• With a given set of securities infinite number of portfolios can be
constructed. The expected returns and variances of returns for each
possible portfolio must be computed.
• Identification of the efficient portfolios is also a complex procedure.
• Due to these difficulties, Markowitz model has found little use in practical
applications

2
Index Models

• Simplification is achieved through Index models , which are of

two types, Single Index and Multi Index
• All stocks are affected by movements in the economic factors.
• Suppose we summarize all the economic factors by one
macroeconomic indicator, and it moves the whole of the stock
market.
• Beyond this common effect, all remaining uncertainty in stock
returns is firm specific.
• Thus , holding period return on security i

3
Index
Models

• = expected return of the security at the beginning of the period

• = impact of unanticipated macro events on the securities returns
during the period
• = impact of the unanticipated firm-specific events during the
period, which by definition must average out to zero

4
Single factor Model

• Different firms have different sensitivities to macroeconomic

events .
• Thus, if we denote unanticipated components of the macro factor
by F, denote responsiveness of security ‘ i’to macro events by Beta
(𝛽𝑖 )then the macro component of the security “i’, = 𝛽𝑖 F and then,

for stock returns

However, a more realistic decomposition of security returns would
require more than one factor

5
Index
Model
• A factor model is of little use without specifying a
way to measure the factor.
• One reasonable approach is to assert that the rate of
return on a broad index of securities ( such as S&P
500) is a valid proxy for the common macro factor .
• Which is called a single index model because it uses
the market index to proxy for the common or
systematic factor

6
 Index Models

According to the Index model we can separate the actual or

realized rate of return on a security into , systematic and firm
specific components in a similar manner

we can write the holding period excess return on a stock as;

We can re write the equation by substituting R=excess return over

risk free rate
=stocks expected return if the market excess return is zero

7
Index
Models

Each security has two sources of risk market/systematic risk reflected

in , and firm specific risk reflected in

If variance is , then we can break the variance into two components

attributable to the uncertainty of the common macroeconomic factor ,

the variance attributable to firm specific uncertainty
the covariance between and is zero , because firm specific risk is
independent of the movements in the market.
Thus, the variance ,
8
Risk and covariance

• Variance of the rate of return on security i, equals the sum of

the variances due to the common and firm specific
components

• Covariance ,
• α‘s are constant their covariance with any variable is zero,
firm-specific terms are assumed uncorrelated with market and
with each other. Only source of covariance between the returns
of the two stocks derives from their common dependence on
the common factor
• Cov=

9
Single-Index Model

• The simplification derived from the Index model is not without

costs
• The classification of uncertainty into macro and micro risk
oversimplifies the real- world uncertainty. for example: this
ignores the industry events without substantially effecting the broad
economy
• Firm specific components of some firms are correlated
• Statistical significance does not always correspond to economic
significance

10
Example
Suppose that the index model for the excess returns of stock A and B is
estimated with the following results

Find the standard deviation of each stock and the covariance between them

11
12
Estimating the Index Model

• Equation suggests actually how we might go about measuring the

market and firm specific risk .
• Suppose that we observe the excess return on the market index
and a specific security over a number of holding periods.
• We can as an example use monthly excess returns on one security
with the market index
• To describe the typical relation between the return on the selected
security and the return on the market index , we can fit a straight
line through this scatter diagram.
• It is clear from the line of best fit that there is a positive relation
between the security’s returns and the markets.

13
Characteristic Line for
GM

14
T-bills, S&P 500 and GM
Stock

15
 Estimating the Index Model
• This is evidence for the importance of broad market conditions on
the performance of the security.

• The slope of the line reflects the sensitivity of the security’s return to
market conditions. A steeper line would imply that security’s rate of
return is more responsive to the market return.

• The scatter diagram also shows that the market conditions are not
the entire story.

• If returns perfectly tracked those in the market , then all return pairs
would lie exactly on the line.

• The scatter of points around the line is evidence that firm specific
events also have a significant impact on the security’s return
16
 The Regression Equation

How do we determine the line of best fit?

Let's, denote the market Index by “” with excess returns of

= standard deviation of σM.

Because the index model is linear , we can estimate the sensitivity

coefficient using a single variable linear regression.

17
 The Regression Equation

We can regress the excess return of a security

= on the excess return of the index

More generally for any stock “i” denote the pair of excess returns
to month t by .
Then the index model can be written as the following regression
equation

To estimate the regression, we can collect a historical sample of

paired observations .
18
 The Regression Equation

• The intercept of this equation is the securities expected excess

return when the market excess return is zero.

• The slope coefficient βi is the security β. It is the security’s

sensitivity to the index. : It is the amount by which the security
return tends to increase or decrease for every 1% increase or
decrease in the return on the index.

• ei is called the residual.

19
Security Characteristic
Line for GM: Summary Output

20
Estimating the Index
Model

Index model is mostly descriptive . It will help us address these

two important questions.

1. What relation might we expect to observe between a stock’s

beta and its expected return
2. What value for alpha should we observe , when markets are
in equilibrium ?

21
The Expected Return Beta
Relationship
• As the expected value of the residual is zero , if we take the expected
values , we can obtain the expected return – Beta relationship of the
single index model

• The second term in the equation tells us that part of a securities risk
Premium is due to the risk premium of the index.

• The market risk premium is multiplied by the relative sensitivity

(beta) of the individual security. We call this the systematic risk
premium , because it is derived from the risk premium that
characterizes the entire market.
22
The Expected Return Beta Relationship

• The remainder of the risk premium is given by the first term in the
equation , Alpha is a non -market premium.

• For example : may be large if your security is underpriced, and

therefore offers an attractive expected return.

• When security prices are in equilibrium, such attractive opportunities

would fade away and , will be driven to zero.

• If managers believe that they can do a superior job of security analysis ,

then they will be confident in their ability to find stocks with non -zero
values for
23
Risk and Co- Variance in the Single Index Model
• One of the problems with the Markowitz model is the large number of
parameter estimates required to implement it.
• The index model simplification vastly reduces the number of parameters
that must be estimated.
Total risk = Systematic Risk + Firm specific risk

The firm specific risk component can be reduced through diversification.

Covariance = Product of Beta’s* Market Index Risk

Correlation=product of correlations with the market index

24
 Exercise
Stock capitalization beta Mean excess SD
Return
A 3000 1.0 10 40%
B 1940 2.0 12 30
C 1360 1.7 17 60

The standard deviation of the market index portfolio is 25%

a) What is the mean excess return of the index portfolio

b) What is the covariance between stock A and stock B
c) What is the covariance between stock B and the Index
d) Break down the variance of Stock B into its systematic and firm
specific components

25
• Total capitalization =3000+1940+1360=6300

• (3000/6300)*10 +(1940/6300)*12 +(1360/6300)* 17=?

• Beta=2, SD=25=.2*25^2=?

• Systematic =.2^2* 25^2=?

• sd^2-systematic=?

26
 Set of estimates needed for the Single Index Model

The set of parameters needed for the single index model are α,β
and σ(e) for the individual securities and the risk premium and
variance of the market portfolio.

The estimates of the α,β and σ(e) are often obtained from
regression analysis of historical data of returns of the security as
well as the market index.

27
 Index Model and Diversification

• In an equally weighted portfolio of n securities ,excess return of

each security is given by
• Similarly for the portfolio
• When the number of stocks included in the portfolio increases, the
portfolio risk attributable to non -market factors become smaller
• Excess returns of the equally weighted portfolio

28
Index Model and Diversification

• Sensitivity to the market = = average of the individual s

• Nonmarket return component = = average of the individual s
• Zero mean variable = = average of the firm specific components
• Portfolio variance =
• The risk that is part of the market risk will remain, regardless of the extent of
portfolio diversification . have zero expected value
• The law of averages can be applied to conclude that when more and more
stocks are added to the portfolio , the firm specific component tend to cancel
out

29
Portfolio Construction using Single Index
Model
• Formation of portfolios with an efficient risk return trade off is our
goal.
• With the simplification offered by the single index model the technique
will be different from what we followed in the Markowitz model
• Index model offers several advantages
• Alpha and Security analysis
• Macro economic and security analysis is done in preparation of the input
list .
• Markowitz model requires estimation of risk premiums for each
security.
• The single index model separates these two quite different sources of
return variation

30
Steps involved in preparing the
input list
1. Macro economic analysis used to estimate the risk premium and the risk
of the market index
2. Statistical analysis is used to estimate the beta coefficients of all
securities and their residual variances
3. Portfolio manager uses these information to establish the expected return
4. Security specific expected return forecasts are derived from various
security valuation models
5. Thus, the Alpha value extracts the incremental risk premium
attributable to private information developed from security analysis.
6. The result is a list of alpha values

31
• In the context of portfolio construction Alpha value is the key variable
that tells us whether a security is a good or a bad buy.
• A positive – alpha security is a bargain and therefore should be
overweighted in the overall portfolio compared to the passive alternative
of using the market index portfolio as the risky vehicle
• A negative alpha security is over-priced and other things equal its
portfolio weights should be reduced
• In more extreme cases the desired portfolio weight might even be
negative ( a short position ) if permitted will be desirable

32
The Index portfolio as an investment asset

• One can take the market index as a passive portfolio that the manager
would select in the absence of security analysis

• Devising an active portfolio, that can be mixed with the index to

provide an even better risk return trade off is also possible

33
The Capital Asset
Pricing Model

34
Introduction

• We have so far discussed the principles of portfolio choice as made by

investors.
• We also considered the significance of the risk- free asset – CAL efficient
frontier , that all investors choose to be on this line.
• The relevant measure of an asset’s risk is its covariance with the market
portfolio of risky assets.
• How do we determine the required rate of return on a risky asset?
• How is an assets risk related to its required rate of return?
• CAPM Provides a framework to determine the required rate of return on
an asset and indicates the relationship between return and risk of an
asset.

35
The CAPM

• CAPM is a set of predictions concerning equilibrium expected

returns on risky assets
• It provides a benchmark rate of return for evaluating possible
investments
• Helps us to make an educated guess as to the expected return
on assets that have not yet been traded in the market- place.
Example : how do we price and IPO?
• Can compare the expected rate of return of an asset with its
required rate of return to determine whether it is fairly valued.

36
Assumptions
CAPM is based on two sets of assumptions;

The first set pertains to investor behavior

a) Investors are rational , mean variance optimizers

b) Their common planning horizon is a single period
c) All investors use identical input lists, an assumption often termed
homogenous expectations. Homogenous expectations are consistent
with the assumption that all relevant information is publicly
available

37
Assumptions

The second set of assumptions pertains to the market setting

(Market Structure)

a) All assets are publicly held and trade on public exchanges

b) Investors can borrow or lend at a common risk- free rate , and
they can take short positions on traded securities
c) No taxes
d) No transaction costs

Assumptions are restrictive. Thus, its important to discuss how the

predictions of the model may change when the assumptions are
relaxed
38
CAPM

• Let’s suppose that each investor uses an input list (expected return and
covariance matrix) to draw an efficient frontier employing all available
risky assets and identifies and efficient portfolio P by drawing the
tangent CAL to the frontier .

• As a result , each investor holds securities in the investible universe with

weights arrived at by the Markowitz optimization process.

• Notice that this framework employs Assumptions 1(a)investors are all

mean variance optimizers ,2(a)all assets trade and therefore can be held
in investors portfolios ,and 2(b) investors can borrow or lend at the risk -
free rate and therefore can select portfolios from the capital allocation
line of the tangency portfolio

39
Capital Market Line

E(rp)
CAL

M
E(Rm)

Rf

σp
σm
40
Capital Market Line

• The CAPM asks what would happen if all investors shared an identical
investable universe and used the same input list to draw their efficient
frontiers . The use of a common input list obviously requires
Assumption 1(c)but notice that it also relies on
• Assumption 1(b), that each investor is optimizing for a common
investment horizon.
• It also assumes that investor choices will not be affected by differences
in Tax rates or trading costs that could affect net rates of return.
Assumptions 2(d) and (c).
• In light of these assumptions , investors would calculate identical
efficient frontiers of risky assets. Facing the same risk-free rate
Assumptions 2 (b) they would then draw an identical tangent CAL and
naturally all would arrive at the same risky portfolio P. All investors
would therefore choose the same set of weights for each risky asset.
What must be these weights?
41
 Capital Market Line
• Because the market portfolio is the aggregation of all these identical risky
portfolios , it too will have the same weights.
• Assumption 2(a) requires that all assets can be traded and included in
investors portfolios.
• Therefore, if all investors choose the same risky portfolio, it must be the
market portfolio, that is the value weighted portfolio of all assets in the
investible universe.
• The Capital Allocation Line , based on each investors optimal risky portfolio
will in fact also be the Capital Market Line .

42
The Efficient Frontier and the Capital Market Line

43
 Why do all investors hold the market
portfolio?
• When we add together all the portfolios held by all the individual investors ,
the value of the aggregate risky portfolio will equal the entire wealth of the
economy.(borrowing and lending cancels out)
• This is the market portfolio M ( entire wealth of the economy)
• The proportion each stock in this portfolio equals the market value of the stock
(Price per share *number of shares outstanding) divided by the total market
value of all stocks .
• Given the assumptions , all the investors will hold the identical risky portfolio.
­ All investors use identical Markowitz Analysis
­ Applied to the same set of securities
­ For the same time horizon
­ Use the same input list

They must arrive at the same optimal portfolio on the efficient frontier identified
by the tangency line from t bills to that frontier.

44
Why do all investors hold the market
portfolio?

• If a particular stock is not included in the market portfolio ,

demand is zero and the price will fall until it is attractive enough
to be held in the optimal portfolio.

• This price adjustment process guarantees that all stocks will be

included in the market portfolio.

• The only issue is the price at which investors will be willing to

include a stock in their optimal risky portfolio.

45
The passive strategy is efficient
• In the simple world of CAPM, M is the optimal tangency
portfolio on the efficient frontier.
• This means that investors can skip the specific analysis and
obtain an efficient portfolio simply by holding the market
portfolio.
• Thus, the passive strategy of investing in the market portfolio is
efficient.
• If all investors would freely choose to hold a common risky
portfolio identical to the market portfolio, they would not
object if all stocks in the market were replaced with shares of a
single mutual fund holding that market portfolio . We
sometimes called this result the mutual fund theorem

46
• The mutual fund theorem is another form of the separation property
discussed earlier.

• Assuming that all investors choose to hold a market index mutual fund we
can separate portfolio selection into two components

• A technological problem; creation of mutual funds by professional managers

• A personal problem that depends on an investor’s risk aversion allocation of

the complete portfolio between the mutual fund and risk-free assets

• In reality, different investment managers do create risky portfolios that

differ from the market index

• However, the practical significance of the mutual fund theorem is that a

passive investor may view the market index as a reasonable first
approximation to an efficient risky portfolio
47
The risk premium of the market portfolio

• We have already discussed how individual investors go about

deciding capital allocation.

• If all investors choose to invest in portfolio M and the risk -free

asset , what can we deduce about the equilibrium risk premium of
portfolio M

• Recall that each individual investor chooses a proportion y

allocated to the optimal portfolio M such that

48
The risk premium of the market portfolio

• In the simplified CAPM economy ,risk free investments involve borrowing

and lending among investors. Any borrowing position must be offset by
lending position of the creditor. This means that net lending and borrowing
across all investors must be zero.

• Substituting the representative investors risk aversion for A, the average

position in the risky portfolio is 100% or y=1

• We find that the risk premium on the market portfolio will be proportional
to its risk and the degree of risk aversion of the investor

49
The risk premium of the market portfolio

Risk premium on individual assets will be proportional to the

risk premium on the market portfolio M, and the extent to
which returns on the stock and the market move together.

50
 Example
A stock index for a fourteen-year period yield the following statistics.
Average excess return = 10.15%,standard deviation = 27.91%,
• If these averages approximated investor expectations for the
period,what would be the average coefficient of risk aversion?
• If the coefficient of risk aversion were 2.5%, what risk premium
would have been consistent with the market's historical standard
deviation?

A= 10.15/(.01*27.91*27.91)=1.3
Risk premium = o.o1*2.5*27.91*27.91= 19.47%

51
 Expected Returns on Individual
securities
• The CAPM is built on the insight that the appropriate risk premium on
an asset will be determined by its contribution to the risk of
investors’ overall portfolios. Portfolio risk is what matters to investors
and is what governs the risk premiums they demand.
• All investors use the same input list, that is the same estimates of
expected returns variances and co variances.
• To measure the contribution to the risk of the overall portfolio from
holding a particular stock by its covariance with the market portfolio
• The calculate the variance of the market portfolio, we use the
bordered covariance matrix with the market portfolio weights.

52
Expected Returns on Individual
securities
• The contribution of one stock to the portfolio variance therefore can be
expressed as the sum of all the covariance terms in the row
corresponding to the stock where each covariance is first multiplied by
both the stock’s weight from its row and the weight from its column
• When there are many stocks in the economy there will be many more
covariance terms than variance terms. Consequently, the covariance of
a particular stock with all other stocks will dominate that stock’s
contribution to total portfolio risk.
• In other words, we can best measure the stock’s contribution to the risk
of the market portfolio by its covariance with that portfolio

53
Expected returns on individual securities

Asset A’ s contribution to variance =

If the covariance is negative, it makes a negative contribution to
the portfolio risk and vice versa.

Contribution of asset A to the risk premium of the market

portfolio is =

The reward to risk ratio for investments in A = A’s contribution to

Risk Premium/A’s contribution to Variance

54
Expected returns on individual securities
• The market portfolio is the tangency portfolio (efficient mean
variance)
• The reward risk ratio for investment in the market portfolio
=Market risk premium / Market variance

often called the market price of risk because it quantifies

the extra return that investors demand to bear portfolio risk.
• For components of the efficient portfolio , such as shares of
A, We measure risk as the contribution to portfolio variance
(covariance with the market).
• In contrast for the efficient portfolio itself, variance is the
appropriate measure of risk
55
• The basic principle of equilibrium is that all investments should offer
the same reward to risk ratio. Then,
• If the ratio were better for one investment than other investors would
rearrange their portfolios, tilting towards the alternative with the better
tradeoff and shying away from the other.
• Such activity would impart pressure on security prices until the ratios are
equalized. Therefore, the reward risk ratios of stock A and the market
portfolio should be equal

56
Expected returns on individual securities

• measures the contribution of stock A to the variance of the

market portfolio as a fraction of the total variance of the market
portfolio.

• The ratio is called Beta (β)

• we can restate the equation as

57
Expected returns on individual securities

• The expected return- Beta relationship is the most familiar expression

of CAPM. This relationship tells us that the total expected rate of
return is the sum of the risk -free rate plus a risk premium .
• The risk premium does not depend on the total volatility of the
investment
• The CAPM predicts that the systematic risk should be priced, but firm
specific risk should not be priced by the market
• We see why the assumption that individuals act similarly are so
useful ie. all hold identical risky portfolio, then everyone will find that
the beta of each asset with the market portfolio equals the assets
beta with his or her own risky portfolio. Then everyone will agree on
the appropriate risk premium on each asset.

58
Expected returns on individual securities
• Even if one does not hold the precise market portfolio , a well
diversified portfolio will be very highly correlated with the market
that a stocks beta relative to the market will still be a useful risk
measure.
• If this relationship holds for an individual asset, it will also hold for
any combination of assets.

Expected return =

Portfolio beta =

This is also true for the market portfolio =

as =1, we can verify by noting that,

59
Expected returns on individual securities
• The weighted average of beta is 1 across all assets .If the market beta is 1
,And the market beta is a portfolio of all assets in the economy , the
weighted average beta of all assets must be 1. Hence Beta s greater
than one are considered aggressive . Betas below 1 can be described as
defensive.
• If everyone knows that a firm is well run , its stock price therefore would
bid up . Security prices reflect public information about a firm's
prospects ,therefore only the risk of the company should affect expected
returns.
• In a rational market investors receive high expected returns only if they
are willing to bear risk
• Investors do not directly observe expected returns on securities. Rather
they observe security prices and bid those prices up or down .Expected
rates can be inferred from the prices investors pay.

60
 Exercise
• Suppose that the risk premium on the market portfolio is estimated at
8% with a standard deviation of 22%
• What is the risk premium on a portfolio invested 25%in Toyota and 75%
in Ford if they have Beta’s of 1.10 and 1.25 respectively ?

Answer
• Portfolio beta = .75*1.25+ .25*1.10 = 1.2125
• Portfolio risk premium = 1.2125*8% = 9.7%

61
The Security Market Line

• We can view the expected return – beta relationship as a reward –

risk equation . The Beta of a security is the best measure of its
risk because beta is proportional to the risk the security
contributes to the optimal risky portfolio.
• Risk averse mean variance investors measure the risk of the
optimal risky portfolio by its variance .
• Hence, we would expect the risk premium on individual assets to
depend on the contribution of the asset to the risk of the portfolio.
• The beta of a stock measures its contribution to the variance of
the market portfolio , thus required risk premium is a function of
beta

62
The Security Market Line

ERi
SML

Rf

Beta
0 1

• The expected return beta relationship can be shown graphically as the Security
Market Line (SML).

• Given the risk of an investment SML provides the required rate of return
necessary to compensate investors for both risk as well as the time value of
money.
63
SML and CML

• It is useful to compare the Security Market Line (SML to the

Capital Market Line (CML)
• CML graphs the risk premiums of efficient portfolios
(consisting of risk-free asset and a risky portfolio)
• SML in contrast graphs individual assets risk premiums as a
function of assets risk.
• The relevant measure of risk for individual assets held as parts of
well diversified portfolios is the contribution of the asset to the
portfolio variance , which is measured by assets beta.
• SML is valid for both efficient portfolios and individual assets

64
 SML

• The SML provides a benchmark for the evaluation of investment

performance. Given the risk of an investment, as measured by its beta,
the SML provides the required rate of return necessary to compensate
investors for risk as well as the time value of money
• Fairly priced assets would plot on the SML (ie. Expected returns are in line
with risk)
• Given the assumptions of the CAPM, all securities must lie on the line at
equilibrium. If SML relation is used as a benchmark to assess fair expected
return on a risky asset, the security analysis is performed to calculate the
return actually expected.
• If a stock is perceived to be a good buy or underpriced ,it will provide an
expected return in excess of the fair return stipulated by the SML.
Underpriced stocks therefore plots above the SML
• Overpriced stocks plot below the SML.

65
How Do We use Expected and Required Rates of Return?
 The stock is fairly priced if the expected return = the required return.
 This is what we would expect to see ‘normally’ or most of the time in an
efficient market where securities are properly priced.

% Return

E(Rs) = R(Rs) 17.6

SML
E(RM)= 14

Risk-free Rate = 6

BM= 1.0 BS = 1.464

66
 SML
• The difference between the fair and actually expected rates of
return on a stock is called a stocks alpha.

• Example If the market return is expected to be 14%a stock has a

beta of 1.2.and the T bill rate is 6%

• The SML would predict an expected return on the stock of 6+

1.2(14-6)=15.6%

• If an investor believes that the stock would provide an expected

return of 17%the implied alpha would be 1.4% one might say that
security analysis is about uncovering securities with positive alphas.

67
Security analysis is performed to calculate the return actually expected.
If a stock is perceived to be a good buy ( or under priced) it will provide an expected return
in excess of the return stipulated by the SML.
E(r)(%)

SML

Stock
17 a
15.6
14 M

B 68
1.0 1.2 1.4
 SML

• The starting point of portfolio management can be a passive market

index portfolio. Later the manager can increase the weight of securities
with excess expected returns and decrease the weight of securities with
negative excess returns

• CAPM can also be used in capital budgeting decisions. CAPM can

provide the required rate of return ( IRR) that project needs to yield
based on its beta for it to be acceptable by the investors.

• Managers can use the CAPM to obtain this cut off internal rate of return
for the project

69
CAPM and the Single Index Market
Single Index Model

States that the realized excess returns on any stock is the sum of the
realized excess return due to market-wide factors, a non- market
premium and firm specific outcomes.

Expected values :
)

The expected return beta relationship in the CAPM

)
70
CAPM and the Single Index Market

Comparing the two models :

• The predictions of the CAPM is for every stock equilibrium value

of alpha is zero.
• Logic of CAPM
Only reason for a stock to provide a premium over the risk- free rate
is the systematic risk for which investor must be compensated.
A positive alpha implies a reward without risk . Investors would
pursue positive alpha stocks and bid up their prices. At higher
prices expected rates of return s may be lower

71
CAPM and the Single Index
Market
• Investors will shun negative alpha stocks driving down their prices
and driving up their expected returns.
• The portfolio rebalancing will take place until all alpha values are
driven to zero
• At this point investors will be content to fully diversify and
eliminate unique risk , that is to hold the broadest possible market
portfolio
• When all stocks have zero alpha s , the market portfolio is the
optimal risky portfolio
• If one estimates the index model regression ,with a market index,
that adequately represents the full market portfolio , estimated
values of alpha should cluster around zero

72
Example

Stock XYZ has an expected return of 12%and risk of β=1. Stock ABC
has expected return of 13% and β=1.5. The markets expected return is
11%, and the risk-free rate is equal to 5%.

• According to CAPM which stock is a better buy?

• What is the alpha of each stock ?

The risk -free rate is 8% and the expected return on the market
portfolio is 16%. A firm considers a project that is expected to have a β
of 1.3

• What is the required rate of return on the project ?

• If the expected IRR of the project is 19%. Should it be accepted?
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• αxyz=12-[15+1.0(11-5)]=1%

• αabc=13-[5+1.5(11-5)]=-1%

• abc is overvalued and plots below SML and xyz is undervalued

and plots above SML

• 8+ 1.3(16-8)=18.4 projects hurdle rate .if the IRR of the project is

19% then it is desirable . Any project with an IRR equal to or less
than 18.4 % should be rejected

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 Extensions of the CAPM

There are two classes of extensions :

• The first attempts to relax the assumptions,

• The second acknowledges the fact that investors worry about

sources of risk other than the uncertain value of their securities such
as unexpected changes in relative prices of consumer goods.

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