asset pricing models: their uses and their limitations - bahattin

CHAPTER 9

The Capital Asset Pricing Model

It is the equilibrium model that underlies all

modern financial theory

Derived using principles of diversification with

simplified assumptions

Markowitz, Sharpe, Lintner and Mossin are

researchers credited with its development

CAPITAL ASSET PRICING MODEL (CAPM)

BAHATTIN BUYUKSAHIN, JHU,

INVESTMENT 2

Individual investors are price takers

Single-period investment horizon

Investments are limited to traded financial assets

There are homogeneous expectations

ASSUMPTIONS: INVESTORS


INVESTMENT 3

Information is costless and available to all

investors

No taxes and transaction costs

Risk-free rate available to all

Investors are rational mean-variance optimizers

ASSUMPTIONS: ASSETS


INVESTMENT 4

All investors will hold the same portfolio for risky

assets – market portfolio, which contains all securities

and the proportion of each security is its market value

as a percentage of total market value held by all investors

includes all traded assets

suppose not: then price… -> included

is on the efficient frontier

asset weights: for each $ in risky assets, how much is in IBM?

for stock i: market cap of stock i / market cap of all stocks

RESULTING EQUILIBRIUM CONDITIONS


INVESTMENT 5

iii

iii

PN

PNw

Risk premium on the market depends on the

average risk aversion of all market participants

Risk premium on an individual security is a

function of its covariance with the market

RESULTING EQUILIBRIUM CONDITIONS

CONTINUED


INVESTMENT 6

FIGURE 9.1 THE EFFICIENT FRONTIER AND

THE CAPITAL MARKET LINE


INVESTMENT 7

MARKET RISK PREMIUM

The risk premium on the market portfolio will be

proportional to its risk and the degree of risk

aversion of the investor:

2

2

( )

where is the variance of the market portolio and

is the average degree of risk aversion across investors

M f M

M

E r r A

A


INVESTMENT 8

The risk premium on individual securities is a

function of the individual security’s contribution

to the risk of the market portfolio

An individual security’s risk premium is a function

of the covariance of returns with the assets that

make up the market portfolio

RETURN AND RISK FOR INDIVIDUAL

SECURITIES


INVESTMENT 9

USING GE TEXT EXAMPLE

Covariance of GE return with the market

portfolio:

Therefore, the reward-to-risk ratio for

investments in GE would be:

1 1

( , ) , ( , )n n

GE M GE k k k k GE

k k

Cov r r Cov r w r w Cov r r

( ) ( )GE's contribution to risk premium

GE's contribution to variance ( , ) ( , )

GE GE f GE f

GE GE M GE M

w E r r E r r

w Cov r r Cov r r


INVESTMENT 10

USING GE TEXT EXAMPLE CONTINUED

Reward-to-risk ratio for investment in market

portfolio:

Reward-to-risk ratios of GE and the market

portfolio:

And the risk premium for GE:

2

( )Market risk premium

Market variance

M f

M

E r r

2

( ) ( ( )

( , )

GE f M f

GE M M

E r r E r r

Cov r r

2

( , )( ) ( )GE M

GE f M f

M

Cov r rE r r E r r


INVESTMENT 11

EXPECTED RETURN-BETA RELATIONSHIP

CAPM holds for the overall portfolio because:

This also holds for the market portfolio:

P

( ) ( ) andP k k

k

k k

k

E r w E r

w

( ) ( )M f M M fE r r E r r


INVESTMENT 12

FIGURE 9.2 THE SECURITY MARKET LINE


INVESTMENT 13

FIGURE 9.3 THE SML AND A POSITIVE-ALPHA

STOCK


INVESTMENT 14

THE INDEX MODEL AND REALIZED RETURNS

To move from expected to realized returns—use

the index model in excess return form:

The index model beta coefficient turns out to be

the same beta as that of the CAPM expected

return-beta relationship

i i i M iR R e


INVESTMENT 15

FIGURE 9.4 ESTIMATES OF INDIVIDUAL

MUTUAL FUND ALPHAS, 1972-1991


INVESTMENT 16

THE CAPM AND REALITY

Is the condition of zero alphas for all stocks as

implied by the CAPM met

Not perfect but one of the best available

Is the CAPM testable

Proxies must be used for the market portfolio

CAPM is still considered the best available

description of security pricing and is widely

accepted


INVESTMENT 17

ECONOMETRICS AND THE EXPECTED

RETURN-BETA RELATIONSHIP

It is important to consider the econometric

technique used for the model estimated

Statistical bias is easily introduced

Miller and Scholes paper demonstrated how

econometric problems could lead one to

reject the CAPM even if it were perfectly valid


INVESTMENT 18

EXTENSIONS OF THE CAPM

Zero-Beta Model

Helps to explain positive alphas on low beta

stocks and negative alphas on high beta stocks

Consideration of labor income and non-traded

assets

Merton’s Multiperiod Model and hedge portfolios

Incorporation of the effects of changes in the

real rate of interest and inflation


INVESTMENT 19

EXTENSIONS OF THE CAPM CONTINUED

A consumption-based CAPM

Models by Rubinstein, Lucas, and Breeden

Investor must allocate current wealth between today’s consumption and investment for the

future


INVESTMENT 20

LIQUIDITY AND THE CAPM

Liquidity

Illiquidity Premium

Research supports a premium for illiquidity.

Amihud and Mendelson

Acharya and Pedersen


INVESTMENT 21

FIGURE 9.5 THE RELATIONSHIP BETWEEN

ILLIQUIDITY AND AVERAGE RETURNS


INVESTMENT 22

THREE ELEMENTS OF LIQUIDITY

Sensitivity of security’s illiquidity to market

illiquidity:

Sensitivity of stock’s return to market illiquidity:

Sensitivity of the security illiquidity to the market

rate of return:

1

( , )

( )

i ML

M M

Cov C C

Var R C

3

( , )

( )

i ML

M M

Cov C R

Var R C

2

( , )

( )

i ML

M M

Cov R C

Var R C


INVESTMENT 23

CAPM: EXAMPLES OF PRACTICAL

PROBLEMS 1

BAHATTIN BUYUKSAHIN, JHU

INVESTMENTS

Question:

The market price of a stock is $40.

Its expected rate of return is 13%.

The risk-free rate is 7%

The market risk premium is 8%.

Suppose its covariance w/ the market portfolio doubles (other variables are unchanged)?

Do you have enough information to find what will be the new price of the stock?

Assume that the stock is expected to pay a constant dividend in perpetuity.

1 /16 /2010 24


INVESTMENTS

Answer:

If the covariance of the security doubles, then so will its beta and its risk premium.

The current risk premium is 6% = 13% - 7%

the new risk premium would be twice as high: 12%

the new discount rate for the security would be 19% = 12% + 7%

If the stock pays a level perpetual dividend, then we know from the original data that:

Price = Dividend/Discount rate => $40 = D/0.13 => D = $5.20.

At the new discount rate of 19%, the stock would be worth only:

$5.20/0.19 = $27.37.

1 /16 /2010 25


PROBLEMS 2


INVESTMENTS

Question:

Consider the following table, which gives a security analyst’s expected return on two

stocks for two particular market returns:

Market Return Aggressive Stock Defensive Stock

-------------------------------------------------------------------------------------------

5% 2% 3.5%

20% 32% 14%

-------------------------------------------------------------------------------------------

What hurdle rate should be used by the management of the aggressive firm for a project

with the risk characteristics of the defensive firm’s stock?

The risk-free rate is 8%.

1 /16 /2010 26


PROBLEMS 3


INVESTMENTS

Answer:

The hurdle rate is determined by the project beta, 0.70, not by the firm’s beta.

The correct discount rate is 11.15%, the fair rate of return on Stock D (defensive). Why?

(a) The beta is the sensitivity of the stock return to the market return movements.

Then beta is the change in the stock return per change in the market return. Therefore:

A = (2 - 32)/(5 - 20) = 2.00

D = (3.5 - 14)/(5 - 20) = 0.70

(d) The defensive stock has a fair expected return of:

E(RD) = 8% + 0.7(12.5% - 8%) = 11.5%,

1 /16 /2010 27


PROBLEMS 4


INVESTMENTS

Question:

Two investment advisors are comparing performance.

One averaged a 19% rate of return and the other a 16% rate of return.

However, the beta of the first investor was 1.5, whereas that of the second was 1.

(a) Can you tell which investor was a better predictor of individual stocks (aside from the

issue of general movements in the market)?

(b) If the T-bill rate were 6% and the market return during the period were 14%, which

investor would be the superior stock selector?

(c) What if the T-bill rate were 3% and the market return were 15%?

1 /16 /2010 28


PROBLEMS 5


INVESTMENTS

Answer:

(a) We know that:

R1 = 19%, R2 = 16%, 1 = 1.5, and 1 = 1.

To tell which investor was a better predictor of individual stocks, we should look at their

abnormal return, which is the ex-post (alpha)

that is, the abnormal return is the difference between

the actual return

and the return predicted by the SML.

Without information about the parameters of this equation (risk-free rate and the market

rate of return) we cannot tell which investor is more accurate.

1 /16 /2010 29


PROBLEMS 6


INVESTMENTS

Answer:

(b) If Rf = 6% and Rm = 14%, then (using the notation of alpha for the abnormal return):

1 = 19% - [6% + 1.5(14% - 6%)] = 19% - 18% = 1%

2 = 16% - [6% + 1(14% - 6%)] = 16% - 14% = 2%.

Here, the second investor has the larger abnormal return

and thus he appears to be a more accurate predictor.

By making better predictions,

the second investor appears to have tilted his portfolio toward underpriced stocks.

1 /16 /2010 30


PROBLEMS 7


INVESTMENTS

Answer:

(c) If Rf = 3% and Rm = 15%, then

1 = 19% - [3% + 1.5(15% - 3%)] = 19% - 21% = -2%

2 = 16% - [3% + 1(15% - 3%)] = 16% - 15% = 1%.

1 /16 /2010


PROBLEMS 8

INDEX MODEL VS. CAPM


INVESTMENTS

Risk CAPM (theoretical, unobservable portfolio)

Index model (observable, “proxy” portfolio)

)(

),(

2M

Mii

R

RRCov

),(),( MiMiiMi ReRCovRRCov

)(0)(0 22MiMi RR

)(

),(

2M

Mii

R

RRCov

1 /16 /2010 32

INDEX MODEL VS. CAPM 2


INVESTMENTS

Beta Relationship

CAPM (no expected excess return for any security)

Index model (average realized alpha is 0)

Fig 10.3

)][(][ fMifi rrErrE

ifMiifi errrr )(

)][(][ fMiifi rrErrE

1 /16 /2010 33

MARKET MODEL


INVESTMENTS

Idea

use realized excess returns

Equivalence

CAPM + Market model = Index model

])[(][ MMiii rErrEr

)][(][ fMifi rrErrE

ifMiifi errrr )(

1 /16 /2010 34

SUMMARY


INVESTMENTS

CAPM

Factor model

Index model

Market model ])[(][ MMiii rErrEr

)][(][ fMifi rrErrE

ifMiifi errrr )(

iiii eFrEr ][

1 /16 /2010 35

CHAPTER 10

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

SINGLE FACTOR MODEL

Returns on a security come from two sources

Common macro-economic factor

Firm specific events

Possible common macro-economic factors

Gross Domestic Product Growth

Interest Rates


INVESTMENT 37

SINGLE FACTOR MODEL EQUATION

ri = Return for security I

= Factor sensitivity or factor loading or factor

beta

F = Surprise in macro-economic factor

(F could be positive, negative or zero)

ei = Firm specific events

( )i i i ir E r F e

i


INVESTMENT 38

MULTIFACTOR MODELS 1


INVESTMENTS

Necessity

CAPM

not practical

Index model

practical

unique factor is unsatisfactory

example: Table 10.2 (very small R2

)

Solution

multiple factors

1 /16 /2010 39

MULTI-FACTOR MODELS 2


INVESTMENTS

Factors in practice

business cycles factors

examples (Chen Roll Ross)

industrial production % change

expected inflation % change

unanticipated inflation % change

LT corporate over LT gvt. bonds

LT gvt. bonds over T-bills

interpretation

residual variance = firm specific risk

1 /16 /2010 40

MULTI-FACTOR MODELS 3


INVESTMENTS

Factors in practice

firm characteristics (Fama and French)

firm size

difference in return

between firms with low vs. high equity market value

proxy for business cycle sensitivity?

market to book

difference in return

between firms with low vs. high BTM ratio

proxy for bankruptcy risk?

1 /16 /2010 41

MULTIFACTOR MODELS 4

Use more than one factor in addition to market

return

Examples include gross domestic product,

expected inflation, interest rates etc.

Estimate a beta or factor loading for each

factor using multiple regression.


INVESTMENT 42

MULTIFACTOR MODEL EQUATION

ri = E(ri) + GDP GDP + IR IR + ei

ri = Return for security I

GDP= Factor sensitivity for GDP

IR = Factor sensitivity for Interest Rate

ei = Firm specific events

i

i

i

i


INVESTMENT 43

MULTIFACTOR SML MODELS

E(r) = rf + GDPRPGDP + IRRPIR

GDP = Factor sensitivity for GDP

RPGDP = Risk premium for GDP

IR = Factor sensitivity for Interest Rate

RPIR = Risk premium for Interest Rate

i i

i

i


INVESTMENT 44

ARBITRAGE PRICING THEORY (APT)


INVESTMENTS

Nature of arbitrage

APT

well-diversified portfolios

individual assets

APT vs. CAPM

APT vs. Index models

single factor

multi-factor

1 /16 /2010 45

ARBITRAGE PRICING THEORY

Arbitrage - arises if an investor can construct a

zero investment portfolio with a sure profit

Since no investment is required, an investor can

create large positions to secure large levels of

profit

In efficient markets, profitable arbitrage

opportunities will quickly disappear


INVESTMENT 46

APT & WELL-DIVERSIFIED PORTFOLIOS

rP = E (rP) + PF + eP

F = some factor

For a well-diversified portfolio:

eP approaches zero

Similar to CAPM,


INVESTMENT 47

FIGURE 10.1 RETURNS AS A FUNCTION OF

THE SYSTEMATIC FACTOR


INVESTMENT 48

FIGURE 10.2 RETURNS AS A FUNCTION OF

THE SYSTEMATIC FACTOR: AN ARBITRAGE

OPPORTUNITY


INVESTMENT 49

FIGURE 10.3 AN ARBITRAGE OPPORTUNITY


INVESTMENT 50

FIGURE 10.4 THE SECURITY MARKET LINE


INVESTMENT 51

APT applies to well diversified portfolios and not

necessarily to individual stocks

With APT it is possible for some individual stocks

to be mispriced - not lie on the SML

APT is more general in that it gets to an

expected return and beta relationship without

the assumption of the market portfolio

APT can be extended to multifactor models

APT AND CAPM COMPARED


INVESTMENT 52

MULTIFACTOR APT

Use of more than a single factor

Requires formation of factor portfolios

What factors?

Factors that are important to performance of

the general economy

Fama-French Three Factor Model


INVESTMENT 53

TWO-FACTOR MODEL

The multifactor APR is similar to the one-factor case

But need to think in terms of a factor portfolio

Well-diversified

Beta of 1 for one factor

Beta of 0 for any other

1 1 2 2( )i i i i ir E r F F e


INVESTMENT 54

EXAMPLE OF THE MULTIFACTOR APPROACH

Work of Chen, Roll, and Ross

Chose a set of factors based on the ability of

the factors to paint a broad picture of the

macro-economy


INVESTMENT 55

ANOTHER EXAMPLE:

FAMA-FRENCH THREE-FACTOR MODEL

The factors chosen are variables that on past evidence seem to predict

average returns well and may capture the risk premiums

Where:

SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess

of the return on a portfolio of large stocks

HML = High Minus Low, i.e., the return of a portfolio of stocks with a high

book to-market ratio in excess of the return on a portfolio of stocks with a

low book-to-market ratio

it i iM Mt iSMB t iHML t itr R SMB HML e


INVESTMENT 56

THE MULTIFACTOR CAPM AND THE APM

A multi-index CAPM will inherit its risk factors

from sources of risk that a broad group of

investors deem important enough to hedge

The APT is largely silent on where to look for

priced sources of risk


INVESTMENT 57

asset pricing models: their uses and their limitations - bahattin

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