Capital Asset Pricing Theory[1]Capem

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MS 60:Financial Markets & Investment Analysis Week VI:

The Capital Asset Pricing Model

Copyright Prentice Hall Inc. 1999. Author: Nick Bagley


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Key Topics to Cover

-The Capital Asset Pricing Model in Brief

- Determining the Risk Premium on the MarketPortfolio

- Beta and Risk Premiums on IndividualSecurities

-Using the CAPM in Portfolio Selection

-CAPM & Valuation


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INTRODUCTION ( cont)

Key question it seeks to answer: Whatwould risk premiums on securities be inequilibrium if people had the same set of

forecasts of expected returns and risks andall chose their portfolios optimally accordingto the principles of diversification

Market does not reward investors for holdinginefficient portfolios, hence rewards only asecuritys contribution to risk of an efficiently

diversified portfolio


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BETA () & RISK PREMIUMS ON

SECURITIES (cont)

CAPM:E(rj) = rf + j*( E(rm) rf ), rj = return on asset i, Rf is risk free rate, rm

is the return on the market and Bj is the correlation between the returnon asset j and the markets return.

Formula for risk premium in CAPM =

E(rj) rf = j*( E(rm) rf )

This is called the Security Market Line (SML)

Slope of the SML is the risk premium on the market portfolio. Hence if

the risk premium is 0.8 then E(rj)rf = 0.8* j

According the the CAPM, in equilibrium, the risk premium on anyasset is equal the product of

b(or Beta)

the risk premium on the market portfolio


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Comment: b = 1

A security with a b = 1 on average risesand falls with the market

a 10% (say) unexpected rise (fall) in themarket return premium will, on average, resultin a 10% rise (fall)in the securitys returnpremium


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Comment: b 1

A security with a b 1 on average risesand falls more than the market

With a b = 1.3, a 10% (say) unexpected rise

(fall) in the market return premium will, onaverage, result in a 13% rise (fall) in thesecuritys return premium

Such a security is said to be aggressive


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Comment: b 1

A security with a b 1 on average risesand falls less than the market

With a b = 0.7, a 10% (say) unexpected rise

(fall) in the market return premium will, onaverage, result in a 7% rise (fall) in thesecuritys return premium

Such a security is said to be defensive


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Security Market Line

The plot of a securitys risk premium (or sometimes

security returns) against security beta

Note that the slope of the security market line is themarket premium

By CAPM theory, all securities must fall precisely onthe SML (hence its name)


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Security Market LineWith A Zero-Beta Portfolio

E(R)

E(Rm)

bi

SML

M

0.0 1.0

E(Rz)

E(Rm) - E(Rz)


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Security Market Line MarketPortfolio

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Beta (Risk)

ExpectedRiskP

remium


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Relationship Between

Systematic Risk and Return Sharpe and Cooper: positive, but non-linear

Douglas: intercept higher than the risk-free rate

Miller and Scholes: possible error in Douglasfindings

Black, Jensen, and Scholes: positive linear

relationship between monthly excess return andportfolio beta

Fama and McBeth: supported the CAPM withthe intercept equal to the RFR


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Relationship Between

Systematic Risk and Return

Effect of skewness on the relationship

preference for high risk and returns

Effect of size, P/E and leverage Effect of book-to-market value

The Fama-French Study


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Observation

All securities, (not just efficient portfolios)plot onto the SML, if they are correctlypriced according to the CAPM


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The Beta of a Portfolio

When determining the risk of a portfolio

using standard deviation results in a formulathats quite complex

using beta, the formula is linear

i

rirnrrrwrwrw innnwwww bbbbb ...

212211 21...

2

1

,

,1

2

... 22211

ji

jirrji

ni

rirwrwrw jiinnwww


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Computing Beta

Here are some useful formulae forcomputing beta

fM

fr

i

M

Mii

M

MiMi

M

Mi

Mii

rri

b

bb

,

2

,

2

,

,

CAPM & PORTFOLIO


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CAPM & PORTFOLIO

SELECTION

CAPM provides a rationale for a simple passive portfoliostrategy:

Diversify your holdings of risky assets in the proportions ofthe market portfolio

Mix this portfolio with the risk-free asset to achieve adesired risk-reward combination

Provides a basis of determining, given a desired rate of return,what is the necessary risk that an investor has to accept

Provides a risk-reward benchmark to evaluate investmentadvisors, i.e. relate returns on portfolio to its riskiness. UseSML


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Using the CAPM in PortfolioSelection

Whether or not CAPM is a valid theory,indexing is attractive to investors because

historically it has performed better than mostactively managed portfolios

it costs less to implement that active

management


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The Portfolio Manager

The further a well diversified portfolio

consistently lies above (below) the SML,the better (worse)the fund managersperformance


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Valuation Using CAPM

Beta may be used to obtain the discount factor/Required Returnon Capital for a new project as follows:

Assume Patty Shop is financed by 20% short-term debt, and80% equity, and its b is 1.3 (assume debt is risk-free)

Its optimal capital structure is 40% (risk-free) debt, and 60%equity


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VALUATION & REGULATING RATESOF RETURN -USES

From Discounted cash flow valuation models:

P0(Equity) = d1 / ( r g) where, r, can be expressed as: r = rf +*((rm)rf)

Cost of equity capital: E(ri) = rf + i*(rm)rf)

CAPM can be used to determine a fair rate of return Assume the market rate is 15%, and the risk-free rate is 5%

Lets First Compute the Beta for Patty Shop as follows:

04.1

0*20.03.1*80.0

bond

company

company

bondequityequitycompany ww

b

b

bbb


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Valuation Using CAPM

The Beta of Patty Shop is equal to the beta of the newProject

To find the required return on the new project, apply theCAPM

%4.1505.015.004.105.0

fmf rrrKnew b


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MODIFICATION & ALTERNATIVES

TO CAPM Why deviations to simple CAPM model:

Poor data

Market imperfections Need more realistic model ( Intertemporal CAPM )

Alternative models:Arbitrage Pricing Model(APT)

Use a number of variables, e.g. inflation, economic growth

etc. to derive a model based on more complex variables


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Relaxing the Assumptionsof the CAPM

Heterogenous expectations If all investors have different expectations about

risk and return, each would have a unique CML

and/or SML, and the composite graph would bea band of lines with a breadth determined by thedivergence of expectations

Planning periods

CAPM is a one period model, and the periodemployed should be the planning period for theindividual investor, which will vary by individual,

affecting both the CML and the SML


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Criticism of CAPM by Richard Roll

Limits on tests: only testable implicationfrom CAPM is whether the market portfoliolies on the efficient frontier

Range of SMLs - infinite number ofpossible SMLs, each of which produces a

unique estimate of beta


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Market efficiency effects - substituting aproxy, such as the S&P 500 creates twoproblems

Proxy does not represent the true marketportfolio

Even if the proxy is not efficient, the market

portfolio might be


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Conflicts between proxies - differentsubstitutes may be highly correlatedeven though some may be efficient and

others are not, which can lead todifferent conclusions regarding betarisk/return relationships

So, CAPM is not testable - but it still hasvalue and must be used carefully

Stephen Ross devised an alternativeway to look at asset pricing - APT


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Arbitrage Pricing Theory - APT

Arbitrage is a process of buying a lowerpriced asset and selling a higher pricedasset, both of similar risk, and capturing thedifference in arbitrage profits

The general arbitrage principle states thattwo identical securities will sell at identicalprices

Price differences will immediately disappearas arbitrage takes place


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Arbitrage Pricing Theory- APT


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Three major assumptions:1. Capital markets are perfectlycompetitive

2. Investors always prefer more wealthto less wealth with certainty

3. The stochastic process generating

asset returns can be expressed as alinear function of a set of Kfactors orindexes


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NibbbERi ikikiii to1for2211

assetsofnumbererror)(randomreturns'assetoneffectuniquea

assetsallofreturnstheinfluencesmean that zeroawithindexesorfactorscommonofseta

indexorfactorcomonainmovementstoreturnss'assetinreaction

changeszerohaveindexes orfactorstheallifassetforreturnexpected

periodtimespecifiedaduringassetonreturn

Ni

KKib

iEiRi

i

k

ik

i


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Roll-Ross Study

1. Estimate the expected returns and thefactor coefficients from time-series data onindividual asset returns

2. Use these estimates to test the basiccross-sectional pricing conclusion impliedby the APT


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Extensions of theRoll-Ross Study

Cho, Elton, and Gruber examined thenumber of factors in the return-generatingprocess that were priced

Dhrymes, Friend, and Gultekin (DFG)reexamined techniques and theirlimitations and found the number of factors

varies with the size of the portfolio


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The APT and Anomalies

Small-firm effect

Reinganum - results inconsistent with the APT

Chen - supported the APT model over CAPM

January anomaly

Gultekin - APT not better than CAPM

Burmeister and McElroy - effect not captured by

model, but still rejected CAPM in favor of APT

APT and inflation

Elton, Gruber, and Rentzler - analyzed real returns


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The Shanken Challenge toTestability of the APT

If returns are not explained by a model, it is notconsidered rejection of a model; however if the factors doexplain returns, it is considered support

APT has no advantage because the factors need not beobservable, so equivalent sets may conform to differentfactor structures

Empirical formulation of the APT may yield differentimplications regarding the expected returns for a given

set of securities Thus, the theory cannot explain differential returns

between securities because it cannot identify the relevantfactor structure that explains the differential returns


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Alternative Testing Techniques

Jobson proposes APT testing with amultivariate linear regression model

Brown and Weinstein propose using abilinear paradigm

Others propose new methodologies

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