1 Finance School of Management Chapter 13: The Capital Asset Pricing Model Objective The Theory of the CAPM Use of CAPM in benchmarking Using CAPM to determine

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FinanceFinance School of Management School of Management

Chapter 13: The Capital Asset Chapter 13: The Capital Asset Pricing ModelPricing Model

Objective• The Theory of the CAPM

• Use of CAPM in benchmarking• Using CAPM to determine

correct rate for discounting

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Chapter 13 ContentsChapter 13 Contents

The Capital Asset Pricing Model in Brief Determining the Risk Premium on the Market Portfolio Beta and Risk Premiums on Individual Securities Using the CAPM in Portfolio Selection Valuation & Regulating Rates of Return Modifications and Alternatives to the CAPM

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IntroductionIntroduction

CAPM is a theory about equilibrium prices in the markets for risky assets.

It is important because it provides– A justification for the widespread practice of passive

investing called indexing, and

– A way to estimate expected rates of return for use in evaluating stocks and projects.

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The Capital Asset Pricing Model in BriefThe Capital Asset Pricing Model in Brief

CAPM is an equilibrium theory based on the theory of portfolio selection.

The basic question:

What would risk premiums on securities be in equilibrium if people had the same set of forecasts of expected returns and risks, and all chose their portfolios optimally according to the principles of efficient

diversifications?

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Assumptions of CAPMAssumptions of CAPM

Assumption 1 (homogeneous in information processing)

Investors agree in their forecasts of expected rates of return, standard deviation, and correlations of the risky securities.

Assumption 2 (homogeneous in behavior)

Investors generally behave optimally according to the theory of portfolio selection.

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Intuitive of CAPM Intuitive of CAPM

All the investors will allocate their investments between the riskless asset and the same tangent portfolio.

In equilibrium, the aggregate demand for each security is equal to its supply.

The only way the asset market can clear is if the relative proportions of risky assets in tangent portfolio are the proportions in which they are valued in the market place, i.e. the market portfolio.

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Expected Return (%)

rf

The Capital Market Line (CML)The Capital Market Line (CML)

Standard Deviation

Lending

Borrowing

M

CML

Market Portfolio

●

E(rM)- rf

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Efficient Risk-RewardEfficient Risk-Reward

M

fMf

rrrr

])(E[)(E

In equilibrium, any efficient portfolio should be a combination of the market portfolio and the riskless asset.

The best risk-reward depends on how much the market-related risk a portfolio bears.

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Determining the Risk Premium on the Determining the Risk Premium on the Market PortfolioMarket Portfolio

The equilibrium risk premium on the market portfolio is the product of

– variance of the market, σ2M

– weighted average of the degree of risk aversion of holders of risk, A

2)( MfM ArrE

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Example: To Determine ‘A’Example: To Determine ‘A’

0.220.0

06.014.0

)(E)(E

,06.0,20.0,14.0)(E

2

22

A

rrAArr

rr

M

fMMfM

fMM

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Contribution of Security Contribution of Security ii to the to the Market RiskMarket Risk

i

iiMM rxr )(E)(E i

iMiMM x 2

iMM

fMfi

rrrr

2

])(E[)(E

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Team WorkTeam Work

To express the covariance between a risky security and the tangent portfolio, Cov(ri, rT) ,

in our example in Chapter 12, as the functions of rf , E[ri], E[rT], and σT .

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Security Market Line (SML)Security Market Line (SML)

MMi

iMiMx 1

2MiMiM

])(E[)(E fMiMfi rrrr

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Security Market Line (SML)Security Market Line (SML)

Risk =

Expected ReturnSML

rf

Market Portfolio

E(rM)

1

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A Simple Derivation of CAPMA Simple Derivation of CAPM

21pprUMax

Utility maximization – risk tolerance, risk-adjusted expected return

n

ifiifp rrwrr

1)(

i j

ijjip ww 2

16



jjijfi

i

wrrw

U0

2)(

j

fijij rrw )(2

)(2

tan RrΣw 1fr

17



)(2

)(fi

k

j

kjij rrw

MMj

fik

k

k

j

kj

k

kij

w

rrIwI

)()(2

1)( )(

– For investor k

– Aggregation

1( )

2iM jM ij M i fj

w r r

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iMM

fi rr 2

22M

MfM rr

)( fMiMfi rrrr

2MiMiM

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The Value Form of CAPMThe Value Form of CAPM

f

MfMMiii r

rrrVVV

1

/)]~[E)(~,~

(Cov]~

[E 21,1,

0,

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Security Prices

10

20

30

40

50

60

70

0.000 0.083 0.167 0.250 0.333 0.417 0.500 0.583 0.667 0.750 0.833 0.917 1.000

Years

Va

lue

Market_Price Stock_Z_Price

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Table of PricesTable of Pricesmonth Mkt_Price Z_Price hpr_Mkt hpr_Z

0 50.00 30.00 hpr_Mkt hpr_Z annual_cont_mktannual_cont_zreg_line1 55.84 33.87 11.68% 12.90% 132.55% 145.56% 176.82%2 52.87 33.65 -5.32% -0.64% -65.59% -7.75% -64.54%3 58.19 39.19 10.07% 16.47% 115.15% 182.98% 155.62%4 60.33 41.30 3.66% 5.38% 43.19% 62.90% 67.97%5 56.97 38.93 -5.57% -5.74% -68.71% -70.89% -68.35%6 51.52 34.20 -9.56% -12.15% -120.56% -155.40% -131.50%7 52.80 35.88 2.47% 4.91% 29.32% 57.54% 51.08%8 55.04 38.24 4.24% 6.56% 49.83% 76.22% 76.06%9 55.76 40.64 1.32% 6.28% 15.70% 73.08% 34.48%

10 62.20 46.26 11.55% 13.83% 131.12% 155.46% 175.09%11 56.84 41.01 -8.62% -11.34% -108.23% -144.43% -116.49%12 55.30 39.54 -2.71% -3.58% -32.93% -43.78% -24.76%

an_an_fact 1.105934 1.318151 mu 10.07% 27.62%an_cont_rate 0.10069 0.27623 sig 0.259099 0.325796

rho 0.968777beta 1.218157

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Regression of Returns of Z on Market

-200%

-150%

-100%

-50%

0%

50%

100%

150%

200%

-150% -100% -50% 0% 50% 100% 150%

Market Return

Ret

urn

on

Z

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Model and Measured Values of Model and Measured Values of Statistical ParametersStatistical Parameters

m m z z

modl 15% 20% 12% 25% 90% 1.13

Meas 10% 26% 28% 33% 97% 1.22

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Security Market Line Market Portfolio

-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)

Exp

ecte

d R

isk

Pre

miu

m

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The Beta of a PortfolioThe Beta of a Portfolio

When determining the risk of a portfolio – using standard deviation results in a formula that’s quite

complex

– using beta, the formula is linear

i

iinnp wwww ...2211

2

1

,1

2 2

jiijiiji

niiip www

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Return-Generating ProcessReturn-Generating Process

ifMiifi rrrr

Where

0E,0E iiMr

E 0 ( )i j i j

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MispricedMispriced

overpiced

undepicedrr

i

ieiii ,0

,0

＜＞

SMLReturn

.

rf

Market Portfolio

1Risk = i

)( fMiMfi rrrr

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Risk DecompositionRisk Decomposition

Total risk for a security = Market risk + Unique risk Since the unique risk can be diversified out, the

market compensates only for the market-related risk.

i i m i2 2 2 2

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Using the CAPM in Portfolio SelectionUsing the CAPM in Portfolio Selection

Passive portfolio management– Proxy for market portfolio

– indexing

Active portfolio management– Positive ALPHA

– Beat the market

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The Market Portfolio and Index FundsThe Market Portfolio and Index Funds

The most important implication of CAPM, that the market portfolio is an efficient portfolio, forms the theoretical basis for constructing proxies of the market portfolio—the index portfolio and index funds.

However, structuring an index portfolio from the primitive securities and making adjustments are inefficient and costly.

The efficient and cheap instruments, index futures contracts were not available until 1982.

In the U.S. market, the market value of outstanding index funds was only 6 million dollars in 1971.

After ten years, the value increased to 10 billion dollars. In 1992, the value increased to 270 billion dollars with about

1/3 pension funds being in the state of indexing.

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Valuation and Regulating Rates of ReturnValuation and Regulating Rates of Return

Discounted Cash Flow Valuation Models

– Suppose you are considering investing in a new project in the same industry of Betaful Corp.

– The market rate is 15%, and the risk-free rate is 5%.

– The beta of Betaful’s stock is 1.3.

– The capital structure of Betaful: 80% of equity, 20% of bond.

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Compute the beta of Betaful’s operations

04.1

0*20.03.1*80.0

bond

company

company

bondequityequitycompany ww


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Beta of Betaful’s operations is equal to the beta of the new project.

Applying the CAPM to find the required return on the new project.

%4.15

05.015.004.105.0

fMf rrrk


Cost of capital

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Assume that you start up a company which is just a vehicle for the new project, 60% capital is financed by issuing equity and 40% by issuing bond.

The beta of your unquoted equity is

73.1

0*40.0*60.004.1

bond

equity

equity

bondequityequitycompany ww


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Your company is all-equity financed.

Your company has an expected dividend of $6 next year, and that it will grow annually at a rate of 4% for ever, the value of a share is

63.52$04.0154.0

610

gk

DP


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Modifications and Alternatives to CAPMModifications and Alternatives to CAPM

Empirical testing of CAPM Modifications

– The proxy of market portfolio

– Market imperfection

– Multifactor Intertemporal CAPM (ICAPM): beta, sensitivity to changes in interest rates and in consumption good prices

Alternatives– Arbitrage Pricing Theory (APT)

Documents

1 Finance School of Management Chapter 13: The Capital Asset Pricing Model Objective The Theory of the CAPM Use of CAPM in benchmarking Using CAPM to determine