chap010

McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved

CHAPTER

10 Return and RiskThe Capital Asset

Pricing Model (CAPM)

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved

McGraw-Hill/Irwin

Chapter Outline10.1 Individual Securities10.2 Expected Return, Variance, and Covariance10.3 The Return and Risk for Portfolios10.4 The Efficient Set for Two Assets10.5 The Efficient Set for Many Assets10.6 Diversification: An Example10.7 Riskless Borrowing and Lending10.8 Market Equilibrium10.9 Relationship between Risk and Expected

Return (CAPM)


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• An individual who holds one security should use expected return as the measure of the security’s return. Standard deviation or variance is the proper measure of the security’s risk.

• An individual who holds a diversified portfolio cares about the contribution of each security to the expected return and the risk of the portfolio.


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10.1 Individual Securities

• The characteristics of individual securities that are of interest are the:– Expected Return– Variance and Standard Deviation– Covariance and Correlation (to another

security or index)

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CHAPTER

10

10.2Expected Return, Variance,

and Covariance


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• Expected Return• Variance (Standard Deviation): the

variability of individual stocks.

Variance

σA2 = Expected value of (RA-RA)2

Standard Deviation = σA


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• Covariance and Correlation : the relationship between the return on one stock and the return on another.– σAB = Cov (RA, RB)

=Expected value of 【 (RA-RA)*(RB-RB) 】

– ρAB= Corr(RA, RB)= σAB / (σA * σB)


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10.3 The Return and Risk for Portfolios

• How does an investor choose the best combination or portfolio of securities to hold?

• An investor would like a portfolio with a high expected return and low standard deviation of return.


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The Expected Return on a Portfolio

• It is simply a weighted average of the expected returns on the individual securities.

Expected return on portfolio = XARA + XBRB


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The Variance of the Portfolio

• Var (portfolio)

= XA2σA

2 + 2XAXB σAB + XB2 σB

2

• The variance of a portfolio depends on both the variances of the individual securities and the covariance between the two securities.


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Diversification Effect

• As long as ρ<1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities.


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An Extension to Many Assets

• The diversification effect applies to a portfolio of many assets.


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10.4 The Efficient Set for Two Assets

• There are a few important points concerning Figure 10.3: – The diversification effect occurs

whenever the correlation between the two securities is below 1.

– The point MV represents the minimum variance portfolio.

– An individual contemplating an investment in a portfolio faces an opportunity set or feasible set.


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– The curve is backward bending between the Slowpoke point and MV.

– The curve from MV to Supertech is called the efficient set.


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• Figure 10.4 shows that the diversification effect rises as ρ declines.

• Efficient sets can be calculated in the real world.

• In Figure 10.5, the backward bending curve is important information. Some subjectivity must be used when forecasting future expected returns.


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10.5 The Efficient Set for Many Securities

Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.

retu

rn

P

Individual Assets


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The Efficient Set for Many Securities

The section of the opportunity set above the minimum variance portfolio is the efficient frontier or fficient set.

retu

rn

P

minimum variance portfolio

efficient frontier

Individual Assets


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10.6 Diversification: An Example

• Suppose that we make the following three assumptions: – All securities posses the same variance. – All covariances in table 10.4 are the same.– All securities are equally weighted in the

portfolio.


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• The variances of the individual securities are diversified away, but the covariance terms cannot be diversified away.

• There is a cost to diversification, so we need to compare the costs and benefits of diversification.


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Total Risk

• Total risk = systematic risk (portfolio risk) + unsystematic risk

• The standard deviation of returns is a measure of total risk.

• For well-diversified portfolios, unsystematic risk is very small.

• Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.


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Systematic Risk

• Risk factors that affect a large number of assets

• Also known as non-diversifiable risk or market risk

• Includes such things as changes in GDP, inflation, interest rates, etc.


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Unsystematic (Diversifiable) Risk

• Risk factors that affect a limited number of assets

• Also known as unique risk and asset-specific risk

• Includes such things as labor strikes, part shortages, etc.

• The risk that can be eliminated by combining assets into a portfolio


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10.7 Riskless Borrowing and Lending

• A risky investment and a riskless or risk-free security.

• The Optimal Portfolio – Capital Market Line(CML)


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– Separation Principle: The investor makes two separate decisions: • Calculates the efficient set of risky assets,

represented by curve XAY in F.10.9. He then determines point A, which represents the portfolio of risky assets that the investors will hold.

• Determines how he will combine point A, his portfolio of risky assets, with the riskless asset. His choice here is determined by his internal characteristics, such as his ability to tolerate risk.


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10.8 Market Equilibrium


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Definition of the Market-Equilibrium Portfolio

• In a world with homogeneous expectations, all investors would hold the portfolio of risky assets represented by point A. It is the market portfolio.

• A broad-based index is a good proxy for the highly diversified portfolios of many investors.

• The best measure of the risk of a security in a large portfolio is the beta of the security.


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The Formula for Beta

)(

)(2

,

M

Mii R

RRCov


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Definition of Risk When Investors hold the Market Portfolio

• Beta measures the responsiveness of a security to movements in the market portfolio.

• F.10.10 tells us that the return on Jelco are magnified 1.5 times over those of the market.


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• While very few investors hold the market portfolio exactly, many hold reasonably diversified portfolios. These portfolios are close enough to the market portfolio so that the beta of a security is likely to be a reasonable measure of its risk.


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10.9 Relationship between Risk and Expected Return (CAPM)

• Expected Return on the Market:

• Expected return on an individual security:

PremiumRisk Market FM RR

)(β FMiFi RRRR

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.


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Expected Return on a Security

• This formula is called the Capital Asset Pricing Model (CAPM):

)(β FMiFi RRRR

• Assume i = 0, then the expected return is RF.• Assume i = 1, then Mi RR

Expected return on a security

=Risk-

free rate+

Beta of the security

×Market risk

premium


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Expected Return on Individual Security (1/2)

• Capital-Asset-Pricing Model(CAPM) : R = RF + β * (RM - RF) (10.17)

– Linearity: In equilibrium, all securities would be held only when prices changed so that the SML became straight.

– Portfolios as well as securities. • The beta of the portfolio is simply a weighted

average of the securities in the portfolio.


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Expected Return on Individual Security (2/2)

– F.10.11 differs from F.10.9 in at least two ways:

• Beta appears in the horizontal axis of F.10.11, but

standard deviation appears in the horizontal axis of

F.10.9.

• The SML(F.10.11) holds for all individual securities

and for all possible portfolios, whereas

CML(F.10.9) holds only for efficient portfolios.

26

25

40

CML

Var(RP) = σp2 = a2σm

2

σp = a*σm

a = σp / σm (代入 E(RP) 中 )

E(RP) = (1-a) Rf + a*E(Rm) , a>0

E(RP) = Rf + (σp / σm)*[E(Rm)-Rf]

= Rf + {[E(Rm)-Rf] / σm } * σp

riskless asset market portfolio


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• Var (portfolio)

= XA2σA

2 + 2XAXB σAB + XB2 σB

2

=XA2σA

2 + 2XAXB ρAB σA * σB + XB2 σB

2

(ρAB = σAB / σA * σB )

=XA2σA

2 + 2XAXB σA * σB + XB2 σB

2

(when ρAB =1)

= (XAσA + XB σB) 2

• Standard Deviation = (XAσA + XB σB)

11

10


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CML

Var(RP) = σp2 = a2σm

2

σp = a*σm

a = σp / σm (代入 E(RP) 中 )

E(RP) = (1-a) Rf + a*E(Rm) , a>0

E(RP) = Rf + (σp / σm)*[E(Rm)-Rf]

= Rf + {[E(Rm)-Rf] / σm } * σp

riskless asset market portfolio


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βp=σpm

σm2

=E{ [W1R1+W2R2–W1E(R1)–W2E(R2)]*[Rm–E(Rm)]}

σm2

=E{ [W1(R1–E(R1) +W2(R2–E(R2)]*[Rm–E(Rm)]}

σm2

=E [W1(R1–E(R1)]*[Rm–E(Rm)]

σm2 +

E [W2(R2–E(R2)]*[Rm–E(Rm)]

σm2

= +σm

2W1

σ1m

σm2

W2

σ2m

= W1 β1 + W2 β2


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E(RA) = 15.0%

E(RZ) = 8.6%

E(RP) = 0.5*15.0% + 0.5*8.6%

= 11.8%

Beta of Portfolio:

= 0.5*1.5 + 0.5*0.7

= 1.1

Under the CAPM, the E(RP) is

E(RP) = 3% + 1.1*8.0% = 11.8%


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σp2 = E [RP–E(RP)]2

= E [(W1R1+W2R2) – (W1E(R1) + W2E(R2))]2

= E [(W1(R1 – E(R1)) + W2(R2 – E(R2)]2

=

=

W12σ1

2 + W22σ2

2 + 2W1W2E [(R1 – E(R1))*(R2 – E(R2)]

W12σ1

2 + W22σ2

2 + 2W1W2 σ12

= Σ Wi2σi

2 + Σ Σ WiWj σiji=1

2

i=1

2

j=1

2

i≠j

W12σ1

2

W22σ2

2

W1W2 σ12

W2W1σ21

設 Portfolio中只有兩種資產

設 Portfolio中只有兩種資產


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σp2 = Σ Xi

2σi2 + Σ Σ XiXj σij

i=1

N

i=1

N

j=1i≠j

N

= NN2

1Var + N(N – 1)

N2

1COV

N1

Var +N(N – 1)

N2COV =

N1

Var + = COV(N – 1)

N

= COV (when N→∞)

(10.10)


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Portfolio Risk and Number of Stocks

Nondiversifiable risk; Systematic Risk; Market Risk

Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk

n

In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not.

Portfolio risk

chap010

Economy & Finance

x b r b

r b r b ab

er p er p

portfolio w

e r p er p

x b ab

w i w j ij

a0 er p