ALL RIGHTS RESERVED No part of this document may be reproduced without written approval from Limkokwing University of Creative Technology 1-1 Chapter 4

ALL RIGHTS RESERVEDNo part of this document may be reproduced without written approval from Limkokwing University of Creative Technology

1-1

Chapter 4

Portfolio Theory & Capital Asset Pricing Model


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1. How do we measure the risk/return tradeoff for an individual security?• We calculate the expected return on the

security and the standard deviation/variance of this return

2. How does the risk/return relationship change when we combine two different securities into a portfolio? • This depends on the covariance between the

returns on the two securities • We can use this to calculate the feasible set

and the efficient frontier of the risk/return outcomes


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4. What happens when we combine many different securities into a portfolio? • The risk of the portfolio is more dependent

on the covariance between the securities than on their variances

• The total risk of any individual security can be separated into its portfolio (systematic) risk and its idiosyncratic (diversifiable) risk


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5. What happens when we are able to combine a portfolio of risky assets with risk-free bonds? • We find there is a single optimal portfolio of

risky assets to hold; this is the same for all investors if they have homogeneous expectations

• Individual investors satisfy their risk preference by choosing how much to invest in this market portfolio relative to the riskless asset


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6. How do we measure the risk of an individual security when investors hold a diversified portfolio of securities (the market portfolio)? • Because investors are well-diversified, they

do not care about the diversifiable risk of an individual security

• Instead, they care about what an individual security contributes to the risk of the market portfolio

• This contribution is measured by the security’s beta


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7. How can we use all of this to determine the return investors require from a security given its risk (beta)? • The expected return on the market as a

whole can be separated into the risk-free rate and a risk premium

• The expected return on any individual security is the risk-free rate, plus the market risk premium times the security’s beta

• This relationship between risk and return is known as the capital asset pricing model (CAPM)


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Expected Return, Variance, and Covariance

Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund.

Rate of ReturnScenario Probability Stock fund Bond fund

Recession 33.3% -7% 17%Normal 33.3% 12% 7%

Boom 33.3% 28% -3%


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Stock fund Bond Fund

Rate of Squared Rate of Squared

Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00%Normal 12% 0.01% 7% 0.00%Boom 28% 2.89% -3% 1.00%Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2%


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%11)(

%)28(31%)12(3

1%)7(31)(

S

S

rE

rE


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%7)(

%)3(31%)7(3

1%)17(31)(

B

B

rE

rE


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%24.3%)7%11( 2


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%01.%)12%11( 2


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%89.2%)28%11( 2


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%)89.2%01.0%24.3(3

1%05.2


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0205.0%3.14


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The relationship between the returns on the two funds is measured by their covariance:

i

BiBSiSiBSSB RRRRwRRCov )()(),(


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Like the variance of an individual security’s return, the covariance between the returns of two securities is difficult to interpret because it is in squared units.

The correlation coefficient normalizes this measure by dividing the covariance by the standard deviations of the two securities:

The correlation coefficient always lies between -1 and +1

)()(

),(),(

BS

BSBSSB RSDRSD

RRCovRRCorr


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The Return and Risk for PortfoliosStock fund Bond Fund



Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.


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The Return and Risk for Portfolios

Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.160%Normal 12% 7% 9.5% 0.003%Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

SSBBP rwrwr

%)17(%50%)7(%50%5


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%)7(%50%)12(%50%5.9

SSBBP rwrwr


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%)3(%50%)28(%50%5.12

SSBBP rwrwr


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The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

%)7(%50%)11(%50%9

)()()( SSBBP rEwrEwrE


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Notice, however, that the variance of the portfolio is not equal to the weighted average of the variances:

0136.00067.0%500205.0%50 2Pσ

This is because the two funds do not move in lock step with one another.


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Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.


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The Efficient Set for Two Assets

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)Po

rtfol

io R

etur

n

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%

50.00% 3.08% 9.00%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

We can consider other portfolio weights besides 50% in stocks and 50% in bonds …


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The Efficient Set for Two Assets

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

Portf

olio R

etur

n

% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These comprise the

efficient frontier.


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Two-Security Portfolios with Various Correlations

100% bonds

retu

rn

100% stocks

= 0.2

= 1.0

= -1.0


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Portfolio Risk/Return Two Securities: Correlation Effects

Relationship depends on correlation coefficient

-1.0 < < +1.0

The smaller the correlation, the greater the risk reduction potential

If= +1.0, no risk reduction is possible


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Portfolio Risk as a Function of the Number of Stocks in the Portfolio

Nondiversifiable risk; Systematic Risk; Market Risk

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

n

Thus diversification can eliminate some, but not all of the risk of individual securities.

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


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The Efficient Set for Many Securities

retu

rn

P

retu

rn

P

Individual Assets

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


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retu

rn

P

minimum variance portfolio

Individual Assets

Given the opportunity set we can identify the minimum variance portfolio


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retu

rn

P

minimum variance portfolio

efficient frontier

Individual Assets

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


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Optimal Risky Portfolio with a Risk-Free Asset

100% bonds

100% stocks

retu

rn

efficient frontier

rf

In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills


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Market Equilibrium

retu

rn

P

efficient frontier

rf

CML

100% bonds

100% stocksM

With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors.


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The Separation Property

retu

rn

P

efficient frontier

rf

MCM

L

The Separation Property states that the market portfolio, M, is the same for all investors—they can separate their risk aversion from their choice of the market portfolio.


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retu

rn

P

efficient frontier

rf

CML

M

Investor risk aversion is revealed in their choice of where to stay along the capital allocation line—not in their choice of the line.


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Market Equilibrium

100% bonds

100% stocks

rf

retu

rn

Balanced fund

CML

Just where the investor chooses along the capital market line depends on his risk tolerance. The big point is that all investors have the same CML.


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Market Equilibrium

100% bonds

100% stocks

rf

retu

rn

Optimal Risky Porfolio

CML

All investors have the same CML because they all have the same optimal risky portfolio given the risk-free rate.


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100% bonds

100% stocks

rf

retu

rn

Optimal Risky Porfolio

CML

The separation property implies that portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML.


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Optimal Risky Portfolio with a Risk-Free Asset

100% bonds

100% stocks

retu

rn

First Optimal Risky Portfolio

Second Optimal Risky Portfolio

CML 0 CML 1

0fr

1fr

By the way, the optimal risky portfolio depends on the risk-free rate as well as the risky assets.


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Definition of Risk When Investors Hold the Market Portfolio

Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta () of the security.

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

)(

)(2

,

M

Mii R

RRCov


1-42

Estimating b with regression

Sec

urity

Ret

urns

Sec

urity

Ret

urns

Return on Return on market %market %

RRii = = ii + + iiRRmm + + eeii

Slope = Slope = iiChara

cteris

tic Line

Characte

ristic

Line


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Estimates of for Selected Stocks

Stock Beta

Bank of America 1.55

Borland International 2.35

Travelers, Inc. 1.65

Du Pont 1.00

Kimberly-Clark Corp. 0.90

Microsoft 1.05

Green Mountain Power 0.55

Homestake Mining 0.20

Oracle, Inc. 0.49


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The Formula for Beta

)(

)(2

,

M

Mii R

RRCov

Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio.


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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 an Individual Security

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

)(β FMiFi RRRR

• If i = 0, then the expected return is RF.• If i = 1, then

Expected return on a security

=Risk-

free rate+

Beta of the security

×Market risk

premium


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Relationship Between Risk & Expected Return

Exp

ecte

d re

turn

)(β FMiFi RRRR

FR

1.0

MR

)(β FMiFi RRRR


1-48

Relationship Between Risk & Expected Return

Exp

ecte

d re

turn

%3FR

%3

1.5

%5.13

5.1β i %10MR

%5.13%)3%10(5.1%3 iR


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Summary and Conclusions

This chapter sets forth the principles of modern portfolio theory.

The expected return and variance on a portfolio of two securities A and B are given by

ABAB2

BB2

AA2P w2w)σ(w)σ(wσ

)()()( BBAAP rEwrEwrE

• By varying wA, one can trace out the efficient set of portfolios. We graphed the efficient set for the two-asset case as a curve, pointing out that the degree of curvature reflects the diversification effect: the lower the correlation between the two securities, the greater the diversification.

• The same general shape holds in a world of many assets.


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retu

rn

P

rf

MM

CML

The efficient set of risky assets can be combined with riskless borrowing and lending. In this case, a rational investor will always choose to hold the portfolio of risky securities represented by the market portfolio.

• Then with borrowing or lending, the investor selects a point along the CML.


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The contribution of a security to the risk of a well-diversified portfolio is proportional to the covariance of the security's return with the market’s return. This contribution is called the beta.

• The CAPM states that the expected return on a security is positively related to the security’s beta:

)(

)(2

,

M

Mii R

RRCov

)(β FMiFi RRRR

Documents

ALL RIGHTS RESERVED No part of this document may be reproduced without written approval from Limkokwing University of Creative Technology 1-1 Chapter 4