chapter 11home.kelley.iupui.edu/rtperry/f523s2012/module04/chap1… · ppt file · web...

Return and Risk: The Capital Asset Pricing Model (CAPM)

Chapter 11

11-2

Calculate expected returns Understand covariances, correlations, and

betas Describe the impact of diversification Explain the systematic risk principle Plot the security market line Comprehend the risk-return tradeoff Utilize the Capital Asset Pricing Model

Key Concepts and Skills

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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)

11-4

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 FundRecession 33.3% -7% 17%Normal 33.3% 12% 7%Boom 33.3% 28% -3%

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Stock Fund Bond FundRate of Squared Rate of Squared

Scenario Return Deviation Return Deviation Recession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000Boom 28% 0.0289 -3% 0.0100Expected return 11.00% 7.00%Variance 0.0205 0.0067Standard Deviation 14.3% 8.2%

Expected Return

%11)(

%)28(31%)12(3

1%)7(31)(

S

S

rE

rE

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Variance

)0289.0001.0324(.310205.

0324.%)11%7( 2

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Standard Deviation

0205.0%3.14

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Covariance

Stock BondScenario Deviation Deviation Product WeightedRecession -18% 10% -0.0180 -0.0060Normal 1% 0% 0.0000 0.0000Boom 17% -10% -0.0170 -0.0057 Sum -0.0117 Covariance -0.0117

Deviation compares return in each state to the expected return.

Weighted takes the product of the deviations multiplied by the probability of that state.

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Correlation

998.0)082)(.143(.

0117.

),(

ba

baCov

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Correlations Interactive WSJ

http://online.wsj.com/article_email/SB10001424052748703859204575525941694618092-lMyQjAxMTAwMDEwMzExNDMyWj.html

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The Return and Risk for Portfolios

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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PortfoliosRate of Return

Scenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012

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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Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviationRecession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012


Portfolios

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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Portfolios

The variance of the rate of return on the two risky assets portfolio is

BSSSBB2

SS2

BB2P )ρσ)(wσ2(w)σ(w)σ(wσ

where BS is the correlation coefficient between the returns on the stock and bond funds.

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Portfolios

Observe the decrease in risk that diversification offers.

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

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Definition:

◦ The efficient set, or efficient frontier, is a graphical representation of a set of possible portfolios that: Minimize risk at specific return levels; and, Maximize returns at specific risk levels.

The Efficient Set

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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)

Por

tfol

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% 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%

The Efficient Set

100% stocks

100% bonds

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

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Portfolios with Various Correlations

100% bonds

retu

rn

100% stocks

= 0.2 = 1.0

= -1.0

Relationship depends on correlation coefficient-1.0 < < +1.0

If = +1.0, no risk reduction is possible

If = –1.0, complete risk reduction is possible

11-19

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

retu

rn

P

minimum variance portfolio

efficient frontier

Individual Assets

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In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills. Now investors can allocate their money across the T-bills and a balanced mutual fund.

Impact of a Riskless Security

100% bonds

100% stocks

rf

retu

rn

Balanced fund

CapitalMark

et Line

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Risk: Systematic 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. CREDIT CRISIS!

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Risk: Unsystematic 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.

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Diversification and Portfolio Risk Diversification can substantially reduce the

variability of returns without an equivalent reduction in expected returns.

This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another.

However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion.

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Diversifiable Risk The risk that can be eliminated by

combining assets into a portfolio Often considered the same as unsystematic,

unique, or asset-specific risk If we hold only one asset, or assets in the

same industry, then we are exposing ourselves to risk that we could diversify away.

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Total Risk Total risk = systematic 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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Portfolio Risk and Number of Stocks

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

Nondiversifiable risk; Systematic Risk; Market Risk

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

Portfolio risk

n

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Risk When Holding the Market Portfolio

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

Beta measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk).

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

)()(

2,

M

Mii R

RRCov

b

11-30

Estimating b with RegressionSe

curi

ty R

etur

ns

Return on market %

Ri = a i + biRm + ei

Slope =biCharacte

ristic

Line

11-31

Risk and Return (CAPM)

Expected Return on the Market:

• Expected return on an individual security:PremiumRisk Market FM RR

)(β FMiFi RRRR

Market Risk PremiumThis applies to individual securities held within well-diversified portfolios.

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Expected Return on a SecurityThis formula is called the Capital Asset Pricing Model (CAPM):

)(β FMiFi RRRR

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

Expected return on a security

= Risk-free rate + Beta of the

security × Market risk premium

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Relation Between Risk and Return

11-34

Relationship Between Risk & Return

Expe

cted

re

turn

b

%3FR

%3

1.5

%5.13

5.1β i %10MR%5.13%)3%10(5.1%3 iR

11-35

Summary statements regarding risk and return for a portfolio. The expected return on a portfolio is always a

weighted average of the expected returns on the portfolio's components.

The risk of a portfolio's return, as measured by standard deviation, is generally less than the weighted average of the risks of the portfolio’s components.

Risk generally declines when new assets are added to a portfolio. Risk declines more: ◦ The lower the new asset's standard deviation, ◦ The lower the correlations between the new asset's

payoffs and payoffs to the various existing assets. Beta measures the amount of Portfolio Risk

chapter 11home.kelley.iupui.edu/rtperry/f523s2012/module04/chap1… · ppt file · web...

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