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RETURN AND RISK:

The Capital Asset Pricing Model

(BASED ON RWJJ CHAPTER 11)

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

Know how to calculate expected returns Understand covariance, correlation, and betas Understand the impact of diversification Understand the systematic risk principle Understand the security market line (SML) Understand the risk-return tradeoff Be able to apply the capital asset pricing mod-

el (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.

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Expected Return

3

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Variance

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Standard Deviation

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Covariance

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

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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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Portfolios

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

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Portfolios

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

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Portfolios

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

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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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

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The Efficient Set for Many Securities

Consider a world with many risky assets; we can still identify the opportuni-ty 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

Individual Assets

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Risk: Systematic and Unsystematic 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.

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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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

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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.

Riskless Borrowing and Lending

In addition to stocks and bonds, consider a world that also has a risk-free security like a T-bill.

Now investors can allocate money across the T-bills and one of the funds that offers a risk return combo’ on the efficient frontier.

rf

retu

rn

Efficient fund 1

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Riskless Borrowing and Lending

In addition to stocks and bonds, consider a world that also has a risk-free security like a T-bill.

Now investors can allocate money across the T-bills and one of the funds that offers a risk return combo’ on the efficient frontier.

rf

retu

rn

Efficient fund 1

Can borrow at risk-free rate,

invest own money and borrowed

money to get to these combinations

of risk & return

Riskless Borrowing and Lending

In addition to stocks and bonds, consider a world that also has a risk-free security like a T-bill.

Now investors can allocate money across the T-bills and one of the funds that offers a risk return combo’ on the efficient frontier.

rf

retu

rn

Efficient fund 1Efficient fund 2

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Riskless Borrowing and Lending

In addition to stocks and bonds, consider a world that also has a risk-free security like a T-bill.

Now investors can allocate money across the T-bills and one of the funds that offers a risk return combo’ on the efficient frontier.

rf

retu

rn

Efficient fund 1Efficient fund 2

Effic. Fund 3

Riskless Borrowing and Lending

In addition to stocks and bonds, consider a world that also has a risk-free security like a T-bill.

Now investors can allocate money across the T-bills and one of the funds that offers a risk return combo’ on the efficient frontier.

rf

retu

rn

Balanced fund

Capital Market

Line

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Market Equilibrium

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. Just where the investor chooses along the Capital Market Line depends on his risk tolerance. The big points are that all investors have the same CML and use the same ‘M’.

retu

rn

P

rf

M

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

() 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.

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Estimating with Regression

Sec

uri

ty R

etu

rns

%

Return on market %

Ri = i + iRm + ei

Slope = i

Estimating of the Market

Ret

urn

on

mar

ket

%

Return on market %

The beta of the market is 1

Slope =

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Estimating of Risk-Free Asset

Ris

k-f

ree

Ret

. %

Return on market %

The beta of a risk-free asset is 0

Slope =

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Risk and Return (CAPM)

Expected Return on the Market:

• Expected return on an individual security:

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):

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

Expected return on a security

=Risk-

free rate+

Beta of the security

×Market risk

premium

Ticky-tack Technically,Should be Required

Return

Examples: RM = 8.0% & RF = 2.0%

Market Risk Premium = 6.0% Thought of as the extra “reward” (extra return) that

you should be able to require for taking on however much market risk the market has

Boeing has a β = 1.1 How much “extra” return? 1.1 x ( 8.0% − 2.0% ) E(rBOE) = 2.0% + 1.1 x ( 8.0% − 2.0% ) = 8.6%

Axiall has a β = 2.4 How much “extra” return? 2.4 x ( 8.0% − 2.0% ) E(rAXI) = 2.0% + 1.1 x ( 8.0% − 2.0% ) = 16.4%

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Relation Between Risk and Return

SECURITYMARKET

LINE

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 if: the new asset's standard deviation is lower, the correlations between the new asset's payoffs and the

various existing assets’ payoffs are lower.

Beta measures the amount of Portfolio Risk