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Topic 4 (Ch. 9) The Capital Asset Pricing Model (CAPM)

The CAPM

The market portfolio

The capital market line

The risk premium on the market portfolio

Expected returns on individual securities

The security market line Some extensions of the CAPM

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The CAPM

An equilibrium model specifying the relationship between risk and expected return on risky assets.

Assumptions:• Investors are price-takers (i.e. their trades do

not affect security prices).• Investors have a single-period investment

horizon.

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• Investments are limited to publicly traded financial assets (e.g. stocks & bonds), and to risk-free borrowing or lending arrangements.

• Investors pay no taxes on returns and no transaction costs (commissions and servicecharges) on trades in securities.

• All investors are rational mean-variance optimizers.

• All investors have the same expectations (i.e. identical estimates of expected returns, variances, and covariances among all assets).

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Model implication 1: The market portfolio

All investors will choose to hold a portfolio of risky assets in proportions that duplicate representation of the assets in the market portfolio (M), which includes all traded assets.

The proportion of each asset in the market portfolio equals the market value of the asset divided by the total market value of all assets.

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Why?

All investors arrive at the same determination of the optimal risky portfolio, the portfolio on the efficient frontier identified by the tangency line from T-bills to that frontier.

As a result, the optimal risky portfolio of all investors is simply the market portfolio.

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Model implication 2: The capital market line (CML)

Not only will the market portfolio be on the efficient frontier, but it also will be thetangency portfolio to the optimal capital allocation line (CAL) derived by every investor.

As a result, the capital market line (CML), the line from the risk-free rate through the market portfolio, M, is also the best attainable capital allocation line.

All investors hold M as their optimal risky portfolio, differing only in the amount invested in it versus in the risk-free asset.

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Mutual fund theorem:

Since all investors choose to hold a market index mutual fund, we can separate portfolio selection into 2 components:

• a technological problem: creation of market index mutual funds by professional managers;

• a personal problem: depends on an investor’s risk aversion, allocation of the complete portfolio between the mutual fund and risk-free assets.

Note: In reality, different investment managers do create risky portfolios that differ from the market index (in part due to the use of different input lists in the formation of the optimal risky portfolio).

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Model implication 3: Risk premium on M

The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the representative investor:

where

E(rM): expected return on M

: variance of M

: average degree of risk aversion across

investors

2)( MfM ArrE

2M

A

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Recall: Each individual investor chooses a proportion y, allocated to the optimal portfolio M, such that:

Risk-free investments involve borrowing and lending among investors. Any borrowing position must be offset by the lending position of the creditor. This means that net borrowing and lending across all investors must be zero, and thus the average position in the risky portfolio is 100%, or y = 1.

Setting y = 1 and rearranging, we obtain the risk premium on the market portfolio.

2

)(

M

fM

A

rrEy

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Model implication 4:

Expected returns on individual securities

The contribution of an asset to the risk of the market portfolio:

Risk-averse investors measure the risk of the optimal risky portfolio (i.e. the market portfolio) by its variance.

We would expect the reward, or the risk premium on individual assets, to depend on the contribution of the individual asset to the variance of the market portfolio.

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To calculate the variance of the market portfolio, we use the following bordered covariance matrix with the market portfolio weights:

Portfolio 　 　 　　 　 　　 　

Weights w1 w2 . . wGE . . wn

w1 Cov(r1,r1) Cov(r1,r2) . . Cov(r1,rGE) . . Cov(r1,rn)

w2 Cov(r2,r1) Cov(r2,r2) . . Cov(r2,rGE) . . Cov(r2,rn)

. . . . . . . . .

. . . . . . . . .

wGE Cov(rGE,r1) Cov(rGE,r2) . . Cov(rGE,rGE) . . Cov(rGE,rn)

. . . . . . . . .

. . . . . . . . .

wn Cov(rn,r1) Cov(rn,r2) . . Cov(rn,rGE) . . Cov(rn,rn)

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The contribution of GE’s stock to the variance of the market portfolio:

The covariance of GE with the market portfolio is proportional to the contribution of GE to the variance of the market portfolio.

In other words, we can measure an asset’s contribution to the risk of the market portfolio by its covariance with the market portfolio.

)],cov(...),cov(...),cov([ 11 nGEnGEGEGEGEGE rrwrrwrrww

),cov(),cov(),cov(11

MGEGE

n

kkkGEGE

n

kkGEkGE rrwrwrwrrww

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• The reward-to-risk ratio for investments in GE:

portfoliomarket the of variance the to oncontributi sGE'

portfoliomarket the of premium risk the to oncontributi sGE'

),(

)(

),(

])([

MGE

fGE

MGEGE

fGEGE

rrCov

rrE

rrCovw

rrEw

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The market portfolio has a risk premium of E(rM)

- rf and a variance of , for a reward-to-risk ratio of:

This ratio is called the market price of risk, because it quantifies the extra return that investors demand to bear portfolio risk (i.e. tells us how much extra return must be earned per unit of portfolio risk).

2M

2

)(

M

fM rrE

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In equilibrium, all investments should offer the same reward-to-risk ratio.

If the ratio were better for one investment than another, investors would rearrange their portfolios, tilting toward the alternative with the better trade-off and shying from the other.

Such activity would impact on security prices until the ratios were equalized.

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In equilibrium, the reward-to-risk ratio of GE stock must equal that of the market portfolio.

Otherwise, if the reward-to-risk ratio of GE > the market’s, investors can increase their portfolio reward for bearing risk by increasing the weight of GE in their portfolio.

Until the price of GE stock rises relative to the market, investors will keep buying GE stock.

The process will continue until stock prices adjust so that reward-to-risk ratio of GE equals that of the market.

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The same process, in reverse, will equalize reward-to-risk ratios when GE’s initial reward-to-risk ratio < that of the market portfolio.

])([),(

)(2 fMM

MGEfGE rrE

rrCovrrE

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),(

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MGE

fGE rrE

rrCov

rrE

fMM

MGEfGE rrE

rrCovrrE )(

),()(

2

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The ratio measures the contribution of GE stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio.

The ratio is called beta and is denoted by .

The expected return-beta relationship:

More generally, for any asset i:

2/),cov( MMGE rr

])([)( fMGEfGE rrErrE

])([)( fMifi rrErrE

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Note: Risk-averse investors measure the risk of the optimal risky portfolio by its variance. We would expect the reward (the risk premium on individual assets) to depend on the contribution of the individual asset to the risk of the portfolio.

The beta of an asset measures the asset’s contribution to the variance of the market portfolio as a fraction of the total variance of the market portfolio. Hence we expect, for any asset, the required risk premium to be a function of beta.

The CAPM confirms this intuition: the security’s risk premium is directly proportional to both the beta & the risk premium of the market portfolio; that is, the risk premium equals .])([ fM rrE

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If the expected return-beta relationship holds for any individual asset, it must hold for any combination of assets.

Suppose that some portfolio P has weight wk for stock

k, where k takes on values 1, …, n.

Then:

where E(rP)

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This result has to be true for the market portfolio itself:

Note:

Þ The market beta (i.e. the weighted average beta of all assets) is 1.

Þ Betas > 1 are considered aggressive in that investment in high-beta assets entails above-average sensitivity to market swings.

Betas < 1 can be described as defensive.

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1),(

2

2

2

M

M

M

MMM

rrCov

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Model implication 5:

The security market line (SML)

The expected return-beta relationship can be portrayed graphically as the security market line (SML).

Because the market beta is 1, the slope is the risk premium of the market portfolio.

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M

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Compare SML to CML

The CML graphs the risk premiums of efficient portfolios (i.e., portfolios composed of the market and the risk-free asset) as a function of portfolio standard deviation.

This is appropriate because standard deviation is a valid measure of risk for efficiently diversified portfolios that are candidates for an investor’s overall portfolio.

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The SML, in contrast, graphs individual asset risk premiums as a function of asset risk.

The relevant measure of risk for individual assets held as parts of well-diversified portfolios is not the asset’s standard deviation or variance.

It is, instead, the contribution of the asset to the portfolio variance, which we measure by the asset’s beta.

The SML is valid for both efficient portfolios and individual assets.

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Uses of the SML

The SML provides a benchmark for the evaluation of investment performance.

Given the risk of an investment, as measured by its beta, the SML provides the required rate of return necessary to compensate investors for risk.

Because the SML is the graphic representation of the expected return-beta relationship, “fairly priced” assets plot exactly on the SML; that is, their expected returns are commensurate with their risk.

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The difference between the fair and actually expected rates of return on an asset is called the asset’s alpha, denoted .

e.g. The market return is expected to be 14%, a stock has a beta of 1.2, and the T-bill rate is 6%.

The SML would predict an expected return on the stock of 6 + 1.2(14 – 6) = 15.6%.

If one believed the stock would provide an expected return of 17%, the implied alpha would be 1.4% (= 17% - 15.6%).

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The CAPM is also useful in capital budgeting decisions.

For a firm considering a new project, the CAPM can provide the required rate of return that the project needs to yield, based on its beta, to be acceptable to investors.

Managers can use the CAPM to obtain this cutoff internal rate of return (IRR), or “hurdle rate” for the project.

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Q: The risk-free rate is 6% and the expected return

on the market portfolio is 14%.

A firm considers a project that is expected to

have a beta of 0.6.

a. What is the required rate of return on the

project?

b. If the expected internal rate of return of the

project is 19%, should it be accepted?

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The CAPM tells us that an acceptableexpected rate of return for the project is:

which becomes the project’s hurdle rate.

If the internal rate of return of the project is 19%, then it is desirable.

Any project with an rate of return 10.8% should be rejected.

%8.10)614(6.6])([ fMf rrEr

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The CAPM and the Index Model

Recall:

For any asset i and the (theoretical) market portfolio, the CAPM expected return-beta relationship is:

Recall:

The index model (in excess return form):

or

iMiii eRR

])([)( fMifi rrErrE

ifMiifi errrr )(

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From the index model, the covariance between the returns on stock i and the market index:

Notes: We can drop i from the covariance terms because i is

a constant and thus has zero covariance with all variables.

The firm-specific or nonsystematic component is independent of the marketwide or systematic component (i.e. Cov(ei, RM) = 0).

2

),(),(

),(),(

Mi

MiMMi

MiMiMi

ReCovRRCov

ReRCovRRCov

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The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship, except that we replace the (theoretical) market portfolio of the CAPM with the well-specified and observable market index.

2),(

M

Mii

RRCov

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If the index M in the index model represents the true market portfolio, we can take the expectation of each side of the index model:

A comparison of the index model relationship to the CAPM expected return-beta relationship shows that the CAPM predicts that i should be zero for all

assets.

The alpha of a stock is its expected return in excess of (or below) the fair expected return as predicted by the CAPM.

If the stock is fairly priced, its alpha must be zero.

])[()( fMiifi rrErrE

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Thus, if we estimate the index model for several firms, using the index model as a regression equation, we should find that the ex post or realized alphas (the regression intercepts) for the firms in our sample center around zero.

The CAPM states that the expected value of alpha is zero for all securities, whereas the index model representation of the CAPM holds that the realized value of alpha should average out to zero for a sample of historical observed returns.

Empirical example:

Burton Malkiel examines the alpha values for a large number of equity mutual funds.

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The distribution of alphas is roughly bell shaped, with a mean that is slightly negative but statistically indistinguishable from zero.

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The market model:

The market model states that the return “surprise” of any security is proportional to the return surprise of the market, plus a firm-specific surprise:

If the CAPM is valid:

and

])([)( fMifi rrErrE

0i

iMMifMifi erErrrErr )]([)])(([

iMMiii erErrEr )]([)(

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The market model equation becomes identical to the index model.

Thus, the terms “index model” and “market model” are used interchangeably.

ifMifi errrr )(

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Some Extensions of the CAPM

Risk-free lending but no risk-free borrowing Zero-beta portfolio:

Every portfolio on the efficient frontier (except for the global minimum-variance portfolio) has a “companion” portfolio on the bottom half (the inefficient part) of the minimum-variance frontier with which it is uncorrelated.

Because the portfolios are uncorrelated, the companion portfolio is referred to as the zero-beta portfolio of the efficient portfolio.

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Efficient portfolios and their zero-beta companions:

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From any efficient portfolio such as P draw a tangency line to the vertical axis.

The intercept will be the expected return on portfolio P’s zero-beta companion portfolio, denoted Z(P).

The horizontal line from the intercept to the minimum-variance frontier identifies the standard deviation of the zero-beta portfolio.

Notice that different efficient portfolios such as P and Q have different zero-beta companions.

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The market portfolio M has a companion zero-beta portfolio on the minimum-variance frontier, Z, with an expected rate of return E(rZ).

Note that since M and Z are uncorrelated, we have: cov[rM, rZ] = 0.

The zero-beta CAPM:

(i.e. rf has been replaced by E(rZ))

Note: The zero-beta CAPM is still valid when (i) there is no risk-free asset at all; or (2) the borrowing rate is higher than the lending rate.

)]()([),cov(

)()(2 ZMM

MiZi rErE

rrrErE

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Liquidity and the CAPM

Liquidity refers to the cost and ease with which an asset can be converted into cash (i.e. sold).

Recall one assumption of the CAPM that all trading is costless. In reality, no security is perfectly liquid, in that all trades involve some transaction cost.

Investors prefer more liquid assets with lower transaction costs, so that all else equal, relatively illiquid assets trade at lower prices (i.e. the expected return on illiquid assets must be higher). Thus, an illiquidity premium must be impounded into the price of each asset.

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Trading costs:

• Commission.

• Bid-asked spread:

bid price: the price at which a dealer is willing to purchase a security.

asked price: the price at which a dealer will sell a security.

bid-asked spread: the difference between a dealer’s bid and asked price.

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A simple example:

• Consider a world in which market risk premium is ignored. Thus, the expected rate of return on all securities will equal the risk-free rate (rf).

• Assume that there are only two classes of securities: liquid (L) and illiquid (I).

The liquidation cost cL of a class L stock to an investor with an investment horizon of h years will reduce the per-period rate of return by cL/h.

Thus, if you intend to hold a class L security for h periods, your expected rate of return net of transaction costs is rf - cL/h.

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Class I assets have higher liquidation costs cI that reduce the per-period return by cI/h, where cI > cL.

Thus, if you intend to hold a class I security for h periods, your expected rate of return net of transaction costs is rf - cI/h.

These net rates of return would be inconsistent with a market in equilibrium, because with equal gross rates of return (rf) all investors would prefer to invest in zero-transaction-cost asset (the risk-free asset).

As a result, both class L and class I stock prices must fall, causing their expected returns to rise until investors are willing to hold these shares.

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Suppose, therefore, that each gross return is higher by some fraction of liquidation cost.

Specifically, assume that the gross expected return on class L stocks is rf + xcL and that of class I stocks is rf

+ ycI.

The net rate of return on class L stocks to an investor with a horizon of h:

(rf + xcL) - cL/h = rf + cL(x - 1/h).

The net rate of return on class I stocks:

(rf + ycI) - cI/h = rf + cI(y - 1/h).

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Now we can determine equilibrium illiquidity premiums.

For the marginal investor with horizon h, the net return from class I and L stocks is the same:

rf + cL(x - 1/h) = rf + cI(y - 1/h)

The expected gross return on illiquid stocks:

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cc

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I

L

)(1

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Recall that the expected gross return on class L stocks is rL = rf + cLx.

Thus, the illiquidity premium of class I versus class L stocks is:

As expected, equilibrium expected rates of return are bid up to compensate for transaction costs.

)(1

LILI cch

rr

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If we allow for market risk premium, we would find that the illiquidity premium is simply additive to the risk premium of the usual CAPM.

Thus, we can generalize the CAPM expected return-beta relationship to include a liquidity effect:

where f(ci) is a function of trading costs ci that measures the effect of the illiquidity premium given the trading costs of security i.

The usual CAPM equation is modified because each investor’s optimal portfolio is now affected by liquidation cost as well as risk-return considerations.

)(])([)( ifMifi cfrrErrE

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Trading frequency may well vary inversely with trading costs.

An investor who plans to hold a security for a given period will calculate the impact of illiquidity costs on expected rate of return; illiquidity costs will be amortized over the anticipated holding period.

Investors who trade less frequently thus will be less affected by high trading costs.

The reduction in the rate of return due to trading costs is lower, the longer the security is held.

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Hence, in equilibrium, investors with long holding periods will, on average, hold more of the illiquid securities, while short-horizon investors will more strongly prefer liquid securities.

This “clientele effect” mitigates the effect of the bid-ask spread for illiquid securities.

The end result is that the illiquidity premium should increase with the bid-ask spread at a decreasing rate.

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The Relationship Between Illiquidity and Average Returns