The Capital Asset Pricing Model (Ch7)

1. The Capital Asset Pricing Model

This chapter has one of the most important models in investment modeling.It addresses the question of what is a reasonable price for an asset.The same model also gives some very good investment advice.The results of the chapter build upon the Markowitz mean-variance portfolio theory of Chapter 6.

Assumptions

Everyone is a mean-variance optimizer, as in Chapter 6.

Everyone assigns to returns ofall availablemarket assets the same mean values, variances, and covariances (available from Morning Star, for example).

There is a unique risk-free rate of borrowing and lending available to all, with no transaction costs.

Under the above assumptions, everyone would rely on theone-fund theorem .They would buy a single (efficient) fund of risky assets, and then also borrow or lend at the risk-free rate.The mix of the fund and the risk-free asset would depend on a investors attitude towards risk.

Basic Question: What is the one fund (shared by all investors) ?

Example of theMarket Fund , after Table 7.1.

Suppose there are only three risky assets in the market, as follows:

The table illustrates the definitions of capitalization (sharesprice) and capitalization weights.

The weights are NOT the same as the relative shares in the market.

Each weight is the ratio of the capital value of the asset to the total market capital value.

If you invested in this market fund, you would use the weights in the last column; a $1,000 investment would give you

[(3/20)1000]/6 = 25 shares of Mahler Inc

[(3/10)1000]/4 = 75 shares of Mozart Inc.

[(11/20)1000]/5.5 = 100 shares of Verdi, Inc.

Theactual market fundwould be comprised ofeveryinvestment asset available.Just as in the example, each investment weight would be the ratio of the capital value of the asset to the total market capital value.

Funds similar to the market fund actually exist, and are calledindex funds .Of course, they do not include every available asset, but they may well include 500 or more.Index funds typically out-perform most actively managed funds.Vanguard Securities offers many such funds (e.g. Index Trust 500)

Index funds have been somewhat resisted by the financial world, including many private investors.The first reaction is usually that surely an actively managed fund will give better performance.The facts, however, are otherwise.About 90% of actively managed funds perform worse than the S&P 500.You can also check performance with the Morning Star services.

Equilibrium Arguments

If everyone buys just one fund, and their purchases add up to the market, then that one fund must be the market as well.

The fund must contain shares of every stock in proportion to that stocks representation in the entire market.

If everyone else solved the one-fund problem, we would not need to.

Suppose everyone elsesolves the mean-variance portfolio problem with their common estimates, and places orders in the market to acquire their portfolios.This solution isefficient , because it minimizes the total variance of the return.

Equilibrium Arguments (Contd)

If the orders placed do not match what is available, the prices must change.Prices of assets under heavy demand will go up, prices of assets under light demand will go down.The price changes will affect the estimates of asset returns directly.Therefore, investors will recalculate their optimal portfolios.The process continues until demand exactly matches supply; that is, until there is an equilibrium.

Extra Comments

Everyone would buy just one portfolio in this idealized world, and it would be the market portfolio.

Prices adjust to drive the market to efficiency.

After the adjustments, the portfolio will be efficient, so we would not need to make the calculations.

This argument is most plausible when applied to assets traded repeatedly over time, which is certainly the case with the stock market.

The fact that index funds perform well provides some verification that the equilibrium argument is plausible.

Final Word: If you have better information than your rivals, you will want your portfolio to include relatively large investments in the stocks you think are undervalued.In a competitive market you are unlikely to have a monopoly of good ideas.In that case, there is no reason to hold a different portfolio of common stocks from anybody else.In other words, you might just as well hold the market portfolio.That is why many professional investors invest in a market-index portfolio, and why most others hold well-diversified portfolios.

The Capital Market Line

M is the market portfolio

M = ( M , E{r M })

E{r M } - r f

E{r} = r f+--------------

The line shows the efficient set , starting at the risk-free point, and passing through the market portfolio.

The CML shows the relation between the expected rate of return and the risk of return (as measured by the standard deviation), for efficient assets or portfolios of assets.

The CML is also called thepricing line .Prices should adjust so that efficient assets fall on this line. equationsw.r.t.

Example 7.1

r f= 6%, E{r M } = 12%, M= 15%.

John Eager wants to retire in 10 years.For this he needs $1,000,000.He currently has $1,000.At the market rate, it would take about 60 years for $1,000 to grow into $1,000,000.If he can get 100% return each year he concludes he will grow $1,000 into $1,000,000 in 10 years.

Example 7.1 (Contd)

The CML relates the expected rr ofany efficient portfolioto its standard deviation.Another step beyond the CML is to show how the expected rate of return ofany individual assetrelates to its individual risk.That is what the capital asset pricing model does.

Example(We use t instead of ).

The equation for the standard deviation of the rate of the return combining the market portfolio with any asset i is

x = f(t) = [t 2 i 2+ 2 t (1-t) iM+ (1-t) 2 M 2 ]

The expected return for this combination becomes

y = g(t) = t E{r i } + (1-t) E{r M }

Note f(0) = M , g(0) = E{r M } .

Example (Contd)

We thus have

dy/dt = g (t) = E{r i } - E{r M }

dx/dt = f (t) = [t i 2+ (1-2 t) iM+ (t-1) M 2 ]/f(t)

Note, when t = 0,

f (t)| t=0= ( iM- M 2 )/f(0) = ( iM- M 2 )/ M

Example (Contd)

CAPM Fundamental Result

If the market portfolio M is efficient, then the expected return E{r i } ofanyasset i satisfies

E{r i } r f= i(E{r M } r f )

i= iM / M 2 .

Proof of CAPM .

For any t, consider the portfolio consisting of a portion t invested in asset i and a portion 1-t invested in the Market Portfolio M.(t < 0 corresponds to selling short the asset.)

The expected rate of return of this portfolio is

y = g(t) = t E{r i } + (1-t) E{r M }

The standard deviation of the rate of return is

x = f(t) = [t 2 i 2+ 2 t (1-t) iM+ (1-t) 2 M 2 ]

As t varies, the values (f(t), g(t)) trace out a curve in the expected return-sd diagram, as shown below.

In particular, the point on the curve (f(0),g(0)) for

t = 0 corresponds to the market portfolio M.

This curve cannot cross the capital market line.If it did, the portfolio corresponding to a point above the capital market line would violate the definition of the capital market line as being the efficient boundary of the feasible set.Hence as t passes through zero,the curve must be tangent to the capital market line at M .This tangency is the condition that we exploit to derive the formula.

The tangency condition can be translated into the condition that the slope of the curve (f(t), g(t)) is equal to the slope of the capital market line at the point M, where t = 0.

If we solve this equation for E{r i } we get

E{r i } = r f+ [(E{r M } r f )/ M 2 ] iM

= r f+ i(E{r M } r f ).

where i= iM / M 2.

Equivalently, we have E{r i } - r f= i(E{r M } r f ).

iis called thebetaof asset i.Sometimes the subscript is omitted.

The Morning Star service estimates betas.

Since E{r i } - r f= i(E{r M } r f ), E{r i } - r fis called theexpected excess rate of return of asset i .It is the amount by which the rate of return is expected to exceed the risk-free rate.

(E{r M } r f ) is called theexpected excess rate of return of the market portfolio .

THE CAPM says the expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio.The constant of proportionality factor is .

Because i= iM / M 2 , it is anormalized version of the covariance of an asset with the market portfolio . The excess rate of return for the asset is directly proportional to its covariance with the market.

Generally speaking, we expect aggressive assets/companies or highly leveraged companies to have high betas.Conservative companies whose performance is unrelated to the general market behavior are expected to have low betas.We expect that companies in the same business will have similar beta values.

Bottom Line

The CAPM changes our concept of the risk of an asset fromto .It is still true that, overall, we measure the risk of aportfolioin terms of .But this does not translate into a concern for the s ofindividual assets .For those, the proper measure is their s.

After Table 7.2.Betas and Sigmas for Some U.S. Companies (1979)

The concept of beta is well-accepted.

Various financial service organizations (e.g., Morning Star) provide beta and other estimates.

Estimates may be based on 6 to 18 months of weekly values.

Companies in the same business should have similar betas: compare, for instance, JC Penny with Sears Roebuck, and Exxon with Standard Oil of California.

We never know the beta and sigma values we only haveestimatesof them.

Generally speaking, we expect aggressive companies or highly leveraged companies to have high betas, whereas conservative companies whose performance is unrelated to the general market behavior are expected to have low betas.Also we expect that companies in the same business will have similar, but not identical, beta values.

Facts about covariances:

cov{U + V, Z} = cov{U,Z} + cov{V,Z}

cov{a U + b V, Z} = cov{a U,Z} + cov{b V,Z}

= a cov{U,Z} + b cov{V,Z}

Beta of a Portfolio

Suppose a portfolio P has 2 assets with returns r 1 , r 2and weights w 1 , w 2 .Let r Mdenote the market return.We know the portfolio return is r = w 1r 1+ w 2r 2 .Let Pdenote the beta of the portfolio (ratio of the portfolio covariance with the market and M 2).

P= cov{r ,r M } / M 2= cov{ w 1r 1+ w 2r 2 , r M } / M 2

= cov{ w 1r 1 , r M } / M 2+cov{w 2r 2 , r M }/ M 2

= w 1cov{r 1 , r M } / M 2+ w 2cov{r 2 , r M }/ M 2

= w 1 1+ w 2 2

Beta of a Portfolio (Contd)

The two dimension formulageneralizes to n assets:

P= cov{r ,r M } / M 2= w 1 1+ w 2 2++ w n n .

The portfolio beta is just the weighted average of the betas of the individual assets in the portfolio .The weights are those that define the portfolio.

Risk neutral portfolio: P= 0 .

Under the equilibrium conditions assumed by the CAPM, any asset should fall on the SML.

The SML expresses the reward-risk structure of assets according to the CAPM, and emphasizes that the risk of an asset is a function of its covariance with the market or, equivalently, a function of its beta.

Systematic Risk

There are several types of risk with an investment:

- systematicrisk

- nonsystematic ,idiosyncratic , orspecificrisk.

The systematic risk is risk associated with the market as a whole.The second type of risk is uncorrelated with the market, and can be reduced by diversification.

We can use the CAPM to quantify these two types of risks.

Consider the equation

r i= r f+ i(r M- r f ) + i (***)

To begin with, we view this equation simply as a definition of the random variable i .Namely,

i = r i [ r f+ i(r M- r f )]

The CAPM provides some information on i .Note firstthat

Let us show that

i 2 var(r i ) = i 2 M 2+ var( i )

We can write

r i= ir M+ i+ k,

where k is a constant.

Implications. For asset i, its risk is the sum of

(1) i 2 M 2 , thesystematicrisk, and

(2) var( i ), thenonsystematicorspecificrisk.

The systematic risk is the risk associated with the market as a whole.It cannot be reduced by diversification, because every asset with nonzero beta contains this risk.

The specific risk is uncorrelated with the market andcan be reduced by diversification .

It is the systematic risk, measured by beta,that is most important.It directly combines with the systematic risk of other assets.

There is a limit to how much diversification can achieve in reducing risk.

The Capital Market Line

M is the market portfolio

M = ( M , E{r M })

E{r M } - r f

E{r} = r f+--------------

For asset i, its risk is the sum of

(1) i 2 M 2 , thesystematicrisk, and

(2)var( i ), thenonsystematicorspecificrisk.

If it hasonly systematic risk , then its standard deviation is

i= i M .

Now considerother fundswith the samevalue, i , as asset i.The CAPM implies all these funds have an expected return of

r f+ i(E{r M } - r f )

But this is the expected return of asset i, E{r i }.Suppose these other assets have nonsystematic risk.Then each will have avariance, for some , of

i 2 M 2+ var( ) > i 2 M 2= i 2 .

Assets with the same beta as asset i, but which also have systematic risk, have the same expected return as asset i but do not fall on the CML.

Bottom Line. The horizontal distance of an asset point from the CML is a measure of the nonsystematic risk of the asset.

CAPM & Investment Implications

A CAPM purist is one who completely believes the CAPM theory as applied to publicly traded securities.A purist would just purchase the market fund and some risk-free securities (e.g., U.S. Treasury bills), adjusting the relative investment in the two according to her/his tolerance for risk.

Individual investors cannot easily purchase the market fund.They can, however, purchase an index fund.These funds allocate their investments in order to duplicate the portfolio of a major stock market index, such as the S&P 500 or the Wiltshire 2,000.

The CAPM requires the assumption that everyone has identical information about the expected returns and variance of returns of all assets.The assumption is certainly open to criticism.

Therefore, akey questionfor an investor is the following:

Do I possess superior information to that required by the Markowitz model and the CAPM?

With superior information, it is likely that one can do better.

Few people quarrel with the idea that investors require some extra return for taking on risk ....

Investors do appear to be concerned principally with those risks that they cannot eliminate by diversification.

The CAPM captures these ideas in a simple way.That is why many financial managers find it the most convenient tool for coming to grips with the slippery notion of risk.

Tests of the CAPM

There are two problems with the CAPM.

- First, it is concerned withexpected returns , whereas we can observe onlyactual returns .Stock returns reflect expectations, but they also embody lots of noise the steady flow of surprises that gives many stocks standard deviations of 30 or 40 percent per year.

- Second, the market portfolio should compriseall risky investments.... Most market indexes contain only a sample of common stocks.

A classic paper by Fama and MacBeth avoids the main pitfalls that come from having to work with actual rather than expected returns.Fama and MacBeth (Risk, Return and Equilibrium: Empirical TestsJ. of Political Economy , 81, 607-636 ,May, 1973) grouped all New York Stock Exchange stocks into 20 portfolios.They then plotted the estimated beta of each portfolio in one 5-year period against the portfolios average return over asubsequent5-year period.1 Figures 8-10 show what they found.You can see that the estimated beta of each portfolio told investors quite a lot about its future return.

If the CAPM is correct, investors would not have expected any of these portfolios to perform better or worse than a comparable package of Treasury bills and the market portfolio.Therefore, the expected return on each portfolio, given the market return, should plot along the sloping lines in Figures 8 10.Notice that the actual returns do plot roughly along those lines.

1 Fama and MacBeth first estimated the beta of each stock during one period and then formed (the 20) portfolios on the basis of these estimated betas.Next they reestimated the beta of each portfolio by using the returns in the subsequent period.This ensured that the estimated betas for each portfolio were largely unbiased and free from error.Finally, these portfolio betas were plotted against returns in an even later period.

Many institutional portfolios (pension funds, mutual funds) now have their performance evaluated using the CAPM framework.

The following example illustrates the evaluation ideas and the use the CAPM.

Example, ABC fund

According to the CAPM, the value of J should be zero when true expected returns are used.Hence J measures, approximately, how much the performance of ABC has deviated from the theoretical value of zero.A positive value of J presumably implies that the fund did better than the CAPM prediction (but of course we recognize that approximations are introduced by the use of a finite amount of data to estimate the important quantities.

The Jensen index can be indicated on the security market line (Figure 7.5 a).

Sharpe Index (Note. Figure 7.5 in the text has mistakes)

The slope of the heavy dashed line is the

Sharpe Index (Ratio)

CAPM as a Pricing Formula

Example 7.5 (Contd)

Certainty Equivalent Form of the CAPM

Certainty Equivalent Form of the CAPM (Contd)

Practical Implication.If we want the CAPM for two assets, and have the CAPM for each, we can get the CAPM for the two in total by adding the certainty equivalent forms for each, or equivalently, using the latter equation.

The reason for linearity can be traced back to the principle of no arbitrage .... This linearity of pricing is therefore a fundamental tenet of financial theory (in the context of perfect markets) ....

Example 7.7(Certainty Equivalent version of Example 7.5)

A fund

invests 10% of its money at r f= 0.07

invests 90% of its money at the market rate,

E{r M } = 0.15.

Its expected return in a year is 0.10.07 + 0.90.15 = 0.142

The epected value of a $100 share in a year will be 100 1.142 = $114.20.

Thevalue is 0.100 + 0.91.0 = 0.9and P=90

Certainty Equivalent Version

P = (1 + r f ) -1 [E{Q} - (cov(Q,r M )/ M 2)(E{r M } r f )]

= (1 + r f ) -1 [E{Q} - P)(E{r M } r f )]

= (1.07)1 [114.20- 900.08] = $100

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The Capital Asset Pricing Model (Ch7)