*A Derivation of the Capital Asset Pricing Model*

I. Establish the Objects of Choice: Mean versus Variance

__Theme__: Investors are risk averse. They measure reward using expected return and risk using variance.

__Underlying assumptions__: The mean-variance assumption can hold only if (a) all investors have quadratic utility function or (b) returns are normally distributed.

__Implication__: Portfolio A with higher expected return and the same variance as portfolio B will be preferred to B

*II. Benefits of Diversification*

For any desired level of risk (s) there exists a portfolio of several assets which yields a higher expected return than any individual security

E(R_{p}) = S w_{i}E(R_{i})

s^{2}_{p}= S S w_{i}w_{j}Cov_{ij}

__Efficient portfolios__: maximize returns for any level of risk.

__Implications__: (a) Everybody should diversify (b) Investors should try to identify and hold efficient portfolios (c) This method has very heavy computational requirements.

*III. The Single Index Model: The Logical Limit of Diversification*

__Assumptions__: (a) Riskfree lending and borrowing (b) Markets which are frictionless - there are no transactions costs (c)Homogeneous expectations

__Implications__: (1) The risky portfolio than when combined with the riskless asset maximizes returns is the market portfolio. (2) Everybody holds some combination of the market portfolio and the risky asset. How much of each is held will be a function of the investor's risk aversion.(3) Since all investors hold the same market portfolio it must contain all assets in the economy in proportion to their value.

*IV. The Risk of an Individual Asset*

Step 1: Individuals diversify and hold portfolios

Step 2: The risk of a security is the risk it adds to the portfolio

Step 3: Everybody holds the market portfolio

Step 4: The risk of a security is the risk that it adds to the market portfolio.

Step 5: The covariance between an asset "i" and the market portfolio (Cov_{im}) is a measure of this added risk. The higher the covariance the higher the risk.

Step 6: This measure can be standardized by dividing by the market variance. b = Cov_{im}/ s^{2}_{m}.

(1) Of all the the zero-beta portfolios this has the minimum variance

(2) The separation principle applies here with the two portfolios, the market portfolio and the zero-beta portfolio, i.e. all investors hold combinations of the two.

(3) The expected return on any security can be expressed as a linear function of its beta.

E(R

where E(R

(a) There is a piecewise linear relationship between expected return and beta for efficient portfolios.

(b) Efficient portfolios with the riskfree asset lie along the segment RfT and those containing only risky assets lies along the segment TMC.

The separation principle still holds but,

(a) Investors hold different portfolios of risky assets depending upon the portfolios of non-marketable assets that they possess.

(b) The market price of risk includes the variance of the market and the covariance between the market portfolio and the portfolio of non-marketable assets

E(Ri) = a + b

where d

R

Model: To get strong conclusions we have to assume that all investors have a certain class of utility functions (Constant Absolute risk aversion) and complete markets (At least as many independent securities as states).

Issue 1: The CAPM can never be tested because the market portfolio can never be observed

Central to the CAPM is the concept of a market portfolio which includes every asset in the economy. To test the CAPM therefore one has to observe and be able to measure this efficient market portfolio. If one cannot do so one cannot test the CAPM. One cannot use of an inefficient portfolio like the S&P 500 or the NYSE 2000 or even every stock in the economy to estimate betas and test for linearity (like all the studies have done) because

(a) The betas measured against an inefficient portfolio are meaningless measures and cannot be used to accept or reject the CAPM which is really a theory about betas measured against the efficient market portfolio

(b) For every inefficient portfolio there exists a set of betas which will satisfy the linearity condition.

The noisiness in beta estimates and the fact that the CAPM yields expected returns for individual assets over the long term makes it difficult to test the CAPM by trying to relate expected returns on individual assets (such as stocks) to their betas. What most tests of the CAPM do instead is to look at portfolios of stocks, based upon betas, and then compare these betas to expected returns in the next time period.

(a) the number of common factors that appeared to affect asset prices over the period for which the data is available

(b) the betas of each asset relative to each factor, again using the same data

(c) the "risk premiums" associated with each factor

These factor betas and factor premiums are then used, in conjunction with a riskfree rate to get an expected return for an asset.

The more difficult question is deciding which financial variables to use in explaining returns. The best place to start is to look at the empirical evidence that has been accumulated over time on market efficiency and the CAPM. This evidence suggests that

- Low market capitalization stocks seem to earn higher returns, on average, than high market capitalization stocks

- Low PE, PBV and PS ratio stocks seem to earn higher returns, on average, than high PE, PBV and PS ratio stocks

- High dividend yield stocks seem to earn higher returns, on average, than low dividend yield stocks

While the initial regression may include all of these variables, many of these variables tend to be correlated with each other. Thus, low PE stocks tend to also be low PBV ratio stocks which pay high dividends. In the interests of efficiency (and to prevent problems in the regression from independent variables being correlated with each other), it makes sense to use the measure that is most highly correlated with returns and drop the others. Thus, the use of price to book value ratios by Fama and French.

Corporate bonds have some upside potential, but it is limited by the fact that bonds can at best become default-free. Thus, the upside potential for a AA rated bond is fairly limited. Consequently, the risk measure that we have to use has to be a downside risk measure, which is what default risk and ratings measure. Clearly, the lower the rating of a bond, the greater the upside potential, and thus, the greater the likelihood that we can estimate bond betas and expected returns on them. For a junk bond, for instance, it may be possible to estimate a beta like a stock beta and get an expected return from it.