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Lecture 6: CAPM & APT

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- The following topics are covered:
- Deriving CAPM
- Extensions of CAPM
- Roll’s critique
- APT

L6: CAPM & APT

- Investors are risk-averse individuals who maximize the expected utility of their wealth
- Investors are price takers and have homogeneous expectations about asset returns that have a joint normal distribution
- When all individuals have homogeneous expectations, the market portfolio must be efficient

- There exists a risk-free asset such that investors may borrow or lend unlimited amount at a risk-free rate.
- The quantities of assets are fixed. Also all assets are marketable and perfectly divisible.
- Asset markets are frictionless. Information is costless and simultaneously available to all investors.
- There are no market imperfections such as taxes, regulations, or restriction on short selling.

L6: CAPM & APT

- If market portfolio exists, the prices of all assets must adjust until all are held by investors. There is no excess demand.
- The equilibrium proportion of each asset in the market portfolio is
- (6.1)

- A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation:
- (6.2)
- (6.3)

- A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation:
- Find expected value and standard deviation of with respect to the percentage of the portfolio as follows.

L6: CAPM & APT

- Evaluating the two equations where a=0:
- The slope of the risk-return trade-off:
- Recall that the slope of the market line is:
;

- Equating the above two slopes:

L6: CAPM & APT

Extensions of CAPM

- No riskless assets
- Forming a portfolio with a% in the market portfolio and (1-a)% in the minimum-variance zero-beta portfolio.
- The mean and standard deviation of the portfolio are:
- The partial derivatives where a=1 are:
- ;
- ;

- Taking the ratio of these partials and evaluating where a=1:
- Further, this line must pass through the point and the intercept is . The equation of the line must be:

L6: CAPM & APT

- The existence of nonmarketable assets
- E.g., human capital; page 162

- The model in continuous time
- Inter-temporal CAPM

- The existence of heterogeneous expectations and taxes

L6: CAPM & APT

- Test form -- equation 6.36
- the intercept should not be significantly different from zero
- There should be one factor explaining return
- The relationship should be linear in beta
- Coefficient on beta is risk premium

- Test results – page 167
- Summary of the literature.

L6: CAPM & APT

- Roll (1977) : page 174
- We are looking at an efficient index, rather than the market portfolio.

L6: CAPM & APT

- Assuming that the rate of return on any security is a linear function of k factors:
Where Ri and E(Ri) are the random and expected rates on the ith asset

Bik = the sensitivity of the ith asset’s return to the kth factor

Fk=the mean zero kth factor common to the returns of all assets

Є=a random zero mean noise term for the ith asset

- We create arbitrage portfolios using the above assets. I.e.,
- No wealth
- Having no risk and earning no return on average

L6: CAPM & APT

- There exists a set of k+1 coefficients, such that,
- (6.57)

- If there is a riskless asset with a riskless rate of return Rf, then b0k =0 and Rf =
- (6.58)

- In equilibrium, all assets must fall on the arbitrage pricing line.

L6: CAPM & APT

- Page 182
- Empirical tests
- Gehr (1975)
- Reinganum (1981)
- Conner and Korajczyk (1993)

L6: CAPM & APT