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MIT Sloan School of Management

Finance Theory I 15.401 CDScott Joslin Spring 2008

Problem Set 5

(Due: 5pm, Friday, April 25, 2008)

1. In 2006, the rate of return on short-term government securities was about 5%. Supposethe expected rate of return on a portfolio with a beta of 1 is 12%. According to CAPM:

(a) What is the expected rate of return on the market portfolio?

(b) What are the expected rate of returns on stocks with = 0.5 and 1.5?

(c) Suppose you consider buying a stock for $100. The stock is expected to pay $5dividend next year and you expect it to sell then for $102. You calculate the ofthe stock as 0.5. Is the stock overpriced or underpriced?

2. (Investment in negative stocks) Suppose that there are only two stocks in theeconomy with the following return moments:

Mean Return (%) Standard Deviation (%)Stock 1 8% 21%Stock 2 2% 35%

The returns of the two stocks have a correlation of -.7. Assume that the risk-free rateis 4%. Also, assume that the investors only care about means and standard deviationsof returns and that the CAPM holds.

(a) Compute the Sharpe ratio for Stock 1.

(b) Plot the Sharpe ratio, as a function ofw, for a portfolio investing a fraction w inStock 1 and 1 w in Stock 2.

(c) Which weight gives the highest Sharpe ratio?

(d) Which weight corresponds to the tangent portfolio?

(e) Which weight corresponds to the market portfolio?

(f) What is the expected return, rM, of the market portfolio?

(g) Show that Stock 2 is a negative stock in the following two ways:

i. Compute = cov(r2, rM)/2

M. Use the fact that rM = w

r1 + (1 w

)r2,where w was found in (d), and that cov(x + y, z) = cov(x, z) + cov(y, z)

ii. Solve the CAPM equation r2 = rf + (rM rf) for .

Discussion: Notice that this problem shows that it may be beneficial to invest innegative stocks with low returns. Here the return of Stock 2 is worse than the riskfree rate, even though it is risky. However, we can improve our Sharpe ratio from thevalue in (a) to the value in (c) by holding a long position in this inferior stock.

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3. (Violations of the CAPM) In this problem, we will show that when the APT holds,the assumptions and results of the CAPM may be be violated.

Suppose that the APT holds and there are two underlying risk factors, f1, f2. Theyhave moments given below:

Mean (%) Standard Deviation (%)Factor 1 (f1) 3% 15%

Factor 2 (f1) 2% 20%

The factors are uncorrelated. In the economy there are only two stocks with returns:

r1 = f1 + f2

r2 = 2f1

The market capitalization of the first stock is $20M and the second stock is $80M. The

risk-free rate is 4.7% and is in zero net supply.(a) Compute the mean and standard deviation of the market return, as well as the

returns r1 and r2.

(b) Compute the of Stock 2.

(c) Compute the quantityrf + 2(rM rf)

That is, compute the return for Stock 2 if the CAPM held.

(d) Plot the portfolio frontier generated by Stocks 1 and 2. On the plot, mark themarket portfolio and draw a line connecting it to the risk-free rate. Is the market

portfolio the tangency portfolio?(e) In light of the answer to part (d), can it be that all investors hold mean-variance

efficient portfolios? Explain.

Discussion: In the CAPM, covariation with the market is the only risk that peopledemand compensation for bearing. In this APT economy, covariance with the two riskfactors is what matters. That is, there is are additional risks that people demand apremium for bearing. So if a portfolio looks good from a mean-variance stand point,it may be correlated with some risks that investors dont like to bear and they maytherefore require a higher premium to hold the portfolio.

Observe that the mean return for Stock 2 according to the CAPM is differentthan the actual mean return for Stock 2.

Observe that the market portfolio is not the tangent portfolio.

This means the market portfolio is not mean-variance efficient.

This means that some investors must hold portfolios that are not mean-variance efficient.

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