Created 7/1/1996

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Part A (20 minutes): Identifications. Write one sentence on each.

1. What is an annuitized annual cost?

- A way of expressing the present-value cost of a durable capital good as an annual sum--an annuity. Useful for comparing the real costs of capital goods with different lifetimes.

2. What is a certainty-equivalent value?

- In this course we have calculated the value of cash flows that bear systematic risk by discounting them at the appropriate "risky" interest rate. An alternative is to calculate the value of cash flows by, first, calculating the certainty-equivalent value of the distribution of payments in year t, and then discounting the certainty-equivalent values back to the present at the risk-free rate.

3. What is a semiannual coupon payment?

- Most bonds pay interest twice a year--half the face value of the bond interest every six months. Thus the effective annual yield of a bond selling at par is a bit more than the face value of its yield.

4. What is a variance?

- A measure of risk. The ex-ante expected value of the squared deviation between the actual return and the previously-anticipated return.

5. What is a preferred equity share?

- A security that carries neither (a) the rights to control over the firm held by common stock, nor (b) the right to sue the firm if it fails to make interest payments held by bonds.

Part B (20 minutes): Net Present Values. Show your work.

1. The net present value at a discount rate of 12% per year of an investment made by spending $1,000,000 this year on a portfolio of stocks with an initial dividend next year of $100,000, and an expected rate of dividend growth thereafter of 4% per year.

- P=D/(r-g) - 1000000=100000/(.12-.04) - 1000000 = 1250000-1000000 = 250000

2. The net present value today at a discount rate of five percent per year of a stock that has expected next year's earnings of $5 a share, and a present value of growth opportunities of $50 a share.

- 50 - 5/.05 = $100 a share

3. The net present value today, at a discount rate of six percent per year, of a convertible bond that pays $1 a year for the next twelve years, and then converts into a stock that in year 13 pays a $1 dividend, which grows by 2% per year thereafter.

- [1/.06][1-1/(1.06^12)]+[1/(1.06^12)][1/(.06-.02)] = 16.67[1-1/2]+(1/2)(25) = 8.33+12.5 = $20.83

4. The net present value at a discount rate of 8% per year of an investment made by spending $1,000,000 this year on a portfolio of stocks with an initial dividend next year of $50,000, and an expected rate of dividend growth thereafter of 3% per year.

- -1000000 + 50000/(.08-.03) = 0

5. The expected profit at a discount rate of 5% per year of a state lottery that offers a prize of infinity--payable at a rate of $1,000,000 a year to the winner and then to his or her heirs and assigns forever--and that expects to sell 18 million one-dollar tickets to the lottery.

- -1000000/.05 + 18000000 = -$2000000

Part C (20 minutes): Risk, Return, and the Capital Asset Pricing Model. Show your work.

1. Suppose we have a risk-free rate of 5% per year, and a market rate of 10% per year. Suppose that the standard deviation of the annual return on the market portfolio is 20% per year, the standard deviation of the annual return on Abacus Corporation stock is 80% per year, and that the correlation between the market's and Abacus's return is 0.5. Abacus's dividend is expected to be $15 a share next year, and to remain constant thereafter. What is the value of this stock?

- Covariance of abacus with the market is .8(.5)(.2)=.08; Variance of market is .04; beta of abacus is 2; required rate of return is .15; price is $100.

2. Suppose that the Calvert Corporation's stock has a 20% chance of going up by 40% in the next year, a 40% chance of going up by 5%, and a 40% chance of falling by 20%. What is its expected annual return over the next year? If the risk-free rate is 5% per year and the market rate is 10% per year, and if the Calvert Corporation pays no dividend, what is its beta?

- Expected return = .40(.05) = 2% per year; this is an equilibrium only if beta = -.6

3. Suppose the covariance between a stock's return and the market return is 0.4; suppose the standard deviation of the market return is 40% per year. What is the stock's beta?

- beta = .4/(.4^2)=2.5

4. Suppose that the risk-free rate is 5% per year, and the market rate
is 10% per year. Suppose, further, that the assumptions of the CAPM are
satisfied. Note that holdings of gold pay no dividends *ever*. What
must the beta of investments in gold be if gold is to sell for a positive
price, and if it is not to be "irrational" for an investor to
hold it as part of a diversified portfolio?

- gold must have a beta of -1, or its value will be zero (or the CAPM does not hold for gold).

Part D (20 minutes): Further Topics.

1. Suppose we have three portfolios, plus a risk-free asset that pays an expected return of 3% per year with a standard deviation of zero.

- A offers an expected return of 15% per year, with a standard deviation of 30% per year.
- B offers an expected return of 10% per year, with a standard deviation of 20% per year.
- C offers an expected return of 5% per year, with a standard deviation of 10% per year.

Suppose that a client asks you to choose a portfolio with an expected return of 39% per year, and as low a standard deviation as possible. Which of A, B, and C do you include in the client's portfolio? Why? What standard deviation can you achieve?

- A combined with the riskless asset gets you the best risk-return tradeoff; borrow an amount equal to 200% of your wealth at the risk-free rate, and invest all 300% of your wealth in portfolio A, and you achieve a standard deviation of 90% per year.

Created 7/1/1996

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