EXERCISE A1Due: March 31, 2015

PART 1 - BASICQUESTION 1Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of 0.5. The alternative risk-free investment in T-bills pays 6% per year.

a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?

E[R] = 6%+8%= 14% = (0.5 70, 000 + 0.5 200, 000)/P 1 1.14 = 135, 000/ P P = 135, 000 /1.14 P = 118,421.05

b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?

E[R] = 135000 118421, 0526/ 118421, 0526 = 14%

c. Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay?

E[R] = 6%+12%= 18% = (0.5 70, 000 + 0.5 200, 000)/P 11.18= 135,000/PP=135,000/ 1.18P = 114,406.78

d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

The higher the risk premium, the lower the price.

QUESTION 2Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. Assume that investors have a utility function given by U = E(r) 0.005A2, where A is the coefficient of absolute risk aversion. What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills?

Portfolio U= E(r) - 0.005A2 U= 12%- 0.005A*0.182 TBills U= E(r) - 0.005A2 U= 7%- 0.005A*02= 7%

7%=12%-0.005A*0.182 A= 308.6419

The investor will prefer the portfolio if A 49900*1.06=52894E(u)= 0.001*ln(52894)+0.999*ln(252894)=12,43CE= exp (12.43)=250196.02b. Its full value.Investment = 49800 => 49800*1.06=52788E(u)= 0.001*ln(52788)+0.999*ln(252788)=12,438CE= exp (12,438)=252205.62c. 1 times its value.Investment = 49700 => 49700*1.06=52682E(u)= 0.001*ln(52682)+0.999*ln(252682)=12,438CE= exp (12,438)=252205.62

PART 3 - ADVANCED

QUESTION 7Consider a two-date (one-period) economy. Define the inter-temporal utility function over consumption as V(c1,c2) = U(c1) + U(c2).

a. Show mathematically that the agent with this utility function will always prefer a smooth consumption stream to a more variable one with the same mean, that is,

if .

b. Verify this result with the utility function

QUESTION 8Suppose that a risk-averse individual can only invest in two risky securities A and B, whose future returns are described by identical but independent probability distributions. How should he/she allocate his/her given initial wealth (normalized to 1 for simplicity) among these two assets so as to maximize the expected utility of next period's wealth?

QUESTION 9Consider the portfolio choice problem of a risk-averse individual with a strictly increasing utility function. There is a single risky asset, and a risk-free asset. Formulate an investor's choice problem and comment on the first-order conditions. What is the minimum risk premium required to induce the individual to invest all his wealth in the risky asset? (Find your answer in terms of his initial wealth, absolute risk aversion coefficient, and other relevant parameters.)

Hint: Take a Taylor series expansion of the utility of next period's random wealth.