Question: Assume investors are choosing their portfolios according to the CAPM. Suppose the risk free net interest rate is 1%.

1. Assume investors are choosing their portfolios according to the CAPM. Suppose the risk free net interest rate is 1%. The mean net return on the NYSE (a proxi for the market portfolio) is 16%. The standard deviation of the NYSE index is 1%.

a) Suppose one of your clients has preferences represented by the following utility in the asset return, r:

U(r) = r - r2.

She has an endowment of £100,000. How much would you recommend her to invest in the NYSE and how much in the risk-free asset?

b) Consider the case in which the risk free net interest rate is now 2%. The mean net return on the NYSE now is 17%. The standard deviation of the NYSE index is 1%. Compute the optimal investment in such a case. How does it compare to your previous answer? Explain.

c) Finally, consider the case in which the risk free net interest rate is 1%. The mean net return on the NYSE is 17%. The standard deviation of the NYSE index is 1%. Compute the optimal investment in such a case. How does it compare to your previous answers? Explain.

2. Consider assets 1 and 2 having the following distribution of returns:

P(r1                                                                                     =  1 and r2 = 0.15) = 0.1, P(r1                                = 0.5 and r2 = 0.15) = 0.8, P(r1                                = 0.5 and r2 = 1.65) = 0.1.

(a) Calculate the mean, variance and covariance of these two assets.

(b) Assuming that only assets 1 and 2 exist in the market, what is the frontier of all possibile portfolios in (µ, 2)? Write down the equation of the frontier and calculate the values of (µ, 2) corresponding to the percentages 0%, 25%, 50%, 75% and 100% invested in asset 1.

(c) Which portfolios belong to the e cient frontier? That is, which portfolios have the lowest variance for a given level of expected return?

(d) Show that asset 1 mean-variance dominates asset 2 in all but one of the e cient portfolios. Give an intuitive explanation.

3. Consider an agent with DARA preferences. Suppose he only has access to a financial market with 5 risky assets. He chooses his optimal allocation, (!1, !2, !3, !4,!5). Suppose now he becomes poorer. Would the weights (!1, !2, !3, !4,!5) change? Suppose, now, that asset 5 can be considered as a risk-free asset. Would your answer change?

4. Consider an economy with complete markets. There are 3 states of nature (!1,!2,!3) and 3 Arrow securities. Specifically, at time 0 investors can buy these Arrow securities at the following prices:

t = 0                                                                            t = 1

Price                                                                          !1       !2 !3 pc(1) = 0.4                                                                              1       0 0 pc(2) = 0.2                                                                              0       1 0 pc(3) = 0.3                                                                              0       0 1

a) Consider a bond that o↵ers the payo↵ 1 in any state of nature. What is the price of this bond?

b) Consider a stock that o↵ers the payo↵ 1 in state 1, 3 in state 2 and 6 in state 3. What is the price of this stock?

c) What is the gross risk free return in this economy?

d) What is the price of the risk free return in this economy?

5. Consider an endowment economy where a representative consumer may buy shares of equity. The consumer’s preferences for consumption at time t are represented by the following utility function:

U(Ct) = 5+10logCt.

The consumer maximizes Et P1  i(5 + 10logCt+i) subject to a budget con-straint given by

Ct = DtSt +PtSt  PtSt+1,

where St represents the shares of equity holdings at time t, Pt is the price of equity at time t, and Dt is the dividend per share at time t.

Furthermore, suppose that the economy-wide resource constraint is given by

Ct = Dt,

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and that the following condition is satisfied: limj!1 Et j                                                                                                                                                                             t+j = 0.

(i) Write down the Bellman equation and derive the consumer’s first order condition for consumption. Interpret the expression that you obtain.

(ii) Use the consumer’s first order condition, the dynamic budget constraint and the resource constraint to derive an expression for the equity price Pt as a function of the asset’s dividends.

(iii) Explain intuitively why an increase in expectations of future dividends does or does not affect the price of the asset. Would your answer change if the

(iv) Repeat the same analysis (all three points) for the utility function U(Ct) =  Ct 1 and for U(Ct) = Ct.

(v) Now generalize your comments to points (iii) and (iv) for any coe cient

of risk aversion ✓ lower, equal or higher than 1. [For a real world interpretation, see Cochrane, Chapter 2, Problem 1 (page 46).]