1. Suppose an investor has preferences represented by the utility function
w1−γ
u(w) = . 1 − γ
and has wealth w0.
In the market there are two assets: a risky asset whose return can be either low, r1, or high, r2, with probabilities π and 1 − π; a risk-free asset with return rf . Suppose r1 < rf < r2.
a) derive the demand for the risky asset (checking for interior and for corner solutions, if any);
b) Discuss how the demand changes with a change in welfare;
c) Discuss how the demand changes with a change in rf .
2. Consider an economy with one risk-free asset whose net return is rf = 0.05 and one risky asset whose return is either r1 = 0.01 with probability π or r2 = 0.10 with probability 1 − π.
a) For which values of π does a risk neutral investor invest in the risky asset? How much would they invest?
b) For which values of π does a risk averse investor invest in the risky asset? How much would they invest?
c) Suppose an investor has preferences represented by the natural logarithm utility function, u(c) = log(c), and initial wealth equal to 10. Which share of his wealth does he invest in the risky asset? Suppose his wealth is 100. Which share of his wealth does he invest in the risky asset? Explain.
3. Consider an economy with a risk-free asset whose net return is rf , and one risky asset whose net return, r, is normally distributed, with mean E(r) and variance σ2. An investor has preferences represented by the following utility function:
u(w) = − 1
γ
e−γw.
The investor makes investment decisions in period 0 and only consumes in period 1.
a) Does this investor have decreasing, constant or increasing absolute risk aversion? decreasing, constant or increasing relative risk aversion?
b) Explain why this investor’s demand of the risky asset is equal to that of an investor with mean-variance utility u(w) = E(w) − γ V ar(w).
c) The net return of the risky asset in this economy can be written as r = x−p , where x is the payoff paid by the asset (e.g., the dividend) in period 1 and p is the price paid in period 0. Assume there is a fixed supply a¯ of the asset. Derive the equilibrium price of the risky asset, p, as a function of the expected payoff, the risk-free return, the coefficient of risk aversion and the variance of the risky return.
d) Using the result obtained in c, explain the effect of a reduction in the risk free rate on the price of the risky asset. (Note that this is equivalent to the effect of a central bank’s decision to lower interest rates on the stock market index.)
4. Consider two securities with net rates of return ri and rj. Suppose that these two securities have identical expected rates of return and identical variance. The correlation coefficient between ri and rj is ρ. Find the weighted portfolio that achieves the minimum variance. How do the weights depend on ρ?
5. Consider a two-period economy in which investors have pref- erences represented by:
ln(ct) + βE[ln(ct+1)].
The subjective discount factor is β = 0.9. The supply of the risky asset is a¯ = 1. Households have an endowment et = 10 at time t; at time t +1 their endowment et+1 is equal to 6 with probability 2/3 and 9 with probability 1/3.
In the economy there is an asset paying a payoff, xt+1, equal to 0 with probability 1/3 or to 6 with probability 2/3. Specifically, xt+1 and et+1 have the following joint distribution:
|
|
xt+1 = 0
|
xt+1 = 6
|
|
|
et+1 = 6
|
1/3
|
1/3
|
2/3
|
|
et+1 = 9
|
0
|
1/3
|
1/3
|
|
|
1/3
|
2/3
|
1
|
a) Compute the price of this asset;
b) Suppose the subjective discount factor were β = 0.8 instead of β = 0.9; would the price be lower or higher than that computed for question “a”?
c) Consider another asset with the same payoffs, 0 and 6, but with the following joint distribution:
|
|
xt+1 = 0
|
xt+1 = 6
|
|
|
et+1 = 6
|
1/2
|
0
|
1/2
|
|
et+1 = 9
|
0
|
1/2
|
1/2
|
|
|
1/2
|
1/2
|
1
|
Find the price of this asset and explain why it is higher or lower than (or equal to) the price you computed for question “a”.
d) What is the expected return of the first asset? What is the risk-free grossreturn, Rf, in this economy? Explain your results.