Suppose there are two assets available to an investor. One is risk-free and has a return of 3 percent. The other is risky and has an expected return of 8 percent and a variance of 0.05. The investor’s utility is given by U(r) = (2/3) E(r) − (1/2) AV aσ(r) and his risk aversion coefficient is 3.
A. The investor is trying to decide what fraction of his wealth he will invest in the risky asset. Write down the investor’s maximization problem.
B. Take the first order conditions for the investor’s problem. Find a formula for y∗ , the optimal fraction of wealth that the investor will invest in the risky asset.
C. Find the value of y ∗ given the characteristics of both assets and the risk-aversion of the investor.
Suppose now that the investor’s utility was given by U(r) = E(r) − (1/2) Aσ(r) where A is his risk aversion coefficient and σ is the standard deviation of returns. Suppose investor cannot borrow at any risk-free rate.
D. Fix a level of utility for this investor and use it to draw an indifference curve on a graph that has expected return on the vertical axis and standard deviation on the horizontal axis.
E. Write down a rule about how the investor will choose what fraction of wealth he will invest in the risky asset and which fraction in the risk-free asset.
F. Apply the rule you wrote above to find the value of y ∗ given the characteristics of both assets and the risk-aversion of the investor.