Consider an optimal portfolio problem

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Consider an optimal portfolio problem. Suppose there are two assets: a safe one and a risky one. Without loss of generality, suppose that the net rate of return on the safe asset is 0. The rate of (random) return on the risky asset is denoted by a random variable ˜ z. The average return on the risky asset is assumed to be higher than that on the safe asset, that is, E(˜ z) > 0. Consider an investor with concave (Bernoulli) utility function u(x) who has initial wealth w to invest. Let α ≥ 0 and β = w−α ≥ 0 denote the amounts of wealth invested in the risky and safe asset. 1. Show that the optimal demand for the risky asset, denoted by α∗, is strictly positive. Then, we take the interior solution following, that is, α∗ ∈ (0,w), therefore FOC is su?cient and necessary for optimality.

2. If the investor exhibits DARA, show that α∗ increases with w. That is, the risky asset is a normal good.

3. Denote γ∗ = α∗/w. Show that γ∗ decrease with w if the investor exhibits IRRA. (This result implies that the elasticity of the demand for risky asset to wealth is smaller than 1, therefore is a necessary good.)

4. Now consider another investor, named Frank, who is more risk averse in the sense of more concave utility function. Show that Frank demands less risky asset than α∗.