An individual with unit initial wealth will construct a portfolio consisting of fractions 1 − α of a risk-free asset and α of a risky asset. The risk-free asset has return rate rf , a known constant.
The risky asset has return rate r1, a random variable with expected value rbar1 and variance σ1^2 . The return rate for the portfolio is the random variable r = (1 − α)rf + αr1.
Assume utility function U(x) = x − (1/3)x^2 for the individual.

a) Find formulas for E[r] and for E[r^2] in terms of α, rf , rbar1, and σ1^2.
b) After one year the utility of the wealth is U(1 + r). Write out the formula for E[U (1 + r)] = E[ 1 + r −(1/3)(1 + r)^2] in terms of α, rf , rbar1 and σ1^2 .
c) If rf = (1/40), rbar1 = (1/20) and σ1 = (1/5), find α for the portfolio of maximum expected utility.