Suppose you have an initial wealth equal to W0 = 4000. The risk-free rate is 3% p.a. The stock market offers an expected rate of return of 8% with a standard deviation of 12%. Your preferences could be best described by the quadratic utility function u(W) = aW bW^2 , where the parameters are a = 15 and b = 0.001.

a) Suppose you invest 50% of your wealth into the risk-free account and the rest into the stock market. Calculate the expected final wealth and the standard deviation of the final wealth.

b) Note that the quadratic utility function has a representation with respect to expected final wealth and the standard deviation of final wealth , which is E[u(W (W , W )] = aW b( 2 + 2 W ).

Calculate the expected utility of (i) a portfolio completely invested in the risk-free rate, and (ii) a portfolio completely invested in the stock market. Compare both results and give an interpretation!

c) Draft the overall situation in a -diagram in terms of final wealth. Show the budget constraint, the indifference curve, and the optimal portfolio.