An umbrella salesman expects profits of $1000 if the summer is rainy, and $500 if it is sunny. The summer will be rainy with probability p. He can also (if he wishes) invest any amount $K in a sunglasses store, which pays a return rate of r_R < 0 if the summer is rainy, r_S > 0 if the summer is sunny.

(a) Write out his expected utility if initial wealth is w₀, he invests $K in the sunglasses store, and utility from wealth w is u(w), via:

i. Calculate his final wealth, w^rainy , in a rainy summer. (Hint: his umbrella business earns $1000 in this case, and the $K in the sunglasses store earns a profit of $K times the rainy-summer return rate).

ii. Calculate his final wealth, w^sunny, in a sunny summer.

iii. Use your answers to (i) and (ii) to find expected utility.

(b) If r_R = -(1-p) and r_S = p, show that a risk-averse umbrella will choose $K so that final wealth does not depend on weather, i.e. w^rainy = w^sunny .

(c) What investment $K achieves the outcome in (b) (wealth independent of weather), and what is the variance of his total wealth?