Hi, I dont quite understand how the following solution was derived. The question is trying to provide an example how the expected utility is a weighted average of the individual outcomes. Question: State an expression for the expectation of the next period utility of investor X, assuming he invests a proportion 'a' in Equity (which pays -4% or +8% with respective probabilities 1/4 and 3/4) and the rest in a non-interest-bearing bank account. He has the utility function U(w) = log(w) Solution: E[U(w)] = 0.25{log((1-0.04a)w)} +0.75{log((1+0.08a)w)} My confusion lies with why is it not 0.25{log (0.96aw)} + 0.75 {log(1.08aw)}? Why is this not the right way to think about it. Would be grateful for an explanation to the solution. Thank you!
Unfortunately your formulation won't work. Looks what happens to your formula when a=0. With proportion a held in Equity A and proportion (1-a) in the bank (with zero return), the wealth of the investor is either: aw(1.08) + w(1-a) or aw(0.96) + w(1-a). The first terms come from Equity A yielding either 8% or -4% and the second terms come from the proportion held in the bank. Simplifying these expressions gives either: w(1+0.08a) or w(1-0.04a) as required. Hope that helps.
