Question on page 5 of Chapter 2 on course notes

[Millie]

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!

 

[Steve Hales]

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.