I do not seem to understand where the figures in the first line of the solution, given in the revision notes comes from. i.e. E[U(W)] = 0.5U{0+1.5(1-a)w_0} + 0.5U{4aw_0+1.5(1-a)w_0} Where does the 0,1.5,4 and 1.5 come from? Would be very grateful for some help on this.
The numbers represent the amount of money you get back in each of the "failure" and "success" states, if you invest a*w0 in A and (1-a)*w0 in B. The terms in the first curly bracket represents the money obtained in the failure state. If you invest a*w0 in A, which produces a return of -100% (ie you lose all your money), then you end up with 0. If you invest (1-a)*w0 in B, which produces a return of +50%, then you end up with your original investment plus 50% extra, ie 1.5*(1-a)*w0. So, your total wealth in the failure state is 0 + 1.5*(1-a)*w0. The second curly bracket is the money obtained in the success state. If you invest a*w0 in A, which produces a return of +300% (ie you get your original investment back, plus a profit equal to three times that amount), then you end up with 4*a*w0. If you invest (1-a)*w0 in B, which produces a return of +50%, then you again end up with 1.5*(1-a)*w0. So, your total wealth in the failure state is 4*a*w0 + 1.5*(1-a)*w0. Hope this makes sense.
Hey thanks Whippet1!! Your explanation makes perfect sense! Just to check if one of the returns in the table was 0, what happens? E.g. If you invest a*w0 in A, which produces a return of 0% (ie no return), then you end up with 1, your original investment, right?
Yes, you just get your original investment back, without any gain or loss. So, an investment of a*w0 would pay back a*w0.
