109 Sept 2002 Q1(b)

[lucky999]

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.

 

[Whippet1]

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.

 

[lucky999]

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?

 

[Whippet1]

Yes, you just get your original investment back, without any gain or loss.

So, an investment of a*w0 would pay back a*w0.