Resume the assumption that Mr Somebodys risk preferences are represented by the expected utility function ()=, and continue to assume that =0.25 . Suppose that a second asset becomes available: call it Beta. Betas return is equal to x, with probability , and 4, with probability 1, where x is a non-negative real number. The following table summarises the relevant information.

= (b) Resume the assumption that Mr Somebody's risk preferences are represented by the expected utility function u(w) = Vw, and continue to assume that p = 0.25. Suppose that a second asset becomes available: call it Beta. Beta's return is equal to x, with probability p, and 4, with probability 1-2, where x is a non-negative real number. The following table summarises the relevant information. State 1 of the world (prob.p) State 2 of the world (prob. 1-p) O 16 Alpha's returns (per invested) Beta's returns (per invested) x 4 Mr Somebody has now the opportunity to either invest 100% of their wealth in Alpha, invest 100% of their wealth in Beta, or diversify their portfolio by investing fraction of their wealth in Alpha and a fraction of their wealth in Beta. Specifically, let 1 [0,1] denote the fraction of Mr Somebody's wealth invested in the Alpha asset and 1-2 the fraction of Mr Somebody's wealth invested in the Beta asset. As indicated in the table above, continue to assume that there are only two relevant states of the world (state 1 and state 2). i. Work out Mr Somebody's optimal diversification strategy 1* as a function of x. ii. For what values of x, would Mr Somebody like to invest 100% of their wealth in the Alpha asset? And in the Beta asset? iii. Calculate the correlation coefficient between the returns of the Alpha asset and the returns of the Beta asset in two cases: (1) when x=0 and (2) when x=20. Compare the correlation coefficients calculated in these two cases with the optimal diversification strategy you derived in a previous sub-question. Are these related in a way that you expect? Comment. In answering the above make sure you fully explain your reasoning