An individual investing has a utility function U-aW-bW² where W is wealth and a,b are constant parameters. She can hold a portfolio of the following two assets: cash that has a net interest rate of 0, or a zero dividend share which has a price of 1 today and will have a price of P₁ next period in state 1 with probability 0.5 and P₂ in state 2 with probability 0.5. Assume that P₁>1 and P₂<1. The individual also receives other income (that is added to wealth at t+1) in state 1 of zero and in state 2 of X₂. The individual has a Wealth of W₁ at time t and wishes to maximise expected utility over a single period. (a) Solve for the individual's optimal portfolio in terms of a, b, W, P₁, P2 and X2. (15 marks) (b) Discuss the relationship between S (the share of wealth held in the share) and W, and X2 in the context of hedging income risk through portfolio management (10 marks)
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