5. Suppose you initial worth is $100. You are given two investment opportunities. Investment A: A risk-free investment that earn r% in 1 year. Investment B: A risky investment that earn 20% with probability 0.6, or loss 20% with probability 0.4. Your utility function is u(x) = ln(x), where x is your worth level.

(a) You are required to invest all your $100 into one of the investments. What is the value of r such that you will be indifferent between the two investments?

(b) Given that r = 2 (2% risk free rate). Suppose that you can allocate certain proportion of your $100 worth into the above two investment. For example, you may invest $60 in investment A and the remaining $40 in investment B. What is the optimal investment mix such that your expected utility is maximized?

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