Assume there is a riskless treasury security that pays a rate of return equal to r; . In addition, there is one risky security (e.g., S&P 500 index fund) that pays a stochastic rate of return equal to 1. The individual's initial wealth is Wo. At the beginning of the period, the individual has to decide how much, in dollar amount, denoted by A, to invest in the risky security. VII. (2 points) Write down the end-of-period wealth, W, in terms of rf, , W, and A. VIII. (2 points) Suppose that the individual has utility function U (W) = ln(). She/he chooses A to maximize the expected utility at the end of the period. Write down the first-order condition of this problem. IX. (3 points) Suppose that i= 4r; with probability Fry with probability Based on your answer from the previous question, what is the optimal A? Assume there is a riskless treasury security that pays a rate of return equal to r; . In addition, there is one risky security (e.g., S&P 500 index fund) that pays a stochastic rate of return equal to 1. The individual's initial wealth is Wo. At the beginning of the period, the individual has to decide how much, in dollar amount, denoted by A, to invest in the risky security. VII. (2 points) Write down the end-of-period wealth, W, in terms of rf, , W, and A. VIII. (2 points) Suppose that the individual has utility function U (W) = ln(). She/he chooses A to maximize the expected utility at the end of the period. Write down the first-order condition of this problem. IX. (3 points) Suppose that i= 4r; with probability Fry with probability Based on your answer from the previous question, what is the optimal A