1. Let in and i, denote the interest rate in the first and second year of a two-year investment. The rates i, and is are independent random variables with the following distributions: P(i = -2%) = 0.10, P(1 = 3%) = 0.55, P(61 = 7%) = 0.35, P(i, = -4%) = 0.20, P(1 = 4%) = 0.50, P(i, = 8%) = 0.30. Let S, denote the accumulated amount at the end of the two years for an initial invest- ment of E1. (a) [11 marks] i. Identify all possible values of S, together with the probability each value occurs and calculate E[S_] and Var(S.). ii. An investor is targeting an accumulated amount which is at least 10% larger than his initial investment. Calculate the following probabilities. A. The probability that the investor achieves the target. B. The probability that the investor makes a loss at the end of the two years. Suppose now that (1+1)) and (1+12) are independent, identically distributed, lognormal random variables with parameters / and o'. (b) [5 marks] Find the values of the parameters a and a? for (1 + i, ) and (1 + 12) knowing that E[S,] and Var($2) are equal to the values calculated in part (@). Consider now an investment of Ca-1 units of capital at the start of the k-th year for n years, 1 S k S n, where the interest rates i, in each year are independent random variables. (c) [9 marks] i. Let C_1 = 1 for all years 1