1. Consider a 2-year investment with annual effective rates i1, i2, in each year respectively. Assume throughout that the rates are independent and they have the same distribution. Let S denote the value at time t = 2 years of this 2-year investment with initial capital C = 100. These assumptions (a) [5 marks] Assume that the distribution of ij is given by Pli1 = 0.02) = 1/4, P(i = 0.03) = 3/4 Find the distribution of S2. (b) [6 marks] Under the assumptions of (a) calculate the expected value and the vari- ance of S, and show your workings. () [3 marks] Now assume instead that 1+1 In N(, 02) for some parameters H>0,0 > 0. What is the distribution of S2? Explain your answer. (d) [5 marks] Under the assumptions of (c), assume further that E[11] = 0.02, Var[in] = 0.04. Find u and op (e) [6 marks] Under the assumptions of (c) consider also a random variable Y, inde- pendent of (11.12), such that In(100) + 2 PY +24) exp = 1/3, PY In(100) + 2 1+ PCY exp = 2/3 o o Find the probability that the value of the investment after 2 years is at least Y and show your workings. You may use that (1) = 0.84134, where 0(x) = P(Z 0,0 > 0. What is the distribution of S2? Explain your answer. (d) [5 marks] Under the assumptions of (c), assume further that E[11] = 0.02, Var[in] = 0.04. Find u and op (e) [6 marks] Under the assumptions of (c) consider also a random variable Y, inde- pendent of (11.12), such that In(100) + 2 PY +24) exp = 1/3, PY In(100) + 2 1+ PCY exp = 2/3 o o Find the probability that the value of the investment after 2 years is at least Y and show your workings. You may use that (1) = 0.84134, where 0(x) = P(Z