Problem 3 [15 marks]

An investor in the stock market purchased a single share of stock for each of 50 different companies. Suppose each stock is worth $20. Every day, each stock goes up by $1 with probability p or down by $1 with probability (1 ? p). Note that the stocks are independent of each other, and each day is independent of the others as well. Denote the value of the k-th stock after T days (where T is a constant) by STk (for k = 1, . . . , 50). The initial value of the portfolio of the investor is

K 50 ?S0k =?20=2050=1,000. k=1 k=1

(a) What is the probability that a share of stock (of any company) was worth $23 after 5 days if it was worth $26 after 10 days? [5 marks]

(b) What is the expected value and the standard deviation of the value of the investor's portfolio after T days? Your answer should be in terms of T and p. Assume that T ? 20 days. [5 marks]

(c) Is it more advantageous for the investor to invest in 50 shares of the same company, or keep his current strategy of buying a single share of stock for 50 different companies? Explain your answer by comparing the expected value and variance of the portfolio values after T days. Again, you can assume that T ? 20 days. [5 marks]

Problem 3 [15 marks] An investor in the stock market purchased a single share of stock for each of 50 dierent companies. Suppose each stock is worth $20. Every day, each stock goes up by $1 with probability p or down by $1 with probability (1 p). Note that the stocks are independent of each other, and each day is independent of the others as well. Denote the value of the k-th stock after T days (where T is a constant) by S; (for k = 1, . . . ,50). The initial value of the portfolio of the investor is K 50 255\" = 220 = 20 x 50 = 1,000. (a) What is the probability that a share of stock (of any company) was worth $23 after 5 days if it was worth $26 after 10 days? [5 marks] (b) What is the expected value and the standard deviation of the value of the investor's portfolio after T days? Your answer should be in terms of T and p. Assume that T S 20 days. [5 marks] (c) Is it more advantageous for the investor to invest in 50 shares of the same company, or keep his current strategy of buying a single share of stock for 50 dierent companies? Explain your answer by comparing the expected value and variance of the portfolio values after T days. Again, you can assume that T S 20 days. [5 marks]