mathematical finance
4. (8 pt) In anticipation of the new iPhone and Galaxy Note, you decide to invest some of your $100,000 savings in the Apple and Samsung stocks. The current stock prices are $112.71 for Apple and $1550.00 for Samsung. To decide how much to invest in each stock, you ask your trusted investment advisor, Rupert, how he sees the stock prices in one moth. He tells you that the following 4 scenarios are equally likely to happen: (i) Both stocks will go up by 10%; (ii) Both will go down by 10%; (iii) Apple will go up by 10% while Samsung will go down by 5%; or (iv) Samsung will go up 10% while Apple price doesn't change. (a) (2 pt) Suppose that you buy 500 shares of apple and 20 shares of Samsung and keep the remainder of your money in your current account (which pays zero interest rate). What is the mean and variance of your portfolio return? (b) (2 pt) Suppose that you decide to spend all of your wealth in the two stocks. How many shares of each stock should you buy (or short-sell) such that your portfolio return has the minimum possible variance? (c) (1 pt) After examining years of daily data on hundreds of US equities, you discover the following relationship between the monthly returns of 3 stocks, A, B, and C: 1 RA = RP+RC 400 What should the effective annual risk free rate be? Why? (Hint: Find a risk-free portfolio of A, B, and C.) (d) (3 pt) Let R1 and Ry be the daily return of two equities, and let 141, M2, ci and oz be the mean and variance of the returns. Find a relationship between R and R, if the returns are perfectly correlated. Determine the risk free rate. (Hint: Recall that two random variables R1 and R2 are perfectly (anti-)correlated if and only if R1 a x R2 +b, for some constants a and b.) 4. (8 pt) In anticipation of the new iPhone and Galaxy Note, you decide to invest some of your $100,000 savings in the Apple and Samsung stocks. The current stock prices are $112.71 for Apple and $1550.00 for Samsung. To decide how much to invest in each stock, you ask your trusted investment advisor, Rupert, how he sees the stock prices in one moth. He tells you that the following 4 scenarios are equally likely to happen: (i) Both stocks will go up by 10%; (ii) Both will go down by 10%; (iii) Apple will go up by 10% while Samsung will go down by 5%; or (iv) Samsung will go up 10% while Apple price doesn't change. (a) (2 pt) Suppose that you buy 500 shares of apple and 20 shares of Samsung and keep the remainder of your money in your current account (which pays zero interest rate). What is the mean and variance of your portfolio return? (b) (2 pt) Suppose that you decide to spend all of your wealth in the two stocks. How many shares of each stock should you buy (or short-sell) such that your portfolio return has the minimum possible variance? (c) (1 pt) After examining years of daily data on hundreds of US equities, you discover the following relationship between the monthly returns of 3 stocks, A, B, and C: 1 RA = RP+RC 400 What should the effective annual risk free rate be? Why? (Hint: Find a risk-free portfolio of A, B, and C.) (d) (3 pt) Let R1 and Ry be the daily return of two equities, and let 141, M2, ci and oz be the mean and variance of the returns. Find a relationship between R and R, if the returns are perfectly correlated. Determine the risk free rate. (Hint: Recall that two random variables R1 and R2 are perfectly (anti-)correlated if and only if R1 a x R2 +b, for some constants a and b.)
