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a. (10 points) Let P1 and P2 be two portfolios whose returns have a joint normal distribution with means 1 and 2, standard deviations 1 and 2, and correlation . Suppose the initial investments are S1 and S2. Show that VaR(P1+P2) VaR(P1)+VaR(P2) under joint normality of the returns. b. i. ( 2 points) Suppose that stock A sells at $85 per share and stock B at $35 per share. A portfolio has 300 shares of stock A and 100 of stock B. What are the weight w and 1w of stocks A and B in this portfolio? ii. (3 points) More generally, if a portfolio has N stocks, if the price per share of the j-th stock is Pj, and if the portfolio has nj shares of stock j, then find a formula for wj as a function of n1,,nN and P1,,PN. iii. (5 points) Let RP be a return on some type on a portfolio and let R1,,RN be the same type of returns on the assets in this portfolio. Is RP=w1R1+ +wNRN true if RP is a net return? Is this equation true if RP is a gross return? Is it true if RP is a log return? Justify your answers. a. (10 points) Let P1 and P2 be two portfolios whose returns have a joint normal distribution with means 1 and 2, standard deviations 1 and 2, and correlation . Suppose the initial investments are S1 and S2. Show that VaR(P1+P2) VaR(P1)+VaR(P2) under joint normality of the returns. b. i. ( 2 points) Suppose that stock A sells at $85 per share and stock B at $35 per share. A portfolio has 300 shares of stock A and 100 of stock B. What are the weight w and 1w of stocks A and B in this portfolio? ii. (3 points) More generally, if a portfolio has N stocks, if the price per share of the j-th stock is Pj, and if the portfolio has nj shares of stock j, then find a formula for wj as a function of n1,,nN and P1,,PN. iii. (5 points) Let RP be a return on some type on a portfolio and let R1,,RN be the same type of returns on the assets in this portfolio. Is RP=w1R1+ +wNRN true if RP is a net return? Is this equation true if RP is a gross return? Is it true if RP is a log return? Justify your answers