(Calculating portfolio variances) Marge owns three stocks, Apple (AAPL), Google
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Question
(Calculating portfolio variances) Marge owns three stocks, Apple (AAPL), Google (GOOG) and Facebook (FB). She expects the price per share of each stock one month from now to be 120, 60, and 60 dollars, respectively. An analysis of the returns to holding these three stocks shows that the monthly standard deviation of the price per share for each stock is 10, 8, and 8 dollars, respectively. This same analysis also concludes the covariance between the price per share of Apple stock and the price per share of Google stock is -36 (dollars squared), between Apple and Facebook is +24, and between Google and Facebook is +19. Answer the following questions assuming that Marge owns 200 shares of Apple, 100 shares of Google, and 50 shares of Facebook.
(a) Compute the expected value of Marge’s portfolio one month from now.
(b) Compute the standard deviation of the value of her portfolio one month from now.
(c) It turns out that Marge’s sister, Maggie, owns 200 shares of Apple, 50 shares of Google, and 100 shares of Facebook. Assuming that Marge and Maggie share the same expectations about the performance of each stock, compute the expected value and standard deviation of Maggie’s portfolio one month from now.
(d) Who do you think owns a better portfolio? Explain.
Explanation / Answer
A) Calculating Portfolio Variances. In this problem, we are asked to find the expected value and variance of a portfolio of stocks. To do that we will repeatedly use the formula for the expected value of a sum of random variables, and the formula for the variance of the sum of random variables. (a) To compute the expected value, we express the value of the portfolio in terms of the value of the Apple shares, the value of the Google shares, and the value of the Facebook shares. Mathematically, let PAAPL be the price of one share of Apple one month from now, PGOOG be the price of one share of Google one month from now, and PFB be the prices of one share of Facebook one month from now. Then, the total value of Marge’s portfolio in one month is: Total Value of Portfolio=200 × PAAPL + 100 × PGOOG + 50 × PFB. This now implies that: E(Total Value of Portfolio) = E(200 × PAAPL + 100 × PGOOG + 50 × PFB) = (200 × $120 + 100 × $60 + 50 × $60) = $24, 000 + $6, 000 + $3, 000 = $33, 000.
B) To calculate the standard deviation, we will first need to calculate the variance in the total value of the portfolio. In the process we will use the following fact. The variance of a sum of three random variables equals the sum of the variances of the three plus the 2 times the covariance of each pair of random variables. The trick in this problem was to derive this formula by applying the formula for a variance of a sum of two random variables twice (in succession). So, to get the variance of Marge’s portfolio, we have: Var(Value of Portfolio) = Var(200 × PAAPL + 100 × PGOOG + 50 × PFB) = Var(200×PAAPL)+Var(100×PGOOG)+Var(50×PFB)+ 2×Cov(200PAAPL,100PGOOG)+2×Cov(200PAAPL,50PFB)+ 2×Cov(100PGOOG,50PFB) = 40000Var(PAAPL)+10000Var(PGOOG)+2500Var(PFB)+40000Cov(PAAPL,PGOOG) +20000Cov(PAAPL,PFB)+10000Cov(PGOOG,PFB) = 40000 × 100 + 10000 × 64 + 2500 × 64 + 40000 × (36) + 20000 × 24 + 10000×19 = 4, 030, 000 (dollars)2 and the standard deviation = sqrt(4, 030, 000) = 2007.49 dollars.
C) For Marge’s sister, Maggie, the calculations are pretty much the same. It should come as no surprise that the expected value of Maggie’s portfolio, one month from now, is exactly the same as that of Marge’s portfolio = $ 33, 000 dollars. The variance of Maggie’s portfolio, one month from now, is a little higher = $ 5, 230, 000 (dollars)2 and the corresponding standard deviation is $2286.92 dollars. (d) There isn’t very much difference between the performances of the two portfolios. However, Marge’s portfolio gives the same expected return, with a slightly lower standard deviation; that is, she exposes herself to less risk. In that sense, her portfolio performs better than Maggie’s.
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