1. Prove that E(x−μx)2 = E(x2) – E2(x).
2. An investor has an opportunity to buy stock in two publicly traded companies: Avvoltoio Airlines and Unctuous Energy. If the investor puts his money in a stock, and the company does well, he earns a dividend of $10. If the company does not do well, he earns $2. Avvoltoio tends to do well when oil prices are low; Unctuous tends to do well when oil prices are high. The returns are therefore negatively correlated.
Dividends have the following probability distribution.
Unctuous Dividend
U = 2 U = 10
Avvoltoio A = 2 .2 .3
Dividend A = 10 .3 .2
a. For each stock, compute the mean dividend and the variance of the dividend. Are you computing a population mean, or a sample mean?
b. Compute the covariance of the returns, Cov(A, U), as well as their correlation, rAU. Explain why these are negative.
c. Compute E[A | U = 2] and E[A | U = 10].
Explain in economic terms why E[A | U = 2] > E[A | U = 10].
d. Suppose the investor decides to diversify by investing equally in each of the two stocks. Write down the distribution of dividends for the 2-stock portfolio. (Hint: Since half the money is invested in each stock, the most each stock can pay is $5.) Compute the mean and the variance of the distribution. Is the diversified portfolio preferable? Explain.