Question

Portfolios X is the yield per dollar of stock in company EX. X is approximately normally distributed with mean 0.04 and standard deviation 0.01. W is the yield per dollar of stock in company WAY. W is approximately normally distributed with mean 0.05 and standard deviation 0.02. (X and W are in dollars, i.e. the mean yields are 0.04 and 0.05 dollars, respectively.) X and W are independent. Consider three alternative portfolios. You want to choose the portfolio that has the highest probability of yielding more than 5 dollars.

Portfolio 1. Buy 200 dollars of EX. The yield on this portfolio is denoted Y1.

Portfolio 2. Buy 200 dollars of WAY. The yield on this portfolio is denoted Y2.

Portfolio 3. Buy 100 dollars EX and 100 dollars of WAY. The yield on this portfolio is denoted Y3.

a) Calculate the probability that the yield will be more than 5 dollars with portfolio 1. Four decimals HINT: The question is not asking for the value of the portfolio, but rather the yield on it. If you buy $1 of stock in EX, the yield on that “portfolio” would be X dollars. If you buy $2 of stock in EX, the yield on that portfolio would be 2X dollars, and so forth. Therefore, the yield on the portfolio is a linear transformation of the yield on $1 of stock in EX. Use the results we have about the mean and variance of a linear transformation.

b) WAY’s stock has a higher mean than EX’s stock, leading you to ask if you should choose portfolio 2 instead. Calculate the probability that the yield will be more than 5 dollars with portfolio 2. Four decimals

c) Portfolio 3 involves diversification. Perhaps diversifying will increase the probability of obtaining a yield of more than 5 dollars. Calculate the probability that the yield will be more than 5 dollars with portfolio 3. Four decimals HINT: Now the yield on the portfolio is a linear combination of the yield on $1 of stock in EX and the yield on $1 of stock in WAY. Use the results we have about the mean and variance of a linear combination.

d) Q is the yield per dollar of stock in company QUEUE. Q is approximately normally distributed with the same mean as W and the same standard deviation as W (Q’s mean is 0.05 and its standard deviation is 0.02). However, X and Q have a covariance of minus 0.0001. You are wondering whether you should replace the 100 dollars of WAY with 100 dollars of QUEUE in portfolio 3. Recalculate the probability that the yield will be more than 5 dollars with portfolio 3 (with stock in QUEUE instead of WAY). Four decimals

Answer 1