The return on an investment is the gain (or loss) on the investment divided by i

Question

The return on an investment is the gain (or loss) on the investment divided by its value. For example, if you buy a share for $100 and the price of the stock increases to $110, your return on the stock is: Return = (110 - 100)/100 = 0.10, or 10% In addition to the return on an individual stock, you can calculate the return on a portfolio (group) of stocks. The return on a portfolio of stocks that represents the entire market is called the market return. Investors don't know the return on a stock in advance, so the return on a stock or a portfolio of stocks is a random variable. Define random variable Y as the return on ABC stock and random variable X as the market return. Imagine that the ABC stock return is linearly related to the market return. In addition, in a given time period, events specific to ABC that are unknown in advance might affect the return on ABC stock. These ABC-specific factors are measured by the random variable Z. The following equation describes the relationship among the random variables Y, X, and Z: Y = 0.005 + 1.1X + Z Assume that Z and X are independent because events specific to ABC are unrelated to the market return. The expected value of X is mu_X = 0.087, or 8.7%, and the standard deviation of X is sigma_X = 0.09, or 9%. The expected value of Z is mu_Z = 0, and the standard deviation of Z is sigma_Z = 0.27, or 27%. As a risk-averse investor, you are interested in the standard deviation of the return on an investment in ABC as well as the expected value of its return. The standard deviation of its return is a measure of the risk associated with an investment in ABC. The following steps will help you compute the expected value mu_Y and the standard deviation sigma_Y of ABC stock returns. Consider the random variable W = 0.005 + 1.1X. The expected value of W is ___, the variance of W is _____, and the standard deviation of W is ____. Observe that Y = W + Z. Note that random variables W and Z are independent because random variables X and Z are independent. The expected value of Y is ______, the variance of Y is _____, and the standard deviation of Y is _____.

Explanation / Answer

Expectation Var = SD^2 SD X 0.087 0.0081 0.09 given Z 0 0.0729 0.27 given W 0.1007 0.009801 0.099 calculated below Y 0.1007 0.0827 0.2876 calculated below W = 0.005+1.1X E(W) = 0.005 + 1.1*E(X) E(W) = =0.005+1.1*0.087 0.1007 V(W) = 0 + 1.1^2*V(X) =1.1^2*0.0081 0.0098 SD(W) = SQRT(V(W)) =SQRT(0.009801) 0.0990 Y = W + Z E(Y) = E(W)+E(Z) =0.1007+0 0.1007 V(Y) = V(W+Z) V(W)+V(Z)+COV(WZ) V(W)+V(Z) =0.009801+0.0729 0.0827 SD(Y) = SQRT(V(Y)) =SQRT(0.082701) 0.2876

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