Question

An investor puts $15,000 into each of four stocks, labeled A, B, C, and D. The table shown below contains the means and standard deviations of the annual returns of these four stocks.
Stock Mean Annual Return Standard Deviation of Annual Return
A 0.15 0.05
B 0.18 0.07
C 0.14 0.03
D 0.17 0.06

a. Assuming that the returns of these four stocks are independent of each other, find the mean and standard deviation of the total amount that this investor earns in one year from these four investments.
b. Now assume that the returns of the four stocks are no longer independent of one another. Specifically, the correlations between all pairs of stock returns are given in the table below.
Correlations Stock A Stock B Stock C Stock D
Stock A 1.00 0.5 0.80 -0.55
Stock B 0.50 1.00 0.60 -0.30
Stock C 0.80 0.60 1.00 -0.75
Stock D -0.55 -0.30 -0.75 1.00

Find the mean and standard deviation of the total amount that this investor earns in one year from these four investments
c. Compare the results of Question “a” and “b”. Explain the differences in your answer.
d. Continue to assume that the returns of the four stocks are no longer independent of one another, and the correlations between all pairs of stock returns are as given in Question “b”. Now, suppose that this investor decides to place $20,000 each in stocks B and D, and $10,000 each in stock A and C. Find the mean and standard deviation of the total amount that this investor earns in one year from these four investments.
e. How do the mean and standard deviation of the total amount that this investor earns in one year change as the $60,000 in cash available is reallocated for investment? Provide an intuitive explanation for the changes you observe here.

To answer these questions, we will use some concepts from finance and statistics. Let's go step by step.

a. To find the mean of the total amount that the investor earns in one year, we need to calculate the weighted average of the mean returns of each stock. We can calculate it using the formula:

Mean of total amount = (Weight of stock A * Mean return of stock A) + (Weight of stock B * Mean return of stock B) + (Weight of stock C * Mean return of stock C) + (Weight of stock D * Mean return of stock D)

In this case, the investor puts an equal amount of $15,000 into each of the four stocks, so the weights are 0.25 for each stock.

Mean of total amount = (0.25 * 0.15) + (0.25 * 0.18) + (0.25 * 0.14) + (0.25 * 0.17)

To find the standard deviation of the total amount, we need to calculate the weighted standard deviation of each stock. We can calculate it using the formula:

Standard deviation of total amount = sqrt((Weight of stock A * (Standard deviation of stock A)^2) + (Weight of stock B * (Standard deviation of stock B)^2) + (Weight of stock C * (Standard deviation of stock C)^2) + (Weight of stock D * (Standard deviation of stock D)^2))

Using the provided values, you can substitute the values into the formula to find the standard deviation of the total amount.

b. When the returns of the four stocks are no longer independent, we need to consider the correlations between them. The formula to calculate the standard deviation of the total amount is modified to account for the covariance of the stocks.

Standard deviation of total amount = sqrt((Weight of stock A * (Standard deviation of stock A)^2) + (Weight of stock B * (Standard deviation of stock B)^2) + (Weight of stock C * (Standard deviation of stock C)^2) + (Weight of stock D * (Standard deviation of stock D)^2) + 2 * ((Weight of stock A * Weight of stock B * Standard deviation of stock A * Standard deviation of stock B * correlation coefficient between stock A and stock B) + ...)

You can use the provided correlation table to calculate the standard deviation of the total amount.

c. The results in parts a and b will differ because in part b, we consider the correlations between the stocks. When the stocks are independent, the correlation coefficient between any two stocks is zero. However, when the stocks are not independent, the correlation coefficient affects the standard deviation of the total amount. This means that the risk (measured by standard deviation) of the portfolio can be higher or lower depending on the correlations.

d. To find the mean and standard deviation of the total amount with the given investment allocations, we can use the same formula as in part b, but with the updated weights.

Mean of total amount = (Weight of stock A * Mean return of stock A) + (Weight of stock B * Mean return of stock B) + (Weight of stock C * Mean return of stock C) + (Weight of stock D * Mean return of stock D)

Standard deviation of total amount = sqrt((Weight of stock A * (Standard deviation of stock A)^2) + (Weight of stock B * (Standard deviation of stock B)^2) + (Weight of stock C * (Standard deviation of stock C)^2) + (Weight of stock D * (Standard deviation of stock D)^2) + 2 * ((Weight of stock A * Weight of stock B * Standard deviation of stock A * Standard deviation of stock B * correlation coefficient between stock A and stock B) + ...)

Substitute the provided values and calculate the mean and standard deviation.

e. The mean and standard deviation of the total amount earned will change as the cash is reallocated for investment because the weights of the stocks are changing. When more money is allocated to certain stocks and less to others, it affects the overall mean and standard deviation of the portfolio. Generally, increasing the allocation to higher-performing and less volatile stocks can lead to higher returns and lower risk. Conversely, decreasing the allocation to lower-performing and more volatile stocks can lead to lower returns and higher risk. The specific changes in mean and standard deviation will depend on the characteristics of the stocks and the correlations between them.