Given the following information, what is the standard deviation of the returns on a portfolio that is invested 40 percent in Stock A, 35 percent in Stock B, and the remainder in Stock C? Rate of Return is State Occurs

State of Economy Probability of State of economy Stock A Stock B Stock C

Normal .65 14.3% 16.7% 18.2%
Recession .35 -9.8% 5.4% -26.9%

a. 12.72 percent
b. 14.07 percent
c. 1.41 percent
d. 7.41 percent
e. 11.86 percent

The correct option is e. 11.86 percent.

Explanation:

Note: The data in this question are merged together and they are therefore sorted before answering the question. See the attached pdf for the full question with the sorted data.

The standard deviation of the returns on a portfolio can now be calculated using the following steps:

Step 1: Calculation of expected returns under each state of the economy

This can be calculated using the following formula:

Expected return under a state of the economy = (Percentage invested in Stock A * Return of Stock A under the state of the economy) + (Percentage invested in Stock B * Return of Stock B under the state of the economy) + (Percentage invested in Stock C * Return of Stock C under the state of the economy) …………… (1)

Substituting the relevant values into equation (1), we have:

Expected return under Normal = (40% * 14.3%) + (35% * 16.7%) + (25% * 18.2%) = 0.16115

Expected return under Recession = (40% * (-9.8%)) + (35% * 5.4%) + (25% * (-26.9%)) = -0.08755

Step 2: Calculation of expected return of the portfolio

This can be calculated using the following formula:

Portfolio expected return = (Probability of Normal Occurring * Expected Return under Normal) + (Probability of Recession Occurring * Expected Return under Recession) …………………. (2)

Substituting the relevant values into equation (2), we have:

Portfolio expected return = (0.65 * 0.16115) + (0.35 * (-0.08755)) = 0.074105

Step 3: Calculation of the variance of the returns on the portfolio

This can be calculated using the following formula:

Variance of the portfolio = (Probability of Normal Occurring * (Expected Return under Normal - Portfolio expected return)^2) + (Probability of Recession Occurring * (Expected Return under Recession - Portfolio expected return)^2) …………………….. (3)

Substituting the relevant values into equation (3), we have:

Variance of the portfolio = (0.65 * (0.16115 - 0.074105)^2) + (0.35 * (-0.08755 - 0.074105)^2) = 0.014071259475

Step 4: Calculation of the standard deviation of the returns on the portfolio

This can be calculated using the following formula:

Standard deviation of the portfolio = Variance of the portfolio^0.5 (4)

Substituting the variance of the portfolio obtained in step 3 into equation (4), we have:

Standard deviation of the portfolio = 0.014071259475^0.5 = 0.118622339696197, or 11.8622339696197%

Rounding to 2 decimal places, we have:

Standard deviation of the portfolio = 11.86%

This implies the standard deviation of the returns on the portfolio is 11.86%.

Therefore, the correct option is e. 11.86 percent.

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