Answer:
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.
