Calculating Returns and Standard Deviations (LO1, CFA2) Consider the following information:
a. Calculate the expected return for the two stocks. (Do not round intermediate calculations. Enter your answers as o percent rounded to 2 decimal pleces.)
b. Caleulate the standard deviation for the two stocks. (Do not round your intermediate caiculations. Enter your answers as a percent rounded to 2 decimal places.) Problem 11-8 Calculating Returns and Standard Deviations (LO1, CFA2) Consider the following information: o. Calculate the expected return for the two stocks, (Do not round intermediate calculations, Enter your onswers os o percent rounded to 2 decimal places.) b. Calculate the standard deviation for the two stocks. (Do not round your intermediate calculations. Enter your answers os o percent rounded to 2 decimal places.)
The Correct Answer and Explanation is:
To calculate the expected return and standard deviation for two stocks, we need to follow the formulas and process for both metrics.
Part (a): Calculating the Expected Return
The expected return for a stock is calculated as the weighted average of the returns in each state of the economy, based on the probabilities of each state occurring. The formula for expected return ((E(R))) is:
[
E(R) = \sum (P_i \times R_i)
]
Where:
- (P_i) is the probability of state (i),
- (R_i) is the return in state (i).
Let’s assume we have the following data for Stock A and Stock B:
- State 1: Probability = 0.4, Return for Stock A = 10%, Return for Stock B = 12%
- State 2: Probability = 0.6, Return for Stock A = 14%, Return for Stock B = 8%
For Stock A:
[
E(R_A) = (0.4 \times 10) + (0.6 \times 14) = 4 + 8.4 = 12.4\%
]
For Stock B:
[
E(R_B) = (0.4 \times 12) + (0.6 \times 8) = 4.8 + 4.8 = 9.6\%
]
Part (b): Calculating the Standard Deviation
The standard deviation of returns measures the variability or risk of a stock. It is calculated by first finding the variance, which is the weighted average of squared deviations from the expected return, and then taking the square root of the variance.
The formula for the standard deviation ((\sigma)) is:
[
\sigma = \sqrt{\sum P_i \times (R_i – E(R))^2}
]
For Stock A:
- Calculate the deviations from the expected return (12.4%):
- Deviation for State 1: (10 – 12.4 = -2.4)
- Deviation for State 2: (14 – 12.4 = 1.6)
- Square the deviations:
- ((-2.4)^2 = 5.76)
- (1.6^2 = 2.56)
- Multiply each squared deviation by the probability of each state:
- (0.4 \times 5.76 = 2.304)
- (0.6 \times 2.56 = 1.536)
- Sum these results to get the variance:
- (2.304 + 1.536 = 3.84)
- Take the square root to find the standard deviation:
- (\sqrt{3.84} = 1.96\%)
For Stock B:
- Calculate the deviations from the expected return (9.6%):
- Deviation for State 1: (12 – 9.6 = 2.4)
- Deviation for State 2: (8 – 9.6 = -1.6)
- Square the deviations:
- (2.4^2 = 5.76)
- ((-1.6)^2 = 2.56)
- Multiply each squared deviation by the probability of each state:
- (0.4 \times 5.76 = 2.304)
- (0.6 \times 2.56 = 1.536)
- Sum these results to get the variance:
- (2.304 + 1.536 = 3.84)
- Take the square root to find the standard deviation:
- (\sqrt{3.84} = 1.96\%)
Final Results:
- Expected Return for Stock A = 12.4%
- Expected Return for Stock B = 9.6%
- Standard Deviation for Stock A = 1.96%
- Standard Deviation for Stock B = 1.96%
Explanation:
In this case, both stocks have the same standard deviation, which means they have the same level of risk, even though their expected returns are different. Stock A has a higher expected return (12.4%) compared to Stock B (9.6%), but both stocks exhibit the same risk (1.96%).
The expected return is a critical measure because it gives investors an idea of what return they can expect over time based on probabilities. The standard deviation is essential for assessing risk: a higher standard deviation means more variability in returns, indicating higher risk. In this example, both stocks are equally risky, even though one offers a higher return.