Final answer:
The standard deviation of the expected returns is 11.73%.
Step-by-step explanation:
To find the standard deviation of the expected returns, we can calculate the weighted average of the returns for each scenario and then calculate the standard deviation using the formula for weighted standard deviation.
Let's calculate the weighted average returns:
- Expected return in a normal economy (15%) * Probability of a normal economy (50%): 0.15 * 0.5 = 0.075
- Expected return in a boom (20%) * Probability of a boom (25%): 0.2 * 0.25 = 0.05
- Expected return in a recession (-10%) * Probability of a recession (25%): -0.1 * 0.25 = -0.025
The weighted average return is the sum of these values: 0.075 + 0.05 - 0.025 = 0.1 (or 10%).
To calculate the standard deviation, we need to find the variance of each scenario:
- Normal economy: (0.15 - 0.1)^2 * 0.5 = 0.0025 * 0.5 = 0.00125
- Boom: (0.2 - 0.1)^2 * 0.25 = 0.01 * 0.25 = 0.0025
- Recession: (-0.1 - 0.1)^2 * 0.25 = 0.04 * 0.25 = 0.01
The variance is the sum of these values: 0.00125 + 0.0025 + 0.01 = 0.01375.
Finally, the standard deviation is the square root of the variance: sqrt(0.01375) = 0.1173 (or 11.73%).