Answer: equal to 2.5%.

## Explanation **Target semideviation** (also called downside deviation) measures the dispersion of returns below a specified target or minimum acceptable return. The formula for target semideviation is: $$\sigma_{target} = \sqrt{\frac{\sum_{i=1}^{n} \min(R_i - T, 0)^2}{n-1}}$$ Where: - $R_i$ = individual return - $T$ = target return - $n$ = number of observations **Key Insight:** When the target return is set equal to the mean return ($T = \mu$), the target semideviation will be **less than or equal to** the standard deviation. This is because: 1. Standard deviation considers deviations from the mean in **both directions** (positive and negative) 2. Target semideviation only considers deviations **below the target** (negative deviations from the target) **In this specific case:** - Mean return ($\mu$) = 2.2% - Target return ($T$) = 2.2% - Standard deviation ($\sigma$) = 2.5% Since the target equals the mean, the target semideviation will be **less than** the standard deviation because: - Standard deviation includes all deviations from the mean (both above and below) - Target semideviation only includes deviations below the mean - By definition, the sum of squared deviations below the mean is less than the total sum of squared deviations from the mean **Mathematically:** $$\sigma_{target}^2 = \frac{\sum (R_i - \mu)^2 \cdot I(R_i < \mu)}{n-1}$$ $$\sigma^2 = \frac{\sum (R_i - \mu)^2}{n-1}$$ Since $\sum (R_i - \mu)^2 \cdot I(R_i < \mu) \leq \sum (R_i - \mu)^2$, it follows that $\sigma_{target} \leq \sigma$. **Therefore, the correct answer is A: less than 2.5%.** **Note:** The target semideviation would only equal the standard deviation if all returns were below the target, which is not the case for a normally distributed portfolio with mean equal to the target.

Author: LeetQuiz .