Explanation

D is correct. The standard deviation of losses for each individual loan is calculated as:

$\sigma = \sqrt{p - p^2} \left[ L(1 - R) \right]$

Where:

• $p$ = probability of default = 0.04
• $L$ = exposure at default = 500,000
• $R$ = recovery rate = 0.3

$\sigma = \sqrt{0.04 - 0.04^2} \left[ 500,000 * (1 - 0.3) \right]$ $\sigma = \sqrt{0.04 - 0.0016} \left[ 500,000 * 0.7 \right]$ $\sigma = \sqrt{0.0384} \left[ 350,000 \right]$ $\sigma = 0.19596 * 350,000 = 68,585.71$

The standard deviation of losses on the portfolio of $n$ loans as a percentage of its size is then calculated as:

$\alpha = \frac{\sigma \sqrt{1 + (n - 1)\rho}}{L \sqrt{n}}$

Where:

• $n$ = number of loans = 30
• $\rho$ = average pairwise default correlation = 0.4

$\alpha = \frac{68,585.71 \sqrt{1 + (30 - 1) * 0.4}}{500,000 \sqrt{30}}$ $\alpha = \frac{68,585.71 \sqrt{1 + 29 * 0.4}}{500,000 * 5.4772}$ $\alpha = \frac{68,585.71 \sqrt{1 + 11.6}}{2,738,612.5}$ $\alpha = \frac{68,585.71 \sqrt{12.6}}{2,738,612.5}$ $\alpha = \frac{68,585.71 * 3.5496}{2,738,612.5}$ $\alpha = \frac{243,500}{2,738,612.5} = 0.08890 \text{ or } 8.9\%$

This calculation accounts for the correlation between defaults in the portfolio, which increases the portfolio standard deviation compared to uncorrelated defaults.