To determine the 95% confidence interval for the surplus value, we need to first calculate the expected surplus at the end of the year and the standard deviation (volatility) of the surplus.

1. Expected Surplus ($E[S]$):

• Assets end-of-year value: $E[A] = A_0 \times (1 + \mu_A) = 396 \times (1 + 0.08) = 396 \times 1.08 = 427.68$ million
• Liabilities end-of-year value: $E[L] = L_0 \times (1 + \mu_L) = 308 \times (1 + 0.10) = 308 \times 1.10 = 338.80$ million
• Expected Surplus $E[S] = E[A] - E[L] = 427.68 - 338.80 = 88.88$ million

2. Standard Deviation of the Surplus ($\sigma_S$): The variance of the surplus is given by: $\sigma_S^2 = (A_0 \sigma_A)^2 + (L_0 \sigma_L)^2 - 2 \rho (A_0 \sigma_A)(L_0 \sigma_L)$

• Dollar Volatility of Assets = $396 \times 0.30 = 118.8$ million
• Dollar Volatility of Liabilities = $308 \times 0.14 = 43.12$ million
• $\rho = 0.63$

$\sigma_S^2 = (118.8)^2 + (43.12)^2 - 2 \times 0.63 \times 118.8 \times 43.12$ $\sigma_S^2 = 14113.44 + 1859.3344 - 6454.698 = 9518.076$ $\sigma_S = \sqrt{9518.076} \approx 97.56 \text{ million}$

3. Confidence Interval (95%): For a 95% confidence interval in a normal distribution, we use $z = 1.96$. $\text{CI} = E[S] \pm 1.96 \times \sigma_S$ $\text{Margin of Error} = 1.96 \times 97.56 = 191.218 \text{ million}$

• Lower Bound: $88.88 - 191.218 = -102.338$ million (approx. -102.34 million)
• Upper Bound: $88.88 + 191.218 = 280.098$ million (approx. 280.1 million)

The 95% confidence interval is (-102.34 million, 280.1 million).