To determine the Expected Default Frequency (EDF), we first calculate the Distance to Default (DD) using the Merton Model framework. The default point (DP) is typically calculated as short-term debt plus half of long-term debt.

Step 1: Calculate Default Point (DP) $DP = \text{Short-Term Debt} + 0.5 \times \text{Long-Term Debt} = 6 + 0.5 \times 4 = £8 \text{ billion}$

Step 2: Calculate Distance to Default (DD) Because we are calculating the physical real-world probability of default (EDF), we use the expected return on assets ($\mu = 8\%$) rather than the risk-neutral risk-free rate.

$DD = \frac{\ln(V / DP) + (\mu - 0.5 \times \sigma^2)T}{\sigma \sqrt{T}}$

Given:

• $V = 25$
• $DP = 8$
• $\mu = 0.08$
• $\sigma = 0.45$
• $T = 1$

$DD = \frac{\ln(25 / 8) + (0.08 - 0.5 \times 0.45^2) \times 1}{0.45 \times \sqrt{1}}$ $DD = \frac{\ln(3.125) + (0.08 - 0.10125)}{0.45}$ $DD = \frac{1.13943 - 0.02125}{0.45} = \frac{1.11818}{0.45} \approx 2.4848$

Step 3: Calculate EDF The Expected Default Frequency is the standard normal cumulative distribution function evaluated at $-DD$: $EDF = N(-2.4848)$

Using standard normal probability tables, $N(-2.48) \approx 0.00657$ or $0.657% $.

Step 4: Determine Rating An EDF of 0.657% falls within the 0.40% – 0.72% range, which corresponds to the BBB-/BB rating class.