At the 99% confidence level, the default loss level has a default probability, Tt, of 0.01. A default loss level of 0.01 corresponds to -2.33 on the standard normal distribution ($\Phi^{-1}(0.01) = -2.33$).

The relationship between the default loss level and the given market return is based on the Vasicek single-factor model:

$x(m) = \rho(m) = \Phi\left(\frac{k - \beta m}{\sqrt{1 - \beta^2}}\right)$

Where:

• $\Phi^{-1}(x)$ is the inverse cumulative standard normal distribution at the target probability
• $k$ is the default threshold = -2.33
• $\beta$ is the correlation to the market factor = 0.8
• $m$ is the realized market value

$\Rightarrow \Phi^{-1}(x) = \frac{k - \beta m}{\sqrt{1 - \beta^2}}$ $\Rightarrow \Phi^{-1}(0.01) = \frac{-2.33 - 0.8m}{\sqrt{1 - 0.8^2}}$ $\Rightarrow -2.33 = \frac{-2.33 - 0.8m}{\sqrt{1 - 0.64}}$ $\Rightarrow -2.33 = \frac{-2.33 - 0.8m}{0.6}$ $\Rightarrow -2.33 \times 0.6 = -2.33 - 0.8m$ $\Rightarrow -1.398 = -2.33 - 0.8m$ $\Rightarrow 0.8m = -2.33 + 1.398$ $\Rightarrow 0.8m = -0.932$ $\Rightarrow m = -1.165$