### Loss distribution

Let \( L_{i} \left( S \right) \) the loss of a name i given the realization of the systemic factor S. Then:

$$ L_{i} \left( S \right)= P\left[ {D_{i} \mid S} \right] \; E_{i} \; LGD_{i} $$

Where E represents the exposure to the name and LGD (=1-Recovery Rate) represents the Loss Given Default. Assuming m names in a portfolio, the conditional loss of the portfolio is the sum of the losses of all names in the portfolio:

$$ L \left( S \right)=\sum_{i=1}^{m} { P\left[ {D_{i} \mid S}\right] \; E_{i} \; LGD_{i}}$$

And assuming each name has an E of 1, a deterministic LGD, and all names have the same LGD and the \( P\left[ {D \mid S}\right] \) (making use of the homogeneity assumption), the expression simplifies considerably:

$$ L \left( S \right)=\sum_{i=1}^{m} { P\left[ {D \mid S}\right] \; 1 \; LGD }=LGD \sum_{i=1}^{m} { P\left[ {D \mid S} \right]}=LGD \; d \left( S \right) $$

Where \( d \left( S \right) \) represents the number of defaults given S.

Let \( l \left( S \right) \) represent the portfolio loss rate (loss/exposure, where exposure = m as each name has exposure of 1) given the realization of the systemic factor S:

$$ l \left( S \right)=\frac{L \left( S \right)}{m}=\frac{LGD \; d \left( S \right)}{m} $$

We now proceed to calculate the probability distribution of the portfolio loss rate conditional on the realization of the systemic factor S:

$$ P\left[ l \left( S \right)<=x \right]=P\left[ \frac{LGD \; d \left( S \right)}{m}<=x \right]= P\left[ d \left( S \right)<= \frac{ m x}{LGD} \right] $$

\( d \left( S \right) \), number of defaults conditional on the realization of the systemic factor S follows a binomial distribution with p equal to \( P\left[ {D \mid S}\right]\) because the defaults of individual names conditional on the systemic factors are independent. Thus:

$$ P\left[ l \left( S \right)<=x \right]= P\left[ d \left( S \right)<= \frac{ m x}{LGD} \right]=\sum_{k=1}^{\frac{ m x}{LGD}}{ {m \choose k} {\left( P\left[ {D \mid S}\right] \right) }^{k} {\left( 1-P\left[ {D \mid S}\right] \right)}^{m-k} } $$

Now the assumption of large portfolio implies means \( m \rightarrow \infty \), and we know that the binomial distribution converges to normal distribution with mean mp and variance mp(1-p)for large m, thus:

$$ P\left[ l \left( S \right)<=x \right]= N\left[ \frac{\frac{ m x}{LGD}-m P\left[ {D \mid S}\right]} {\sqrt {m P\left[ {D \mid S}\right] \left(1-P\left[ {D \mid S}\right] \right)} } \right] $$ $$ P\left[ l \left( S \right)<=x \right]= N\left[ \frac{\sqrt{m}}{\sqrt {P\left[ {D \mid S}\right] \left(1-P\left[ {D \mid S}\right] \right)} } \left( \frac{ x}{LGD}-P\left[ {D \mid S}\right] \right) \right] $$ $$ $$Making use of the large portfolio assumption, we get:

$$ \lim_{m \to \infty} P\left[ l \left( S \right)<=x \right] = \left\{ \begin{array}{rl} N\left[ \infty \right] = 1 & \mbox{if} & \frac{ x}{LGD} > P\left[ {D \mid S}\right]\\ N\left[ 0 \right] = \frac{1}{2} & \mbox{if} & \frac{x}{LGD}=P\left[ {D \mid S}\right]\\ N\left[ -\infty \right] = 0 & \mbox{if} & \frac{x}{LGD} < P\left[ {D \mid S}\right]\\ \end{array} \right. $$

Essentially it means that as the portfolio becomes very large, the idiosyncratic risk vanishes, and we are just left with the systemic risk. Gross loss conditional on S becomes equal to the conditional PD, and as result, the probability of loss less than (greater than) this conditional PD is 1 (0). In terms of the realised value of the systemic factor, the relevant domain (which gives non zero probability of loss) is:

$$ \frac{x}{LGD}>P\left[ {D \mid S}\right] \Rightarrow \frac{x}{LGD}>N\left[ \frac{Threshold_{i}-\sqrt{\rho} S}{\sqrt{1-\rho} } \right] $$ $$ \Rightarrow S >-\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}}$$

We can now calculate the unconditional loss distribution by integrating out S, through an application of expectation under the probability density of S:

$$ P\left[ l <=x \right]= E \left[ P\left[ l \left( S \right)<=x \right] \right] = \int_{-\infty}^{\infty}{ P\left[ l \left( S \right)<=x \right] d N\left[ S \right]} $$ $$ P\left[ l <=x \right]= \int_{-\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}}}^{\infty}{ 1 d N\left[ S \right]} $$ $$ P\left[ l <=x \right]= N\left[ \infty \right]-N\left[ -\frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] $$ $$ P\left[ l <=x \right]= 1-1+N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] $$ $$ P\left[ l <=x \right]= N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] $$

### Loss density and quantile

Having derived the loss distribution function, we can calculate the loss density using straight forward calculus:

$$ P\left[ l =x \right]= \frac{d P\left[ l <=x \right] }{d x} =\frac{d}{d x} N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] $$ $$ = n\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{d}{d x} N^{-1} \left[ \frac{x}{LGD} \right] $$ $$ = n\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right] \frac{\sqrt{1-\rho}}{\sqrt{\rho}} \frac{1}{n\left[ N^{-1} \left[ \frac{x}{LGD} \right] \right] LGD } $$

And we can calculate quantile or Value at Risk at \( \alpha \) level using straight forward algebra:

$$ P\left[ l <=x \right]=\alpha $$ $$ N\left[ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} \right]=\alpha $$ $$ \frac{\sqrt{1-\rho} N^{-1} \left[ \frac{x}{LGD} \right]-Threshold_{i}}{\sqrt{\rho}} = N^{-1} \left[ \alpha \right] $$ $$ { N^{-1} \left[ \frac{x}{LGD} \right]} = \frac{ Threshold_{i}+ \sqrt{\rho} N^{-1} \left[ \alpha \right]}{\sqrt{1-\rho}} $$ $$ x = LGD \; N\left[\frac{ Threshold_{i}+ \sqrt{\rho} N^{-1} \left[ \alpha \right]}{\sqrt{1-\rho}}\right] $$