Credit Risk Management

Unexpected Loss

We have already calculated the Expected loss in case of a default. But the amount of loss, i.e. \(\tilde L\) is a random variable. So the amount of loss may vary from EL. This is measured as Unexpected Loss. Unexpected loss can be defined as-

\[\begin{aligned} Unexpected\ Loss &= s.d(\tilde L) \\ &= \sqrt{Var(\tilde L)} \\ &= \sqrt{Var(EAD . SEV . L)} \\ \end{aligned}\]

Now,

\[\begin{aligned} Var(\tilde L) &= Var(EAD . SEV . L) \\ &= EAD^2. Var(SEV.1_D) \qquad we\ have\ taken\ EAD\ as\ a\ constant \\ &= EAD^2. [E(SEV^2.1_D^2) - E^2(SEV.1_D)] \\ &= EAD^2. [E(SEV^2). E(1_D^2) - E^2(SEV).E^2(1_D)] \qquad since\ SEV\ and\ D\ are\ independent\ random\ variables \\ &= EAD^2. [(E(SEV^2).DP - LGD^2.DP)-(LGD^2.DP^2-LGD^2.DP)] \\ &= EAD^2. [Var(SEV).DP + LGD^2.DP.(1-DP)] \end{aligned}\]

Hence, \(UL = EAD.\sqrt{Var(SEV).DP + LGD^2.DP.(1-DP)}\).

Now a bank’s portfolio contains lending loans to many customers. Let is define each customer’s loss variable by- \(\tilde L_i\) for i = 1(1)m.
Then portfolio loss variable is defined as-

\[\tilde L_{PF} = \sum_{i=1}^{m}\tilde L_i\]

Now we have already defined \(\tilde L\) as- \(EAD.SEV.L\). So-

\[\begin{aligned} \tilde L_{PF} &= \sum_{i=1}^{m}\tilde L_i \\ &= \sum_{i=1}^{m}EAD_i.SEV_i.L_i \end{aligned}\] Then Expectation and variance of \(\tilde L_{PF}\) is-

\[\begin{aligned} E(\tilde L_{PF}) &= \sum_{i=1}^{m}EAD_i.LGD_i.DP_i \\ Var(\tilde L_{PF}) &= \sum_{i=1}^{m}\sum_{j=1}^{m}EAD_i.EAD_j.LGD_i.LGD_j.\rho_{ij}.\sqrt{DP_i.(1-DP_i).DP_j.(1-DP_j)} \end{aligned}\]

where \(\rho_{ij} = corr(L_i,L_j)\).

Economic Capital

So far we have learned so far that the bank should hold some capital cushion against unexpected losses due to a default. But- \(P[(\tilde L - EL)> UL]\) is still quite high. To tackle this problem economic Capital was introduced. If we a little bit into statistics, for a prescribed Level of confidence- \(\alpha\), Economic Capital is defined as- Expected loss of the portfolio subtracted from \(\alpha\)-th quantile of the portfolio loss.i.e. \[EC_\alpha = q_\alpha - EL_{PF}\]

where,

\(q_\alpha\) = \(\alpha\)-th quantile of the portfolio loss.

\(EL_{PF}\) = Expected loss of the portfolio.

The following figure is very important. Given default has occurred, this is the loss distribution of the portfolio. If you have read a little bit about different kind of continuous probability distribution functions, you can see- this loss distribution has a lot of similarity with Log-normal distribution and Chi-square distribution.

Monte Carlo Simulation Technique:

Economic Capital is very easy to estimate. First we will simulate portfolio losses using Monte Carlo simulation technique. Suppose they are- \(L^{[1]}_{PF},L^{[2]}_{PF},... L^{[n]}_{PF}\). Then we can define the Empirical distribution function of Loss distribution as- \[F(x) = \frac{1}{n}\sum_{j=1}^n1_{[0,x]}(L^{[j]}_{PF})\] Now arrange the simulated portfolio losses in ascending order. The order statistic is denoted by- \(L^{[i_1]}_{PF}\leq L^{[i_2]}_{PF} \leq... \leq L^{[i_n]}_{PF}\). The \(\alpha\)-th quantile of the empirical distribution function can be defined as- \[\begin{equation} \hat q_\alpha = \begin{cases} \alpha \tilde L_{PF}^{i_{[n\alpha]}}+(1-\alpha)\tilde L_{PF}^{i_{[n\alpha]+1}} & \text{if $n\alpha$ $\in$ N} \\ \tilde L_{PF}^{i_{[n\alpha]}} & \text{if $n\alpha \not \in$ N} \\ \end{cases} \end{equation}\]

where \([n\alpha] = \min(k \in {(1,2,...,n)}: n\alpha \leq k)\). Once \(q_\alpha\) is estimated using \(\hat q_\alpha\), \(EC_\alpha\) can be defined as-

\[EC_\alpha = \hat q_\alpha - \frac{1}{n}\sum_{j = 1}^{n}\tilde L_{PF}^{[j]}\]

Analytical Approximation:

In analytical approximation, we consider a parametric distribution to represent portfolio loss distribution. Since loss distribution only takes positive value, i.e in mathematical term- the domain of the loss distribution function is non-negative part of the real line, we will take those parametric distributions having non-negative support, for example- Chi square distribution.

Then estimate the parameters of the assumed distribution for the portfolio’s expected loss and unexpected loss. Now once we have estimated the parameters, we can easily find the \(\alpha\)-th quantile and that leads us to the Analytically approximated value of Economic Capital with level of confidence \(\alpha\).

Example-

In Analytical approximation what we do is, suppose we are given a portfolio with Expected loss = 0.3% and unexpected loss = 0.225%. The percentage of portfolio loss distribution is denoted by- \(X\). We assume that \(X\) follows beta distribution, i.e. \[X \sim \beta(a,b)\] Then we have- \[\begin{aligned} &E(X) = 0.003\\ &Var(X) = 0.00225^2 \end{aligned}\]

\(X\) has a probability density function-

\[\begin{aligned} & f_X(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}\quad \text{with x $\in$ [0,1]} \\ & \\ & E(X) = \frac{a}{a+b} \\ & Var(X) = \frac{ab}{(a+b)^2(a+b+1)} \end{aligned}\]

Now we have two equations- \[\begin{aligned} & E(X) = \frac{a}{a+b} = 0.003\\ & Var(X) = \frac{ab}{(a+b)^2(a+b+1)} = 0.00225^2 \end{aligned}\] From these two equations we can estimate two unknowns- a & b. In this case we have- \(a = 1.76944\) & \(b = 588.045\). Pretty easy huh!!

Conclusion:

This concludes our introduction part of the Credit risk management, why it is necessary. From now we will move little bit towards the connection between Risk in finance and Brownian motion, Stochastic process etc. This is very easy. I hope we have a lots of fun.