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"Credit Risk"


Financial institutions are increasingly measuring and managing the risk from credit exposures at the portfolio level. The traditional binary classification in "good" and "bad" credits is not sufficient anymore, any credit can potentially become "bad" over time given the influences go towards an undesired direction.

But what is the credit risk of a given portfolio? Here comes one of many possible definitions:

Definition (Credit Risk)
Credit Risk is the distribution of financial losses due to unexpected changes in the credit quality of a counterparty.

The distribution of credit losses is complex. At its center stands the probability of default ( = likelihood of any type of failure to honor financial agreements) but we also have to address credit exposures ( = how large will the outstanding obligation be, if default occurs?) and recovery rates ( = what fraction of the exposure may be recovered?). We define each as random processes. The Credit Loss distribution is then defined as an aggregation of these three distributions.

Let us describe the Credit Loss distribution more formally. Therefore we have to introduce some notation:



The portfolio loss D(T) being T the last time unit in our fixed time horizon is the sum of the individual losses Dj(T): [B is the set of obligors]



The individual credit losses Dj(T) are unknown in advance. Because the sum of random variables is a random variable too the full portfolio credit loss is a random variable also. The portfolio's credit loss distribution is the CDF of this random variable: (CDF=Cumulative Probability Distribution)

F(x) = Prob [ D(T) <= x ].

What we haven't modeled were the correlations between the different Dj(T). The correlation parameters typically have a strong influence on the tail of the Credit Loss distribution - and thus on the Value-at-Risk.

Portfolio Credit Loss Distributions have some interesting features:
  1. The distribution is not symmetrical,
  2. The distribution is highly skewed, and
  3. The distribution has heavy tails.


Above, we set up a stochastic model for calculating the Credit Loss Distribution. But how do we estimate the Credit Risk in practice?

Often financial institutions calculate the Credit Value-at-Risk (CVaR). This calculation assigns a single number to a portfolio, and this number is seen as the Economic Credit Capital.

By making the assumptions:
  • The Probability of Default (within the time horizon T) Q(T) is for each credit the same, and
  • The pair wise correlation rho of any two credits are always identical
we can calculate the CVaR for a confidence level X. That means we can be X% sure that the Credit Loss in T periods will not exceed the CVaR. A formula for calculating CVaR is given by



where



and

L = size of the portfolio,
R = recovery rate, and
rho = estimated correlation.
X = confidence (if you want to calculate a CVaR with the confidence level of 99,9 %, X must be set equal to 0,999)



This formula for V(X,T) was first established by Vasicek in "Probability of Loss on a Loan Portfolio".


For internal purposes quite a lot of banks have established their own methodology of calculating ECC. Here we will discuss CreditRisk+, a purely actuarial model where the risk drivers are the expected default rates.

In this model default is modeled as a stopping time random variable and follows a Poisson distribution.

We make the following assumptions:
  • The likelihood of default of a loan is in each period of time the same,
  • The number of defaults that occur in any given period is independent of the number of defaults that occur in any other period

Under these assumptions, the PDF for the number of defaults in a given period of time (say one year) is given by a Poisson distribution:


Exhibit CR.1: The Poisson Distribution

where

  • n is the number of defaults in the holding period

  • lambda is the average number of defaults in the holding period
In order to derive the Loss distribution the exposures (net of the corresponding recovery rates) are divided into bands. Now let L be the unit of exposure. Each band j (j=1..m) has am average common exposure of vj = L * j. Each band j is seen as an independent portfolio Denote by kA the expected loss of obligor A in units of exposure, kA = vA/L.Then zj, the expected loss in band j (expressed in units of L) can be written kj = vj * lambdaj or lambdaj = kj / vj.

To analyze the distribution of losses (for each band j) we use a probability generating function Gj for the portfolio defined in terms of an auxiliary variable z ( abs(z) <= 1)



The probability generating function for the entire portfolio is (independence between obligors !!!)



Now we get as the Loss distribution for the entire portfolio:



That means that the Loss distribution is "recovered" by taking derivatives from G.

CVaR is now easily derived by first computing the, say, 99 % - percentile and then substracting the expected loss from this number.

The graph below shows a typical Loss distribution.


Exhibit CR.2: A typical (Credit) Loss distribution

The principal limitations of this model is that it ignores market and migration risk by focusing only on default events.