Objective

An extension of OPAS is proposed that takes into account credit risk in addition to the market risk.

Outline of the procedure

The quantity central for the evaluation of the returns and risks of possible portfolios is a sample of return vectors  resulting from the Monte-Carlo simulation of the underlyings and approximating the joint probability density of the returns of all assets at the reallocation date. On this basis, marginal return distribution  can be calculated, where is either a single asset return or the portfolio return.

To take into account credit risk, we add a module for evaluating the default probability  of the different assets  (cf. Fig.). Credit risk information is taken into account in the single asset return distribution  in the following way:

.

Here,  is the price of the asset under consideration minus the recovered value in the case of default,  is the -function, and the approximate equality is valid under the assumption that . Thus, credit risk essentially shows up in the return probability density as an additional peak at  the lost in the case of default. For clarity of the argument we assume that the recovery rate is known exactly. In the actual implementation the -function will be replaced by a more realistic distribution of recovered values, and the subsequent equations for mean, variance, covariance and value-at-risk accordingly adapted.

Credit risk simulation

Credit risk simulation is based on the number of counter parties.

The model consists of two parts:

·         Simulation of default probabilities

·         Modelling of recovery rate distribution in the case of default.

For the second part we suggest density functions as mentioned in ref. [1].

For the first part, two different approaches are possible: Either a CreditMetrics like approach as described in ref. [1] (Markov model for the dynamics of credit rating changes) or using KMV’s EDF measure [2]. In either case default correlations will be added using the results of correlation results of the OPAS simulator. In a second step, the default simulation can be improved using Olsen and  technology.

Influence on mean vector and covariance matrix

Since mean vector and covariance matrix are calculated from the joint return distribution, inclusion of credit risk contribution is straight forward. However, it is interesting to get an idea of the order of magnitude of the expected contribution.

Under the assumption that all counter parties are AAA rated and there are no correlations, the probability that at least one counter parties defaults in a portfolio of  100 counter parties with reallocation horizon of one month amounts to about , i.e., 1/1000 of a percent. Under the same assumptions one gets for BBB rated counter parties a total default probability of the order of 1%. Thus, the correction of the mean vector is of the order of 0.01 percent in the case of AAA rated counter parties and of the order of 10% in the case of BBB rated counter parties. The influence on the volatility is much larger (since it is quadratic in the return and therefore much more sensitive to particularly large contributions). The respective estimates are of the order of 1% for AAA ratings and 1000% for BBB ratings.

Mean and variance  of asset  and covariance between assets  and  can be calculated explicitly. The corresponding expressions are:

,

Here, , , and  are mean, variance and covariance without the credit risk contribution,  is the probability that asset  defaults and asset  does not,  is the probability that asset  defaults and asset does not, and  is the probability that both asset defaults. These terms are related to the single asset defaulting probabilities by  where  may be  or . Notice that the covariance correction starts with a second order term in the defaulting probabilities while the variance correction starts with a first order term. If both assets are issued by the same counter party the defaulting probabilities simplify to  and , such the covariance reduces to

,

now starting with a first-order correction term as the variance correction.

Influence on VaR

The effect of credit risk on the VaR is much smaller than its effect on the variance. Indeed, the 99% VaR is defined by

,

.

This expression makes sense only if . For AAA rated counter parties,  is of the order of , i.e., 1/1000 of a percent. The inequality is thus fulfilled and, moreover,  is so small that the VaR will be increased by only an unperceptible amount. For BBB rated counter parties, on the other hand, one gets , such that the validity of the inequality in no longer guaranteed. Even if it were satisfied, the additional contribution is likely to cause a dramatic increase of the 99% VaR, but the increase of the 95% VaR may still remain moderate compared to the increase of the variance.

Detail about the proposed formalism

In the present state market risk is estimated by means of a Monte-Carlo simulation of the underlyings of the assets under consideration such that at reallocation time  a sample of  points of the joint return distribution of the set of assets is available,

 

where  is the return vector of the th simulation path.

Credit risk shows up in the return probability density as an additional contribution with returns equal to the negative price of the defaulting assets. Thus, it can be taken into account by the following modification:

where is the vector of prices minus recovered values at reallocation time,  is a diagonal matrix whose th element is 0 or 1 depending on whether or not the th asset defaults,  is the number of possible combinations of defaulting assets, and the  are the probabilities attributed to these events. In principle the index  runs over all combinations . Since there are too many, most of them occurring only with very small probabilities, an appropriate cut-off criterium has to be defined. A possible choice is to take into account terms up to the  counter party risk, where  corresponds to the lowest order correlations used above for the covariance calculation.

References

[1] J.P. Morgan, CreditMetrics-- Technical Document. New York, 1997.

[2] KMV, Modeling Default Risk. San Francisco, 1999.