Capital allocation for credit portfolios with kernel estimators

Abstract

Determining the contributions of sub-portfolios or single exposures to portfolio-wide economic capital for credit risk is an important risk measurement task. Often, economic capital is measured as the Value-at-Risk (VaR) of the portfolio loss distribution. For many of the credit portfolio risk models used in practice, the VaR contributions then have to be estimated from Monte Carlo samples. In the context of a partly continuous loss distribution (i.e. continuous except for a positive point mass on zero), we investigate how to combine kernel estimation methods with importance sampling to achieve more efficient (i.e. less volatile) estimation of VaR contributions.

Keywords:

• Corporate risk management
• Copulas
• Applications to credit risk
• Applications to default risk

Acknowledgements

The opinions expressed in this paper are those of the author and do not necessarily reflect the views of Lloyds Banking Group.

Notes

† See Tasche (2004) or Tasche (2006) for notable exceptions.

† Instead of trying to determine VaR contributions in a continuous or semi-continuous setting, some risk managers use contributions to expected shortfall. This approach is very fruitful and has some other advantages; see Kalkbrener et al. (2004) and Merino and Nyfeler (2004).

‡ Kalkbrener (2005) considers relation (3b) in a more general context. He calls it ‘linear aggregation’.

§ That is, VaR_P,α(hL) = hVaR_P,α(L) for positive h.

† This problem can be avoided by using the risk measure Expected Shortfall (see, e.g., Acerbi and Tasche 2002) instead of VaR. With the definition of the Expected Shortfall slightly simplified for practical purposes, equation (7) then reads

† We may assume , since for real-world portfolios the case seems very unlikely.

‡ In general, itself will have to be estimated. Thus, at first glance, it seems strange to choose it as a basis for finding Q. However, in a first step, for instance, it can be replaced by a rough estimate and be refined in further stages of the estimation procedure.

† The new sample need not necessarily be generated by a new Monte Carlo simulation. Alternatively, as done in section 5, the new sample can be created from the previous sample by substituting from (21a) for (S ₁, …, S _k ). In this case, T ₂ = T ₁.

‡ For instance, by ordering the pairs (L ^(t), R ^(t)) in descending order according to the L component and selecting the largest t with . Then take L ^(t) as the estimator.

† The R-script for the calculations can be downloaded from http://www-m4.ma.tum.de/pers/tasche/.

‡ For further reduction of the estimation variance, the standard deviation and sample size of the positive losses for the bandwidth (both for standard MC as well as for importance sampling) according to (10) are not estimated but calculated numerically. Under assumption 5.1 this can be done exactly for the standard deviation of the positive losses and approximately for the sample size of the positive losses, by approximating the distribution of the number of defaults via moment matching by a negative binomial distribution.

§ See Tasche (2006, section 4) for the motivation for this definition and the definition of marginal diversification indices. The diversification indices are calculated on an unexpected loss (UL) basis, in accordance with definition (2a) of economic capital. Note that the value of the portfolio-wide diversification index depends upon whether the portfolio is decomposed into assets or into sectors since, in the latter case, the diversification potential is larger.

† The coefficient of variation of a sample is defined as the ratio of the sample standard deviation and the sample mean.