ABSTRACT: Most credit portfolios contain obligor concentration risk and yet international bank regulatory capital rules and many industry models assume perfect diversification. Multiple methods are available to calculate the approximate capital needs of a concentrated credit portfolio, but many of these involve advanced mathematical arguments, substantial computation time, and fail to clearly identify the most important credits causing concentration risk. In this article, I illustrate three approaches for calculating loss distributions and value-at-risk capital requirements. Of these the large exposure approach proposed by Kupiec (2015) is especially easy to implement. It produces accurate estimates of the economic capital required for a concentrated portfolio and immediately identifies the obligors most responsible for generating concentration risk.
I. Introduction
Diversification is perhaps the only “free lunch” offered in well-functioning financial markets. By merely selecting investments in the correct proportions, a financial institution can reduce unnecessary risk exposures and reduce the economic risk capital it needs to fund operations. Yet, despite the unquestioned benefits of holding a diversified portfolio, the literature on measuring credit risk diversification, to me at least, seems underdeveloped and mathematically daunting.
International bank regulators have focused on credit risk and regulatory capital for the last 15 years, and so one might think there would be useful guidance in the Basel capital regulations. However hard you look, you will be disappointed. For example, on concentration risk, Basel II only offers the following wisdom: “Risk concentrations are arguably the single most important cause of major problems in banks.”
Maybe it is unfair to be so hard on the Basel Committee on this point, but then again, maybe not. If concentration is indeed the biggest risk facing banks, and the benefits to reducing concentration risk are available at minimal cost, why haven’t international bank regulators devoted more time and energy on developing practical concentration risk methodologies that will improve all banks’ safety and soundness? Perhaps one reason is that credit risk concentration is complex and can stem from many sources.
One source that gives rise to concentration risk is omitted common factors that drive defaults. If the true default correlation is driven by 3 factors, and a default correlation model assumes there is only one factor, any measurements produced by the model will have unmeasured concentration risk against the two missing factors. Another source of concentration risk could arise from the choice of an incorrect factor probability distribution (or so-called copula model) for default correlation. Use of the wrong distributional assumption (e.g. Gaussian instead of Student-t) will generate an unmeasured concentration of portfolio defaults in the tail of the portfolio’s loss distribution. Concentration risk can also arise because of unmeasured correlation between credits’ conditional probability of default and their loss given default. Unless it is accounted for, such a correlation could produce unmeasured concentrations in the tail of the loss distribution. The final source of concentration risk I will mention is obligor exposure concentrations—a portfolio with large unhedged exposures to individual borrowers.
All of these sources of concentration risk are potentially important, and there are various ways a risk manager might deal with each of them. In this article I will focus on methods to measure the most straight-forward source of concentration risk in a credit portfolio―obligor concentration risk. Obligor concentration risk arises because individual borrowers have different loan sizes and loss rates. Obligor concentration risk is pervasive in practice and yet there is no widely accepted technique for measuring its impact on credit portfolio loss distributions.
A search of the literature will find various “granularity adjustments” that can be used in conjunction with the alternative credit default models including Basel II and other frameworks. These adjustments involve complex mathematical approximations and offer no intuitive appeal―at least not to me.
An alternative approach is to approximate the entire conditional loss distribution using a generalization of the loss-bucketing approach proposed in CDO and n-th to default CDS pricing models of Andersen, Sidenius and Basu (2003), and Hull and White (2004). This approaches make good intuitive sense, and it can be very accurate, but it requires a significant amount of numerical computation.
A third alternative is to use a new “largest exposure approach” for approximating the critical values of the portfolio loss distribution developed in Kupiec (2015). This approach is very simple and accurate and the answers are very intuitive. It also directly identifies the obligors that are the biggest source of concentration risk. Regardless of which method you prefer, the “bad news” is that properly accounting for obligor concentration risk has the potential to substantially increase the economic capital requirements for many portfolios.
An outline of the article follows. The influences of common factors that drive default correlations must first be removed before calculating the impact of obligor concentrations on the loss distribution and economic capital for a credit portfolio. There are various ways to model default correlation, but I will focus on the approach pioneered by Vasicek (1987, 1991) and adopted by the Basel Committee on Banking Supervision for use in Basel II. Section II discusses default correlation and the calculation of conditional default probabilities. The Basel II assumes that idiosyncratic default risk is fully diversified. Section III discusses alternative ways of measuring the true idiosyncratic risk that remains in credit portfolios because of obligor concentration risk and the associated methods for estimating value-at-risk economic capital. Section IV concludes.
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