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Abstract
In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, that is, what is the maximum and minimum value-at-risk (VaR) of a portfolio of risky loans that can be justified given a certain amount of available information. Puccetti and Rüschendorf [2012a. “Computation of Sharp Bounds on the Distribution of a Function of Dependent Risks”. Journal of Computational and Applied Maths 236, 1833–1840] and Embrechts, Puccetti, and Rüschendorf [2013. “Model Uncertainty and VaR Aggregation”. Journal of Banking and Finance 37, 2750–2764] propose the rearrangement algorithm (RA) as a general method to approximate VaR bounds when the loss distributions of the different loans are known but not their interdependence (unconstrained bounds). Their numerical results show that the gap between worst-case and best-case VaR is typically very high, a feature that can only be explained by lack of using dependence information. We propose a modification of the RA that makes it possible to approximate sharp VaR bounds when besides the marginal distributions also higher order moments of the aggregate portfolio such as variance and skewness are available as sources of dependence information. A numerical study shows that the use of moment information makes it possible to significantly improve the (unconstrained) VaR bounds. However, VaR assessments of credit portfolios that are performed at high confidence levels (as it is the case in Solvency II and Basel III) remain subject to significant model uncertainty and are not robust.
ORCID
Carole Bernard http://orcid.org/0000-0002-3996-9260
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. It is well known that VaR does not capture tail risk and does not reward diversification. Its main challenger is tail-value-at-risk (TVaR), also known as expected shortfall, which in essence is the mean of the tail distribution and which addresses several shortcomings of VaR. However, TVaR has its own deficiencies. For example, Kou and Peng (Citation2014a) show that it is more sensitive to model mispecification than VaR. As a better alternative for VaR, they propose the median shortfall which is the median of tail loss distribution; see also the discussion in Kou and Peng (Citation2014b). In this paper, we focus on VaR, which is the preferred reference measure for risk quantification and (regulatory) solvency assessment for banks and insurers.
2. The properties of the different industry models are further discussed in Section 6.1. We apply them in the numerical Section 6.2.
3. The use of multivariate normal models is often based on the (wrong) intuition that correlations are enough to model dependence. It is, however, clear that correlations only are not enough to model dependence, as a number (i.e. the correlation) can never be sufficient to describe the complex interaction between variables unless additional assumptions are made (see, e.g. Embrechts, Puccetti, and Rüschendorf Citation2013). This fallacy may then also partially explain why the KMV model has gained so much support in the industry.
4. In the presence of inequality constraints, the bounds on VaR will become wider as compared to a situation in which all moments are assumed to be known with certainty.
5. See, for example, Acerbi (Citation2002) and Rockafellar and Uryasev (Citation2000).
6. The traditional way to describe dependence is to use copulas. Indeed, Sklar's theorem states that for a multivariate vector , it holds that
for some suitable chosen vector
in which the
are uniformly distributed. The joint distribution of
is called a copula.
7. Note that we use the notation instead of
. The reason is that we represent risks as columns of a matrix and it is more natural to express the corresponding matrix as
rather than
8. More precisely, in this context, the appropriate notion to indicate that a sum is flatter than another one is the so-called convex order. In Bernard, Rüschendorf, and Vanduffel (CitationForthcoming), it is shown that maximum VaR bounds are obtained if one can create a dependence among the risks (restricted to the domain ) that makes the sum minimum in the sense of convex order (see also Bernard, Jiang, and Wang Citation2013). For
, this result goes back to Rüschendorf (Citation1982).
9. Note indeed that minimizing the variance of a sum requires that each component has minimum correlation with the sum of all other components.
10. See also Remark 4.2 in Bernard, Rüschendorf, and Vanduffel (CitationForthcoming) where the same idea appears in the context of a constraint on the portfolio variance.
11. Algorithm 1 of Section (3.2) can essentially be applied too (i.e. after removing the last column of M), but doing so is unnecessary and does not yield the sharp bounds in general. By contrast, when the portfolio is inhomogeneous, it makes sense to apply Algorithm 1 to obtain the approximate minimum variance portfolio.
12. The basic reason is that in the case of (scaled) Bernoulli distributions, knowledge of pairwise correlations and single default probabilities implies knowledge of the (pairwise) joint default probabilities, and thus also knowledge of the distribution of partial sums that only involve two components. Note that the probabilities that three or more loans default together cannot be determined based on single default probabilities and default correlations alone.
13. The first moment requirement amounts to a condition of the type constant, which implies that l and q are uniquely determined.
14. We discretized the originally provided PDs by expressing them in basis points (e.g. 0.0312% became 0.03%).
15. In each of the following examples, another portfolio model is used as the benchmark for discussing model uncertainty.
16. They have been computed under the specification of the Credit Risk model.