Giovanni Puccetti

An Academic Response to Basel 3.5

Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.

Bounds for Functions of Dependent Risks

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the structure of dependence of the random vector is available, we provide a bound on the distribution function of the sum of risks which we prove to be better than the one generally used in the literature.

Computation of sharp bounds on the distribution of a function of dependent risks

We propose a new algorithm to compute numerically sharp lower and upper bounds on the distribution of a function of d dependent random variables having fixed marginal distributions. Compared to the existing literature, the bounds are widely applicable, more accurate and more easily obtained.

Multivariate comonotonicity

In this paper we consider several multivariate extensions of comonotonicity. We show that naive extensions do not enjoy some of the main properties of the univariate concept. In order to have these properties, more structures are needed than in the univariate case.

Bounds for the sum of dependent risks having overlapping marginals

We describe several analytical and numerical procedures to obtain bounds on the distribution function of a sum of n dependent risks having fixed overlapping marginals. As an application, we produce bounds on quantile-based risk measures for portfolios of financial and actuarial interest.

Extremal Dependence Concepts

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges; recent developments in extremal dependence concepts have drawn a lot of attention in probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in the literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is actually focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements and this not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way and this implies several advantages on the approximation of solutions in a broad class of optimization problems and allows a deeper understanding of dependence itself.

Asymptotic Equivalence of Conservative Value-at-Risk- and Expected Shortfall-Based Capital Charges

We show that the conservative estimate of the value-at-risk (VaR) for the sum of d random losses with given identical marginals and finite mean is equivalent to the corresponding conservative estimate of the expected shortfall in the limit, as the number of risks becomes arbitrarily large. Examples of interest in quantitative risk management show that the equivalence also holds for relatively small and inhomogeneous risk portfolios. When the individual random losses have infinite first moment, we show that VaR can be arbitrarily large with respect to the corresponding VaR estimate for comonotonic risks if the risk portfolio is large enough.

The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables

We propose a new algorithm to compute numerically the distribution function of the sum of d dependent, non-negative random variables with given joint distribution.

VaR Bounds for Joint Portfolios with Dependence Constraints

Abstract Based on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a general subset of the domain of the distribution function of a risk portfolio. The newly provided VaR bound can be interpreted as a comonotonic VaR computed at a distorted confidence level and its quality is illustrated in a series of examples of practical interest.

Reduction of Value-at-Risk Bounds via Independence and Variance Information

We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve those available in the literature that are based on the sole knowledge of the marginal distributions. When the variance of the joint portfolio loss is small enough, further improvements can be obtained.