Robust risk aggregation

The series of working papers contains research results on robust risk aggregation. A typical question in this area is to determine worst-case values of a risk measure, such as the Value-at-Risk (VaR) or the Expected Shortfall (ES, also known as TVaR, CVaR and AVaR), for a portfolio of risks with known marginals and unknown dependence. The problems here are motivated by the statistical challenges arising from estimating and modeling the dependence structure in practice. Robust risk aggregation (such as finding VaR bounds) often involves finding the smallest convex-order element in an aggregation set, and many results in the working paper series on Joint mixability and negative dependence will be useful. Numerical approximations can be carried out via the Rearrangement Algorithm. Some basic terminology is listed below.

Some particular results are:

A review article on the topic and its implications to financial regulation is WP03. A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

We study the risk aggregation problem in dimension two under the additional constraint that one risk is no larger than the other. This problem turns out to be closely related to the directional optimal transport in WP01 of Optimal transport and its applications. In particular, we obtain sharp bounds on tail risk measures such as VaR, as well as bounds on the distribution of the total risk. The journal version is published in Insurance: Mathematics and Economics (2022).

We investigate ordering relations between two aggregation sets for which the sets of marginals are related by two simple operations: distribution mixtures and quantile mixtures. As a general conclusion from our results, more "homogeneous" marginals lead to a larger aggregation set, and thus more severe model uncertainty. Among other results, we obtain an order relation on VaR under quantile mixture for marginal distributions with monotone densities. The journal version is published in Mathematical Finance (2022).

We obtain new (semi-analytical) bounds on worst-case quantile (VaR) aggregation under dependence uncertainty, based on an RVaR inf-convolution formula obtained in WP02 of Risk sharing, allocations, and equilibria. The so-called convolution bound is so far the best on this problem, and it can be a standard tool in the future for risk aggregation problems for VaR and RVaR. The journal version is to appear in Operations Research (2024).

This paper is a follow-up of WP05 and WP11. The notion of conditional joint mixability is introduced to analytical results on robust risk aggregation for distortion risk measures. Different from the setting of WP05 and WP11, the number of risks in the portfolio is finite in this paper. This paper is cross-listed as WP11 of Joint mixability and negative dependence. The journal version is published in Finance and Stochastics (2019).

This paper is a study of risk aggregation for VaR, ES and RVaR (see WP02 of Risk sharing, allocations, and equilibria), where the marginal distributions are not assumed known. Instead, available information is the mean and the variance, as well as symmetry and/or unimodality of each risk. The journal version is published in SIAM Journal on Financial Mathematics (2018).

This paper is a follow-up of WP05 by studying heterogeneous risk aggregation for distortion risk measures. An asymptotic equivalence is established for heterogeneous portfolios (Theorem 3.5). The journal version is published in Mathematical Finance (2018).

This paper studies robust risk aggregation for factor models, i.e., the conditional distributions of risks given a common factor are known but not the dependence structure. The journal version is published in Finance and Stochastics (2017).

This paper studies robust risk aggregation for collective risk models, i.e., the number of risks in the portfolio is random. Main results are ES bounds (Theorem 3.3) and the VaR-ES asymptotic equivalence (Theorem 4.1). The journal version is published in ASTIN Bulletin (2017).

This paper contains results on the limit of the diversification ratio under dependence uncertainty for heavy-tailed distributions. Main results are Theorems 3.2 and 3.3 which give an explicit form of the above limit. The journal version is published in Extremes (2016).

This paper contains the most advanced result (Theorem 3.1) on the smallest convex-order element in a heterogeneous aggregation set. Under some conditions satisfied by monotone densities, the solution can be explicitly constructed via a system of ODE, which relies on results from on WP07 of Joint mixability and negative dependence. The journal version is published in Scandinavian Actuarial Journal (2016).

This paper studies robustness issues in risk aggregation. In particular, it contains the VaR-ES asymptotic equivalence in heterogeneous setting (Theorem 3.3). An application of this result further shows that VaR has larger uncertainty spread in risk aggregation than ES (Theorem 4.1). The journal version is published in Finance and Stochastics (2015).

This paper studies the risk aggregation problem for distortion risk measures, and for this purpose the extreme-aggregation measure is introduced. The main result is the asymptotic equivalence of a distortion risk measure and its smallest dominating coherent risk measure (Theorem 3.2, homogeneous setting). The journal version is published in SIAM Journal on Financial Mathematics (2015).

This paper introduces the notion of an extremely negatively dependent (END) sequence, i.e., an identically distributed sequence whose partial sum does not diverge. The main result is the existence and a construction of an END sequence for any marginal distribution with finite mean (Theorem 2.2). Applying the result leads to the VaR/ES asymptotic equivalence (Theorem 3.4 and Corollary 3.7, homogeneous setting). The journal version is published in Journal of Multivariate Analysis (2015).

This paper is a review article on robust risk aggregation problems and their implications to Basel 3.5, a big regulatory debate in 2013. We advocate ES over VaR based on recent results on risk aggregation; later, ES was confirmed to replace VaR by the Basel Committee on Banking Supervision in 2016. The journal version is published in Risks (2014).

This paper is the first systemic study of aggregation sets. An aggregation set does not always have a smallest convex-order element (Example 3.1). For a homogeneous aggregation set, under some conditions (satisfied by e.g., a decreasing density), the smallest convex-order element is explicitly constructed (Theorems 3.1-3.3), extending results in WP01 of Joint mixability and negative dependence. A useful relationship between VaR bounds and tail distributions is obtained (Theorem 4.6). The journal version is published in Insurance: Mathematics and Economics (2014).

In this paper, the term joint mixability is coined. The main results are sharp bounds on the distribution function and VaR of aggregate risk with homogeneous marginals and monotone densities (Theorem 3.6 and Corollary 3.7). This paper is cross-listed as WP03 of Joint mixability and negative dependence. The journal version is published in Finance and Stochastics (2013).