## Welcome to Ruodu Wang's Homepage Lhasa, China, 2019

## Working Paper Series

Axiomatic theory of risk measures Risk sharing, allocation, and equilibria
Joint mixability Robust risk aggregation
Risk management with risk measures E-values: Theory and methods

# 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 will be useful. Numerical approximations can be carried out via the Rearrangement Algorithm. Some basic terminology is listed below.

• Dependence uncertainty: a joint model with known marginals and unknown dependence
• An aggregation set: the set of all possible distributions of the sum with specified marginals
• A homogeneous aggregate risk: one with identical marginals; otherwise heterogeneous
• The worst-case risk measure: the maximum value that a risk measure takes over an aggregation set
• Asymptotic equivalence: the ratio between two worst-case risk measures converging to 1 as n → ∞
• Some particular results are:

• Worst-case VaR: homogeneous (WP01, WP02), heterogeneous (WP07), limits (WP08)
• Worst-case distortion risk measures (WP05, WP11, WP13)
• Asymptotic equivalence: VaR (WP04, WP06), distortion risk measures (WP05, WP11)
• Other models: collective risk models (WP09), factor models (WP10), marginal uncertainty (WP12)
• 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.

WP17 Risk aggregation under dependence uncertainty and an order constraint
(by Yuyu Chen, Liyuan Lin, Ruodu Wang)

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 WP14. 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).

WP16 Ordering and inequalities for mixtures on risk aggregation
(by Yuyu Chen, Peng Liu, Yang Liu, Ruodu Wang)

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).

WP15 Convolution bounds on quantile aggregation
(by Jose Blanchet, Henry Lam, Yang Liu, Ruodu Wang)

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.

WP14 The directional optimal transport
(by Marcel Nutz, Ruodu Wang)

We introduce the directional optimal transport problem where origins on the real line can only be transported to destinations to the right of the origins. We obtain an optimal coupling for supermodular costs, which admits manifold characterizations: geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel. The journal version is published in Annals of Applied Probability (2022).

WP13 Dual utilities on risk aggregation under dependence uncertainty
(by Ruodu Wang, Zuoquan Xu, Xunyu Zhou)

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. The journal version is published in Finance and Stochastics (2019).

WP12 Worst-case Range Value-at-Risk with partial information
(by Lujun Li, Hui Shao, Ruodu Wang, Jingping Yang)

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).

WP11 Asymptotic equivalence of risk measures under dependence uncertainty
(by Jun Cai, Haiyan Liu, Ruodu Wang)

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).

WP10 Risk bounds for factor models
(by Carole Bernard, Ludger Rüschendorf, Steven Vanduffel, Ruodu Wang)

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).

WP09 Collective risk models with dependence uncertainty
(by Haiyan Liu, Ruodu Wang)

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).

WP08 Diversification limit of quantiles under dependence uncertainty
(by Valeria Bignozzi, Tiantian Mao, Bin Wang, Ruodu Wang)

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).

WP07 General convex order on risk aggregation
(by Edgars Jakobsons, Xiaoying Han, Ruodu Wang)

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. The journal version is published in Scandinavian Actuarial Journal (2016).

WP06 Aggregation-robustness and model uncertainty of regulatory risk measures
(by Paul Embrechts, Bin Wang, Ruodu Wang)

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).

WP05 How superadditive can a risk measure be?
(by Ruodu Wang, Valeria Bignozzi, Andreas Tsanakas)

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).

WP04 Extreme negative dependence and risk aggregation
(by Bin Wang, Ruodu Wang)

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).

WP03 An academic response to Basel 3.5
(by Paul Embrechts, Giovanni Puccetti, Ludger Rüschendorf, Ruodu Wang, Antonela Beleraj)

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).

WP02 Risk aggregation with dependence uncertainty
(by Carole Bernard, Xiao Jiang, Ruodu Wang)

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. 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).

WP01 Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities (by Ruodu Wang, Liang Peng, Jingping Yang)

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. The journal version is published in Finance and Stochastics (2013).