Measures for risk, dependence and diversification

Abstract

Two primary tasks in quantitative risk management are measuring risk and managing risk. Risk measures and dependence modeling are important tools for assessing portfolio risk, which have gained much interest in the literature of finance and actuarial science. The assessment of risk further serves to address risk management problems, such as portfolio optimization and risk sharing.

Value-at-Risk (VaR) and Expected Shortfall (ES) are the most widely used risk measures in banking and insurance regulation. The Probability Equivalent Level of VaR-ES (PELVE) is a new risk metric designed to bridge VaR and ES. In Chapter 2, we investigate the theoretical properties of PELVE and address the calibration problem of PELVE, that is, to find a distribution model that yields a given PELVE. Joint mixability, dependence of a random vector with a constant sum, is considered an extreme negative dependence as it represents a perfectly diversified portfolio. Chapter 3 explores the relationship between joint mix and some negative dependence notions in statistics. We further show that the negatively dependent joint mix plays a crucial role in solving the multi-marginal optimal transport problem under the uncertainty in the components of risks.

Diversification is a traditional strategy for mitigating portfolio risk. In Chapter 4, we employ an axiomatic approach to introduce a new diversification measurement called the diversification quotient (DQ). DQ exhibits many attractive properties not shared by existing diversification indices in terms of interpretation for dependence, ability to capture common shocks and tail heaviness, as well as efficiency in portfolio optimization. Chapter 5 provides some technical details and illustrations to support Chapter 4. Moreover, DQ based on VaR and ES have simple formulas for computation. We explore asymptotic behavior of VaR-based DQ and ES-based DQ for large portfolios, the elliptical model, and the multivariate regular varying (MRV) model in Chapter 6, as well as the portfolio optimization problems for the elliptical and MRV models.

Counter-monotonicity, as the converse of comonotonicity, is a natural extreme negative dependence. Chapter 7 conducts a systematic study of pairwise counter-monotonicity. We obtain its stochastic representation, invariance property, interactions with negative association, and equivalence to joint mix within the same Fr ́echet class. We also show that Pareto-optimal allocations for quantile agents exhibit pairwise counter-monotonicity. This finding contrasts sharply with traditional comonotonic allocations for risk-averse agents, inspired further investigation into the appearance of pairwise counter-monotonic allocation in risk-sharing problems. In Chapter 8, we address the risk-sharing problem for agents using distortion riskmetrics, who are not necessarily risk-averse or monotone. Our results indicate that Pareto-optimal allocations for inter-quantile difference agents include pairwise counter-monotonicity. Chapter 9 further explores other decision models in risk-sharing that exhibit pairwise counter-monotonicity in optimal allocations. We introduce a counter-monotonic improvement theorem – a converse result to the widely used comonotonic improvement theorem. Furthermore, we show that pairwise counter-monotonic allocations are Pareto optimal for risk-seeking agents, Bernoulli utility agents, and rank-dependent expected utility agents under certain conditions.

Besides the studies of two extreme negative dependencies, we expand our analysis to dependence modeling through Pearson correlation and copula. In Chapter 10, we characterize all dependence structures for a bivariate random vector that preserve its Pearson correlation coefficient under any common marginal transformations. For multivariate cases, we characterize all invariant correlation matrices and explore the application of invariant correlation in sample duplication. Chapter 11 discusses the selection of copulas when marginals are discontinuous. The checkerboard copula is a desirable choice. We show that the checkboard copula has the largest Shannon entropy and carries the dependence information of the original random vector.