DEFAULT RISK AND DIVERSIFICATION: THEORY AND EMPIRICAL IMPLICATIONS

Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of "diversifiable default risk." The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999) . We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.     

OPTIMAL CAPITAL AND RISK TRANSFERS FOR GROUP DIVERSIFICATION

Diversification is at the core of insurance and other financial business. It constitutes an important issue in the preparation of the new Solvency II framework for the regulation of European insurance undertakings. In this paper, we propose a conceptual framework for a legally enforceable capital and risk transfer which optimally accounts for the designated group diversification benefits. We also provide a consistent valuation principle which is compatible with any prior valuation method. This makes our framework fully flexible and universally applicable. A first simple numerical example illustrates the practicability of our proposal.     

SHAREHOLDER RISK MEASURES

The aim of this paper is to put forward a new family of risk measures that could guide investment decisions of private companies. But at the difference of the classical approach of Artzner, Delbaen, Eber, and Heath and the subsequent extensions of this model, our risk measures are built to reflect the risk perception of shareholders rather than regulators. Instead of an axiomatic approach, we derive risk measures from the optimal policies of a shareholder value‐maximizing company. We study these optimal policies and the related risk measures that we call shareholder risk measures. We emphasize the fact that due to the specific corporate environment, in particular the limited shareholders' liability and the possibility to pay out dividends from cash reserves, these risk measures are not convex. Also, they depend on the specific economic situation of the firm, in particular its current cash level, and thus they are not translation invariant. This paper bridges the gap between two important branches of mathematical finance: risk measures and optimal dividends.     

ASYMPTOTIC EQUIVALENCE OF RISK MEASURES UNDER DEPENDENCE UNCERTAINTY

In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.     

ONE-PARAMETER FAMILIES OF DISTORTION RISK MEASURES

This paper introduces parametric families of distortion risk measures, investigates their properties, and discusses their use in risk management. Their derivation is based on Kusuoka's representation theorem of law invariant and comonotonically additive coherent risk measures. Our approach is to narrow down a tractable class of risk measures by requiring their comparability with the traditional expected shortfall. We make numerical comparison among them and propose a method of estimating the value of the distortion risk measures based on data. Their use and interpretation in risk management will also be discussed.     

RISK MEASURES ON AND VALUE AT RISK WITH PROBABILITY/LOSS FUNCTION

We propose a generalization of the classical notion of the V@R_λ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The V@R_λ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on .     

DISTRIBUTION-INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY

In the first part of the paper, we characterize distribution-invariant risk measures with convex acceptance and rejection sets on the level of distributions. It is shown that these risk measures are closely related to utility-based shortfall risk.
In the second part of the paper, we provide an axiomatic characterization for distribution-invariant dynamic risk measures of terminal payments. We prove a representation theorem and investigate the relation to static risk measures. A key insight of the paper is that dynamic consistency and the notion of "measure convex sets of probability measures" are intimately related. This result implies that under weak conditions dynamically consistent dynamic risk measures can be represented by static utility-based shortfall risk.     

CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY

A new class of risk measures called cash subadditive risk measures is introduced to assess the risk of future financial, nonfinancial, and insurance positions. The debated cash additive axiom is relaxed into the cash subadditive axiom to preserve the original difference between the numéraire of the current reserve amounts and future positions. Consequently, cash subadditive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented, and in such contexts cash additive risk measures cannot be used. Several representations of the cash subadditive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sublinear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash subadditive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash subadditive risk measure.     

A SHORT NOTE ON SECOND-ORDER STOCHASTIC DOMINANCE PRESERVING COHERENT RISK MEASURES

We improve results on law invariant coherent risk measures satisfying the Fatou property due to Kusuoka (2001; Adv. Math. Econ . 3, 83–95) by considering risk measures which are in addition second order stochastic dominance preserving. In particular, we derive a representation result for such risk measures.     

AN OLD-NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and puts it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the φ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.     

OVERLAPPING SETS OF PRIORS AND THE EXISTENCE OF EFFICIENT ALLOCATIONS AND EQUILIBRIA FOR RISK MEASURES

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short‐selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no‐arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow–Debreu equilibria.     

MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set‐valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set‐valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set‐valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set‐valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set‐valued risk measures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).     

COHERENCE AND ELICITABILITY

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile‐based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law‐invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.     

BETTER THAN DYNAMIC MEAN‐VARIANCE: TIME INCONSISTENCY AND FREE CASH FLOW STREAM

As the dynamic mean‐variance portfolio selection formulation does not satisfy the principle of optimality of dynamic programming, phenomena of time inconsistency occur, i.e., investors may have incentives to deviate from the precommitted optimal mean‐variance portfolio policy during the investment process under certain circumstances. By introducing the concept of time inconsistency in efficiency and defining the induced trade‐off, we further demonstrate in this paper that investors behave irrationally under the precommitted optimal mean‐variance portfolio policy when their wealth is above certain threshold during the investment process. By relaxing the self‐financing restriction to allow withdrawal of money out of the market, we develop a revised mean‐variance policy which dominates the precommitted optimal mean‐variance portfolio policy in the sense that, while the two achieve the same mean‐variance pair of the terminal wealth, the revised policy enables the investor to receive a free cash flow stream (FCFS) during the investment process. The analytical expressions of the probability of receiving FCFS and the expected value of FCFS are derived.