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Two by-now folkloric results in the theory of risk sharing are that (i) any feasible allocation is convex-order-dominated by a comonotonic allocation; and (ii) an allocation is Pareto optimal for the convex order if and only if it is comonotonic. Here, comonotonicity corresponds to the no-sabotage condition, which aligns the interests of all parties involved. Several proofs of these two results have been provided in the literature, mostly based on the comonotonic improvement algorithm of Landsberger and Meilijson (1994) and a limit argument based on the Martingale Convergence Theorem. However, no proof of (i) is explicit enough to allow for an easy algorithmic implementation in practice; and no proof of (ii) provides a closed-form characterization of Pareto optima. In this paper, we provide novel proofs of these foundational results. Our proof of (i) is based on the theory of majorization and an extension of a result of Lorentz and Shimogaki (1968), which allows us to provide an explicit algorithmic construction that can be easily implemented. In addition, our proof of (ii) leads to a crisp closed-form characterization of Pareto-optimal allocations in terms of alpha-quantiles (mixed quantiles). An application to collaborative insurance, or decentralized risk sharing, illustrates the relevance of these results.

This paper studies the quantile risk-sharing rule introduced in Denuit, Dhaene & Robert (2022). This rule is not actuarially fair, but instead satisfies another type of fairness, which is comparable with “solvency fairness” in classical centralized insurance. New properties are investigated and an axiomatic theory is developed for the quantile risk-sharing rule, which allows for a deeper understanding of its proper use. The axiomatic characterization of the quantile risk-sharing rule is based on aggregate and comonotonicity-related properties of risk-sharing rules.

This paper offers a systematic treatment of risk-sharing rules for insurance losses, based on a list of relevant properties. A number of candidate risk-sharing rules are considered, including the conditional mean risk-sharing rule proposed in Denuit and Dhaene (2012) and the newly introduced quantile risk-sharing rule. Their compliance with the proposed properties is established. Then, methods for building new risk-sharing rules are discussed. The results derived in this paper are shown to be helpful in the development of peer-to-peer insurance (or crowdsurance), as well as to manage contingent risk funds where a given budget is distributed among claimants.

The computation of various risk metrics is essential to the quantitative risk management of variable annuity guaranteed benefits. The current market practice of Monte Carlo simulation often requires intensive computations, which can be very costly for insurance companies to implement and take so much time that they cannot obtain information and take actions in a timely manner. In an attempt to find low-cost and efficient alternatives, we explore the techniques of comonotonic bounds to produce closed-form approximation of the risk measures for variable annuity guaranteed benefits. The techniques are further developed in this paper to address in a systematic way risk measures for death benefits with the consideration of dynamic policyholder behavior. ispartof: FEB Research Report AFI_1598 nrpages: 31 status: published

In this paper, we introduce two classes of indices which can be used to measure the market perception concerning the degree of dependency that exists between a set of random variables, representing di¤erent stock prices at a xed future date. The construction of these measures is based on the theory of comonotonicity. Both types of herd behavior indices are model-free and risk-neutral, derived from available option data. Depending on its particular de nition, each index represents a particular aspect of the market sentiment concerning future co-movement of the underlying stock prices.

In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors X = (X1;X2; : : : ;Xn) and Y = (Y1; Y2; : : : ; Yn) with equal expected utilities (or distorted expectations) of the sums X1 +X2 +: : :+Xn and Y1 +Y2 +: : :+Yn for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is X d=Y . The results in this paper can be considered as generalizations of the results of Cheung (2010), who presents necessary conditions related to the distribution of X1 + X2 + : : : + Xn for the random vector X = (X1;X2; : : : ;Xn) to be comonotonic. ispartof: FEB Research Report AFI_1377 pages:1-23 status: published

Distorted expectations can be expressed as mixtures of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of mixtures of quantiles. ispartof: FEB Research Report AFI_1273 pages:1-11 status: published

We introduce a new and easy to calculate measure for systemic risk in financial markets. This measure is baptized the Herd Behavior Index (HIX). It is model-independent and forward looking, based on observed option data. In order to determine the degree of systemic risk or herd behavior in a financial market one should compare the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The Herd Behavior Index (HIX) is defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance of this extreme market situation. Using the theory of comonotonicity, the extreme situation can easily be backed out of the observed option quotes. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days implied Herd Behavior Index for the Dow Jones Industrial Average, covering the period January 2003 to October 2009. ispartof: FBE Research Report AFI_1157 pages:1-31 status: published

ispartof: Tijdschrift voor Economie en Management issue:2 pages:265-278 status: published

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makeham's law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion. ispartof: DTEW Research Report 0523 pages:1-26 status: published

For a series of cash flows, its stochastically discounted or compounded value is often a key quantity of interest in finance and actuarial science. Unfortunately, even for most realistic rate of return models, it may be too difficult to obtain analytic expressions for the risk measures involving this discounted sum. Some recent research has demonstrated that in the case where the return process follows a Brownian motion, the so-called comonotonic approximations usually provide excellent and robust estimates of risk measures associated with discounted sums of cash flows involving log-normal returns. In this paper, we derive analytic approximations for risk measures in case one considers the continuous counterpart of a discounted sum of log-normal returns. Although one may consider the discrete sums as providing a more realistic situation than its continuous counterpart, considering in this case, the continuous setting leads to more tractable explicit formulas and may therefore provide further insight necessary to expand the theory and to exploit new ideas for later developments. Moreover, the closed-form approximations we derive in this continuous set-up can then be compared more effectively with some exact results, thereby facilitating a discussion about the accuracy of the approximations. Indeed, in the discrete setting, one always must compare approximations with results from simulation procedures which always give rise to room of debate. Our numerical comparisons reveal that the comonotonic 'maximal variance' lower bound approximation provides an excellent fit for several risk measures associated with integrals involving log-normal returns. Similar results as we derive here for continuous annuities can also be obtained in case of continuously compounding which therefore opens a roadmap for deriving closed-form approximations for the prices of Asian options. Future research will also focus on optimal portfolio slection problems. ispartof: DTEW Research Report 0518 pages:1-18 status: published

In this paper, we consider different approximations for computing the distribution function or risk measures related to a sum of non-independent lognormal random variables. Approximations for such sums, based on the concept of comonotonicity, have been proposed in Dhaene et al. (2002a,b). These approximations will be compared with two well-known moment matching approximations: the lognormal and the reciprocal Gamma approximation. We find that for a wide range of parameter values the comonotonic lower bound approximation outperforms the two classical approximations. ispartof: DTEW Research Report 0418 pages:1-13 status: published

We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskless and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of 'constant mix' portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari's dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically. ispartof: DTEW Research Report 0408 pages:1-44 status: published

In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. (2003). ispartof: DTEW Research Report 0416 pages:1-33 status: published

In this paper we present a simple static super-hedging strategy for the payoff of an arithmetic Asian option in terms of a portfolio of European options. Moreover, it is shown that the obtained hedge is optimal in some sense. The strategy is based on stop-loss transforms and is applicable under general stock price models. We focus on some popular Lévy models. Numerical illustrations of the hedging performance are given for various Lévy models calibrated to market data of the S&P 500. ispartof: DTEW Research Report 0365 pages:1-17 status: published

Renshaw and Verrall (1994) specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations. ispartof: DTEW Research Report 0332 pages:1-21 status: published

In this paper, we investigate the notion of dependency between risks and its effect on the related stop-loss premiums. The concept of comonotonicity, being an extreme case of dependency, is discussed in detail. For the bivariate case, it is shown that, given the distributions of the individual risks, comonotonicity leads to maximal stop-loss premius. Some properties of stop-loss order preserving premium principles are considered. A simple proof is given for the sub-additivity property of Wang's premium principle. ispartof: DTEW Research Report 9721 pages:1-17 status: published