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We consider the optimal risk transfer from an insurance company to a reinsurer. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some crucial distinctions. In particular, the insurance company's surplus is here (as is routinely the case) approximated by a Brownian motion, as opposed to the geometric Brownian motion used to model assets in finance. Furthermore, risk exposure is dialled 'down' via reinsurance, rather than 'up' via risky investments. This leads to interesting qualitative differences in the optimal designs. In this paper, using the martingale method, we derive the optimal design as a function of proportional, non-cheap reinsurance design that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider several realistic constraints on the terminal value: a strict lower boundary, the probability (Value at Risk) constraint, and the expected shortfall (conditional Value at Risk) constraints under the $ \mathbb {P} $ P and $ \mathbb {Q} $ Q measures, respectively. In all cases, the optimal reinsurance designs boil down to a combination of proportional protection and option-like protection (stop-loss) of the residual proportion with various deductibles. Proportions and deductibles are set such that the initial capital is fully allocated. Comparison of the optimal designs with the optimal portfolios in finance is particularly interesting. Results are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal decision making of an insurer towards a new insurable business, whose risk is independent of the existing risk faced by the insurer. We assume that the insurer, with the objective of maximizing the expected utility of its final wealth, together with the solvency constraint and the availability of reinsurance as a risk transfer mechanism, is deciding if it is viable to underwrite a new insurance business risk. If this new business is underwritten, it is shown that a stop-loss reinsurance contract is optimal when the solvency risk is quantified by the conditional value at risk. If the regulatory regime changes to the value at risk, the optimal reinsurance form becomes more complicated; it can be either stop-loss or two-layer under the assumption that the new risk has a strictly decreasing probability density function. Numerical examples are provided to illuminate the insurer's decision making and the optimal form of the reinsurance strategy. [ABSTRACT FROM AUTHOR]

In this paper, we study two optimisation settings for an insurance company, under the constraint that the terminal surplus at a deterministic and finite time T follows a normal distribution with a given mean and a given variance. In both cases, the surplus of the insurance company is assumed to follow a Brownian motion with drift. First, we allow the insurance company to pay dividends and seek to maximise the expected discounted dividend payments or to minimise the ruin probability under the terminal distribution constraint. Here, we find explicit expressions for the optimal strategies in both cases, when the dividend strategy is updated at discrete points in time and continuously in time. Second, we let the insurance company buy a reinsurance contract for a pool of insured or a branch of business. We set the initial capital to zero in order to verify whether the premia are sufficient to buy reinsurance and to manage the risk of incoming claims in such a way that the desired risk characteristics are achieved at some terminal time without external help (represented, for instance, by a positive initial capital). We only allow for piecewise constant reinsurance strategies producing a normally distributed terminal surplus, whose mean and variance lead to a given Value at Risk or Expected Shortfall at some confidence level α. We investigate the question which admissible reinsurance strategy produces a smaller ruin probability, if the ruin-checks are due at discrete deterministic points in time. [ABSTRACT FROM AUTHOR]

The assessment of portfolio risk is often explicitly (e.g. the Basel III square root formula) or implicitly (e.g. credit risk models) driven by the marginal distributions of the risky components and their correlations. We assess the extent by which such practice is prone to model error. In the case of n = 2 risks, we investigate under which conditions the unconstrained Value-at-Risk (VaR) bounds (which are the maximum and minimum VaR for S = ∑ i = 1 n X i when only the marginal distributions of the X i are known) coincide with the (constrained) VaR bounds when in addition one has information on some measure of dependence (e.g. Pearson correlation or Spearman's rho). We find that both bounds coincide if the measure of dependence takes value in an interval that we explicitly determine. For probability levels used in risk management practice, we show that using correlation information has typically no effect on the highest possible VaR whereas it can affect the lowest possible VaR. In the case of a general sum of n ⩾ 2 risks, we derive Range Value-at-Risk (RVaR) bounds under an average correlation constraint and we show they are best-possible in the case of a sum of n ⩾ 3 standard uniformly distributed risks. [ABSTRACT FROM AUTHOR]

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment will be studied under a general framework. Indeed, there is no specific type of reinsurance contract, there is no specific dynamics of the involved financial instruments and the financial market does not have to be free of frictions. On the other hand, it will be pointed out how the optimal combination above may provide us with new premium principles making the insurer global risk vanish. The risk will be managed with a coherent risk measure, and the new premium principles will seem to reflect several properties, which are desirable from both the analytical and the economic perspectives. From the analytical viewpoint, the premium principles will be continuous, homogeneous and increasing. From the economic viewpoint, the premium principles will lead to cheaper prices with respect to both the insurance market and the financial one. In other words, the premium principles will make the insurer more competitive in prices under a null risk. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Several methods preventing unbounded optimization problems will warrant special attention, and one particular case will be more thoroughly studied, namely, the combination of the Black–Scholes–Merton pricing model with the conditional value at risk. [ABSTRACT FROM AUTHOR]

The Value-at-Risk (VaR) of comonotonic sums can be decomposed into marginal VaRs at the same level. This additivity property allows to derive useful decompositions for other risk measures. In particular, the Tail Value-at-Risk (TVaR) and the upper tail transform of comonotonic sums can be written as the sum of their corresponding marginal risk measures. The other extreme dependence situation, involving the sum of two arbitrary counter-monotonic random variables, presents a certain number of challenges. One of them is that it is not straightforward to express the VaR of a counter-monotonic sum in terms of the VaRs of the marginal components of the sum. This paper generalizes the results derived in [Chaoubi, I., Cossette, H., Gadoury, S.-P. & Marceau, E. (2020). On sums of two counter-monotonic risks. Insurance: Mathematics and Economics92, 47–60.] by providing decomposition formulas for the VaR, TVaR and the stop-loss transform of the sum of two arbitrary counter-monotonic random variables. [ABSTRACT FROM AUTHOR]

In this paper, we define a new multivariate conditional Value-at-Risk (MCVaR) risk measure. This MCVaR considers both individual risks and the aggregate risk of a portfolio, but prioritizes the aggregate risk. The new MCVaR risk measure is based on the minimization of the expectation of a multivariate loss function, which balances the shortfall and surplus risks of the aggregate risk and the individual risks in an overall risk of a portfolio. It is shown that the MCVaR risk measure holds the properties of positive homogeneity, translation invariance, subadditivity, and monotonicity under certain conditions. Numerical examples of the MCVaR risk measure are presented to illustrate the effect of dependence among individual risks on the MCVaR. [ABSTRACT FROM AUTHOR]

The Bowley solution refers to the optimal pricing density for the reinsurer and optimal ceded loss for the insurer when there is a monopolistic reinsurer. In a sequential game, the reinsurer first sets the pricing kernel, and thereafter the insurer selects the reinsurance contract given the pricing kernel. In this article, we study Bowley solutions under asymmetric information on the insurer's risk preferences where the identity of the insurer is unknown to the reinsurer. By assuming that the insurer adopts a Value-at-Risk measure or a convex distortion risk measure, the optimal pricing kernel for the insurer and the optimal ceded loss function for the reinsurer are determined. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In light of the richness of their structures in connection with practical implementation, we follow the seminal works in economics to use the principal–agent (multidimensional screening) models to study a monopolistic reinsurance market with adverse selection; instead of adopting the classical expected utility paradigm, the novelty of our present work is to model the risk assessment of each insurer (agent) by his value-at-risk at his own chosen risk tolerance level consistent with Solvency II. Under information asymmetry, the reinsurer (principal) aims to maximize his average profit by designing an optimal policy provision (menu) of 'shirt-fit' reinsurance contracts for every insurer from one of the two groups with hidden characteristics. Our results show that a quota-share component, on the top of simple stop-loss, is very crucial for mitigating asymmetric information from the insurers to the reinsurer. [ABSTRACT FROM AUTHOR]

Accurate estimation of value-at-risk (VaR) and assessment of associated uncertainty is crucial for both insurers and regulators, particularly in Europe. Existing approaches link data and VaR indirectly by first linking data to the parameter of a probability model, and then expressing VaR as a function of that parameter. This indirect approach exposes the insurer to model misspecification bias or estimation inefficiency, depending on whether the parameter is finite- or infinite-dimensional. In this paper, we link data and VaR directly via what we call a discrepancy function, and this leads naturally to a Gibbs posterior distribution for VaR that does not suffer from the aforementioned biases and inefficiencies. Asymptotic consistency and root-n concentration rate of the Gibbs posterior are established, and simulations highlight its superior finite-sample performance compared to other approaches. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Firstly, we consider different risk measures in order to determine the solvency capital requirement of a pension fund. Secondly, we illustrate the impact of the time horizon of long-term guarantee products on these capital. We consider a financial market modelled by a common Black–Scholes–Merton model. We neglect the mortality and underwriting risks by assuming that the pension fund is fully hedged against these risks, which allows us to keep understandable and tractable formulæ (the longevity risk will be a part of future researches). A portfolio is built in this market according to different strategies and the pension fund offers a fixed guaranteed rate on a certain time horizon. We begin with well-known static risk measures (value at risk and conditional tail expectation measures) and then we consider their natural dynamic generalization. In order to be time consistent, we consider their iterated versions by a backward iterations scheme. Within the dynamic setting, we show that solvency capital can be expensive and that attention must be paid to the safety level considered. [ABSTRACT FROM PUBLISHER]

We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve those available in the literature that are based on the sole knowledge of the marginal distributions. When the variance of the joint portfolio loss is small enough, further improvements can be obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we extend the concept of mutual exclusivity proposed by [Dhaene, J. & Denuit, M. (1999). The safest dependence structure among risks.Insurance: Mathematics and Economics25, 11–21] to its tail counterpart and baptize this new dependency structure as tail mutual exclusivity. Probability levels are first specified for each component of the random vector. Under this dependency structure, at most one exceedance over the corresponding Value-at-Risks (VaRs) is possible, the other components being zero in such a case. No condition is imposed when all components stay below the VaRs. Several properties of this new negative dependence concept are derived. We show that this dependence structure gives rise to the smallest value of Tail-VaR (TVaR) of a sum of risks within a given Fréchet space, provided that the probability level of the TVaR is close enough to one. [ABSTRACT FROM PUBLISHER]

This article investigates optimal reinsurance treaties minimizing an insurer’s risk-adjusted liability, which encompasses a risk margin quantified by distortion risk measures. Via the introduction of a transparent cost-benefit argument, we extend the results in Cui et al. [Cui, W., Yang, J. & Wu, L. (2013). Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles.Insurance: Mathematics and Economics53, 74–85] and provide full characterizations on the set of optimal reinsurance treaties within the class of non-decreasing, 1-Lipschitz functions. Unlike conventional studies, our results address the issue of (non-)uniqueness of optimal solutions and indicate that ceded loss functions beyond the traditional insurance layers can be optimal in some cases. The usefulness of our novel cost-benefit approach is further demonstrated by readily solving the dual problem of minimizing the reinsurance premium while maintaining the risk-adjusted liability below a fixed tolerance level. [ABSTRACT FROM PUBLISHER]

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal form of reinsurance from the perspective of an insurer when he decides to cede part of the loss to two reinsurers, where the first reinsurer calculates the premium by expected value principle while the premium principle adopted by the second reinsurer satisfies three axioms: distribution invariance, risk loading, and preserving stop-loss order. In order to exclude the moral hazard, a typical reinsurance treaty assumes that both the insurer and reinsurers are obligated to pay more for the larger loss. Under the criterion of minimizing value at risk (VaR) or conditional value at risk (CVaR) of the insurer's total risk exposure, we show that an optimal reinsurance policy is to cede two adjacent layers, where the upper layer is distributed to the first reinsurer. To further illustrate the applicability of our results, we derive explicitly the optimal layer reinsurance by assuming a generalized Wang's premium principle to the second reinsurer. [ABSTRACT FROM AUTHOR]

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided. [ABSTRACT FROM PUBLISHER]

Ever since the first introduction of the expected discounted penalty function (EDPF), it has been widely acknowledged that it contains information that is relevant from a risk management perspective. Expressions for the EDPF are now available for a wide range of models, in particular for a general class of Lévy risk processes. Yet, in order to capitalize on this potential for applications, these expressions must be computationally tractable enough as to allow for the evaluation of associated risk measures such as Value at Risk (VaR) or Conditional Value at Risk (CVaR). Most of the models studied so far offer few interesting examples for which computation of the associated EDPF can be carried out to the last instances where evaluation of risk measures is possible. Another drawback of existing examples is that the expressions are available for an infinite-time horizon EDPF only. Yet, realistic applications would require the computation of an EDPF over a finite-time horizon. In this paper we address these two issues by studying examples of risk processes for which numerical evaluation of the EDPF can be readily implemented. These examples are based on the recently introduced meromorphic processes, including the beta and theta families of Lévy processes, whose construction is tailor-made for computational ease. We provide expressions for the EDPF associated with these processes and we discuss in detail how a finite-time horizon EDPF can be computed for these families. We also provide numerical examples for different choices of parameters in order to illustrate how ruin-based risk measures can be computed for these families of Lévy risk processes. [ABSTRACT FROM PUBLISHER]

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies. [ABSTRACT FROM AUTHOR]

In this paper, we establish several relations between convex order, variance order, and comonotonicity. In the first part, we extend Cheung (2008b) to show that when the marginal distributions are fixed, a sum with maximal variance is in fact a comonotonic sum. Thus the convex upper bound is achieved if and only if the marginal variables are comonotonic. Next, we study the situation where besides the marginal distributions; the variance of the sum is also fixed. Intuitively one expects that adding this information may lead to a bound that is sharper than the comonotonic upper bound. However, we show that such upper bound does not even exist. Nevertheless, we can still identify a special dependence structure known as upper comonotonicity, in which case the sum behaves like a convex largest sum in the upper tail. Finally, we investigate when the convex order is equivalent to the weaker variance order. Throughout this paper, interpretations and significance of the results in terms of portfolio risks will be emphasized. [ABSTRACT FROM AUTHOR]

The Danish pension market of life-cycle products have expanded considerably since its introduction in the beginning of the millennium. The market is maturing and pensioners have the choice between a wide area of different products. It is therefore about time that financial insurance technology is developed to guide the performance measurement of available products. In this paper we develop a simple first version of such a method and we investigate life-cycle products recommended on the web of the four biggest commercial Danish pension companies on one day in February 2007. All considered products are outperformed by trivial benchmark products with constant stock proportion over time. Our approach is the following: for each life-cycle product we first find a trivial benchmark product with the same longterm risk and then we compare the long-term return of the two equivalent products. We primarily consider value at risk (VaR) and tail VaR as risk measures, but we also include a study where the fair value of an interest guarantee is used as risk measure. We consider both long-term mean returns and long-term median returns. We hope that our new method will be regarded as a first step toward a scientifically based ranking of the quality of pension products. [ABSTRACT FROM PUBLISHER]

A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the 'folded' distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated. [ABSTRACT FROM AUTHOR]