OPTIMAL REINSURANCE FROM THE PERSPECTIVES OF BOTH AN INSURER AND A REINSURER

2015 ◽  
Vol 46 (3) ◽  
pp. 815-849 ◽  
Author(s):  
Jun Cai ◽  
Christiane Lemieux ◽  
Fangda Liu
Keyword(s):  
Wide Class ◽  
Value At Risk ◽  
Risk Measures ◽  
Convex Combination ◽  
Expected Value ◽  
Point Of View ◽  
The Other ◽  
Optimal Reinsurance ◽  
Optimal Form ◽  
Conflicting Interests

AbstractOptimal reinsurance from an insurer's point of view or from a reinsurer's point of view has been studied extensively in the literature. However, as two parties of a reinsurance contract, an insurer and a reinsurer have conflicting interests. An optimal form of reinsurance from one party's point of view may be not acceptable to the other party. In this paper, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the Value-at-Risk (VaR) risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interests of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance forms in a wide class of reinsurance policies and under the expected value reinsurance premium principle. These optimal reinsurance forms are more complicated than the optimal reinsurance contracts from the perspective of one party only. The proposed models can also be reduced to the problems of minimizing the VaR of one party's loss under the constraints on the interests of both the insurer and the reinsurer.

OPTIMAL REINSURANCE FROM THE VIEWPOINTS OF BOTH AN INSURER AND A REINSURER UNDER THE CVAR RISK MEASURE AND VAJDA CONDITION

2021 ◽  
pp. 1-29
Author(s):  
Yanhong Chen
Keyword(s):  
Loss Function ◽  
Value At Risk ◽  
Numerical Study ◽  
Risk Measure ◽  
Convex Combination ◽  
Weighting Factor ◽  
Loss Functions ◽  
Conditional Value At Risk ◽  
Optimal Reinsurance ◽  
The Impact

ABSTRACT In this paper, we study the optimal reinsurance contracts that minimize the convex combination of the Conditional Value-at-Risk (CVaR) of the insurer’s loss and the reinsurer’s loss over the class of ceded loss functions such that the retained loss function is increasing and the ceded loss function satisfies Vajda condition. Among a general class of reinsurance premium principles that satisfy the properties of risk loading and convex order preserving, the optimal solutions are obtained. Our results show that the optimal ceded loss functions are in the form of five interconnected segments for general reinsurance premium principles, and they can be further simplified to four interconnected segments if more properties are added to reinsurance premium principles. Finally, we derive optimal parameters for the expected value premium principle and give a numerical study to analyze the impact of the weighting factor on the optimal reinsurance.

scholarly journals Optimal Limited Stop-Loss Reinsurance under VaR, TVaR, and CTE Risk Measures

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Xianhua Zhou ◽  
Huadong Zhang ◽  
Qingquan Fan
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Loss Ratio ◽  
Optimal Reinsurance ◽  
Numerical Examples ◽  
Optimal Values ◽  
Stop Loss ◽  
Conditional Tail Expectation ◽  
Reinsurance Market

This paper aims to provide a practical optimal reinsurance scheme under particular conditions, with the goal of minimizing total insurer risk. Excess of loss reinsurance is an essential part of the reinsurance market, but the concept of stop-loss reinsurance tends to be unpopular. We study the purchase arrangement of optimal reinsurance, under which the liability of reinsurers is limited by the excess of loss ratio, in order to generate a reinsurance scheme that is closer to reality. We explore the optimization of limited stop-loss reinsurance under three risk measures: value at risk (VaR), tail value at risk (TVaR), and conditional tail expectation (CTE). We analyze the topic from the following aspects: (1) finding the optimal franchise point with limited stop-loss coverage, (2) finding the optimal limited stop-loss coverage within a certain franchise point, and (3) finding the optimal franchise point with limited stop-loss coverage. We provide several numerical examples. Our results show the existence of optimal values and locations under the various constraint conditions.

scholarly journals Pareto-Optimal Reinsurance Revisited: A Two-Stage Optimisation Procedure Approach

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ying Fang ◽  
Lu Wang ◽  
Zhongfeng Qu ◽  
Wenguang Yu
Keyword(s):  
Value At Risk ◽  
Integral Form ◽  
Expected Value ◽  
Value Premium ◽  
Pareto Optimal ◽  
Two Stage ◽  
Optimal Reinsurance ◽  
Incentive Compatible ◽  
Ex Post ◽  
Optimisation Procedure

In this paper, based on the Tail-Value-at-Risk (TVaR) measure, we revisit the Pareto-optimal reinsurance policies for the insurer and the reinsurer via a two-stage optimisation procedure. To reduce ex-post moral hazard, we assume that reinsurance contracts satisfy the principle of indemnity and the incentive compatible constraint which have been advocated by Huberman et al. (1983). We show that the Pareto-optimal reinsurance policy exists if the reinsurance premiums can be expressed as an integral form. The proposed class of premium principles encompasses the net premium principle, expected value premium principle, TVaR premium principle, generalized percentile premium principle, and so on. We further use the TVaR premium principle and the expected value premium principle as examples to illustrate the two-stage optimisation procedure by deriving explicitly the Pareto-optimal reinsurance policies. We extend the results by Cai et al. (2017) when the expected value premium principle is replaced by the TVaR premium principle.

Optimal Reinsurance for Variance Related Premium Calculation Principles

2010 ◽  
Vol 40 (1) ◽  
pp. 97-121 ◽  
Author(s):  
Manuel Guerra ◽  
Maria de Lourdes Centeno
Keyword(s):  
Numerical Computation ◽  
Point Of View ◽  
Adjustment Coefficient ◽  
Optimal Arrangement ◽  
Optimal Reinsurance ◽  
Optimal Form ◽  
Retention Level ◽  
Premium Calculation ◽  
Stop Loss ◽  
Increasing Function

AbstractThis paper deals with numerical computation of the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk and the reinsurance loading is an increasing function of the variance.We compare the optimal treaty with the best stop loss policy. The optimal arrangement can provide a significant improvement in the adjustment coefficient when compared to the best stop loss treaty. Further, it is substantially more robust with respect to choice of the retention level than stop-loss treaties.

OPTIMAL REINSURANCE DESIGN WITH DISTORTION RISK MEASURES AND ASYMMETRIC INFORMATION

2021 ◽  
pp. 1-23
Author(s):  
Tim J. Boonen ◽  
Yiying Zhang
Keyword(s):  
At Risk ◽  
Asymmetric Information ◽  
Value At Risk ◽  
Functional Form ◽  
Risk Measures ◽  
Risk Measure ◽  
Incentive Compatibility ◽  
Optimal Reinsurance ◽  
Net Profit ◽  
Distortion Risk Measures

ABSTRACT This paper studies a problem of optimal reinsurance design under asymmetric information. The insurer adopts distortion risk measures to quantify his/her risk position, and the reinsurer does not know the functional form of this distortion risk measure. The risk-neutral reinsurer maximizes his/her net profit subject to individual rationality and incentive compatibility constraints. The optimal reinsurance menu is succinctly derived under the assumption that one type of insurer has a larger willingness to pay than the other type of insurer for every risk. Some comparative analyses are given as illustrations when the insurer adopts the value at risk or the tail value at risk as preferences.

scholarly journals Multivariate Fréchet copulas and conditional value-at-risk

2004 ◽  
Vol 2004 (7) ◽  
pp. 345-364 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Convex Combination ◽  
Conditional Value At Risk ◽  
Risk Allocation ◽  
Risk Capital ◽  
Multivariate Distributions ◽  
Multivariate Statistical ◽  
New Family

Based on the method of copulas, we construct a parametric family of multivariate distributions using mixtures of independent conditional distributions. The new family of multivariate copulas is a convex combination of products of independent and comonotone subcopulas. It fulfills the four most desirable properties that a multivariate statistical model should satisfy. In particular, the bivariate margins belong to a simple but flexible one-parameter family of bivariate copulas, called linear Spearman copula, which is similar but not identical to the convex family of Fréchet. It is shown that the distribution and stop-loss transform of dependent sums from this multivariate family can be evaluated using explicit integral formulas, and that these dependent sums are bounded in convex order between the corresponding independent and comonotone sums. The model is applied to the evaluation of the economic risk capital for a portfolio of risks using conditional value-at-risk measures. A multivariate conditional value-at-risk vector measure is considered. Its components coincide for the constructed multivariate copula with the conditional value-at-risk measures of the risk components of the portfolio. This yields a “fair” risk allocation in the sense that each risk component becomes allocated to its coherent conditional value-at-risk.

scholarly journals Comparing Risk Adjusted Premiums from the Reinsurance Point of View

1998 ◽  
Vol 28 (2) ◽  
pp. 221-239 ◽  
Author(s):  
João Manuel Andrade e Silva ◽  
Maria de Lourdes Centeno
Keyword(s):  
Expected Value ◽  
Point Of View ◽  
The Other ◽  
Adjustment Coefficient ◽  
Proportional Hazard ◽  
Premium Calculation ◽  
Stop Loss

AbstractIn this paper we compare, from the point of view of reinsurance, the several risk adjusted premium calculation principles considered in Wang (1996b). We conclude that, with the exception of the proportional hazard (PH) premium calculation principle, all the others behave in a way similar to the expected value principle. We prove that the stop loss reinsurance premium when calculated using the PH premium principle gives a higher premium than any of the other transforms, provided that the priority is big enough. We observe a similar behaviour with respect to excess of loss reinsurance in all the examples given.We also study the behaviour of the adjustment coefficient, both from the insurer's and the reinsurer's point of view as functions of the priority, when the PH principle is used as opposed to the expected value principle.

Optimal expected utility risk measures

2018 ◽  
Vol 35 (1-2) ◽  
pp. 73-87 ◽  
Author(s):  
Sebastian Geissel ◽  
Jörn Sass ◽  
Frank Thomas Seifried
Keyword(s):  
At Risk ◽  
Expected Utility ◽  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Point Of View ◽  
Financial Loss ◽  
Certainty Equivalents ◽  
Average Value ◽  
Average Value At Risk

AbstractThis paper introduces optimal expected utility (OEU) risk measures, investigates their main properties and puts them in perspective to alternative risk measures and notions of certainty equivalents. By taking the investor’s point of view, OEU maximizes the sum of capital available today and the certainty equivalent of capital in the future. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility functionuhas constant relative risk aversion. We present several different risk measures that can be derived with special choices ofuand illustrate that OEU is more sensitive than value at risk and average value at risk with respect to changes of the probability of a financial loss.

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