scholarly journals SOME DISTRIBUTIONAL PROPERTIES OF A CLASS OF COUNTING DISTRIBUTIONS WITH CLAIMS ANALYSIS APPLICATIONS

2013 ◽  
Vol 43 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jae-Kyung Woo
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Negative Binomial ◽  
Claims Analysis ◽  
Explicit Formulas ◽  
Distributional Properties ◽  
Special Cases ◽  
Aggregate Claims ◽  
Stop Loss

AbstractWe discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

scholarly journals PENGUKURAN RISIKO PADA RETENSI OPTIMAL UNTUK REASURANSI STOP LOSS DENGAN VALUE AT RISK

2012 ◽  
Vol 4 (1) ◽  
Author(s):  
Agustina Sunarwatiningsih ◽  
Yuciana Wilandari ◽  
Agus Rusgiyono
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Stop Loss

Return-Risk Evaluation of Investments

2020 ◽  
pp. 161-177
Author(s):  
Paul Weirich
Keyword(s):  
At Risk ◽  
Probability Distribution ◽  
Value At Risk ◽  
Risk Evaluation ◽  
Expected Return ◽  
Factors Affecting ◽  
Special Cases ◽  
Mean Risk

In finance, a common way of evaluating an investment uses the investment’s expected return and the investment’s risk, in the sense of the investment’s volatility, or exposure to chance. A version of this method derives from a general mean-risk evaluation of acts, under the assumption that only money, risk, and their sources matter. Although the method does not require a measure of risk, finance investigates measures of risks to assist evaluations of risks. An investment creates possible returns, and the variance of the probability distribution of their utilities is a measure of the investment’s risk. This measure neglects some factors affecting an investment’s risk, and so is satisfactory only in special cases. Another measure of risk is known as value-at-risk, or VAR. It also neglects some factors affecting an investment’s risk, and so should be restricted to special cases.

scholarly journals Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

2019 ◽  
Vol 181 (2) ◽  
pp. 473-507 ◽  
Author(s):  
E. Ruben van Beesten ◽  
Ward Romeijnders
Keyword(s):  
At Risk ◽  
Error Bounds ◽  
Value At Risk ◽  
Risk Measure ◽  
Mixed Integer ◽  
Conditional Value At Risk ◽  
Two Stage ◽  
Special Cases ◽  
Integer Recourse ◽  
Mean Risk

Abstract In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds.

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 Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (1) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

scholarly journals Analytical Bounds for two Value-at-Risk Functionals

2002 ◽  
Vol 32 (2) ◽  
pp. 235-265 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Decision Makers ◽  
Conditional Value At Risk ◽  
General Interest ◽  
Dominance Order ◽  
Risk Capital ◽  
Time Period ◽  
Aggregate Loss ◽  
Stop Loss

AbstractBased on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interest.

MULTIVARIATE GEOMETRIC TAIL- AND RANGE-VALUE-AT-RISK

2019 ◽  
Vol 50 (1) ◽  
pp. 265-292
Author(s):  
Klaus Herrmann ◽  
Marius Hofert ◽  
Mélina Mailhot
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Distribution Functions ◽  
Special Cases ◽  
Dimensional Distribution

AbstractA generalization of range-value-at-risk (RVaR) and tail-value-at-risk (TVaR) for d-dimensional distribution functions is introduced. Properties of these new risk measures are studied and illustrated. We provide special cases, applications and a comparison with traditional univariate and multivariate versions of the TVaR and RVaR.

scholarly journals The Pareto-Optimal Stop-Loss Reinsurance

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Haiyan You ◽  
Xiaoqing Zhou
Keyword(s):  
At Risk ◽  
Linear Combination ◽  
Value At Risk ◽  
Optimal Parameter ◽  
Insurance Industry ◽  
Risk Measure ◽  
Pareto Optimal ◽  
Stop Loss ◽  
Parameter Values

Reinsurance plays a role of a stabilizer of the insurance industry and can be an effective tool to reduce the risk for the insurer. This paper aims to provide the optimal reinsurance design associated with the stop-loss reinsurance under the criterion of value-at-risk (VaR) risk measure. In this paper, the probability levels in the VaRs used by the both reinsurance parties are assumed to be different and the optimality results of reinsurance are derived by minimizing linear combination of the VaRs of the cedent and the reinsurer. The optimal parameter values of the stop-loss reinsurance policy are formally derived under the expectation premium principle.

scholarly journals Conditional value-at-risk bounds for compound Poisson risks and a normal approximation

2003 ◽  
Vol 2003 (3) ◽  
pp. 141-153 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Conditional Value At Risk ◽  
Insurance Risk ◽  
Simple Method ◽  
Compound Poisson ◽  
Loss Distributions ◽  
Mean And Variance ◽  
Aggregate Claims ◽  
Risk Bounds

A considerable number of equivalent formulas defining conditional value-at-risk and expected shortfall are gathered together. Then we present a simple method to bound the conditional value-at-risk of compound Poisson loss distributions under incomplete information about its severity distribution, which is assumed to have a known finite range, mean, and variance. This important class of nonnormal loss distributions finds applications in actuarial science, where it is able to model the aggregate claims of an insurance-risk business.

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