scholarly journals Analytical Evaluation of Economic Risk Capital for Portfolios of Gamma Risks

2001 ◽  
Vol 31 (1) ◽  
pp. 107-122 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Value At Risk ◽  
Decision Makers ◽  
Dominance Order ◽  
Economic Risk ◽  
Risk Capital ◽  
Exact Probability ◽  
Multivariate Risk ◽  
Time Period ◽  
Stop Loss ◽  
Analytical Expressions

AbstractBased on the notions of value-at-risk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. 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). Quantitatively, RaC is equal to VaR plus an additional stop-loss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additive for comonotonic risks, which is an important extremal situation encountered in the modeling of dependencies in multivariate risk portfolios. Numerical illustrations for portfolios of gamma distributed risks follow. As a result of independent interest, new analytical expressions for the exact probability density of sums of independent gamma random variables are included, which are similar but different to previous expressions by Provost (1989) and Sim (1992).

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.

scholarly journals Sharing Risk – An Economic Perspective

2009 ◽  
Vol 39 (2) ◽  
pp. 591-613 ◽  
Author(s):  
Andreas Kull
Keyword(s):  
Value At Risk ◽  
Risk Sharing ◽  
Risk Measures ◽  
Insurance Industry ◽  
Risk Measure ◽  
Capital Allocation ◽  
Risk Capital ◽  
Risk Class ◽  
Capital Costs ◽  
Risk Transfer

AbstractWe revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (‘risk-based supervision’) pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structures.

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

Selecting the Best Alternative Based on Its Quantile

2020 ◽  
Author(s):  
Demet Batur ◽  
F. Fred Choobineh
Keyword(s):  
Value At Risk ◽  
Decision Makers ◽  
Correct Selection ◽  
Zone Size ◽  
Indifference Zone ◽  
Probability Of Correct Selection ◽  
A Value ◽  
Conscious Decision ◽  
Target Probability ◽  
Sequential Procedures

A value-at-risk, or quantile, is widely used as an appropriate investment selection measure for risk-conscious decision makers. We present two quantile-based sequential procedures—with and without consideration of equivalency between alternatives—for selecting the best alternative from a set of simulated alternatives. These procedures asymptotically guarantee a user-defined target probability of correct selection within a prespecified indifference zone. Experimental results demonstrate the trade-off between the indifference-zone size and the number of simulation iterations needed to render a correct selection while satisfying a desired probability of correct selection.

On risk measuring in the variance-gamma model

2018 ◽  
Vol 35 (1-2) ◽  
pp. 23-33 ◽  
Author(s):  
Roman V. Ivanov
Keyword(s):  
Value At Risk ◽  
Hypergeometric Functions ◽  
Risk Measures ◽  
Random Variables ◽  
Gamma Model ◽  
Variance Gamma ◽  
Stochastic Volatilities ◽  
Investment Returns ◽  
Variance Gamma Model ◽  
Analytical Expressions

AbstractIn this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

ON SOME PROPERTIES OF TWO VECTOR-VALUED VAR AND CTE MULTIVARIATE RISK MEASURES FOR ARCHIMEDEAN COPULAS

2014 ◽  
Vol 44 (3) ◽  
pp. 613-633 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Value At Risk ◽  
Integral Transform ◽  
Risk Measures ◽  
Usual Sense ◽  
Translation Invariance ◽  
Archimedean Copulas ◽  
Coherent Risk ◽  
Multivariate Risk Measures ◽  
Stop Loss ◽  
Conditional Tail Expectation

AbstractWe consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. and Di Bernardino, E. (2013) Journal of Multivariate Analysis, 119, 32–46; Cousin, A. and Di Bernardino, E. (2014) Insurance: Mathematics and Economics, 55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.

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.

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