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 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).

Comparison of market risk models with respect to suggested changes of Basel Accord

2014 ◽  
Vol 64 (Supplement-2) ◽  
pp. 257-274
Author(s):  
Eliška Stiborová ◽  
Barbora Sznapková ◽  
Tomáš Tichý
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Risk Estimation ◽  
Market Risk ◽  
Capital Requirements ◽  
Internal Models ◽  
Conditional Value At Risk ◽  
Risk Capital ◽  
Basel Accord ◽  
Capital Charge

The market risk capital charge of financial institutions has been mostly calculated by internal models based on integrated Value at Risk (VaR) approach, since the introduction of the Amendment to Basel Accord in 1996. The internal models should fulfil several quantitative and qualitative criteria. Besides others, it is the so called backtesting procedure, which was one of the main reasons why the alternative approach to market risk estimation — conditional Value at Risk or Expected Shortfall (ES) — were not applicable for the purpose of capital charge calculation. However, it is supposed that this approach will be incorporated into Basel III. In this paper we provide an extensive simulation study using various sets of market data to show potential impact of ES on capital requirements.

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.

A Fourier approach to the computation of conditional value-at-risk and optimized certainty equivalents

2014 ◽  
Vol 16 (6) ◽  
pp. 3-29 ◽  
Author(s):  
Samuel Drapeau ◽  
Michael Kupper ◽  
Antonis Papapantoleon
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Conditional Value At Risk ◽  
Certainty Equivalents
Sign in / Sign up

Export Citation Format

Share Document