How to Get Bounds for Distribution Convolutions? A Simulation Study and an Application to Risk Management

Durrleman, Valdo; Nikeghbali, Ashkan; Roncalli, Thierry

doi:10.2139/ssrn.1032554

Cited by 7 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let X-{X x ,..., X n ) be a portfolio of multivariate losses, where the marginal losses X t have distributions F ( {x), i = 1,..., n. Suppose one is interested in the maximum VaR and CVaR for the aggregate loss S(X) = Xi + ... + X n whenever XeD(F u ..., F n ) belongs to the set of all multivariate losses with given marginals Fj(x). The evaluation of the maximum VaR of a portfolio with fixed margins is related to Kolmogorov's problem treated among others in Makarov(1981), Ruschendorf(1982), Frank et al(1987), Denuit et al(1999), Durrleman et al(2000), Luciano and Marena(2001) The result (5.1) yields a simple recipe for the calculation of the maximum CVaR for the aggregate loss of portfolios given incomplete information about the marginal losses, say X, eD l m :-Z>4((-°°, °°); lA, -,/4)> ™ e {2,4}, i = 1,..., n, for which it is known that the corresponding stop-loss ordered extremal random variables XJ™l msa have absolutely continuous distributions (Section 3 for m = 2, Section 4, proof of Theorem 4.2 for m = 4).…”

Section: The Maximum Cvar For Portfolios Under Incomplete Informationmentioning

confidence: 99%

Analytical Bounds for two Value-at-Risk Functionals

Hürlimann¹

2002

ASTIN Bull.

View full text Add to dashboard Cite

Based 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.

show abstract

Section: The Maximum Cvar For Portfolios Under Incomplete Informationmentioning

confidence: 99%

Analytical Bounds for two Value-at-Risk Functionals

Hürlimann¹

2002

ASTIN Bull.

View full text Add to dashboard Cite

show abstract

“…In contrast to this, the maximum VaR of a portfolio with fixed marginal losses is not attained at the portfolio with mutual components. This assertion is related to Kolmogorov's problem treated by Makarov [38], Rüschendorf [45], Frank et al [22], Denuit et al [13], Durrleman et al [18], Luciano and Marena [37], Cossette et al [10], and Embrechts et al [19]. In the comonotonic situation, one has with (3.19) only the additive relation…”

Section: The Linear Spearman Copulamentioning

confidence: 89%

Multivariate Fréchet copulas and conditional value‐at‐risk

Hürlimann¹

2004

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

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.

show abstract

“…It has by now become common practice to measure the riskiness of a portfolio 2 An early contribution to the subject explicitly referred to stock indices (to which our application will refer too) is Peirò (1994). With reference to VaR, see, for instance, Duffie and Pan (1997).…”

Section: Portfolio Var For Linearly Correlated Returnsmentioning

confidence: 99%

“…As for the third approach, due to the fat-tailed nature of the marginal returns, put into evidence by the analysis in figures (1), (2) and 3, we assumed that marginal returns were distributed according to a Student's t:…”

Section: Qq-plotmentioning

confidence: 99%

“…In the financial practice, the problem is usually solved either by assuming that returns are normally distributed, or by resorting to historical or Monte Carlo simulation. As for the first approach, recent financial research has cast more and more doubts on the assumption of unconditional and joint normality: it is by now well known that financial returns usually present leptokurtic distributions, with higher peaks and fatter tails than the normal 2 . As for the other approaches (both historical and Monte Carlo), simulation is computationally intensive; Monte Carlo, in addition, may be affected by model misspecification.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Value at Risk Bounds for Portfolios of Non-Normal Returns

Luciano

Marena

2001

SSRN Journal

View full text Add to dashboard Cite

This paper studies Value at Risk (VaR) bounds for sums of stochastically dependent random variables, i.e. portfolios of correlated financial assets. The bounds hold under no restrictions on the dependence or on the marginal distributions of returns. An improvement of the bounds is given for positive (quadrant) dependent rvs. Both sets of bounds are computed for portfolios of 6 international indices. Backtesting confirms the usefulness of the approach, even with respect to other shortcuts, such as the normality assumption. For small portfolios, bounds are not over conservative.

show abstract

How to Get Bounds for Distribution Convolutions? A Simulation Study and an Application to Risk Management

Cited by 7 publications

References 22 publications

Analytical Bounds for two Value-at-Risk Functionals

Analytical Bounds for two Value-at-Risk Functionals

Multivariate Fréchet copulas and conditional value‐at‐risk

Value at Risk Bounds for Portfolios of Non-Normal Returns

Contact Info

Product

Resources

About