Grzegorz Darkiewicz

Can a Coherent Risk Measure Be Too Subadditive

We consider the problem of determining appropriate solvency capital requirements for an insurance company or a financial institution. We demonstrate that the subadditivity condition that is often imposed on solvency capital principles can lead to the undesirable situation where the shortfall risk increases by a merger. We propose to complement the subadditivity condition by a regulators condition. We find that for an explicitly specified confidence level, the Value-at-Risk satisfies the regulators condition and is the most efficient capital requirement in the sense that it minimizes some reasonable cost function. Within the class of concave distortion risk measures, of which the elements, in contrast to the Value-at-Risk, exhibit the subadditivity property, we find that, again for an explicitly specified confidence level, the Tail-Value-at-Risk is the optimal capital requirement satisfying the regulators condition.

On the Distribution of Life Annuities with Stochastic Interest Rates

In the traditional approach to life contingencies only decrements are assumed to be stochastic. In this contribution we consider the distribution of a life annuity (and a portfolio of life annuities) when also the stochastic nature of interest rates is taken into account. Although the literature concerning this topic is already quite rich, the authors usually restrict themselves to the computation of the first two or three moments. However, if one wants to determine e.g. capital requirements using more sofisticated risk measures like Value-at-Risk or Tail Value-at-Risk, more detailed knowledge about underlying distributions is required. For this purpose, we propose to use the theory of comonotonic risks developed in Dhaene et al. (2002a and 2002b), which has to be slightly adjusted to the case of scalar products. This methodology allows to obtain reliable approximations of the underlying distribution functions, in particular very accurate estimates of upper quantiles and stop-loss premiums. Several numerical illustrations confirm the very high accuracy of the methodology.

Bounds for stop-loss premiums of stochastic sums (with applications to life contingencies)

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makehams law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion.

Distortion risk measures for sums of random variables

SummaryWhen we consider random couples (X1,Y1) and (X2,Y2), both elements ofR(FX, FY), relative riskiness of the sumsSi=Xi+Yiresults from dependency structure between the summands. In this paper we investigated the relation between a measure of risk for sums of random variables derived from distortion functions and traditional measures of dependencies like Pearson’sr, Spearman’sρ and Kendall’sτ. In the general case we proved that there is no relation between distortion risk measures and Pearson’sr. We also showed that for many classes of distortion risk measures (non-concave distortion risk measures, Tail Value-at-Risk, proportional Hazard Transform, beta distortion risk measures and many others) the same holds true additionally for Spearman’sρ and Kendall’sτ. These findings aim to illustrate the problem of defining what the right measure of dependency is, and that risk measures widely used in practice are not always consistent with traditional measures of dependencies.ZusammenfassungBeim Betrachten von Paaren von Zufallsvariablen, (X1,Y1) und (X2,Y2), beide Elemente vonR(FX, FY), wird deutlich, dass das relative Risiko der SummenSi=Xi+Yisich aus der Abhängigkeitsstruktur zwischen den Summanden ergibt. In dieser Studie untersuchen wir die Beziehung zwischen einem aus Verzerrungsfunktionen hergeleiteten Risikomaß für die Summen von Zufallsvariablen auf der einen Seite, und traditionellen Abhängigkeitsmaßen wie Pearsonsr, Spearmansρ und Kendallsτ auf der anderen Seite. Wir zeigen auf, dass im Allgemeinfall kein Zusammenhang besteht zwischen dem Verzerrungsrisikomaß und Pearsonsr. Desweiteren zeigen wir, dass für viele Gruppen von Verzerrungsrisikomaßen (nicht-konkave Verzerrungsrisikomaße, Tail Value-at-Risk, proportionaler Hazard Transform, Beta Verzerrungsrisikomaße und viele andere), außerdem das gleiche für Spearmansρ and Kendallsτ gilt. Diese Ergebnisse veranschaulichen zum einen, dass das richtige Abhängigkeitsmaß schwer definierbar ist und zum anderen, dass die weit verbreiteten Risikomaße nicht immer übereinstimmen mit traditionellen Abhängigkeitsmaßen.