Dependence Uncertainty for Aggregate Risk: Examples and Simple Bounds

Abstract

Over the recent years, numerous results have been derived in order to assess the properties of regulatory risk measures (in particular VaR and ES) under dependence uncertainty. In this paper we complement this mainly methodological research by providing several numerical examples for both homogeneous as well as inhomogeneous portfolios. In particular, we investigate under which circumstances the so-called worst-case VaR can be well approximated by the worst-case (i.e. comonotonic) ES. We also study best-case values and simple lower bounds.

Appendix

See (Figs. 15, 16, 17, 18 and 19).

Fig. 15

The fitted values of $\lambda (\widetilde{{\text {M}}}_\alpha )$, $\alpha =0.99$ obtained by regression for different tail-weight levels $\widetilde{{\text {M}}}_\alpha$, for three collections of portfolios: either only Gamma, Lognormal, or Generalized Pareto marginal distributions

Fig. 16

Fitted curve $\delta _\alpha =1-\widetilde{d}^{\,\lambda }$, $\alpha =0.99$ for tail-heaviness level $\widetilde{{\text {M}}}_\alpha =1.4$, superimposed on scatterplots of the true $\delta _\alpha (S_d)$ of the sampled portfolios. Portfolios with only Gamma marginal distributions

Fig. 17

Fitted curves $\delta _\alpha =1-\widetilde{d}^{\,\lambda }$, $\alpha =0.99$ for tail-heaviness levels $\widetilde{{\text {M}}}_\alpha =1.6,2.0,2.2$, superimposed on scatterplots of the true $\delta _\alpha (S_d)$ of those sampled portfolios with tail-heaviness within $10\,\%$ of these levels. Portfolios with only Lognormal marginal distributions

Fig. 18

Fitted curves $\delta _\alpha =1-\widetilde{d}^{\,\lambda }$, $\alpha =0.99$ for tail-heaviness levels $\widetilde{{\text {M}}}_\alpha =1.6,2.0,2.6$, superimposed on scatterplots of the true $\delta _\alpha (S_d)$ of those sampled portfolios with tail-heaviness within $10\,\%$ of these levels. Portfolios with only Generalized Pareto marginal distributions

Fig. 19

On the horizontal axis $\underline{{\text {ES}}}_\alpha (S_d)$ of the sampled portfolios as approximated by the RA (with discretization parameter $N=10^5$ and stopping condition $\varepsilon =10^{-4}$). Vertical axis the difference between the non-sharp bound (11) and the RA bound. For each portfolio the values in $[\mathbb {E}[S_d],{\text {ES}}^+_\alpha (S_d)]$ are normalized to lie within [0, 1]. The color corresponds to the dimension from $d=2$ (dark) to $d=15$ (light). The marginal dfs are sampled from the families given above each plot. For LogN and Gamma distributions, light lower tails are possible, so (11) is away from sharpness and gives a worse bound than the RA. For Pareto, (11) always gives a better bound due to heavy upper tails. For LogN and Pareto mixed, either of the two bounds may be sharper