PELVE from a regulatory perspective

Abstract

Under Solvency II, the Value-at-Risk (VaR) is applied, although there is broad consensus that the Expected Shortfall (ES) constitutes a more appropriate risk measure. Moving towards ES would necessitate specifying the corresponding ES level. The recently introduced Probability Equivalent Level of VaR and ES (PELVE) determines this by requiring that ES equals the prescribed VaR for a given future payoff, reflecting the situation of an individual insurer. We incorporate the regulator's perspective by proposing PELVE-inspired methods for multiple insurers. We analyze existence and uniqueness of the resulting ES levels, derive expressions for elliptically distributed payoffs and establish limit results for multivariate regularly distributed payoffs. A case study highlights that the choice of method is crucial when payoffs arise from different distribution families. We provide recommendations which of our PELVE-inspired methods are most appropriate in certain scenarios.

Figures (15)

• Figure 1: Solid lines in the left plot are the functions $f$ and $g$. The red and green dashed lines are the a.e. defined maps $t\mapsto -f(t)-tf^{\prime}(t)$ and $t\mapsto -g(t)-tg^{\prime}(t)$. The orange dotted vertical line is the chosen level $\lambda=\frac{1}{3}$. The gray vertical lines indicate the interval over which the objective of the MSE-PELVE remains constant; compare with the right-hand side.
• Figure 2: Graph of the function $c\mapsto \frac{\varphi(\Phi^{-1}(c\lambda))}{c\lambda}$, where $\lambda=0.05$. The solid blue line is $\mathop{\mathrm{\Pi}}\nolimits^{\mathop{\mathrm{Sys}}\nolimits}_{\lambda,g}(X)$ for $X\sim \text{N}(\mu,\sigma)$ and $g(x)=\max\{0,x\}$, where $\mu = 0.75$ and $\sigma = 0.4$. The blue dashed line refers to the value $\frac{\mu}{\sigma}$. The red solid line is $\mathop{\mathrm{\Pi}}\nolimits_{\lambda}(X)$, while the red dashed line refers to the value $-\Phi^{-1}(\lambda)$.
• Figure 3: Boxplot of simulated equity capitals of six insurers. The left-hand side are the simulated values and the right-hand side are the pseudo-log transformed ($f(x) = \text{sign}(x)\log_{10}(1+|x|)$) values.
• Figure 4: Histograms of the simulated equity capitals of insurers $1$, $2$, $4$ and $6$.
• Figure 5: PELVE curves of individual insurers for different VaR levels.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Corollary 2.7
• Proposition 2.8
• Corollary 2.9
• Definition 3.1