Robust risk evaluation of joint life insurance under dependence uncertainty

Abstract

Dependence among multiple lifetimes is a key factor for pricing and evaluating the risk of joint life insurance products. The dependence structure can be exposed to model uncertainty when available data and information are limited. We address robust pricing and risk evaluation of joint life insurance products against dependence uncertainty between two lifetimes. We first show that, for some class of standard contracts, the risk evaluation based on a distortion risk measure is monotone with respect to the concordance order of the underlying copula. Based on this monotonicity, we then study the most conservative and anti-conservative risk evaluations for this class of contracts. We prove that the bounds for the mean, Value-at-Risk and Expected Shortfall are computed by combinations of linear programs when the uncertainty set is defined by a norm-ball centered around a reference copula. Our numerical analysis reveals that the sensitivity of the risk evaluation against the choice of the copula differs depending on the risk measure and the type of the contract, and our proposed bounds can improve the existing bounds based on the available information.

Figures (4)

• Figure 1: Histograms of simulated payoffs ($L$) for the four contracts (F2DA, S2DA, F2DI and S2DI) under the survival Gumbel copula with $\tau = 0.49$. Vertical lines indicate the true values for $\mathbb{E}[L]$ (solid), VaR at 99% (dashed), and ES at 97.5% (dotted). The number of Monte Carlo replications is $10^5$.
• Figure 2: Bounds on the Expectation (E), 99% VaR, and 97.5% ES for the four contract types, plotted as a function of the $\mathcal{L}^1$ distance $\varepsilon$ from a reference survival Gumbel copula with $\tau = 0.49$ (denoted by $C^{\text{ref}}$). The solid black lines represent the bounds for a given $\varepsilon$. The horizontal dashed lines show the values obtained under $C^{\text{ref}}$ (red), the independence copula (dark green), the improved Fréchet-Hoeffding (FH) bounds given Kendall’s tau $\tau = 0.49$ (blue), the improved FH bounds given values $Q=C^{\text{ref}}$ on a subset $\mathcal{S}'=\{(u_m, v_m)\in \mathcal{S}: 0.2 \leqslant u_m,v_m \leqslant 0.8\}$ (orange), and the standard FH bounds (purple).
• Figure 3: Bounds on the Expectation (E), 99% VaR, and 97.5% ES for the four contract types, plotted as a function of the $\mathcal{L}^\infty$ distance $\varepsilon$ from a reference survival Gumbel copula with $\tau = 0.49$. For the (quasi-) copulas in comparison, see the caption of Fig. \ref{['fig:bounds:L1']}.
• Figure 4: Empirical quantile curves of the payoffs for various copulas in comparison. The quantiles are computed based on simulated samples with the number of Monte Carlo replications $10^5$.

Theorems & Definitions (15)

• Definition 1: Monotonicity of payoff
• Proposition 1: Formulas and monotonicity of distortion risk measures
• Remark 1: Improved Fréchet bounds
• Remark 2: Choice of $\varepsilon$
• Proposition 2: Reformulation of $\overline \varrho_h(\mathcal{C}_{\mathcal{S},\varepsilon}(C^{\operatorname{ref}}))$ and $\underline \varrho_h(\mathcal{C}_{\mathcal{S},\varepsilon}(C^{\operatorname{ref}}))$
• Corollary 3: Reformulation of $\overline \varepsilon(\mathcal{C})$
• Proposition 4: VaR bounds on $\mathcal{C}_{\mathcal{S},\varepsilon}(C^{\operatorname{ref}})$
• Proposition 5: ES bounds on $\mathcal{C}_{\mathcal{S},\varepsilon}(C^{\operatorname{ref}})$
• Lemma 6: The case of $\mathcal{L}^1$- or $\mathcal{L}^{\infty}$-norm
• proof : Proof of Proposition \ref{['prop:monotonicity:distortion']}