Model Combination in Risk Sharing under Ambiguity

TL;DR

The paper addresses risk sharing under model ambiguity with multiple loss models by introducing a chi-squared divergence penalty that aggregates across reference models. It extends monotone mean-variance preferences to a multi-model setting and achieves time-consistency by enlarging the state space with auxiliary processes Z^β, yielding a tractable solution. The insurer’s optimal contract α^* and the optimal measure Q^* are derived in closed form under a Cramér-Lundberg loss model, and the wealth process X^* is shown to be affine in the auxiliary Z^* with a variance penalty governed by θ; a verification theorem ensures optimality. The approach is illustrated with a Spanish auto insurance data application, showing variance reduction through model combination and a θ- and η-dependent pricing dynamic, and it highlights the framework’s potential extensions to broader stochastic settings and divergences.

Abstract

We consider the problem of an agent who faces losses in continuous time over a finite time horizon and may choose to share some of these losses with a counterparty. The agent is uncertain about the true loss distribution and has multiple models for the losses. Their goal is to optimize a mean-variance type criterion with model combination under ambiguity through risk sharing. We construct such a criterion using the chi-squared divergence, adapting the monotone mean-variance preferences of Maccheroni et al. (2009) to the model combination setting and exploit a dual representation to expand the state space, yielding a time consistent problem. Assuming a Cramér-Lundberg loss model, we fully characterize the optimal risk sharing contract and the agent's wealth process under the optimal strategy. Furthermore, we prove that the strategy we obtain is admissible and that the value function satisfies the appropriate verification conditions. Finally, we apply the optimal strategy to an insurance setting using data from a Spanish automobile insurance portfolio, where we obtain differing models using cross-validation and provide numerical illustrations of the results.

Figures (4)

• Figure 1: Scatterplots of parameters estimated from the Spanish auto insurance data set. The parameters for the ${\mathbb{P}}_k$ models, $k=1,\ldots,100$, estimated by repeated cross-validation with 50% of the data, are in black. The parameters for ${\mathbb{P}}_C$, estimated from the full data set, are in red.
• Figure 2: Kernel density estimates of the distribution of the insurer's terminal wealth $X_T$ if the insurer does not engage in risk sharing (grey) or executes the optimal risk sharing strategy with $\theta =0.02$ (blue) $\theta =0.01$ (orange), and $\theta =0.005$ (turquoise) under the probability measures ${\mathbb{P}}_C$ and ${\mathbb{Q}}^*$. The other parameters are $\eta=0.12$, $c=5{,}550$, $\pi_k = 1/ 100$ for $k = 1,\ldots, 100$, $\pi_C=0$, $x_{0} = 5{,}000$, and $T=5$.
• Figure 3: Paths of $X^*_t$ and two selected $Z^*_{k,t}$ for $t \in [0,5]$ under the reference measure ${\mathbb{P}}$. The parameters are $\theta = 0.01$, $\eta=0.12$, $c=5{,}550$, $\pi_k = 1/ 100$ for $k = 1,\ldots, 100$, $\pi_C=0$, $x_{0} = 5{,}000$, and $T=5$. The black lines are the mean (middle), the mean plus standard deviation of the paths above the mean (upper), and the mean minus standard deviation of the paths below the mean (lower).
• Figure 4: The left panel shows ${\mathbb{E}}^{{\mathbb{P}}_C}[Y^\eta_T]$ as a function of $\eta$ for three values of $\theta$: $\theta = 0.02$ (dark blue), $\theta = 0.01$ (orange) and $\theta = 0.005$ (turquoise). The optimal $\eta$, found numerically, is denoted by the point. The right panel shows the optimal $\eta^*$ as a function of $\theta$. The other parameters are $\pi_k = 1/ 100$ for $k = 1,\ldots, 100$, $\pi_C=0$, $c=5{,}550$, $y_{0} = 5{,}000$, and $T=5$.

Theorems & Definitions (41)

• Remark 2.1
• Definition 2.4: Admissible risk sharing strategies
• Definition 2.5
• Definition 2.6: Admissible compensators
• Lemma 2.7
• proof
• Corollary 2.8
• Proposition 3.2
• proof
• Proposition 3.3