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Optimal risk mitigation by deep reinsurance

Aleksandar Arandjelović, Julia Eisenberg

TL;DR

The paper addresses optimal risk mitigation for an insurer facing claims and market-driven surplus fluctuations over a finite horizon. It introduces a neural-network-based framework to learn proportional reinsurance policies that maximize a blended objective $\beta \cdot \mathbb{E}[u(X_n^b)] - (1-\beta) \cdot \mathbb{P}(\min_{0\le i\le n} X_i^b < 0)$ by replacing the ruin indicator with a surrogate loss $g_\gamma$ and solving via empirical risk minimization. The authors provide theoretical justification for NN policy approximation and demonstrate the approach on a OU-perturbed Cramér--Lundberg model, showing meaningful trade-offs between utility and ruin probability and revealing complex, non-linear policy structures under parameter variation. The work offers a versatile framework for risk-averse reinsurance design with potential extensions to richer models and distributional constraints, enabling practical, data-driven risk management.

Abstract

We consider an insurance company which faces financial risk in the form of insurance claims and market-dependent surplus fluctuations. The company aims to simultaneously control its terminal wealth (e.g. at the end of an accounting period) and the ruin probability in a finite time interval by purchasing reinsurance. The target functional is given by the expected utility of terminal wealth perturbed by a modified Gerber-Shiu penalty function. We solve the problem of finding the optimal reinsurance strategy and the corresponding maximal target functional via neural networks. The procedure is illustrated by a numerical example, where the surplus process is given by a Cramér-Lundberg model perturbed by a mean-reverting Ornstein-Uhlenbeck process.

Optimal risk mitigation by deep reinsurance

TL;DR

The paper addresses optimal risk mitigation for an insurer facing claims and market-driven surplus fluctuations over a finite horizon. It introduces a neural-network-based framework to learn proportional reinsurance policies that maximize a blended objective by replacing the ruin indicator with a surrogate loss and solving via empirical risk minimization. The authors provide theoretical justification for NN policy approximation and demonstrate the approach on a OU-perturbed Cramér--Lundberg model, showing meaningful trade-offs between utility and ruin probability and revealing complex, non-linear policy structures under parameter variation. The work offers a versatile framework for risk-averse reinsurance design with potential extensions to richer models and distributional constraints, enabling practical, data-driven risk management.

Abstract

We consider an insurance company which faces financial risk in the form of insurance claims and market-dependent surplus fluctuations. The company aims to simultaneously control its terminal wealth (e.g. at the end of an accounting period) and the ruin probability in a finite time interval by purchasing reinsurance. The target functional is given by the expected utility of terminal wealth perturbed by a modified Gerber-Shiu penalty function. We solve the problem of finding the optimal reinsurance strategy and the corresponding maximal target functional via neural networks. The procedure is illustrated by a numerical example, where the surplus process is given by a Cramér-Lundberg model perturbed by a mean-reverting Ornstein-Uhlenbeck process.
Paper Structure (6 sections, 2 theorems, 13 equations, 5 figures, 1 table)

This paper contains 6 sections, 2 theorems, 13 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model description
  3. Ruin probability and binary classification problems
  4. Algorithmic reinsurance policies
  5. Numerical experiments
  6. Conclusions

Key Result

Proposition 2.4

Given $b \in \mathcal{A}$, let $(g_{n})_{n \in \na}$ be a uniformly bounded sequence of functions such that, $\PP$-almost surely, $g_{n}(F_{b}(Y)) \to \mathds{1}_{(-\infty,0)}(F_{b}(Y))$. Then,

Figures (5)

  • Figure 1: Surrogate loss functions $g_{\gamma}$ for various choices of $\gamma$.
  • Figure 2: Expected surrogate loss $\mathbb{E}[g_{\gamma}(F_{b}(Y))]$ for various parametrizations of the surrogate loss function in the case of no reinsurance ($b \equiv 1$). The dotted horizontal line corresponds to the ruin probability ($\approx 34.1 \%$).
  • Figure 3: Optimal retention levels $\overline{b}(x)$ for different values of $\beta$. The base model assumes $\beta=0.4$.
  • Figure 4: Trade-off between expected utility of terminal wealth and survival probability obtained through optimal algorithmic reinsurance polices for different choices of $\beta \in [0,1]$. The star denotes the corresponding values obtained when no reinsurance is available.
  • Figure 5: Optimal retention levels for time-varying exposure.

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Definition 3.1: Deep feedforward neural network
  • Definition 3.2: Algorithmic reinsurance policy
  • Theorem 3.3
  • proof
  • ...and 2 more