Optimal dividends for a NatCat insurer in the presence of a climate tipping point

TL;DR

The paper investigates optimal dividend strategies for NatCat insurers facing a climate tipping point that irreversibly alters claim dynamics. It advances a two-stage, Markovian framework by modeling claims with a compound Cox process and a tipping time $\omega\sim\text{Erlang}(k,\xi)$, solving a sequence of auxiliary two-dimensional stochastic control problems in surplus $x$ and intensity $\lambda$ using viscosity solutions of the HJB equation. A finite-grid numerical scheme is developed, with proven uniform convergence to the continuous problem, enabling backward induction across tipping-point stages to handle multiple deterioration events. Numerical experiments show that observable tipping and fair premium adaptation can yield upward potential for shareholders despite increased catastrophe risk, with barrier and two-band strategies adapting to the evolving risk regime. The work contributes a practical, tractable approach to dividend optimization under non-stationary NatCat risk, bridging theory and climate-inspired risk dynamics.

Abstract

We study optimal dividend strategies for an insurance company facing natural catastrophe claims, anticipating the arrival of a climate tipping point after which the claim intensity and/or the claim size distribution of the underlying risks deteriorates irreversibly. Extending earlier literature based on a shot-noise Cox process assumption for claim arrivals, we show that the non-stationary feature of such a tipping point can, in fact, be an advantage for shareholders seeking to maximize expected discounted dividends over the lifetime of the portfolio. Assuming the tipping point arrives according to an Erlang distribution, we demonstrate that the corresponding system of two-dimensional stochastic control problems admits a viscosity solution, which can be numerically approximated using a discretization of the current surplus and the claim intensity level. We also prove uniform convergence of this discrete solution to that of the original continuous problem. The results are illustrated through several numerical examples, and the sensitivity of the optimal dividend strategies to the presence of a climate tipping point is analyzed. In all these examples, it turns out that when the insurance premium is adjusted fairly at the moment of the tipping point, and all quantities are observable, the non-stationarity introduced by the tipping point can, in fact, represent an upward potential for shareholders.

Figures (12)

• Figure 7.1: Action regions (dark) and no-action regions (light) of the optimal strategy as a function of $\lambda$ and $x$ for each of the states in Example 1
• Figure 7.2: Value functions $V^i(x,\lambda)$ as a function of $x$ and $\lambda$ for each State $i$ in Example 1
• Figure 7.3: Comparison of the value function $V^2(x,\lambda_{\text{av}})$ (solid) and $V_{\text{CL}}(x,\lambda_{\text{av}})$ (dashed) as a function of $x$ (left), $V^2(x,\lambda)-V^{\text{NT}}(x,\lambda)$ (middle) and $V^1(x,\lambda)-V^{\text{NT}}(x,\lambda)$ ()right) in Example 1
• Figure 7.4: Action regions (dark) and no-action regions (light) of the optimal strategy as a function of $\lambda$ and $x$ for each of the states in Example 2
• Figure 7.5: Value functions $V^i(x,\lambda)$ minus $x$ as a function of $x$ and $\lambda$ for each State $i$ in Example 2

Theorems & Definitions (12)

• Proposition 2.1
• Lemma 2.2
• Proposition 2.3
• Definition 3.1
• Proposition 3.1
• Proposition 3.2
• Corollary 3.3
• Proposition 4.1
• Proposition 4.2
• Proposition 5.1