Index insurance under demand and solvency constraints

TL;DR

This paper analyzes the feasibility of index insurance under solvency and mutualization constraints, proposing a rigorous expected-utility framework and identifying when index products compete with traditional indemnity coverage. It derives sufficient conditions under exponential utility, including a solvency-based threshold on the number of policyholders, and introduces a hybrid indemnity-index product to exploit the strengths of both approaches. A practical cyber-insurance illustration demonstrates how accumulation risk and model-based index payouts affect demand and solvency, and shows how machine-learning methods (e.g., RF, XGBoost) improve index payoff estimation and reduce basis risk. The work offers a pathway to viable index insurance by balancing demand, pricing, and reserve requirements, and it provides a concrete algorithm for identifying cases where index payments are preferable in a portfolio.

Abstract

Index insurance is often proposed to reduce protection gaps, especially for emerging risks. Unlike traditional insurance, it bases compensation on a measurable index, enabling faster payouts and lower claim management costs. This approach benefits both policyholders, through quick payments, and insurers, through reduced costs and better risk control due to reliable data and robust statistical estimates. An important difference with the concept of Cat Bonds is that the feasibility of such coverage relies on the possibility of mutualization. Mutualization, in turn, is achieved only if a sufficiently high number of policyholders agree to subscribe. The purpose of this paper is to introduce a model for the demand for index insurance and to provide conditions under which the solvency of the portfolio is achieved. From these conditions, we deduce a product that combines index and traditional indemnity insurance in order to benefit from the best of both approaches. We illustrate our results with a practical example involving the design of an index insurance product in the field of cyber insurance.

Figures (8)

• Figure 1: Losses $Y$ against duration of interruption $T$ with a distinction between policyholders whose backup plans were activated and those whose backup plans failed to kick-in. We observe an increasing relationship between losses and duration of interruption
• Figure 2: Estimation of $\alpha\rightarrow \log \Psi_Y(\alpha)/\alpha$ (exponential premium). The left point corresponds to the value of risk aversion corresponding to an exponential premium equal to $\pi_Y.$ The right point corresponds to the case where $\pi_Y$ increases by 40%.
• Figure 3: Number of policyholders $n$ in the population of size $N$ who prefer index insurance, as a function of the delay in compensation $\tau$ of the competing indemnity-based insurance product. The values of $n$ are represented for five index payout models namely a linear model (LM), a neural network model (NN), a random forest model (RF), a regression tree model (RT) and an extreme gradient boosting model (XGB). The reference values are computed for $\varepsilon = 0.5\%$
• Figure 4: Number of policyholders $n$ in the population of size $N$ who prefer index insurance, as a function of the mean risk aversion in the population $\Bar{\alpha} = \alpha_- + 1/\lambda$ (panel (a)) and the loading factor of index insurance $\theta$ (panel (c)). The reference values are computed for $\varepsilon = 0.5\%$. Panel (b) shows a plot of expected utility against risk aversion for index and indemnity-based insurance.
• Figure 5: Maximum loading factor of index insurance $\theta^{\text{max}}= \eta_{\frak{e}}(\alpha, \beta)\beta^{-1}$ (panels (a), (b) and (c)) and proportion of compensations for which index insurance is preferable $p_{\frak{e}}(\alpha,\beta)$ (panels (d), (e) and (f)) as functions of the parameter $\frak{e}$ for various values of $\beta$. The value of $\theta_Y$ is also plotted in panels (a), (b), and (c) for comparison.

Theorems & Definitions (8)

• Remark 2.1
• Proposition 3.1
• Corollary 3.2
• Proposition 3.3
• Proposition 3.4
• Remark 3.5
• Proposition 3.6
• Lemma 6.1