Understanding Pure Excess-of-Loss Reinsurance

• Pure excess-of-loss reinsurance is a non-proportional risk transfer mechanism where losses above a defined retention level are ceded to a reinsurer.
• The contract employs a per-claim deductible structure that simplifies premium calculations and optimizes risk management through variance reduction.
• Optimal strategies for this reinsurance type are derived using mean-variance, HJB, and distortion risk minimization frameworks, offering quantifiable benefits in dynamic settings.

Pure excess-of-loss reinsurance is a class of non-proportional reinsurance contracts under which the insurer cedes the portion of each individual claim above a pre-specified retention level to the reinsurer. The contract is characterized by its per-claim deductible structure, i.e., for each claim $Z$, the insurer retains $\min\{Z, m\}$ and the reinsurer indemnifies $(Z - m)^+$, where $m$ is the retention limit. The analytical structure and unique optimality of pure excess-of-loss reinsurance have been rigorously established under various optimal control criteria in risk theory—most notably under mean-variance preferences, HJB/PDE-based stochastic control frameworks, and distortion risk minimization settings (Li et al., 2017).

1. Mathematical Formulation and Contract Structure

Let $Z$ be the claim size random variable with cumulative distribution function $F$. In a pure excess-of-loss treaty with retention level $m \geq 0$, the ceded and retained losses per claim are

$$\text{Ceded loss: } (Z - m)^+, \qquad \text{Retained loss: } Z \wedge m.$$

The aggregate insurer loss is the sum of the retained amounts, plus the premium paid to the reinsurer for the excess layer. Under the expected value premium principle with loading $\rho$, the total reinsurance premium per claim is $(1+\rho)\mathbb{E}[(Z-m)^+]$.

Within multi-period models and stochastic factor settings, the retention $m$ may be allowed to be adapted, i.e., $m_t$ can depend on time and potentially on exogenous risk/environmental factors (Brachetta et al., 2019). In all cases, the defining property is the per-claim application of the deductible.

2. Optimality in Stochastic Control and Mean-Variance Criteria

The rigorous derivation of the unique equilibrium nature of pure excess-of-loss reinsurance arises in the study of time-inconsistent mean-variance reinsurance-investment problems, as addressed by Li, Li, and Young (Li et al., 2017).

The insurer's surplus process is modeled, for example, as a spectrally negative Lévy process: $$\mathrm{d}U_t = c\,\mathrm{d}t + \sigma_1\,\mathrm{d}B^{(1)}_t - \int_0^\infty z\, N(\mathrm{d}z, \mathrm{d}t),$$ with premium $c$ computed under the expected value principle, and with additional investment in risky and risk-free assets.

Under a mean-variance objective,

$$J^u(x, t) = \mathbb{E}_{x,t}\left[X^u_T\right] - \frac{\gamma}{2} \mathrm{Var}_{x,t}\left[X^u_T\right],$$

the unique equilibrium reinsurance strategy is to cede for each claim of size $z$,

$$\ell^*(z,t) = \min\left\{ \frac{\eta}{\gamma}e^{-r(T-t)}, ~ z \right\}.$$

That is, the optimal policy is pure excess-of-loss reinsurance with a deterministic, time-dependent deductible/attachment point, uniquely determined as the maximizer of the strictly concave local control problem within the HJB framework (Li et al., 2017).

3. Extensions: Stochastic Environments, Utility, and Multiple Lines

The optimality of pure excess-of-loss contracts persists under stochastic environmental drivers and for various objective criteria:

• In stochastic factor models where claim arrival intensity and claim size distribution are affected by an exogenous factor $Y_t$, and the insurer maximizes the expected exponential utility of terminal wealth, the optimal retention $u^*(t, y)$ solves a strictly convex variational inequality (Brachetta et al., 2019):

$$\left. \frac{\partial q}{\partial u}(t, y, u) = \lambda(t, y)\exp\left(\eta u e^{r(T-t)}\right)\bar{F}(u,y) \right|_{u=u^*(t,y)}.$$

Under the expected-value premium principle, this leads to a closed-form, deterministic $u^*_{\mathrm{EVP}}(t)$.

• In multi-line environments or networks with dependent claim arrival (e.g., common shock models), dynamic programming again yields a unique, pointwise optimal per-claim cap structure—i.e., the per-claim control for each line is of excess-of-loss form: $\mathcal{H}_i^*(u, x) = \min\{x, d_i^*(u)\}$ (Han et al., 2020).
• In dividend optimization models for collaborating business lines, the optimal reinsurance strategy is pure excess-of-loss, both in bounded and unbounded dividend rate regimes, with the optimal deductible being a (possibly reserve-dependent) function but always appearing inside the per-claim minimum function (Boonen et al., 14 Nov 2025).

4. Variational, Distortion, and Risk-Minimization Criteria

Within the unified distortion risk/distortion premium framework (Huang et al., 2018), the optimal ceded loss function is often shown to be increasing and convex, with the pure excess-of-loss (or "stop-loss" in the limiting aggregate sense) being the unique solution for most admissible distortion risk/premium combinations. The solution is characterized by a threshold equation for the retention $d^*$,

$$\frac{g_\pi(t^*)}{g_\rho(t^*)} = \frac{\lambda}{(1-\lambda)(1+p)},$$

where $g_\pi, g_\rho$ are distortion functions for the premium and risk measures, respectively.

Special cases recover Value-at-Risk and Tail Value-at-Risk retention formulae. Explicit formulas and closed-form inversions exist for many such settings, where the minimizer always occurs at an attachment point defining a pure excess-of-loss layer.

5. Statistical Estimation and Empirical Properties

Modern treatments additionally address the estimation of the optimal excess-of-loss retention from data. The sample-based estimator $\hat d^*_{N,\rho_N}$ is constructed by substituting empirical moments into the closed-form first-order condition characterizing the optimal deductible: $$\left(d - \hat\mu_1(d)\right)^2 - \left(\frac{\sqrt{N}\rho_N}{\Phi^{-1}(p)}\right)^2\left[\hat\mu_2(d) - \hat\mu_1(d)^2\right] = 0,$$ with $$\hat\mu_1(d) = \frac{1}{N} \sum_{i=1}^N (X_i \wedge d)$$. Asymptotic normality results hold under decreasing loading or alternative premium principles (e.g., standard deviation or Sharpe-ratio) (Aboagye et al., 30 Apr 2024). Simulation studies and analyses of real datasets confirm the practical accuracy and robustness of these estimators.

Optimal $d^*$ increases with the premium loading, and decreases with the quantile/risk aversion level, in keeping with risk management intuition. Pure excess-of-loss contracts avoid implausible properties such as the zero-insolvency probability associated with stop-loss aggregate treaties.

6. Dynamic and Economic Implications

The economic rationale for the universal optimality of pure excess-of-loss reinsurance is that it maximally reduces loss variance per unit premium cost. In dynamic control settings, the per-claim deductible structure is superior to any hybrid or proportional scheme for variance reduction, and uniquely optimal under mean-variance, exponential utility, and many distortion-based criteria. As the insurer's capital position improves, optimal deductible levels rise, indicating a de-risking regulatory/solvency regime for highly capitalized insurers (Boonen et al., 14 Nov 2025). In fully dynamic or impulse control settings, explicit QVI analysis justifies this policy as minimizing long-run capital injection costs (Luo et al., 2011).

The retention is sensitive to parameters reflecting risk aversion, market loadings, claim severity distributions, and the cost of capital injections. Extensions to time-dependent, state-dependent, and multi-line regimes, as well as nonparametric empirical estimation, all preserve the pure excess-of-loss characterization as the uniquely optimal risk transfer structure across a wide spectrum of actuarial optimization problems (Li et al., 2017).