Equilibrium Reinsurance & Investment Strategy

• Equilibrium reinsurance and investment strategies are frameworks that simultaneously optimize risk transfer and capital allocation using a mean–variance criterion.
• They employ extended Hamilton–Jacobi–Bellman equations and game-theoretic models to achieve time-consistent, state-independent decision-making.
• Optimal policies feature excess-loss (deductible) reinsurance and wealth-independent investment allocations that adjust according to market parameters and risk aversion.

Equilibrium reinsurance and investment strategy encompasses the simultaneous optimization of risk transfer via reinsurance and capital allocation in financial markets, under various performance objectives and in diverse stochastic environments. The recent literature rigorously characterizes these strategies within dynamic frameworks that often feature time inconsistency, market incompleteness, game-theoretic interactions, and advanced premium calculation principles. This article provides a comprehensive account of the mathematical structures, equilibrium concepts, key optimization frameworks, solution methodologies, and implications for real-world insurance and reinsurance market design.

1. Mean–Variance Equilibrium and Time-Inconsistency

The equilibrium approach to reinsurance–investment typically originates from a mean–variance (MV) objective, acknowledging its inherent time-inconsistency—i.e., the BeLLMan principle fails, and future selves may optimally wish to deviate from decisions made at earlier times. This dynamic inconsistency is addressed by reframing the problem as a noncooperative intrapersonal game. For a surplus process $X^u_T$ with initial state $(x, t)$ and control $u$, the MV performance criterion is

$$J^u(x,t) = \mathbb{E}_{x,t}[X_T^u] - \frac{\gamma}{2}\,\operatorname{Var}_{x,t}[X_T^u], \quad (x,t) \in \mathbb{R}\times[0,T].$$

Time-consistent equilibrium strategies are defined such that no infinitesimal local deviation improves performance. This is formalized through extended Hamilton–Jacobi–BeLLMan (HJB) equations featuring additional nonlinearity to enforce dynamic consistency (Li et al., 2017).

2. Structure of Equilibrium Reinsurance and Investment Strategies

A distinguishing feature of the equilibrium solution under the mean–variance criterion, when the surplus is modeled by a spectrally negative Lévy process and the reinsurance premium uses the expected value principle, is the explicit feedback form:

• Excess-Loss (Deductible) Reinsurance is Uniquely Optimal: For each claim of size $z$ at time $t$,

$$\ell^*(z, t) = \frac{\eta}{\gamma} e^{-r(T-t)} \wedge z,$$

where $\eta$ is the reinsurer’s safety loading and $\gamma$ is the insurer’s absolute risk aversion. For claim sizes smaller than the deductible, the insurer retains all risk. The strategy is state-independent (Li et al., 2017).

• Wealth-Independent Investment Allocation: The optimal dollar investment in the risky asset is

$$\pi^*(t) = \frac{\mu - r}{\gamma\, \sigma_2^2\, e^{-r(T-t)} - \rho\, \sigma_1 \sigma_2},$$

with financial market coefficients ($\mu$, $r$, $\sigma_1$, $\sigma_2$, $\rho$) as defined in the surplus and risky asset dynamics. The absence of $x$ in $\pi^*$ reflects constant absolute risk aversion and the mean–variance criterion (Li et al., 2017).

• Value Function: The equilibrium value function is affine in $x$,

$V(x, t) = e^{r(T-t)} x + B(t),$

with the deterministic function $B(t)$ solving an associated ODE with $B(T) = 0$ (Li et al., 2017).

3. Extended HJB Equations and Claimwise Optimization

The explicit equilibrium is derived by translating the time-consistent problem into an extended HJB equation. The infinitesimal generator, for test function $\phi(x, t)$, is

$$\begin{aligned} &\mathcal{A}^{\ell, \pi}\phi(x,t) = \phi_t(x,t) + \left[ r x + (\mu - r)\pi + \int_0^\infty \left( (\theta - \eta)z + (1 + \eta)\ell(z, t) \right) \nu(dz) \right] \phi_x(x,t) \ &\qquad + \frac{1}{2} \left( \sigma_1^2 + 2\rho\sigma_1\sigma_2\pi + \sigma_2^2\pi^2 \right) \phi_{xx}(x, t) + \int_0^\infty \left( \phi(x - \ell(z, t), t) - \phi(x, t) \right) \nu(dz) \end{aligned}$$

and the equilibrium problem involves maximizing, at each instant,

$$\mathcal{A}^{\ell, \pi} V(x,t) - \frac{\gamma}{2} \mathcal{A}^{\ell, \pi}[g^2(x,t)] + \gamma g(x,t) \mathcal{A}^{\ell, \pi} g(x,t) = 0,$$

with $g(x, t)$ the conditional expectation of terminal wealth. Due to the linearity in the reinsurance premium under the expected value principle, the maximization over $\ell$ for each $z$ is independent (claim-by-claim); the optimal form is always excess-of-loss (Li et al., 2017).

4. Role of the Premium Principle and Risk Aversion

The expected value premium principle yields linearity in the cession (claim) size, and thus the cost and expected value contribution of ceding a loss are proportional. This property is pivotal: the optimal retention is always a deductible (excess-loss) policy, independent of the surplus level $x$. Any other admissible strategy (e.g., proportional reinsurance) fails to be optimal within the class $\ell(z, t) \in [0, z]$ for all $z$ and $t$.

Parameter sensitivities check:

• Deductible size: Increases with $\eta$, decreases with $\gamma$, and decays exponentially with the risk-free rate $r$ and time-to-horizon $T-t$.
• State independence: Retention and investment do not depend on the current surplus, a property directly linked to constant absolute risk aversion and the mean–variance or exponential utility specification.
• Investment–Reinsurance Independence: The investment allocation $\pi^*(t)$ does not interact with the reinsurance retention—reinforcing that, under these assumptions, reinsurance and investment are decoupled at equilibrium (Li et al., 2017).

5. Implications, Robustness, and Extensions

The core result establishes excess-loss reinsurance as uniquely optimal in equilibrium under the MV criterion and expected value premium principle, providing a performance guarantee (time-consistency) not held by pre-commitment solutions. Extensions beyond the expected value premium (such as variance or Wang’s premium principle) or under utility functions with state-dependent risk aversion yield markedly different results—a plausible implication is that the universality of the excess-loss form may not hold in such generalized settings.

Further, in competitive or regime-switching environments, or in multi-agent games where Nash equilibria must be computed numerically (see (Bui et al., 2018)), the structure of equilibrium strategies can depend on interactions, information, or market regime, possibly involving a mix of proportional and excess-of-loss contracts, but the explicit analytic structure in the univariate mean–variance and expected value setting remains a benchmark.

6. Solution Approach and Implementation Guidance

To implement the equilibrium reinsurance-investment strategy in a real-world setting, one requires:

• Specification of model parameters: surplus dynamics (via claim process $\nu(dz)$), market parameters ($r$, $\mu$, $\sigma_1$, $\sigma_2$, $\rho$), premium loadings ($\theta$ for primary, $\eta$ for reinsurance), and risk aversion $\gamma$.
• For each $z$ and $t$, compute the deductible

$d^*(t) = \frac{\eta}{\gamma} e^{-r(T-t)},$

set $\ell^*(z, t) = d^*(t) \wedge z$.

• At each $t$, set the risky asset investment as

$$\pi^*(t) = \frac{\mu - r}{\gamma\, \sigma_2^2\, e^{-r(T-t)} - \rho\, \sigma_1 \sigma_2}.$$

• The value function is constructed as above.
• In discretized, regime-switching, or non-smooth environments, numerical solution of the associated extended HJB equation (using Markov chain approximation, backward value iteration, and numerical maximization) is required (Bui et al., 2018).

Computational resource requirements are modest for univariate, Markovian settings, while higher-dimensional or game-theoretic problems (multiple insurers, stochastic regimes) necessitate substantial simulation or Markov chain approximation techniques.

7. Limitations and Case-Specific Considerations

The current equilibrium characterizations presuppose:

• Surplus modeled as a spectrally negative Lévy process
• Reinsurance premium calculated via the expected value principle
• Mean–variance (or, by approximation, exponential utility) criterion
• Absence of features such as capital injections, transaction costs, or dynamic constraints

Departure from these core assumptions (alternative premium principles, model uncertainty, state-dependent risk aversion, or multi-line risk) may significantly alter strategic structure and solution tractability.

---

In summary, the equilibrium reinsurance and investment strategy, under the mean–variance criterion and expected value premium principle, is characterized by state-independent, deductible reinsurance (excess-loss) and a wealth-independent investment rule in the risky asset, both explicitly specified in terms of the model’s structural parameters. The extended HJB approach and game-theoretic equilibrium definition ensure time-consistency and preclude profitable short-term deviation by future selves, offering a theoretically rigorous and implementationally feasible solution to the dynamic, optimization-driven design of reinsurance and investment policies in insurance markets (Li et al., 2017).