Hybrid defined contribution pension funds occupy an awkward middle ground in retirement finance. In a pure defined contribution plan, members contribute at a fixed rate and their retirement benefit depends entirely on how investments perform over a working life that can span twenty to forty years. In a defined benefit plan, the benefit is guaranteed and the sponsor absorbs the investment risk. Hybrid schemes blend both: contributions are fixed as in a DC plan, but the fund carries stochastic liability dynamics of the kind traditionally associated with DB schemes. Because members can observe performance and reallocate their contributions toward funds that are doing better, managers face competitive pressure that distorts how they invest.

Most existing models treat a single pension manager optimizing in isolation, or they model liabilities as smooth diffusions – continuous random walks that never leap. Reality is less polite. Pension obligations can jump discontinuously when regulators revise benefit rules, when longevity shocks force actuarial reassessments, when bulk member transfers occur, or when discount rates are reset. The Botswana team models these shocks explicitly: each fund’s liabilities follow a jump-diffusion process in which jumps arrive according to a finite-activity Poisson random measure, and each jump size is drawn from an exponential distribution truncated to the interval between zero and one. The truncation reflects the dominant direction of actuarial risk – regulatory changes, court judgments, and longevity shocks almost exclusively increase obligations rather than reduce them.

The mathematical machinery rests on a carefully chosen state variable. Rather than tracking absolute surplus – assets minus liabilities – the model tracks the exact asset-to-liability ratio, the funding ratio that pension practitioners use to gauge solvency. A ratio at or above one means the fund is fully funded; below one signals underfunding. Using the Itô-Lévy formula, the authors derive the exact dynamics of this ratio without any proportional surplus approximation, producing a homogeneous stochastic differential equation. When a liability jump of size z occurs, assets are momentarily unaffected while liabilities rise proportionally, so the funding ratio is multiplied by one over one plus z – a proportional reduction of z over one plus z, which is smaller than the shock itself. If liabilities double, the funding ratio halves.

Into this setting the authors place two competing managers, each maximizing constant relative risk aversion utility of their terminal funding ratio relative to the competitor’s. This pure relative-performance objective is the game-theoretic engine: manager a’s welfare depends on the ratio of their funding ratio to manager b’s, and vice versa. Applying the Hamilton-Jacobi-Bellman dynamic programming approach to the coupled system yields Nash equilibrium portfolio strategies and value functions in closed form – a rarity in jump-diffusion games, where closed-form solutions are typically out of reach.

The headline structural result is an invariance theorem. The Nash equilibrium portfolio weights turn out to be completely independent of the liability jump parameters – neither the jump arrival intensity nor the mean jump size enters the equilibrium formula. In the symmetric case, where both managers face identical liability-correlation parameters, the equilibrium weight collapses to the ratio of excess return to variance, the familiar Merton-like quantity determined solely by the market’s risk-reward profile. Neither manager’s risk aversion matters. The mechanism is an exact cancellation: each manager’s reaction function has a competitive amplification slope that grows with risk tolerance, while the intercept falls with risk aversion, and in equilibrium these two effects offset precisely. The managers remain genuinely risk-averse – their value functions retain full dependence on risk aversion through curvature and welfare terms – but the portfolio weights themselves behave as if they were risk-neutral. This invariance holds only under specific conditions: pure relative utility, CRRA preferences, a single shared risky asset, and constant coefficients. Relax any one, and risk aversion re-enters through a cross-liability correction term that appears in the asymmetric case.

Competition itself is unambiguously risk-amplifying. Comparing the Nash equilibrium against the single-agent benchmark, where each manager optimizes absolute funding-ratio utility without regard to the rival, the equilibrium weight strictly exceeds the solo optimum for every risk aversion level above one. The amplification gap widens as risk aversion increases: the most risk-averse managers, who would invest most conservatively in isolation, are pushed furthest from their solo positions by competitive pressure. Under the paper’s benchmark calibration – a risk-free rate of four percent, an expected risky return of twelve percent, and twenty percent volatility – the equilibrium weight equals two, meaning each manager borrows an amount equal to the entire fund value and invests twice the fund’s assets in the risky asset, holding a short position in the risk-free asset to finance the leverage.

The welfare results carry the sharpest economic sting. Two loading functions, absent from all prior work, capture how liability jumps affect each manager’s expected utility. A manager’s own jump intensity and jump size both reduce welfare: more frequent or larger shocks to one’s own liabilities drag down the funding ratio and with it the relative-performance payoff. But the competitor’s jumps work in the opposite direction. When the rival fund’s liabilities jump more often, the rival’s funding ratio decays faster, improving one’s own relative standing even though one’s own fund is untouched. Numerical illustrations confirm the theory: as the competitor’s jump intensity rises from one-half to three, the opponent’s value function collapses from roughly negative zero-point-nine to nearly negative seven hundred, while the beneficiary’s welfare improves monotonically. This creates a strategic incentive for each manager to benefit from the competitor’s exposure to adverse shocks – a spillover entirely invisible in single-manager frameworks.

The implications reach into pension governance and regulation. If relative-performance evaluation, whether explicit benchmarking or implicit peer comparison, induces managers to take more risk than is individually optimal, then regulators observing excessive risk-taking may need to ask whether peer-comparison incentives rather than individual risk appetite are the true driver. The spillover result adds a systemic dimension: funds with lower liability jump exposure effectively profit from competitors’ adverse shocks, an incentive structure that only becomes visible once competition is modeled explicitly. The authors note that their model imposes no leverage or short-sale constraints, whereas real pension regulation often limits leverage – incorporating such constraints would require numerical methods beyond the current scope. Future extensions include inflation-indexed assets, relaxed risk-aversion restrictions through alternative jump distributions, and allowing managers to invest in different risky assets, which would introduce additional asymmetry into the reaction functions. For now, the message is stark: in the competitive arena of pension management, the game itself – not the shocks – determines how much risk everyone takes, but the shocks decide who wins.

Subject of Research: Stochastic differential game theory applied to competing hybrid pension fund managers facing jump-diffusion liability risk

Article Title: A stochastic differential game between two hybrid pension fund managers under a jump-diffusion liability process

Article References: Keganneng, O., Chikodza, E., & Basimanebotlhe, O. (2026). A stochastic differential game between two hybrid pension fund managers under a jump-diffusion liability process. Frontiers in Applied Mathematics and Statistics, 12, Article 1931310. https://doi.org/10.3389/fams.2026.1931310

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1931310

Keywords: pension funds, Nash equilibrium, stochastic differential game, jump-diffusion, defined contribution, liability risk, CRRA utility, Hamilton-Jacobi-Bellman, funding ratio, competitive risk-taking, welfare spillover, mathematical finance