A game between two hybrid pension managers under jump-diffusion liabilities: competition pushes risk-taking to K=(μ−r)/σ², while own liability jumps lower welfare and the rival's jumps raise it

Under the exact asset-to-liability ratio formulation, the Nash equilibrium portfolio weights of the two competing managers are independent of the liability jump parameters (λi, μi) and of the risk-free rate r (which cancels algebraically in the ratio dynamics); in the symmetric liability-correlation case ρaηa=ρbηb the weights further reduce to K=(μ−r)/σ², independent of both managers' risk aversion γa and γb. Earlier competing-DC-pension Nash equilibria such as Guan and Liang's were obtained under smooth diffusion liabilities and an additive contribution structure; this paper replaces smooth diffusion liabilities with a jump-diffusion process and uses the exact ratio rather than an approximating surplus, yielding a homogeneous SDE that eliminates the additive auxiliary function of the surplus-based ansatz, and it establishes the jump-parameter and risk-aversion invariance of the equilibrium weights. Supported by the exact homogeneous ratio dynamics of Lemma 1, the individual optimal strategy of Proposition 1, the Nash equilibrium pair of Proposition 2, existence and uniqueness in Proposition 3, and the risk-aversion invariance of Proposition 5, with proofs in Appendices A, C, D and F; Table 2 shows numerically that the Nash weight stays exactly 2.0000 as γa varies over (1,2) with γb=1.5 fixed.

Competition strictly amplifies risk-taking relative to the single-agent benchmark: in the symmetric case the Nash weight K=(μ−r)/σ² exceeds the single-agent Merton benchmark θi=K/γi, and the amplification gap grows as risk aversion increases; in the asymmetric case ρaηa≠ρbηb, risk aversion enters through a cross-liability correction term in Θi and may reduce or even reverse this amplification. Single-manager pension optimisation work such as Josa-Fombellida and Rincón-Zapatero contains no competitor reaction; under a pure relative-performance CRRA objective this paper shows the reaction-function slope (γi−1)/γi∈(0,1) and the intercept θi=K/γi cancel exactly at equilibrium, and it characterises the cross-liability correction under asymmetry. Corollary 1 gives the amplification gap and states it is positive under γi>1 and μ−r>ρiσηi, and Remark 6 gives the asymmetric correction growing with γa−1; Table 2 shows θa=K/γa decreasing in γa while the Nash weight stays at 2.0000, with the gap rising from 0.1682 at γa=1.1 to 0.8763 at γa=1.9.

Liability jumps do not alter equilibrium portfolio weights but materially affect welfare through two loading functions: a higher own jump intensity λi or a larger own mean jump size 1/μi lowers own welfare, whereas a higher opponent jump intensity λb raises manager a's welfare, producing a positive cross-player spillover. The authors state that loading functions capturing own-jump welfare costs and cross-player spillovers are absent from prior work; the mechanism comes from the pure relative-performance objective, since a's welfare depends on Ya(T)/Yb(T) and the opponent's liability jumps push Yb down, improving a's relative standing. Proposition 4 gives the welfare exponent κi and the loading functions, Lemma 3 gives their sign for γi∈(1,2), and Propositions 6, 7 and 8 give ∂Vi/∂λi<0, ∂Vi/∂(1/μi)<0 for μi≥2, and ∂Va/∂λb>0 respectively; Table 4 shows Va falling in λa with a steeper decline for γa closer to 2, and Table 6 shows Va rising from −3.410 to −0.044 while Vb falls from −3.410 to −696.01 as λb goes from 1 to 3.

Under the benchmark parameters the model yields an interpretable numerical picture: a leveraged Merton ratio of 2.0000 (with a risk-free allocation of 1−K=−1.0000), a single-agent optimum of about 1.3333, and a manager with higher jump intensity whose funding ratio decays faster and falls behind in the mean funding-ratio comparison. These numbers place the closed-form results on concrete parameters (Table 1: r=0.04, μ=0.12, σ=0.20, T=20, γa=γb=1.5, λa=λb=1.0, μa=μb=2.0) and explain why E[Ya]/E[Yb]=exp((δa−δb)t) is used instead of E[Ya/Yb] for the comparison. Table 1 lists the benchmark parameters and notes that market parameters follow standard calibrations, liability parameters are consistent with Guan, and jump parameters are set to yield economically realistic actuarial shock frequencies and severities; Figures 5 and 6 show the ratio rising above unity when λa=0.5<λb=1.0 and falling below unity when λa=2.0>λb=1.0.

The results are aimed at readers studying competitive investment strategies of hybrid DC pension funds, and apply in a setting where liabilities evolve as a finite-activity jump-diffusion with truncated exponential jump amplitudes, contributions are proportional to fund value, and both managers invest in the same risky asset while maximising CRRA utility of the terminal funding ratio relative to the competitor's; within this setting the closed-form Nash equilibrium weights, the welfare exponent and the two loading functions can be used directly to compare symmetric and asymmetric liability-correlation cases and as an analytical starting point for regulatory discussion distinguishing individual risk appetite from peer-comparison incentives. The authors note that future work could incorporate inflation-indexed assets, relax the restriction on the risk aversion parameter, and allow managers to invest in different risky assets.

The loaded text is an incomplete version: many equations, some table columns and the specific values in Figures 1–6 are not fully rendered, so the complete expressions for the equilibrium weights, the explicit integral forms of the loading functions and the exact plotted trajectories cannot be checked item by item here; in addition, the welfare conclusions rely on conditions such as γi∈(1,2) and μi≥2, Table 5 shows non-monotone behaviour at μa=1.5 and μa=1.0 which lie outside the domain of Proposition 7, and the authors leave downward jumps with z∈(−1,0), inflation-indexed assets and asset specialisation to future work, so conclusions in those directions await further study.