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

1 Introduction

Hybrid defined contribution (DC) pension funds, which blend the fixed-contribution structure of DC schemes with liability features traditionally associated with defined benefit plans, are an increasingly important part of retirement planning globally [1–3]. In a DC pension plan, the contribution rate is fixed before retirement but the retirement benefit depends on investment performance over the accumulation phase. This differs from a defined benefit (DB) plan, where the benefit is fixed and the sponsor bears the investment risk. Because DC pension plans run over a typical working life of twenty to 40 years [4, 5], the choice of investment strategy is of central importance for members. Such hybrid plans combining DB-style liability dynamics with the investment flexibility of DC schemes have become increasingly prevalent [6–10].

The single-manager DC pension problem has attracted substantial attention. Early contributions by Vigna and Haberman [11] and Haberman and Vigna [12] characterized optimal strategies and quantified risk. Guaranteed-return variants were treated by Boulier et al. [13] under stochastic interest rates and by Deelstra et al. [14] with a stochastic floor guarantee. The long-run effects of inflation and interest rates are examined in Cairns et al. [4], Gerrard et al. [15], and Vigna and Haberman [11], and stochastic salaries and contributions are treated in Josa-Fombellida and Rinc{́o}n-Zapatero [16, 17]. Liability-driven investment, where the fund tracks its obligations, forms a related strand [18–28]. Jump-diffusion liability models, which accommodate sudden obligation shocks, are studied in Gao [29] and Mudzimbabwe [30], and Guan and Liang [31] address stochastic interest rates with mean-reverting returns.

The papers reviewed above each study a single pension fund manager optimizing in isolation. In practice, however, pension fund managers compete with each other. Members compare funds and reallocate contributions toward higher-performing ones, creating incentives for managers to outperform their competitors. This competition distorts each manager's portfolio choice. Portfolio games between two competing investors were first studied by Browne [32]. Basak and Makarov [33] studied the case where investors care about their wealth relative to a competitor, and relative performance concerns in asset management are also studied in Cuoco and Kaniel [34] and Basak and Makarov [35]. Competition between insurance companies as a stochastic game is studied in Bensoussan et al. [36]. In the pension setting, Guan and Liang [37] derived the Nash equilibrium investment strategies for two DC pension funds competing under inflation risk, giving the first closed-form Nash equilibrium for competing DC pension funds. Subsequent contributions address model uncertainty [38, 39], ambiguity aversion [40], community pension competition [41], and jump risk in pension games [42, 43]. The present paper is most closely related to Josa-Fombellida and Rincon-Zapatero [42] and Rincón-Zapatero and Josa-Fombellida [43]: we extend their setting by adopting the exact asset-to-liability ratio formulation (eliminating the surplus approximation), a truncated exponential jump structure that yields explicit welfare loading functions, and deriving the risk-aversion and jump-parameter invariance of the Nash equilibrium weights.

This paper extends the work of Guan and Liang [37] by adding liability jump risk. Three features distinguish our model from the closest related work. Guan and Liang [37] study two DC pension funds under inflation risk with smooth diffusion liabilities and an additive contribution structure; we replace smooth diffusion liabilities with a jump-diffusion liability process and use the exact asset-to-liability ratio rather than an approximating surplus. Basak and Makarov [33, 35] study relative performance in asset management with specialized assets and no liabilities; our setting adds stochastic liabilities with jump risk, a constant relative risk aversion (CRRA) objective on the funding ratio, and an explicit pension management interpretation. Josa-Fombellida and Rincón-Zapatero [16, 17] study single-manager pension optimisation; we study the two-manager competitive game. These differences have two mathematical consequences. The exact ratio formulation yields a homogeneous stochastic differential equation (SDE), eliminating the additive auxiliary function that appears in the surplus-based ansatz of Guan and Liang [37]. The jump-liability structure produces two welfare loading functions, capturing own-jump welfare costs and cross-player spillovers, absent from all prior work. Each pension manager faces liabilities that jump at random times, for example owing to a regulatory change, a longevity shock, or a bulk member transfer. These sudden jumps cannot be captured by a smooth diffusion model alone. Each manager maximizes CRRA utility of the terminal funding ratio relative to the competitor's. Dynamic programming then yields Nash equilibrium strategies in closed form. We show that the equilibrium portfolio weights are invariant to the liability jump parameters. In the symmetric liability-correlation case, they are also invariant to both managers' risk aversion, being determined solely by K = (μ−r)/σ²; in the asymmetric case, risk aversion enters through a cross-liability correction term. Competition makes both managers take more risk than they would on their own, and this effect is stronger for more risk-averse managers. We also show that a manager's welfare falls when their own liabilities jump more, but rises when the other manager's liabilities jump more, a spillover that reflects the relative-performance objective. All results are illustrated by numerical examples.

The rest of this paper is organized as follows. Section 2 presents the financial market and pension management structure. Section 3 derives the Nash equilibrium strategies. Section 4 presents sensitivity analysis and numerical illustrations. Section 5 concludes. Proofs of Lemmas 1, 2 are in Appendices A, B. Proofs of Propositions 1, 2 are in Appendices C, D respectively; the Verification Theorem is proved in Appendix E. Remaining proofs (Propositions 3, 5, 6–8, Corollary 1, and Lemma 3) are in Appendix F.

2 The financial market and pension management

We model a hybrid DC pension fund in which the contribution rate is fixed (DC feature) while the liability process evolves stochastically with both a diffusion and a jump component (DB-style liability dynamics). Throughout this section we work on a filtered complete probability space (Ω,F,{Ft}t∈[0,T],ℙ) satisfying the usual conditions (right-continuity and completeness of the filtration), where [0, T] is a fixed finite time horizon. All processes are assumed to be well-defined and {Ft}-adapted. Solutions to SDEs driven by a Brownian motion and a finite-activity Poisson random measure have cádlág (right-continuous with left limits) sample paths [44], a property used when evaluating the value function at the terminal time T.

2.1 The financial market

The financial market consists of one risk-free asset and one risky asset. Trading is frictionless: there are no transaction costs or taxes, and short selling is permitted.

The price of the risk-free asset B(t) satisfies

dB(t)=rB(t)dt, B(0)=1,(1)

where r>0 is the constant risk-free interest rate (we show in Lemma 1 that r cancels from the funding-ratio dynamics, so it does not appear in any equilibrium formula). The price S(t) of the risky asset satisfies

dS(t)S(t)=μdt+σdW(t), S(0)=1,(2)

where μ>r and σ>0 are positive constants and W(t) is a standard Brownian motion. The ratio (μ−r)/σ², equal to the Sharpe ratio divided by σ, determines the Nash equilibrium portfolio weight K derived in Section 3. Both managers invest in the same risky asset S(t), in contrast to Basak and Makarov [35] who assume asset specialization with each manager restricted to a different stock. Our single-asset specification isolates the effect of liability jump risk from portfolio diversification considerations; allowing asset specialization would conflate the two channels. There are two pension fund managers in the market, indexed by i∈{a, b}, and both have access to B(t) and S(t).

2.2 The pension management

The hybrid pension fund can be viewed as a saving vehicle for retirement. Members make regular contributions to the fund throughout their working lives, and the accumulated wealth is invested in the financial market to build retirement income.

Let X_*i*(t) be the gross asset value of fund i at time t. The manager allocates a proportion π_*i*(t)∈ℝ of gross assets to the risky asset S(t), so the dollar allocation in the risky asset is π_*i*(t)X_*i*(t) and the remaining proportion 1−π_*i*(t) is held in the risk-free asset B(t). Since π_*i*∈ℝ, values π_*i*>1 correspond to a leveraged position (borrowing at rate r to invest more than 100% of assets in the risky asset) and π_*i* <0 corresponds to a short position; short selling and leverage are thus both permitted. The proportional contribution rate c_*i*>0 is the rate at which members' contributions flow into the fund as a fraction of current fund value; this differs from Guan and Liang [37] who use an additive inflation-indexed contribution c_*i*I(t)dt. The gross asset dynamics, used in Lemma 1 to derive the funding-ratio SDE (after which X_*i* does not appear explicitly), are:

{dXi(t)=Xi(t)[r+πi(t)(μ−r)+ci]dt+πi(t)Xi(t)σdW(t), Xi(0)=xi>0.(3)

The fund also faces stochastic liability risk. Let Z_*i*(t) be a standard Brownian motion driving the liability diffusion, correlated with the asset Brownian motion W(t) via d〈W,_{Z_*i*〉t} = ρ_*i*dt with ρ_*i*∈(−1, 1). The liability L_*i*(t) satisfies

{dLi(t)=(r+ℓi)Li(t)dt+ηiLi(t)dZi(t)+Li(t-)∫01zÑi(dt,dz),Li(0)=li>0,(4)

where ℓ_*i*∈ℝ is the excess liability drift above the risk-free rate r (ℓ_*i*>0 in practice since pension obligations typically grow faster than the risk-free rate; ℓ_*i* <0 is theoretically permissible), η_*i*>0 is the liability diffusion volatility, Li(t-):=lims↑tLi(s) is the left limit before a jump, and Ñ_*i*(dt, dz): = N_*i*(dt, dz)−ν_*i*(dz)dt is the compensated Poisson random measure with Lévy measure ν_*i* on (0, 1) [42, 44]. The drift form r+ℓ_*i* follows Pan and Xiao [45]; since r cancels in the ratio dynamics Equation 7, only ℓ_*i* affects the equilibrium. Jump sizes z∈(0, 1) represent proportional liability increases; the compensated measure ensures Ñ_*i* is a martingale.

To make the correlation structure of Z_*i* explicit, we use the orthogonal decomposition

Zi(t)=ρiW(t)+1-ρi2W~i(t),(5)

where W~a(t), W~b(t), and W(t) are standard Brownian motions on (Ω,F,ℙ) with d〈W,W~i〉t=0 and

d〈W~a,W~b〉t=ρabdt, ρab∈(-1,1).

This gives the cross-covariation d〈Za,Zb〉t=(ρaρb+(1-ρa2)(1-ρb2)ρab)dt=:ρZdt, where ρ_*Z* captures the dependence between the two liability Brownian motions. Hence Z_*a* and Z_*b* are mutually independent if and only if ρ_*Z* = 0, which holds when ρab=-ρaρb/(1-ρa2)(1-ρb2); in particular, in the symmetric calibration ρ_*a* = ρ_*b* we set ρ_*ab* = 0 so that ρ_*Z* = ρ_*a*ρ_*b*. The full covariance matrix of (W, Z_*a*, Z_*b*) is

(1ρaρbρa1ρZρbρZ1),

which is positive semi-definite whenever ρa2+ρb2+ρZ2-2ρaρbρZ≤1. We impose this condition throughout; the benchmark parameters of Table 1 satisfy it (the left-hand side equals 0.172 <1).

Table 1. Benchmark parameters {r: risk-free rate; μ: expected return on risky asset; σ: volatility; T: horizon; η: liability diffusion; ρ_*i*: asset–liability corr.; ρ_*ab*: residual liability–liability corr.; ρZ=ρaρb+(1-ρa2)(1-ρb2)ρab: total liability–liability corr.; c: contributions; ℓ: excess liability drift; λ: Lévy measure weight [effective arrival rate λ(1-e-μi)]; μ_*i*: jump rate}.

Market parameters (r, μ, σ) follow standard calibrations in the pension finance literature [33, 37]; liability parameters (η_*i*, ρ_*i*, ℓ_*i*, c_*i*) are consistent with Guan [37]; jump parameters (λ_*i*, μ_*i*) are set to yield economically realistic actuarial shock frequencies and severities. All other parameters held fixed in each sensitivity experiment.

Assumption 1 (Finite-activity jump-diffusion). For i∈{a, b}, at each Poisson event the liability L_*i* increases by a random fraction J_*k*∈(0, 1), where J_*k* follows an exponential distribution truncated to (0, 1) with rate parameter μ_*i*; the integration variable z in Equation 4 below has the same distribution under ν_*i*(dz). The corresponding Lévy measure on (0, 1) is

νi(dz)=λiμie-μizdz, z∈(0,1), λi,μi>0,(6)

where λ_*i*>0 is the Lévy measure weight parameter (the effective Poisson arrival rate is λi(1-e-μi), the total mass of ν_*i*) and 1/μ_*i* approximates the mean proportional liability increase (the exact mean is E[Jk]=(1-(1+μi)e-μi)/(μi(1-e-μi)); at the benchmark μ_*i* = 2 the approximation gives 0.500 versus the exact value 0.344, an overestimate of 45%; 1/μ_*i* is used for economic interpretation only and does not enter any formula). After each jump the funding ratio becomes Yi(t-)/(1+Jk)>0, preserving positivity.

Following Chen et al. [46], the funding ratio is a more meaningful solvency indicator than the absolute surplus X_*i*−L_*i* because it enables fair comparison across funds of different sizes.

Definition 1 (Asset-to-liability ratio). The asset-to-liability ratio of fund i is

Yi(t):=Xi(t)Li(t), Yi(0)=yi:=xili>0.

A value Y_*i*(t)≥1 indicates a fully-funded (or over-funded) position; Y_*i*(t) <1 indicates underfunding; Y_*i*(t) = 1 means assets exactly cover liabilities. The numerical results in Section 4 show mean Y_*i* trajectories relative to the fully-funded threshold Y_*i* = 1.

Lemma 1 (Exact funding-ratio dynamics). Under Definition 1, the asset-to-liability ratio Y_*i* satisfies the exact, homogeneous dynamics

{dYi(t)=[πi(t)(μ-r)+ci-ℓi+ηi2-πi(t)σηiρi-Γi]Yi(t)dt +πi(t)σYi(t)dW(t) -ηiYi(t)dZi(t)-Yi(t-)∫01z1+zÑi(dt,dz),Yi(0)=yi=xi/li>0,(7)

where Γi:=∫01z1+zνi(dz)>0 (under Assumption 1, Γi=λiγ̄i where γ̄i:=∫01z1+zμie-μizdz).

Proof. See Appendix A.

The drift in Equation 7 groups naturally as: [π_*i*(μ−r)] (investment return) +[c_*i*−ℓ_*i*] (net cash flow) +[ηi2-πiσηiρi] (liability-correlation adjustment) −Γ_*i* (jump compensator). Every coefficient scales with Y_*i*, confirming the dynamics are homogeneous and exact; no proportional surplus approximation is made.

Remark 1. The jump term in Equation 4 captures sudden discrete increases in the fund's obligations whose sample paths are discontinuous: the diffusion term η_*i*L_*i*dZ_*i*(t), which produces continuous paths, cannot represent such shocks. Examples include regulatory benefit revisions, longevity shocks, bulk member transfers, and discount rate resets [4]. As shown in Equation 7, each such event multiplies Y_*i* by 1/(1+z): since L_*i* increases proportionally by z (i.e., Li→Li(t-)(1+z)) while X_*i* is unaffected, Y_*i* = X_*i*/L_*i* becomes Yi(t-)/(1+z). The proportional reduction in Y_*i* is therefore z/(1+z)∈(0, 1), which is smaller than the liability shock z itself since z/(1+z) <z for z>0. For example, if liabilities double (z = 1), the funding ratio halves (Y_*i*→Y_*i*/2), a reduction of z/(1+z) = 1/2 per unit Y_*i*.

Remark 2 (Sign of jump sizes). The model restricts jump sizes to z∈(0, 1), so all liability jumps are upward (Li(t)>Li(t-)). This reflects the dominant direction of actuarial risk in pension funds: regulatory changes, longevity shocks, and court judgments almost exclusively increase obligations. Downward liability revisions (z∈(−1, 0)) are theoretically permissible—for example a beneficial actuarial assumption revision or a members' buyout that extinguishes obligations—but are substantially rarer in practice. Extending the jump domain to z∈(−1, 0) would require the loading functions Φi± to be re-evaluated over the negative domain and the integrability condition γ_*i*∈(1, 2) to be re-examined; we leave this extension to future work.

We call π_*i*(·) an admissible strategy if:

• π_*i*(t) is progressively measurable w.r.t. (Ω,F,{Ft}t∈[0,T],ℙ).
• 𝔼[∫0T(πi(t)2+1)Yi(t)2dt]<+∞, ensuring square-integrability of both the asset diffusion π_*i*σY_*i*dW and the liability diffusion η_*i*Y_*i*dZ_*i*.
• 𝔼[∫0T∫01|Yi(t-)z1+z|2νi(dz)dt]<+∞, ensuring the compensated jump integral is square-integrable.
• The SDE Equation 7 has a unique strong solution for (t₀, y_*i*)∈[0, T] × (0, +∞). By Lemma 1 this holds for all constant strategies; for general admissible strategies it follows from the standard theory of SDEs with finite-activity jumps and locally Lipschitz, linearly growing coefficients [47].

As proved in Appendix A, Y_*i*(t)>0 a.s. for all t∈[0, T]. Denote the set of admissible strategies by Ai.

Throughout this paper all model coefficients (r, μ, σ, ρ_*i*, η_*i*, ℓ_*i*, c_*i*, λ_*i*, μ_*i*) are assumed deterministic and constant. This is essential for the closed-form Nash equilibrium: stochastic or time-varying coefficients would break the homothetic structure and require numerical methods. Note that the risk-free rate r cancels algebraically in the ratio dynamics of Equation 7, so it does not appear as a free parameter.

3 The competition

3.1 The competition between pension managers

In practice, pension fund managers compete for contributors. Members compare funds and allocate contributions to better-performing ones, so each manager has an incentive to outperform the other. Each manager therefore evaluates terminal funding ratio relative to that of the competitor. The utility function of pension manager i is the standard CRRA utility

Ui(x)=x1-γi1-γi, γi>0,γi≠1,(8)

which is strictly increasing and strictly concave. The optimisation goal of pension manager a is

maxπa∈Aa𝔼[Ua(Ya(T)Yb(T))],(9)

where Y_*a*(T)/Y_*b*(T) is the terminal funding-ratio of manager a relative to manager b; manager b's objective is obtained by switching subscripts. We adopt the pure relative-performance objective following Guan and Liang [37], adapted to the asset-to-liability ratio rather than real wealth. We assume γ_*i*∈(1, 2) and γ_*a*+γ_*b*>1 throughout.

Remark 3. The restriction γ_*i*∈(1, 2) serves three roles. First, γ_*i*>1 ensures the reaction-function coefficient (γ_*i*−1)/γ_*i*>0 (competitive amplification, Corollary 1). Second, γ_*i* <2 ensures the integrand of Φi+ in Equation 20 is strictly negative on (0, 1) (Lemma 3), which is the key sign condition driving the welfare results of Section 4; note that convergence of Φi+ holds for any finite γ_*i* since the integration domain (0, 1) is bounded and (1+z)γi-1 is bounded on [0, 1] for all finite γ_*i*, so the condition γ_*i* <2 is required for the sign of Φi+, not for integrability. Third, γ_*i*>1 ensures Φi+<0 via the monotonicity argument in Lemma 3. The condition γ_*a*+γ_*b*>1 guarantees a unique Nash equilibrium (Proposition 3).

Since both objectives are coupled through the funding-ratio objective, we seek Nash equilibrium strategies.

Definition 2 (Nash equilibrium). A pair of portfolio strategies (πa,πb)∈Aa×Ab constitutes a Nash equilibrium if πa*(t) is manager a's best response to πb(t) and πb(t) is manager b's best response to πa(t). That is, πa(t) and πb*(t) solve the following problems simultaneously:

maxπa∈Aa𝔼[Ua(Ya(T)Yb(T))], maxπb∈Ab𝔼[Ub(Yb(T)Ya(T))],

where Ya(T) and Yb(T) are the terminal funding ratios of manager a and manager b under strategies πa(t) and πb(t), respectively.

To characterize the Nash equilibrium we introduce the value function of manager a:

Va(t,ya,yb):=supπa∈Aa𝔼t,ya,yb[Ua(Ya(T)Yb(T))],(10)

where E_{t,_*y**a*, y_*b*}[·] denotes expectation under the law of (Y_*a*, Y_*b*) started at (y_*a*, y_*b*) at time t. Since all model coefficients are deterministic constants, the state process (Y_*a*, Y_*b*) is a time-homogeneous Markov process and this expectation is well-defined. The value function satisfies the terminal condition

Va(T,ya,yb)=(yayb)1-γa1-γa.(11)

Under E_{T,_*y**a*, y_*b*}, the process starts at (y_*a*, y_*b*) at time T, so Y_*a*(T) = y_*a* and Y_*b*(T) = y_*b* a.s. (Poisson measures have no atoms at fixed deterministic times, so no jump occurs at T a.s.), making U_*a*(Y_*a*(T)/Y_*b*(T)) = U_*a*(y_*a*/y_*b*) a constant independent of π_*a*. Since Aa is non-empty (the zero strategy π_*a*≡0 is admissible), the supremum of a constant equals that constant. Switching subscripts a and b yields the value function V_*b*(t, y_*b*, y_*a*) and its terminal condition for manager b.

3.2 The Nash equilibrium strategies

Throughout we restrict attention to classical C^{1, 2, 2} solutions within the homothetic class {f(t)(ya/yb)1-γa/(1-γa)}; existence within this class is established constructively in Appendix D. Viscosity solutions or solutions outside this class are not considered. For a control pair (πa,πb), the infinitesimal generator Lπa,πb acting on Va∈C1,2,2([0,T]×(0,∞)2) is defined by

Lπa,πb∗Va:=Vya[πa(μ−r)+ca−la+ηa2−πaσηaρa−Γa]ya +Vyb[πb∗(μ−r)+cb−lb+ηb2−πb∗σηbρb−Γb]yb +12(σ2πa2+ηa2−2ρaσηaπa)ya2Vyaya +12(σ2(πb∗)2+ηb2−2ρbσηbπb∗)yb2Vybyb +(σ2πaπb∗−ρaσηaπb∗−ρbσηbπa+ηaηbρZ)yaybVyayb +​∫01​[Va​(t,ya1+z,yb)−Va+yaz1+zVya]νa(dz) +​∫01​[Va​(t,ya,yb1+z)−Va+ybz1+zVyb]νb(dz).(12)

Switching subscripts a and b gives Lπb,πa*Vb. The dynamic programming principle [47, 48] then yields the Hamilton–Jacobi–Bellman (HJB) equation.

Lemma 2 (HJB Equation). Suppose Va∈C1,2,2([0,T)×(0,∞)2) and satisfies the dynamic programming principle for the controlled jump-diffusion Equation 7 with opponent strategy πb* fixed. Then V_*a*(t, y_*a*, y_*b*) satisfies the HJB equation

∂tVa+supπa∈AaLπa,πb*Va=0,(13)

subject to the terminal condition Equation 11. The HJB equation for manager b is obtained by switching subscripts a and b throughout.

Proof. See Appendix B.

From Equation 13, the optimal feedback strategy follows by applying the first-order condition to the supremum in Equation 13 with respect to π_*a*. Let θi:=(μ-r)/(γiσ2) denote the Merton ratio for manager i [the ratio K/γ_*i* where K = (μ−r)/σ² is the symmetric Nash equilibrium weight]. In the ratio model Equation 7 the liability-correlation term ρ_*i*ση_*i* cancels from the Merton benchmark because the drift term −π_*i*ση_*i*ρ_*i* in Equation 7 shifts the first-order condition. We define the effective intercept for manager a:

Θa:=μ-rγaσ2+(γa-1)(ρaηa-ρbηb)γaσ =θa+(γa-1)(ρaηa-ρbηb)γaσ.(14)

Switching subscripts a and b yields Θ_*b*.

Proposition 1 (Individual optimal strategy). Given that manager b plays a constant strategy πb(t)=π^b∈ℝ, the optimal feedback strategy of pension manager a is the constant

πa(t)=π^a, π^a=Θa+γa-1γaπ^b,(15)

where Θ_*a* is defined in Equation 14. The dollar allocation in the risky asset under the ratio model Equation 7 is π^aYa(t)La(t), where π^a is the fraction of the asset-to-liability ratio invested. Switching subscripts a and b yields the corresponding optimal strategy for manager b.

Proof. See Appendix C. Global optimality over all πa∈Aa follows from Theorem 1 (Appendix E).

Proposition 1 shows that each manager's optimal response has two terms: Θ_*a*, the effective intercept [reducing to θa=(μ-r)/(γaσ2) when ρ_*a*η_*a* = ρ_*b*η_*b*], and the strategic complement (γa-1)/γa·π^b*, a competitive amplification term.

Proposition 2 (Nash equilibrium pair). The Nash equilibrium pair (π^a,π^b) for the two pension fund managers, whose existence and uniqueness are established in Proposition 3, is:

π^a*=[γaΘa+(γa-1)Θb]γbγa+γb-1,(16)

π^b*=[γbΘb+(γb-1)Θa]γaγa+γb-1,(17)

with equilibrium portfolio fractions πi(t)=π^i (constant), so the dollar allocation in the risky asset under the ratio model is π^i*Yi(t)Li(t), and the corresponding value functions

Va(t,ya,yb)=eκa(T-t)1-γa(yayb)1-γa,Vb(t,yb,ya)=eκb(T-t)1-γb(ybya)1-γb,(18)

where κ_*a* and κ_*b* are given by Equation 19 evaluated at the Nash equilibrium controls.

Proof. See Appendix D (Proof of Proposition 2).

The strategies Equations 16, 17 depend only on the diffusion parameters (μ, σ, ρ_*i*, η_*i*, γ_*i*), not on the jump parameters (λ_*i*, μ_*i*) or the risk-free rate r (which cancels algebraically in the ratio dynamics Equation 7).

Proposition 3 (Existence and uniqueness). Under Assumption 1 and the conditions γ_*i*∈(1, 2), γ_*a*+γ_*b*>1, the Nash equilibrium (πa,πb)∈Aa×Ab in the sense of Definition 2 exists and is unique among constant portfolio fractions within the homothetic value function class Appendix Equation 24. The equilibrium strategies are the constant portfolio fractions Equations 16, 17, and the associated value functions are Equation 18.

Proof. See Appendix F.

Corollary 1 (No-competition benchmark). The single-agent problem maxπa∈AaE[Ua(Ya(T))] (no relative performance concern) has the optimal constant strategy

π^isolo=μ-r+(γi-1)ρiσηiγiσ2=μ-rγiσ2+(γi-1)ρiηiγiσ,

which exceeds θi=(μ-r)/(γiσ2) by (γ_*i*−1)ρ_*i*η_*i*/(γ_*i*σ). In the symmetric case ρ_*a*η_*a* = ρ_*b*η_*b*, the Nash equilibrium satisfies π^i=K>π^isolo for all γ_i*∈(1, 2), establishing that competition strictly amplifies risk-taking. The amplification gap is

K-π^isolo=μ-rσ2-μ-r+(γi-1)ρiσηiγiσ2 =(γi-1)(μ-r-ρiσηi)γiσ2>0,

since γ_*i*>1 and μ−r>ρ_*i*ση_*i*. In the asymmetric case ρ_*a*η_*a*≠ρ_*b*η_*b*, the cross-liability correction in Θ_*i* may reduce or reverse this amplification for the disadvantaged manager.

Proof. See Appendix F.

The following proposition derives the welfare exponent κ_*i* and the associated jump loading functions.

Proposition 4 (Value function and welfare exponent). The welfare exponent κ_*a* in Equation 18 is given by

κa:=(1-γa)[π^a(μ-r)+ca-ℓa+ηa2-π^aσηaρa-Γa] -(1-γa)[π^b(μ-r)+cb-ℓb+ηb2-π^bσηbρb-Γb] -12γa(1-γa)Aa+(1-γa)(2-γa)2Ab -(1-γa)2[σ2π^aπ^b-ρaσηaπ^b-ρbσηbπ^a+ηaηbρZ] +(1-γa)[λaΦa+-λbΦa-],(19)

where Ak:=σ2(π^k)2+ηk2-2ρkσηkπ^k. Switching subscripts a and b yields κ_*b*. Here Φa+ and Φa- are two jump loading functions depending on manager a's risk aversion γ_*a* and on the respective jump size parameters. Φa+ is the own jump loading function defined by

Φa+:=∫01[(1+z)γa-1-11-γa+z1+z]μae-μazdz,(20)

and Φa- is the opponent jump loading function defined by

Φa-:=∫01[1-(1+z)1-γa1-γa+z1+z]μbe-μbzdz.(21)

Switching subscripts a and b yields Φb-, which uses μ_*a* and γ_*b*. Both Φa+ and Φa- depend only on γ_*a* and on μ_*a* or μ_*b* respectively, not on the state (y_*a*, y_*b*).

Proof. See Appendix D.

Remark 4 (Derivation of the loading functions Equations 20, 21). These integrands arise from evaluating the jump integrals in the generator Appendix Equation 24 under the value-function conjecture Equation 24.

Own-jump integral (Equation 20). When fund a's liability jumps by z∈(0, 1), the funding ratio falls: y_*a*→y_*a*/(1+z) while y_*b* is unaffected, so ξ → ξ/(1+z). Evaluating V_*a* after the jump:

Va(t,ya1+z,yb)=fa1-γa(ξ1+z)1-γa=faξ1-γa1-γa(1+z)γa-1.

The compensation term in the generator is yaz1+zVya=z1+zfaξ1-γa (using Vya=faξ1-γa/ya). The integrand of the own-jump term in Equation 12 is therefore:

Va(t,ya1+z,yb)-Va+yaz1+zVya=faξ1-γa[(1+z)γa-1-11-γa+z1+z],

Integrating against νa(dz)=λaμae-μazdz and factoring out faξ1-γa gives the own-jump contribution λafaξ1-γaΦa+, with Φa+ exactly as in Equation 20.

Opponent-jump integral (Equation 21). When fund b's liability jumps by z, the ratio rises: y_*b*→y_*b*/(1+z) while y_*a* is unaffected, so ξ → ξ(1+z). Evaluating V_*a* after the jump:

Va(t,ya,yb1+z)=faξ1-γa1-γa(1+z)1-γa.

The compensation term is ybz1+zVyb=-z1+zfaξ1-γa (using Vyb=-faξ1-γa/yb). The integrand of the opponent-jump term is:

Va(t,ya,yb1+z)-Va+ybz1+zVyb=faξ1-γa[(1+z)1-γa-11-γa-z1+z].

Since (1+z)1-γa-1=-(1-(1+z)1-γa), this equals -faξ1-γa times the integrand of Φa- in Equation 21. Integrating against ν_*b*(dz) gives the opponent-jump contribution -λbfaξ1-γaΦa-.

Both integrands differ in the sign of their first term because an own jump reduces ξ (bad for a) while an opponent jump raises ξ (good for a), reversing the welfare direction. Since Φa-<0, the term -(1-γa)λbΦa-<0 reduces κ_*a* and hence raises V_*a*: the factor (1+z)1-γa<1 (since 1−γ_*a* <0) makes V_*a* less negative, a welfare improvement, the opposite of the own-jump factor (1+z)γa-1>1.

Remark 5 (Single-agent limit). Setting π^b=0 in the reaction function Equation 15 gives π^a=Θa, which equals θa=(μ-r)/(γaσ2) in the symmetric case ρ_*a*η_*a* = ρ_*b*η_*b*. This differs from the true single-agent optimum π^asolo=(μ-r+(γa-1)ρaσηa)/(γaσ2) because the reaction function is derived from the ratio-utility Nash problem, not from the single-agent problem with utility U_*a*(Y_*a*(T)). Note also that setting π^b=0 is distinct from taking γ_b → ∞ in the Nash formula Equation 16; the two limits are not equivalent in general. Economically, π^b=0 represents deliberate passivity: manager b exits the risky market and manager a optimizes against a non-investing competitor. By contrast, γ_*b* → ∞ represents an infinitely risk-averse competitor who remains in the game. In the symmetric case ρ_*a*η_*a* = ρ_*b*η_*b*, taking γ_*b* → ∞ gives Θb→(μ-r)/σ2=K>0, so π^b*→K>0: an infinitely risk-averse manager still holds equities because competitive pressure prevents full withdrawal.

Remark 6 (Competitive amplification and cross-liability asymmetry). In the symmetric case ρ_*a*η_*a* = ρ_*b*η_*b*, the Nash weight K = (μ−r)/σ² exceeds the true single-agent optimum π^isolo=(μ-r+(γi-1)ρiσηi)/(γiσ2) by exactly (γi-1)(μ-r-ρiσηi)/(γiσ2), which grows with γ_*i*, so more risk-averse managers experience the largest competitive amplification. In the asymmetric case ρ_*a*η_*a*≠ρ_*b*η_*b*, the cross-liability correction in Θ_*i* (see Equation 14) may reduce or reverse the amplification for the disadvantaged manager. Explicitly, substituting Equation 14 into Equation 16 gives

π^a*=K+(γa-1)(ρaηa-ρbηb)γaσ,(22)

where K = (μ−r)/σ². The correction is positive when ρ_*a*η_*a*>ρ_*b*η_*b* and grows with γ_*a*−1, so more risk-averse managers are more sensitive to liability-correlation asymmetry. Setting ρ_*a*η_*a* = ρ_*b*η_*b* in Equation 22 recovers π^a=K. Note that compared to the standard no-liability Nash model, the liability-correlation terms ρ_iη_i* cancel from K in the ratio formulation, appearing only in the asymmetric correction and in the solo benchmark.

Theorem 1 (Verification). Let the dynamics Equation 7 hold. Suppose Vi∈C1,2,2([0,T)×(0,∞)2) satisfies the HJB equation Equation 13 under the opponent strategy π-i* and the terminal condition Equation 11, and πi∈Ai attains the supremum in Equation 13 given the opponent plays π-i. Then V_*i* is manager i's value function and πi* is manager i's optimal strategy given π-i. When both hold simultaneously, (πa,πb*) is a Nash equilibrium in the sense of Definition 2.

Proof. See Appendix E.

4 Sensitivity analysis

In this section we present numerical results illustrating the analytical findings of Section 3. Since the Nash equilibrium portfolio fractions π^i* are deterministic constants (Proposition 5), mean funding-ratio trajectories are computed from the exact closed-form expression E[Y_*i*(t)] = y_*i*exp(δ_*i*t), where δ_*i* is the deterministic drift in Equation 7 evaluated at the equilibrium controls. All welfare sensitivity operates through the welfare exponent κ_*i* and the jump loading functions Φi+ and Φi- in Equations 20, 21. The net cash-flow differential c_*a*−c_*b* and liability-drift differential ℓ_*a*−ℓ_*b* both enter κ_*a* through the drift terms in Equation 19: a higher own contribution rate c_*a* or lower own liability drift ℓ_*a* raises κ_*a* and thereby reduces the value function (since 1/(1−γ_*a*) <0). Benchmark parameters are in Table 1; we work throughout under γ_*i*∈(1, 2). We set ρ_*Z* = ρ_*a*ρ_*b* throughout (the case W~a⊥W~b), consistent with the symmetric calibration ρ_*a* = ρ_*b*.

Under these parameters, the Merton ratio is θa=θb=(μ-r)/(γσ2)=1.3333 and the Nash equilibrium weight is π^i=(μ-r)/σ2=K=2.0000 (Proposition 5 below), independent of γ_i, ρ_i, and η_i*. Since K = 2.0000>1, this is a leveraged position: each manager borrows an amount equal to the fund value at rate r and invests twice the fund assets in the risky asset, so the remaining allocation to the risk-free asset is 1−K = −1.0000. This is consistent with π_*i*∈ℝ as defined in Section 2. Note that the true single-agent optimum π^isolo=(μ-r+(γi-1)ρiσηi)/(γiσ2)=1.3833 at γ_*i* = 1.5 differs from the Merton ratio θ_*i* = 1.3333 by the liability-correlation correction (γ_*i*−1)ρ_*i*η_*i*/(γ_*i*σ) = 0.05; the competitive amplification gap K-π^isolo=0.6167 is measured against the solo optimum, not the Merton ratio. The asymmetric figure parameters γ_*a* = 1.5, γ_*b* = 1.8 with ρ_*a*η_*a* = ρ_*b*η_*b* also give π^a=π^b=2.0000, confirming risk aversion invariance.

4.1 Nash equilibrium strategies

4.1.1 Risk aversion invariance

We first establish a structural property of the Nash equilibrium weights with respect to the risk aversion parameters γ_*i*.

Proposition 5 (Risk aversion invariance of Nash weights). In the symmetric case ρ_*a*η_*a* = ρ_*b*η_*b*, the Nash equilibrium portfolio fractions Equations 16, 17 are independent of both risk aversion parameters γ_*a* and γ_*b*:

π^i*=K:=μ-rσ2, i∈{a,b}.

In the asymmetric case ρ_*a*η_*a*≠ρ_*b*η_*b*, the Nash weights depend on (γ_*a*, γ_*b*) through the cross-liability correction in Θ_*i* Equation 14, and risk-aversion invariance does not hold in general.

Proof. See Appendix F.

Remark 7 (Risk-neutrality-like equilibrium in the symmetric case). When ρ_*a*η_*a* = ρ_*b*η_*b*, Proposition 5 shows that the Nash equilibrium portfolio weight K = (μ−r)/σ² is determined solely by the excess return per unit of variance and does not depend on γ_*i*, ρ_*i*, or η_*i*. This symmetry condition means both managers face identical liability-correlation parameters; when it fails the cross-liability correction (γ_*i*−1)(ρ_*a*η_*a*−ρ_*b*η_*b*)/(γ_*i*σ) in Θ_*i* reintroduces γ_*i*-dependence.

Within the symmetric case the γ-cancellation mechanism is as follows. Each manager's reaction function Equation 15 has slope (γ_*i*−1)/γ_*i*∈(0, 1), reflecting strategic complementarity: a more risk-tolerant competitor induces more risk-taking. Simultaneously, the intercept Θi=θi=(μ-r)/(γiσ2) falls with γ_*i*, reflecting that a more risk-averse manager would invest less in isolation. In the Nash equilibrium these two effects cancel exactly: the competitive amplification ((γ_*i*−1)/γ_*i*)·K exactly offsets the reduction in θ_*i* relative to K, leaving π^i=K for all γ_i*∈(1, 2).

The weight K = (μ−r)/σ² coincides with the myopic Merton portfolio of a log-utility (γ = 1) investor and with the risk-neutral optimum. It is not, however, a consequence of risk neutrality: the managers remain risk-averse (γ_*i*>1) and their value functions retain full γ_*i*-dependence through the pre-factor 1/(1−γ_*i*), the curvature exponent (1−γ_*i*), and the welfare exponent κ_*i*. The invariance is a structural feature of the pure relative-performance specification and depends on (i) pure relative utility, (ii) CRRA preferences, (iii) a single shared risky asset, and (iv) constant coefficients. Relaxing any one of these conditions, for example by using partial relative utility α_*i*∈(0, 1), asset specialization, or asymmetric liability risk (ρ_*a*η_*a*≠ρ_*b*η_*b*), generally reintroduces γ_*i*-dependence into the equilibrium weights.

Table 2 illustrates this result numerically. For all values of γ_*a*∈(1, 2), holding γ_*b* = 1.5 fixed, the Nash weights remain exactly π^a=π^b=2.0000, while the Merton ratio θ_*a* = K/γ_*a* varies with γ_*a* and is strictly below the Nash weight, confirming the amplification established in Corollary 1.

Table 2. Nash equilibrium weights and competitive amplification gap as γ_*a* varies, with γ_*b* = 1.5 fixed and symmetric benchmark.

The Merton ratio θ_*a* = K/γ_*a* decreases with γ_*a*, the Nash weight is invariant at π^a=2.0000 (Proposition 5), and the gap π^a-π^asolo=(γa-1)(μ-r-ρση)/(γaσ2) strictly increases with γ_*a*: more risk-averse managers experience the largest competitive amplification.

Table 2 reveals two properties. First, the Merton ratio θ_*a* = K/γ_*a* is strictly decreasing in γ_*a*: higher risk aversion reduces the solo benchmark. Second, the Nash weight K = 2.0000 is strictly above the solo optimum π^asolo in every row, with the gap (γa-1)(μ-r-ρση)/(γaσ2) widening as γ_*a* increases. Competition therefore amplifies risk-taking most strongly for the most risk-averse manager.

Having established that portfolio weights are invariant to the risk-aversion and jump parameters (γ_*i*, λ_*i*, μ_*i*) in the symmetric case (while still depending on the market parameters μ, r, σ, ρ_*i*, η_*i*), we characterize the welfare exponent κ_*i*. While π^i* depends neither on risk aversion nor jump parameters (in the symmetric case), the value function V_*i* depends on both through κ_*i*. The following results characterize how κ_*i* responds to the jump parameters (λ_*i*, μ_*i*), using the jump loading functions Φi+ and Φi-.

4.1.2 Sign of the jump loading function

Lemma 3. For γ_*i*∈(1, 2): Φi+<0, Φi-<0, and both integrals converge.

Proof. See Appendix F.

4.1.3 Theoretical welfare sensitivity

The key partial derivatives follow directly from Equation 19. Noting that Γi:=λiγ̄i where γ̄i:=∫01z1+zμie-μizdz is independent of λ_*i*, differentiating Equation 19 with respect to λ_*a* and λ_*b* gives:

∂κa∂λa=(1-γa)(Φa+-γ̄a), ∂κa∂λb=(1-γa)(γ̄b-Φa-).

Switching subscripts a and b yields the corresponding derivatives for manager b. Because Va=eκa(T-t)/(1-γa)·ξ1-γa and (1−γ_*a*) <0, expected utility increases when κ_*a* decreases and decreases when κ_*a* increases. This sign relationship is central to interpreting the propositions below.

Proposition 6 (Own jump intensity). For γ_*i*∈(1, 2): ∂V_*i*/∂λ_*i* <0. More frequent own liability jumps decrease welfare.

Proof. See Appendix F.

Proposition 7 (Mean jump size). For γ_*i*∈(1, 2) and μ_*i*≥2: ∂V_*i*/∂(1/μ_*i*) <0. A larger mean jump size decreases welfare.

Proof. See Appendix F.

Proposition 8 (Cross-player spillover). For γ_*a*, γ_*b*∈(1, 2): ∂V_*a*/∂λ_*b*>0. More frequent opponent jumps increase manager a's welfare. Switching subscripts a and b yields the corresponding result for manager b.

Proof. See Appendix F.

Remark 8. The mechanism is the relative performance objective: manager a's welfare is determined by Y_*a*(T)/Y_*b*(T). When fund b's liability jumps increase (λ_*b*↑), Y_*b* is reduced more frequently on its path to T, improving manager a's relative standing even though Y_*a*(T) is unaffected. This creates a strategic incentive: each manager benefits from the competitor's exposure to more frequent liability jumps.

Table 3 summarizes all sensitivity results.

Table 3. Sign of welfare and portfolio sensitivity to jump parameters (γ_*a*, γ_*b*∈(1, 2); for the mean jump size row, μ_*a*, μ_*b*≥2).

Own jumps reduce manager a's welfare and help manager b; opponent jumps do the reverse. Portfolio weights are invariant to the jump parameters (λ_*i*, μ_*i*) and, in the symmetric case, to risk aversion γ_*i*; they continue to depend on the market parameters (μ, r, σ, ρ_*i*, η_*i*). Switching subscripts a and b yields the corresponding results for manager b.

4.1.4 Numerical results

Table 4 reports V_*a*(0, 1, 1) as λ_*a* increases, for three risk aversion levels. In each column expected utility decreases as λ_*a* rises, consistent with Proposition 6. The decline is steeper for γ_*a* closer to 2, reflecting the growing sensitivity of Φa+ to risk aversion.

Table 4. V_*a*(0, 1, 1) as a function of λ_*a* for γ_*a*∈{1.2, 1.5, 1.8}, with T = 20.

Benchmark with γ_*b* = 1.5, λ_*b* = 1, μ_*a* = μ_*b* = 2, π^a=π^b=2.0000. Welfare decreases with λ_*a* in every column, consistent with Proposition 6. The decline is steeper for larger γ_*a*, reflecting the dependence of Φa+ on risk aversion.

4.1.5 Welfare vs. mean jump size

Table 5 fixes γ_*a* = γ_*b* = 1.5, λ_*a* = λ_*b* = 1 and decreases μ_*a* (increasing the mean jump size 1/μ_*a*). For μ_*a*≥2 (the domain of Proposition 7), V_*a* decreases as 1/μ_*a* grows, consistent with that proposition. The rows μ_*a* = 1.5 and μ_*a* = 1.0 lie outside the proposition's domain (μ_*i*≥2) and illustrate the non-monotone behavior noted in the proof. V_*b* increases for moderate values of 1/μ_*a* consistent with Proposition 8, with the effect reversing at the extreme μ_*a* = 1.0. At symmetry μ_*a* = μ_*b* = 2, welfare levels are equal.

Table 5. V_*a* and V_*b* as mean jump size 1/μ_*a* increases (μ_*a* decreases), with T = 20.

Benchmark with γ_*a* = γ_*b* = 1.5, λ_*a* = λ_*b* = 1, μ_*b* = 2, π^a=π^b=2.0000. At symmetry μ_*a* = μ_*b* = 2, welfare levels are equal. For μ_*a*≥2, larger own jumps decrease V_*a* consistent with Proposition 7; the rows μ_*a* = 1.5 and μ_*a* = 1.0 lie outside that proposition's domain. V_*b* increases for μ_*a*≥2 via an analogous cross-player mechanism (Proposition 8 states the result in terms of λ_*b*; the mean jump size effect here is the analogous welfare spillover with μ_*a* playing the role of the opponent's jump parameter); non-monotone behavior at μ_*a* <2 lies outside that proposition's domain and is noted in the text.

4.1.6 Cross-player spillover

Table 6 holds λ_*a* = 1 fixed and varies λ_*b*. As λ_*b* increases, V_*a* increases while V_*b* decreases, consistent with Proposition 8. The column ΔV_*a* = V_*a*(λ_*b*)−V_*a*(1) quantifies the welfare gain to manager a from the opponent's jumps.

Table 6. Cross-player effect: V_*a* increases and V_*b* decreases as λ_*b* rises, with λ_*a* = 1 fixed, T = 20.

ΔV_*a* = V_*a*(λ_*b*)−V_*a*(1). Benchmark with γ_*a* = γ_*b* = 1.5, μ_*a* = μ_*b* = 2, π^a=π^b=2.0000. Fewer opponent jumps hurt manager a (ΔV_*a* <0 at λ_*b* = 0.5); more opponent jumps help manager a (ΔV_*a*>0), consistent with Proposition 8.

4.1.7 Nash equilibrium strategies over time

The effect of competition is shown by comparing the Nash equilibrium (pure relative performance, Corollary 1) with the no-competition benchmark in which each manager optimizes absolute funding-ratio utility independently. Figure 1 shows manager a investing only the solo fraction π^asolo=1.3833 (close to the Merton ratio θ_*a* = 1.3333). Figure 2 overlays the Nash equilibrium, revealing the competitive amplification gap π^a*-π^asolo=0.6167.

Figures 3, 4 show the time evolution of the equilibrium dollar allocations for managers a and b over [0, 20] years at the benchmark parameters with γ_*a* = 1.5, γ_*b* = 1.8, and λ_*a* = λ_*b* = 1. Since π^i*=2.0000>1, both managers hold short positions in the risk-free asset to finance leveraged risky investment, analogous to the negative cash positions documented in Guan et al. [38]. Note that our model does not impose leverage or short-sale constraints; in practice, pension regulation often limits leverage, and incorporating such constraints would require numerical methods beyond the scope of this paper. The portfolio fraction π^i* is constant, but the dollar allocation π^iYi(t)Li(t) varies over time as the funding ratio evolves: the decaying dollar amounts visible in the figures reflect funding-ratio dynamics, not dynamic rebalancing of the portfolio fraction. Both allocations decay monotonically as the expected funding ratio declines under the net liability drift ℓ_i*>0.

Figures 5, 6 plot the ratio of mean funding ratios E[Y_*a*]/E[Y_*b*] for two jump intensity configurations. We use E[Y_*a*]/E[Y_*b*] rather than the economically more fundamental quantity E[Y_*a*/Y_*b*] because the latter requires Monte Carlo simulation (since Y_*a* and Y_*b* are correlated jump-diffusions with no closed-form joint distribution), whereas E[Y_*a*]/E[Y_*b*] = exp((δ_*a*−δ_*b*)t) follows from the closed-form drift formula and permits exact comparison. The two quantities share the same ordinal ranking: E[Y_*a*/Y_*b*]>1⇔E[Y_*a*]/E[Y_*b*]>1 by Jensen's inequality when Y_*a* and Y_*b* are positively correlated, so the figures correctly identify the outperforming manager. When λ_*a* = 0.5 <λ_*b* = 1.0 (Figure 5), E[Y_*a*] stays persistently above E[Y_*b*] and the ratio rises above unity. When λ_*a* = 2.0>λ_*b* = 1.0 (Figure 6), manager a's funding ratio decays faster and the ratio falls below unity. Both results are consistent with Proposition 6: more frequent own liability jumps accelerate funding-ratio decay, so the manager with the higher λ_*i* falls behind.

5 Conclusion

This paper studies competition between two hybrid DC pension fund managers. The state variable is the asset-to-liability ratio Y_*i* = X_*i*/L_*i*, whose dynamics are derived exactly via the Itô–Lévy formula; no proportional surplus approximation is made and the ratio SDE is both exact and homogeneous. Each manager incorporates relative performance (the ratio of own terminal funding ratio to the competitor's) into a CRRA objective function. Liability risk enters through both a diffusion component and a finite-activity jump-diffusion with truncated exponential jump amplitudes. Dynamic programming yields explicit Nash equilibrium strategies in closed form, with optimality established by Theorem 1. In the symmetric liability-correlation case (ρ_*a*η_*a* = ρ_*b*η_*b*), competition strictly amplifies risk-taking: the Nash equilibrium weight K = (μ−r)/σ² exceeds the single-agent Merton benchmark for every level of risk aversion greater than one, with the amplification gap growing as risk aversion increases, and the equilibrium weight is completely invariant to both risk aversion and jump parameters. In the asymmetric case, equilibrium weights depend on risk aversion through a cross-liability correction term in Θ_*i*, and risk-aversion invariance no longer holds. Own liability jumps reduce a manager's welfare, while the opponent's jumps generate a positive welfare spillover, reflecting the relative-performance structure of the objective.

These results carry implications for pension fund governance and regulation. The competitive amplification finding suggests that relative-performance evaluation, whether explicit or implicit through peer comparison, induces DC pension managers to take on more risk than is individually optimal. Regulators who observe excessive risk-taking may therefore need to consider whether peer-comparison incentives, rather than individual risk appetite, are the primary driver. The welfare spillover result implies that funds with lower liability jump exposure benefit from competitors' adverse shocks, creating systemic incentives that are invisible in single-manager frameworks.

Several extensions are deferred to future work. First, incorporating inflation-indexed assets, as in the single-manager framework of Guan and Liang [38], would add realism for long-horizon pension funds. Second, the restriction on the risk aversion parameter could be relaxed by adopting a truncated jump distribution that avoids the integrability constraint at the boundary. Third, allowing managers to invest in different risky assets would introduce additional asymmetry in the reaction functions and is a natural next step.

Generative AI statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fams.2026.1931310/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Said Hamadene, Le Mans Université, France

Reviewed by: Farai Julius Mhlanga, University of Limpopo, South Africa

Qicai Li, Nanjing Normal University, China

Liu Shengda, Chinese Academy of Sciences (CAS), China