Optimal management of DB pension fund under both underfunded and overfunded cases

Abstract

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, a zero-coupon bond and a stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyzes to illustrate the manager's economic behaviors.

Keywords:

• Risk management
• DB pension plan
• solvency risk
• expected utility
• efficient frontier

Submission Classifications:

• IB13
• IE13
• IB43

JEL Classifications:

• G22
• G11
• C61

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 91G05
• 91G10
• 91B05

Acknowledgments

The authors thank the members of the group of Mathematical Finance and Actuarial Science at the Department of Mathematical Sciences, Tsinghua University for their feedback and useful conversations.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is important to note that $P(t)$ represents the benefit rate at time t. As $M^{\prime }(\cdot )$ is the probability density function of the ages of workers and the integral of $M^{\prime }(\cdot )$ from m to d is 1, NC can be regarded as the weighted average of expected discounted values of future benefit rates, and is a ratio. Since $M(\cdot )$ represents the cumulative distribution function of the ages of workers, applying integration by parts reveals that the integrand of AL is equivalent to integrating $P(\cdot )$ and subsequently multiplying it by the density function $M^{\prime }(\cdot )$. This process performs a weighted average, representing the expected discounted value of all future benefit payments owed to the pension plan participants, which shows that AL is a quantity value.

2 In our model, $P(t)$ and $C(t)$ represent the benefit rate and contribution rate, respectively. In a small time horizon $[t,t+\mathrm{\Delta t}]$, the accumulated benefits and contributions are $P(t)\mathrm{\Delta t}$ and $C(t)\mathrm{\Delta t}$, respectively. The accumulated benefits $P(t)\mathrm{\Delta t}$ are not available at time t and affect the fund wealth at time $t+\mathrm{\Delta t}$. At the beginning of the time horizon $[t,t+\mathrm{\Delta t}]$, the available fund wealth is $F(t)$ and the manager invests it in the financial market. So we have $F(t)=u_0(t)+u_B(t)+u_S(t)$. Then from time t to $t+\mathrm{\Delta t}$, $F(t+\mathrm{\Delta t})-F(t)=u_0(t)\frac{S_0(t+\mathrm{\Delta t})-S_0(t)}{S_0(t)}+u_B(t)\frac{B_K(t+\mathrm{\Delta t})-B_K(t)}{B_K(t)}+u_S(t)\frac{S(t+\mathrm{\Delta t})-S(t)}{S(t)}+(C(t)-P(t))\mathrm{\Delta t}.$ Letting $\mathrm{\Delta }t\to 0$, Equation (11(11) $\mathrm{d}F(t)=u_0(t)\frac{\mathrm{d}{S}_0(t)}{S_0(t)}+u_B(t)\frac{\mathrm{d}{B}_K(t)}{B_K(t)}+u_S(t)\frac{\mathrm{d}S(t)}{S(t)}+(C(t)-P(t))\mathrm{d}t.$(11) ) holds.

3 Based on Equation (19(19) $\rho (t)=\exp (-{\int }_0^{t}r(s)\mathrm{d}s+\mathrm{kt}-\frac{{\lambda }_r^2+{\lambda }_S^2}{2}t-{\lambda }_r{W}_r(t)-{\lambda }_S{W}_S(t)),0⩽t⩽T,$(19) ), we observe that $\mathrm{\beta \rho }(T)⩾0,a.s.$ for $\beta ⩾0$. As a result, $Z^{\ast }(\beta )$ is well-defined.

Additional information

Funding

The authors acknowledge support from the National Natural Science Foundation of China (Grant Nos. 12371477, 11901574, 12271290, 11871036), the MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD910003), and the National Social Science Fund of China (20AZD075).