Premium Valuation of the Pension Benefit Guaranty Corporation with Regime Switching

Peng Li,Zhixin Yang,Lin Xie,Wei Wang,Xiaofeng Zong

doi:10.1155/2021/9966515

Abstract

The Pension Benefit Guaranty Corporation (PBGC) provides insurance coverage for single-employer and multiemployer pension plans in private sector. It has played an important role in protecting the retirement security for over 1.5 million people since it was established about half a decade ago. PBGC collects insurance premiums from employers that sponsor insured pension plans for its coverage and receives funds from pension plans that it takes over. To address the issue of underfunded plans that the PBGC has, this work studies how to evaluate risk-based premiums for the PBGC. Inspired by a couple of existing work in which the premature termination of pension fund and distress termination of sponsor assets are analyzed separately, our work examines the two types of terminations under one framework and considers the occurrence of each termination dynamically. Given that market regime might have a big impact on the dynamics of both pension fund and sponsor’s assets, we thus formulate our model using a continuous-time two-state Markov chain in which bull market and bear market are delineated. We thus formulate our model using a continuous-time two-state Markov Chain in which bull market and bear market are delineated. In other words, the pension fund and sponsor assets are market dependent in our work. Given that this additional uncertainty described by regime switching makes the market incomplete, we therefore utilize the Esscher transform to determine an equivalent martingale measure and apply the risk neutral pricing method to obtain the closed-form expressions for premium of PBGC. In addition, we carry out numerical analysis to demonstrate our results and observe that premium increases according to the retirement benefit irrespective of the type of terminations. In comparison to the case of early distress termination of sponsor assets, the premium goes up more quickly when premature termination of pension funds occurs first due to the fact that pension fund is the first venue of retirement security. Furthermore, we look at how the premium changes with respect to other key parameters as well and make some detailed observations in the section of numerical analysis.

Highlights

Different from defined contribution pension plans, where employees themselves bear the investment risk and where employees are not sure about the amount of benefit they would receive after retirement, sponsors of defined benefit plans offer their employees a definite amount of benefit by the time of retirement, regardless of the performance of the underlying investment pool
For any plans covered by Pension Benefit Guaranty Corporation (PBGC), sponsor of the plan pays premium to the PBGC
In [6], it is assumed that PBGC is the first line of defense for the deficit of pension fund

Summary

Introduction

Different from defined contribution pension plans, where employees themselves bear the investment risk and where employees are not sure about the amount of benefit they would receive after retirement, sponsors of defined benefit plans offer their employees a definite amount of benefit by the time of retirement, regardless of the performance of the underlying investment pool. E major problem of their work is that the maturity of the PBGC’s insurance is assumed to be known and the fact that the pension fund will be terminated prematurely due to underfunding has not been taken into consideration. Because the focus of this work is on discussing how the PBGC charges the premiums under different scenario cases, we define the benefit that a typical beneficiary expects to receive similar to that in [10] and express it as below:. E Markov chain takes different values when the market is in different regimes With this practical assumption, the risk-free interest rate and expected rate of return are functions of the market regime α(t). To move forward, we will first find the risk neutral probability measure for the dynamic system with the help of Esscher transform

Esscher Transform under Regime Switching

Scenario Case Analysis

Further Remarks

Proof of Theorem 1

Proof of Corollary 1