INTRODUCTION

Andreas Reichlin

Asset Liability Management for Defined-Benefit Pension Funds

Problem description

A defined-benefit pension fund offers predefined retirement benefits to its members. The pension promise is typically related to the final salary achieved just prior to retirement. The expected present value of the benefits of the pensioners and the present employees based on the salary history and salary projections up to retirement represents the pension fund's liabilities. The pension fund's assets correspondingly consist of its accumulated financial wealth (past contributions and investment earnings) and the expected present value of future contributions. The goal of pension fund management is to optimally allocate pension fund assets relative to long-term liabilities such that pension fund assets consistently outperform them and create the lowest pension funding cost possible for a given level of risk. Important decisions concern the level of contributions, the investment strategy and the benefit policy. Asset liability management (ALM) provides the mechanics for these decisions. Typically, it makes use of mathematical models that integrate both sides of the balance sheet of a pension fund and demonstrate the consequences of decisions on assets and liabilities.

Key elements of ALM are the choice of objectives, the analysis of the available ALM instruments, and the examination of the regulatory requirements. To derive a well-defined objective function, the pension fund has to identify its stakeholders and to analyze their objectives, i.e. their risk-return trade-off. Generally, the stakeholders are the sponsoring company, the employees, and the pensioners. Their objectives might be conflicting. The ALM instruments are the contribution policy, the investment policy and the benefit policy. The level of freedom in varying these instruments differs significantly. The freedom in choosing an appropriate level of contributions is considerable. However, the overall flexibility may be limited by the employer's objective to keep the volatility in contributions low or by a maximum contribution level stipulated by the regulating authorities. As far as the investment strategy is concerned, flexibility is even greater although constraints on certain asset classes have to be considered in many countries. In contrast, the benefits being guaranteed, the freedom in varying the benefit policy is limited. Regulatory requirements that are relevant for ALM are the regulation of investments, the regulation of funding and the regulation of the ownership of surplus as well as the responsibility to eliminate unfunded liabilities. They must be considered simultaneously when determining the optimal ALM policy as they constrain the optimal choices.

Moreover, ALM has to take account of the uncertainties inherent in the valuation of assets and liabilities. Whereas financial assets can be valued at market prices, the value of the outstanding contributions and the value of the liabilities have to be estimated. Their calculation requires assumptions on macroeconomic, company- and member-specific, as well as demographic variables. Thus, any decision on the optimal ALM policy is based on assumptions regarding these variables.

An ALM model attempts to cover these aspects. In the literature static and dynamic ALM models are distinguished depending on the treatment of time. In static ALM models the planning horizon is considered to be a single period. The optimal ALM policy is determined at the beginning of the period and remains fixed until the end of the period. Thus, it only affects the objective in that period. Typically, static ALM models focus on the derivation of an optimal investment policy. The contribution policy is exogenous. The models range from traditional mean-variance optimization models taking account of liabilities (e.g. Sharpe and Tint (1990), Ezra (1991)) to models that include stochastic simulations of assets and liabilities (e.g. Hardy (1993), Booth and Ong (1994)). Static ALM models have the advantage of being practical and mathematically relatively simple.

In contrast, dynamic ALM models divide the planning horizon into multiple time periods (discrete time) that can be infinitely small (continuous time). In a dynamic context, the pension fund is concerned both with its current optimal ALM policy and the optimal ALM policies in all future periods of the planning horizon. Assuming that the periods are not independent of each other, any decision made in one specific period has consequences in subsequent periods. These consequences must be considered when making decisions. The models can generally be divided into two categories: Discrete time stochastic programming models (e.g. Cariño et al (1994), Dert (1995)) and continuous time stochastic optimal control models (e.g. Haberman and Sung (1994), Boulier, Trussant, and Florens (1995), Cairns (1999)). Whereas the former are in practice successfully implemented by institutional investors to address real world features, the latter are often used in academic studies. Dynamic ALM models describe the problem set of pension funds more realistically than static models and lead to different policies. They generally result in lower costs of funding and smaller probabilities of underfunding. However, they are mathematically and computationally more complex.

In this paper the focus is on dynamic ALM models. The optimal ALM policy is analyzed in a continuous time stochastic optimal control context. Based on the standard work of Haberman and Sung (1994), Boulier, Trussant, and Florens (1995), and Cairns (1999) a dynamic economic model that allows to simultaneously determine the optimal investment and contribution policy is derived. The benefit policy is assumed to be exogenous. The standard model is extended in several ways: First, instead of minimizing the contribution and the solvency risk, the expected present value of the surplus is maximized for a given level of risk. Second, both assets and liabilities are modeled endogenously. Third, the impact of adjusting retirement benefits to inflation on the optimal contribution and investment policy is discussed. Finally, exogenous or endogenous minimum and/or maximum funding limits are considered. They are crucial for the control of a pension fund.

Within this modeling framework the following questions concerning an optimal ALM policy are addressed explicitly:

Fixed versus flexible contributions

The analysis is done under the assumption that unexpected changes in contributions cause costs. Do pension funds that manage contributions actively, i.e. that adjust contributions contingent on the prevailing state, perform better than pension funds that leave contributions fixed? Does this hold - if at all - if certain conditions are met?

Simultaneous management of contributions and investments

Assume that a pension fund disposes of two policy instruments, of the contributions and of the investment strategy. What is an optimal ALM policy if both are determined simultaneously taking into account that unexpected changes in contributions cause costs? What is the role of objectives in selecting an optimal ALM policy? How do the results vary if liabilities are modeled exogenously or endogenously? How should the pension fund adjust the ALM policy if there is a surplus or an unfunded liability?

Indexation of retirement benefits

Assume that a pension fund adjusts retirement benefits to inflation. What is an optimal ALM policy, consisting of the contribution and the investment policy? How does it differ from the one derived under the assumption that retirement benefits are independent of inflation? Does the optimal ALM policy depend on the availability of an appropriate asset to hedge inflation?

Management of the funding level

Assume that the funding level, i.e. the ratio of assets to liabilities, is regulated by means of minimum and maximum funding limits. The funding limits may be self-imposed by the pension fund or stipulated by the regulating authorities. What conditions have to hold such that the regulated pension fund performs better than the unregulated one if an exogenous minimum and maximum funding limit is imposed and if interventions at the funding limits are costly? What are these conditions in case of a single exogenous minimum funding limit? What are optimal minimum and maximum funding limits if they can be determined endogenously?

Outline

The purpose of Chapter 2 is to discuss the characteristics of a defined-benefit pension fund, to derive the economic fundamentals of ALM and to introduce the analysis performed in Chapter 4. First, the main institutional features of defined-benefit pension funds are presented and the determinants of their assets and liabilities are derived. Moreover, the questions of ownership of surplus and responsibility to eliminate an unfunded liability are addressed. Then, key elements of ALM are discussed. These concern the objectives the pension fund aims to achieve, the available ALM instruments and the regulatory requirements. Finally, the features of a general ALM model are presented and the analysis performed in Chapter 4 is outlined.

Chapter 3 lays the mathematical grounds for ALM modeling and provides a survey of the literature on ALM for pension funds. Economics, actuarial research and operations research are covered. The survey focuses especially on the economic literature but tries to cover the most important articles of the other fields as well. The chapter starts with a short introduction to static and dynamic modeling. By analyzing the single period and multiperiod consumption-investment problem, the major differences between these two modeling approaches are explored. Then, the focus is turned to ALM. An overview of the literature on static ALM models is given, and two representative approaches are presented in more detail: Mean-variance surplus optimization and surplus optimization with shortfall constraints. Then, the literature on dynamic ALM models is covered, and two representative approaches are discussed: Discrete time stochastic programming and continuous time stochastic optimal control. Finally, the pros and cons of static and dynamic ALM models are examined.

Chapter 4 is the core of this paper. A dynamic economic model to analyze the optimal ALM policy is derived in a continuous time stochastic optimal control context. The chapter is structured as follows: First, the theoretical ground for active management of the contribution policy is laid. Assuming that unexpected changes in contributions cause costs, it is examined under what conditions - if at all - pension funds that manage contributions actively perform better than pension funds that leave contributions fixed. Second, two models to simultaneously determine the optimal contribution and investment policy of a defined-benefit pension fund are presented. Their main differences concern the objective function and the way liabilities are modeled. The first model relies on the work done by Haberman and Sung (1994), Boulier, Trussant, and Florens (1995), Cairns (1999) and others. It is assumed that the pension fund aims at minimizing risk, namely the risk of unexpected changes in contributions (contribution risk) and the risk of falling short of the liabilities (solvency risk), which are exogenous. In the second model, the standard model structure is extended by assuming that the pension fund follows a different objective, namely maximizing the expected present value of the surplus. Costs incurred by unexpected changes in contributions are taken into account. They reduce the expected present value of the surplus. Moreover, the liabilities of the pension fund are modeled endogenously. The optimal contribution and investment policy is analyzed in both models, and the results are compared. In the analysis, particular emphasis is placed on the control of the optimal policies on the basis of an observed deficit or surplus. Third, the optimal contribution and investment policy is derived in the case where the pension fund adjusts retirement benefits to inflation. To compare the results with the ones derived in step two, the same objective function as in the standard model is chosen, i.e. minimization of the contribution and of the solvency risk. The correlation between inflation and the asset returns is explicitly taken into account, and it is analyzed whether the optimal policies depend on the availability of an appropriate asset to hedge inflation. Finally, the management of the funding level by means of minimum and/or maximum funding limits is examined. Three regulatory concepts are distinguished, a two-sided regulated funding level with exogenous minimum and maximum funding limits, a funding level that is regulated by a single exogenous minimum funding limit, and a two-sided regulated funding level with endogenous minimum and maximum funding limits. Costs related to interventions at the funding limits are taken into account. The results are compared to the ones derived for an unregulated pension fund.

Chapter 5 concludes the paper. The main results are summarized and some suggestions for further research are given.

Fenster schliessen