Optimal Design of the Attribution of Pension Fund Performance to Employees

• Author: Heinz Müller,David Schiess
• Published date: 01 June 2014
• DOI: http://doi.org/10.1111/j.1539-6975.2013.01516.x
• Date: 01 June 2014

©

DOI: 10.1111/j.1539-6975.2013.01516.x

431

OPTIMAL DESIGN OF THE ATTRIBUTION OF PENSION

FUND PERFORMANCE TO EMPLOYEES

Heinz M ¨

uller

David Schiess

ABSTRACT

The article analyzes risk sharing in a defined contribution pension fund in

continuous time. According to a prespecified attribution scheme, the interest

rate paid on the employees’ accounts is a linear function of the fund’s invest-

ment performance. For each attribution scheme, the pension fund maximizes

the expected utility and the employees derive utility from their savings ac-

counts. It turns out that all Pareto-optimal attribution schemes are character-

ized by the same optimal participation rate. Wederive the total welfare gain

that installs from replacing no participation with optimal participation. This

welfare gain can be quantified and is substantial for reasonable parameter

values.

INTRODUCTION

In pension finance one must distinguish between defined benefit and defined con-

tribution plans. In a defined benefit plan, benefits are defined in advance and the

plan sponsor and the employees must adjust their contributions. In the literature,

optimal investment strategies for defined benefit plans are treated by Haberman and

Sung (1994), Boulier, Trussant, and Florens (1995), Cairns (2000), and Josa-Fombellida

and Rincon-Zapatero (2006) among others. Furthermore, the articles by Wilkie(1985),

Sharpe and Tint (1990), Keel and M¨

uller (1995), and Leippold, Trojani, and Vanini

(2004) deal with portfolio optimization in the asset–liability context. Since the plan

sponsor has to bear substantial financial and actuarial risk and since the valuation

of accrued retirement benefits of employees changing their plan is difficult, defined

benefit plans have become less and less popular. In a defined contribution plan in

the narrow sense, the accrued retirement benefits of employees are invested and the

positive or negative fund return is fully attributed to the employees’ accounts. At

retirement each employee obtains the final amount from his account either as a cash

payment or a corresponding pension. However, there is a wide span of different

Heinz M¨

uller and David Schiess are at Group for Mathematics and Statistics, University of

St. Gallen, Bodanstrasse 6, 9000 St. Gallen, Switzerland. The authors can be contacted via

e-mail: heinz.mueller@unisg.ch and david.schiess@unisg.ch, respectively. We would like to

thank Dr.Roger Baumann for his valuable input and the two anonymous referees for their very

constructive advice. We also gratefully acknowledge financial support from the University of

St. Gallen’s research program “Wealth and Risk.”

The Journal of Risk and Insurance, 2013, Vol. 81, No. 2, 431–468

432 THE JOURNAL OF RISK AND INSURANCE

defined contribution plans. On the one hand, there exist pure defined contribution

plans like the U.S. 401(k) plans where the employee is free to choose an individ-

ual investment policy. The pension fund consequently bears no financial risk at all

and acts only as a broker between employees and financial and life insurance mar-

kets. Hence, employees may be considered as individual investors. Authors dealing

with this topic in continuous time are Merton (1969), Adler and Dumas (1983), and

Gao (2008, 2009); in discrete time there are articles by Solnik (1978) and Haberman

and Vigna (2002) among others. On the other hand, there exist defined contribution

pension plans with certain guaranteed benefits. The plan sponsor bears substantial

financial risk in order to protect employees against fluctuations of financial markets.

For example, in the mandatory part of Swiss pension funds, the interest paid on the

employees’ accounts may depend on the financial situation and the recent perfor-

mance of the fund, but it has a lower bound fixed by central authorities. As this

minimum interest rate is typically above the riskless interest rate we will model the

rate attributed to the employees’ accounts as the riskless interest rate plus a premium.

However, on top of this, the majority of the Swiss pension funds let their employees

participate in the fund’s investment performance in some way. The overmandatory

part allows them to go below the minimum rate in case of bad investment perfor-

mances. Inspired by these facts, we will assume that the total rate attributed to the

employees’ accounts consists of a premium in addition to the riskless interest rate

and participation in the fund’s investment performance. At retirement, the accounts

are transformed into benefit streams. Of course, in such a framework the value of the

assets may differ from the present value of the liabilities. In the literature, Boulier,

Huang, and Taillard (2001) and Deelstra, Graselli, and Koehl (2003, 2004) analyze

a defined contribution plan providing a guarantee. This article concentrates on the

optimal design of the attribution of the pension fund performance to its employ-

ees. If a pension plan increases the funding ratio (ratio of the value of assets to the

present value of net obligations) during prospering financial markets and decreases

it in declining financial markets, then a risk transfer between different generations

of employees can be established. This idea was formalized by Baumann and M ¨

uller

(2008). In their model they introduced an intertemporal risk transfer by assuming

that the rate attributed to the employees’ accounts depends on the funding ratio but

not directly on the fund’s recent investment performance. In this way the attribution

of investment performance is flattened over time. They showed that in such a frame-

work the pension fund can choose investment policies such that all employees would

be worse off if they acted as individual investors. Furthermore, there are articles

deriving optimal investment strategies for defined contribution plans that put the

funding ratio into the focus of the optimization like Browne (1997), Denzler, M¨

uller,

and Scherer (2001), and M ¨

uller and Baumann (2006). In contrast to these articles,

the liabilities in the papers by M ¨

uller and Baumann (2008) and Baumann and M ¨

uller

(2008) are not exogeneous. Baumann and M ¨

uller (2008) model the return attributed to

the employees’ accrued retirement benefits as a function of the funding ratio. In this

article, we analyze the dependency on the fund’s investment performance. A similar

idea has already been pursued in the work of Baumann (2005). In this article, however,

we get closed-form solutions by restricting the fund’s performance attribution to lin-

ear schemes. Substantial welfare gains result even for this special class of attribution

schemes.

OPTIMAL DESIGN OF THE PENSION FUND’S PERFORMANCE ATTRIBUTION 433

This article deals with financial risk sharing between the defined contribution plan

sponsor and the employees in continuous time. We assume that the rate attributed

to the employees’ accounts consists of a premium in addition to the riskless interest

rate and a linear participation in the fund’s investment performance. The attribution

scheme is part of the institutional setup and can be characterized by the premium and

the participation rate. One of the main purposes of the article is to find Pareto-optimal

attribution schemes. Given an attribution scheme, the pension fund maximizes its

expected utility by choosing a corresponding investment policy. Hence, for each

attribution scheme a corresponding expected utility is obtained. This defines the

indirect utility function of the pension fund on the set of attribution schemes. Given the

investment policies of the pension fund, one may calculate the expected utility of the

employees for each attribution scheme. In this way,the indirect utility function of the

employees can be derived as well. The set of Pareto-optimal attribution schemes

are calculated and it is shown that all Pareto-optima are characterized by the same

participation rate. Since the optimal participation rate is constant, there is a conflict

of interests about the premium only. The specific choice of the premium depends on

the interaction of employees, the firm or institution attached to the pension fund,

and the regulator. On the one hand, the premium should attain a level such that an

employee is better off as a member of the fund than as an individual investor using

the Merton solution. On the other hand, the premium should be low enough to ensure

financial stability of the pension fund. We show that such levels for the premium can

be attained for reasonable parameter values. Furthermore, a formula for the optimal

participation rate is derived. Afterward, we define the welfare of the pension fund

and the employees, respectively, by making use of appropriate certainty equivalents.

Finally, a comparison of optimal participation and no participation shows that there

is a substantial welfare gain for reasonable parameter values.

The article is organized as follows. The second section presents the model setup by

defining the financial market as well as the objective function of both the pension

fund and the employees. Furthermore, linear attribution schemes are introduced and

the evolution of assets and liabilities are developed, which taken together yield the

dynamics of the funding ratio. In the third section, we maximize the expected utility

of the pension fund for a given attribution scheme, which leads to a constant optimal

investment strategy. We discuss the optimal investment rule and close the section

with the derivation of the optimally controlled funding ratio dynamics. In the fourth

section, we derive the indifference curves of the pension fund and the employees,

respectively, which leads to a discussion of risk-adjusted rates of return. The fifth

section then presents the result that there is a unique Pareto-optimal participation

rate α∗∈(0, 1). The Pareto-optimal investment strategy is derived by exploiting this

participation rate. The section closes with a discussion of the parameter sensitivities

of optimal participation and investment. The sixth section then narrows the set of

Pareto-efficient attribution schemes by imposing individual rationality of the em-

ployees and financial stability of the pension fund. The seventh section first defines

the welfare of the pension fund and the employees by making use of appropriate

certainty equivalents. Following this, we derive the welfare gain that results if no

participation is replaced with optimal participation. It is shown that this welfare gain

is substantial for reasonable parameter values. Finally,the eighth section summarizes

and presents the main conclusions.