Net Contribution, Liquidity, and Optimal Pension Management

• DOI: http://doi.org/10.1111/jori.12072
• Author: Sang‐youn Roh,Changki Kim,Changhui Choi,Bong‐Gyu Jang
• Published date: 01 December 2016
• Date: 01 December 2016

NET CONTRIBUTION,LIQUIDITY,AND OPTIMAL PENSION

MANAGEMENT

Changhui Choi

Bong-Gyu Jang

Changki Kim

Sang-youn Roh

ABSTRACT

This article presents an optimal portfolio balancing strategy for a pension

fund manager in the presence of fixed and proportional transaction costs

with respect to stock trades and changes in net contribution. An analytic

solution to the one-period problem is presented and a heuristic method for a

multiperiod problem is developed. For reasonably calibrated parameters, we

find that our numerical results explain the actual asset allocation schemes of

some internationally renowned pension funds. Moreover, we show that net

contribution and liquidity have significant impacts on the optimal asset

allocation of a pension fund.

INTRODUCTION

During the recent global financial crisis, one of the most severe crises to afflict the

global financial markets, internationally renowned pension service providers (PSPs)

showed different reactions to the market turmoil: some PSPs did not alter their asset

allocation, whereas others actively adjusted their portfolios to adapt to the adverse

market conditions.

Changhui Choi is at the Department of Financial Strategy, Korea Insurance Research Institute.

Bong-Gyu Jang is at the Department of Industrial and Management Engineering, POSTECH.

Changki Kim is at Korea University Business School, Korea University. Sang-youn Roh is at the

Performance Evaluation Team, National Pension Research Institute. The authors can be

contacted via e-mail: cchoi@kiri.or.kr, bonggyujang@postech.ac.kr, changki@korea.ac.kr, and

riskhunter@nps.or.kr, respectively. We thank journal editor Keith J. Crocker and the

anonymous referee for the careful reading of our manuscript and the valuable comments.

This work was supported by the National Research Foundation of Korea Grant funded by the

Korean Government (NRF-2014S1A3A2036037) and by Basic Science Research Program

through the National Research Foundation of Korea (NRF) funded by the Ministry of

Education, Science, and Technology (NRF-2012R1A1A2038735, NRF-2013R1A2A2A03068890).

© 2015 The Journal of Risk and Insurance. 83, No. 4, 913–948 (2016).

DOI: 10.1111/jori.12072

913

When a financial market experiences a slump, investment returns on stocks (risky

assets) can fall below returns on bonds (risk-free assets). From a myopic perspective,

liquidating risky assets and investing funds in risk-free assets may seem more

profitable. Yet, according to our observations, risky asset to total asset ratios of some

large-scale PSPs experienced marginal change of only a few percent (see Table D1 in

Appendix D). These observations represent the motivation for this research, which

focuses on the investment strategies of large-scale PSPs. This research develops

theoretical asset allocation schemes to examine the investment strategies of large-

scale PSPs.

To lay the theoretical foundations of PSP portfolio management, this article constructs

a portfolio balancing strategy that maximizes the expected value of CRRA (constant

relative risk aversion) utility function in discrete time where assets consist of one risk-

free asset and one risky asset and proportional and fixed costs are paid for risky asset

trades.

1

CRRA criterion may be better suited for normal periods than turbulent periods.

However, this research attempts to analyze the asset allocation behaviors of PSPs

during both normal and turbulent periods using the widely used CRRA framework.

Within this framework, this research provides a multiperiod optimal portfolio

balancing strategy that maximizes the expected utility of the last period while

considering periodic (negative or positive) cash flows into the funds at the beginning

of each period. For PSPs, these cash flows are usually the net contribution, defined as

the difference between the contributions from participating members and the annuity

(benefits) payments to members. It is possible to consider net liability instead of net

contribution. For example, D’Arcy, Dulebohn, and Oh (1999) study an alternative

approach to the optimal management of a pension fund that considered the current

pension obligation (such as liabilities for current employees and retirees) and the

respective growth rates of pension expenses and the tax base.

Additionally, the impact of market liquidity on the optimal asset portfolio strategies

of some large-scale PSPs is examined using the bid–ask spreads as the proxy for the

1

According to Martellini and Milhau (2008), a series of papers (Leland, 1980; Benninga and

Blume, 1985; Franke and Subrahmanyam, 1998) examined the investor preferences and market

characteristics that would support the idea of including derivatives features in insurance and

investment portfolios. The authors mostly find that holding such derivatives payoffs can be

justified under usually severe forms of market incompleteness and/or the presence of

background risk. This finding is supported by five PSPs that carry a small amount of financial

derivatives (the value is usually less than 0.1 percent of the total assets). As discussed in Leland

(1980), many authors consider maximizing the terminal wealth and minimizing the

probability of insolvency as appropriate goals for DC (defined-contribution) pension plans

and DB (defined-benefit) pension plans. However, a PSP that manages a DB pension plan can

still pursue long-term wealth maximization if there is little chance of insolvency, which is the

case for many large public PSPs. van Binsbergen and Brandt (2007) and Martellini and Milhau

(2008) refer to cases where maximizing the expected terminal funding ratio (asset/liability) (or

terminal net asset (asset-liability)) is used as an objective for DB pension plans.

914 THE JOURNAL OF RISK AND INSURANCE

proportional transaction cost, although bid–ask spread could understate the true

transaction costs for large investors such as pension funds (Marshall, 2006).

2

Korn (1998) demonstrates that a fixed transaction cost can be easily incorporated into

a continuous time model because this type of cost is incurred whenever an investor

rebalances a portfolio. However, incorporating the fixed cost into a discrete time

portfolio optimization problem is complex because an investor may or may not

rebalance her portfolio at a certain period, in which case the fixed cost is a binary

variable. When a binary variable exists, a terminal wealth maximization problem has

multiple discontinuous points complicating the application of traditional methods.

This article proposes a resolution to this issue by separating the problem into two

parts: one part assumes no trading and the other part assumes trading a positive

amount of risky assets.

Portfolio optimization is one of the most actively studied subjects in finance.

3

Whereas studies concerning optimal portfolio rebalancing in continuous time

2

According to Sarr and Lybek (2002), liquidity measures include the following: transaction cost

measures, volume-based measures, price-based measures, market-impact measures, and

other econometric techniques. Plerou, Gopikrishnan, and Stanley (2005) analyze market

liquidity using bid–ask spread. Wei and Zheng (2010) measure the liquidity of individual

equity options using the bid–ask spread, which can be expressed as the absolute value or the

ratio of a gap between the bid price and the ask price. For example, Marshall and Young (2003)

calculate the bid–ask spread using every Wednesday’s closing bid and ask prices. According

to Marshall (2006), an order-based measure such as the bid–ask spread is effective for

measuring the liquidity of small investors; however, it is not effective for measuring the

liquidity of larger investors. Marshall insists that weighted order value can compensate for

the weak points of traditional liquidity proxies by incorporating the bid–ask spread and the

market depth; weighted order value is one of the liquidity proxies used by Aitken and

Comerton (2003).

3

Merton (1971) obtains a closed-form solution for a two-asset portfolio optimization problem in

the absence of transaction costs. Following this paper, Cvitanic

´and Karatzas (1992), Xu and

Shreve (1992), and He and Pearson (1993) analyze portfolio optimization with strategic

constraints. Davis and Norman (1990), Magill and Constantinides (1976), Taksar, Klass, and

Assaf (1988), and Jang et al. (2007) examine portfolio optimization problems with proportional

stock trading cost. Constantinides (1979) shows that the no-trading region (NTR) is a convex

cone for an investor with the power utility and a proportional transaction cost. An

approximate solution to this problem is provided in Constantinides (1986), and Gennotte and

Jung (1994) numerically identify the approximate boundary values of NTR for portfolio

optimization with a finite terminal date. Moreover Eastham and Hastings (1988) extend the

problem by considering both fixed and proportional transaction costs in their impulse control

approach, whereas Grossman and Vila (1992) study optimal dynamic trading with leverage

constraints. The approach employed by Eastham and Hastings (1988) is improved further by

Korn (1998), who proposes using an optimal stopping criteria method. Shreve and Soner (1994)

apply the theory of viscosity solution to Hamilton–Jacobi–Bellman (HJB) equations. Dumas

and Luciano (1991) provide an exact solution to the portfolio choice problem for a CRRA

investor who pays a proportional cost in an infinite horizon. More recently Zakamouline

(2002) proposes a solution to finite-horizon portfolio optimization in continuous time with

both fixed and proportional costs using a Markov approximation. FinallyØksendal and Sulem

(2002) consider the portfolio choice problem for a CRRA investor who pays both proportional

NET CONTRIBUTION,LIQUIDITY,AND OPTIMAL PENSION MANAGEMENT 915