Chapter 2 investment decision in defined contribution pension schemes incorporating incentive mechanism

Th e framework used by Battocchio and Menoncin 2004 is modifi ed to incorporate risks from the fi nancial market and background risks in describing the infl ation rate and labor income u

Investment Decision in Defined Contribution

Pension Schemes

Incorporating Incentive Mechanism

Bill Shih-Chieh Chang

and Evan Ya-Wen Hwang

2.5 Asset Allocation for General Form 59

In t his st udy, we investigate the portfolio selection problem with

incen-tive m echanism (i.e., bo nus f ees a nd d ownside pena lty) i ncorporated into defi ned contribution (DC) pension schemes Th e framework used by Battocchio and Menoncin (2004) is modifi ed to incorporate risks from the

fi nancial market and background risks in describing the infl ation rate and labor income uncertainties through stochastic processes In order to properly evaluate the fi nancial impact of incentive structures on fund management,

a performance-oriented arrangement induced by bonus fees and downside penalty is i ntroduced Th e fund managers are rewarded with bonus fees for their superior performance, while downside penalty is also imposed on them once their performance is below the specifi ed benchmark

In o rder t o sc rutinize t he g eneral pa ttern o f f und dy namics u nder performance-oriented arrangement, a st ochastic control problem is for-mulated Th en, the optimization algorithm is employed to solve the asset allocation p roblem t hrough dy namic p rogramming F inally, n umerical illustrations are shown and results are summarized as follows:

1 A fi ve-fund separation theorem is derived to characterize the ment strategy Th e fi ve funds are the myopic market portfolio, the hedge portfolio for t he st ate va riables, t he hedge portfolio for t he infl ation risk, the hedge portfolio for the labor income uncertainties, and cash Except cash, all funds are dependent on the incentive setup When performance-oriented arrangement is taken into account, the fund managers tend to increase the holding in risky asset

2 When incentive mechanisms are incorporated, the settlement of egated ma nagement contract is v ital since t he setup could signifi -cantly aff ect the fund dynamics Our fi nding is consistent with the conclusion put forward by Raghu et al (2003) Our numerical results show that performance-oriented arrangements dominate the invest-ment discretion in fund management Hence, an incentive program has t o be c arefully i mplemented i n o rder t o ba lance t he r isk a nd reward in fund management for DC pension

del-Keywords: Defi ned c ontribution, b ackground r isks, s tochastic

control, bonus fee, downside penalty

2.1 INTRODUCTION

Th e investment strategy of pension funds has a profound eff ect on global capital markets Th ey aff ect the development of fi nancial innovation, the behavior of security prices, and rates of return In recent years, with an increase i n t he percentage of population t hat comes u nder pension age worldwide, pens ion-related t opics ha ve t aken o n n ew s ignifi cance and much attention has been focused on the implementation of a better invest-ment program for the aging society In 1990, the U.S government began

to implement the defi ned contribution (DC) pension plans Over the last two decades, DC pension plans, such as the 401(K) plan, have been the pri-mary engine of growth in the U.S private pension market (see Lachance

et al 2003) In view of the improved mortality rates, other countries such

as Germany, the United Kingdom, Australia, and India have also started

to implement DC pension plans

On July 1, 2005, the Labor Pension Act (LPA) was enacted in Taiwan, establishing a new, portable, defi ned contribution scheme for employees

Th e Taiwan government replaced the defi ned benefi t (DB) pension plans with DC pension plans, while all employees in Taiwan were given the option

to enroll in the LPA or remain with the old DB pension system under the Labor Standards Law Under the old pension system, employees receive a lump sum pension at the end of their employment term Employees are eligible to apply for retirement aft er having been employed at a company for 25 years Alternatively, an employee can also retire at age 55 as long as

he has worked for the same company for at least 15 years Companies erally pay 2%–15% of an employee’s monthly wages for each month the employee served, capped at 45 months However, many workers in Taiwan are not eligible to receive a pension since they do not always remain with the same company for at least 15 years

gen-While the benefi t design and contribution arrangement of the DC plans vary be tween co untries, t he n ewly enac ted DC labo r pens ion sch emes adopt the delegated management schemes Th e new LPA creates a labo r pension fund, which is made up of individual pension accounts for each employee who enrolls Employers were required to apply for enrollment

in L PA b y July 15, 2 005 E mployer en rollment r equired a n a pplication for labo r pens ion co ntributions, t he r eport o f h is o r h er labo r pens ion

contributions and a copy of the company’s registration license All ers are required to contribute at least 6% of an employee’s monthly salary toward the personal pension account Employees can also contribute up to 6%; the amount contributed will be deducted from the employee’s taxable annual income Once eligible to receive a pension under LPA, an employee will receive their pension funds on a quarterly basis.

employ-As a st imulus for the fund manager to act in the best interest of the plan participants, the prudent man rule is usually adopted in fund man-

agement A ccording to t he de fi nition in Wikipedia, t he p rudent ma n

rule means to observe how men of prudence, discretion, and intelligence manage t heir own a ff airs, n ot i n r egard t o spec ulation, but i n r egard

to t he per manent d isposition o f t heir f unds, co nsidering t he p able income, as well as the probable safety of the capital to be invested

rob-Th e fund investment mandate is oft en modifi ed with a certain incentive mechanism co nsisting o f bo nus f ees a nd d ownside pena lty S ince t he performance of fund growth aff ects h eavily t he pens ion w ealth o f the plan participants, it is obligatory for the fund to implement a certain downside protection mechanism For example, in Taiwan, the return of the pension fund cannot be less than the interest rate of a 2 y ear fi xed deposit Incorporating bonus fees and downside penalty in investment mandate may a lso c ause f und ma nagers to de viate f rom t heir d iscre-tionary behaviors Th erefore, in order to quantify the impact of a given incentive mechanism on pension fund dynamics, explicit solutions and numerical results are explored

Previous studies focusing on DB areas can be f ound in Bowers et al (1982), M cKenna ( 1982), Sha piro ( 1977, 1 985), O ’Brien ( 1986, 1 987), Racinello (1988), D ufresne (1988, 1 989), Haber man (1992, 1 993, 1 994), Haberman and Sung (1994), Janssen and Manca (1997), Haberman and Wong (1997), Chang and Cheng (2002) among others On the other hand, the investment risks including interest rate risk and market risk that had been a ssumed b y t he p lan spo nsor u nder t he DB promise i s g radually transferred to the worker in DC plans due to the severe longevity risk in aging society (Bodie 1990) Th us, the investment decision is critical for the

DC scheme Moreover, the DC scheme is accumulated through the annual salary-related contributions, and hence, the long-term fi nancial strategy will signifi cantly aff ect the fund performance Th erefore, both the uncer-tainties of labor income and the infl ation rate, also known as background risks, proposed by Menoncin (2002) are employed in our model Brinson

et al (1991) have shown convincingly that the allocation of investment

funds to asset categories is far more important than the selection of vidual securities within each asset category Hence, in this study, the back-ground r isks generated w ithin t he pension scheme a re i ncorporated to explore the optimal investment strategy for the DC plan.

indi-Th e rest of this chapter is organized as follows In Section 2.2, the ture related to the infl ation risk, the uncertainty of labor income, and the incentive mechanism is reviewed In Section 2.3, the general framework and t he fi nancial ma rket st ructure a re introduced Th en, t he stochastic optimal co ntrol p roblem i s f ormulated Th e o ptimization a lgorithm i s employed to derive the explicit solution in Section 2.4 In Section 2.5, how the bonus fees and the downside penalty infl uence the investment discre-tions of the fund manager is explicitly discussed Finally, Section 2.6 pro-vides a conclusion and summarizes this study

litera-2.2 LITERATURE REVIEW

2.2.1 Uncertainties of Infl ation and Salary

When the labor income uncertainty is incorporated into the investment decision, i t co uld s ignifi cantly infl uence t he h olding pos ition o f r isky assets due to the attained age of the plan member A trade-off between the capital gain in the fi nancial market and the expected discounted value of future labor income, that is, human resource, becomes crucial in lifestyle investment decision Hence, by diversifying among stocks and bonds, a more stable and effi cient portfolio can be created Campbell and Viceira (2002) suggest that investors not only own tradable fi nancial asset as part

of their total wealth portfolio, but they also own a va luable asset that is not readily tradable, which is labor income Imrohoroglu et al (1995) and Huang et a l (1997) investigate t he impact of salary under certain rates

of return Th en, Campbell et al (2001) consider the long-run pattern of lifetime savings and portfolio allocation in the presence of income and the rate of return uncertainty and with various pension arrangements Under

no circumstances do they consider the impact of the varying degrees of imperfection in annuity markets On t he contrary, they do consider the

fi xed costs of entering the equity market

Campbell and Viceira (2002) fi nd t hat t he existence of other income prospects tends to substitute for bonds in the investor portfolio Hence,

a relatively young investor w ith ex tensive f uture earning prospects w ill tend to possess a higher proportion of stocks than does an investor at a later stage of his or her working life However, this eff ect is reduced if the

income prospects are uncertain In line with the literature on background risks, the investor becomes in eff ect more risk-averse to market risks and, hence, b uys f ewer st ocks V iceira ( 2001) o ptimizes t he i nter-temporal investment-consumption policy of an investor who has an uncertain sal-ary In his model, labor income follows a geometric process and any sav-ings out of labor i ncome a re i nvested i n t he portfolio Th e single r isky asset a lso follows a pos sibly correlated geometric process Viceira fi nds that the ratio of portfolio wealth to labor income is stationary, and using

a log-linear a pproximation, he der ives a n optimal portfolio policy t hat has a constant stock proportion Moreover, he also fi nds that when the sal-ary risk is independent of the asset return risk, employed investors hold

a larger fraction of their savings in the risky asset than retired investors Koo (1998) and Heaton and Lucas (1997) also derive optimal consumption and portfolio policies with stochastic wage Koo uses a continuous-time model and shows that the optimal level of risk-taking is lower in the pres-ence of an uninsurable labor income risk Heaton and Lucas, in an infi nite horizon model, do not fi nd any signifi cant eff ect of labor income risk on portfolio composition

As for infl ation risk, when a l onger time horizon is considered, this risk becomes signifi cant Since the pension fund is a long-term plan, the managers sh ould ma nage t he i nfl ation r isk a nd e stablish t he o ptimal strategy to resist infl ation u ncertainties Modigliani a nd Cohn (1979), Madsen (2002), and Ritter and Warr (2002) have shown that stock mar-ket investors suff er from infl ation i llusion Menoncin (2002) considers both the salary uncertainty and the infl ation risk to analyze the port-folio problem of an investor maximizing the expected exponential util-ity of his or her terminal real wealth In his model, t he investor must cope with both a set of stochastic investment opportunities and a set of background risks Given that the market is complete, an explicit solu-tion can be obtained When the market is incomplete, an approximated solution is recommended Contrary to other exact solutions obtained in the literature, all the related results are obtained allowing the stochastic infl ation risk and without specifying any particular functional form for the va riables i n our problem Moreover, i n Battocchio a nd Menoncin (2004), an optimal investment strategy is derived according to the uncer-tainties of salary and infl ation risk However, these works did not refl ect the actual delegated management plan with incentive mechanism in DC pension schemes

2.2.2 Incentive Mechanism

Most o f t he l iterature i n pens ion r esearch f ocuses o n i mplementing a better benefi t scheme, while studies on the fi nancial impact on incentive mechanism are scarce Th e original motivation of performance-oriented arrangement in the fund investment mandate is to control the fund man-ager behaviors within a certain risk tolerance Th e forms of bonus struc-ture can be varied, such as fi xed-dollar fees, asset-based fees, and incentive fees (Eugene and Mary 1987) Under fi xed-dollar fees, the money man-ager would receive a fi xed amount of management fees regardless of the performance of t he ma naged f und For a sset-based fees, t he ma nager’s fees vary with the value of the fund Incentives fees are contingent upon the performance of the managed fund Generally speaking, the incentive mechanisms for the fund manager include the penalty for underperfor-mance and bonus for outstanding performance In our model, when the fund growth shows superior performance to the benchmark portfolio, the fund manager is rewarded with bonus fees, while he is also facing a certain downside penalty if the fund shows underperformance results

Richard and Andrew (1987) suggest that incentive fees off er a wa y of improving the relationship between money managers and plan sponsors However, the incentive fee contracts have to be set properly and setting the parameters is important Mark (1987) use the call option to price the incen-tive fees and fi nd that the value of this option depends on (1) the spread between the standard deviations of the fund portfolio and the benchmark portfolio, (2) the correlation between them, (3) the value of the managed fund, (4) the manager’s percentage participation in incremental return, and (5) the measurement period Because the manager could control factors (1) and (2), the setting of incentive fees contract would infl uence the invest-ment decisions of fund managers Lawrence and Stephen (1987) claim that

it is important to choose the parameters especially for the benchmark folio, and Richard and Andrew (1987) propose that this portfolio should

port-be able to represent the manager’s typical investment style In the model setting, we assume that the benchmark rate is a positive constant but the performance mechanism is related to the value of management asset Th us, our model is a time-dependent benchmark portfolio

Raghu et al (2003) simulate the delegated investment decisions under

fi ve types of incentive mechanisms Th ey show the effi cacy of the incentive contracts in improving the welfare of investors Edwin et al (2003) inves-tigate the investment behavior of mutual fund managers under incentive

fees R oy a nd W illiam (2007) per form a s imilar st udy f or h edge f und managers Both studies fi nd that the managers would increase the risk of portfolio when the return rate is below the benchmark rate because they consider the limited-liability incentive forms In this chapter, we use the combined form of asset-based and target-based incentive fee mechanisms

Th e target-based form means that when the performance exceeds the get, the manager would receive the incentive fees On the other hand, the manager has to make up for the shortage when the performance is below the benchmark; t herefore, t his i s a n u nlimited l iability i ncentive form Moreover, the amounts of bonus fees and downside guarantees are related

tar-to the value of fund, so i t is a k ind of asset-based incentive fees In our study, the fi nancial infl uence of diff erent bonus fees and downside penalty set is fully explored

2.3 PROPOSED MODEL

First, a time-varying opportunity set in the fi nancial market is introduced and the fund wealth process of DC pens ion scheme is formulated Our research broadened the attention from the risks in the fi nancial markets

to those outside the fi nancial markets that are referred to as background risks Back ground va riables c an be t he i nvestor’s wa ge process a nd t he contributions to and withdrawals from a pension fund Menoncin (2002) models t he back ground r isks a s a se t of stochastic va riables i n a nalyz-ing the portfolio problem By i nserting the infl ation risk that aff ects the growth rate of an investor’s wealth, Menoncin derives an exact solution

to the optimal portfolio problem when the fi nancial market is complete Menoncin also suggests an approximated general solution if the market is incomplete

2.3.1 Financial Market and Fund Dynamics

We assume that the fi nancial market is arbitrage free, incomplete, and

con-tinuously open over the investment time horizon [0, T], where T denotes

the ter minal d ate o f ma nagement co ntract Th e i ndependent Wien er

processes zr(t) and zm(t) represent the interest rate risk and market risk, respectively Th ey are defi ned on a p robability space (Ω, F, P), in which

P is the real-world probability and F = {F(t)} t∈[0,T] is the fi ltration that resents the information structure assumed to be generated by Brownian motion and satisfying the usual conditions

rep-Let r(t) be the interest rate at time t Actually, we can simulate the value

of r(t) by calibrating the trading information of the fi xed income

securi-ties However, due to the limited trading volume in Taiwan treasury bond

in the fi xed income market, model calibration merits further investigation; and hence, a one-factor spot interest rate model is employed We assume

directly that r(t) follows the Vasicek model (1977) Under the real-world probability measure P, the process r(t) satisfi es the dynamics

mea-Th ere are three investment vehicles in the fi nancial market Th e fi rst

underlying asset is cash, S0(t), which pays the instantaneous interest rate

without any default risk and the price process is expressed as the following stochastic diff erential equation:

=

0 0

d ( ) ( )d( )

λr represents the risk premium of interest rate risk

Th e duration of B K (t) is fi xed with K, so it is easy for application Moreover,

in asset management, manager could use cash and zero coupon bond to replicate the rolling bond fund

Th e other risky asset is the stock index fund, S(t), whose dynamic

pro-cess is given by

volatility w hose ri sk s ource does not b elong to zr and zm Th is

non-hedgable risk is called zL(t) that is independent of zr(t) and zm(t) Moreover,

µL(t) i s t he d rift term of labor income process, and we assume it to be

constant to simplify the derivation Next, we assume that each employee contributes a co nstant proportion, γ, of his or her labor income into his personal account

Th en, we introduce the other background risk, the infl ation risk We use t he co nsumption p rice i ndex (CPI) t o r epresent t he i nfl ation rate Hence, we present the stochastic partial diff erential equation describing the evolution of CPI

dP dt d ( )z t d ( )z t d ( )z t

Similarly, C PI process i s a ff ected by zr(t), zm(t), a nd zL(t) I n p articular,

we call FN the nominal fund and F the real fund According to the Fisher

equation (1930), we can write (Battocchio and Menoncin 2002)

In the above conversion equation, when we want to convert nominal fund into real fund wealth, we need to incorporate the diff erence that is caused

by change in infl ation Note that the diff erence is in the form of dP/P, so

the diff erence is related only to the increasing rate of infl ation For plicity, in Equation 2.6 we assume that the increasing rate of infl ation is

sim-just a constant Th er efore, P is not a state variable when we derive the

γ dL represents the contribution to the pension fund

e1 denotes the incentive fee ratio when the fund return is positive

e2 den otes t he pa rtial fl oor p rotections wh en t he f und r eturn i s negative

In Equation 2.8, we can see that the fund manager must charge ment incentive fees or face loss compensation, which correlates with the fund’s performance In other words, if the fund performance is good, the fund manager should charge higher management incentive fees Equation 2.8 can be viewed as an option problem; however, it is too complicated and diffi cult to solve In order to simplify the problem, we assume that

manage-e1 = e2 = e Simultaneously, we substitute Equation 2.7 into Equation 2.8, the accumulated real fund wealth process at any time t ∈ [0, T] can be

written in a reduced form as follows:

K B

−σ

2.4 ASSET ALLOCATION FOR RESTRICTED FORM

Since the fund manager’s attitude to risk varies, exponential utility tion is employed to measure investor’s satisfaction with wealth accumula-tion Th e goal of the fund manager is to construct an optimal investment strategy to maximize the expected utility value of the terminal wealth.2.4.1 Stochastic Optimal Control

func-Th e stochastic optimal control problem is written as follows:

G G

to the Hamiltonian equation and obtain the optimal weight *w G

In order to illustrate the optimal behavior, we adopt the results of Markus and William (2004) and rewrite Equation 2.12 as follows:

Th e vectors ′w , ′ M w , and ′ P w are two-dimensional with elements that sum L

to 1, a nd ′w is of d imension 2 × 2 w ith row elements t hat su m to 1; Y

A; B; C; and D are real constants Th e optimal portfolio consists of fi ve

single portfolios: t he ma rket portfolio ′w , t he hedge portfolio for t he M

state variables ′w , the hedge portfolio for the infl ation risk ′ Y w , the hedge P

portfolio for the salary uncertainty ′w , and cash Th us, we can state the L

W e J We should note that this is a speculative

component proportional to both the portfolio Sharpe ratio and the inverse of the Arrow–Pratt risk aversion index In other words, this portfolio’s investment weight will be infl uenced by the fund man-ager’s risk aversion index

2 Th e second ter m denotes a st ate va riable hedge portfolio (i.e., t he interest r ate a nd labo r i ncome u ncertainties) Th is component

provides a detailed mutual fund in the capital market to hedge the uncertainties.

3 Th e t hird a nd f ourth co mponents a re en thralling F or t he back ground risks (labor income uncertainty and infl ation risk), there exist

-no perfect hedging instruments in the fi nancial markets However, the third and fourth portfolios show how background risks can be hedged in the capital market and these components are preference-free components depending only on the diff usion terms of assets and background variables

According to our fi ve separated mutual funds, a pens ion fund manager who plans to hedge the market risk, interest rate risk, infl ation rate risk, and labor income uncertainty should invest the wealth in the following

C e

4 Th e salary uncertainty hedge portfolio ′w with level L − γ

−

N(1 )

L D

Note t hat e is commonly between zero and one Th er efore, 1/(1 − e) i s

greater than one Th us, when we take the bonus fee and downside penalty into account, we fi nd t hat t he weights in risky assets increase In other words, the levels invested in the market portfolio, the state variable hedge portfolio, the infl ation hedge portfolio and the salary uncertainty hedge portfolio increase; while the level invested in cash decreases Th is seems rational and reasonable Since the fund manager charges the management incentive fee from the pension account and the management fee ratio is positively co rrelated w ith t he pens ion f und p reference ( i.e., t he f und’s return), it is necessary to pay extra money in the hedging components

In the fi nancial literature, researchers commonly use separability condition

to solve this PDE Accordingly, following the previous works in Battocchio and Menoncin (2004), our value function is assumed to be g iven by the

product of two terms: an increasing and concave function of the wealth W,

and an exponential function depending on time and interest rates Th en,

the value function J and utility function can be written as follows:

2

( , ) 1

( ; , ) ( )( )

h t W

ν β

⎪

Th en, aft er complicated derivation, we could derive the optimal portfolio

as follows (see Appendix A)

g s s s

e L

dem-Figure 2 1 p lots t he o ptimal po rtfolio h oldings o f c ash, st ocks, a nd nominal bonds as a function of investment horizon, that is, 30 years We