CHAPTER Investment Decision in Defined Contribution Pension Schemes Incorporating Incentive Mechanism Bill Shih-Chieh Chang and Evan Ya-Wen Hwang CONTENTS 2.1 I ntroduction 2.2 L iterature Review 2.2.1 Uncertainties of Inflation and Salary 2.2.2 I ncentive Mechanism 2.3 Pr oposed Model 2.3.1 Financial Market and Fund Dynamics 2.3.2 Back ground Risks 2.3.3 F und Dynamics 2.4 Asset Allocation for Restricted Form 2.4.1 Stochastic Optimal Control 2.4.2 An Exact Solution 2.4.3 N umerical Illustrations 41 43 43 45 46 46 48 49 50 50 54 54 39 © 2010 by Taylor and Francis Group, LLC 40 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 2.5 Asset Allocation for General Form 2.5.1 Optimal Investment Decision 2.5.2 F inancial Implication 2.6 C onclusion Appendix References 59 60 62 63 65 I n t his st udy, we investigate the portfolio selection problem with incentive m echanism ( i.e., bo nus f ees a nd d ownside pena lty) i ncorporated into defined contribution (DC) pension schemes The framework used by Battocchio and Menoncin (2004) is modified to incorporate risks from the financial market and background risks in describing the inflation rate and labor income uncertainties through stochastic processes In order to properly evaluate the financial impact of incentive structures on fund management, a performance-oriented arrangement induced by bonus fees and downside penalty is i ntroduced The f und ma nagers a re rewarded w ith bonus fees for their superior performance, while downside penalty is also imposed on them once their performance is below the specified benchmark In o rder t o sc rutinize t he g eneral pa ttern o f f und dy namics u nder performance-oriented a rrangement, a st ochastic control problem i s formulated Then, 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: A five-fund separation theorem is derived to characterize the investment st rategy The five f unds a re t he myopic ma rket portfolio, t he hedge po rtfolio f or t he st ate va riables, t he h edge po rtfolio f or t he inflation 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 When incentive mechanisms are incorporated, the settlement of delegated ma nagement co ntract i s v ital s ince t he se tup co uld s ignificantly affect the fund dynamics Our finding is consistent with the conclusion put forward by Raghu et al (2003) Our numerical results show that performance-oriented arrangements dominate the investment 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 © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 41 Keywords: Defined c ontribution, b ackground r isks, s tochastic control, bonus fee, downside penalty 2.1 INTRODUCTION The investment strategy of pension funds has a profound effect on global capital markets They affect the development of financial innovation, the behavior of security prices, a nd rates of return In recent years, w ith a n increase i n t he per centage o f po pulation t hat co mes u nder pens ion a ge worldwide, pens ion-related t opics ve t aken o n n ew s ignificance and much attention has been focused on the implementation of a better investment program for the aging society In 1990, the U.S government began to implement the defined contribution (DC) pension plans Over the last two decades, DC pension plans, such as the 401(K) plan, have been the primary 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, defined contribution scheme for employees The Taiwan government replaced the defined benefit (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 t he old pension system, employees receive a lump su m pension at t he end of t heir employment ter m Employees a re eligible to apply for retirement after 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 generally pay 2%–15% of a n employee’s monthly wages for each month t he employee served, capped at 45 months However, many workers in Taiwan are not eligible to receive a pension since they not always remain with the same company for at least 15 years While the benefit 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 t he delegated ma nagement schemes The new LPA creates a labo r pension fund, which is made u p of individual pension accounts for each employee who en rolls Employers were required to apply for en rollment in L PA b y J uly 5, 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 © 2010 by Taylor and Francis Group, LLC 42 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling contributions and a copy of the company’s registration license All employers 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 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 management A ccording to t he de finition in Wikipedia, t he p rudent ma n rule means to observe how men of prudence, discretion, and intelligence manage t heir o wn a ffairs, n ot i n r egard t o spec ulation, b ut i n r egard to t he per manent d isposition o f t heir f unds, co nsidering t he p robable income, as well as the probable safety of the capital to be invested The fund investment mandate is often modified 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 affects 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 f und cannot be l ess t han t he interest rate of a y ear fi xed deposit Incorporating bonus fees and downside pena lty in investment mandate ma y a lso c ause f und ma nagers t o de viate f rom t heir d iscretionary behaviors Therefore, in order to quantify the impact of a given incentive mechanism on pension fund dynamics, explicit solutions and numerical results are explored Previous st udies focusing on DB a reas c an be f ound i n B owers e t a l (1982), M cKenna ( 1982), Sha piro ( 1977, 985), O ’Brien ( 1986, 987), Racinello ( 1988), D ufresne ( 1988, 989), Haber man ( 1992, 993, 994), Haberman a nd Sung (1994), Janssen a nd Ma nca (1997), Haber man a nd 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 D B p romise i s g radually transferred to the worker in DC plans due to the severe longevity risk in aging society (Bodie 1990) Thus, the investment decision is critical for the DC scheme Moreover, the DC scheme is accumulated through the annual salary-related contributions, a nd hence, t he long-term financial strategy will significantly affect the fund performance Therefore, both the uncertainties of labor income and the inflation 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 © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 43 funds to asset categories is far more important than the selection of individual securities within each asset category Hence, in this study, the background r isks g enerated w ithin t he pens ion sch eme a re i ncorporated t o explore the optimal investment strategy for the DC plan The rest of this chapter is organized as follows In Section 2.2, the literature related to the inflation risk, the uncertainty of labor income, and the incentive mechanism is reviewed In Section 2.3, the general framework and t he financial ma rket st ructure a re i ntroduced Then, t he st ochastic optimal co ntrol p roblem i s f ormulated The 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 influence the investment discretions of the fund manager is explicitly discussed Finally, Section 2.6 provides a conclusion and summarizes this study 2.2 LITERATURE REVIEW 2.2.1 Uncertainties of Inflation and Salary When the labor income uncertainty is incorporated into the investment decision, i t co uld s ignificantly influence 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 financial market and the expected discounted value of future labor income, that is, human resource, becomes crucial in lifestyle investment dec ision Hence, by d iversifying a mong stocks a nd bonds, a more stable and efficient portfolio can be c reated Campbell and Viceira (2002) suggest that investors not only own tradable financial asset as part of t heir total wealth portfolio, but t hey a lso own a va luable asset t hat is not readily tradable, which is labor income Imrohoroglu et al (1995) and Huang e t a l (1997) i nvestigate t he i mpact o f s alary u nder c ertain r ates of return Then, C ampbell et a l (2001) consider t he 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 t hey consider t he impact of t he varying degrees of imperfection in a nnuity ma rkets On t he contrary, t hey consider t he fi xed costs of entering the equity market Campbell a nd V iceira (2002) find t hat t he ex istence of other i ncome prospects tends to subst itute for bonds i n t he i nvestor portfolio Hence, a r elatively y oung i nvestor w ith ex tensive f uture e arning p rospects 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 effect is reduced if the © 2010 by Taylor and Francis Group, LLC 44 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling income prospects are uncertain In line with the literature on background risks, the investor becomes in effect 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 salary In his model, labor income follows a geometric process and any savings o ut o f labo r i ncome a re i nvested i n t he po rtfolio The s ingle r isky asset a lso f ollows a pos sibly co rrelated g eometric p rocess V iceira finds that the ratio of portfolio wealth to labor income is stationary, and using a l og-linear a pproximation, h e der ives a n o ptimal po rtfolio po licy t hat has a constant stock proportion Moreover, he also finds that when the salary 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 w ith stochastic wage Koo uses a co ntinuous-time model and shows that the optimal level of risk-taking is lower in the presence of an uninsurable labor income risk Heaton and Lucas, in an infinite horizon model, not find any significant effect of labor income risk on portfolio composition As for inflation r isk, when a l onger t ime horizon is considered, t his risk becomes significant Since the pension fund is a long-term plan, the managers sh ould ma nage t he i nflation r isk a nd e stablish t he o ptimal strategy t o r esist i nflation u ncertainties M odigliani a nd C ohn (1979), Madsen (2002), and Ritter and Warr (2002) have shown that stock market i nvestors su ffer f rom i nflation i llusion Me noncin (2002) c onsiders both t he s alary u ncertainty a nd t he i nflation r isk to a nalyze t he portfolio problem of an investor maximizing the expected exponential utility of h is or h er ter minal real wealth I n h is m odel, t he i nvestor must cope with both a set of stochastic investment opportunities and a set of background risks Given that the market is complete, an explicit solution 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 inflation risk and without specifying any particular functional form for the va riables i n o ur p roblem M oreover, i n Ba ttocchio a nd M enoncin (2004), an optimal investment strategy is derived according to the uncertainties of salary and inflation risk However, these works did not reflect the actual delegated management plan with incentive mechanism in DC pension schemes © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 45 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 benefit scheme, while studies on the financial impact on incentive mechanism are scarce The original motivation of performance-oriented arrangement in the fund investment mandate is to control the fund manager behaviors within a certain risk tolerance The forms of bonus structure can be varied, such as fixed-dollar fees, asset-based fees, and incentive fees (Eugene a nd Ma ry 1987) Under fi xed-dollar fees, t he money ma nager would receive a fi xed amount of management fees regardless of the performance o f t he ma naged f und F or a sset-based f ees, 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 t he f und ma nager i nclude t he pena lty for u nderperformance 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 a nd A ndrew (1987) su ggest t hat i ncentive fees offer 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 incentive fees and find 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) t he measurement per iod B ecause t he ma nager could control fac tors (1) and (2), the setting of incentive fees contract would influence the investment decisions of fund managers Lawrence and Stephen (1987) claim that it is important to choose the parameters especially for the benchmark portfolio, and Richard and Andrew (1987) propose that this portfolio should 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 Thus, our model is a time-dependent benchmark portfolio Raghu et al (2003) simulate the delegated investment decisions under five types of incentive mechanisms They show the efficacy of the incentive contracts in improving the welfare of investors Edwin et al (2003) investigate the investment behavior of mutual fund managers under incentive © 2010 by Taylor and Francis Group, LLC 46 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling fees R oy a nd W illiam ( 2007) per form a s imilar st udy f or h edge f und managers Both studies find 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 The target-based form means that when the performance exceeds the target, 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 ben chmark; t herefore, t his i s a n u nlimited l iability i ncentive f orm Moreover, the amounts of bonus fees and downside guarantees are related to t he va lue of f und, so i t is a k ind of a sset-based i ncentive fees I n our study, the financial influence of different bonus fees and downside penalty set is fully explored 2.3 PROPOSED MODEL First, a time-varying opportunity set in the financial market is introduced and t he f und wealth process of DC pens ion scheme i s formulated O ur research broadened the attention from the risks in the financial markets to those outside the financial markets that are referred to as background risks Back ground va riables c an be t he i nvestor’s wa ge p rocess 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 o f st ochastic va riables i n a nalyzing t he portfolio problem By i nserting t he inflation r isk t hat a ffects the growth rate of an investor’s wealth, Menoncin derives an exact solution to the optimal portfolio problem when the financial 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 financial market is arbitrage free, incomplete, and continuously open over the investment time horizon [0, T], where T denotes the ter minal d ate o f ma nagement co ntract The i ndependent Wien er processes zr(t) a nd zm(t) represent t he i nterest rate r isk a nd ma rket r isk, respectively They a re defined on a p robability space (Ω, F, P), i n w hich P is the real-world probability and F = {F(t)}t∈[0,T] is the filtration that represents the information structure assumed to be g enerated by Brownian motion and satisfying the usual conditions © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 47 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 fixed income securities 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 t hat r(t) follows t he Vasicek model (1977) Under t he real-world probability measure P, the process r(t) satisfies the dynamics dr (t ) = a (b − r (t ))dt + σr dz r (t ) (2 1) where a, b, and σr are positive constants The short rate r(t) is mean reverting, implying that for t going to infinity, the expected interest rate would be close to t he va lue b Moreover, t he strength of t his attraction is measured by a There a re t hree i nvestment vehicles i n t he financial ma rket The first underlying asset is cash, S 0(t), which pays the instantaneous interest rate without any default risk and the price process is expressed as the following stochastic differential equation: dS0 (t ) = r (t )dt S0 (t ) (2.2) Then, t he st ochastic p rocess o f t he r olling bo nd f und BK(t) ( Rutkowski 1999) is as follows: dBK (t ) − e − aK = (r (t ) + σ KB λ r ) dt − σ KB dz r (t ), σ KB = σr BK (t ) a (2.3) where σKB den otes t he v olatility m easuring h ow t he i nterest r ate v olatility affects the bond λr represents the risk premium of interest rate risk The duration of BK (t) is fixed 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 The other risky asset is the stock index fund, S(t), whose dynamic process is given by © 2010 by Taylor and Francis Group, LLC 48 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling dS(t ) = (r(t ) + σSr λ r + σSm λ m )dt + σSr dzr (t ) + σSm dz m (t ) S(t ) (2.4) where σSr and σSm are positive, indicating that the volatility scale factors are affected by interest rate risk and financial market risk, respectively λm represents the risk premium of financial market risk in addition to interest rate risk 2.3.2 Background Risks We first ex press t he dy namic p rocesses o f t he back ground r isks w ithin the plan scheme The dynamic evolution of the aggregated labor incomes from c ontributions i s for mulated s ince t he e mployee m ust c ontribute a proportion of his or her labor income to the fund dL(t ) = µL (t )dt + σLr dzr (t ) + σLm dz m (t ) + σL dz L (t ) L(t ) (2.5) where σLr and σLm are the volatility factors that are affected by the interest rate and the market risk, respectively Moreover, σL ≠ is a non-hedgable volatility w hose ri sk s ource d oes n ot b elong t o z r an d z m This nonhedgable risk is called z L(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 Then, we i ntroduce t he other back ground r isk, t he i nflation r isk We use t he co nsumption p rice i ndex ( CPI) t o r epresent t he i nflation rate Hence, we present t he stochastic pa rtial d ifferential equation describing the evolution of CPI dP = µ π dt + σπr dzr (t ) + σπm dz m (t ) + σπ dz L (t ) P (2.6) Similarly, C PI p rocess i s a ffected 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) dW = dWN − WN © 2010 by Taylor and Francis Group, LLC dP P (2.7) 56 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Market portfolio State variable hedge portfolio Stock Bond −0.005 −0.01 0 10 20 30 Inflation hedge portfolio 10 20 30 Labor income hedge portfolio –0.2 0.8 Weight −0.015 −0.4 0.6 −0.6 0.4 0.2 −0.8 10 20 Investment horizon (year) 30 −1 10 20 30 FIGURE 2.2 Separation of four fund effects in optimal portfolio selections and their behaviors in each component over time (solid line, the weight of stock index fund and gray dashed line, the weight of rolling bond fund) find that the optimal investment decision is short-selling cash and buying stocks and bonds during the early period, and then decreasing the holdings of risky assets when the terminal date draws near Figure 2 c onfirms t he sepa ration o f f our f und effects i n o ptimal portfolio sel ections a nd t heir beha viors i n e ach co mponent o ver t ime The ma rket portfolio s shown a dec reasing t rend for stock i ndex a nd bond fund holdings due to the utility maximization principle In contrast, the state variable hedge portfolio shows a ste ady pattern for the optimal weight for bond f und a nd stock i ndex holdings To hedge t he r isk f rom state variables, the investment strategy needs to hold a fixed proportion of bond fund and also reduce the holding of the stock index In the inflation hedge portfolio, the investors are required to hold a high proportion of the stock index, up to 80%, to hedge the inflation risks, while only a small proportion of the bond fund is sufficient in the hedge portfolio However, in the labor income hedge portfolio, the investor should short-sell his or her stock index and the bond portfolio in order to preserve the salary uncertainty over his or her investment horizon © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 57 Stock −1 10 15 Bond 20 30 IHP LIHP MP SVHP Weight 25 −1 10 15 20 Investment horizon (year) 25 30 FIGURE 2.3 Weights o f s tock i ndex a nd b ond f or s eparated m utual f unds (bold solid line, market portfolio; solid line, state variable hedge portfolio; gray dashed line, inflation hedge portfolio; and gray dotted line, labor income hedge portfolio) In Figure 2.3, the weights of the stock index and bond in the entire optimal portfolios and the weights for the separated mutual funds are shown for illustration As can be seen, the inflation hedge portfolio constitutes the overwhelming proportion (75%) of t he optimal portfolios On t he other nd, when the time is close to the end of the investment horizon, the state variable hedge portfolio, market myopic portfolio, and labor income hedge portfolio play only minor parts in the optimal portfolio selection As for bond fund, these results indicate that the inflation hedge portfolio (around 35%) constitutes the largest proportion of all long-term financial portfolios In the beginning of the investment period, the myopic portfolio is the main proportion of bond fund However, the market myopic portfolio and labor income hedge portfolio play only minor roles in the optimal portfolio selection However, t he o ptimal w eights ( Equation 14) a re r elative t o t he fund wealth WN a nd t he labo r i ncome L Therefore, in each simulation (different market condition), the investment strategy is diverse We perform 10,000 s imulations t o find t he t rends o f o ptimal i nvestment po rtfolios Figure 2.4 displays the largest, medium, and lowest weights of cash, stock index f und, a nd rolling bond f und We find t hat t he f und ma nager s © 2010 by Taylor and Francis Group, LLC 58 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Cash −5 10 15 20 25 30 Stock Min −5 10 15 Medium Max 20 25 30 10 15 20 Investment horizon (year) 25 30 Bond Weight −5 FIGURE 2.4 Optimal portfolio holding distributions of cash, stocks, and nominal bonds given time horizon T = 30 years (bold solid line, minimum weights; solid line, medium weights; and gray dashed line, maximum weights) to short-sell cash i n e very ma rket s ituation a nd d uring t he e arlier per iod (T = t o T = 0), t he o ptimal w eights a re m ore v olatile The optimal decision suggests t hat investors should hold risky assets a nd decline t he weights with time Similarly, the optimal weights of stock index fund and rolling bond fund are more volatile during the earlier period In Figure 2.5, we present the investment trends of stock index fund and rolling bond fund in the market portfolio, state variable hedge portfolio, and s alary u ncertainty h edge po rtfolio The i nvestment p roportions o f the i nflation hedge portfolio a re not presented here because t he weights are constant and the values are the same as those in Figure 2.3 From these plots, we find that investors have to adjust the optimal weights of underlying assets according to different financial market situations For both the stock index fund and rolling bond fund, the volatility of market portfolio is larger than that of state variable hedge portfolio and labor income hedge portfolio However, these volatilities will further decrease when the time horizon is close to the terminal investment date A reasonable explanation is that the objective of market portfolio is to seek the highest return Hence, © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 59 Market portfolio (stock) Market portfolio (bond) 5 10 20 State variable hedge portfolio (stock) ×10−3 20 30 −0.012 10 20 30 −0.013 Labor income hedge portfolio (stock) −0.5 −0.5 −1 10 20 Investment horizon (year) Min 10 20 30 Labor income hedge portfolio (bond) Weight Weight 10 State variable hedge portfolio (bond) −1.5 −0.011 −1 30 30 −1 −1.5 Medium 10 20 Investment horizon (year) 30 Max Weight distributions of stock index and bond for separated mutual funds ( bold s olid l ine, m inimum w eights; g ray d ashed l ine, me dium w eights; solid line, maximum weights Left: the weights of stock index fund and Right: the weights of rolling bond fund) FIGURE 2.5 its results show much volatility related to the financial variation However, the other hedge portfolios have shown relatively stable property As se en i n Figure 5, t he optimal i nvestment weights a re i mportant and fund managers should not make the same allocation decision in every financial ma rket co ndition I nvestors co uld ad just t he p roportions o f underlying assets to optimize the expected terminal wealth according to our closed-form solution 2.5 ASSET ALLOCATION FOR GENERAL FORM In this section, we seek to obtain the solution for the investment problem in t he general form, t hat i s, e1 ≠ e2, i n order to i nvestigate t he financial impact d ue t o t he bo nus f ee a nd d ownside pena lty i n per formanceoriented arrangement Moreover, when e1 = e2 and p1 = p2 = 0, it means that the f und sponsors reward and pena lize t he managers at equal standard © 2010 by Taylor and Francis Group, LLC 60 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling However, the distribution of market return is not symmetrical; thus, the performance co ntract s t o se t d ifferent ben chmark r ates a nd pa rticipated rates of bonus fees as well as downside penalty 2.5.1 Optimal Investment Decision The optimal problem is reset as follows: dS dS dB ⎤ ⎡ dWN = WN ⎢(1 − w S − w B ) + w S + w B K ⎥ + γ dL S0 S BK ⎦ ⎣ ⎛⎡ dS dS dB ⎤ ⎞ − e1WN max ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p1 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ ⎛⎡ dS dS dB ⎤ ⎞ − e2WN ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p2 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ (2.15) Note t hat it i s a ssumed t hat t here ex ist i n t he i nvestment ma ndate t wo benchmarks p1 an d p2 i n t riggering t he bo nus f ees a nd d ownside pen alty This kind of structural setup is intended to be similar to the contract requirement, that is, according to the Taiwan labor pension fund management regulation, the return rate cannot be less than the interest rate of year fi xed deposit In this regard, when the fund performance is better than p1, the fund manager is entitled to get the bonus fee; otherwise, the manager is required to reduce the management fee due to downside penalty In our performance mechanism, the optimal problem contains two kinds of financial options and the explicit solution is sometimes hard to find Thus, in this section, the optimization method is employed to solve the problem In case (2.15), we find t hat t he f und manager takes greater risks in seeking higher return since the fund manager is required to guarantee t he m inimum r eturn r ate p2, wh ich sh ows t hat t he f und spo nsor need not worry about the downside risk of the investment Thus, the optimal a sset a llocation i s i nvestigated f rom t he v iew o f t he pens ion f und manager The optimal investment problem becomes ⎛⎡ dS dS dB ⎤ ⎞ dV = e1WN max ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p1 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ ⎛⎡ dS dS dB ⎤ ⎞ + e2WN ⎜ ⎢(1 − w S − w B ) + w S + w B K ⎥ , p2 ⎟ S0 S BK ⎦ ⎠ ⎝⎣ © 2010 by Taylor and Francis Group, LLC (2.16) Investment Decision in Defined Contribution Pension Schemes ◾ 61 where V denotes the fund surplus, that is, the fund manager is rewarded through obtaining t he bonus fee when t he f und per formance i s be tter than p1 On the other hand, the management fee of the fund manager has to be reduced due to the downside penalty when his or her performance is worse than the investment benchmark p2 This is a combined structure of a sset-based a nd t arget-based i ncentive m echanisms I n co mputation, the MATLAB® p rogram i s w ritten t o a pply t he p roposed o ptimization method i n computing t he optimal i nvestment weights i n Equation 16 In each scenario, 50,000 realizations are simulated and t he short-selling restriction is a lso employed The trade-off parameters e2 and the performance b enchmark p2 a re a ssumed t o be % a nd %, r espectively The investment time horizon in our illustration is set to be 10 years In Figure 2.6, the optimal multi-period investment strategies are illustrated given different e1 and p1 As seen in Figure 2.6a through c, the optimal p1 = 3% e1 = 0.7% 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 10 (a) Proportion of wealth 0.8 0.6 0.4 0.2 0 (d) (e) 10 (c) 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 10 10 (b) p1 = 5% 0.8 e1 = 0.5% p1 = 4% 10 Investment horizon (year) Stock Bond Cash 10 (f ) Optimal p ortfolio h oldings o f c ash, s tocks, a nd n ominal b onds given tim e h orizon T = y ears u nder i ncentive p rograms ( bold s olid l ine, weights of stock index fund; gray dot line, weights of rolling bond fund; gray dashed line, weights of cash (a) e1 = 0.7% and p1 = 3%, (b) e1 = 0.7% and p1 = 4%, (c) e1 = 0.7% and p1 = 5%, (d) e1 = 0.5% and p1 = 3%, (e) e1 = 0.5% and p1 = 4%, and (f) e1 = 0.5% and p1 = 5%) FIGURE 2.6 © 2010 by Taylor and Francis Group, LLC 62 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling weights with increasing p1 under e1 is 0.7% In Figure 2.6d through f, e1 is 0.5% and p1 is rising from 3% to 5% The bo ld so lid l ine r epresents t he optimal investment weights of t he stock index f und The red dashed line denotes the proportion of fund in cash The third line illustrates the weights of rolling bond fund First, Figure shows t hat t he f und ma nager w ill i ncrease t he holding of stock index f und as t he investment horizon approaches maturity A p robable ex planation i s t hat u nder t he o ptimal i nvestment dec ision, the performance of fund will be better than the benchmark; hence, the manager increases the risk of portfolio in order to seek higher bonus fees Comparing Figure 2.1 with Figure 2.4 shows that the optimal investment strategy i s t o h old m ore r isky a ssets s ince t he w eights i n t hese figures exceed 100% However, in Figure 2.6, we set t he short-selling constrain; thus the holding weight of stock index fund is close to at maturity date Second, the weights of cash are small because its return rate is low Fund managers w ould n ot p refer t o h old c ash bec ause t hey ve t o m eet t he requirement of minimum guarantees p2 However, Figure 2.6 shows certain diverse characteristics of the investment behaviors In Figure 2.6a through c, the weights of stock index fund (bold solid line) in the beginning are decreasing from 0.88 to 0.58 when p1 increases On the other hand, the allocation in Figure 2.6d through f show that when e1 decreases to 0.5%, the fund manager would hold less stock index fund from 0.48 to 0.62 in the beginning with increasing p1 This is interesting that the settlement of bonus fees would affect the investment behaviors o f f und ma nagers I n o ther w ords, t his p henomenon i mplies that the settlement of incentive mechanism contract is important on the delegated management contract because the parameters would affect the investment behaviors of fund managers 2.5.2 Financial Implication In Section 2.4.3, the optimal investment decisions are simulated under certain constrains (e1 = e2 = e and p1 = p = 0) The f und ma nager w ill increase t he h olding i n r isky a sset U nder t he d ownside p rotection arrangement, a fund holder will tend to increase the risk profi le of fund portfolio I n Section 5.1, m ore r ealistic per formance m echanisms including e1 ≠ e2 an d p1 ≠ p2 a re i nvestigated M oreover, R aghu e t a l (2003) conclude that there exists the agency problem between the fund managers and the plan participants In Section 2.5.1, we try to investigate the financial influence of performance mechanisms on the optimal © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 63 investment d ecisions The a sset a llocation p roblem i n Section 4.3 i s simplified t o d erive t he e xplicit s olution The o ptimal so lution va ries according to t he va rious scenarios of t he fi nancial ma rket In order to explore t he realistic impact of t he incentive mechanism, t he optimization p rogram i s i mplemented t o a pproximate t he o ptimal i nvestment weights through simulations For t he DC pens ion f und ma nagement, t he se tup i n Section 5.1 i s more p ractical a nd v ital The o ptimal i nvestment dec isions o f f und managers u nder per formance m echanisms a re i nvestigated O ur m odel extends the previous research through implementing the unlimited liability downside protection The unlimited liability is incorporated since the limited mechanisms would motivate the fund manager to increase the risk profile of portfolio after a period of poor performance (Edwin et al 2003) Moreover, t he labo r pens ion p lan i mplemented i n Taiwan i ncludes a lso this kind of guarantee arrangement We find that under this performance contract, the benchmark rate (p1) and the participated rate (e1) of bonus fees would change the investment behavior of the fund manager This is consistent with the conclusion made by Raghu et al (2003), that the performance would be influenced by the commission rate That is, the participated rate of bonus fees would affect the investment behaviors However, Raghu et al (2003) propose that the efficacy of limited incentive is better than unlimited contract Moreover, t he optimal i nvestment dec ision i s a nalyzed a nnually The performance of fund is measured and the fund managers are also rebalancing their asset class every year Mark (1987) discovers that the length of the time horizon is very crucial During the shorter period, the performance contract would not identify whether t he success of t he f und performance is the true investment ability or pure luck of the fund manager Lawrence a nd S tephen (1987) a lso su ggest t hat t he proper per formance index should employ the moving year time period 2.6 CONCLUSION In this study, we investigate the asset allocation issue for DC labor pension f und t hat considers not only t he ma rket r isk a nd i nterest r isk but also t he u ncertainties f rom labo r i ncomes, t he i nflation r isk, a nd t he incentive scheme We fi nd that if the fund manager would like to maximize t he expected exponential utility of his or her terminal wealth, he can adopt the mutual fund separation theorem through five components in its optimal asset allocation Hence, the optimal investment behaviors © 2010 by Taylor and Francis Group, LLC 64 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling of the pension fund managers are characterized by the relative weights among t he sepa rated m utual f unds acco rding t o t heir p reference, t he fi nancial market, and the influential factors With both the financial a nd background risks incorporated, pension fund ma nagers a re recommended to consider t he short-term f und performance a nd t he h edge r equirements s imultaneously B ecause back ground risks cannot be controlled by fund managers, a comprehensive dynamic framework is formulated to describe the decision-making process A s t he r esults sh ow, t he dy namic po rtfolio o f t he r estricted f orm that ma ximizes t he ex pected u tility o f t he p lan pa rticipant co nsists o f five components: the market portfolio, the state variables hedge portfolio, the inflation hedge portfolio, the salary uncertainty hedge portfolio, and cash By solving explicitly the optimal portfolio problem, the numerical results i ndicate t hat t he i nflation h edge po rtfolio co nstitutes t he o verwhelming proportion of stocks in the optimal portfolios In addition, the inflation hedge portfolio and the state variable hedge portfolio constitute the overwhelming proportions of bond holdings This shows t hat longterm investors should hedge inflation rate risk by holding the stock index In addition, these investors should respond to the inter-temporal hedging demands in the financial markets by increasing the average allocation to their bond fund To understand the roles of these components, it is necessary to explore the economic interpretations by solving the dynamic optimization problems With respect to the most common approach used in the literature, the i ncorporation o f t he labo r i ncome a nd i nflation r isks a llows u s t o characterize the general pattern of the optimal strategy The results indicate that the inflation hedge portfolio constitutes the main proportion of the optimal stock portfolios, while in the earlier stage the market portfolio makes up t he larger part of t he stock index f und However, in t he labor income h edge po rtfolio, t he i nvestor sh ould sh ort-sell h is o r h er st ock index and the bond portfolio in order to preserve the salary uncertainty over his or her investment horizon Finally, t he optimal a sset a llocation st rategy i s solved for t he general incentive mechanism The optimal behaviors of the fund managers alter according to various parameter settings within the incentive mechanism Our results are also consistent with the findings of Richard and Andrew (1987) and Lawrence and Stephen (1987), who confirm that the incentive setting is essential in the delegated management contract (Lawrence and Stephen 1987, Richard and Andrew 1987) © 2010 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 65 APPENDIX A H* is as follows: H * = µ′ν JW + JW [WN (r − re − µ π ) + γLµ L − ( A′ + B′)Γ ′(ΓΓ ′)−1 M] + tr(Ω′ΩJ νν ) − J ( J W )2 M ′(ΓΓ ′)−1 M − W M ′(ΓΓ ′)−1 ΓΩJ νW JWW JWW + ( A′ + B′)( I − Γ ′(ΓΓ ′)−1 Γ)ΩJ vW − 1 J ν′ W ΩΓ ′(ΓΓ ′)−1 ΓΩJ νW + JWW ( A′ + B′)( I − Γ ′(ΓΓ ′)−1 Γ)( A + B), JWW (2.17) where we denote A = FNΦ, B = γLΛ, and that I is the identity matrix Then, substituting Equation 2.13 into Equation 2.17, we obtain J(t;W, v) ht + H* = 0,h(T, v(T) ) = a nd after dividing by J, we can write Equation 2.17 in the following way: = ht + µ ν′ hν + UW ⎡WN (r − re − µ π ) + γLµ L − ( A ′ + B ′)Γ ′(ΓΓ ′)−1 M ⎤ ⎦ U ⎣ 1 (UW )2 (U )2 M ′(ΓΓ ′)−1 M − W M ′(ΓΓ ′)−1 ΓΩhν + tr(Ω ′Ω(hνν + hνhν′ )) − UWWU 2 UWWU 1 (UW )2 hν′ Ω ′Γ ′(ΓΓ ′)−1 ΓΩhν + (A ′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )Ωhν − 2 UWW U + UWW ( A′ + B ′ )(I − Γ ′(ΓΓ ′ )−1 Γ )( A + B) U We h ave t hat UW U is β2 and (UW )2 UWWU is 1, and then substitute these two values into the above equation Therefore, the HJB equation can be written as follows: = ht + [µ′ν − M ′(ΓΓ ′)−1 ΓΩ + β2 ( A′ + B′)(I − Γ ′(ΓΓ ′)−1 Γ )Ω]hν + tr(Ω′Ωhνν ) − hν′ Ω′Γ ′(ΓΓ ′)−1 ΓΩhν + β2[WN (r − re − µ π ) + γLµ L − ( A′ + B′)Γ ′(ΓΓ ′)−1 M ] 1 − M ′(ΓΓ ′)−1 M + β22 ( A′ + B′)(I − Γ ′(ΓΓ ′)−1 Γ )( A + B) 2 © 2010 by Taylor and Francis Group, LLC 66 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling This kind of partial differential equation can be solved using the Feynman– Kac t heorem, a nd so w e can find t he f unctional for m of h(v;t), which is ⎡ T ⎤ given by h(ν; t ) = Ε t ⎢ g (v(s), s)ds ⎥ , where ⎣ t ⎦ ∫ dν(s) = [µ ′ν − M ′(ΓΓ ′)−1 ΓΩ + β2 ( A ′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )Ω]′ ds + Ω(v(s), s)′ dZ ν(s) = ν(s), g (ν(t ), t ) = [WN (r − re − µ π ) + γLµ L − ( A′ + B ′)Γ ′(ΓΓ ′)−1 M ] − 1 M ′(ΓΓ ′)−1 M ] + β2 ( A′ + B ′)(I − Γ ′(ΓΓ ′)−1 Γ )( A + B) β2 Finally, the optimal portfolio is written as follows: 1 1 (ΓΓ ′)−1 M − (ΓΓ ′)−1 ΓΩ ⋅ β2 WN (1 − e) β2 WN (1 − e) wG* = − − ∫ T t ∂ Ε t [ g (ν(s), s)]ds ∂ν 1 (ΓΓ ′)−1 ΓΦ − γL(ΓΓ ′)−1 ΓΛ WN (1 − e) 1−e APPENDIX B Now we are interested in the second component wG*(2) of the optimal portfolio, which is the state variable hedge portfolio We follow Battocchio and Menoncin (2004) t o d erive wG*(2) S ince t he ter m M′(ΓΓ′)−1 M es not rely on the state variables, this term is deleted We rearrange wG*(2) as the following equation: wG*(2) = (ΓΓ ′)−1 ΓΩ⋅ WN (1 − e) ∫ T t ∂ Et [Q1 + Q2 ]ds, ∂ν where Q1 = [WN (r − re − µ π )] − WN Φ′Γ′(ΓΓ′)−1 M + β2WN2 σ2π Q2 = γLµ L − γLΛ′Γ′(ΓΓ′)−1 M + β2[−2γLWN σL σ π + γ L2 σ2L ] © 2010 by Taylor and Francis Group, LLC (2.18) Investment Decision in Defined Contribution Pension Schemes ◾ 67 Next, we derive the derivative of the last term in Equation 2.18: ⎡ ∂ ⎤ ⎢ ∂r(t ) Et [Q1 + Q2 ]⎥ ⎡ (1 − e)WN (t ) ⎤ ⎢ ⎥=⎢ ⎥ 2 −1 ⎢ ∂ E [Q + Q ]⎥ ⎢⎣γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t ) + β2 γ σ L Et [L]⎥⎦ t ⎢⎣ ∂L(t ) ⎥⎦ In t he above equation, we have to compute t he ex pected va lue of t he ∼ modified process of labor incomes, Et[L], which is called the modified real contribution First, we need to compute the following matrix product: −1 −1 ⎡ ⎤ ⎣ − M ′(ΓΓ ′) ΓΩ + β2 (WN Φ ′ + γLΛ ′)(I − Γ ′(ΓΓ ′) Γ )Ω ⎦ ′ For simplicity, we assume that Γ′(ΓΓ′)−1Γ = I Then, the above equation is equal to the first term, and we can write it as ′ ⎡ w1 ⎤ ⎡⎣ − M ′(ΓΓ′)−1 ΓΩ ⎤⎦ = ⎢ ⎥ ⎣ Lw2 ⎦ where w1 and w2 are given by w1 ≡ σr λ r , w2 ≡ −2σ Lm σSr λ r σSm − σ Lm λ m + σLr λ r Thus, we can get the modified differential of the state variables ∼ v (s) as follows: ⎡ dr ⎤ ⎡a(b − r ) − w1 ⎤ ⎡ σr dt + ⎢ ⎢ ⎥=⎢ ⎥ w ⎣ σLr ⎣⎢dL L ⎦⎥ ⎣ µL − ⎦ σLm ⎡ dz ⎤ ⎤⎢ r ⎥ dz σL ⎦⎥ ⎢ m ⎥ ⎢ dz L ⎥ ⎣ ⎦ In particular, for s < t, the solutions of the interest rate process and the modified labor income process are r (s) = r (t )ea(t − s ) + ab − w1 (1 − ea(t − s ) ) + σr e −as a s ∫e aτ t dzr (τ) ⎡⎛ 1 ⎞ L(s) = L(t )exp ⎢⎜ µ L − w2 − σ2Lr − σ2Lm − σ2L ⎟ (s − t ) ⎝ 2 ⎠ ⎣ ⎤ + σLr (zr (s) − zr (t )) + σLm (z m (s) − z m (t )) + σL (z L (s) − z L (t ))⎥ ⎦ © 2010 by Taylor and Francis Group, LLC 68 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Then, according to the boundary equation (v∼(s) = v(s) ), we can obtain ∼ the expected value Et[L (s)] = L(t)eR(s−t), where 1 ⎞ ⎛ R(s − t ) = ⎜ µ L − w2 − σ2Lr − σ2m − σ2L ⎟ (s − t ) ⎝ 2 ⎠ + σ Lr (z r (s) − z r (t )) + σ Lm (z m (s) − z m (t )) + σ L (z L (s) − z L (t )) Thus, the integral term of wG*(2) in the optimal portfolio becomes ∫ T t (1 − e)WN (t ) ⎡ ⎤ ⎢ ⎥ −1 ∂ Et [Q1 + Q2] ds = ⎢ γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t )⎥ ∂ν ⎢ ⎥ 2 R ( s −t ) ⎢⎣ + β2 γ σ L L(t )e ⎥⎦ In the end, we could get the solution of wG*(2) as wG*(2) (1 − e)WN (t ) ⎡ ⎤ ⎢ ⎥ − (ΓΓ ′)−1 ΓΩ⋅ ⎢ γ µ L − γΛ ′Γ ′(ΓΓ ′) M − β2 γσ L σ πWN (t )⎥ = WN (1 − e) ⎢ ⎥ 2 R ( s −t ) ⎢⎣ + β2 γ σ L L(t )e ⎥⎦ 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Menoncin (20 02) are employed in our model Brinson et al (1991) have shown convincingly that the allocation of investment © 20 10 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution. .. investment horizon © 20 10 by Taylor and Francis Group, LLC Investment Decision in Defined Contribution Pension Schemes ◾ 57 Stock −1 10 15 Bond 20 30 IHP LIHP MP SVHP Weight 25 −1 10 15 20 Investment horizon