Optimal Pension Funding Incorporating Stochastic Simulations and Dynamic Programming

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(1)~~~~~~ M~~ M-M(~~A+A~) Asia Pacific Management Review, Vol.4. Optimal Pension Funding Incorporating. Stochastic Simulations and Dynamic. Programming *. 4mM_~_~~A~~~~~_~~~~_ 5*- ±1tt Shih-Chieh Chang ~ iLjEi~~ *-*JRF1t*J£gH~F1t* ~. Department of Risk Management and Insurance, National Chengchi University. .fiij. ~. *-XFJTJtiLzi!1:f.£~M.:#UjUt ~ ~JtJHti;-fh ~:{ti!1tJl'.~. , JlJ YA i.k:Jt l!l Jt~1fi!1:f.it~~ .It ...... ~ ••• _oM*.~~JlJ~8~*,~~~.Jt~~i!1:f.£~~~~*.*~.*~.~~~JlJ~~ ~M~1hhR.. 0 #*-~1i:: 'f '. M.~~~lJ ~~JtJlJ##iil~:{ti!1tlt1~~nt. '. £#M.ifU~Jit~AjJ ~t~afJf~Jt. ~m~.#T'~JlJ.~~~~~**~:{ti! ••• _'~~~~~.AIi!1:f..~~~AT~~ff~ ~oX'f.~~~~.~Jt.~~.a#~fJf~~M*o. Hm~:i!1:f.£~~~'M.~~~'.~~~o. ABSTRACT A simulation-based mechanism combining dynamic optimization is constructed to decide the optimal funding policy of the defined-benefit pension scheme. The results show a significant advantage and flexibility of this approach in projecting the optimal financial status over the traditional deterministic pension valuation. In this study, the valuation mechanism is formulated as a stochastic control system and the optimal funding strategies are calculated through dynamic programming under the simulated workforce and specified constraints. Taiwan public employees retirement system is studied for illustration. This article outlines the procedure of constructing the proposed mechanism and comments on the empirical findings. Keywords: pension valuation; stochastic control; dynamic programming .. • The author wishes to thank research assistant Hsin-yi Cheng for carrying out the programming works and the kind assistance from Tai-PERS Supervisory and Management Board..

(2) 2. I. Introduction Some traditional problems in pension valuation are conveniently viewed in terms of dynamic and stochastic processes. We assume that, at any time, each plan participant is in one of a number of plausible working statuses. Changing working status of the plan member may have direct financial impact on the expected of future cash flows. In a recent study (Frees et aI., 1997), a forecasting model has been constructed to assist policy-makers in projections of the Social Security system in United States. This forecasting model emphasizes on using statistical techniques to explore characteristics of the data. Based on the forecasts of plan assets, the policy maker could monitor the plan financing and amend the funding policy. In this study, a different perspective based on the stochastic simulations of the workforce and the optimal funding strategy are considered. Two types of risks concerning the stability and security of funding are introduced (see Haberman and Sung, 1994): the contribution rate risk and the solvency risk, which have characterized the trade off in plan decision process. A performance measurement associated with these two risks was then constructed to derive the optimal contributions subject to specific constraints through dynamic optimization. Based on the optimal funding policy, the policy-maker could attain the plan's target fmancing status by amending the funding schedule. In this study, the similar methodology is adopted and a solvency monitoring technique combining stochastic simulations and dynamic optimization is proposed to decide the optimal funding strategy of the defined benefit pension scheme. The accrued liabilities are estimated based on the realistic plan members' turnovers, while the plan investment performance is projected according to its stochastic returns. The results provide significant value added to the traditional pension valuation. Furthermore, stochastic simulation is used to explicitly characterize the participants' population based on its turnover and the investment return adjusted by its asset return associated with this scheme. The contribution rate risk and the solvency risk are minimized through dynamic optimization. Traditional valuation techniques, however, can merely examine the plan status at a specific valuation date.. Stochastic simulation and dynamic optimization are two of the best tools in policy-making, which can help the plan sponsor amend the benefit scheme or funding policy. Stochastic simulations using time as the operational parameter in Bacinello (1988) are performed to obtain the best estimates of the projected workforce, while the projected cash flows are scrutinized through dynamic simulations. An extensive review of past pension cost analyses can be found in Shapiro (1985). Since pension plans are almost subject to some degree of interventions in their financial consideration. This has been much more the case since the substantial legislation was enacted in recent years. The proposed projection techniques can provide valuable inputs for planning decision. This study emphasizes the importance of optimal control applied to pension research in optimizing the plan financial condition. Similar researches can be found in: Bowers et al. (1982), O'Brien (1986, 1987), Dufresne (1988,1989), Haberman (1992,1993,1994), Daykin et al. (1994), Haberman and Sung (1994), Haberman and Wong (1997), Cairn and Parker (1997), Runggaldier (1998), Sch I (1998) and others. A discussion of a rigorous and tractable stochastic model for pension fund can be found in Janssen and Manca (1997). Our proposed pension financing basically involves projecting a series of cash flows for the future years through stochastic simulations according to the probabilistic experience set of actuarial assumptions. Then the optimal contributions are estimated subject to the specific performance measure. A brief summary of the advantages of this approach is listed below: 1. Through high-speed computers, the plan sponsor can forecast the plan's future cash flows and the turnover of the future workforce, which helps the fund management. 2. The optimal contribution can be estimated under systematic scenario based on specific plan investment and recruiting strategies. 3. The optimal funding strategy and funding status of the plan can be estimated under specific performance measurement implemented through a computerized system. 4. Running an extensive set of scenarios will shed light on the interaction between the plan liability and the investment performance.. a.

(3) ~~:t~. LTe IIp ng. : Optimal. Pension Funding Incorporating Stochastic Simulations and Dynamic Programming. 5. This approach will produce optimal solutions and is capable of inputting specific funding intervention.. he ire. II. Data Description. :ed iU'e. An be iU'e. leir the in :an nal the be. t7), 94), /4), 97), A stic and :ing for ons of ons nce this. In this research, the Taiwan public employees retirement system (Tai-PERS) is studied and evaluated through the proposed dynamic optimization. In reality, fluctuations of the inflation rates, rate of returns, rate of increase in salaries and demographic factors subjected to recent economic conditions need to be taken into account when considering cost allocation and projection of the plan. Due to recent global financial crisis and the highly uncertainty of future economic predictions in the financial market, fund investment strategy and asset-liability related topics have taken on new significance and much attention has been focused on the implementation of a better monitoring system to oversee the entire pension. 3. risks. Tai-PERS is a large public retirement system that is designed to provide retirement and ancillary benefits to all government employees. There are 271,215 active members under the current benefit scheme. The present funding policy requires each member to contribute 2.8% of his covered salary while the sponsor of the plan contributes 5.2% of the plan member's covered payroll monthly to a public trust fund. A sample of 3,823 participants is used to evaluate the performance of the proposed approach. The average age of employee in our sample is 42.99 years and the average year of service is 15.6 years. Generally speaking, the workforce structure of these samples were relatively older than the overall population. Accordingly, the contribution rates obtained from the sample vary from that for Tai PERS. The statistical summary of the sample is listed below.. Table I. Basic Statistics of the Sample. 1sor the the Ider ~Ian. of :ific gha IS. shed lility. This system is a contributory defined benefit plan since the participant's retirement benefits are calculated according to the length of his service and his final salary upon retirement. To protect the retirement benefits against post-retirement inflation, the retirees have the option of a lump-sum retirement payment, monthly pension with the cost of living adjustments, or a mix between lump-sum payment and monthly pension.. A service table, which reflects all causes of member's turnover, is constructed based on the experience data collected from July 1, 1995 to June 30, 1996. Owing to the limitation of the data collection, further updating this table is necessary. In Section 3, the proposed procedure within this framework is formulated. In Section 4, the optimal contributions are estimated under the given performance measurement and the associated.

(4) 4. stochastic cash flows are performed. Section 5 presents the model validation justifications and the fmal results of optimal plan status of Tai-PERS. Concluding comments are given in Section 6.. III. The Dynamic Procedure A simulation-based optimization is proposed to assist policy-makers to evaluate the financing efficiency of the plan. Since management of the dynamic cash flows is critical to financial soundness of the plan, the model has emphasized on two elements that significantly affect the plan's fmancial balance. One is the annual optimal contribution of the plan members and sponsor, and the other is the fund actuarial status for the active members and pensioners. Every year, the cash inflow from plan members, plan sponsor and the investment returns should sustain the annual normal cost of this plan. Complexity of the actuarial valuation is sometimes viewed as another element of cost. Hence, the proposed simulation-based optimization procedure is. built into a user-oriented computerized system to reduce duplication of efforts. We use the Visual Basic 5.0 program in the calculation. Implementing the proposed approach into a computerized system can achieve more efficient policy-making. A dynamic approach is proposed to decide the pension funding subject to specific constraints. Under certain risk criteria given in the estimation, an optimal contribution can be obtained. In our study, the size constrained population assumption is used to project the plan workforce. In the open system, to complete the specification of the model we must assume how new entrants are allocated to various states and age intervals (see Bartholomew, 1982). In our study, the new entrants are placed in the lowest job rank. The future plan workforce are predicted based on a size constrained assumption. Based on the previous experience of the new entrants into the system, a simplified dynamic model is assumed in this study to perform the simulations. The recruitment distributions of these new entrant are given as follows.. Table 2. Recruitment Distribution age. 20-24. 25-29. 30-34. 35-39. 40-44. percentage. 20%. 60%. 18%. 1%. 1%. Based on the given assumptions, the projection. shown in Figure 1.. of workforce demographics is given in Table 3 and. Future workforce for active membeu ( ] 00 Simulltion_). " ---6-Ap(7' pa-centile) .a····"At:e<'Oporoenlile) ........ Ap(l' pCI"!:enlile). .,. .. Figure 1. Workforce Projection.

(5) : Optimal Pension. ~~±~. Funding Incorporating Stochastic Simulations and Dynamic Programming. 5. Table 3. Workforce Demographics Future workforce between 1998-2017 active employees. year. number. 1998 1999. 3751 3750. age (50 percentile). 40.30. 101. 2.68%. 41.61. 116. 3.\0% 3.51%. 2002. 3754. 42.91. 132. 2003. 3754. 42.20. 147. 3.90%. 2004. 3756. 43.50. 162. 4.31%. 2005. 3754. 44.79. 176. 4.69%. 2006. 3741 3739. 44.16 45.60. 196. 5.25%. 217. 5.80%. 2009. 3731 3737. 46.52. 241 261. 6.99%. 3740. 47.95. 278. 7.44%. 3741. 47.44. 2012. 3726. 47.00. 295 317. 8.50%. 2013. 3710. 345. 9.29%. 2014. 3708. 48.65 48.47. 371. 10.02%. 2015. 3684. 48.48. 407. 11.05%. 2016. 3691. 48.69. 438. 11.85%. 2017. 3681. 47.07. 471. 12.81%. contribution at time t;. risk weighted ratio at time t;. {NC,} 'EN. =. {F, }'EN. pension fund asset at time t;. 11. =. =. 6.47%. 2010. {VI} 'EN = discount factor at time t;. {AL'}'EN. 45.07. 2011. {i,} 'EN = return rate of pension fund at time t;. =. 2.15%. 3746. N = the set of chosen time frame;. =. 1.62%. 61 81. 40.98. 3755. The actuarial notations used in stochastic simulation and fund recursive relationship are given as follows:. {[3,} 'EN. 39.67. percentage by active employees. 2000. 2008. =. number. 2001. 2007. {C,} 'EN. pensioner. normal cost at time t;. accrued liability at time t;. target fund ratio.. The procedure of our approach is constructed as follows:. 7.87%. '"",. ""[. ""~. "~. 1. ~. of dynamic processes using Bernoulli trials. The working status of each member is simulated according to the decrement probabilities from the service table. Let Ex., denote the working status of the employee age x between t and t+ 1 year in the future. The process is shown as follows.. (a) Simulate Ex,! by generating a pseudo-random number from Bernoulli (p;t», where p;t) is obtained from the service table.. (b) If Ex,l. =1. (i.e., this member is in working. status), then go to step (e).. (c) If Ex". =0. (i.e., this member is not in working. status), then a pseudo-random number of multinomial distribution is generated from Multi. q~d). q~'). q;') q;'). (-W-'w'w'w) qx. qx. qx. where. qx. q;') = q;d) + q;n + q;/) + q~'). The superscript (i). Step 1: The future information ofthe active and inactive members in the system is simulated through a series. stands for disability; (d) for death; (I) for layoff and withdrawal and (r) for retirement. Let Rx, denote the living status of the retiree aged x at.

(6) 6. time t and Bx., denotes the option of his retirement benefits. If Rx.,. =. Step 2:. 1 (i.e., the member. The contributions are assumed to be paid at the beginning of each year in proportion to the covered payroll of each active employee. Constant rate is paid by the active employees, whereas the remaining part is paid by the government. The major cash flows are defined as following: Benefit payments B, including, disabled, death,. is retiring), a pseudo-random number Bx., is generated from Bernoulli (bx ). to simulate the. chosen benefit program where hx., is estimated from past experience. If Rx., = 0, the member is dead and the procedure ends. (d) If Bx., = 1 (i.e., the retiree chooses the monthly annuity), the living status. F.,I. layoff, retirement benefits and withdrawal refund.. (also see Chang, 1998). Contributions C, from active employees, new. is generated from. Bernoulli( qx+,) where qx+, is estimated from the. entrants and the employer.. Returns R, from pension fund and the accumulated. post-employment life table of the retiree. If Bx., = 0 (i.e., the retiree choose the lump sum. fund asset F,.. payment), the benefit payment is then computed.. (e) Let. The Monte Carlo simulations can be carried out by. repeating Step 1 and 2. The flow chart of the open. group simulations is shown in Table 4.. t = t + 1.. Table 4. Flow Chart of the Open Group Simulations Turnover simulations of active and inactive members in the valuation year.. ---.. Adding new entrants according the number of turnovers in the valuation year.. Calculating projected. ---+ cash flows.. I. T the next year. the year simulated from Step (2) and plus the investment return:. Step 3: Based on the projected benefits, C, is estimated through minimizing the performance measure defined in Equation (1) from present to time T - 1 . Detailed clarification and .explanation of this performance measure will be discussed in Section 4.. F I + l = (F, + C I. -. B I )(1 + i l. ),. i l oc N(9,cr 2 ). where the real return i l between time t and. r =. (CO,..,CT _ I ). E{Irv,(I-S.-)2 + V'+IP'+I(1- ~)2]} NC, T]AL'+I. (1). ,=0. During the year between time t and t + 1 the fund balance will increase by the contribution C, and decreare by the benefit outgo B, at the beginning of. (2). t. +1. are assumed to be a sequence of independent and identically distributed normal variables with mean e and variance cr 2 • The return rates used in computing the accrued liability are simulated based on the model updated by the past investment performance. In this study, a simplified approach is used to model the flat term structure. For valuation and funding.

(7) ~~:t:~. : Optimal. Pension Funding Incorporating Stochastic Simulations and Dynamic Programming. purpose, relative conservative and optimistic assumptions are employed to investigate the financial impact on the future pension cost.. IV. The Model. 7. financial condition of the plan each year. Since bias may exist owing to variation in various realizations, we summarize the statistics of these estimates based on the simulations. For a particular j-th simulation, the projected benefit at time t+ 1 is denoted by j B,+I . From these simulations, a median estimate JBI + 1 is. After the results from these simulations are obtained, the benefit payments are estimated and recorded in the database. In our approach, time is assumed to be the operational parameter and the above steps are repeated as the fund development in time. We intend to forecast dynamically the overall. used to estimate the contribution at time t+ 1. The cash flows of the benefit payments, normal costs and the accrued liability in 20-years are projected. Then the optimal contributions are computed through the optimization procedure.. Tumovcrstatistk:s (mean) ilr 1998 to 2017 (open gmup simulations m- 100). (person) 180. 160. 140. 120. 80. lJ. rrr{[JI. 100. fffl111f'. 60. -:il. 40. -. 20. f[i ~. !. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Figure 2. Turnover Statistics. In Figure 2, the averaged number of turnover using 100 simulations is shown. The number of the retiree has an increasing trend after year 2010, while the number of withdrawal for other causes remains steady. In order to determine the parameters that best fit the data, a performance measure, J'T-I' is defined to measure the performance of the estimation between time 0 and time T-1. The performance measure is a nonnegative function based on the sum of weighted discounted squared deviations. The parameter estimates are determined by minimizing the combined performance measure in Equation (1). VI (1. -. '-s'-)2. NC. rate risk, while. is used to denote the contribution. I. VI+I~I+I(l- ~+I 1+1. )2. is used to denote. the solvency risk where. ~I+I. is the relative weight in. measuring the fund financial stability at time t. The relative ratios are used in measuring the discounted quadratic deviation over the chosen time horizon. The advantage of using this criterion is in decision making process, which is clearly requisite in predicting the performance ratio by years. Then the optimization could be formulated as: min. CU""CT. or. T _1. I. T-I. (3). =. C. min E~)v (1- -'-)' + v ~ -, .. 0 3 NC s HI HI. CU"",C T ' I. F. (1- _'_+1_)' I H. ..,AL.r+'. 1 I. where HI is the cr - field generated by {F;" ... ,F,}. {F,},.N are assumed to follow a first order Markov. process, which is the case that Tai-PERS internally.

(8) 8. evaluate its fmancial status annually. Then. (4). boundary condition in the above equation which means no bias at time t = T . That is, the long term financial status of this pension plan is assumed to have no risk incurred. Then we could solve the recursive relationship for a, (t) and a2 (t) for. o.:;, t .:;, T . Given By defming V,(F,) as:. J{(Jf)=. C. T-I. ~Y+v ~ (1. minVGv(l. c;,c;.:'''''~ S. s+II-1+1. ---;:;(;. F s+1 ¥IF,]. 1)4~/. (5). '. 2v 2v ~ K =__ 1 + 1+11-'/+1. F we have V,(F,) =. 1. TI 2AL2 1+1. +2a (t+l)K 1. '. the recursive relationship for al (t) and a2(t) are. . C mmE{v (1- - ')2 + V c,. lHC 1. lV'. F. A (I __'_+'_)2 IF} 1+,1-'1+' ~AT I. NC. '. (6). solved by the following equations.. I~~+l. I. + E{V,+, (F,+,) I F,} then C, can be estimated by induction, To solve the. (8). (Hamilton-Jacboi-) Bellman equation (for example, see Merton, 1990 or Fleming and Rishel, 1975), we assume that V, (F,) = a,(t)F,2 + a2(t)F, + a3(t) for all Equation (6) could be written as reC,). t,O':;' t':;' T.. which is a second-order function of C,. Then the optimal contribution condition a (t + 1) >. ,. C,. satisfying the unique VI. «(1 + 9)2 + cr 2)NC/. + vl +,!),+, (T]ALI+Y. is obtained in Equation (7).. c =[2v,. + 2~+II3I+IH _ 2( ~+II3/+1. NC;. I. flAI;+1. v. The Results and Analysis. Tl2 A~+I. + ~(t + 1»K(F, - B,) -Clz(t + 1)H]. (7). 2v 2v~K /[_1 + 1+11-'/+1 + 2~(t + 1)K] NC; Tl2 AEI+I where. We. set. a,(T) = a2(T) = a3(T) = 0. as. the. We now use Tai-PERS to illustrate the results. As stated in previous sections, Tai-PERS presents a challenge for the pension actuary because of the complexity of its possible cash flow. There are certainly many directions in order to perform a reasonably realistic description of the actual behavior of a pension plan. Due to the specific and complex statutory requirements of Tai-PERS, we shall consider generalization where the return rate is described by a stochastic process. The actuarial.

(9) ~:t~. : Optimal. Pension Funding Incorporating Stochastic Simulations and Dynamic Programming. factors are reduced in performing the pension valuation. This restriction is more a reflection of our interest than an ambition to cover each detail. So the simplified procedure aims to reduce the computer run times and the outputs of the financial information stored in the database. Thus if we assume that: Population: Tai-PERS service table based on 1995 1996; 1989 Taiwan Standard Ordinary Experience Life Table (1998) is used for the retiree's life table; Actuarial Cost Method: Entry age normal (EAN) cost method (see Anderson, 1992); Salary scale and inflation rate: 3.5% for annual salary increase and 3% for annual inflation rate; Interest rate: a flat term structure model is assumed in discounting the cash flow. 8% is given for we assume performance valuation, i.e., v,. = v' = (1.08f/, i.e., a constant discount rate. assumption; Target Fund ratio: 1'] = 100% for every year; Risk measurement weight:. ~,=. 9. Fund return rate: e = 10% and 0'2 = 0.04% . The actuarial factors are given above III performing the optimization. This restriction is more a reflection of our interest than an ambition to cover each possible scenario. So the given values aims to illustrate the results based on our baseline estimates. With the above assumptions, the estimated actuarial accrued liabilities, normal costs and benefit payments are simulated based on Steps 1 and 2. We assume that the fund provides benefits for 3,823 employees in Tai-PERS and the initial fund is set to be 373,211,585 NT dollars. This corresponds to 1.41 % of the 271,215 plan participants in 1996. The future optimal fund status are forecasted through the recursive relation in Equation (8) and (9). With the boundary condition being to have no risk at the end of the forecast period, the explicit optimal contributions could be projected by recursively computing estimates C, using Equation (7).. 100% for every year;. (%). lS% , . - - - - - - - . - - - - ... -. 300/.. 25%. 20%. -+-Optimal Contribullon Rate. (6~). --M-Optimal Contnbution Rale. (IO~.). -6-0pl;mll Contribution Rale f'%-). IS%. *;; g. 100LI~~--~~~~~~--~~~~~~~--~~~~. ~_ ~ ~ _ _. §M 8 8N §N 8N 8N ~M §N N8 §N N. N. N. N. ~N ~ ~ N N. (year). Interest RalC:O.08: Retum-llDN(lO%,O.04%); Risk Wcighr.:L Target Fund Ratio: I00%. Figure 3. Optimal Contribution Rates in 1997-2016 under Various Fund Return Rate Scenarios. Figure 3 compares the optimal contribution rates from 1997 to 2016 under various fund return scenarios. It shows that the estimates of the contribution are decreasing by years. Since theEAN cost method is used in this study, the fund accrued. liabilities from the simulations have much larger values than the estimates from other cost methods. Hence the optimal contribution rates computed from the dynamic programming are the most conservative among other actuarial cost methods.. ,. '..

(10) 10. , l ei. ,... ...........••••...•.................._ ....... Figure 4. Optimal Contribution Ratios and Fund Ratios in 1997-2016 under Various Fund Return Rates. Jr. Figure 4 compares the optimal contribution ratios and optimal fund ratios from 1997 to 2016 under various fund returns. It shows that the estimated contribution ratios vary by years, suggesting that the variable contribution ratios are influenced by the fund return rates significantly. The optimal fund ratios are gradually increasing given return = 10% by years from 1998 to 2017, while show decreasing trends given return set at 6% and. 8%.. The optimal fund ratios significantly deviate from 1 between 1997 and 2017 might be partially due to the assigned return rates. Further analysis is needed to explore this. In this study, a set of given actuarial assumptions are chosen to illustrate the optimal procedure. By varying the assumptions, the plan manager could foresee the optimal plan financial status.. .... ....,.\. .............................................................................................................. u. \. ,."\. ,.\\. :: ~\. )&-., '" ,. ~. \ \ \. " ~, \. ·•· . . ~I. .........,...,a.. ·..·... ........,1. ·.6-· ... ........,11. <>'_',..102. ". Figure 5. Optimal Contribution Ratios and Fund Ratios in 1997-2016 under Various Risk Weight Measurements. Figure 5 compares the optimal contribution rates under various risk weights. The risk ratios are modified to reflect the user's subjective risk measurement in performing the optimization. It shows that the optimal fund ratios increase with an. increase in risk weight measurements, while the optimal contribution rate ratios have shown a similar pattern. Through minimizing the performance measure, the explicit fund information could be obtained..

(11) ~i:i:~. :. Optimal Pension Funding Incorporating Stochastic Simulations and Dynamic Programming. .... QtnCblriUJ:nRbc. 11. Q:Iin1IIRnlRIi:J. 1.1,. .,). ,..-.~~•.....-./. "~f"r~......./ .......··~····· ~..-......./'·1, ::......_ .......... f/ ...~..-... ~ . .•.,......_,... ....T... I\nd~ __ T.... RIIII~. ··.··T..... RnII~. 0.7. ~l. i. 0.1' . i ; ; ~ i i ! iiI! ! l. ~ ~. gi. ~. i t-). ;;!. ............. ~IIIII...Of!Ii; ... YoIIiIkI. ~ ~ ~ ~. ! ~ i ~ ! ! ~ ~ g§ ~ i. . . . . . . .IU~.......aDN'IOOIoO'~IIi*.",....,l. E ~(lGI"l. Figure 6. Optimal Contribution Ratio and Fund Ratios in 1997-2016 under Various Target Fund Ratios Figure 6 compares the optimal contribution rates under various target fund ratios. The target fund ratios are given at various levels with respect to the attained accrued liabilities to reflect the user's subjective management requirements. It shows that the optimal contribution rate ratios increase at higher target fund ratios, while the optimal fund ratios have shown decreasing trends in Figure 6.. 3. . VI. Conclusions. 4. . Most models of pension valuation have been based on deterministic approaches. While a dynamic stochastic model in control framework is constructed. This paper has described algorithms that are used to compute the optimal funding strategy. One purpose of this paper is to provide a flexible mechanism which allows the plan manager to incorporate his own performance measurement in the decision processes. The approach in this model has integrated two relatively recent innovations in pension valuation: the stochastic simulations in predicting the factors in the performance function and the dynamic programming in calculating the optimal funding strategies. This approach could be adopted in assisting decision making and acted as a benchmark to compare the funding schedule from the many actuarial cost methods employed in traditional valuation.. References 1. Anderson, A. W., Pension Mathematics for. 2. . 5. . 6. . 7. . 8. . Actuaries, 2nd edition. ACTEX Publications, Winsted, Connecticut, 1992. Bacinello, A. R., "A Stochastic Simulation Procedure for Pension Scheme.," Insurance: Mathematics and Economics, 7, 1988, pp.l53-161. Bartholomew, D. J., Stochastic Models for Social Processes, 3rd Edition. John Wiley & Sons, 1982. Bowers, N. L., Gerber, H. U., Hickman, 1. c., Jones, D. A. and Nesbitt, C. 1., "Notes on the Dynamics of Pension Funding.," Insurance: Mathematics and Economics, 1, 1982, pp.261-270. Cairn, A. J. and Parker, G., "Stochastic Pension Fund Modeling.," Insurance: Mathematics and Economics, 21, 1997, pp.43-79. Chang, S. c., "Building Dynamic Valuation System for Taiwan Public Employees Retirement System.," Proceedings of 1998 Taiwan Public Employees Pension Fund Symposium, Taipei, 1998. (in Chinese) Daykin, C. D., Pentikainen, T. and Pesonen, M., Practical Risk Theory for Actuaries, Monographs on Statistics and Applied Probability 53. Chapman and Hall, London, U.K. 1994. Dufresne, D., "Moments of Pension Fund Contributions and Fund Levels when Rates.

(12) 12. I I. I I. '~. I. I ... ~. I'. of Return are Random.," Journal of the Institute of Actuaries, 115, 1988, pp.535544. 9. Dufresne, D., "Stability of Pension Systems when Rates of Return are Random.," Insurance: Mathematics and Economics, 6, 1989, pp.129-134. 10. Felming W. H. and Rishel R. W., Deterministic and Stochastic Optimal Control. Springer-Verlag, New York, 1975. 11. Frees, E., Kung, Y. C., Rosenberg, M., Young, V. and Lai, S. W., "Forecasting Social Security Actuarial Assumptions.," North American Actuarial Journal, 1, No.4, 1997, pp.49-82. 12. Haberman, S., "Pension Funding with Time Delays: A Stochastic Approach.," Insurance: Mathematics and Economics, 11, 1992, pp.l79-89. 13. Haberman, S., "Pension Funding with Time Delays and Autoregressive Rates of Investment Return.," Insurance: Mathematics and Economics, 13, 1993, pp.45-56. 14. Haberman, S., "Autoregressive Rates of Return and the Variability of Pension Contributions and Fund Levels for a Defined Benefit Pension Scheme.," Insurance: Mathematics and Economics, 14, 1994, pp.219-240. 15. Haberman, S. and Sung, J. H., "Dynamic. Approaches to Pension Funding.,". Insurance: Mathematics and Economics, 15,. 1994, pp.151-162.. 16. Haberman, S. and Wong, L. Y., "Moving. Average Rates of Return and the. Variability of Pension Contributions and. Fund Levels for A Defined Benefit Pension. Scheme.," Insurance: Mathematics and. Economics, 20, 1997, pp.l15-135.. 17. Janssen, J. and Manca, R., "A Realistic. Non-homogeneous Stochastic Pension. Fund Model on Scenario Basis.,". Scandinavian Actuarial Journal, 2, 1997,. 18.. 19.. 20.. 21.. 22.. 23.. 24.. pp.I13-37.. Merton, R. C., Continuous-Time Finance,. Cambridge MA and Oxford UK, Blackwell, Inc., 1992. O'Brien, T., "A Stochastic-dynamic Approach to Pension Funding.," Insurance: Mathematics and Economics, 5, 1986, pp.141-46. O'Brien, T., "A Two-parameter Family of Pension Contribution Functions and Stochastic Optimization.," Insurance: Mathematics and Economics, 6, 19&7, pp.129-134. Runggaldier, W. J., "Concept and Methods for Discrete and Continuous Time Control under Uncertainty.," Insurance: Mathematics and Economics, 22, 1998, pp.25-39. Scha 1, M., "On piecewise Deterministic Markov Control Processes: Control of Jumps and of Risk Processes in Insurance" Insurance: Mathematics and Economics, 22, 1998, pp.75-91. Shapiro, A. F., "Contributions to the Evolution of Pension Cost Analysis.," Journal of Risk and Insurance, 52, 1985, pp.81-99. Taipei Life Insurance Association, Taiwan Standard Ordinary Experience Mortality and Lapse Rate Report, 1998..

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