From a literature review, we find similar studies using stochastic simulations. Stochastic simulations using time as the operational parameter in Bacinello (1988) are performed to obtain the best estimates of the projected work-force, while the projected cash flows are scrutinized through dynamic simulations. An extensive review of past pension cost analyses can be found in Shapiro (1985). Early works can be found in Winklevoss (1982) who developed the pension liability and asset simulation model (PLASM) to evaluate the pension financing. The pro-jection techniques can provide valuable inputs for the planning decisions, such as funding policy and investment strategy.
This study emphasizes the importance of pension cost analysis in optimizing the plan financial condition. Similar researches can be found in: Bowers et al. (1982), Winklevoss (1982), McKenna (1982), Shapiro (1977,1985), O’Brien (1986,1987), Bacinello (1988), Dufresne (1988,1989), Haberman (1992,1993,1994), Daykin et al. (1994), and Haberman and Sung (1994). 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 discounted cash flows for the future years through stochastic simulations according to the probabilistic experience set of actuarial assumptions. Then the optimal contribution rates are estimated subject to the given performance measure. A brief summary of the advantages of this approach is listed below:
(i) Detailed demographics of the future workforce can be projected from this model.
(ii) Through high speed computers, the plan sponsor can forecast the plan’s future cash flows, which helps the fund management.
(iii) The optimal contribution rates can be estimated under various scenarios based on specific plan strategies. (iv) The optimal funding and actuarial status of the plan can be estimated under risk measurement implemented
through a computerized system.
2. Data description
In this research, the Taiwan public employees retirement system (Tai-PERS) is studied and evaluated through the proposed dynamic optimization procedure. 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. In recent years, with an increase in the percentage of population that comes under pension age in Taiwan, pension related topics have taken on new significance and much attention has been focused on the implementation of a better retirement system for the aging society.
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. This system is a de-fined benefit (DB) 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 adjust-ments (COLA), or a mix between lump-sum payment and monthly pension. Since there is no clear relevance between the contribution and the benefit payments, the plan solvency risk is solely the responsibility of the government.
A sample of 3,823 participants is used to evaluate the performance of the proposed approach. A service table is constructed based on the experience data collected from 1 July 1995 to 30 June 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 various risk measurements and the associated stochastic cash flows are performed. Section 5 presents the model validation justifications and the final results of optimal plan status of Tai-PERS. Concluding comments are given in Section 6.
3. The dynamic procedure
A simulation-based optimization procedure 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 financial 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. This approach is capable of inputting user-specified inflation rates, fund return rates and demographic assumptions to compute the projected cash flows. 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. The procedure of our approach is constructed as follows:
Step 1. The future information of the active and inactive members in the system is simulated through a series of dynamic processes using Bernoulli trials. The working status of each member is simulated according to the decrement probabilities from the service table. LetEx,t denote the working status of the employee agex between
t and t + 1 year in the future. The process is shown as follows:
(i) SimulateEx,t by generating a pseudorandom number from Bernoulli (px(τ)), wherepx(τ)is obtained from the service table.
(ii) IfEx,t = 1 (i.e., this member is in working status), then go to step (v).
(iii) IfEx,t = 0 (i.e., this member is not in working status), then a pseudorandom number of multinomial distribution is generated from Multi q (d) x qx(τ) , qx(i) qx(τ) , qx(l) qx(τ) , qx(r) qx(τ) !
whereqx(τ) = qx(d) + qx(i)+ qx(l)+ qx(r). The superscript (i) stands for disability; (d) for death; (l) for layoff and withdrawal and (r) for retirement. LetRx,tdenote the living status of the retiree agedx at time t and Bx,t denote the option of his retirement benefits. IfRx,t = 1 (i.e., the member is retiring), a pseudorandom number
Bx,t is generated from Bernoulli (bx,t) to simulate the chosen benefit program wherebx,t is estimated from past experience.
(iv) IfBx,t = 1 (i.e., the retiree chooses the monthly annuity), the living status Px,t is generated from Bernoulli
(qx+t) where qx+t is estimated from the annuity table of the retiree. IfBx,t = 0 (i.e., the retiree choose the lump-sum payment), the benefit payment is then computed.
(v) Lett = t + 1.
Step 2. The contributions are assumed to be paid at the beginning of each year in proportion to the covered payroll of each active employee. A 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:
(i) Benefit functionsBt including withdrawal, disabled, death, layoff and retirement benefits. (ii) ContributionsCtfrom active employees and new entrants.
(iii) ReturnsRt from pension fund and the accumulated random fund assetFt. The process is shown as follows:
(i) Set timet = 0 at the plan valuation date. (ii) Use the simulated database from Step 1. (iii) Compute the benefits paymentBtat timet.
Fig. 1. Diagram of the procedures in Step 2.
(iv) Compute the normal costNCt at timet.
(v) Compute the accrued liabilityALt+1at timet + 1. (vi) Lett = t + 1.
A simple line diagram is plotted in Fig. 1 for clarification.
The Monte Carlo simulations can be carried out by repeating Steps 1 and 2. Tables 1–3 provide the detailed procedure of this algorithm in Steps 1 and 2.
Step 3. Based on the projected benefits,Ct is estimated through minimizing the performance measure defined in Eq. (1) from present to timeT − 1. Detailed clarification and explanation of this performance measure will be discussed in Section 4. Γ (C0, . . . , CT −1) = E (T −1 X t=0 " vt 1− Ct NCt 2 + vt+1βt+1 1− Ft+1 ηALt+1 2#) . (1)
The actuarial notations and fund recursive relationship are given as follows: N the set of chosen time span
{it}t∈N return rate of pension fund at timet
{Ct}t∈N contribution at timet
{vt}t∈N discount factor at timet
{βt}t∈N risk weighted ratio at timet
{NCt}t∈N normal cost at timet
Table 1
Flow chart of the open group simulations
Table 2
Table 3
Flow chart of the open group simulations in (2)
{ALt}t∈N accrued liability at timet
η target fund ratio
During the year between timet and t + 1 the fund balance will increase by the contribution Ct and decrease by the benefit outgoBt at the beginning of the year simulated from Step 2 plus the investment return
Ft+1= (Ft+ Ct − Bt)(1 + it), it ∼ IIDN(θ, σ2) (2)
where the real returnit between timet and t + 1 are assumed to be a sequence of independent and identically distributed normal (IIDN) variables with meanθ and variance σ2.
4. The proposed model
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. A simpler procedure focusing on individual proposed by Bacinello (1988) is not used since we intend to forecast dynamically the overall 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 particularjth simulation, the projected benefit at time t + 1 is denoted byjBt+1. From these simulations, a median estimatej ˆBt+1that satisfies Pr(jBt+1 ≤j ˆBt+1) ≤ 50%, J ={set of all simulations} is used to estimate the contribution at timet + 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.
In order to determine the parameters that best fit the data, a performance measure,0ΓT −1, is defined to measure the adequacy of the estimation between time 0 and timeT − 1. The performance measure is a nonnegative function based on the sum of discounted squared deviations. The parameter estimates are determined by minimizing the combined performance measure in Eq. (1).
Similar to Haberman and Sung (1994), vt(1 − Ct/NCt)2 is used to denote the contribution rate risk, while
vt+1βt+1(1 − Ft+1/ALt+1)2is used to denote the solvency risk whereβt+1 is the relative weight in measuring the fund financial stability at timet. 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 Ct,... ,CT −1t0T −1= min Ct,... ,CT −1E T −1X s=t " vs 1− Cs NCs 2 + vs+1βs+1 1− Fs+1 ηALs+1 2 Ht # , (3)
whereHt is theσ -field generated by {F0, . . . , Ft}. {Ft}t∈N are assumed to follow a first-order Markov process, which is the case that Tai-PERS internally evaluate its financial status annually. Then
E T −1X s=t " vs 1− Cs NCs 2 + vs+1βs+1 1− Fs+1 ηALs+1 2 Ht # = E T −1X s=t " vs 1− Cs NCs 2 + vs+1βs+1 1− Fs+1 ηALs+1 2 Ft # . (4) By definingVt(Ft) as Vt(Ft) = min Ct,... ,CT −1E T −1X s=t " vs 1− Cs NCs 2 + Vs+1βs+1 1− Fs+1 ηALs+1 2 Ft # , (5) we have Vt(Ft) = minC t E ( vt 1− Ct NCt 2 + vt+1βt+1 1− Ft+1 ηALt+1 2 |Ft+ Vt+1(Ft+1)| Ft ) (6)
thenCt can be estimated by induction. To solve the Bellman equation (Aström, 1970), we assume thatVt(Ft) =
a1(t)Ft2+ a2(t)Ft+ a3(t) for all t, 0 ≤ t ≤ T . Eq. (6) could be written as 0(Ct) which is a second-order function ofCt. Then the optimal contribution ˆCtsatisfying the unique solution is obtained in Eq. (7).
ˆCt=2vt/NCt + 2vt+1βt+1H/ηALt+1− 2(vt+1βt+1/η 2AL2 t+1+ a1(t + 1))K(Ft− Bt) − a2(t + 1)H 2vt/NCt+ 2vt+1βt+1K/η2AL2 t+1+ 2a1(t + 1)K , (7)
where
H = 1 + θ, K = H + σ2.
We seta1(T ) = a2(T ) = a3(T ) = 0 as the boundary condition in the above equation which means no bias at time
t = T . That is, the long term financial status of this pension system is assumed to have no risk incurred. Then we
could solve the recursive relationship fora1(t) and a2(t) for 0 ≤ t ≤ T . Given
Dt = NC2vt t + 2vt+1βt+1H ηALt+1 + 2vt+1βt+1K η2AL2 t+1 Bt+ 2a1(t + 1)KBt− a2(t + 1)H, Et = −2vt+1βt+1K η2AL2 t+1 − 2a1(t + 1)K, Gt = NC2vt t + 2vt+1βt+1K η2AL2 t+1 + 2a1(t + 1)K,
the recursive relationship forat(t) and a2(t) are solved by the following equations:
a1(t) = vtE 2 t G2 tNCt2+ vt+1βt+1 K η2AL2 t+1 1+Et Gt 2 + a1(t + 1)K 1+Et Gt 2 , (8) a2(t) = 2Et GtNCt Dt GtNCt − 1 − 2vt+1βt+1ηALH t+1 1+Et Gt +2vt+1βt+1η2ALK2 t+1 1+Et Gt D t Gt − Bt + 2a1(t + 1)K 1+ Et Gt D t Gt − Bt +a2(t + 1)H 1+Et Gt . (9)
5. Results and analysis
We now use Tai-PERS to illustrate these results. 3,823 samples are collected for the calculations. The actuarial assumptions are:
– Population: Tai-PERS service table based on 1995–1996; 1989 Taiwan Standard Ordinary life table (1989 TSO) for the retiree’s annuity table.
– Actuarial cost method: Entry age normal (EAN) cost method.
– Salary scale and inflation rate: 3.5% for annual salary increase and 3% for annual inflation rate.
– Interest rate: 6% for pension valuation, i.e., we assume vt = vt = (1.06)−t, i.e., a constant discount rate assumption.
– Target fund ratio:η = 75% for every year.
– Risk measurement weight:βt = 60% for every year. – Fund return rate:θ = 10% and σ2= 0.04%.
With the above assumptions, the estimated actuarial accrued liabilities, normal costs and benefit payments are simulated based on Steps 1 and 2. The numerical values of the optimal contributions are listed in Table 4.
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 is forecasted through the recursive relation in Eqs. (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
Table 4
Optimal contributions of Tai-PERS
1997 1998 1999 2000 2001 2002 2003 2004 Fund asset 373,211,585 600,381,624 861,349,176 1,120,815,583 1,339,070,206 1,765,948,737 2,092,758,922 2,476,541,889 Optimal 275,496,575 256,618,424 253,850,803 246,475,587 238,192,682 277,859,062 224,117,339 219,592,961 contributions Benefits outgo 106,636,560 48,948,364 89,903,272 95,409,192 67,699,656 76,361,992 65,474,592 102,758,288 Accrued liability 585,530,240 851,652,800 1,152,353,024 1,461,198,464 1,775,253,248 2,101,083,136 2,454,753,024 2,813,157,888 Normal cost 264,658,176 254,203,072 250,750,880 247,360,208 243,759,120 240,339,776 238,243,632 235,766,864 Covered payroll 1,091,617,528 1,136,869,006 1,166,532,793 1,195,472,961 1,224,048,954 1,260,401,090 1,291,833,377 1,324,803,499 Fund return rate 10.76% 6.60% 9.32% 10.00% 12.51% 9.14% 10.00% 10.00%
Ft/ηALt 0.850 0.940 0.997 1.023 1.051 1.121 1.137 1.174 Ct/NCt 1.041 1.010 1.012 0.996 0.977 0.948 0.941 0.931 Optimal 25.24% 22.57% 21.76% 20.62% 19.46% 18.08% 17.35% 16.58% contribution rate 2005 2006 2007 2008 2009 2010 2011 2012 Fund asset 2,852,714,280 3,293,112,124 3,641,364,756 4,073,949,110 4,563,110,444 5,013,138,724 5,476,497,663 6,060,102,605 Optimal 215,734,838 210,697,528 208,076,022 206,097,227 203,120,348 203,325,805 203,156,242 202,318,111 contributions Benefits outgo 134,871,760 193,478,128 161,202,912 209,269,152 186,510,336 222,970,272 225,303,168 255,283,040 Accrued liability 3,183,145,472 3,578,989,056 3,953,183,488 4,343,038,464 4,712,074,752 5,146,928,640 5,565,712,384 5,987,062,784 Normal cost 233,146,752 231,159,440 227,970,288 225,249,024 221,934,336 219,645,904 216,970,176 214,121,360 Covered payroll 1,358,531,004 1,384,319,519 1,413,676,091 1,435,214,137 1,472,414,452 1,496,865,737 1,521,840,935 1,549,543,771 Fund return rate 12.26% 10.00% 10.46% 12.09% 9.46% 9.67% 11.11% 10.00%
Ft/ηALt 1.195 1.227 1.228 1.251 11.291 1.299 1.312 1.350 Ct/NCt 0.925 0.911 0.913 0.915 0.915 0.926 0.936 0.945 Optimal 15.88% 15.22% 14.72% 14.36% 13.80% 13.58% 13.35% 13.06% contribution rate 2013 2014 2015 2016 2017 Fund asset 6,607,851,586 6,985,743,560 7,400,130,518 7,998,144,864 8,309,446,988 Optimal 202,729,988 202,641,694 202,119,295 200,590,284 contributions Benefits outgo 340,608,224 384,232,000 422,253,632 510,401,472 564,830,528 Accrued liability 6,427,823,616 6,821,858,816 7,183,346,176 7,431,084,544 7,703,323,648 Normal cost 212,082,800 209,096,448 206,210,160 202,645,072 199,309,184 Covered payroll 1,570,998,111 1,584,815,608 1,584,733,735 1,598,323,058 1,607,790,314 Fund return rate 7.97% 8.76% 11.39% 8.08%
Ft/ηALt 1.371 1.365 1.374 1.435 1.438 Fund return mean:10%
Fund return variance:0.04%
Ct/NCt 0.956 0.969 0.980 0.990 Discount rate:6%
Risk weight:0.6
Optimal 12.90% 12.79% 12.75% 12.55% Fund ratio:75%
contribution rate
computing estimatesCt using Eq. (7). Fig. 2 shows the simulated benefit payments, normal costs, accrued lia-bilities between 1998 and 2017 based on 50 dynamic simulations. The benefit payments and accrued lialia-bilities are increasing by years while the normal costs are decreasing. In Table 2, the results show that the contribution ratios are increasing through the years (i.e., 1.041% in 1997) and finally approach the target fund ratio at 0.99% in 2017. The contributed rates that are obtained by dividing the total contribution by covered payroll are high-lighted by the dotted line. It shows that contribution rates are decreasing by years from 25.24% in 1997 to 12.55% in 2017.
Fig. 2. Continued....
Fig. 3 compares the contribution ratios and optimal fund ratios from 1997 to 2016. It shows that the estimated contribution ratios vary by years, suggesting that the variable contribution ratios are influenced by the demographic assumptions based on the simulations. The optimal fund ratios are gradually increasing by years from 85% in 1998 to 143.5% in 2017. The optimal fund ratios significantly deviating from 1 between 1997 and 2017 might be partially due to the assigned risk measurement weightβ = 0.6 since the weight assigned in contribution rate risk is larger than that in solvency risk.1 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.
Fig. 2. (a) Simulated benefit in 1998–2017 (open dynamical model, number of simulations= 50); interest rate= 6%; salary increase rate= 3.5%. (b) Simulated accrued liability in 1998–2017 (open dynamical model, number of simulations= 50); interest rate= 6%; cost method: EAN. (c) Simulated normal cost in 1998–2017 (open dynamical model, number of simulations= 50); interest rate= 6%; cost method: EAN.
Fig. 3. Optimal fund ratio and contribution ratio in 1998–2016.
Fig. 4 compares the optimal contribution rates under various risk measurements. The risk ratios are modified to reflect the user’s subjective risk measurement in performing the optimization. It shows that the contribution rates increase with an increase in risk measurement. Through minimizing the performance measure, the explicit fund information could be obtained from Eq. (1).
Fig. 4. Optimal fund ratio in 1998–2017 under various risk weight.
Fig. 5. Optimal fund ratio in 1998–2017 under various target fund ratio.
Fig. 5 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 requirement. It shows that the fund ratios are increasing at higher target fund ratios, while the trend of increases in fund ratios have shown a similar pattern both in Figs. 4 and 5. This occurs because we set a constant mean of the return rate at 10% for the fund in 1997–2017.
6. Conclusions
The purpose of this paper is to provide a flexible method which allows the plan manager to incorporate his own risk measurement in funding policy. The approach in this model has integrated two relatively recent innovations in pension financing: the stochastic simulations in estimating the factors in the objective function and the dynamic programming in calculating the optimal contributions. This study could be adopted in assisting policy making and as a benchmark to compare the results from the many actuarial cost methods used in plan valuation.
The approach proposed in this paper has linked the possible scenarios and equation-type optimal solutions together, which provide an alternative way to decide funding policy. The stochastic modeling of the liabilities may provide helpful guidance on investment strategy for proper assessment of the trade-off between various risks. Hence the future fund financing could be explored in detail using this approach. Since controlling the pension fund solvency and keeping contribution at a steady level are two main goals in pension financing, the proposed method proves helpful in achieving these goals.
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