Dynamic Accumulation Model for the Second Pillar of the Slovak Pension System

Transcription

1 UDC: (437.6) JEL classification: C1, E27, G11, G23 Keywors: ynamic stochastic programming; fune pillar; utility function; Bellman equation; Slovak pension system; risk aversion; pension portfolio simulations Dynamic Accumulation Moel for the Secon Pillar of the Slovak Pension System SoÀa KILIANOVÁ Igor MELICHERâÍK Daniel EVâOVIâ* 1. Introuction Before January 0, pensions in Slovakia were operate by the unfune pay-as-you-go system. Mainly because of high unemployment an low contributions pai on behalf of the unemploye by the government as well as a high rate of contribution evasions, the system generate eficits. The emography crisis was suppose to generate further pressure on the balance of the pay-as-you-go system. In April 03 the government passe the Principles of Pension Reform in the Slovak Republic. The goals of the pension reform were to secure a stable flow of high pensions to the beneficiaries, an sustainability an overall stability of the system. Corresponing legislation, as passe in December 03, establishe a system base on three pillars: 1. the manatory non-fune first pillar (pay-as-you-go pillar), 2. the manatory fully fune secon pillar, 3. the voluntary fully fune thir pillar. The contribution rates were set for the first pillar at 19.7 % (ol age 9 %, isability an survival 6 % an reserve fun 4.7 %) an for the secon pillar at 9 %. The total rate is about 0.7 % higher than the ol one. A thorough escription of the Slovak pension reform with calculations of the balance of the pension system an expecte level of pensions in the new system can be foun in (Golia, 03), (Melicherãík Ungvarsk, 04), (Thomay, 02). Compare to Polan an Hungary, the Slovak secon pillar is more substantial. Contribution rates are higher in Slovakia compare to 7.3 % in Polan an 6 % (with a possible future increase to 8 %) in Hungary. A thorough escription of the pension reforms in Hungary an Polan can be foun in (Benczúr, 1999), (Chlon et al., 1999), (Fultz, 02), (Palacios Rocha, 1998) an (Simonovits, 00). * Corresponing author: Igor Melicherãík (igor.melichercik@fmph.uniba.sk) All authors: Department of Applie Mathematics an Statistics, Faculty of Mathematics, Physics an Informatics, Comenius University, Bratislava, Slovak Republic Acknowlegment: We are inebte to Prof. P. Brunovsk for his valuable comments an ieas that significantly improve the quality of this paper. This work was supporte by Grant VEGA 1/3767/ Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

2 TABLE 1 Limits for Investment for the Pension Funs Fun type Stocks Bons an money market instruments Growth Fun up to 80 % at least % Balance Fun up to 0 % at least 0 % Conservative Fun no stocks 0 % The savings in the secon pillar are manage by pension-asset aministrators. Each pension-asset aministrator manages three funs: Growth Fun, Balance Fun an Conservative Fun, each of them with ifferent limits for investment (see Table 1). At the same instant savers may hol assets in one fun only. In the last 1 years preceing retirement, the saver may not hol assets in the Growth Fun an in the last seven years all assets must be in the Conservative Fun. Even with these restrictions the contributors have some space for iniviual ecisions with regar to which fun is optimal in a specific situation (the age of the contributor, the save amount, the past performance of the pension funs). The aim of this paper is to stuy whether the above restrictions for the funs can be illustrate by a mathematical moel an to calculate optimal strategies for switching between the pension funs (Growth, Balance an Conservative) keeping in min the risk preferences of the contributors. Our moel inicates that aopte pension-fun regulations can be supporte by means of a ynamic accumulation moel. The paper is organize as follows: In Section 2 we present a simple example supporting the iea of graual ecreasing of the risk when saving for the future pension. We also give a motivation for stuying the ynamic accumulation moels. Section 3 contains the formulation of the ynamic stochastic programming accumulation moel an the numerical scheme for fining a solution of this moel. In Section 4 we present the calculate results an we iscuss the sensitivity of fun-switching strategies with respect to various scenarios of evelopment of financial markets, wage growth evelopment as well as iniviual risk preferences. At the en of the section we compare ynamic an static strategies using the mean-variance framework. The last section contains final remarks an conclusions. 2. First Run a Risk, then Secure Savings Pension funs usually hol portfolio consisting of bons an equities. Limits for their weights in the portfolio may iffer across countries. In Slovakia, each pension company manages three funs: Growth Fun, Balance Fun an Conservative Fun, each of them with ifferent limits for investment (see Table 1). As alreay mentione in the Introuction, instant savers may hol assets in one fun only an they may not hol assets in the Growth Fun in the last 1 years preceing retirement. Moreover, all assets shoul be hel in the Conservative Fun in the last seven years preceing retirement. The intention of these restrictions an government regulations was to lower the risk of the value of savings falling substantially shortly before retirement. Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

3 TABLE 2 Risk-Return Analysis of Different Strategies No. r 1 r E(M) M E(M) M E(M) 2 M 1 % % 0 % 0 % % % % % % % % % % 0 % % % % 0 % % 2..0 % % con. % con % For the sake of simplicity let us consier a plain two-perio moel of saving (the length of each perio is one year). At the beginning of each year the saver eposits an amount A. Suppose that the returns of the eposits are r 1 an r 2 in the first an secon year, respectively. The save amount M after two years is M = A(1 + r 1 )(1 + r 2 ) + A(1 + r 2 ) = A(2 + r 1 )(1 + r 2 ) Then for the sensitivity of M with respect to r 1 an r 2 we have M M = A(1 + r 2 ), = A(2 + r 1 ) r 1 r 2 an therefore M M > (1) r 2 r 1 for realistic asset returns r 1 an r 2. This is in accorance with intuition that the save amount is more sensitive to later returns than to earlier ones. If the iniviual mae just a single contribution at the start of his/her working career, the impact on his/her final pension wealth woul be the same regarless of whether the asset-price fall occurre early in life or just before retirement. But if a series of contributions throughout one s life is mae, a fall in assets value early in life oes not affect the future contributions, i.e. only part of one s future pension wealth is affecte, while if it occurs close to retirement it affects all past accumulate contributions an returns on them, i.e. most of one s pension wealth. Let us consier two funs: 1. a risky fun with normally istribute return with average of % an stanar eviation of %, 2. a secure fun with a certain return of %. Suppose that the saver eposits one unit in the first an one unit in the secon perio. Table 2 an Figure 1 emonstrate a risk-return analysis of five ifferent strategies. Strategy 1 assumes that in both perios the savings are investe in the secure fun. Strategy 2 is the most risky one in both perios the savings are investe in the risky fun. This strategy has the highest expecte value of the savings at the en of the secon perio E(M) but also the highest stanar eviation M. To ecrease the risk, Strategies 3 an 4 invest in the secure fun in one of the perios. Accoring to (1) the level of final savings is more sensitive to the secon-perio asset returns. Therefore the risk (see the last three 08 Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

4 FIGURE 1 Risk-Return Analysis of Different Strategies (2) E(M) 2.2 () (4) 2.2 (3) 2.1 (1) 0 1 σ M % columns of the Table 2) connecte with Strategy 4 (first year secure fun, secon year risky fun) is higher than the risk connecte with Strategy 3 (first year risky fun, secon year secure fun). Strategy is a ynamic strategy where in the first year the savings are eposite in the risky fun an the ecision in the secon year is conitional: if the return in the first year is more then 1 % then the secure fun is chosen in the secon perio; otherwise the risky fun is chosen again. Compare to Strategy 4, this strategy is more efficient (see Table 2 an Figure 1). Hence, by using close-loop strategies, both the risk an return parameters coul be improve. 3. The Dynamic Stochastic Programming Accumulation Moel Suppose that the future pensioner eposits once a year a -part of his/her yearly salary w t in a pension fun j 1, 2,..., m. Denote by s t, t = 1, 2,...T the accumulate sum at t where T is the expecte retirement. Then the buget-constraint equations rea as follows: s t+1 = s t (1 + r tj ) + w t+1, t = 1, 2,..., T 1 (2) s 1 = w 1 where r tj is the return of the fun j, in the perio [t, t + 1). When retiring, the pensioner will strive to maintain his/her living stanar at the level of the last salary. From this point of view, the save sum s T at the of retirement T is not precisely what the future pensioner cares about. For a given life expectancy, the ratio of the cumulative sum s T an the yearly salary w T, i.e. T = s T /w T is more important. Using the quantity t = s t /w t one can reformulate the buget-constraint equation (2): t+1 = F t ( t, j), t = 1, 2,..., T 1 (3) 1 = where 1 + r t j F t (, j) = +, t = 1, 2,..., T 1 (4) 1 + t Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

5 an t enotes the wage growth efine by the equation w t+1 = w t (1 + t ) Suppose that each year the saver has the possibility to choose a fun j (t, I t ) 1, 2,..., m, where I t enotes the information set consisting of the history of returns r j t, t = 1, 2,..., t 1, j 1, 2,..., m an the wage growth t, t = 1, 2,..., t 1. Now suppose that the history of the wage growth t, t = = 1, 2,..., T 1 is eterministic an the returns r tj are assume to be ranom an they are inepenent for ifferent s t = 1, 2,..., T 1. Then the only relevant information is the quantity t. Hence, j(t, I t ) j(t, t ). One can formulate a problem of ynamic stochastic programming: max E(U( T )) () j with the following recurrent buget constraint: t+1 = F t ( t, j(t, t )) t = 1, 2,..., T 1 (6) 1 = where the maximum is taken over all non-anticipative strategies j = j(t, t ). Here U stans for a given preferre utility function of wealth of the saver. Using the law of iterate expectations E(U( T )) = E(E(U( T ) I t )) = E(E(U( T ) t )) we conclue that E(U( T ) t ) shoul be maximal. Let us enote Then by using the law of iterate expectations we obtain the Bellman equation V t () = max E(U( T ) t = ) (7) j E(U( T ) t ) = E(E(U( T ) t+1 ) t ) V t () = max E[V t+1 (F t (,j))] = E[V t+1 (F t (,j(t, )))] (8) j 1, 2,..., m For t = 1, 2,..., T 1, where V T () = U(). Using (8) the optimal feeback strategy j(t, t ) can be foun backwars. This strategy gives the saver the ecision for the optimal fun for each t an level of savings t. Suppose that the stochastic returns r tj are represente by their ensities f tj. Then equation (8) can be rewritten in the form V t () = max E[V t+1 (F t (,j))] j 1, 2,..., m 1 + r = max V t+1 + f t j (r)r j 1, 2,..., m R 1 + t Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

6 1 + t 1 + t = max V t+1 (y)f t j (y ) 1 y j 1, 2,..., m R 1 + t 1 + t y (9) R = V t+1 (y)f t j(t,) (y ) 1 where the substitution y = (1+r)(1+ t ) 1 + has been use an R enotes the set of real numbers. 3.1 The Constant Relative Risk Aversion (CRRA) Utility Function An important part of the problem () (6) is the choice of the utility function U. The utility function varies across the investors an represents their attitue to the risk. A key role in efining the utility function is playe by the coefficient of relative risk aversion C(x) = xu (x)/u (x). Constant relative risk aversion implies that people hol a constant proportion of their wealth in any class of risky assets as their wealth varies see e.g. (Frien Blume, 197), (Pratt, 1964) an (Young, 1990). In this case the utility function is of the form U(x) = Ax 1 C + B if C > 1 U(x) = A ln(x) + B if C = 1 U(x) = Ax 1 C + B if C < 1 () where A, B are constants an A > 0. One can easily prove that, concerning the problem () (6), the utility function is invariant to positive affine transformations, i.e. U an K.U + L are equivalent. In our case, constant relative risk aversion implies that the utility functions U() an U( ) where is a constant lea to the same strategies. We use the constant relative risk aversion (CRRA) utility function 1 U() = ( ) 1 a 1 (11) 1 a where a > 0 is the constant coefficient of relative risk aversion. Using = 1/12 the utility function is steeper for reasonable values an the numerical proceure is more stable. Problem () (6) then maximizes the expecte utility of savings (compare to the last yearly salary) corresponing to 1/12 of the yearly benefits (i.e. the benefits for one month). It is clear that maximizing monthly benefits or yearly benefits shoul lea to the same strategy an therefore we can utilize the CRRA utility function. The coefficient of relative risk aversion a plays an important role in many fiels of economics. There is a consensus toay that the value shoul be less than see e.g (Mehra Prescott, 198). In our typical results we consiere values close to 9. It shoul probably be lower for lower equity premium. However, our goal was to formulate the mathematical moel an to Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

7 manage the numerical proceure. The reaer can change the parameters an calibrate the moel in his/her own way. 4. Pension Portfolio Simulations. Numerical Experiments The purpose of this section is to present the results of pension portfolio simulations. The numerical approximation scheme is iscusse in the Appenix. The output of the numerical coe is a matrix of size (T = ) (k = 0) allowing us to browse between ifferent years (rows) t an ifferent levels of (columns). At a given cell of the table we can rea the name of the fun (j = 1,...,m) which has to be chosen. Plots of compute output matrices ajuste to the omain (,t), t (0,T), ( min,t/2) are epicte in this section. Our results will be summarize in graphical plots of the so-calle optimal choice function j = j(t,) as well as several tables iscussing compute results of optimization. The role of the optimal choice function j = j(t,) is to provie information when to switch between ifferent funs for a given level of the ratio of save money an wages. We focus on two basic questions an problems: 1. what the regions of constant values of j(t,) are; 2. what the path of expecte values of t is. Before presenting results of the simulation we have to iscuss input ata such as, e.g., fun structures an characteristics, an the wage growth. Concerning the structure of funs we consier the present situation in the Slovak Republic. Accoring to the aopte government regulation, there are three funs (i.e. m = 3). Namely, the Growth, Balance an Conservative Funs (see Table 1). Hereafter, we shall suppose that these three funs are constructe from stocks (S) an secure bons (B) where stocks are represente by the S&P Inex (Jan 1996 June 02) with average return r s = = 0.28 an stanar eviation s = whereas the secure bons are represente by -year US government bons (Jan 1996 June 02) with the average return r b = an stanar eviation b = Using the historical ata, the estimate of correlation between stocks an bons is Stochastic asset returns are assume to have normal istributions. 2 We shall suppose that the structure of funs (F 1 = Growth Fun, F 2 = Balance Fun, F 3 = Conservative Fun) of the secon pension pillar in the Slovak Republic is as follows: F 1 = 0.8 S B F 2 = 0. S + 0. B F 3 = B (12) 1 We consiere the estimate asset returns only for illustration of the moel capability. We o not have any ambition to estimate future asset returns. However, in Section 4.3 we present a sensitivity analysis for ifferent asset returns. 2 The normal istribution is a simplification. There is well-known empirical evience that stock returns exhibit asymmetry an heavy tails. However, the moel presente in Section 3 allows ifferent istributions. 12 Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

8 TABLE 3 Data Use for Computation. Fun Returns an Their Stanar Deviations (left), Expecte Wage Growth for the Perio 06 0 (right) Fun Return StDev F 1 r 1 = = 0.1 F 2 r 2 = = F 3 r 3 = = Perio Wage growth (1 + t ) Both returns r i an stanar eviations i, i = 1, 2, 3 of the above funs can be easily calculate from returns r s, r b, stanar eviations s, b an the estimate correlation (see Table 3). Accoring to the Slovak legislature, the percentage of salary transferre each year to a pension fun is 9 %. The law sets aministrative costs of the secon pillar at 1 % of the monthly contribution an 0.07 % of the monthly asset value (i.e % p.a.). Therefore, we consiere effective contributions = 8.91% (= 9%*0.99). The value 0.84 % was subtracte from the asset returns in Table 3. We assume the perio of saving to be T = years. The ata for the expecte wage growth are taken from the Slovak Savings Bank (SLSP). 3 The values are shown in Table Description of Compute Results an Simulations In Figure 2 we present a typical result of our analysis with the coefficient of relative risk aversion a = 9. It contains three istinct regions in the (,t) plane etermining the optimal choice j = j(,t) of a fun epening on t [1, T 1] an the average save-money-to-wage ratio [ min, max]. t For practical purposes we chose 4 min = an max t = t/2 for t 1. In each year t = 1,..., T 1 we invest the save amount of money s t uniquely corresponing with t in one of the funs j = 1, 2, 3 epening on the compute optimal value j = j(, t). In the first year of saving we take 1 = min. The curvilinear soli line in Figure 2 represents the path of the mean wealth E( t ), obtaine by,000 simulations an here we use a = 9. Notice that, for t > 1, the ratio t is a ranom variable epening on (in our case normally istribute) ranom returns of the funs an on the compute optimal fun choice matrix j(,t ), t < t. The ashe curvilinear lines correspon to E( t ) t intervals where t is the stanar eviation of the ranom variable t. In Table 4 we present the mean final wealth E( T ) as well as the so-calle switching-s for mean path E( t ), t [1, T 1] an the intervals (in brackets) of switching s for one stanar eviation of the mean path. 3 The ata were provie by the analysts of SLSP: Martin Barto an Juraj Kotian. 4 min = because 8.91 % is the effective 2 n pillar contribution rate. Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

9 FIGURE 2 Regions of Optimal Choice an the Path of Average Save-Money-to-Wage Ratio (a = 9) FIGURE 3 Cumulative Distribution Function 1 F (left), Histogram of Simulations an Density Function (right) Sample Mean E( T ) = 4.28 an Stanar Deviation of T = TABLE 4 Summary of Computation of the Mean Save-Money-to-Wage Ratio t an Switching Times (a = 9) Mean Switch Switch E( T ) F 1 F 2 F 2 F (12 16) 33 (32 3) The normalize histogram of the simulate final wealth is very similar to a normal istribution, as can be seen in Figure 3. In the next sections we focus on the sensitivity of results when some parameters change. 4.2 Sensitivity Analysis for Varying Risk Aversions Let us consier ifferent risk aversion parameters a in the utility function: a = 3,, 8, 9,. It shoul be obvious that increasing risk aversion leas to a choice of a less risky fun. Inee, base on our computations, one can observe that increasing a (increasing risk aversion) causes that 14 Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

10 FIGURE 4 Sensitivity of Regions of Optimal Choice with Respect to Different Risk-Aversion Values of the Parameter a = 3,, 8, a = 3 a = a = 8 a = TABLE Results for Fixe Wage Growths, Fixe Returns an Stanar Deviations (see values in Table 3), Different Risk Aversion Parameter a a Mean Switch Switch E( T ) F 1 F 2 F 2 F (32 38) never ( 26) never (13 18) 37 (36 38) (11 1) (29 32) the switches between funs are shifte to an earlier, i.e. we switch from F 1 to F 2 sooner, as well as from F 2 to F 3. Obviously, for higher values of the risk aversion parameter a we obtain lower levels of the final wealth. Results for the experiments are isplaye in Figure 2, Figure 4, Table 4 an Table. The relation between ifferent values of risk aversion parameter a (0,2) an the final mean wealth to last wage ratio is shown in Figure. We can see that the curve can be ivie into three segments where the kinks separate ranges of the parameter a for which there are no switches, one switch, an two switches between funs in the optimal strategy. One can see that results partially in accorance with legal regulations are reache for a = 9. This value is relatively high see e.g. (Mehra Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

11 FIGURE Relation between the Risk Aversion Parameter a an the Level of Mean Save Wealth E( T ) 7 6 E() a Prescott, 198). In the next section we show that the results are highly sensitive to asset returns an for lower stock returns the typical value of a shoul be lower. 4.3 Sensitivity Analysis for Various Stock an Bon Returns Now, let us examine the impact of the change in returns of funs on the optimal strategy an results. One can expect that if, for example, the return of stocks becomes higher, it will be more favorable to stay in F 1 or F 2 for a longer perio. In our computations, we first fix the bon return an increase/ecrease the stock return (a = 9 an other parameters are fixe). This change is mirrore in the returns of the funs F 1 an F 2. The results obtaine show that a higher return of stocks implies a later switch from more risky to less risky funs. The wealth in the final perio of savings is higher too. Seconly, we fix the stock return an increase/ecrease the bon return. A higher return of bons implies an earlier switch from more risky to less risky funs. For an overview of all results, see Figure 6 an Table Sensitivity Analysis with Respect to Varying Wages Finally, we consier ifferent wage-growth rates. The intuition says that one can expect lower save-money-to-wage ratio t for higher wage growth. 6 To examine the influence of this parameter on results, we consiere the wage growth being raise (uniformly for all perios) an lowere by one percentage point. We enote by (+1pp) ( ( 1pp) ) the wage growth evelopment erive from Table 3 where t has been increase by 1 pp (ecrease by 1 pp) for each of five perios in Table 3. As we can see in Figure 7 an Table 7, a higher wage growth leas to a lower wealth to last wage ratio, guie by shifting the switch-s to later moments. We have merely varie means of returns istributions up or own, but kept stanar eviations unchange. As a result, the coefficient of variation (= stanar eviation/mean) changes. If riskiness change proportionately with returns, the results woul iffer much less. 6 Although this increases the contributions (contribution rate unchange), there is a steeper wage profile an hence lower savings to last wage ratio. 16 Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

12 FIGURE 6 Sensitivity of Regions of Optimal Choice for Various Expecte Values of Stock an Bon Returns a) lower stock return r s = b) higher stock return r s = an fixe bon return r b = c) lower bon return r b = ) higher bon return r b = an fixe stock return r s = 0.28 TABLE 6 Results for Fixe Wage Growths, Fixe a = 9, Fixe Stanar Deviations 1 = 0.1, 2 = , 3 = an Various Bon an Stocks Returns r b an r s, respectively Stock & Bon Fun Mean Switch Switch returns returns E( T ) F 1 F 2 F 2 F 3 r s = 0.28 r 1 = r b = r 2 = (12 16) (32 3) r 3 = r s = r 1 = r b = r 2 = (7 9) (19 23) r 3 = r s = r 1 = never r b = r 2 = (16 ) r 3 = r s = 0.28 r 1 = never r b = r 2 = (17 22) r 3 = r s = 0.28 r 1 = r b = r 2 = (6 8) ( 24) r 3 = Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

13 FIGURE 7 Sensitivity of Regions of Optimal Choice for Various Wage Growth Scenarios lower wage growth ( 1pp) higher wage growth (+1pp) TABLE 7 Results for Fixe Returns an Stanar Deviations (see values in Table 3), Fixe a = 9, an Different Wage-Growth Rates Wage Mean Switch Switch growth E( T ) F 1 F 2 F 2 F 3 ( 1pp). 12 ( 14) 32 ( 34) (12 16) 33 (32 34) (+1pp) (14 18) 34 (33 36) 4. The Comparison of Dynamic an Static Strategies One can think about static strategies where the instants when a contributor switches between the funs are etermine at the beginning of the saving. The most risk-averse contributor always eposits the savings in the Conservative Fun. The least risk-averse investor contributes to the risky funs as long as the law permits it: in the first 2 years of saving to the Growth Fun (the total perio of saving years suppose), in the next eight years to the Balance Fun an in the last seven years to the Conservative Fun. To compare the performance of ynamic an static strategies we have chosen two representatives of the static ones: 1. The most risky (accepting the legal regulations) strategy with switching s 2 (F 1 F 2 ) an 33 (F 2 F 3 ). 2. The strategy with switching s 14 an 33 similar to a typical representative of ynamic strategies with the risk aversion parameter a = 9. In Figure 8 one can see the average t evelopment an E( t ) t intervals for chosen static strategies. The strategy with switching s 14 an 33 has the same E( T ) = 4.67 comparing to a ynamic one with a = 8 but significantly higher the stanar eviation of T, T = 1.41 (compare to 1. for the ynamic strategy with a = 8). A mean-variance analysis of ynamic strategies with ifferent risk aversion (represente by the curve-efficient frontier) an the two static ones is epicte in Figure 9. The static strategies are clearly inefficient. 18 Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

14 FIGURE 8 Static Strategies the Development of Savings a) switching s 2, 33 b) switching s 14, 33 FIGURE 9 Mean-Variance Analysis of Dynamic an Strategic Strategies E (T) σ. Conclusions We have presente a ynamic accumulation moel for etermining optimal switching strategies for choosing pension funs with ifferent risk profiles. It turne out that ynamic strategies coul be more efficient compare to static ones. The results of simulations of a mathematical moel have illustrate that graual ecreasing of the risk (incorporate in the corresponing legislation) is reasonable an can be supporte by means of a ynamic accumulation moel. The resulting strategies epen on iniviual risk preferences of the future pensioners represente by their iniviual utility functions. In accorance with common intuition, higher wage growth implies lower performance of the fune pillar relative to the pay-as-you- -go pillar. Since it is very ifficult to preict the future asset returns, the results were calculate for various means of asset returns istributions. Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

15 Appenix Numerical Approximation Scheme In this section we iscuss the numerical approximation scheme we use in our pension portfolio simulations. The principal ifficulty in computing the Bellman integral (9) is ue to significant oscillations in the integran function. More precisely, it may attain both large values as well as low values of the orer one. Therefore a scaling technique is neee when computing the integral (9). The iea of scaling is rather stanar an is wiely use in similar circumstances. Let H t () be any boune positive function for t = 1, 2,..., T. We scale the function V t by H t, i.e. we efine a new auxiliary function W t () = H t ()V t () Clearly, the original function V t () can be easily calculate from W t () as follows: V t () = W t ()/H t (). Then, for each step t from t = T own to t = 2 we have W T () = H T ()V T () an W t 1 () = H t 1 ()V t 1 () 1 + t = max H t 1 ()V t (1 + r) + f t j (r)r (13) j 1,2,...,m R H t 1 ()W t (1 + r) t = max f j t (r)r j 1,2,...,m R H t (1 + r) t H t 1 ()W t (y) 1 + t 1 + = max j f t t (y t) 1 y j 1,2,...,m H t (y) R It is worthwhile to note that any choice of the family H t, t = 1,..., T, of positive boune scaling function oes not change the result. It may however significantly improve the stability of numerical computation. In orer to capture both large an small values of V t we recursively efine the scaling functions H t, t = T, T 1,..., 2, 1, epening on the previously compute solution V t+1 as follows: 1 1 H T =, an H t = for t= T 1,..., 1 (14) 1 + V 2 T 1 + V 2 t+1 In our algorithm we compute values of the function W t = W t () for iscrete values of from the epenent interval ( min, t/2), where we use min = The upper boun t/2 has been chosen with respect to maximal expecte values of the savings-to-yearly-salary ratio. In each level t = T own to t = 1 we choose a uniform spatial iscretization of the interval ( min, t/2) consisting of k = 0 mesh points. Stochastic fun r tj returns were assume to have normal istributions with ensities f tj having constant in- means r j an stanar eviations j, j = 1,..., m. In orer to compute numerically the Bellman type integral with normal istribution ensities f tj we use the Simpson rule with 11 equiistant gri points covering the essential interval ( r j 3 j, r j + 3 j ). Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã

16 REFERENCES BENCZÚR, P. (1999): Changes in the Implicit Debt Buren of the Hungarian Social Security. National Bank of Hungary, Working Paper, August CHLON, A. GÓRA, M. RUTKOWSKI, M. (1999): Shaping Pension Reform in Polan: Security Through Diversity. The Worl Bank, Social Protection Discussion Paper Series, August FRIEND, I. BLUME, M. E. (197): The Deman for Risky Assets. The American Economic Review, vol. 6, 197, no., pp FULTZ, E. (02): Pension Reform in Central an Eastern Europe. Vol. 1. Buapest, ILO, 02. GOLIA, P. (03): Pension Calculations for the PAYG an the Fune Pension System in Slovakia. Acaemia Istropolitana Nova, Professional Programme in Applie Economics an Finance, August 03. MEHRA, R. PRESCOTT, E. (198): The Equity Premium: a Puzzle. Journal of Monetary Economics, vol. 1, 198, pp MELICHERâÍK, I. UNGVARSK, C. (04): Pension Reform in Slovakia: Perspectives of the Fiscal Debt an Pension Level. Finance a úvûr - Czech Journal of Economics an Finance, vol. 4, 04, no. 9-, pp PALACIOS, R. ROCHA, R. (1998): The Hungarian Pension System in Transition. In: L. Bokros J. Dethier (es.): Public Finance Reform uring the Transition: The Experience of Hungary. The Worl Bank, Washington, D.C., 1998, pp PRATT, J. W. (1964): Risk Aversion in the Small an in the Large. Econometrica, vol. 32, 1964, no. 1-2, pp SIMONOVITS, A. (00): Partial Privatization of a Pension System: Lessons from Hungary. International Journal of Development, vol. 12, 00, pp THOMAY, M. VEJNA, I. ORAVEC, J. (02): Koncepcia reformy ôchokového systému. The F. A. Hayek Founation, Bratislava, June 02. YOUNG, H. P. (1990): Progressive Taxation an Equal Sacrifice. The American Economic Review, vol. 80, 1990, no. 1, pp SUMMARY JEL classification: C1, E27, G11, G23 Keywors: Bellman equation; ynamic stochastic programming; fune pillar; pension portfolio simulations; risk aversion; Slovak pension system; utility function A Dynamic Accumulation Moel for the Secon Pillar of the Slovak Pension System Soňa KILIANOVÁ Igor MELICHERČÍK Daniel ŠEVČOVIČ: All authors: Department of Applie Mathematics an Statistics, Faculty of Mathematics, Physics an Informatics, Comenius University, Bratislava Corresponing author: Igor Melicherčík (igor.melichercik@fmph.uniba.sk) Since January 0, pensions in Slovakia are operate by a three-pillar system as propose by the Worl Bank. This paper concentrates on the manatory, fully fune secon pillar. The authors present a ynamic accumulation moel for etermining the optimal switching strategy among pension funs with ifferent risk profiles. The resulting strategies epen on the iniviual risk preferences of future pensioners. The authors results illustrate that graual ecreasing risk while amassing savings for a pension is rational. Furthermore, the authors present several simulations of optimal fun-switching strategies for various moel parameter settings. Finance a úvûr Czech Journal of Economics an Finance, 6, 06, ã