Predicting the development of voluntary pension funds in serbia by applying the mathematical method of linear regression

Nội dung text: Predicting the development of voluntary pension funds in serbia by applying the mathematical method of linear regression

1. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 PREDICTING THE DEVELOPMENT OF VOLUNTARY PENSION FUNDS IN SERBIA BY APPLYING THE MATHEMATICAL METHOD OF LINEAR REGRESSION Ivan D. Radojkovic “Dunav” voluntary pension fund management company, Nis, Serbia Boban Gajic “Dunav insurance”, a.d.o. Belgrade, Serbia Branislav Randjelovic University of Nis, Faculty of EE, Nis, Serbia ABSTRACT The introduction of private pension funds, which operate with the state pension fund, is the essence of the reform of the pension system in Serbia. Private pension funds are based on voluntary benefits. Thus, the functioning of the pension system takes place in three interconnected processes: payments to a voluntary pension fund, investment of free funds, and ultimately programmed payments – pensions. Stability in the voluntary pension funds and predictability of payments allow the quality of investment portfolio to be formed and achieve a long-term yield of investment. In this work we implement the method of least square approximation for data processing and a mathematical method of linear regression, which give a link between the observed size, in our case, the number of fund members, the average salaries in Serbia and the size of Fondex, and to be used to predict the number of fund members depending on other sizes. Based on the data obtained by approximation function we can estimate number of fund members, in dependence of average salary and size of FONDEX. Keywords: pension system, voluntary pension funds, linear regression JEL: C38, G11, G23, J32 1. INTRODUCTION The reform of the pension system in Serbia, which has been going on for years, is yielding results because the state participates in the financing of pensions, where the average state pension was 26,738.00 RSD in February 20181and 28,216.00 RSD in February 20192.Mandatory and voluntary pension insurance operates in Serbia. The pay as you go financing system can work well if the national economy is on the rise and when the number of employees is significantly higher than the number of retirees. If there is no economic self- sustainability of the public pension fund, financed according to the pay as you go principle, the state inevitably intervenes as a financier using general budget funds, and if they are insufficient, it uses special taxes on tobacco, alcohol, gasoline, luxury goods, etc. (Jelena Kočović, Predrag Šulejić, Tatjana Rakonjac Antić 2010, p.493 [1]). Private pension funds function as a fully funded financing system, often called a capital accumulation system or a system of capitalized funds. Basically, the amount of pension compensation depends on the amount of accumulated premiums (contributions) and the return on invested premiums (contributions). (Jelena Kočović, Predrag Šulejić, Tatjana Rakonjac Antić, Osiguranje 2010, p.493 [1]). At the end of the fourth quarter of 2019, 201,587 users3 were in the accumulation phase. It should be noted that the membership in the fund is divided into two phases - the accumulation phase (the period in which the funds are paid) and the withdrawal phase (the period when the member withdraws 1 June 16, 2018). 2 March 07, 2020). 3 p.9
2. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 the accumulated funds)4. The stability of inflows into voluntary pension funds and the predictability of payments enable the formation of a quality investment portfolio and the realization of a long-term return on investment. (Ivan Radojković, Boban Gajić 2017, p.34 [7]). The strategic goal in this area is to introduce a healthy multi-pillar pension system.(Ivan Radojković 2012, p.41 [2]). 2. CHARACTERISTICS AND SIGNIFICANCE OF VOLUNTARY PENSION FUNDS The enactment of the Law on Voluntary Pension Funds and Pension Plans, adopted in September 2005 - which entered into force on 1 April 2006, with its first amendment on 7 May 2011 - provided the legal framework for pension reform. in Serbia. This law introduces the third pillar of pension insurance. Private pensions are completely independent of state pensions and are based on the principle of personal accounts. The funds of the private pension fund are invested in financial instruments that provide portfolio optimization, ie. give the best ratio of investment risk and rate of return. Voluntary pension fund funds are invested in accordance with the following investment principles prescribed by law: 1) the principle of security, which is achieved by investing in securities of issuers with a high rating; 2) the principle of portfolio diversification, which is achieved by investing in various financial instruments (government bonds, corporate bonds, treasury bills, shares, bank deposits, mortgage bonds, etc.). By applying different quantitative methods, horizontal diversification is performed, ie.the selection of specific securities within different types of instruments on offer. The most important issuers of financial instruments are the state, commercial banks, companies, and local self-government; 3) the principle of maintaining liquidity, which is achieved by investing in securities that can be quickly sold and bought at a stable price. The fund's goal is to have a sufficient percentage of liquid financial instruments in its portfolio to be able to meet its obligations at any time5. Articles 31, 32, 33 and 34 of the Law on Voluntary Pension Funds and Pension Plans ("Official Gazette of RS", no. 85/2005, 31/2011), precisely define where the assets of a voluntary pension fund can be invested. The members of the Fund themselves choose the Fund to which they will pay the money, the manner and amount of payment, as well as the manner of payment of the pension. There are currently four voluntary pension fund management companies operating in Serbia6, which manage seven voluntary pension funds7. Fund members can start withdrawing funds at the age of 53 or 58, depending on when they joined the Fund8. The data in the table indicate solid returns of funds, which indicates that the funds place the collected funds well. Table following on the next page 4 p.12 5 še/investiciona-politika/Investiciona načela (accessed April 27, 2017). 6 May 10 2017). 7 May 10, 2017) 8 10, 2017).
3. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Table 1: Number of members, assets and rates of return for voluntary pension funds operating in Serbia9 (Source: Statistical Anex of NBS for December 2019.) Fund Members Assets (in millions of RSD) Yield (2019) Generali Basic 46535 13.075,8 9,14% Generali Index 4966 1.095,6 8,34% Raiffaisen Future 35064 5.459,9 4,87% Raiffaisen Euro Future 4464 225 2,91%10 DDOR GarantEkvilibrio 53517 6.050,3 5,63% DDOR GarantŠtednja 19287 1.328,4 7,85% DUNAV 87195 18.010,5 6,80% Yield rates of voluntary pension funds are also favorable if the exchange rate movements during the last year are taken into account. On January 3, 2019, 1 euro amounted to 118.3439 dinars11 and on December 31, 2019, 1 euro amounted to 117.592812. As a percentage, the fall of the euro is 0.64%, while the annual inflation in 2019 was 1.9%13. Based on the information from Table 2, positive trends can be observed in the growth of the Fund's net assets as well as in the number of beneficiaries. The influence of various factors in society on the development of pension funds, as well as the possibility of predicting development in this domain, are the subject of a number of papers from different countries and parts of the world, on which we based our research in this paper (H.C. Benediktsson, T.T. Gerbertssonand J.M. Orszag, 2001 [3];D. Blake, 2004 [8];S. Chlon, 2002 [9]; W.L. Dellvaand G.T.Olson, 1998 [10];R. Ottenand D. Bams, 2002 [11]; A.F.M.Shamsuddin, 2001 [14]; A. Kabašinkskas, K. Šutiene, M. Kopaand E. Valakevičius, 2017 [16]; C. Marti, J.C. Matallin and M.A.Fernandez, 2009 [17], J.Bikker, O.W.Steenbeek and F.Torrachi, 2011 [20], Ch.Cheng and F.Uzelac, 2016 [21], W.Gerke, F.Mager, T.Reinschmidt, C.Schmieder, 2008 [22]). Table following on the next page 9 (accessed March 10, 2020) 10 (accessed March 10, 2020) 11 March 08, 2020). 12 March 08, 2020). 13 March 08, 2020).
4. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Table 2: Key indicators of voluntary pension funds in Serbia development14 Indicators Companies for Voluntary Members Contracts Net estate of managing pension funds in voluntary pension funds millions funds RSD15 2010 6 8 166780 220451 9.862,7 2011 6 9 174868 234405 12.452.3 2012 5 9 179823 240369 16.011,3 2013 4 6 183508 244462 19.007,7 2014 4 6 187997 252072 23.565,3 2015 4 7 190492 258680 28.874,8 2016 4 7 183553 250460 32.790,1 2017 4 7 185445 253900 36.200,0 2018 4 7 192295 261726 40.185,0 2019 4 7 201587 275833 45.245,5 3. LINEAR REGRESSION By the term linear regression (O.David, 2017 [4]) we mean any modelling of the relationship between a quantity, which we call a dependent variable (we can denote it by ) and one or more quantities, which we call independent variables (we can denote them by 1, 2 푛,), so that that model is a linear dependence on independent variables16,forms = 1 1 + 2 2 + ⋯ + 푛 푛 + , where 1, 2 푛, are real numbers. If the dependence is on several independent variables, the process is called multiple linear regression. If the dependence of the variable on only one independent variable = + , then it is a simple linear regression. Linear regression is easy to use in practical applications, because models that linearly depend on their unknown parameters are easier to model than models with nonlinear dependence on parameters. Most applications of linear regression fall into one of the following two types: • If the goal is prediction, linear regression is used to determine the predictive model according to the considered data set of values of dependent and independent quantities. When the appropriate model is obtained, then the corresponding value of the dependent variable can be determined for some new values of the independent variable . • If the goal of regression analysis is to quantify the strength of the relationship between the dependent variable and each of the independent variables 1, 2 푛. In this paper, we will use the first approach, using an approximation procedure, known as the least squares method (discrete mean square approximation). The least squares method (discrete mean square approximation) (G.V. Milovanović 1991 [5])belongs to the so-called the best approximations, ie.approximation methods in which the criterion is the minimization of the error according to one of the norms. 14 . 15Source: National Bank Serbia 16 (accessed April 20, 2020).
5. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Specifically, this is the norm 퐿2, ie. the total sum of the squares of the errors in the approximation nodes is minimized (G.V. Milovanović, M.A. Kovačević, 1991 [6]). We also have had, on our minds, some results of previous considerations regarding predictions and risk models (T.C.Wong, C.H.Hui, C.F.Lo, 2010 [18], A.Amendola, M.Restaino, L.Sensini, 2011 [19]). 4. MAIN RESULTS In this paper, we applied the method of linear regression to the given data from the following table, which shows the values of the average salary in Serbia, the value of FONDEX, as well as the number of fund members in a period of 5 years (2015-2019). Table 3: Data for the Republic of Serbia for the period 2015-2019 Year Average salary FONDEX Fund members 2019 54.908,25 3064.86 201587 2018 49.642,59 2862.92 192295 2017 47.887,67 2713.39 185445 2016 46.836,75 2592.50 183553 2015 44.436,50 2407.45 190490 We first applied a simple linear regression, to determine the dependence of the number of fund members on the average salary, and then a simple linear regression to arrive at a relationship between the number of fund members and FONDEX. Finally, we received a complex, multiple linear regression, to determine the dependence of the number of fund members on both the average salary and the FONDEX. The obtained dependences are given in the next two sections, together with the prediction tables. 4.1. Simple linear regression Based on the data from Table 3, where we take the number of users for the independent variable ( ), and as the dependent variable ( ) the average salary, we will apply the procedure of forming a linear regression approximation function, form φ1(), x=+ a bx by the method of least squares . We will start from the initial condition, that the error of approximation in the nodes is equal to zero, ie. φ1( xkk )== y ( x )( k 1, , n ) . We have system of linear equations ỡ φ(54.908250)= a + b ì 54.908250 = 201587 ù 1 ù φ1(49.642590)= a + b ì 49.642590 = 192295 ù ớ φ(47.887670)= a + b ì 47.887670 = 185445 ù 1 ù φ(46.836750)= a + b ì 46.836750 = 183553 ù 1 ùợ φ1(44.436500)= a + b ì 44.436500 = 190490
6. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 If we transform this system into a matrix form, we have ộựộự154.908250201587 ờỳờỳ ờỳờỳ149.642590192295 ờỳờỳ ộựa ờỳờỳ147.887670185445ì=ờỳ ờỳờỳ ờỳởỷb ờỳờỳ146.836750183553 ờỳờỳ x ờỳờỳởỷởỷ144.436500190490 A b Table 4: Prediction of the number of fund members depending on average salary Avarage salary in Serbia Fund members 50.000,00 194966 52.500,00 204753 55.000,00 214541 57.500,00 224329 60.000,00 234117 62.500,00 243904 65.000,00 253692 67.500,00 263480 70.000,00 273268 and we solve, by means of Aì x =ịì bAA/ =TTT A xA b ộự- 790.405006 ()/ATTTTT ()() Aì xA =ị=ì= bA AxA AA 11 b ờỳ ởỷờỳ3915.118438 So the linear regression function for this case is φ1( x )= - 790.405066 + 3915.118438 x . Using the obtained function, we can make a prediction of the number of fund members, depending on further possible changes in the average salary, which is shown in Table 4. On the other hand, if we take the number of users for the independent variable ( ) and FONDEX as the dependent variable ( ), then we apply the procedure of forming a linear regression approximation function, form φ2 (), x=+ a bx by the method of least squares.
7. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 We will start with the initial condition, that the error of approximation in the nodes is equal to zero, ie. φxyxkn2 ()()(1, ,)kk== ỡ φ(3064.86)= a + b ì 3064.86 = 201587 ù 2 ù φ2 (2862.92)= a + b ì 2862.92 = 192295 ù ớ φ(2713.39)= a + b ì 2713.39 = 185445 , ù 2 ù φ(2592.50)= a + b ì 2592.50 = 183553 ù 2 ùợ φ2 (2407.45)= a + b ì 2407.45 = 190490 If we transform this system into a matrix form, we have ộựộự13064.86201587 ờỳờỳ ờỳờỳ12862.92192295 ờỳờỳ ộựa ờỳờỳ12713.39185445ì=ờỳ , ờỳờỳ ờỳởỷb ờỳờỳ12592.50183553 ờỳờỳ x ờỳờỳởỷởỷ12407.45190490 A b and we solve, by means of Aì x =ịì bAA/ =TTT A xA b ộự- 1164.563403 ()/ATTTTT ()() Aì xA =ị=ì= bA AxA AA 11 b ờỳ ởỷờỳ69.974794 So the linear regression function for this case is φxx2 ( )1164.56340369.974794= -+ Using the obtained function we can make a prediction of the number of fund members, depending on further possible changes of FONDEX, which is shown in the table 5. Table 5: Prediction of the number of fund members depending on FONDEX FONDEX Fund members 3000.00 208760 3050.00 212259 3100.00 215757 3150.00 219256 3200.00 222755 3250.00 226254 3300.00 229752 3350.00 233251 3400.00 236750
8. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 4.2. Complex linear regression Based on the data from Table 3, where we take the number of users for the independent variable ( ), and the average salary value of FONDEX as dependent variables, we will apply the procedure of forming a multiple linear regression approximation function, form φxyabxcy3 (,), =++ usingleast squares method. We will start with the initial conditions, that the error of approximation in the nodes is equal to zero, ie. φxyzxykn3 (,)(,)(1, ,).kkkk == ỡ abc+ì+ì=54.9082503064.86201587 ù ù abc+ì+ì=49.64259029862.92192295 ù ớ abc+ì+ì=47.8876702713.39185445 ù ù abc+ì+ì=46.8367502592.50183553 ù ùợ abc+ì+ì=44.4365002407.45190490 If we transform this system into a matrix form, we have ộựộự154.9082503064.86201587 ờỳờỳ ờỳờỳ149.6425902862.92192295ộựa ờỳờỳ ờỳ ờỳờỳ147.8876702713.39185445ì=ờỳb ờỳờỳ ờỳ ờỳờỳ146.8367502592.50183553ờỳởỷc ờỳờỳ ờỳờỳởỷởỷ144.4365002407.45190490 x A b and we solve, by means of Aì x =ịì bAA/ =TTT A xA b ộự- 600.7212221 ờỳ ()/ATTTTT ()()9158.414506 Aì xA =ị=ì= bA AxA AA 11 b ờỳ ờỳ ờỳởỷ- 93.69044454 So the linear regression function for this case is φ3 ( x , y )= - 600.7212221 + 9158.414506 x - 93.69044454 y Using the obtained function we can make a prediction of the number of fund members, depending on further possible changes in the average salary and FONDEX, which is shown in the following table: Table following on the next page
9. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Table 6: Prediction of the number of fund members depending on average salary and FONDEX Average FONDEX Fund Average FONDEX Fund salary members salary members 50000 3000.00 176249 52500 3000.00 199145 50000 3050.00 171564 52500 3050.00 194460 50000 3100.00 166880 52500 3100.00 189776 50000 3150.00 162195 52500 3150.00 185091 50000 3200.00 157511 52500 3200.00 180407 50000 3250.00 152826 52500 3250.00 175722 50000 3300.00 148142 52500 3300.00 171038 50000 3350.00 143457 52500 3350.00 166353 50000 3400.00 138772 52500 3400.00 161669 50000 3450.00 134088 52500 3450.00 156984 50000 3500.00 129403 52500 3500.00 152299 Table 7: Prediction of the number of fund members depending on average salary and FONDEX Average FONDEX Fund Average FONDEX Fund salary members salary members 55000 3000.00 222041 57500 3000.00 244937 55000 3050.00 217356 57500 3050.00 240252 55000 3100.00 212672 57500 3100.00 235568 55000 3150.00 207987 57500 3150.00 230883 55000 3200.00 203303 57500 3200.00 226199 55000 3250.00 198618 57500 3250.00 221514 55000 3300.00 193934 57500 3300.00 216830 55000 3350.00 189249 57500 3350.00 212145 55000 3400.00 184565 57500 3400.00 207461 55000 3450.00 179880 57500 3450.00 202776 55000 3500.00 175196 57500 3500.00 198092 Table 8:Prediction of the number of fund members depending on average salary and FONDEX Average FONDEX Fund Average FONDEX Fund salary members salary members 60000 3000.00 267833 62500 3000.00 290729 60000 3050.00 263148 62500 3050.00 286044 60000 3100.00 258464 62500 3100.00 281360 60000 3150.00 253779 62500 3150.00 276675 60000 3200.00 249095 62500 3200.00 271991 60000 3250.00 244410 62500 3250.00 267306 60000 3300.00 239726 62500 3300.00 262622 60000 3350.00 235041 62500 3350.00 257937 60000 3400.00 230357 62500 3400.00 253253 60000 3450.00 225672 62500 3450.00 248568 60000 3500.00 220988 62500 3500.00 243884
10. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Table 9: Prediction of the number of fund members depending on average salary and FONDEX Average FONDEX Fund Average FONDEX Fund salary members salary members 65000 3000.00 313625 67500 3000.00 336521 65000 3050.00 308940 67500 3050.00 331836 65000 3100.00 304256 67500 3100.00 327152 65000 3150.00 299571 67500 3150.00 322467 65000 3200.00 294887 67500 3200.00 317783 65000 3250.00 290202 67500 3250.00 313098 65000 3300.00 285518 67500 3300.00 308414 65000 3350.00 280833 67500 3350.00 303729 65000 3400.00 276149 67500 3400.00 299045 65000 3450.00 271464 67500 3450.00 294360 65000 3500.00 266780 67500 3500.00 289676 Table 10: Prediction of the number of fund members depending on average salary and FONDEX Average FONDEX Fund Average FONDEX Fund salary members salary members 70000 3000.00 359417 72500 3000.00 382313 70000 3050.00 354732 72500 3050.00 377628 70000 3100.00 350048 72500 3100.00 372944 70000 3150.00 345363 72500 3150.00 368259 70000 3200.00 340679 72500 3200.00 363575 70000 3250.00 335994 72500 3250.00 358890 70000 3300.00 331310 72500 3300.00 354206 70000 3350.00 326625 72500 3350.00 349521 70000 3400.00 321941 72500 3400.00 344837 70000 3450.00 317256 72500 3450.00 340152 70000 3500.00 312572 72500 3500.00 335468 5. CONCLUSION AND DIRECTIONS OF FURTHER RESEARCH In first two sections of this paper, we analyzed the work of voluntary pension funds in Serbia in terms of financial results. In the continuation of the paper, we modelled the behavior and interdependence of the private pension fund members and two relevant parameters, which are in corelationwith fund members. In the first case, the approximation function, which represents a mathematical model of behavior, that is not relevant enough, because it takes into account only one of the factors, and that is the average salary in the country. In the second case, the approximation function that represents the mathematical model of behavior is more relevant, but still insufficient, because it takes into account only FONDEX as variable. Finally, the last approximation function, a mathematical model in which the number of fund members is modeled over the average salary and size of FONDEX, gives a very precise picture and gives opportunity for good prediction of growth and development of the pension system, number of fund members, and its relevance is based on interaction and influence of two independent factors.
11. Journal of Economic and Social Development (JESD) – Resilient Society - Vol. 8, No. 1, March 2021 Based on the data obtained in Tables 6, 7, 8, 9 and 10, we can be sure of a good estimate obtained by this mathematical model, specifically on the current situation, with the current average salary and the current size of FONDEX. Further research and analysis should consider determining the structure of the population, which could affect the improvement of the performance of voluntary pension funds in Serbia. With these new solutions and a better employment ratio, and with an increase in the number of members who pay contributions to private pension funds (eg that 10% of the population saves in private pension funds, currently about 2%), future retirees in Serbia can expect safer and more certain the future 6. DECLARATION OF INTEREST The authors report no conflict of interest. The authors alone are responsible for the content and writing of the paper. ACKNOWLEDGEMENT: This work is under grants TR-32012 and III-43007 Ministary of Education, Science and Technological development of Republic of Serbia. REFERENCES 1. Jelena Kočović, Predrag Šulejić and Tatjana Rakonjac Antić. 2016. “Osiguranje”, Ekonomski fakultet, Beograd. 2. Ivan Radojković.2012. “Significance and Prospects of Voluntery Pension Funds in Serbia”, Tokovi osiguranja,Vol.3: 41-47 (in Serbian). 3. H.C. Benediktsson, T.T. Gerbertssonand J.M.Orszag.2001.”The charge ratio on individual accounts and investment plans in Iceland”. Applied Economics, 33(9): 79–87. 4. Olive David, “Linear Regression”. 2017. Springer International Publishing 5. Gradimir V. Milovanović. 1991.“Numerical Analysis II”, NaucnaKnjiga, Belgrade, (First Edition 1985, Second Edition 1988, Third Edition 1991), VIII+210 pp. (Serbian); MR 87e: 65001b. 6. Gradimir V. Milovanović, Milan A. Kovačević. 1991.“A Collection of Solutions for Problems in Numerical Analysis”, NaucnaKnjiga, Belgrade, (First edition, Second Edition 1988, Third Edition 1991), IV+278 pp. (Serbian). 7. Ivan Radojković, Boban Gajić. 2017. „Development of Voluntary Pension Funds in Serbia“, Tokovi osiguranja, Beograd 2017, Vol 4 : 33-44(in Serbian) 8. D. Blake. 2004. “The impact of wealth on consumption and retirement behaviour in the UK”. Applied Financial Economics, 14(5):55–76. 9. Agniezcka Chlon. 2002. “Administrative costs of pension funds in Poland in international perspective”, Room Document No. 24, Regional Meeting for the Eastern and Central European Countries, Tallinn, Estonia, 7–8 February 2002. 10. W.L. Dellva and G.T. Olson. 1998. “The relationship between mutual fund fees and expenses and their effects on performance”. The Financial Review, 33: 85–104. 11. R. Ottenand D. Bams. 2002. European mutual fund performance. European Financial Management, 8,pp.75–101. 12. Statistički godišnji bilten za 2016.2017. Republički fond za penzijsko i invalidskoosiguranje, Beograd (in Serbian). 13. Sektor dobrovoljnih penzijskih fondova u Srbiji. 2017. Izveštaj za četvrtotromesečje 2016. Godine, Narodnabanka Srbije (in Serbian). 14. A.F.M. Shamsuddin. 2001. “Public pension and wealth inequality in Canada”. Applied Economics Letters, 8: 315–320. 15. A. Stanković. 2017.“Prezentacija 5-4-2017”, DunavDPF Beograd, “Prezentacija 5-4-2017”
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