NOTIONAL DEFINED CONTRIBUTION ACCOUNTS (NDCs): SOLVENCY AND RISK; APPLICATION TO THE CASE OF SPAIN *

Transcription

1 NOTIONAL DEFINED CONTRIBUTION ACCOUNTS (NDC): SOLVENCY AND RISK; APPLICATION TO THE CASE OF SPAIN * MARÍA DEL CARMEN BOADO-PENAS INMACULADA DOMÍNGUEZ-FABIÁN CARLOS VIDAL-MELIÁ ABSTRACT December 22, 2006 The aim of thi work i twofold, on the one hand, to demontrate the actuarial imbalance of the Spanih penion ytem in it current configuration, and on the other, to meaure the aggregate economic rik to which the penioner would be epoed if it were decided to apply ten formula for the calculation of the retirement penion baed on notional account. Given the uncertainty involved in working with a long term horizon, a model of generation of multi-periodic cenario i ued, baed on the prediction of mean value of Alono and Herce (2003) for the period Thi provide up to ten thouand trajectorie, per each year, of the macroeconomic indice needed to calculate uch parameter a the initial penion, the replacement rate (RR) or the internal rate of return (IRR), and the value-at-rik (VaR) of the penioner. The reult obtained are analyzed in both objective and ubjective term. The main concluion are that, applying the notional philoophy, the epected average RR and IRR would be much lower than thoe obtained under the current rule of the pay-a-you-go ytem. If the projection ued were lightly probable, the penion ytem would build up uch a large additional financial imbalance in the future that it would require either a coniderable reduction in the initial penion or a evere combination of parameter adjutment. From the rik perpective, the preferred formula for a beneficiary mot avere would be thoe baed on future variation in alarie with a penion contant in real term, wherea thoe beneficiarie le avere to rik would prefer formula upplying a lower initial penion which grow in real term in line with future variation in alarie. (JEL: H55, J26). Keyword: Rik averion, Spain, Scenario generation, Retirement, Penion. * Inmaculada Domínguez-Fabián and Carlo Vidal-Meliá gratefully acknowledge the financial upport received from the Miniterio de Trabajo y Aunto Sociale [Minitry of Work and Social Affair] directed to tudie and reearch activitie in the contet of ocial protection (Order TAS/1051/2005), project "Improvement of the equity and financial utainability of Spain' public retirement penion ytem by mean of the employment of notional defined-contribution cheme (NDC)". Maria del Carmen Boado-Pena gratefully acknowledge the financial upport from the Department of Education, Univeritie, and Reearch of the Baque Country Government through a reearcher training grant. The three author are incerely grateful to Pierre Devolver, Ole Settegren, Elieo Navarro-Arriba, Francico Climent-Diranzo and very epecially to Salvador Valdé-Prieto for their comment and uggetion. Preliminary verion of thi paper were preented at the XIV Foro de Finanza in Catellón, (Spain) and at the VIII Congreo Panamericano de Actuario in Bueno Aire (Argentina). Any error that the work may contain are entirely the reponibility of the author. Univerity of the Baque Country, Department of Econometric and Statitic, Faculty of Economic and Buine Science, Avenida del Lehendakari Aguirre 83, Bilbao (Spain). mcboado@hotmail.com. Univerity of Etremadura, Department of Financial Economic and Accounting, Faculty of Buine and Tourim Studie, Avenida de la Univeridad /n, Cácere (Spain). idomingu@guadiana.une.e. Univerity of Valencia, Department of Financial Economic and Actuarial Science, Faculty of Economic, Avenida de lo Naranjo /n, Valencia (Spain). Carlo.Vidal@uv.e. 1

2 NOTIONAL DEFINED CONTRIBUTION ACCOUNTS (NDC): SOLVENCY AND RISK, APPLICATION TO THE CASE OF SPAIN 1. INTRODUCTION In Spain numerou work have been carried out that tudy the uppoed inviability, unutainability, inolvency, and financial inufficiency of the public penion ytem, and that propoe meaure aimed at avoiding the long-term financial inolvency of the ytem. According to Alono and Herce (2003), the Spanih contributory penion ytem i doomed to face eriou financial inadequacy from 2020 onward, with the acceleration of the ageing of the Spanih population and the hrinkage of the margin of it activity. Not even the foreeeable increae in fertility and in migratory flow would add enough human reource to avoid that deterioration. In a imilar line for Ahn et al. (2005), the financial ituation of the penion ytem in Spain i perceivably affected by an advere future demographic ituation. During the firt decade of thi century it will have a mall urplu. However, the deficit during the following decade will be high and will grow with time. In 2050, the deficit will be greater than 6% of the GDP with a probability of 90%, and greater than 15% with a probability of 10%. In the ame year, the accumulated deficit 1 would be between 77% and 260% of the GDP, with a confidence interval of 80%. The tudy conducted by the EU (2005) conclude that the percentage ependiture in contributory penion retirement, orphan, widow(er), relative, etc. on GDP will pa from 8.8% in 2005 to 15.7% in It i neceary to emphaize, however, that thi projection i more optimitic than the previou projection made in 2001 which ituated penion pending in 2050 at 17.3%. On the other hand, not all reearcher agree with the previou diagnoi. Thu, for eample, Del Brio and González (2004) find that almot all the prediction on the olvency of the penion ytem have had numerou error in the pat, and that they have turned out to be eceively peimitic. In their work, they preent projection baed on the oberved flow of new immigrant, concluding that the financial problem of the penion ytem will be deferred beyond the year However, even the official ource, MTAS (2005), admit that in the year 2015 the firt deficit of the ytem will appear for an amount equivalent to 0.04% of the GDP. In the ame year the Reerve Fund will begin to be ued in order to balance the reult of the period , and from 2021 effective deficit would begin to be produced. What i more, they agree with the need to et a new reform proce in motion, with a priority character in the framework of the lat recommendation of the "Pacto de Toledo" [Toledo Pact], although they conider that there i a long enough time period in which to carry out the reform. Vidal-Meliá and Domínguez-Fabián (2006) have trongly recommended the neceity of etablihing a profound overhaul of the Spanih public penion ytem. In the preent work, an alternative reform of the ytem i propoed baed on notional defined- 1 Accumulated deficit mut not be confued with implicit debt. According to the lat forecat carried out by Devea and Devea (2005), the implicit debt at of the overall public penion ytem (general regime, pecial, and paive clae), only for the contingency of retirement, varie between 152% and 214% of GDP, according to the method of etimation. 2

3 contribution cheme (NDC) 2. Thi i the mot important innovation that ha been introduced into the field of public penion in recent year, and thee cheme are already in operation in different countrie 3. Thi model ha been propoed, Holzmann (2006), a a fundamental referent for the future unified penion ytem of the European Union. Devea-Carpio and Vidal-Meliá (2004) tudy retropectively how the introduction of penion formula baed on the notional philoophy would have affected the initial amount of the retirement penion and the internal rate of return of the penion ytem in Spain. Their concluion i that the effect of their implantation would have notably reduced both the amount of the penion and the replacement rate (RR) currently produced by the penion formula baed on the traditional defined benefit. Alo, the epected theoretical real internal rate of return (IRR) of the contribution, auming urvival to retirement age, would have paed from around 5.07% (5.64%, in the cae of women) to le than 2.21% (2.91%, for women) for whichever of the applied formula. Vidal-Meliá et al. (2006) carried out a propective analyi of the penion ytem, in which they quantify the aggregate "economic" rik to which the beneficiary in Spain would be epoed if a ytem of retirement penion baed on notional account were introduced. To thi end they ued the cenario generation technique to make projection of the factor that determine the real epected IRR for the beneficiary a a function of iteen retirement formula obtained from the mot widely accepted notional indice or rate. It wa concluded that the IRR, under whichever of the applied retirement formula, would alo be le than the correponding value under current Spanih legilation, while the mot favourable replacement rate would only jut reach 50.5% for a peron of 65 year and with 40 year of contribution. The preent work, which i directly related to that of Vidal-Meliá et al. (2006), i aimed at perfecting the cenario generation technique ued for the projection of the macroeconomic indice that determine the value of uch variable a the initial penion or IRR, amongt other. The number of poible cenario, per each year, that could arie for each formula of the calculation of the penion i increaed to ten thouand, in order for the reult to gain in robutne. We ue a mean value to obtain the trajectorie of the macroeconomic indice three new, more up-to-date, baic macroeconomic projection Alono and Herce (2003), MTAS (2005) and EU (2005). We then analyze the utility of the penion from both an objective and a ubjective perpective, taking the individual' rik averion into account. Another contribution with repect to the referred work i that we perform a enitivity analyi of the anticipated and unanticipated change in the urvival rate, of the epected average growth, and of the change in the bae macroeconomic projection. The work i organized a follow. Following thi Introduction, Sec. 2 decribe the model, differentiating two part firt, a brief overview of the actuarial concept of the notional account, and econd, detail of the model ued for the projection of the macroeconomic variable that intervene in the notional formula. Section 3 preent the principal aumption, tarting data, and notional account formula ued. Section 4 preent and 2 A Gronchi and Niticò (2006) note, the original idea of the NDC wa preent in two work publihed in the 1960 by Buchanan (1968) and Catellino (1969) which were redicovered in the late Italy (1995), Latvia (1996), Khirghitzia (1997), Poland (1999), Sweden (1999), and Mongolia (2000). According to Williamon (2004), other countrie uch a China and Ruia, are eriouly conidering the introduction of thi ytem of penion. 3

4 analyze the principal reult: RR, average IRR, value-at-rik (VAR) of the IRR, Markowitz ranking of the IRR and the utility of the penion. Section 5 decribe the enitivity analyi of the anticipated and unanticipated change of the urvival rate, the epected average growth, and the change in the bae macroeconomic projection. The work end with the main concluion, future line of reearch, the literature reference, and a technical appendi. 2. THE MODEL A notional account i a virtual account containing the individual contribution of each participant and the fictitiou return that thee contribution generate over the coure of the working life. The return are calculated in accordance with a notional rate that may be the growth rate of the GDP, of average wage, of aggregate wage, of contribution payment, etc. When the individual retire, he or he (henceforth, he) receive a benefit that i derived from the accumulated notional fund, the pecific mortality of the cohort retiring in that year, and the applied notional rate. According to Valdé-Prieto (2000, 2005), the ytem of notional account i a very ueful way in which to minimize the political rik aociated with pay a you go ytem, and increae the financial olvency of the ytem in the long term, although it alo increae the eplicit economic rik affecting the contributor. A Diamond (2006) correctly point out, all the advantage attributed to NDC 4 could be obtained with a well deigned ytem of defined benefit, although of coure thi i preciely the difficulty inherent in uch ytem the eae with which erroneou political deciion convert them into badly deigned ytem. Following the development of Vidal-Meliá et al. (2006), in order to calculate the initial penion of an individual at the age of retirement, in the model of notional account, the contribution made and valued at the date of retirement are made equal to the penion that he i going to receive until hi death, alo valued at the age of retirement, i.e.: K= Notional Accumulated Fund r = e CR CB 1 r i= (1 + r ) = P i r a&& β { r G= Converion Factor [1] where: CR : Contribution rate at age "", defined a the percentage to apply on the bae contribution. CB : Contribution bae for the contingency of retirement at age "". CR CB : Effective contribution for an age "". e : Age of entry to the labour market. 4 See in thi regard the work of Barr and Diamond (2006), Börch-Supan (2006), Lindbeck and Peron (2003), Palmer (2006), Vidal et al. (2004), and Williamon (2004), inter al. 4

5 r : Age of retirement. r i : Notional rate that i applied to capitalize the contribution. P r : Initial penion at the age of retirement. K: Notional accumulated fund when the individual reache the age of retirement. β a& & r : Preent value, at the age r, of a life pre-payable actuarial annuity due of 1 per year, increaing at the accumulative annual rate of β, with a technical interet rate equal to ρ. Thi actuarial annuity i alo the o-called converion factor (G). The retirement penion will be: P K K = Kg [2] a& & G = r β = r with the invere of the converion factor in thi cae being equal to ( 1+ β ) = (1 + ρ), can be epreed a: 1 g =, which, if && β a r P r K = [3] 1+ e r where e j : Life epectancy at retirement age. With the aumption of Equation 3, ince the initial penion i aumed to be lower, it i conidered to be increaed in real term. If a higher initial penion i deired, to maintain actuarial equilibrium between the notional capital and the epected penion the revaluation would obviouly have to be le than in the uppoition hown. Auming that ( 1+ β ) = (1 + α ρ); α > 1, the new penion at the age of retirement, will be: P r, P r < P * r = a& & K β =α ρ r [4] A wa noted above, the equivalence between a ytem of notional account and a ytem of well defined benefit under certain condition wa demontrated in the work of Cichon (1999) and Devolder (2005). Since the principal objective of the preent work wa to meaure what i the aggregate economic rik to which the penioner would be epoed, projection were made of different macroeconomic variable for a tandard individual who enter the labour market in 2005 at e year of age. It i not eay to etablih a long-term macroeconomic cenario in 5

6 view of how the Spanih economy ha changed over the lat fifty year. The aim therefore i not to etimate the future value of the variable, but the conequence for the beneficiary of the introduction of a ytem of notional account in a contet of uncertainty. Following the approach of Devolder (1993), the model ued to get the trajectorie for the different macroeconomic indice i: I t = µ + λ σ [5] t t I where: I t : Macroeconomic inde "I" in period "t" and under cenario "". Two indice are conidered variation of the gro dometic product (VGDP) and variation of the average earning inde (VAEI). µ t : Real mean value of the macroeconomic inde in the period "t". Information baed on the projection of Alono and Herce (2003), MTAS (2005) and EU (2005). λ t : Variable N(0,1) i different for each period "t" and cenario "". σ I Real tandard deviation of each inde, that incorporate pat information of the variable into the model. In order to calculate thi value the real hitorical value of each inde for the period will be taken into account. Thi cenario generation model aume perfect correlation between the indice in the long term, ince the analyi of the erie of real hitorical data of the indice in Spain for how that the correlation between GDP and VAEI for quinquennial data i 0.65, and increae to 0.92 if decennial data are ued. 3. NOTIONAL FORMULAE, DATA, AND ASSUMPTIONS 3.1. NOTIONAL FORMULAE The formulae that will be eplored in order to determine the initial penion and it ubequent variation are ten and are given in Table 1. Table 1: Formulae for the calculation of the initial penion and it ubequent variation Model Notional rate for contribution Notional rate for penion 1 VGDP Contant in real term 2 VAEI Contant in real term 3 VGDP Contant in real term ± differential VGDP 4 VGDP Contant in real term ± differential VAEI 5 VAEI Contant in real term ± differential VGDP 6 VAEI Contant in real term ± differential VAEI 7 VGDP VGDP 8 VGDP VAEI 9 VAEI VGDP 10 VAEI VAEI 6

7 In the ten model analyzed, the notional rate of the contribution i either the variation in GDP (VGDP) or wage variation (VAEI). The bai ued to calculate the initial penion i Formula 4 for the firt i model. In theory, thee model will provide higher initial penion, ince later the penion will be contant in real term. The bai for the calculation of the penion in the remaining model, 7 10, i Formula 3. In thi cae, the initial penion (replacement rate) will be maller, although the penion will be increaed in real term. If the notional rate of the penion include ± differential (Formulae 3-6), thi rate will be adjuted through a poitive or negative difference. For eample, for Model 3, the penion will take the real behaviour of the GDP with repect to the epected behaviour into account (average value of the ued macroeconomic projection) in uch a way that, if the real value i greater or le than the epected value, a poitive or negative difference will be obtained, repectively DATA The projection of the macroeconomic indice given in Table 1 are baed on the cenario generation model decribed above. According to Alono and Herce (2003), MTAS (2005) and EU (2005), the mean value of thee projection have in common that they predict a marked future fall in the rate of GDP growth with repect to the average of the Spanih economy of the period We will take a the bae macroeconomic cenario that of Alono and Herce (2003). The mean cenario depicted by the MTAS (2005) i more peimitic than that of Alono and Herce (2003), and it alo bae the future GDP growth on the baic contribution of employment rather than on productivity growth (wage). The cenario of Alono and Herce (2003) hold that the future bai of GDP growth, given the epected retriction of labour, will be the growth in productivity, which will in turn be tranmitted entirely to wage growth. The projection of the MTAS (2005) i far from complete, and therefore a erie of additional aumption ha to be adopted in order to be able to diaggregate the GDP. It i aumed that the percentage repreented by the productivity in the increae of the GDP in the projection of Alono and Herce (2003) i alo maintained in that of the MTAS (2005). The EU (2005) projection i more optimitic than that of Alono and Herce (2003) for the firt ten year. Subequently, however, a can be een in Figure 1, the forecat growth rate for the GDP and productivity i lower, although the profile are very imilar. Figure 1 how the real hitorical variation of the lat eleven year and the evolution of the macroeconomic variable (VGDP and VAEI) in real term, uing the average value projection Alono and Herce (2003), MTAS (2005), and EU (2005). The mean projection of the tudie conidered reach up to the year The calculation that are to be carried out involve working a a minimum with a horizon of 75 year, ince the maimum age of the PEMF table reache 100 year. It wa aumed that the mean value for the year 2050 will remain contant in ubequent period. 7

8 VGDP 0,05 0,04 Real variation 0,03 0,02 0, Year AH (2003) MTAS (2005) UE (2005) VAEI 0,02 Real variation 0,01 0, Year AH (2003) MTAS (2005) UE (2005) Evolution of the Macroeconomic Variable in Spain ( ) Real variation 0,05 0,04 0,03 0,02 0,01 0,00-0,01-0, Year VGDP Figure 1: Evolution of the macroeconomic variable (VGDP and VAEI) in real term, uing the mean value projection of Alono and Herce (2003), MTAS (2005), and EU (2005), and the real hitorical variation of the lat eleven year. Figure 2 repreent the cenario generated for each of the indice, VGDP and VAEI, and each period, taking a the mean cenario that of Alono and Herce (2003). Thi plot give an idea of of the volatility and the etreme value that the indice can reach over the coure of the projection period. To illutrate thi graphically, Figure 3 how the VGDP and VAEI denity function for three year choen purely for illutrative purpoe 2010, 2030, and The mean cenario wa taken to be that of Alono and Herce (2003). Although one can perceive etreme value in the figure, the probability that they will occur i very low. Thu the VGDP will take value greater than 10% in ome 0.11% of cae and le than -10% in 0.001%. The probabilitie that VAEI will be outide the interval [-10%, 10%] will be 0.07% and 1.55%, repectively. VAEI 8

9 VGDP cenario 0.15 Real value of the VGDP Year VAEI cenario 0.15 Real value of the VAEI Year Figure 2: cenario of VGDP and VAEI for the period , baed on the mean cenario of Alono and Herce (2003). 9

10 Normal denity function fit to VGDP and VAEI for different year VGDP, year 2010 VGDP, year 2030 VGDP, year 2050 VAEI, year 2010 VAEI, year 2030 VAEI, year 2050 Frequency Figure 3: Normal denity function fit to VGDP and VAEI for different year, taking a reference the mean cenario of Alono and Herce (2003) ASSUMPTIONS The working aumption were the following: CR : Contribution rate at age. Thi will be aumed contant and equal to 15% throughout the period. Thi i an approimation, and it i conidered that, according to the data of the Social Security budget, approimately 50% of the common contingencie are aigned to the retirement contingency. The profile of the repreentative individual that i conidered evolve according to the variation of wage in the projection ued, i.e., it will alway be increaing in real term. Neither maimum nor minimum penion are taken into account. It i conidered that wage and contribution bae coincide at all time. e : Age of entering the labour market. It i conidered equal to 25 year. δ : Pure preference rate in time, which reflect the impatience of the penioner to conume. It i conidered contant throughout the period and equal to 2%. Initially, we ue the mortality table of the Spanih population (PEMF-98-99) 5, although ubequently we perform variou enitivity analye taking into account the GRMF-95 table 6 that ome inurance companie apply, and that might better repreent the demographic tendencie of the future Spanih population GRMF-95 (1996): "Mortality probabilitie of the GRMF-95, GKMF-95, and EVK-90 table". Actuario 13,

11 For the determination of the initial penion of the firt i, it i conidered that the notional capital i actuarially equal to a contant pre-payable actuarial annuity at a real interet rate of 3%. For the age of 60, 65, and 70, thi implie a value of G=16.14, 13.94, and 11.65, repectively, conidering the PEMF table. For the determination of the initial penion in the remaining model, the real interet rate i taken to be 1.25%, which implie value of G=19.52, 16.38, and for the age of 60, 65, and 70, repectively. The value i obtained by averaging the life epectancy of men and women. Thi i of great importance in the determination of the amount of the initial penion, and i the denominator in the formula that give the initial penion in each cae, the numerator being the notional capital (K). It i logical that one interet rate i ued in ome of the formulae, and another lower one in other, ince the initial amount of the penion i ubequently increaed differently. 4. RESULTS The reult that are preented below were calculated uing the program Mat-Lab 7, verion 6.0 for Window. The beneficiary i ubject to rik in a much a he doe not know with certainty what will be the IRR of hi contribution, or the replacement rate reached. Thi rik could be qualified a non-diverifiable or ytematic, a it i directly aociated with the overall rik of the economy. The beneficiary' aggregate "economic" rik with repect to the IRR i defined a the poibility that the effective rate of return obtained by the contribution doe not coincide with the epected rate due to the uncertain return on ome economic aet (behaviour of wage, of the GDP, of the contributing population, etc.) that i providing upport to the notional retirement account, and may be een a an indicator of the ytem' financial health. With repect to the RR, it will be the poibility that the RR differ from the epected value due to the uncertain return on the economic aet that i ued to revalue the contribution. The rik that i evaluated i the economic rik (baically from reduction of the GDP or wage growth rate), which i alo dependent on demographic rik (increae in the longevity of the population, fall in the fertility rate or in the activity rate) that affect economic activity and the health of the penion ytem REPLACEMENT RATE (RR) Table 2 give the value of the mean epected replacement rate (RR) for the different formula a a function of the individual' retirement age, and the replacement rate adjutment coefficient (AC) with repect to the retirement age of 65 year. Table 2: Epected mean replacement rate a a function of the average wage, for variou age and AH (2003) projection. Fórmula RR (60) AC RR (65) RR (70) AC 2, 5, % % 87.24% , 3, % % 76.53% , % % 76.37% , % % 67.00% See in thi regard the Web ite of the upplier: 11

12 One oberve that RR increae with increaing retirement age. Thi i becaue the fund accumulated until that age i larger, while there are fewer year in which to receive the retirement penion. Early retirement would apparently be dicouraged, not o much by the dicount with repect to the age of 65 year (adjutment coefficient of 0.723, for Formulae of Table 2), but becaue the RR, already very mall at 65 year, would be even maller at 60 year. The formulae are grouped into four ignificant et a a function of whether the notional rate ued i VGDP or VAEI, or whether the revaluation of the penion i done in real or remain contant. The formula that preent the highet RR value are thoe which are baed on the variation of wage and the amount of the penion remain contant in real term. The RR on the average wage for the age of 65 year, after 40 year of contribution, i 62%, when with the current rule of the Spanih ytem the rate would be over 90%. The meage that i given by thi reult i clear. If the projection ued were even minimally cloe to the truth, the penion ytem would how a very important future imbalance that, to be reolved, would need a coniderable reduction of the initial penion or a combination of evere parameter adjutment. Only for retirement at 70, after 45 year of contribution, doe the RR approach 87% in the bet of cae. An increae of the RR would only be poible with a major etra effort in contribution. If, with the perpective of growth derived from the projection of AH (2003), one wanted to attain replacement rate of 80% of the average wage of all the working life, the contribution rate for Formulae 2 and 10 would be 19.36% and 22.75%, repectively, in any cae far from eaily upported in the Spanih labour market 8. The reult preented in thi ection bring out a manifet problem in the current penion ytem. Neverthele, in order to avoid a lanted analyi that only conider the firt penion, it i neceary to include an indicator the IRR that alo take how the amount of the penion evolve over time into account. 8 It i not difficult to how that the fraction repreented by any replacement rate RR i at a given contribution rate TC i relative to a given replacement rate RR at contribution rate TC i equal to the ratio of the contribution rate (TC i /TC). 12

13 4.2. INTERNAL RATE OF RETURN (IRR) The reult of the epected IRR for each formula will be preented eparately for each retirement age conidered (Table 3, 4, and 5), a well a it deviation and the percentage that thi repreent with repect to the epected IRR. Table 3: Average IRR and tandard deviation of the IRR for men and women of 65 year, AH (2003) projection. Formula IRRM Dev %dev Formula IRRW Dev %dev % % % % % % % % % % % % % % % % % % % % Table 4: Average IRR and tandard deviation of the IRR for men and women of 60 year, AH (2003) projection. Formula IRRM Dev %dev Formula IRRW Dev %dev % % % % % % % % % % % % % % % % % % % % One note immediately the ignificant difference, at the age of 65, which appear between the IRR of men and that of women. The IRR of women i approimately 66% to 106% higher than that of men. Thi i becaue when the retirement penion i being calculated the mean life epectancy between men and women i ued. Comparion of thee reult with thoe obtained relative to the replacement rate how that there i no clear relationhip between the two concept. Furthermore, the model with the greatet IRR for all age, Model 10, had a low replacement rate. Alo, one oberve that, in term of IRR, delaying retirement age i not rewarded. Thi i logical ince, a the amount of the penion depend upon the future growth and the evolution of the demographic parameter, and the future prognoi i unfavourable due to the lowing of the growth of the macroeconomic variable that intervene in the formula, the return on the invetment, IRR, woren with the paage of time. There are marked difference in the epected value of the IRR, on the contrary, the percentage tandard deviation doe not vary notably among the different formulae. It 13

14 eem clear that formulae 10 and 2, both wage-baed, are the mot uitable ince they have a high IRR and a low volatility. The formula that have higher IRR practically coincide for the cae of men and women (there are difference for 70 year of age), but women are ubject to a maller variability of the IRR, o that, beide obtaining a greater average IRR, they are ubject to a far lower rik. Table 5: Average IRR and tandard deviation of the IRR for men and women of 70 year, AH (2003) projection. Formula IRRM Dev %dev Formula IRRW Dev %dev % % % % % % % % % % % % % % % % % % % % The value obtained for the IRR are urpriingly low, epecially compared to the epected return under unchanging legilation in the current ytem which approache 4% for the age of 65 year. But it ha to be taken into account that thi reference 4% i not a net return of an increae in future contribution. Following the argument of Geanakoplo et al. (1998), thi return hould decreae coniderably ince; to maintain the promied benefit and the financial equilibrium of the pay-a-you-go ytem, there would have to be a very notable increae in the contribution relative to wage. The IRR obtained from the ytem of notional account i actuarially fair. Beide, thee figure are not o alarming if they are compared with the average growth of the macroeconomic indice conidered, which are about 1.66% for the increae in GDP or 1.84% for the variation in wage according to the projection of Alono and Herce (2003). Another important quetion i the fact that the IRR decreae with increaing retirement age, and alo the rik to bear i far greater. Therefore, if the analyi made by the contributor-beneficiary i from thi perpective 9, he would not likely be willing to prolong hi tay in the labour market, even though it i conidered, Palmer (1999), that NDC mitigate the effect of diincentive to work that appear with a pay a you go ytem of defined benefit, epecially if that ytem i badly deigned. 9 In making a deciion of whether to take early retirement, the beneficiary will, or hould, at leat conider the following quetion: the amount of the penion he will receive relative to hi need, hi valuation, Valdé- Prieto (2002), of hi poibilitie of leiure, hi tate of health, hi interet in hi work, the poible acce to a better job poition, etc. Palmer (2001) indicate that there i empirical evidence that the individual tend to go into retirement a oon a he i allowed to, o that one mut be very cautiou when it come to etablihing the minimum retirement age. 14

15 4.3. VALUE AT RISK (VAR) Beide the calculation of the deviation of the IRR, another important tool for the meaurement of rik i the value-at-rik (VaR). Thi parameter ha become one of the mot widely ued tool with which to meaure rik by regulator, economic agent and academic. One of the reaon for it popularity i the implicity of the concept and, above all, the intuitivene of it interpretation, it being an etimate of the maimum poible lo for a given time horizon and ignificance level under circumtance conidered a "normal in the market". In the analyi to follow, VaR ε i undertood to be the minimum value of IRR at a determined confidence levelε. Thu, a VaR 0.95 of 0.61% for Formula 2 implie that only 5% of the time (auming a 95% confidence level) in normal condition would the IRR of a man be le than 0.61%. Auming a confidence level of 95%, the following reult are obtained (Table 6). For the men at any age, the formula that would provide the bet value of VaR 0.95 i Formula 2, which ue the VAEI to revalue the contribution, and the penion remain contant in real term. Women would only chooe thi model at the age of 60. They would opt for Formula10 at the age of 65 and 70. It ha to be emphaized that Formula 10 would give a notably lower initial penion replacement rate than Formula 2 at any age for intance 37% a againt 44% at the age of 65 but the penion will be increaed at the ame rate a wage in Formula 10 a againt contancy in real term in Formula 2. The reult eem logical given women' longer life epectancy. Table 6: VaR 0.95 for men and women, variou age and AH (2003) projection. For VarM VarM VarM VarW VarW VarW For For For For For (60) (65) (70) (60) (65) (70) The variation of the retirement age with repect to that conidered ordinary (65 year) alo alter notably the value of VaR Anticipating of the retirement age increae VaR 0.95, and it deferment diminihe it notably. The concluion therefore i that neither in term of VaR 0.95 would delaying the retirement age be rewarded, ince the VaR for an age of retirement of 60 year would be greater than that for 65 year, and thi in turn greater than that obtained for 70 year. In general, the wort model in term of VaR 0.95 i Formula 4, which ue VGDP to revalue the contribution, and VAEI for the penion. Thi i becaue the correponding diperion of IRR around it mean value already mall in itelf, a one can ee in the table of the previou ubection i one of the greatet. 15

16 4.4. MARKOWITZ FUNCTION. In order to carry out an overall analyi of the rik, it i neceary to introduce the ubjectivity of it valuation by the beneficiary through hi rik averion. Analyzing the return in term of Markowitz i equivalent to doing o in term of a utility function. In the preent cae, the reult are preented after applying the theory of Markowitz. 10 Table 7 preent the ranking of the formulae according to the Markowitz function criterion for men and women of 65 year, with different value of rik averion (γ ).Following the method ued by Feldtein and Ranguelova (2001), value of rik averion are ued in relation to both the mean determinant variable (1+IRR) of each formula and to the difference between the formula. The value of γ that are ued depend to a great degree on the relationhip between the value of the formula that one wihe to rank. If the difference are mall, a i the cae with the IRR, then the rik averion γ will take much higher repreentative value. Individual (men and women) neutral to rik or not avere to rik would prefer Formula 10 in which return i rewarded. Thoe individual (men and women) with a more marked degree of rik averion, however, would prefer Formula 2, which i preferred when the VaR 0.95 criterion i applied for men and for women with lower retirement age. Table 7: Ranking of the formula according to the Markowitz function criterion, for men and women of 65 year, with different value of rik averion γ and AH (2003) projection. γ =0 γ =50 γ =100 γ =150 γ =200 γ =300 γ =400 Men Women Table 8 give the preferred formula for each age according to the Markowitz function for men and women, with different value of rik averion (γ ) and projection. The formulae that are found to be preferable for the beneficiary are 10 and 2 the former for thoe individual more avere to rik, and the latter for the neutral or le avere to rik individual. 10 Identical reult were obtained in a calculation of the utility of (1+IRR) through a CRRA utility function 16

17 Table 8: Preferred formula for each age according to the Markowitz function, for men and women, with different value of rik averion ( γ ) and AH (2003) projection. Age γ =0 γ =50 γ =100 γ =150 γ =200 γ =300 γ = M M M W W M UTILITY OF THE PENSION. To complete the rik tudy, the evolution of the penion i analyzed throughout the paive lifetime of the individual in term of utility. Table 9 preent the ranking of the formulae according to the utility of the penion for men and women, with different value of rik averion ( β ) and projection. The value of rik averion ( β ) to be ued vary from 1 to 5. Feldtein and Ranguelova (2001) provide qualitative argument that very high value of rik averion hould not be ued in the application of CRRA (contant relative rik averion) type utility function to the analyi of penion. Table 9: Ranking of the formula according to the utility of the penion, for men and women of 65 year, with different value of rik averion ( β ) and AH (2003) projection. β =0 β =1 β =2 β =3 β =5 Men Women The reult are practically identical to thoe of the previou ubection. The formulae choen are 10 and 2 according to the individual' degree of rik averion. The leat avere individual will chooe Formula 10, and a the individual' degree of averion increae, they would tend toward a preference for Formula 2 (higher initial penion, but contant in real term). Table 10 lit the formula choen for each age according to the utility of the penion, for men and women, with different value of rik averion ( β ) and the AH (2003) bae projection. One oberve that the deciion of individual who are neutral or not very avere to rik doe not vary even though the retirement age i modified. The critical value of 17

18 averion i 2 or 3 the value where the change from Formula 10 to 2 occur. The deciion of very avere individual i unchanged they all chooe Formula 2. Table 10: Preferred formula for each age according to the utility of the penion, for men and women, with different value of rik averion ( β ) and AH (2003) projection. Age β =0 β =1 β =2 β =3 β =5 60 M M M W W W SENSITIVITY ANALYSIS Thi ection decribe a enitivity analyi with repect to change in the urvival rate, the epected average GDP growth derived from the mean macroeconomic projection ued and the change in the bae macroeconomic projection SENSITIVITY OF THE RESULTS WITH RESPECT TO INCREASES IN THE SURVIVAL RATE Thi firt ub-ection decribe the enitivity analyi of the reult with repect to change in the urvival rate. It i a fact that in the lat 50 year there ha been a notable increae in longevity in mot developed countrie, including Spain, Goerlich and Pinilla (2005). According to the mortality table of the INE [Spain' National Intitute of Statitic] of 1950, the life epectancy for men and women at the age of 65 wa and year for men and women, repectively. The INE table of , which are thoe that have been ued in all the foregoing calculation, indicate that, in the intervening approimate 50 year, thee figure have rien by 36% for men and 49% for women. The peron who enter the labour market now, at the age of 25 year, auming retirement at 65, would begin to collect hi penion in the year It i reaonable to think, Diamond (2005), that life epectancy will have further increaed, and mortality table that better reflect thi increae would have to be ued. Alo, one of the fundamental virtue that i claimed for the NDC i that they can deal with demographic and economic change by mean of the deign of the formula they ue to calculate the initial penion ANTICIPATED CHANGES IN THE SURVIVAL RATE It i aumed that the authority governing the penion ytem i capable of anticipating thi increae in longevity 11, and that thi i reflected in a change of the mortality table valid for the calculation of the amount of the initial penion. The mortality table ued are the Swi GRMF-95, which inurance companie in Spain have already been applying for everal year for the commercialization of life annuitie. According to thee table, the life epectancie of men and women at the age of 65 are and year, repectively. If, for the determination of the initial penion at 65, the notional capital i conidered to be actuarially equal to a contant pre-payable annuity at a real interet rate of 3%, a value of G=16.65 i obtained a againt the that wa obtained with the PEMF table; 11 In Sweden and Brazil, there i a proce of automatic annual adjutment of the demographic parameter baed on the oberved urvival rate. In Italy, Brugiavini and Peracchi (2005), the adjutment i made every ten year. To avoid undeired political manipulation, it i conidered more appropriate, Diamond (2005), which the adjutment be made annually with real data intead of with projection. 18

19 thee value are applied to the firt i formulae, which correpond to a greater initial penion, but contant in real term. If a real interet rate of 1.25% i conidered, the value of G with the GR-95 table i intead of Thee are the value applied to the lat four formula which correpond to an initial penion that i maller but that will be increaed in real term. Table 11: Average RR, average IRR, and VaR 0.95 of the IRR with GRMF-95 and with AH (2003) for men and women of 65 year. Comparion with PEMF RR (65) IRRM(65) IRRW(65) VarM(65) VarW(65) For. GR PE For. GR PE GR PE GR PE GR PE % 61.99% % 61.99% % 61.99% % 55.14% % 55.14% % 55.14% % 52.74% % 52.74% % 46.92% % 46.92% The reult preented in Table 11 are revealing. The notional account ytem react to an anticipated change in the urvival rate with an automatic reduction of the replacement rate to try to maintain the actuarial equilibrium between contribution and penion. A can be oberved, the epected average replacement rate would decline in whichever of the cae by around 10%, to, in the bet of the cae, ome 51% of the average wage. The adjutment, however, would be neither complete nor perfect, ince the IRR would increae lightly becaue the reduction in the initial penion would not ufficiently compenate the foreeen increae in longevity. Hence, ome additional fine-tuning would be neceary in order to leave the IRR unchanged. The VaR 0.95 of the IRR would alo increae a a conequence of the foreeen greater longevity UNANTICIPATED CHANGES IN THE SURVIVAL RATE In thi ub-ection, unlike the previou one, it i aumed that the authority governing the penion ytem i unable to anticipate the increae in longevity, or that, while recognizing it, it ha inufficient political courage to apply it immediately, o that the formula for the calculation of the penion would ue the demographic parameter derived from the PEMF table, but the longevity would correpond to the GRMF-95 urvival table. The effect of thi atypical ituation i given in Table 12, and i compared with the initial ituation. The average RR would remain unchanged, but the epected average IRR would increae notably for both men and women by around an additional 0.7%, becaue on average men and women would collect 4 and 7 year more of penion, repectively relative to the initial etimate. Thi would be a particularly critical ituation ince, a oberved by Settergren and Mikula (2005), thee unanticipated change in mortality would lead to the violation of the Samuelon-Aaron rule, ince the implicit average IRR of the ytem, for whichever of the formulae, would urpa the etimated average GDP growth for the period according to the AH (2003) projection, which in all cae i le than 1.8%. 19

20 Table 12: Average RR, average IRR, and VaR 0.95 of the IRR with unanticipated change in the urvival rate and with AH (2003) for men and women of 65 year. Comparion with PEMF RR (65) IRRM(65) IRRW(65) VarM(65) VarW(65) For. GR PE For. GR PE GR PE GR PE GR PE % 61.99% % 61.99% % 61.99% % 55.14% % 55.14% % 55.14% % 52.74% % 52.74% % 46.92% % 46.92% SENSITIVITY OF THE RESULTS WITH RESPECT TO INCREASES IN THE EXPECTED AVERAGE GROWTH OF THE GDP AND ITS COMPONENTS A enitivity analyi wa made of the RR, GDP, and reulting VaR 0.95 with repect to increae of the epected average growth in GDP and it component. Two additional hypothee were conidered: (1) The average growth i 50% greater than the baic aumption, i.e., the value of mean VGDP etimate are multiplied by 1.5. (2) The average growth i 100% greater than the baic aumption, i.e., the value of mean VGDP etimate are multiplied by 2. Table 13: Epected mean replacement rate a a function of the average wage for AH (2003) and their variation for men of 65 year. Formula AH(2003) AH(2003)1.5 AH (2003)2 2, 5, % 73.23% 85.47% 1, 3, % 61.51% 67.88% 9, % 62.31% 72.72% 7, % 52.34% 57.76% Table 13 give the epected mean RR a a function of the average wage for variou projection and their variation. The reult wa a epected RR increae until in ome cae it reache almot 85%. The reaction of the notional account ytem to greater economic growth i an increae in the replacement rate. Thi i becaue, if the longevity rate remain contant on increaing the notional capital from the effect of a greater revaluation of the contribution, the reult i an increae in the initial penion, and conequently the RR i increaed. The effect on the average IRR of men and women (Table 14) and on the VaR 0.95 of the IRR are even clearer than on the RR. With repect to the IRR, one oberve an almot proportional increae in the cae of men, and a notable increae, but not o high, in the cae of women. 20