Pension expectations and reality. What do Italian workers know about their future public pension benefits?

Transcription

1 Pension expectations and reality. What do Italian workers know about their future public pension benefits? Massimo Baldini*, Carlo Mazzaferro $ and Paolo Onofri *University of Modena and Reggio Emilia and Capp $ University of Bologna and Capp Prometeia Abstract We use 6 waves of the Bank of Italy s SHIW to check the evolution of workers expectations on future pension benefits and retirement age from 2000 to Based on these two subjective evaluations, we compute a measure of expected pension benefit and compare it with a true measure of the same variable that we estimate on the basis of the pension rules in each year of the considered time lapse. By comparing subjective and true measures of the variable, we are able to measure the evolution over time of the expectation error and its distribution among different economic and demographic subsets of the population. Finally, we estimate a subjective measure of net social security wealth and the degree of substitution between the subjective measure of social security wealth and the private net worth of workers households, in order to quantify the effects of pension reforms approved in the period considered on saving behavior of workers.

2 1. Introduction The accuracy of workers information over rules of the public pension system should represent an important aim for any Government. A correct knowledge on their future level of pension benefits and of their retirement age would help workers to improve both labour supply and saving choices. This topic is even more important in countries like Italy, where the social security wealth (i.e. the present discounted value of future pension benefits) represents the biggest share of the total wealth for a large majority of households [Mazzaferro and Toso 2009] and where reforms in the public pension system have radically changed both the expected level of future pension benefits and the retirement age. Unfortunately Italian Governments did not undertake in the recent past necessary steps to make individuals aware of the rather complex implications of this reform process. Simplifying drastically one can say that current workers (and future pensioners) will accrue their pension rights with a much less generous (even if more sustainable and homogenous) rule when compared with that of current pensioners. At the same time they will also retire later than current pensioners did. If the first factor should encourage workers to increase personal saving now in order to face a future reduction in the level of social security wealth, the second effects goes in the opposite direction, since it increases the active lifetime and cuts the retirement period. As a matter of fact the future adequacy of public pension benefits and the necessity to increase current saving to compensate their eventual reduction both depend on the relative strength of these two opposite effects. Precisely for this reason the delay of an appropriate information campaign on the likely effects of the reforms on the future level of pension benefits and on the age at which current workers will be allowed to retire appears even more worrying. In this study we describe how information on the future of the public pension system has evolved among Italian workers, using data from the SHIW survey (2000 to 2012). Since 2000 respondents of the survey are asked about their expectations on both the future level of the replacement ratio (i.e. the ratio between the first pension benefit and the last wage) and of the retirement age. These two variables are used in our study to estimate the expected level of the future public pension benefit for workers in the survey from 2000 to In a second step of the study we compute the pension error defined as the difference between the expected value and the true value of the pension benefit, the second variable defined as the pension benefit level computed, at the expected retirement age, on the basis of the current pension rule. We study then the distribution and the evolution of the pension error among social and demographic categories of the surveys population. Using then the expected value of the pension benefit, together with information on lifetime expectation at retirement, we construct a simplified measure of net and gross social security wealth. Finally we study the time evolution of this variable and its degree of substitutability with respect to different measures of private wealth.

3 2. Previous literature 3. Data Since year 2000 individuals participating to the SHIW survey of the Bank of Italy are asked to answer two questions regarding their future pension, namely: i) what is your expected replacement ratio between first year pension benefit and last year earning?; ii) at what age do you expect to retire? Using these two pieces of information we implement a procedure first proposed by Jappelli [1995] to compute the expected value of the pension benefit at the age of retirement for those workers who responded positively both to the first and to the second question. Table 1 reports the total population of the survey, the number of dependent workers and self employed and, among these, the number of those that provide a positive answer both to the first and to the second question above reported. Table 1 Number of selected individuals Year Total Dependent Selected Dep Self Employed Selected SE ,336 6,147 5,405 1,795 1, ,215 5,817 5,784 1,642 1, ,659 5,792 5,779 1,526 1, ,639 5,746 5,720 1,413 1, ,989 5,800 3,791 1, ,918 5,546 3,195 1, ,126 5,383 3,226 1, The sample has a yearly dimension of around 20,000 individuals. Among these individuals we first select those who classify themselves as dependent workers or self-employed and then we drop from the sample all individuals that did not respond at both the selected questions about the pension system. Following this procedure we end up with a population that, as reported in table 2, presents an important discontinuity in correspondence to the year Indeed, starting from this year only individuals who were physically present to the interview are allowed to answer to the two questions. This innovation was not neutral on the composition of the selected population, which appears to be older after 2008.

4 Table 2 Average age of the original and of the selected population Dep Selected Dep Self Selected Self year workers workers employed Employed Some other adjustments were necessary before starting to compute the expected value of future pension benefits. In particular we drop from the sample all individuals that declared they had not previously paid payroll taxes to a pension scheme (1,767 observations) and those older than 70 (167 observations). Finally we adjusted the expected retirement age, imposing that it cannot be lower than 57 and greater than 70. After all these adjustments we end up with a sample composed by 31,665 dependent workers and 8,080 selfemployed. The next steps describe the hypotheses used to estimate the expected value of pension benefits. First, we need to impute to each individual of the selected sample a value of his/her labour income gross of the income tax and of the part of the social security contribution that is paid by the worker. The SHIW survey contains only the information on net incomes, i.e. after the payment of the personal income tax, but for the computation of future pensions we need the data on gross incomes. In order to overcome this shortcoming we moved to another survey of the Italian population, namely the SILC survey. The SILC survey on households living conditions, carried out every year by the Italian national statistical institute, gathers data on both gross and net income. We have therefore performed a regression, on the workers aged between 25 to 65 year in the SILC survey for the year 2012 (containing 2011 incomes) of gross income as a function of net income and a set of personal characteristics (age, gender, dependent or selfemployed, education, number of children in the household, geographic area). The following table shows the results of the regression.

5 Table 3 Regression results of gross income on the SILC survey Coef. Std. Err. t Net income Net income squared Age Age squared Man High school Degree Employee N. children N. children N. children North Centre Constant R 2 =0.97; N. obs We use the coefficients estimated from this regression to impute to each SHIW observation a value for his/her gross income. We also compute on the pseudo panel of the SHIW ( ) different rates of growth of lifetime earnings. To get these rates of growth, we split the sample of workers in the SHIW survey into six groups, resulting from the interaction between gender and three education levels (less than high school, high school, degree). Then for each group regress yearly gross income on age and its square, obtaining a life-cycle profile for earnings. For each individual of the sample, this fitted profile passes through the actual earning of the survey, at the corresponding age. Then we obtain the average growth rate of gross earnings for each group, and depending on the age compute the earning of the last year of work. After all these steps we are able to estimate the expected value of the pension benefit in the first year after retirement for each individual in the sample (P_ex)i as: P_exi = RRi * Y_fini (1) where RRi is the individual expected replacement rate reported in the survey Y_fini is the value of individuals earning the year before retirement The computation of Y_fini is obtained as

6 Y_fini = YLi,t * (1 + mk) (ret i age i ) (2) where YL i, t is the estimated gross earning of individual i at time t (the observation s year) mk is the group specific rate of growth of earning reti is the expected age at retirement for individual i agei is the age of individual i in the year he/she is observed in the survey In fact we project forward the current value of the (estimated) gross earning for a number of years equal to the difference between the expected age of retirement reported in the survey and the current age of each individual. In doing so we hypothesize that all individuals in the sample will not experience periods of unemployment. We also impose different growth rates of earnings, taking into account both gender and educational level (see above). In order to compute the pension error we also need to estimate the true value of the pension benefit for each individual in the sample and then compute the difference between the two levels of benefit. We introduced a number of (necessary) simplifications that allowed us to reach our aim. In particular: i. (True) pension benefit is computed at the expected retirement age. ii. We split our sample into three groups in order to take into account the different phasing in of the NDC system. In particular we distinguish, on the basis of the accrued seniority in 1995, the DB workers (i.e. those that in 1995 had at least 18 years of seniority at work); the mixed workers (i.e. those that in 1995 had less more than 0 years but less than 18 years of seniority at work) and the NDC workers (i.e. those that started to work after 1995). iii. We distinguish only three occupational schemes: private dependent workers, public dependent workers and self-employed. iv. We impose that workers will not experience periods of unemployment. v. We compute the true pension benefit (P_true) according to the rules described in the appendix A. As already mentioned, the pension error is defined as the difference between the level of the pension benefit computed according to the law and the level of the same variable obtained on the basis of equation (1). For each individual in the sample we have then: P_errori = P_truei P_exi (3)

7 4. How Italian workers estimate their future pension benefits As a starting point to interpret our results it is useful to describe the evolution of both the expected replacement ratio and of the expected retirement age over the whole sample. Results, by year, are presented in table 4. Table 4 Average value of the expected replacement ratio and of the expected retirement age Year Replacement Retirement ratio age % % % % % % % 65.1 Figures in the table tell that over the whole period here considered workers in the sample revised substantially their expectations on the future of the public pension system: the expected replacement ratio decreased round by 10%, while the expected retirement age, during the same period, increased by 3.5 years. So at a first glance it seems that, at least on average, the message that in the future the public pension system will not be as generous as it has been in the past was perceived by Italian workers who expect both to receive a lower pension benefit and to retire later. Looking first at the expected replacement ratio it is worth to notice the continuous decrease in the average value reported in the table. Starting from 72.5% in 2000 this indicator reaches the average value of 62.3% twelve years later. The reduction is much intense in the first years of the period and from 2010 to As for the retirement age, differently from the replacement ratio, changes are more concentrated in the second part of the period. In particular from 2010 to 2012 the expected retirement age increases by 1.6 years, nearly half of the total changes. Table 5 and table 6 decompose changes in the expected replacement ratio and in the expected retirement age by different socio-economic subsamples of the population.

8 Table 5 Expected replacement ratio by subsamples of the population. Percentage values Year Priv dep Pub dep Self empl Men wom DB MIX NDC FIR DEG SEC DEG HIGH DEG Even if the reduction in the ratio between first year pension and last year wage is common to all the socioeconomic characteristics here considered, it is worthwhile noticing that some groups appear to be more affected than others. In particular the reduction is stronger among dependent workers than among selfemployed; among future NDC pensioners than among future DB pensioners, among highly educated individuals than among individuals with first level education. Table 6 Expected retirement age by subsamples of the population Year Priv dep Pub dept self emp Men wom DB MIX NDC FIRST DEG SEC DEG HIGH DEG

9 Also in the case of the expected retirement age, the change appears not uniformly spread among the population. In this case women and dependent workers expect a much larger increase in retirement age than men the self-employed. The next step is the analysis of errors made by workers in the computation of their future pension benefits. Before doing that it is however useful to look at data presented in table 7. Here we compare the true replacement ratio, defined as the ratio between P_true and the last gross wage for each individual in the sample, and the expected replacement ratio, already presented and discussed above. Remembering that we compute pension benefits at the expected retirement age and that this variable increases over the whole period considered, it is interesting to note that the reduction in the true replacement ratio associated with the phasing in of the NDC system over the years results not particularly strong and in any case smaller than the one expected on average by workers 1. Table 7 True and expected replacement ratio, given expected retirement age True RR Exp RR From this point of view it is interesting to notice that from 2010 to 2012 (i.e. corresponding to the higher increase in the expected retirement age) there is a noticeable increase in the true replacement ratio, while the expected one continues to decrease. This point appears to us as particularly important: workers seem not to have correctly understood one of the most important mechanism of any pension system (and stronger under NDC): increasing retirement age determines a higher level of the pension benefit and therefore of the replacement ratio. Moving now on the error, we compute its average value over the whole sample as equal to Euro at 2012 prices. The median value equals 752 Euro. 62% of individuals report a positive (or zero) value, while 41% estimate a pension benefit which is smaller than the true value. Remembering that the average value of the 1 The positive relation between the age of retirement and the replacement ratio is consistent both with the DB and with the NDC formulae.

10 estimated pension benefits equals 24,715 Euro, the average percentage error is equal to 6.2%. Average values however do not convey a complete picture of the phenomenon. In fact there is a wide distribution of the error, as the figure 1 shows. Figure 1 Errors distribution over the whole sample. Euro at 2012 prices. Figure 2 shows that there is a positive relation between the sign of the error and the reported value of the replacement ratio. The average value of errors is negative for very low values of the expected replacement ratios. It monotonically increases thereafter reaching a value round to zero for expected replacement ratios between 50% and 70%. For higher values of the expected replacement ratios the error becomes positive and quite large for values higher than 100%.

11 Titolo asse 50,000 40,000 30,000 20,000 10, ,000-20,000-30,000-40,000-50,000 Figure 2 Average errors and expected replacement ratio in the whole sample. Euro at 2012 prices Expected replacement ratio Figure 3 a) and b) Average error by year (a) and expected retirement age (b). Euro at 2012 prices 4,000 3,500 3,000 2,500 ERROR 4,000 3,000 2,000 1,000 2,000 1, , , ,000-1, ,000-3,000-4,000-5,000-6,000 Figure 3a and 3b display respectively the time evolution of the average error in the computation of pension benefits and its relation with the expected retirement age. As for the part a) of the figure, it is particularly clear the continuous downward adjustment of workers expectations. Workers, on average, overestimate their future level of benefits until 2010, but they progressively become less and less and less confident on the adequacy of the future pension benefits. In the last year, 2012, on average workers even expect for themselves a future pension benefit that is lower than what they, on the basis of the current rules and on the expected retirement age, should reasonably expect.

12 Interestingly part b) of the figure reports that individuals who expect to retire earlier than 66 have an estimated value of their pension benefit that is higher than the true one. On the opposite, as retirement age increases beyond this value the difference between the true and the expected value becomes on average negative. Figure 3a and 3b confirm our perception that the positive implications on the replacement ratio of increasing retirement age are still not completely understood by Italian workers. Table 8 reports the evolution over years of the average error for specific subsamples of the whole population. Table 8 Average error for subsamples of the population Euro at 2012 prices. YEAR PRIV DEP PUB DEP SEL EMPL MEN WOMEN DB MIXED NDC FIRST SECOND THIRD ,430 5,816 2,212 3,427 3,084 1,595 4,314 5, ,879 11, ,174 4,436 1,350 1,828 2, ,761 4, ,642 9, ,867 1,293 1,390 2, ,160 3, ,866 10, , , ,545 2, ,200 8, , , ,480-1,017 1,009 6, , ,217-1,309 1,122 1, , ,957 1, , ,000-1, ,148-1,717 2,984 The downward trend in expectations is common to all subgroups. As for the level of the average error it is interesting to notice that it is higher for those with a high level of education, those who will compute their pension benefit under the NDC system, women and public dependent workers. In order to better understand the relationship (if any) between the way in which individuals form their expectations on pension benefits and the dimension of their errors, we introduce three new variables that can contribute to describe the degree of understanding of an individual on his/her perspective pension situation. All three variables split the whole sample into two groups. The first variable distinguishes those who overestimate their future pension benefits with respect to the true one (P_exp P_true > 0) from those who underestimate it (P_exp P_true < 0). We call the first group optimistic. The second variable (Ok_age) splits those who are able to predict correctly their retirement age from those who expect to retire at an age which is not consistent with the (complex) evolution of retirement age during next decades. The third variable (Ok_pension) (arbitrarily) separates those who are able to predict correctly their future pension benefits from those who are not able to predict it. The individual prediction is considered correct if the estimated value of the pension benefit is in the range +/- 20% with respect to the true value computed under the hypothesis described above. Table 9 reports the average value of the error for these three variables.

13 Table 9 Average error and the degree of understanding of the pension system. Euro at 2012 prices. Whole sample. Status Optimistic Ok_pension OK_age No -5,536 2,189 3,087 Yes 6, ,066 The average error on pension computation is about 5,500 Euro in case of underestimation and 6,700 Euro in case of overestimation. Those who are able to predict correctly their future retirement age are also better in the estimation of their pension benefit level (1,066 Euro), with respect to those who expect to retire earlier than they will be allowed (3,087 Euro). Table 10 presents some results in term of the evolution over time of the same variables. Table 10 Average error and the degree of understanding of the pension system over time Percentage of those who respond positively. Year Optimistic Ok_age OK_pension % 79.6% 47.3% % 83.5% 51.9% % 86.4% 50.6% % 85.0% 48.0% % 88.8% 51.0% % 56.6% 51.6% % 36.5% 46.4% As it can be noticed from figures in the table the percentage of individuals who are optimistic about their future level of pension benefit decreases constantly from 2000 to 2010 and more abruptly in the last year considered. The percentage of those who are able to compute their future level of pension benefit remains more or less constant, while those who predict correctly their future retirement age halve in the considered period, with a substantial decrease from 2008 to 2010 and from 2010 to 2012.

14 As a final step we run three probit regressions where the dependent variable is respectively the ability to capture correctly the level of the future pension benefit (Ok_pensione), the retirement age (Ok_eta) and the overestimation of the pension benefit with respect to the true value (Ottimista). Results of the three regressions are reported in table 11 where marginal effects of changes in independent variables on the dependent one are reported. Table 11 Probit regressions (1) (2) (3) ok_pensione ok_eta ottimista income quintile *** *** ** (0.0304) (0.0293) (0.0277) income quintile *** *** *** (0.0324) (0.0301) (0.0286) income quintile *** ** *** (0.0332) (0.0307) (0.0295) income quintile *** *** *** (0.0338) (0.0327) (0.0311) N. years of contributions *** *** *** (0.0020) (0.0018) (0.0017) Male (0.0239) (0.0219) (0.0210) Private employee *** *** *** (0.0241) (0.0243) (0.0224) Public employee *** *** *** (0.0306) (0.0283) (0.0270) High school *** *** (0.0228) (0.0208) (0.0196) Degree *** (0.0333) (0.0314) (0.0308) North *** (0.0239) (0.0219) (0.0210) South ** * (0.0289) (0.0261) (0.0253) Mixed regime *** *** *** (0.0342) (0.0296) (0.0287) NDC regime *** *** *** (0.0541) (0.0501) (0.0476) _cons *** *** (0.0727) (0.0685) (0.0644) N pseudo R

15 The position in the income distribution shows, for all the three variables, a similar positive effect. In general being on a higher position in the income distribution increases, with respect to the first quintile (variable omitted), the ability to predict correctly future values of the pension benefit and the retirement age. Also a higher level of income is associated with an overestimation of the future level of the benefit. Seniority at work has a not uniform effect of the three variables: it increases the probability to correctly predict the retirement age, but It decreases the probability to predict correctly the future level of the pension benefit. Interestingly, a higher seniority at work reduces the probability to overestimate the future pension benefit level. As for the occupational position, being a self employed means a much lower probability to predict the level of the pension benefit and also to overestimate its level. The level of education has a positive effect to the capacity to predict the future retirement age. It also increases the probability to overestimate future pension benefits, but the effect on the probability to correctly predict the future retirement age is not uniform. Finally, being a future DB pensioner implies a higher probability to predict both the level of pension benefit and the retirement age and also to overestimate its level. 5. Social security wealth based on expected pension benefits The value of annuities expected from the pension system constitutes a major part of total household wealth in all developed countries. Any analysis of the accumulation and distribution of wealth, and of its evolution over time, would therefore be misleading without its inclusion. An augmented measure of wealth, defined as the sum of net worth and pension wealth (Davies and Shorrocks, 2000; Wolff, 2005a), should overcome any possible shortcoming. The first component of augmented wealth is net worth, which is equal to the total value of all those assets a household can sell on the market, less any debts. Feldstein (1976) and Wolff (1987) argue that net worth is not a very satisfactory definition of wealth in those countries where there is a mandatory, public pension system. If contributions to a social security scheme are perceived by individuals as a substitute for other forms of lifecycle saving, then a definition of wealth capable of measuring the stock of resources to be used to finance consumption during old age should also take into consideration the present value of future pension entitlements. Jappelli and Modigliani (1998) point out that in any pension scheme, contributions should be regarded as a component of life-cycle savings because they entitle workers to receive a retirement pension in the future. They also point out that pension benefits represent the utilization of pension wealth that was previously accumulated prior to retirement, and should therefore be considered as a decumulation of wealth.

16 In this paper we define social security wealth as the discounted sum of all expected future pension benefits, less the discounted sum of contributions an individual expects to pay between the time of observation and his/her retirement. For each employed individual i observed at time t social security wealth is defined as: (4) where: r is the discount/interest rate P_exi is the pension benefit expected by individual i upon retirement Ck,i is the payroll tax rate of sector k (employed, self employed and quasi self employed) Yi is the gross income of individual i p is the expected year of retirement of individual i, d is the life expectancy at retirement of individual i, r is the discount rate, mk is the group specific real growth rate of earnings In our simulation we impose the following numbers: r = 1.5%; d is taken from the ISTAT dynamic population projection to 2060; Ck,I follows the normative evolution and currently is equal to 33% for employed; 24% for self employed and 26% for quasi self employed. The specific earnings rate of growth are reported in the following table. Table 12 Specific growth rate of earnings. Men women first second third Table 13 reports the average value of the net social security wealth from 2000 to 2012 computed according to equation (4) for the whole population and for subsamples of the population.

17 Table 13 Average value of the net social security wealth for employed. Thousand of Euro at 2012 prices Year Total Priv dep Pub dep Self emp Men Women DB MIXED NDC FIRST SECOND THIRD As it has already been documented, the time evolution of this variable is strongly influenced by the reform process of the last 12 years. On average SSWN decreased from 220 thousand of Euro in 2000 to 161 thousand of Euro in Among occupational categories private dependent workers are the most hidden. Reforms also reduced SSW more for women and highly educated people. During the same period important facts modified also the level and the composition of the net worth of Italian workers, defined as the sum of real and financial wealth, net of any debts. Table 14 Average value of the net worth for employed. Thousand of Euro at 2012 prices year ALL Priv PUB SEL EMPL MEN WOMEN DB MIXED NDC FIRST SECOND THIRD Differently from SSWN the net worth of workers did not decrease in the observed period. Starting from an average value of 255 thousand Euro in 2000, it was equal to 285 thusand of Euro in It reached a maximum of 310 thousand of Euro in Most probably the dynamic of net worth is influenced by the price

18 dynamics of real estate, which after 2010 started to decrease albeit slowly. Even if the level of net worth is quite different among the subgroups reported in the table, its dynamics is quite similar. The next step will be an estimation of the degree of substitution between social security wealth and net worth during the observed period.

19 Appendix A: Computation rules for the pension benefit Below we describe the different rules prevailing under the DB, the mixed and the NDC schemes. The population in the model has been divided into three groups according to seniority in 1995: DB, mixed and NDC. The computation rule for pension benefit of those workers who are under the DB system is summarized by the formula: P DB= r*(n 1W 1 + N 2W 2) (1) where r is an accrual rate, N 1 and N 2 represent the years of contribution before and after 1992 respectively, W 1 and W 2 represent the pensionable earnings used for computing pension installment, for the contributions paid before and after 1992 respectively. The terms r and W in the DB formula vary according to pension scheme and to the amount of pensionable earnings. In particular, W 1 is equal to the last yearly-earning for employees in the public sector; the average of the last five or ten pensionable yearly-earnings for those employed in the private sector and self-employed workers respectively. W 2 is the mean computed over the last ten years of positive earnings for public and private sector employees and over the last 15 years for self-employed workers. The accrual rate r is equal to 2% for the pensionable earnings bracket between 0 and 42,111 Euros (2009 prices) and it decreases with earnings level down progressively to a value of 1.1% for the pensionable earnings bracket over 55,976 Euros (2009 prices). For workers under the mixed regime, the old age pension benefit is determined as the sum of two components: P mixed=p A+P B (2) where the general rule for determining P A is similar to the formula used in the DB regime for the contribution paid before 1995, while the second, P B is computed according to a NDC rule on the contributions paid after Nevertheless, in the mixed regime the pensionable earnings for the contributions paid between 1992 and 1995 is determined differently, as the average yearly earnings indexed to 1% yearly rate according to a simple compounding rule. The P B term of the mixed pension is figured according to the NDC rule of equation (3). Old-age pension in the NDC system is computed as: P NDC=D x * MC (3)

20 where D X is a conversion factor that varies with retirement age (x) so as to guarantee a quasi-actuarial equity between the present value of paid contributions and the present value of expected pension benefits 2. MC is the total of contributions accrued during the whole working life in proportion to gross earnings (33% for employees and 24% for self-employed), capitalized at the rate of growth of nominal GDP. The yearly contribution is computed as a share of the gross wage for employees and gross income for the self-employed. The contribution rate is set at 33% for employees and 24% for self-employed workers. A contributory cap is set at 91,507 Euros (2009 prices). At least five years of contributions are required to claim an old age pension if the corresponding pension installment exceeds the amount of social allowance increased by 20%. The latter condition is not applied for those who will retire after the statutory retirement age. 2 The conversion factor has been computed as the result of the following simplified formula: D X w x l l l v w x l x t t ( 1 i) t 0 x t 0 l l v x t x q v x t a F x t l (1 i) ( t 1) found in Caselli et al (2003) where w is the maximum life span (set equal to 100 years); v lx t l x is the pensioner s probability at age x of being alive at age x + t; i is the annual real discount rate (set equal to 1.5 per cent, assumed to be equal to the long-run annual growth rate of Gross Domestic Product in real terms); β (set equal to 0.54 for a male pensioner and 0.42 for a female one) is the v fraction of the pension paid out the surviving spouse (if there is any); q is the probability of dying between age x + t and age x + t + 1; a F x t l x t is the expected present value of a real annuity of one dollar paid to the surviving spouse (if there is any) after the pensioner s death at age x + t +1.