CHAPTER 22 Public and Private DC Pension Schemes, Termination Indemnities, and Optimal Funding of Pension System in Italy Marco Micocci, Giovanni B Masala, and Giuseppina Cannas CONTENTS 22.1 I ntroduction 22.2 Optimal Private/Public Pension Mix 22.3 Optimal Portfolio Allocation in Occupational Pension Funds 2.3.1 Sensitivity Analysis 22.4 Role of the Termination Indemnity Scheme 2.4.1 Sensitivity Analysis 22.5 C onclusions References 582 585 588 590 591 592 593 94 S oci a l s ecu r it y co nt r ibuti ons o f Italian employees finance a twopillar system: public and private pensions that are both calculated in a DC scheme (funded for the private pension and unfunded for the public 581 © 2010 by Taylor and Francis Group, LLC 582 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling one) In addition to this, a large number of workers have also termination indemnities at the end of their active service In this chapter, we aim at giving an answer to the following questions Are the different flows of contributions coherent with the aim of minimizing the pension risk of the workers? Given the actual percentages of contributions, is the asset allocation of private pension funds optimal? Which percentages would optimize the pension risk management of the workers (considering public pension, private pension, and termination indemnities)? Keywords: Pension f unds, p ublic a nd p rivate p ensions, ass et allocation 22.1 INTRODUCTION A literature that dates back to at least the contributions of Diamond (1965), Samuelson (1975), and Diamond (1977) points out the need for an unfunded pension system to avoid capital overaccumulation This situation could arise in a social system in which only individualistic savings decisions are allowed, and so it is possible to accumulate capital to the extent that the return on capital assets is lower than the growth rate in national income, and the economy becomes dynamically inefficient It is possible to make such a situation better if people save less and consume more As Blanchard and Fischer (1989) and Abel e t al (1989) remarked, it is improbable to sustain a dy namically inefficient economy in the long run, since the owners of the capital are likely to t ransfer t heir c apital t o eco nomies offering h igher r eturns D iamond (1965), E nders a nd L apan (1982), a nd Merton (1983) h ighlighted t hat t he rationale for a public statutory pay-as-you-go pension program (henceforth paygo) depends on the potential for intergenerational risk sharing by means of Pareto improving transfers from the young to the old On t he other nd, t here is a lso a g rowing literature pointing out t he case for funded pension system In their seminal papers, Aaron (1966) and Feldstein (1974, 1996) showed the condition under which, in the long run, funded pension schemes are superior to unfunded schemes It requires the real rate of return on the assets in funded schemes to exceed the real growth rate in the wage bill In this regard, it is well known that the implicit return of t he pa ygo s ystem i s g iven b y t he g rowth o f a ggregate wa ge i ncome, reflecting the combined effect of productivity and labor supply growth So, the real growth rate in the wage bill, in turn, equals the growth rate in the national income, if the share of wages in the national income is constant The higher performance of the funded scheme has both an empirical and a theoretical reason The first one lies in the experienced superior performance © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 583 of the capital market in terms of the rate of return on investment These high real returns available make it more likely that funded pension schemes will be able to deliver the pension promise The theoretical reason for funded pension scheme is that, in the long-run equilibrium, saving via a high-yielding pension fund helps the process of capital accumulation, which, in turn, improves the productivity of workers This has to with the so-called dynamic efficiency of the economy Abel et al (1989) argued that the major Organization for Economic Cooperation and Development (OECD) economies were really dynamically e fficient, a nd F eldstein (1996) a sserted t hat t his i mplied t hat funded pensions were therefore superior to paygo transfer systems and that a paygo system could be regarded as a tax, with a distortive tax wedge The general insight from the earlier literature dealing with pension scheme issues is b iased b y t he g eneral s ole det erministic en vironment t aken in to account Risk and uncertainty did not figure in those arguments Instead, since a stochastic framework has been introduced, literature has later split the case up into a deterministic and a stochastic one Therefore, only in a deterministic dynamically efficient world the steady state return from a f ully funded pension scheme is higher than the steady state return from a paygo scheme and, hence, the society does not want the unfunded component Not surprisingly, projections of deteriorating dependency ratios have led ma ny economies to attempt to derive a politically feasible and maybe even Pareto-optimal transition from a paygo program to a (partly) funded program Deterministic models therefore predict that a funded program is superior to a paygo program in steady state, reflecting that the benefits from a fully funded system depend on the return on financial markets, while in a pay-as-you-go system the relevant variable is the growth of the contribution base, which depends on productivity and labor supply growth In reference to this, see de Menil and Sheshinski (2004), Matsen and Thøgersen (2004), and Bilancini and D’Antoni (2008) As w e ve a lready m entioned, t he s ituation cha nges i f u ncertainty i s introduced A s M erton (1983), M erton e t a l (1987), G ordon a nd Varian (1988), G ale ( 1990), a nd Bla ke ( 2000) ve em phasized, f unding i s n ot a pa nacea The l ifetime e arnings of a co hort a re subject to shocks; t herefore e ach co hort co uld ba lance i ts own e arnings a nd d rawings w ith t hat of its successor, through the intergenerational transfers entailed in a paygo pension sch eme F ollowing t his co nsideration, i n r ecent y ears, a n umber of papers have emphasized the role of social security in providing intergenerational risk sharing with respect to several sources of risk, including return on financial markets, and demographic and productivity shocks See in t his regard Ma rchand e t a l (1996), B elan a nd Pestieau (1999), B oldrin et al (1999), Dutta et al (2000), Miles and Timmerman (1999), Demange and © 2010 by Taylor and Francis Group, LLC 584 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Laroque (1999), (2000a), (2000b), Lindbeck (2000), Demange (2002), Bohn (2001), Bohn (2004), Wagener (2003), Matsen and Thøgersen (2004), Krueger and Kubler (2006), and Ball and Mankiw (2007) The general insight from this literature is that if you take into account that returns, on both paygo and funded systems, are stochastic, a funded program is not always superior to a paygo program in steady state When returns are stochastic and they are not perfectly correlated, it is possible to diversify risk by optimally choosing a mix of unfunded and funded systems Contrary to what happens in a fully deterministic setting, the paygo asset is not necessarily banned from the rational worker’s portfolio: it depends on the covariances between stock returns and the growth of aggregate wage (or gross domestic product (GDP)) income Thus, the paygo system can be seen as a government-created asset that can be used as a hedge to reduce total portfolio risk, or rather it can be seen as a system that allows intergenerational risk sharing providing society with insurance against bad draws in lifetime income.* The issue, as it has been set, can be dealt with a portfolio choice approach to social security design Miles and Timmermann (1999), Dutta et al (2000), Persson (2002), Matsen and Thøgersen (2004), and de M enil, Murtin, and Sheshinski (2006) explicitly use a portfolio approach that treats the pay-asyou-go system and the funded system as financial assets that can help diversify risk This kind of models highlights the optimal mix between a public paygo program and a private retirement saving From a m ore t heoretical perspec tive, t here a re so me m odels ( Sinn, 1999; Von Weizsäcker, 2000; Miles, 2000; Castellino and Fornero, 2000; Menzio, 2000; Perrson, 2002; de Menil and Sheshinski, 2004; Bilancini and D’Antoni, 2008) that analyse, in different contexts, specific circumstances and reasons why a mixed system is preferable to one that relies only on a single component, either paygo or funding For instance, in Von Weizsäcker (2000), the pension mix with a compulsory pay-as-you-go component and a funded private one may be well suited for reducing pension risk due to demographic shocks, if contribution rates are politically determined as a function of the dependency ratio In this case, if you look, for instance, at a sudden increase in population, the dependency ratio declines and so the contribution rate to the pension system also declines This counteracts the increased return to the pay-as-you-go system due to an increased national product The same stabilizing effect of the pay-as-you-go system works if the opposite holds De Menil and Sheshinski (2004) argued that the optimal size of the pension savings’ paygo part and the funded part depends * See Merton (1983), Persson (2002), and Matsen and Thø gersen (2004) © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 585 both on t he stochastic cha racteristics of GDP g rowth a nd t he return to market savings, and on the shape of the utility function of the representative a gent B ilancini a nd D ’Antoni ( 2008) ndle t he o ptimal ch oice between funded and unfunded pension system, taking into account that people care about their consumption relative to others All these models have almost two points in common: • A mixed paygo and funded system may be optimal due to the extent that wages and the return on capital are negatively correlated • The role of paygo pension system is that of a provider of intergenerational insurance 22.2 OPTIMAL PRIVATE/PUBLIC PENSION MIX The Italian pension system is characterized by the fact that workers set aside about 43% (on average) of their income to pension savings This amount comprises 33% of unfunded (public) pension and 10% of the funded one So, the actual split between public and private pension is 77% and 23% Basically, a two-pillar social security system exists The first pillar is represented b y t he st atutory pa y-as-you-go pens ion sch eme (unfunded pen sion) It is characterized by a contribution-based pension formula, following pension reforms introduced in Italy during the 1990s The formula incorporates rough actuarial principles and thus represents a d rastic change from the previous system, based on a defined benefit calculation rule Alongside the public paygo system, a f unded private second pillar has been provided by decree 124/1993 and, afterwards, b y t he st ate law n o 335/1995 It consists of a voluntary occupational pension scheme In add ition, a pa rt of t he Italian employees get a ter mination i ndemnity called trattamento di fine rapporto (henceforth TFR) The TFR is a lump sum severance pay granted by the employer at the moment the active service ends The indemnity due from the employer is calculated as follows The sum, for the entire period of employment, of 7.41% of the leaving indemnity reference salary for each year is diminished by 0.5% for the financing of a l eaving indemnity guarantee fund under the social security administration, Instituto Nationale Previdenza Sociale (INPS), which takes the place of insolvent employers The entitlement i s r evalued o n De cember e ach y ear, e xcluding t he a mount accrued during the year, by 1.50% plus 75% of the inflation rate measured by the national statistical institute (ISTAT) with respect to December of the previous year More formally, the TFR revaluation yearly rate, called δt, is equal to δ t = 1.5% + π t © 2010 by Taylor and Francis Group, LLC 586 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling where πt is inflation rate at time t The recent pension reform (2007) stated that f uture flows o f T FR m ust a utomatically be pa id i nto occ upational pension f unds, in a very conservative investment line.* This entails that, currently, the above-mentioned 10% pension saving set aside for funded pension scheme is composed by the TFR amount (about 6.91%) and by the employer and employee contribution for the remaining part Our first goal is to find t he o ptimal pens ion s avings d istribution between public and private pension systems in Italy, from a representative worker’s po int o f v iew F or t his p urpose, w e ad opt a po rtfolio sel ection approach that allows us to find the optimal split between the paygo part of the pension savings and the funded part We use a simple static model with mean-variance preferences The r epresentative I talian w orker i s defined a s ving t he f ollowing mean–variance utility function: E[U (P )] = E(P ) − γ ⋅ var(P ) where P is t he employee’s total pension per u nit of money contributed P = + w ru + (1 − w) rf γ is a risk-aversion parameter We n ote t hat w is t he u nfunded pension sha re, a nd ru and rf are random variables representing, respectively, the return of the unfunded pension and of the funded one Let E(P) = + w µu + (1 − w) µf and var(P ) = w ⋅σ2u + (1 − w)2 ⋅σ2f + 2w ⋅ (1 − w ) ⋅σ u,f stand for the mean and the variance of a two-asset portfolio, where µu and σ u are, respectively, the mean and the variance of public pension returns, µf and σ2f are the mean and the variance of private pension returns, and σu,f is the covariance between the two random variables ru and rf To find the optimal paygo pension share w*, we maximize the worker’s utility function: γ Max U (P ) = + w ⋅µ u + (1 − w ) ⋅µ f − ⋅ w [ w ⋅σ2u + (1 − w)2 ⋅σ2f + 2w ⋅ (1 − w) ⋅σ u,f ] with < w < * Single employees can always explicitly choose to divert their TFR individually to a personal pension plan or explicitly refuse to accept the transfer of TFR to pension funds and keep it with the employer © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 587 TABLE 22.1 The Composition of the Average Italian Private Pension Fund Asset Class Index Weight (%) Equity Italy Equity EU Equity world MSCI Italy MSCI EU MSCI World ex EMU 2.20 15.30 8.30 Total equities Bond Italia Bond EU bond world 25.80 JP Morgan GBI Italy JP Morgan GBI EU JP Morgan GBI Global Total bonds 26.90 41.70 5.60 74.20 To per form a n umerical a nalysis we utilized t he Italian GDP g rowth rate as a p roxy for ru The return of the funded component rf is approximated by a benchmark of the average Italian occupational pension fund It consists of 74.2% bond and 25.8% equities Table 22.1 shows percentages of t he va rious a sset cla sses i n t he a verage Italian pens ion f und a nd t he indexes used to approximate them All the time series used in this chapter range over the period 1988–2007 and all nominal values have been deflated by regular consumer price index (CPI) figures Table 22.2 shows the empirical estimates of the mean, standard deviation, and covariance of the annualized 20 year real returns of unfunded and funded pension scheme in Italy We specify that, for the public pension, we have made a co rrection of the GDP growth rate to reflect the fact that the conversion factors, used to annuitize the amount accrued from the workers in their active lives, are more convenient than those used by the Insurance Companies to convert the amount accrued for the private pension In other terms, t he conversion factors applied by t he State a re more convenient than the conversion factors applied by the private pensions Using these data, we can calculate the optimal paygo pension share w* As Figure 22.1 illustrates, it varies with the degree of employee’s risk aversion, TABLE 22.2 Financial Technical Bases GDP growth mean Pension fund return mean GDP growth volatility Pension fund return volatility Covariance © 2010 by Taylor and Francis Group, LLC 2.50% 4.08% 2.68% 4.84% 0.02% 588 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 100,000% 90,000% 80,000% 70,000% 60,000% 50,000% 40,000% 30,000% 20,000% 10,000% 0,000% γ ∞ FIGURE 22.1 3γ 2γ γ γ/2 γ/3 γ/6 Risk aversion and optimal public pension quota which is here parameterized by γ As a st arting point, we use a va lue of γ which corresponds to the current split of the pension savings between public (77%) and private part*; other values are obtained varying γ 22.3 OPTIMAL PORTFOLIO ALLOCATION IN OCCUPATIONAL PENSION FUNDS Furthermore, following the same approach used in Section 22.2, we investigate if the average asset allocation between bond and equity in a domestic occupational pension fund is optimal according to the selected utility function For this purpose, the model can be easily modified: E(P ) = + w ⋅µ u + (1 − w) ⋅[α ⋅µ e + (1 − α) ⋅µ b ] var(P ) = w ⋅σ2u + α (1 − w )2 ⋅σ2e + (1 − w )2 ⋅ (1 − α )2 ⋅σ2b + 2w ⋅ (1 − w ) ⋅σ u,e + 2w ⋅ (1 − w) ⋅ (1 − α) ⋅σ u,b + 2α ⋅ (1 − α) ⋅ (1 − w)2 ⋅σ e,b where α is the fraction of the funded pension allocated to equity µe an d µb a re, r espectively, t he r eturns o n eq uities a nd t he r eturns o n bonds, while σ 2e and σ2b are the variances of equities and bond returns σu,e, σu,b, and σe,b are the covariances * This value is about 180 © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 589 TABLE 22.3 Paygo Equity Bond TABLE 22.4 Optimal Quotas of Equities, Bonds, and Risk Aversion within Worker’s Total Pension Equity (%) Bond (%) g γ→∞ 3γ 2γ γ γ/2 γ/3 γ/6 a 3.11 3.68 3.97 4.86a 6.62 8.39 13.68 20.15 19.58 19.29 18.40 16.64 14.87 9.58 The c urrent val ue o f eq uity share co rresponding t o γ is 6% TABLE 22.5 Optimal Quotas of Equities, Bonds, and Risk Aversion within the Asset Allocation of an Italian Pension Fund Equity (%) Bond (%) g γ→∞ 3γ 2γ γ γ/2 γ/3 γ/6 a 13.52 16.00 17.26 21.12a 28.79 36.47 59.49 86.48 84.00 82.74 78.88 71.21 63.53 40.51 The c urrent val ue o f eq uity share co rresponding t o γ, within the pension fund asset allocation, is 25.8% The Covariance Matrix Paygo Equity Bond 0.072% −0.041% 0.037% −0.041% 1.944% 0.168% 0.037% 0.168% 0.134% Given the fi xed value of w corresponding to the c urrent l evel, t he a sset a llocation problem is to choose the optimal level of α Our numerical analysis highlights that in Italy, t he average yearly real y ield on equity in a p rivate pension f und is 6.03%, over t he period 988–2007 O ver t he sa me pe riod, the average yearly y ield on bond a sset cla ss is 41% The co variance ma trix is g iven i n Table 22.3 First o f all , w e in vestigate if t he c urrent asset allocation of the average pension fund is optimal Currently, worker’s pension savings consists 77% of a paygo system and 23% of an o ccupational p ension f und This 23% is co mposed b y eq uities (6%) a nd b onds (17%) Substituting t he val ue o f γ, f ound in the p revious s ection, w hich co rresponds to t he c urrent si tuation, i t t urns o ut t hat the o ptimal le vel o f eq uities in a n Italian pension fund α* should be equal to 4.86% This suggests that the current asset allocation for the funded pension scheme is not the optimal one Tables 22.4 and 22.5 show how the optimal sp lit b etween b onds a nd eq uities va ries wi th dif ferent le vels o f em ployee’s r isk aversion Until now, we have relied on historical means, variances, and correlations as inputs to ou r c alculations a bove, but we c annot © 2010 by Taylor and Francis Group, LLC 590 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling neglect that all these conclusions are sensitive not only to t he spec ified degree of employee’s r isk a version γ, b ut a lso t o t he co rrelation between the random variables involved in our calculations As shown i n t he recent dramatic crisis of the financial markets, correlations can change along the time The following figures show the sensitivity of t he r esults t o t he co rrelation coefficient and to the risk aversion TABLE 22.6 Optimal Quotas of Private Pension Fund and Risk Aversion Correlation (%) Equity Weight (%) 10 20 30 40 50 60 70 80 90 8.45 7.16 6.06 5.11 4.29 3.57 2.94 2.40 2.03 1.58 22.3.1 Sensitivity Analysis We su ppose n ow t hat t he co rrelation between eq uity a nd bo nd is va riable a nd we wa nt to determine t he sensitivity of t he equity weight with respect to correlation The covariance between equity and bond is 0.168% so that the actual correlation is about 33% The results are given in the following Table 22.6 The risk-aversion coefficient takes the value γ = 180 as in the previous analysis See Figure 22.2 for a graphical representation In a further analysis, we allow the risk-aversion coefficient to vary and we consider the sensitivity of the equity weight with respect to both the correlation and the risk-aversion coefficient (Figure 22.3) We can plot the results as a bidimensional surface 0.07 Weight 0.06 0.05 0.04 0.03 0.02 0.01 10 20 30 40 50 60 70 FIGURE 22.2 Equity weight vs correlation between equity and bonds © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 591 Weight 0.02 0.04 0.06 0.08 0.8 250 200 γ 0.6 150 0.2 0.4 elatio rr Co n 100 FIGURE 22.3 Equity weight vs correlation between equity a nd bonds a nd risk aversion 22.4 ROLE OF THE TERMINATION INDEMNITY SCHEME Finally, we aim at finding the percentages that would optimize the pension risk management of the workers, considering the presence of a ter mination indemnity at the end of the career In this way, we pick out employee’s pension optimal asset allocation between three assets, two of which currently exist (paygo and funded pension) and the “dying breed” asset represented by TFR Expected value and variance of worker’s pension become: E(P ) = + x ⋅µ u + y ⋅µ f + z ⋅µ δ var(P ) = x σ2u + y σ2f + z σ2δ + x y σ u,f + x ⋅ z ⋅σ u,δ + y ⋅ z ⋅σ f ,δ with x + y + z = 1, < x < 1, < y < 1, and where µδ is the average yearly TFR revaluation rate, σ 2δ is its variance, σu,δ and σf,δ are the covariances, respectively, between the returns of TFR, public pension, and private pensions We consider the occupational pension fund as using the current average a sset a llocation: 25 8% eq uities a nd 4.2% TABLE 22.7 Financial Technical Bases Regarding bonds Table 22.7 reports the technical basis for TFR TFR Returns 0.61% µδ asset used in our calculations 1.28% Table 22 sh ows t he o ptimal m ix o f p ublic σδ 0.024% pension, private pension, and termination indem- σu,δ 0.020% σ f,δ nity plan when the risk aversion varies © 2010 by Taylor and Francis Group, LLC 592 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 22.8 Optimal Quotas of Public Paygo Pension, Private Pension Fund and Termination Indemnities g Public Pension (%) Pension Fund (%) TFR (%) 52.89 56.07 62.38 68.10 60.66 35.51 34.50 32.52 30.85 39.27 11.60 9.43 5.10 1.05 0.07 2γ γ γ/2 γ/3 γ/6 Also in this case, we perform a sensitivity analysis to describe the relationships between the optimal quotas, the correlations, and the risk aversion of the worker 22.4.1 Sensitivity Analysis We suppose now that the correlation ρf,δ is fi xed, while the other two correlations ρu,f and ρu,δ are variables We want to determine the sensitivity of the optimal TFR weight with respect to both correlations Thi s sensitivity is given by a surface Moreover, we repeat this procedure by changing the value of the correlation ρf,δ (from 10% to 80% with a step of 10%) and we superimpose the surfaces obtained in this way The r isk-aversion coefficient takes the value γ = 180 The starting values of the three correlations are, respectively, ρu,f = 13.3%, ρu,δ = 69.7%, and ρf,δ = 31.8% See Figure 22.4 for a graphical representation 0.6 ρ u,δ 0.4 0.2 0.5 Weight 0.4 0.3 0.2 0.1 0.2 ρu0.4 ,f 0.6 FIGURE 22.4 level of ρf,δ) TFR weight vs correlations (each surface corresponds to a different © 2010 by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 593 0.6 0.4 0.4 ρu,δ Weight 0.3 0.2 0.1 0.2 0.2 ρu,f 0.4 0.6 FIGURE 22.5 TFR weight vs correlations (x and y axes) and risk aversion (each surface corresponds to a different level of γ) In a seco nd step, we fi x t he correlation ρf,δ a nd let t he other t wo correlations ρu,f and ρu,δ vary We repeat the sensitivity analysis changing the risk-aversion coefficient (from 100 to 260 with a step of 20) See Figure 22.5 for a graphical representation 22.5 CONCLUSIONS In t his cha pter, w e i nvestigated t hree t hemes abo ut t he I talian pens ion system First of all, we analyse the current average composition of the pension contribution rate of the Italian employee; in fac t the 77% o f the pension contribution te finances t he p ublic p ension syst em; t his no t o nly has lower expected returns than the private pension one, but also lower risk The two quotas are optimal only if the risk aversion of the worker is equal to a p redetermined le vel; if t he r isk-aversion pa rameter is different, the optimal quotas vary but (and this is an important remark) not so much (as shown in Figure 22.1) The second question is: given the actual level of the two-pillar contribution rates, is the average composition of the Italian pension fund portfolio optimal f rom t he w orker’s u tility po int of v iew? Ob viously, t his a nswer also depends on t he r isk aversion (and on t he correlations a mongst t he various asset classes); starting from the implied risk tolerance of the workers, we discover that the pension fund portfolios are not optimal and that portfolios tend to oversize the equity quotas © 2010 by Taylor and Francis Group, LLC 594 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling At last, we consider a t hree asset portfolio formed by the private pension, the public pension, and the TFR (an Italian termination indemnity that has been cancelled to finance the private pension sector); in this case, we discover that, in some conditions, the role played by TFR is able to improve significantly the trade-off between risk and return; in fact, while in t he ba sic c ase t he t heoretical T FR w eight i s abo ut 9.4%, i t g rows u p when the risk aversion increases Obviously, in this case also, the results depend on the worker risk aversion and on the correlation structure REFERENCES Aaron, H J 1966 The social insurance paradox Canadian Journal of Economics and Political Science 32: 371–374 Abel, A., Mankiw, N G., Summers, L., and Richard, J 1989 Assessing dynamic efficiency Review of Economic Studies 56: 1–19 Ball, L a nd M ankiw, N G 2007 I ntergenerational r isk sha ring in t he sp irit o f Arrow, Debreu, and Rawls, with applications to social security design Journal of Political Economy 115: 523–547 Belan, P and Pestieau, P 1999 P rivatizing s ocial s ecurity: A cr itical ass esment Geneva Papers on Risk and Insurance, Issues and Practice 24: 114–130 Bilancini, E a nd D’Antoni, M 2008 P ensions and intergenerational risk sharing when relative consumption matters Quaderni del dipartimento di Economia Politica, 541, agosto 2008 Blake, D 2000 Does it matter what type of pension scheme you have? 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Taylor and Francis Group, LLC 592 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 22. 8 Optimal Quotas of Public Paygo Pension, Private Pension Fund and Termination Indemnities... by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 587 TABLE 22. 1 The Composition of the Average Italian Private Pension Fund Asset Class Index Weight... by Taylor and Francis Group, LLC Public and Private DC Pension Schemes, Termination Indemnities ◾ 583 of the capital market in terms of the rate of return on investment These high real returns