Willem Heeringa. Optimal Life Cycle Investment with Pay-as-you-go Pension Schemes: A Portfolio Approach

Transcription

1 Willem Heeringa Optimal Life Cycle Investment with Pay-as-you-go Pension Schemes: A Portfolio Approach Discussion Paper February 7, 008

2 Optimal life cycle investment with pay-as-you-go pension schemes: a portfolio approach Willem Heeringa y De Nederlandsche Bank and Netspar February 7, 008 Abstract In this paper we show how pay-as-you-go pension schemes impact on the individual s optimal investment portfolio. Introducing a pay-as-you-go pension scheme implies that human wealth of young generations is transferred to retired generations. As a consequence, individuals will in general invest less conservatively. These portfolio e ects gradually disappear at the end of life. J.E.L Classi cation: H55; D9; G Keywords: Social security; Risk sharing; Portfolio choice I am grateful to comments and suggestions by Robert-Paul Berben, Lans Bovenberg, Peter Broer, Jiajia Cui, Frank de Jong, John Lewis, Job Swank, Coen Teulings and the participants of the 6th SUERF Colloquium on Money, Finance and Demography held October 006 in Lisbon. Henk van Kerkho provided useful research assistance. The usual caveat applies. y Contact information: w.l.heeringa@dnb.nl, De Nederlandsche Bank, Economics & Research Division, P.O. Box 98, 000 AB, Amsterdam, The Netherlands, tel: +3 (0) , fax: +3 (0)

3 Introduction An individual saving for his pension has to decide how to invest his pension savings. A pay-as-you-go (PAYG) pension scheme interferes with this portfolio decision by reducing net wage income when young and providing pension income when retired. In fact, introducing a pay-as-you-go pension scheme implies that human wealth of young generations is partly transferred to retired generations. As noted by Merton (983) and Persson (00), a PAYG-pension scheme may actually be considered as a quasi-asset relevant for the optimal portfolio allocation. The main question we address in this paper is how a PAYG pension scheme impacts on the optimal nancial investment over lifetime. To this end we will analyze how human wealth from (net) wages and pension wealth from PAYG-pension bene ts impact on the optimal allocation of nancial wealth over the individual s lifecycle. The literature on the optimal funding of pensions has mainly focussed on the question whether pensions should be nanced on a funded or a pay-as-you-go basis. However, more recently optimal portfolio theory from nance literature has shed a new light on the issue of PAYG versus funding. Merton (983) already argued that a mix of PAYG and funding might be optimal for reasons of diversi cation. Persson (00) noted that a PAYG-scheme can be considered as an asset in the overall investment portfolio. By formalizing this argument in a two period overlapping generations model, Matsen and Thøgersen (004) demonstrated that even a low-yielding PAYG-scheme can bene t individuals if it contributes to hedge other risks to their lifetime resources. Our paper is in two aspects an extension of the analysis of Bodie et al. (99). The rst extension is that pension wealth from PAYG pension schemes is explicitly taken into account when determining the optimal allocation of - nancial wealth. This matters especially after retirement when an individual s human wealth is fully depleted. Apart from nancial wealth, total wealth then consists of pension wealth from future PAYG pension bene ts. The second extension is the inclusion of in ation as a risk factor. Uncertainty about future in ation implies uncertainty about future real wealth, which is relevant for consumption and investment decisions. Apart from in ation we also include productivity growth as a risk factor in our model. We take the existence of a PAYG pension scheme as given and do not analyse the welfare e ects of a PAYG pension scheme. The paper is organized as follows. The next section presents our basic theoretical model framework. Applying the dynamic programming approach (Campbell and Viceira, 00), we derive the individual s optimal consumption and portfolio allocation over the lifecycle. In section 3, we extend the model with stochastic wage income and analyse how human wealth from stochastic wages matters for the optimal allocation of nancial wealth in complete markets. In section 4, we extend the model with a PAYG pension scheme and analyse how See Matsen and Thøgersen (004) for an overview of the literature. There is a rich literature about the welfare e ects of PAYG schemes, see for instance Bohn (00) and Sánchez-Marcos et al. (006).

4 this impacts on the optimal allocation of nancial wealth. We consider both a de ned contribution and a de ned bene t PAYG-pension scheme and show how pension wealth from PAYG-schemes matters for the optimal allocation of nancial wealth. In section 5, we calibrate the model to quantify the e ects predicted by the theoretical model. The results of the paper are summarized in section 6, while appendices A-G provide the detailed formal derivations as well as a table of notation. Model In this section we explore the lifetime consumption and portfolio choice model of Fischer (975) to derive optimal portfolio investment over the life cycle. Individuals live from time 0 to D (being deterministic) and derive utility from consumption. Initially, we assume that individuals only receive income as a (stochastic) return on tradable assets. In sections 3 and 4, we extend the model with stochastic wages and pay-as-you-go pension bene ts, respectively.. Price dynamics and asset returns Initially, we assume that the individual s investment portfolio can contain three assets: price indexed bonds, equity and nominal bonds. The portfolio can be adjusted instantaneously and costlessly and there are no nonnegativity constraints on the asset holdings. Suppose the behavior of the price level P follows a geometric Brownian motion: dp P = dt + s dz () with being the expected price increase per unit of time 3. Here, dz is a Wiener process with s being the variance of the change in the price level per unit of time. The real return on price indexed bonds (Q ) equals: d (Q =P ) Q =P = r dt () Suppose that real returns on equity also follow a geometric Brownian motion. The expected real return on equity (Q ) equals r. Stochastic changes in the real returns on equity are assumed to be related to productivity growth. On its turn, productivity growth is supposed to be driven by the Wiener process dz. This gives: d (Q =P ) Q =P = r dt + s dz (3) 3 Some models in the literature apply stochastic expected in ation, see for instance Brennan and Xia (00). However, in our model expected in ation is deterministic for reasons of tractability. 3

5 where s denotes the variance of real equity returns per unit of time. The nominal return on nominal bonds (Q 3 ) equals R 3 implying that the real return on nominal bonds can be expressed as (Fischer (974)): where d (Q 3 =P ) Q 3 =P = r 3dt + s 3 dz 3 (4) r 3 = R 3 + s dz 3 = dz s 3 = s. Budget constraints and individual s choice problem Let ; and 3 be the proportions of the portfolio held in price indexed bonds, equity and nominal bonds, respectively. This implies that the stock budget constraint is = (5) The ow budget constraint, describing the change in real wealth (W ), can be de ned as: P dw = 3 i r i W dt i= P Cdt + 3 i s i dz i W (6) with C being consumption per unit of time. Note that the last term on the right-hand side of (6) captures the stochastic change in real wealth due to holdings of equity and nominal bonds. Substituting for from (5) into (6) gives i= dw = [[r + (r r ) + 3 (r 3 r )] W C] dt + [ s dz + 3 s 3 dz ] W (7) Merton (969) demonstrates that the optimal portfolio and consumption problem of individuals living from time 0 to D, is to nd max C; i E 0 Z D 0 U [C (t) ; t] dt (8) subject to (7) and W (0) = W 0 > 0, with C (t) 0 and U () strictly concave in C. Let J [W (t) ; t] denote the derived utility function of this problem: J [W (t) ; t] max C; i E t Z '=D 4 '=t U [C (') ; '] d' (9)

6 subject to the transversality condition J [W (D) ; D] = 0 (0) The Bellman-Dreyfus fundamental equation of optimality yields: 0 = max U [C (t) ; t] + C; i dt E t [dj (W; t)] () The Bellman principle implies that at the optimum, the investor has traded o the value of present and future consumption perfectly. Applying Itô s lemma to J [W (t) ; t] gives: dj [W (t) ; t] = J W dw + J t dt + J W W dw () where subscripts denote partial derivatives. Substituting () into the Bellman equation () and computing the rst-order conditions by taking derivatives with respect to C; and, gives three expressions for optimal consumption and portfolio choice 4 U C (C; t) = J W (3) J W (r r ) (r 3 r ) = J W W W ( ) s ( ) s (4) s 3 J W (r3 r ) (r r ) 3 = J W W W ( ) s 3 ( ) s (5) s 3 subject to the transversality condition J [W (D) ; D] = 0 and with being the instantaneous correlation coe cient between Wiener processes dz and dz (j j<). Equation (3) determines the optimal consumption policy and is known as the envelope condition, stating that at the optimum an extra unit of consumption is as valuable as an extra unit of wealth to nance future consumption. Equations (4) and (5) determine the optimal portfolio allocation to equity and nominal bonds, respectively. For both equations, the coe cient of the rst J term in brackets ( W J W W W ) is the inverse of the degree of relative risk aversion of the individual. Suppose for the moment that = 0. Then the demand for both equity and nominal bonds depends in an intuitively plausible way on their expected excess real return over price indexed bonds (r 3 r respectively r r ) and their return variances (s and s 3). Moreover, for < 0 and s 3 < 0, equity and nominal bonds are substitutes in that the demand for each is negatively related to the expected real return of the other. As equations (3)-(5) depend on the indirect utility function J, they do not o er the complete solution to the model. In order to nd a complete solution, we will now consider a speci c utility function with constant relative risk aversion (CRRA): 4 See appendix A for the formal derivation. 5

7 [C (t)] U [C (t) ; t] = exp [ t] where is the rate of time preference. As a trial solution for the indirect utility function we take: (6) J [b (t) ; W (t) ; t] = b (t) exp ( t) [W (t)] (7) By substituting the trial solution (7) into the Bellman-Dreyfus equation of optimality (), we nd that b (t) must satisfy the following ordinary di erential equation: db dt = ( ) b + bx (8) where X is a constant 5. A solution to (8) satisfying the transversality condition (J [b (D) ; W (D) ; D] = 0) is: 0 b (t) ( h ) t exp X D i X A (9) Optimal consumption over time can now be expressed as a function of real wealth: C (t) = [b (t)] while the optimal portfolio shares are now equal to: W (t) (0) (r r ) = ( ) + (r 3 r ) s ( ) s 3 = (r r ) (r 3 r ) ( ) s ( ) s s 3 3 = (r3 r ) (r r ) ( ) s 3 ( ) s s 3 (r + r 3 r ) ( ) s s 3 () () (3) Here, is determined as the residual investment following from () and (3). The interpretation of the optimal portfolio shares in case of CRRA-utility is the same as the general case ((4)-(5)). However, it is important to note that the optimal portfolio shares are now independent of the level of W. This implies that the optimal allocation of total real wealth is constant over lifetime. We will explore this result in sections 3 and 4, when we extend the analysis with human wealth from wages and pension wealth from pension bene ts. 5 See appendix B for the derivation. 6

8 3 Wage income So far we have assumed that (gross) returns from assets are the only source of income. As an extension, we now also introduce wage income from inelastically supplied labour 6. Moreover, suppose the individual retires at the statutory retirement age T (< D). The present value of future wages can be considered as human wealth (Bodie et al. (99)). This implies that total wealth can be de ned as the sum of nancial wealth F and human wealth from wages H w : W (t) = F (t) + H w (t) (4) If we manage both to value this human wealth as well as to nd a replicating portfolio in terms of nancial assets, we can determine the optimal portfolio allocation of ( nancial) wealth by using equations ()-(3). We suppose that real wages are subject to both in ationary and productivity risks. Hence, we suppose that real wages w follow the geometric Brownian motion dw w =!dt + s wdz + s w dz (5) with s w being the variance of the change in real wages per unit of time related to price changes and s w being the variance of the change in real wages per unit of time related to productivity changes. In order to value the human wealth from this wage income, we now apply a similar approach as taken by Bodie et al. (99). Proposition The real value of human wealth from wages (H w ) equals the present value of future wages discounted at a risk-adjusted discount rate w : with H w [w(t); t] = w (t) w exp w bt t (6) bt = max [t; T ] w = r + w (r r ) + w (r 3 r )! w = s w s 3 Proof. See appendix C. w = s w s here. 6 The option to substitute labour for leisure (Bodie et al. (99)) is not taken into account 7

9 In (6), the discount factor w is equal to the real return on price indexed bonds plus the risk-adjusted equity and nominal bond premiums respectively, minus the real wage drift. Guiding this risk adjustment are the factors w and w, where w re ects the relative sensitivity of real wages for in ationary shocks and w the relative sensitivity of real wages for productivity shocks. At birth human wealth is at its maximum, while it is gradually depleted thereafter. Note that H w = 0 once the individual is retired (i.e. for t > T, bt equals t). Given the expression we derived to value human wealth, we now need a technique to nd equivalents of this value in terms of nancial assets. An applicable technique is contingent claim analysis (Merton, 990). Proposition The value of human wealth from wages H w is economically equivalent to a portfolio with w H w invested in equity, w H w invested in nominal bonds and ( w w ) H w invested in price indexed bonds. Proof. See appendix D. Armed with propositions and and applying equation (4), we can now express the explicit euro investment in equity (Z Q (t)) as: Z Q (t) = W (t) w H w (t) = F (t) + ( w ) H w (t) (7) where the explicit investment in equity equals the desired gross investment minus the implicit investment via human wealth from wages. Equivalently, the explicit euro investment in nominal bonds (Z Q3 (t)) can be expressed as: Z Q3 (t) = 3 W (t) w H w (t) = 3 F (t) + ( 3 w ) H w (t) (8) Finally, the explicit euro investment in price indexed bonds (Z Q (t)) can be expressed as: Z Q (t) = F (t) + ( ( w w )) H w (t) (9) Hence, the explicit investment in price indexed bonds equals the desired gross investment minus the implicit investment via human wealth from wages. This implies that the optimal portfolio shares of nancial wealth F (t) invested in equity, nominal and price indexed bonds respectively (denoted by b, b 3 and b respectively) can be expressed as b (t) Z Q (t) F (t) = + ( w ) Hw (t) F (t) (30) 8

10 and b 3 (t) = Z Q 3 (t) F (t) = 3 + ( 3 w ) Hw (t) F (t) (3) b (t) Z Q (t) F (t) = + ( ( w w )) Hw (t) F (t) (3) where the second terms on the right-hand side of expressions (30)-(3) allow for an analysis of the incremental e ects of human wealth on the optimal portfolio allocation. Note that the optimal portfolio allocation of nancial wealth is now time-dependent (i.e. upon the value of H w and F ). Hence, in response to changes in either or H w or F, a continuous reallocation of nancial wealth may be required under constant relative risk aversion in order to keep the allocation of total wealth constant over the lifecycle. Early in the lifecycle total wealth consists predominantly of human wealth. Hence, it might be optimal for the individual to borrow early in the lifecycle. We will calibrate the model in section 5 in order to quantify the incremental portfolio e ects as predicted by equations (30)-(3). However, we will rst derive the portfolio e ects of a pay-as-you-go pension scheme. 4 Pay-as-you-go pension scheme In this section we will consider how a PAYG pension scheme impacts on the optimal portfolio derived before. Under a PAYG pension scheme, income from the young is taxed in order to nance an old-age pension bene t for retirees. The present value of future pension bene ts can be considered as a form of positive pension wealth for the individual. Equivalently, the present value of future tax payments can be considered as a form of negative pension wealth for the individual. In fact, introducing a pay-as-you-go pension scheme implies that human wealth of young generations is partly transferred to retired generations. We start by considering a de ned contribution pension scheme in section 4., before proceeding to a de ned bene t pension scheme (section 4.). 4. De ned contribution pension scheme Suppose the government introduces a de ned contribution (DC) PAYG pension scheme providing retirees with a real pension bene t (t). The scheme is - nanced by a xed proportional wage tax. The ow budget constraint of this DC PAYG pension scheme is 9

11 n o (t) (t) = n y (t) w (t) (33) Suppose that the ratio of the number of young persons n y (t) over the number of old persons n o (t) (the so-called support ratio k) 7 remains constant. Using (33), we can now express the real pension bene t as (t) = kw (t) (34) implying that the DC pension bene t depends on current wages. Total wealth now equals the sum of nancial wealth (F DC ), human wealth and net pension wealth, the latter being equal to pension wealth from future pension bene ts (H ) minus the government s tax claim on human wealth (H w ): W (t) = F DC (t) + H w (t) + H (t) H w (t) (35) In order to analyze the impact of the DC pension scheme on total wealth and the optimal allocation of nancial wealth, we need to value pension wealth from pension bene ts. Proposition 3 The value of pension wealth from DC pension bene ts as de ned in (34) H equals the present value of future pension bene ts discounted at the risk-adjusted discount rate w with: H [(t); t] = k w (t) w exp w bt t exp [ w (D t)] (36) bt = max [t; T ] w = r + w (r r ) + w (r 3 r )! w = s w s 3 w = s w s Proof. See appendix E. Note that the development of pension wealth di ers from human wealth over lifetime. After being born, the value of human wealth gradually diminishes until pension age T when it is fully depleted. In the same period, the value of pension wealth gradually increases until it reaches its maximum at retirement age T. After being retired, the value of pension wealth gradually diminishes until t = D when it is fully depleted. 7 Note that the support ratio is the inverse of the so called old-age dependency ratio, being the ratio of number of old over the number of young persons. 0

12 Armed with equation (35) and proposition 3, we can distinguish two potential e ects of the DC PAYG pension scheme on total wealth. First of all, as wage income is taxed individuals may save less over life causing F DC (t) < F (t). Such crowding out of nancial wealth reduces total wealth. Secondly, total wealth changes with the value of net pension wealth (H (t) H w (t)). Using propositions and 3, we can show that net pension wealth at birth is zero if 8 : ( + k ) exp [ w T ] k exp [ w D] = (37) By applying a second order Maclaurin expansion on equation (37), we can derive the following approximation for the equilibrium support ratio k : with k T D T ( w) (38) ( w ) = wt w (T + D) Net pension wealth at birth will be positive for k > k and negative for k < k. Let us now take a closer look at equation (38) by considering the case of a dynamically e cient economy (implying w > 0) with no population growth (implying a constant support ratio k). In this case the support ratio k will be equal to k = T D T as every dying generation is exactly replaced by a new generation. Moreover, we can show that ( w ) > for small values of 9 w, implying k = k < k. Hence, net pension wealth at birth will always be negative in this case. Put di erently, net pension wealth will only be positive when population growth causes k > k > k. In fact, equation (38) re ects the Aaroncondition (Aaron, 966) stating that a PAYG pension scheme in a dynamically e cient economy is only a Pareto improvement if population growth exceeds the (positive) di erence between the return on nancial wealth and real wage growth (i.e. w ). The replicating portfolio of pension wealth can again be determined with contingent claim analysis: Proposition 4 The value of pension wealth from pensions H is economically equivalent to a portfolio with w H invested in equity, w H invested in nominal bonds and ( w w ) H invested in price indexed bonds. Proof. See appendix E. Let DC i denote the optimal fraction of nancial wealth invested in asset i under a DC PAYG pension scheme. Armed with propositions 3 and 4 and equation (35), we can now express the incremental changes in the optimal nancial portfolio due to the introduction of the DC PAYG pension scheme as: 8 See appendix E. 9 Mathematically, ( w ) has a vertical asymptote at w = and a horizontal asymptote at =. Hence, ( w ) > for w < T +D T +D.

13 for DC (t) b (t) = ( ( w w )) [ (t) + (t)] (39) DC (t) b (t) = ( w ) [ (t) + (t)] (40) DC 3 (t) b 3 (t) = ( 3 w ) [ (t) + (t)] (4) (t) = H (t) H w (t) F DC (t) (t) = Hw (t) F DC (t) H w (t) F (t) (4) (43) Equations (39)-(4) show that we can break up the incremental portfolio changes in two terms. In the rst term, (t) re ects the direct e ect of net DC PAYG pension wealth on the optimal composition of nancial wealth. In the second term, (t) re ects the indirect e ect arising in case of crowding out of nancial wealth (i.e. F DC (t) < F (t)). In that case, the incremental portfolio e ects of human wealth as derived in the previous section are ampli ed. We will calibrate the model in the next section in order to quantify these incremental portfolio e ects. 4. De ned bene t pension scheme In this subsection, we will consider a de ned bene t (DB) pension scheme with a xed real pension bene t but a variable tax rate ( (t)). The ow budget constraint of this DB PAYG pension scheme is n o (t) = n y (t) (t) w (t) (44) With a constant support ratio k, the tax rate can be expressed as (t) = kw (t) (45) Hence, for a given real pension bene t, keeping a balanced budget implies that the tax rate under a DB pension scheme becomes a function of the wage level. Apart from creating an asset in the form of pension wealth from pension bene ts, the DB pension scheme also creates a stochastic liability for the young individual in the form of a future tax payments, which can be considered as a government s tax claim on the individual. Unlike a DC pension scheme, the tax claim is no longer a xed proportion of human wealth under a DB pension scheme with stochastic tax rates (see equation (45)). Total real wealth can now be expressed as the sum of nancial wealth (F DB ), human wealth and net pension wealth, being gross pension wealth from DB pension bene ts (H ) minus the government s tax claim on human wealth (H ):

14 W (t) = F DB (t) + H w (t) + H (t) H (t) (46) In order to analyze the impact of the DB-pension scheme on total wealth and the optimal portfolio allocation, we need to value both DB pension wealth as well as the government s tax claim on human wealth. The value of pension wealth from the xed real DB pension bene ts is simply the present value of these bene ts discounted at the riskfree real interest rate: with H (t) = r exp r bt t exp [ r (D t)] (47) bt = max [t; T ] Note that the value of DB pension wealth is deterministic. This also implies that the replicating portfolio of H (t) consists solely out of price indexed bonds. Put di erently, the value of DB pension bene ts H (t) is economically equivalent to a portfolio with H (t) invested in price indexed bonds 0. De ne the individual s instantaneous payable taxes under a DB pension scheme as: Using (45), we can rewrite this as: (t) = (t) w (t) (48) (t) = = k (49) Proposition 5 The value of the government s tax claim under a DB pension scheme (H ) equals the present value of future payable taxes discounted at the discount rate r with H [ (t); t] = kr exp r bt t (50) bt = max [t; T ] Proof. See appendix F. The value of the tax claim is perfectly hedged for changes in the wage level, given that the tax rate is perfectly inversely related to wages (see equation (45)). Put di erently, wage shocks are compensated for by inverse tax rate shocks keeping instantaneous payable taxes constant. 0 See appendix F for the formal proof. 3

15 Armed with equations (46), (47) and (50), we can distinguish two potential e ects of introducing a DB PAYG pension scheme on total wealth. First of all, as wage income is taxed individuals may save less over life causing F DB (t) < F (t). Such crowding out of nancial wealth reduces total wealth. Secondly, total wealth changes with the value of net pension wealth (H (t) H (t)). Using propositions and 5, we can show that net pension wealth at birth is zero if : ( + k ) exp [ r T ] k exp [ r D] = (5) This implies that net pension wealth at birth will be positive for k > k and negative for k < k. However, analogously to the DC-case, we can show that net DB PAYG pension wealth will only be positive if population growth causes k > k > k. Thus, equation (5) again re ects the Aaron-condition (Aaron, 966). The replicating portfolio of the government s tax claim can again be determined with contingent claim analysis: Proposition 6 The the present value of payable taxes under a DB pension scheme ( H ) is economically equivalent to a portfolio with H invested in price indexed bonds. Proof. See appendix F. Armed with propositions 5 and 6 and equation (46), we can now express the incremental changes in the optimal nancial portfolio due to the introduction of the DB PAYG pension scheme as: DB (t) b (t) = ( ) (t) + ( ( w w )) (t) (5) DB (t) b (t) = (t) + ( w ) (t) (53) DB 3 (t) b 3 (t) = 3 (t) + ( 3 w ) (t) (54) with (t) = H (t) H (t) F DB (t) (t) = Hw (t) H w (t) F DB (t) F (t) where DB i denotes the optimal fraction of nancial wealth invested in asset i under a DB PAYG pension scheme. Equations (5)-(54) show that we can break up the incremental portfolio changes in two terms. In the rst term, (t) re ects the direct e ect of net DB PAYG pension wealth on the optimal composition of nancial wealth. In the second term, (t) re ects an indirect See appendix F. See appendix F for the formal derivation. 4

16 e ect arising in case of crowding out of nancial wealth (i.e. F DB (t) < F (t)). In that case, the incremental portfolio e ects of human wealth as derived in the previous section are ampli ed. We will calibrate the model in the next section in order to quantify these incremental portfolio e ects. 5 Calibration In this section we will calibrate the model to get an idea of the quantitative magnitude of the e ects as predicted by the model. Based on the revealing study by Dimson, Marsh and Staunton (00) (DMS), providing long run statistical properties of di erent countries asset market performance, we set the expected real return on price indexed bonds (r ) equal to.5% and on equity (r ) equal to 5.8% with a standard deviation (s ) of %. Also based on DMS, the expected nominal return (R 3 ) on nominal bonds is set equal to 4.%, while expected in ation () is set at %, with a standard deviation (s ) of 5%. This yields an expected real return on nominal bonds (r 3 ) of.4%. The puzzling tendency of real equity returns to covary negatively with in ation has been extensively documented in the literature 3. Marshall (99) estimates the value for the correlation coe cient between real equity returns and in ation ( ) at De Jong (005) estimates annual real wage growth in the Netherlands in the period at.4% with a total standard deviation of 4.5%. Based on these estimates, we set expected real wage growth (!) equal to.4%, s w equal to.3% and s w equal to 3.% (yielding a total standard deviation of 4.5%). The values for the rate of relative risk aversion () and time preference () are derived from the literature review undertaken by Teulings and De Vries (006) (TDV). Following TDV we set the retirement age (T ) at 40 and the time of death (D) at 55. Table : Parameter calibration Asset Expected real return Stand. dev. Price-indexed bonds Equity Nominal bonds In ation Correlationcoe cient ( ) -0.9 Real wages Preferences Risk aversion () 5 Time preference () 0.0 Life cycle Retirement age (T ) 40 Time of death (D) 55 3 See Marshall (99) for an overview of the literature. The negative covariance suggests that supply shocks dominate demand shocks. 5

17 The optimal nancial portfolio in the absence of human and PAYG pension wealth is summarized in table. Approximately a quarter of nancial wealth is invested in price-indexed bonds and equity respectively, and the remaining half in nominal bonds. As demonstrated before, the incremental change in the optimal nancial portfolio in case of human and PAYG pension wealth depends inter alia on the sensitivity of real wages to economic shocks. The relative sensitivity of real wages to in ationary shocks ( w ) equals 0.46, implying that 3 > w. The relative sensitivity of real wages to productivity shocks ( w ) equals 0.9, implying that > w. As a consequence, w w equals 0.35, implying that < w w. The value of w, the discount factor used to calculate inter alia the value of human wealth, equals This implies the economy under consideration is dynamically e cient. Table : optimal portfolio and implied values Optimal portfolio Fraction Price-indexed bonds ( ) 0.8 Equity ( ) 0.6 Nominal bonds ( 3 ) 0.47 Implied values w 0.46 w 0.9 w w 0.35 w 0.03 We ran the model 0000 times to simulate the impact of human and pay-asyou-go pension wealth (both DC and DB) on optimal portfolio and consumption behaviour. We set initial nancial wealth equal to and initial wages equal to 0. Below, we will discuss the average outcomes of our simulations in terms of wealth dynamics, optimal consumption and the incremental change in the optimal nancial portfolio. Figure shows the wealth dynamics when there is only human wealth (section 3). Human wealth is gradually depleted until the retirement age, while - nancial wealth follows the typical hump-shaped pattern as predicted by standard life cycle theory (Ando and Modigliani, 963). As a result, total wealth increases until midlife and is depleted thereafter. Consumption develops smoothly over life. However, at the end of life consumption steeply increases as the remaining nancial wealth (partly being precautionary savings for unexpected shocks) is depleted. Figure shows the incremental e ects on the optimal nancial portfolio due to the inclusion of human wealth in overall wealth (derived in equations (30)- (3)). The fraction of nancial wealth invested in equity and (too a lesser degree) nominal bonds is increased, while the fraction invested in price indexed bonds is decreased. Put di erently, our simulations predict that an individual will invest less conservatively due to human wealth. These portfolio e ects peak at birth, when the ratio of human wealth over nancial wealth is at its maximum, and gradually decline thereafter. 6

18 Figure : Wealth dynamics with human wealth Value Total Human Financial Consumption Human/financial (RHS) Age 0 Figure : Relative change in nancial portfolio due to human wealth Fraction Equity Nominal bonds Price indexed bonds Age Now we will consider the portfolio e ects of a DC PAYG pension scheme as modelled in section 4. with a tax rate of 0% (i.e. = 0:). Assuming no population growth, the support ratio k equals k = T=(D T )=.7. Using equation (37), we can calculate that k =3.8 implying k = k < k : net pension wealth at birth is negative in this case. Figure 3 shows the wealth dynamics over life. As predicted net pension wealth at birth is indeed negative. However, it gradually increases in value until the retirement age and gradually falls thereafter. Financial wealth again follows the typical hump-shaped pattern as predicted by life-cycle theory. However, comparing gures and 3 shows that nancial savings over lifetime fall as a consequence of the DC PAYG pension scheme. Apart from reducing lifetime consumption ( gure 3), this nancial crowding out also implies a reduction in the absolute investments in all three assets ( gure 4). In fact, these reductions mirror the implicit portfolio inherent to the DC PAYG pension scheme. It can be shown that there is complete neutrality if net DC PAYG pension wealth 7

19 Figure 3: Wealth dynamics with DC PAYG pension wealth Value Total Net pension Financial Consumption change (RHS) Θ_ (RHS) Θ_ (RHS) 400 Age 0 is zero at birth (i.e. if k = k ). In that case, the implicit portfolio arising due to the DC PAYG pension scheme is exactly mirrored by the absolute change in the nancial portfolio. Figure 4: Absolute changes in nancial portfolio due to DC PAYG pension scheme Value Equity Price indexed bonds Nominal bonds Age Using the analysis of section 4., we will now consider the relative changes in the nancial portfolio. The term guiding the incremental portfolio e ects in equations (39)-(4) ( (t) + (t)) remains positive over life. Put di erently, the initial direct e ect of negative net DC PAYG pension wealth ( (t)) is dominated by the positive indirect crowding out e ect ( (t)). Consequently, the DC PAYG pension scheme ampli es the incremental portfolio changes related to human wealth. Hence, the fraction of nancial wealth invested in equity and (too a lesser degree) nominal bonds is increased, while the fraction invested in price indexed bonds is decreased (see gure 5). Summarizing, our simulations predict that an individual will invest his nancial wealth less conservatively due 8

20 to the DC PAYG pension scheme. gradually disappear thereafter. These portfolio e ects peak at birth and Figure 5: scheme Relative changes in nancial portfolio due to DC PAYG pension Fraction Equity Nominal bonds Price indexed bonds Age Instead of a DC PAYG pension scheme, we will now consider the portfolio e ects of introducing a DB PAYG pension scheme as modelled in section 4.. More speci cally, we suppose the DB PAYG pension scheme provides retirees with a real pension bene t of (i.e. a replacement rate of 60% in terms of initial real wage). Again assuming no population growth, the support ratio k equals k = T=(D T )=.7. Using equation (5), we can calculate that k =4. implying k = k < k. Hence, net DB pension wealth at birth will be negative in this case. Figure 6 shows the wealth dynamics in case of a DB PAYG pension scheme. Net pension wealth at birth is indeed negative as predicted above. However, it gradually increases in value until the retirement age and gradually falls thereafter. Financial wealth follow the hump-shaped pattern as predicted by life-cycle theory. However, comparing gures and 6 shows that nancial savings over lifetime are reduced as a consequence of the DB PAYG pension scheme. Hence, just as with the DC PAYG pension scheme there is substantial crowding out of nancial wealth. Apart from causing a reduction of lifetime consumption ( gure 6), this crowding out of nancial wealth also imply a change in absolute investments in all three assets (see gure 7). These changes mirror the implicit portfolio inherent to the DB PAYG pension scheme. It can be shown that there is complete neutrality if net DB PAYG pension wealth is zero at birth (i.e. if k = k ). In that case, the implicit portfolio arising due to the DB PAYG pension scheme is exactly mirrored by the absolute change in the nancial portfolio. Using the analysis of section 4., we will now consider the relative changes in the nancial portfolio. Figure 8 shows the changes optimal nancial portfolio due to the introduction of the DB pension scheme as derived in equations (5)-(54). Early in life the fraction of nancial wealth invested in price-indexed bonds and equity is increased, while the fraction invested in nominal bonds is 9

21 Figure 6: Wealth dynamics with DB pension wealth Value Total Net pension Financial Consumption change (RHS) Θ_bar_ (RHS) Θ_bar_ (RHS) 400 Age 0 Figure 7: Absolute changes in nancial portfolio due to DB PAYG pension scheme Value 00 Equity Nominal bonds Price indexed bonds Age decreased. However, for bonds these changes are reversed after a certain age as net pension wealth becomes positive and indirect crowding out e ects gain importance. Consequently, ultimately the individual will invest less conservatively due the the introduction of the DB PAYG pension scheme. 0

22 Figure 8: scheme Relative changes in nancial portfolio due to DB PAYG pension Fraction Nominal bonds Equity Price indexed bonds Age 6 Summary and conclusions In this paper, we have analyzed the impact of a PAYG pension schemes on the optimal allocation of nancial wealth. Our model contains some simpli cations of reality. First of all, demographic risks due to changes in birth rates, longevity or migration are not taken into account. Moreover, the hold-up risk inherent to pay-as-you-go schemes is not included and important PAYG pension parameters as the tax rate (DC), the replacement rate (DB) and the retirement age are exogenously determined which might not be optimal for the individual. Besides, we take the existence of a PAYG pension scheme as given and do not justify it in welfare terms. Finally, we abstain from credit constraints in modelling the optimal investment behaviour of an individual. Although certainly being relevant, including these factors in the model would take us away from the main question we attempt to address in this paper. We have shown that taking into account the replicating portfolios inherent to human wealth and pension wealth impacts on the individual s optimal investment portfolio. Introducing a pay-as-you-go pension scheme implies that human wealth of young generations is partly transferred to retired generations. As a consequence, individuals will in general invest less conservatively. These portfolio e ects gradually disappear at the end of life. As pointed out by Bodie et al. (99), ignoring human wealth constitutes an "omitted variable" problem. The same holds true for pension wealth from PAYG pension schemes. This is not only relevant for individuals saving and investing for their pension, but also for pension funds investing on behalf of the individual.

23 References [] Aaron, H.J. (966), "The social insurance paradox", Canadian Journal of Economic and Political Science, 3, [] Ando, A. and Modigliani (963), F., The Life Cycle Hypothesis of Saving: Aggregate Implications and Tests, American Economic Review, 53, [3] Bodie, Z., R.C. Merton and W.F. Samuelson (99), "Labor supply exibility and portfolio choice in a life cycle model", Journal of Economic Dynamics and Control, 6, [4] Bohn, H. (00), "Social security and demographic uncertainty: The risk sharing properties of alternative policies", in: J.Y. Campbell and M. Feldstein (eds.), Risk Aspects of Investment Based Social Security Reform, The University of Chicago Press, [5] Brennan, M.J. and Y. Xia (00), "Dynamic asset allocation under in ation", Journal of nance, vol. 57, no. 3. [6] Campbell, J.Y. and L.M. Viceira (00), Strategic asset allocation - Portfolio choice for long-term investors, Clarendon Lectures in Economics, Oxford University Press. [7] De Jong, F. (005), Valuation of pension liabilities in incomplete markets, DNB Working paper no. 67. [8] Dimson, E., P. Marsh and M. Staunton (00), Triumph of the optimists, 00 years of global investment returns, Princeton, Princeton University Press. [9] Fischer, S. (974), "The demand for index bonds", MIT Working Paper 3, Cambridge, MA. [0] Fischer, S. (975), "The demand for index bonds", Journal of Political Economy, vol. 83, no. 3, [] Magill, M. and Quinzy, M. (00), Theory of incomplete markets, MIT Press. [] Marshall, D.A. (99), "In ation and asset returns in a monetary economy", The Journal of Finance, vol. XLVII, no. 4. [3] Matsen, E. and O. Thøgersen (004), "Designing social security a portfolio choice approach", European Economic Review, 48, [4] Merton, R.C. (969), "Lifetime portfolio selection under uncertainty: the continuous time case", Review of Economics and Statistics, 5,

24 [5] Merton, R.C. (983), "On the Role of Social Security as a Means for E - cient Risk Sharing in an Economy Where Human Capital is not Tradable", in: Z. Bodie and J.B. Shoven (eds.), Financial Aspects of the US Pension System, Chicago, University of Chicago Press. [6] Merton, R.C. (990), Continuous-time nance, Basil Blackwell, Cambridge, MA. [7] Persson, M. (00), "Five fallacies in the social security debate", in: T. Ihori and T. Tachibanaki (eds.), Social Security Reforms in Advanced Countries, Routledge Press. [8] Sánchez-Marcos, V. and A. R. Sánchez-Martín (006), "Can social security be welfare improving when there is demographic uncertainty?", Journal of Economic Dynamics and Control, Volume 30, Issues 9-0, pp [9] Teulings, C.N. and C.G. de Vries (006), "Generational accounting, solidarity and pension losses", De Economist, volume 54, number. 3

25 A Optimal portfolio allocation Inserting equation () in the Bellman equation (): 0 = max U [C (t) ; t] + C; i dt E t J W dw + J t dt + J W W dw In order to develop E t [], we can use the following multiplication rules for multivariate Itô-processes (Magill and Quinzy, 00): dz i dt = 0 dz i = dt dz dz = dt with being the instantaneous correlation coe cient between Wiener processes dz and dz. Applying these multiplication rules gives: E t [dw ] = [[r + (r r ) + 3 (r 3 r )] W C] dt E t dw = s + 3s s s 3 W dt Inserting gives the Bellman-Dreyfus fundamental equation of optimality: 0 = max C; i U (C; t) + JW [[r + (r r ) + 3 (r 3 r )] W C] + J t + J W W s + 3s s s 3 W First-order conditions: C: : 0 = U C J W JW [[r + (r r ) + 3 (r 3 r )] W C] + J t + J W W s + 3s s s 3 W Implicit solution: = J W r r J W W W s 3 s s 3 s Same for 3 : 0 JW [[r + (r r ) + 3 (r 3 r )] W C] + J 3 + J W W s + 3s s s 3 W 4

26 Implicit solution: 3 = J W r 3 r J W W W s 3 s s 3 s 3 Combining both implicit solutions gives explicit solutions for and 3 : = 3 = J W (r r ) J W W W ( ) s J W (r3 r ) J W W W ( ) s 3 (r 3 r ) ( ) s s 3 (r r ) ( ) s s 3 Finally, using the stock budget constraint (5), we can derive: J W (r r ) = J W W W ( ) s J W (r3 r ) J W W W ( ) s 3 = + J W (r r ) J W W W ( ) s (r 3 r ) ( ) s s 3 (r r ) ( ) s s 3 (r 3 r ) ( ) s s 3 + J W J W W W (r3 r ) ( ) s 3 (r r ) ( ) s s 3 5

27 B Optimal consumption We start with the trial solution: implying: J [W (t) ; t] = b (t) exp ( t) [W (t)] J W = b (t) exp ( t) [W (t)] J W = ( ) J W and: and: J W W = ( ) J ( ) J W W J J W W = ( ) W J t = bt b + ( ) W t J W Inserting J t in the Bellman-Dreyfus fundamental equation of optimality () gives: 0 = exp [ t] [C ] + J W [[r + (r r ) + 3 (r 3 r )] W C bt ] + b + i J W W h( ) s + ( 3) s 3 + ( ) ( 3) s s 3 W Recall: Inserting gives: E t [W t ] = [r + (r r ) + 3 (r 3 r )] W C + ( ) W t J W 0 = exp [ t] [C ] + J W ( ) J C + J W W [r + (r r ) + 3 (r 3 r )] W bt + + ( ) [r + (r r ) + 3 (r 3 r )] J b + i J W W h( ) s + ( 3) s 3 + ( ) ( 3) s s 3 W 6

28 Recall the rst order conditions: 0 = U C J W J W (r r ) = J W W W ( ) s J W (r3 r ) 3 = J W W W ( ) s (r 3 r ) ( ) s s 3 (r r ) ( ) s s 3 implying: C (t) = [J W exp [t]] = 3 = J W A J W W W J W A 3 J W W W for A = A 3 = Inserting C ; ; 3 gives: (r r ) ( ) s (r 3 r ) ( ) s 3 (r 3 r ) ( ) s s 3 (r r ) ( ) s s 3 0 = J W ( ) J bt [J W exp [t]] + J W W r + J W b J W J ( ) J W W W [r + (r r ) A + (r 3 r ) A 3 ] + (J W ) J W W s A + s 3A 3 + A A 3 s s 3 (r r ) A (r 3 r ) A 3 Inserting J W and J W W gives: 0 = ( ) W ( ) ( ) J b t exp [t] + W b s A + s 3A 3 + A A 3 s s 3 (r r ) A (r 3 r ) A 3 + ( ) r + [r + (r r ) A + (r 3 r ) A 3 ] 7

29 Finally, inserting J yields: 0 = ( ) [b] + b t b ( ) Y + r with: Y = s A + s 3A 3 + A A 3 s s 3 (r r ) A (r 3 r ) A 3 Implying: with: db dt = ( ) b + bx and X = + Y C (t) = [b (t)] W r = A Implying: 3 = A 3 = 3 = (A + A 3 ) A solution for the ordinary di erential equation db dt is: 0 h i b (t) ( ) t D exp X A X 0. As b (D) = 0, this solution also satis es the transversality condition J [b (D) ; W (D) ; D] = 8

30 C Valuing wealth H x from income x We will determine the value of human wealth applying a similar approach as Bodie et al. (99). However, we will do this by presenting a general method to determine the wealth of stochastic income x (based on Merton (990), section.) as this allows us to apply this method to other stochastic incomes as well. Suppose x follows a geometric Brownian motion with drift and di usion terms s x dz and s x dz : dx x = dt + s xdz + s x dz with dz and dz being the Wiener processes as de ned before. De ne: allowing us to express dx=x as x = s x s 3 x = s x s Recall: dx x = dt + xs 3 dz + x s dz d (Q =P ) Q =P = r dt + s dz implying: d (Q 3 =P ) Q 3 =P = r 3dt + s 3 dz s dz = d (Q =P ) (Q =P ) s 3 dz = d (Q 3=P ) (Q 3 =P ) r dt r 3 dt Inserting gives: dx d x = dt + (Q =P ) x (Q =P ) d (Q3 =P ) r dt + x (Q 3 =P ) r 3 dt d (Q 3 =P ) = ( x r 3 x r ) dt + x (Q 3 =P ) + d (Q =P ) x (Q =P ) 9

31 Hence, we have found an expression for dx d(q3=p ) x as a function of (Q 3=P ) and d(q =P ) (Q =P ). The next step is to nd an expression for x (t) as a function of (Q 3=P ) and (Q =P ): Proposition 7 x (t) can be expressed as a function of the value of equity and nominal bonds: with x = x (t) = x (0) exp [ x t] x r 3 + x r x ( x ) s 3 x x (Q3 =P ) (t) (Q =P ) (t) (Q 3 =P ) (0) (Q =P ) (0) x ( x ) s + x x (r 3 r + s 3 s ) Proof. We will proof that x (t) is compatible with dx x = dt + xs 3 dz + x s dz as derived above. Applying Itô s lemma to x (t) gives: dx x = d (Q 3 =P ) x (Q 3 =P ) + d (Q =P ) x (Q =P ) + xdt d (Q 3 =P ) d (Q =P ) + x x (Q 3 =P ) (Q =P ) x ( x ) d (Q3 =P ) (Q 3 =P ) x ( x ) d (Q =P ) (Q =P ) We can derive: and: and: d (Q3 =P ) = s (Q 3 =P ) 3dt d (Q =P ) = s (Q =P ) dt Inserting gives: d (Q 3 =P ) d (Q =P ) (Q 3 =P ) (Q =P ) = (r 3r + s 3 s ) dt 30

32 dx x = x r 3 + x r x ( x ) s 3 x ( x ) s + x x (r 3 r + s 3 s ) + x dt + x s 3 dz + x s dz QED Given that we have expressed x (t) as a function of (Q =P ) (t), the next step is to express the wealth of x (t) (i.e. H x (t)) as a function G of (Q 3 =P ) (t) ; (Q =P ) (t) and t: H x (t) = G [(Q 3 =P ) (t) ; (Q =P ) (t) ; t] Suppose the investor receives an instantaneous payout x from H x : dh x = (r H xh x x) dt + H x s H x dz H x + s H x dz H x Applying Itô s lemma on G [(Q 3 =P ) (t) ; (Q =P ) (t) ; t] gives: dg = G (Q3=P )d (Q 3 =P ) + G (Q=P )d (Q =P ) + G t dt + G(Q3=P )(Q 3=P )d (Q 3 =P ) + G (Q=P )(Q =P )d (Q =P ) + G (Q3=P )(Q =P )d (Q 3 =P ) d (Q =P ) recall: d (Q 3 =P ) = s 3 (Q 3 =P ) dt d (Q =P ) = s (Q =P ) dt Combining: dg = d (Q 3 =P ) d (Q =P ) = (r 3 r + s 3 s ) (Q 3 =P ) (Q =P ) dt G (Q3=P )r 3 (Q 3 =P ) + G (Q=P )r (Q =P ) + G t + G (Q 3=P )(Q 3=P )s 3 (Q 3 =P ) + G (Q =P )(Q =P )s (Q =P ) + G (Q3=P )(Q =P ) (r 3 r + s 3 s ) (Q 3 =P ) (Q =P ) + G (Q3=P )s 3 (Q 3 =P ) dz + G (Q=P )s (Q =P ) dz dt Recall we can express: By de nition: dh x = [r H xh x x] dt + s H x H x dz H x + s H x H x dz H x dh x (t) = dg [(Q 3 =P ) (t) ; (Q =P ) (t) ; t] 3

33 implying the equalities: r H xh x G (Q3=P )r 3 (Q 3 =P ) + G (Q=P )r (Q =P ) + G t + G (Q 3=P )(Q 3=P )s 3 (Q 3 =P ) + G (Q =P )(Q =P )s (Q =P ) + G (Q3=P )(Q =P ) (r 3 r + s 3 s ) (Q 3 =P ) (Q =P ) + x s H x H x G (Q3=P )s 3 (Q 3 =P ) dz H x dz s H x H x G (Q=P )s (Q =P ) dz H x dz Now, suppose we construct a zero net investment portfolio with the number of euro s invested in Q (indexed bonds), the number of euro s invested in Q (equity), 3 the number of euro s invested in Q 3 (riskless asset) respectively) and 4 the number of euro s invested in H x. Zero net investment implies: = 0, = 3 4 Now, the real return of this portfolio can be expressed as: d (Q =P ) Q =P + d (Q =P ) Q =P + d (Q 3 =P ) 3 Q 3 =P + dh x + xdt 4 H x = [ (r r ) + 3 (r 3 r ) + 4 (r H x r )] dt + 4 s H x + 3 s 3 dz + 4 s H x + s dz where we used the identities: Zero in ation risk implies dz H x dz dz H x dz while zero productivity risk implies 3 = 4 s H x s 3 No arbitrage implies: = 4 s H x s (r r ) + 3 (r 3 r ) + 4 (r H x r ) = 0 3

34 Note that we have now three equalities and three unknowns (, 3, 4 ). Combining gives: Implying: s 3 (r r ) s H x H x s (r 3 r ) s H x H x = r H x r H xh x s 3 s s 3 (r r ) G (Q=P )s (Q =P ) + s (r 3 r ) G (Q3=P )s 3 (Q 3 =P ) s 3 s = G (Q3=P )r 3 (Q 3 =P ) + G (Q=P )r (Q =P ) + G t + G (Q 3=P )(Q 3=P )s 3 (Q 3 =P ) + G (Q =P )(Q =P )s (Q =P ) + G (Q3=P )(Q =P ) (r 3 r + s 3 s ) (Q 3 =P ) (Q =P ) + x r G () 0 = G (Q3=P )r (Q 3 =P ) + G (Q=P )r (Q =P ) + G t + G (Q 3=P )(Q 3=P )s 3 (Q 3 =P ) + G (Q =P )(Q =P )s (Q =P ) + G (Q3=P )(Q =P ) (r 3 r + s 3 s ) (Q 3 =P ) (Q =P ) + x r G This is the fundamental partial di erential equation (fundamental PDE) G has to satisfy. However, given that H x (t) is by de nition equal to G [(Q 3 =P ) (t) ; (Q =P ) (t) ; t], H x (t) also has to satisfy the fundamental PDE. Proposition 8 with H x [x(t); t] = x (t) x [ exp [ x (D t)]] Proof. Recall: x = r + x (r r ) + x (r 3 r ) with x = implying: x (t) = x (0) exp [ x t] x r 3 + x r x ( x ) s 3 x x (Q3 =P ) (t) (Q =P ) (t) (Q 3 =P ) (0) (Q =P ) (0) x ( x ) s + x x (r 3 r + s 3 s ) H x [(Q =P ) (t) ; (Q 3 =P ) (t) ; t] = x (0) x exp [ x t] [ exp [ x (D t)]] 33 (Q3 =P ) (t) (Q 3 =P ) (0) x x (Q =P ) (t) (Q =P ) (0)