A Certainty Equivalent Valuation of Social Security Entitlements

Transcription

1 A Certainty Equivalent Valuation of Social Security Entitlements Sylvain Catherine * HEC Paris February 13, 2015 Abstract This paper studies how US households should value Social Security entitlements. To do so, I set up a continuous time life-cycle model in which the stock and labor markets are cointegrated and Social Security benefits are wage-indexed. Then, I use data from the Survey of Consumer Finances to calibrate relative risk aversion by matching the evolution of equity holdings over the life-cycle. First, I find that the certainty equivalent of Social Security for working households is 46% lower than the sum of future cash flows discounted at the riskfree rate and negative for young households. Second, at the national scale, and taking into account retirees, the risk-adjusted value of Social Security entitlements is 19.6 trillion dollars, which is 37% lower than the unadjusted value of 31 trillion dollars. My findings suggest that the present value of pension entitlements and the transition cost to a funded system may be largely overestimated. Keywords: Household finance, Social Security, Public liabilities, Portfolio choices, Human capital JEL codes: G11, G18, D91, H55, H06 * sylvain.catherine@hec.edu. 1

2 1 Introduction In most developed countries, Social Security pension entitlements represent one of the largest assets owned by households and an implicit public liability that often exceeds the government debt. A correct valuation of these claims is key to assess the sustainability of highly indebted governments and potential Social Security reforms in aging countries. However, as pension entitlements are not tradable, their exact value is unclear. In 2013, the US Social Security Administration (SSA) valued its obligations towards current workers and retirees at 23.7 trillion dollars, or 141% of GDP. In 2006, valuations of Social Security pension liabilities in western continental Europe ranged from 200% of GDP in Belgium and Spain to more than 300% in France and Italy (Müller et al. (2009)). These estimates assume that expected Social Security cash flows should be discounted at the government bond rate. However, Social Security benefits are very different from government bonds. First, governments can change Social Security rules without formally defaulting. Second, unlike typical bonds, Social Security benefits are indexed on inflation. Third, and most importantly, for a given set of rules, benefits and contributions depend on wages. In short, Social Security benefits are higher in good states of the labor market and must therefore be discounted at a higher rate than government bonds. Hence, valuations that are not properly risk-adjusted may overstate the true value of public liabilities and the transition cost to a funded pension system. To address this issue, I set up a continuous time life-cycle model in which individuals contribute to a pay-as-you-go system similar to the United States Social Security and compute its certainty equivalent. In the model, an agent with constant relative risk aversion (CRRA) receives a stochastic labor income until retirement and chooses how much to consume or save. His financial wealth can be invested in a risk-free asset or in the stock market portfolio. As in Benzoni et al. (2007) (hereafter BCG), the labor and stock markets are cointegrated. This means lies that, at a long-term horizon, the evolution of the national wage index is correlated with the performance of the stock market. The model captures important features of US Social Security. First, individuals pay a 10.4% payroll tax to current retirees. Second, their own pension is computed on the basis of their historical earnings and low wage earners enjoy higher rates of return on their contributions. Third, the value of past earnings is indexed on the evolution of the average national wage. 2

3 In the model, the certainty equivalent of Social Security takes into account three important facts. First, Social Security returns depend on the growth rate of wages at the national level (Samuelson (1958)). This relationship generates a positive correlation between the evolution of the agent s human capital and his expected rate of return on past Social Security contributions. This correlation reduces his valuation of Social Security entitlements. On the other hand, at the idiosyncratic level, negative labor income shocks increase the expected rate of return on past contributions. From the agent s point of view, this redistributive aspect of Social Security provides a valuable insurance against background risk that the market is unlikely to offer (Merton (1983)). Finally, the cointegration documented by BCG implies that Social Security risk can be partially hedged by short-selling the stock market portfolio, which in turn, means that the appropriate discount rate of pension entitlements depends on the equity premium. My main results are as follows. First, for a level of relative risk aversion of γ = 5, the certainty equivalent of Social Security entitlements of current workers is 46% smaller than the sum of future cash flows discounted at the riskfree rate (2%). Second, for workers entering the labor market, the value of Social Security is negative and represents nearly two years of wages. Third, at the national scale, and taking into account retirees, the sum of certainty equivalents is 37% below the unadjusted present value of Social Security obligations. The sum of certainty equivalents can be interpreted as the immediate cost of ending the old-age program with the consent of workers and retirees. The model suggests that the risk-adjusted value is $19.6 trillion dollars, while the adjusted value is $31 trillion dollars. In this case, the size of the risk-adjustment is close to the public debt value. In support of the model and its calibration, I compare its predictions regarding portfolio choices to empirical data from the Survey of Consumer Finances (SCF). In particular, the baseline calibration replicates relatively well the life-cycle pattern of the share of financial wealth invested in equity, that is an equity share which is very low among young and even middle-aged workers and then rises until retirement. Moreover, robustness test show that the size of the risk-adjustment is not overly sensitive to γ because the price of risk is largely determined by the equity premium anyway. The discount rate used by households to value Social Security depends on their age and is the highest around 40 years old. As they get closer to retirement, the risk-adjusted discount rate 3

4 gets closer to the risk-free rate because the long run correlation between stock market returns and the growth of the wage index becomes less of a concern. For workers who just entered the labor market, the cointegration does not really affect the value of Social Security either because they have not accumulated a large amount of entitlements yet. From their point of view, positive news about the future growth the wage index means higher pensions and higher payroll taxes. Nonetheless, young households value Social Security negatively as it forces them to invest in an asset that correlates with the stock and labor markets and yet delivers low expected returns. Around 40 years, households have already contributed for roughly 20 years and the final value of these contributions is still exposed to the long run correlation between the labor and stock markets. The discount rate used by households to value Social Security depends on their age and is the highest around 40 years old. As they get closer to retirement, the risk-adjusted discount rate gets closer to the risk-free rate because the long run correlation between stock market returns and the growth of the wage index becomes less of a concern. For workers who just entered the labor market, the cointegration does not really affect the value of Social Security either because they have not accumulated a large amount of entitlements yet. From their point of view, positive news about the future growth the wage index means higher pensions and higher payroll taxes. Nonetheless, young households value Social Security negatively because it forces them to invest in an asset that correlates with the stock and labor markets and yet delivers low expected returns. Around 40 years, households have already contributed for roughly 20 years and are still young enough for the final value of their contributions to be exposed to the long run correlation between the wage index and the stock market. At that age, the risk-adjusted discount rate is slightly above 4%. Consequences of cointegration of the labor and stock markets on the valuation of pension entitlements have already been explored by Geanokoplos and Zeldes (2010) who find a baseline market value 19% lower than the SSA actuarial estimate using derivative pricing methods. Their risk-adjustment focuses on the relationship between stocks and wages and assumes other risks to have a market price of zero. This approach leaves, for example, unpriced the relationship between pension wealth and human capital that is not captured by their joint comovement with the stock market. As noted by Campbell (1996), labor income represents two thirds of value-added in the 4

5 United States and human capital constitutes probably a similar proportion of the total wealth. Economic theory suggests that positive covariance with human capital should therefore translate into higher risk premia, even if the states of the labor and stock markets were unrelated. This paper proceeds as follows. In Section 2, I further motivate the need for a risk-adjusted valuation by discussing the way Social Security returns are tied to labor and stock markets conditions. The model is described in Section 3 and its calibration is detailed in Section 4. Section 5 describes how the certainty equivalent of Social Security entitlements evolves over the life-cycle, proposes a risk-adjusted nationwide valuation of pension entitlements and compares simulated equity shares to empirical data. Finally, section 6 analyses the sensitivity of the results to the calibration. 2 Social Security, stocks and the wage index In this section, I present key aspects of the United States Old-Age program and discuss two elements of its risk-profile that are of particular importance for valuation: its wage-indexation and the relationship between the wage-index and the stock market. 2.1 Computation and wage-indexation of pension benefits The United States Social Security is a mostly unfunded program in which current workers contributions are directly distributed as pension benefits to current retirees 1. Since the seminal work of Samuelson (1958), economists know that the performance of an unfunded pension system is indexed on the nationwide average wage. Indeed, the return obtained by a generation is defined as the percentage difference between the payroll taxes it receives from its children and the taxes it paid for its own parents. For a constant tax level, this difference is close to the sum of the population and wage growth rates. In the United-States, wage-indexation is explicit and achieved through the computation of benefits. The following describes in a simplified manner the two steps involved in this computation. In a first step, the SSA computes the average indexed yearly earning (AIYE) of the agent. 1 In 2013, the Old-Age and Survivor Insurance (OASI) program paid $ 672 bn in benefits while its total reserves were 2,674 bn, representing only four years of expenditures. 5

6 Indexed yearly earnings are the individual s historical salaries adjusted for growth in the nominal wage index (hence corrected for inflation and growth in real wages). Yearly earnings are capped by an upper limit that has been more than twice the average wage since The AIYE is then defined as the mean of the resulting best 35 years. In a second step, the SSA computes the percentage of the AIYE that the agent receives during her first retirement year. This percentage is determined by a bend point system through which retirees with higher earnings records get a lower share of their AIYE in benefits. Bend points are limits above which retirees get a smaller percentage of their AIYE and which are themselves wage-indexed. More precisely, the first year benefits are the sum of 1. 90% of the share of the AIYE below the first bend point; 2. 32% of the share of the AIYE between the first and second bend points; 3. and 15% of the share of the AIYE above the second bend point. The two bend points being respectively close to 20% and 100% of the wage index, the first year pension benefits B are defined as B = 0.9 AIY E if AIY E/L 1 (T ) < L 1 (T ) AIY E if 0.2 AIY E/L 1 (T ) < L 1 (T ) AIY E if 1 AIY E/L 1 (T ) (1) where L 1 is the nationwide average wage, and T the agent retirement year. After retirement, subsequent benefits are only adjusted for inflation. Figure 1 represents yearly benefits as a function of L 1 (T ) and AIY E and illustrates the fact that Social Security payoffs are a linear function of the average wage index but a concave function of the agent s own earnings record. This concavity acts as a redistributive mechanism by reducing the benefits of high-earners. [ Insert Figure 1 here ] 6

7 2.2 Cointegration between the stock and labor markets Social Security being wage-indexed, any statistical relationship that ties the labor and stock markets should tie the Social Security discount rate to the equity premium. While immediate correlation between wage growth rate and stock market returns is low (Cocco et al. (2005), hereafter CGM), BCG find that over the long term, wages and dividends are cointegrated. Adding more recent data, I replicate their analysis and extend it by considering stock market gains instead of dividends. The ratio of accumulated stock market gains to the wage index appears to be trend stationary, thus confirming the possibility of hedging against long-term Social Security risk by short-selling stocks. In economic terms, the cointegration described by BCG implies that the ratio of wages to dividends tends to revert to an historical mean. Following their methodology, I note ld the historical mean of the difference between the log of the real wage-index (l 1 ) and the log of S&P500 real dividends ( ˆd(t)), and define y as y(t) l 1 (t) ˆd(t) ld (2) The variable y(t) measures whether the wage-to-dividend ratio is above or below its historical mean. If this ratio is indeed mean-reverting, then the dynamics of y(t) should be of the form y(t) = κy(t 1) + ɛ(t) (3) where κ is the speed of mean-reversion and ɛ(t) an error term. BCG test this model on the period and find values of κ ranging between and depending on the specification of the model and the time period. In particular, BCG cannot reject the unit-root hypothesis for the post Second World war period. I compute the average wage as total wages divided by total employees using data from the NIPA tables for the period 2 Using stock market data for the S&P500 compiled by Robert Shiller 3, I run augmented Dickey-Fuller tests on the period and report the 2 These data can be retrieved from Emmanuel Saez website: saez/tabfig2012prel.xls - Table B1. 3 Data available on Robert Shiller s website: shiller/data/ie data.xls 7

8 findings in table 1 (models (1) to (6)). Overall, I find values of κ that are consistent with BCG s previous results. In model (2), I allow for a time trend but find it to be insignificant. Models (3) to (6) allows for autocorrelation or errors until the last lag is found insignificant. I find the Dickey- Fuller p-value to be below 10% in all models, except the one with the statistically insignificant time trend (but the Dickey-Fuller critical values are higher when a time trend is included). [ Insert Table 1 here ] As the main concern of investors are returns rather than dividends, I test the cointegration of stock and labor markets using stock market gains instead of dividends. To measure stock markets gains, I build a series representing the market value of the portfolio of an agent reinvesting dividends in the S&P500 since S(t + 1) = ( 1 + ) P (t) P (t 1) + D(t) S(t) (4) P (t) with S(1929) = 100. Then, I define y S similarly to y, using the log of S instead of the log of dividends. y s (t) l 1 (t) s(t) (5) If the price-to-dividend ratio P/D is stationary, then y s should be trend stationary, its trend reflecting the difference between expected stock returns and the growth rate of the wage index. Models (7) and (8) test this hypothesis. In model (7), I allow for a time trend but no autocorrelation and find a speed of mean-reversion of 0.20 and a p-value of 6.7%. Allowing for a lag term increases the p-value to more than 20%, but the autocorrelation term added in model (8) is not supported by a statistically significant coefficient. 3 Model I model an agent who receives labor earnings until retirement and chooses his level of consumption and the share of her financial wealth invested in the stock market portfolio. Her future labor 8

9 earnings are subject to permanent idiosyncratic and macroeconomic shocks. The key point of the model is that shocks to the stock market have delayed consequences on the labor market through the expected growth rate of wages, and consequently on expected pension benefits. 3.1 Stock and labor markets Financial market The agent allocates her financial wealth between a risky asset, the stock market portfolio, and a risk-free asset delivering an interest rate r. The dividend process D(t) of the risky asset is modeled as a geometric brownian motion: where g D is the expected growth rate of dividends. dd D = g Ddt + σdz 3 (6) The discount factor is assumed to be constant. Thus the stock price is proportional to the dividend level and follows the same Brownian motion. The stock return process includes capital gains and dividends and also follows a geometric brownian motion: ds(t) S(t) = dp (t) + D(t)dt P (t) = µdt + σdz 3 (7) where µ is the stock market expected return. In this framework, the volatility of stock returns and dividends are the same, a simplification previously discussed National average wage To capture the cointegration between stock and labor markets, I use the state-variable y defined in equation (2). Its dynamic is: dy(t) = κy(t)dt + v 1 dz 1 (t) v 3 dz 3 (t) (8) where v 1 represents the standard error of permanent shocks on the average wage and v 3 the degree to which shocks on dividends affect y immediately. By combining (6), (2) and (8) we get the 9

10 following dynamics for the wage index: DL 1 = ( κy(t) + g D σ2 L v (σ v ) 3) 2 dt + v 1 dz 1 (t) + (σ v 3 ) dz 3 (t) (9) 2 The state variable y indicates whether the labor income process is late relative to dividends. As such, it determines the expected growth rate of the wage index. A negative shock on dividends increases y and implies a lower expected growth rate of the wage index. This consequently affects the value of human capital and pension entitlements. The parameter v 3 controls the immediate effects of stock market shocks on the wage index, with v 3 = σ implying, for example, no contemporaneous correlation. 3.2 Agent Labor income The Social Security wealth of a given individual is affected differently by the nationwide average wage and his own position on the wage scale. For this reason, I decompose her wage as the product of two components: L 1 is the average wage in the economy and L 2,i the agent s wage in percentage of L 1. Hence her labor income is L(t) = L 1 (t)l 2 (t) (10) The idiosyncratic component of labor income follows an geometric brownian motion. where α(t) captures the quadratic effect of experience on wages. DL 2 L 2 = α(t)dt + v 2 dz 2 (t) (11) α(t) = α 0 + α 1 t (12) 10

11 3.2.2 Pension Benefits Introducing Social Security into a dynamic programming problem creates some difficulties. Therefore, I simplify it by assuming that benefits are computed on the basis of a complete career. Because the full-career average wage is lower than the mean of the 35 best years, I multiply the result by a coefficient a chosen such that its expectation equals that of the best-35-years AIYE. Therefore, my proxy for the AIYE is AIY E(T ) = a 45 T T 45 L 1 (T ) L 1 (u) L 1(u)L 2 (u)du (13) where L 1 (u)l 2 (u) was the agent s wage at time u and L 1 (T )/L 1 (u) is the indexation coefficient. Within the integral, the L 1 (u) terms cancel each others such that the AIYE can be computed as the product of the average wage at retirement date L 1 (T ) and an historical average of L 2,i. To keep track of historical values of L 2,i, I define a new state variable H as which has the following dynamic: H(t) = a 45 t T 45 L 2 (u)du (14) dh = a L 2(t) dt (15) 45 The yearly pension benefits can be written as a function of the final values of L 1 and H. B = 0.9 AIY E if AIY E/L 1 (T ) < L 1 (T ) AIY E if 0.2 AIY E/L 1 (T ) < L 1 (T ) AIY E if 1 AIY E/L 1 (T ) L 1 (T ) L 1 (T ) if AIY E/L 1 (T ) > 2.5 (16) where AIY E = H(T ) L 1 (T ) (17) and where an AIYE above 2.5 times the wage index does not provide any additional benefits to 11

12 reflect the fact that historical wages are capped by the maximum earning index Objective function The agent maximizes her utility by choosing dynamically her optimal consumption (C) and the share of her financial wealth invested in the market portfolio (π). I explicitly model the working years, between 20 and 65 years old and assume that once retired, the agent face the simpler problem described by Merton (1971), i.e. an agent endowed with some financial wealth and receiving a safe stream of cash flows. Assuming a CRRA utility, her objective function is [ T J(W (t), L 1 (t), L 2 (t), y(t), H(t), t) max E ψu (C(u))1 γ t e [C,π] t 1 γ ] + J(T ) where J(T ) is her expected utility on the day of her retirement, γ her coefficient of relative risk aversion, and ψ her preference for the present. (18) 3.3 Solution method Optimal consumption, portfolio decisions and utility levels are determined by solving the Hamilton- Jacobi-Bellman equation backward from retirement day. Following BCG, I numerically solve the optimal consumption policy using the fact that the first-order condition provides a perfect mapping between optimal consumption and the value function. Further details on the HJB equation and the numerical methodology are provided in appendix. The following focuses on the terminal conditions of the problem, i.e. the agent s behavior after retirement, and the computation of the certainty equivalent of Social Security Retirement years To solve the HJB equation backward, I assume that once retired the agent owns a financial wealth W and receives a safe stream of cash flows from Social Security. Thanks to Merton (1971), we know that the problem of the retired agent has an analytical solution. In particular, in the CRRA case, solutions for optimal consumption and equity share are C (T ) = b(t ) 1 γ W (T ) (19) 12

13 π (T ) = µ r W (T ) γσ 2 W (T ) (20) where W T is total wealth at retirement, including the present value of pension wealth, b(t) and c(t) are values known from the Merton portfolio problem, ( ) 1 e v(r+t t) γ b(t) (21) v v(t) ( ) ϕ (1 γ) (µ r) 2 + r 2γσ 2 γ (22) and R is the number of retirement years. After retirement, benefits are safe and can be discounted at the risk-free rate. Hence, total wealth is W (T ) = W (T ) Certainty equivalents R 0 e ru Bdu (23) Before retirement, total wealth is the sum of three components: financial wealth (W ), human capital (HC) and Social Security money worth (SS). W = W + HC + SS (24) Total wealth can be computed as the financial wealth that would make the agent indifferent to a drop of L 1 and H to zero, a scenario in which she would have no more human capital nor Social Security wealth. Thus, W is solution to the equation J ss (W, 0, L 2, y, 0, t) = J ss (W, L 1, L 2, y, H, t) (25) where J ss and J ss denote the expected utility functions of the agent when Social Security respectively exists and does not exist. Conveniently, the left-hand-side of (25) does not depend on L 2 or y anymore and has an analytical solution provided by Merton (1969). Indeed, for a CRRA utility 13

14 function, we know that J ss (W, 0, L 2, y, 0, t) = e ϕt 1 γ b(t)w 1 γ (26) where b is defined by (22) and (21). By combining (25) and (26), we can compute total wealth as [ ] 1 (1 γ)e ϕt W = J ss 1 γ (W, L 1, L 2, y, H, t) b(t) (27) Similarly, the value of human capital can be defined as the total amount of cash that should be given to an individual so that he remains indifferent to a drop of L 1 to zero when Social Security does not exist. If Social Security does not exist, then the agent does not pay payroll taxes and get a higher average wage. Thus, HC is solution to the equation J ss (W + HC, 0, L 2, y, t) = J ss (W, ) L 1 1 τ, L 2, y, t (28) And, following the same steps as for the computation of W, we get [ )] 1 (1 γ)e ϕt HC = J (W, ss L 1 b 1 τ, L 1 γ 2, y, t W (29) Finally, we can combine equations (24), (27) and (29) to compute the wealth equivalent of Social Security. 4 Calibration In this section, I detail the baseline calibration of the model summarized in Table 2. [ Insert Table 2 here ] 14

15 4.1 Macroeconomy Financial market Historical estimates of the risk-free rate ranges from 0.64% to 3.02% 4. In its valuation, the SSA assumes a relatively high risk-free rate at 2.9%. I set r = 0.02 in the baseline calibration and also reports results for r = and r = The equity premium is µ r = 0.06 and stock market volatility is σ = Wage index In the model, g D determines both the expected growth rate of dividends and that of the average wage. I find L 1 to have grown by 1.1% on average between 1947 and 2011, while the growth rate of D has been 1.8%. The growth rate of wages being the most important parameter in the perspective of this paper, g D is set at 1.2%. The key parameter of the model is κ, the speed of mean-reversion of y. In table 1, estimations of κ range from 0.16 to Using data between 1929 and 2004, BCG estimate the same statistical models with other datasets and find values of κ ranging from 0.05 to 0.26, depending on the model and the time period. Their choice of a value of κ = 0.15 is followed in the baseline calibration. The standard deviation of permanent shocks to the wage-index is v 1 = and the correlation of these shocks with the stock market is assumed to be zero (implying v 3 = σ). Initial values To capture the diversity of macroeconomic perspectives that an agent may face when he enters the labor market, I initiate y = 0 at t = 1000 to generate random starting values at t = 0. I initialize L 1 (0) at $40,000 the level of the average wage around 1970 in 2013 dollars, and W at $50, Agent Labor income α 0 and α 1 are calibrated as in BCG, producing a hump-shaped life-cycle profile for L 2. I set L 2 (0) = 0.6, such that, given α 0 and α 1, L 2 averages 1 over the life-cycle. The agent faces relatively high permanent income shocks (v 2 = 0.15 as in BCG). Preferences Estimates of the psychological discount rate generally lie between 0.3% and 0.4%. I set ϕ = There is little consensus on the degree of relative risk aversion. Mehra and Prescott argue that reasonable values of γ lie between 1 and 10. Following several recent papers in 4 Historical estimates reported by Mehra (2007) using different data sets include 0.64% (Ibbotson, ), 1.31% (Mehra-Prescott, ), 2.68% (Shiller, ) and 3.02% (Siegel, ). 15

16 the portfolio choice literature (Gomes and Michaelides (2005), BCG, Chai et al. (2011)), I choose γ = 5 in the baseline scenario because this value replicate well the observed equity share. Retirement Another important parameter is the number of retirement years. Life-expectancy at age 65 in the United-States is currently estimated between 19 and 20. I thus set R = 20. Financial constraints I assume that the agent cannot short-sell the stock market portfolio nor can she invest borrowed money in equity. Hence 0 π Social Security Computation of benefits Simulations show that the mean of the best 35 years of L 2 is on average 14% higher than the mean L 2 over the whole career. Thus, I set a = 1.14 such that I do not penalize retirees by computing their AIYE using their entire career instead of their best 35 years of earnings. Payroll taxes In order to value of Social Security, I also build a benchmark scenario where benefits do not exist and wages are higher. I note τ the percentage difference in wages between the two scenarios. The value τ depends on the expected payroll tax rate. In 2013, the SSA estimated that the present value of future deficits of the OASI program over the next 75 years represents 2.40 % of taxable payroll 5. On the other hand, 17.3 % of the 2013 OAS benefits (or 2.3 payroll tax percentage points) were allocated to survivors benefits that are not explicitly modeled in this paper. Overall, assuming that those two corrections cancel out, the current tax rate of 10.6 % 6 of the OAS program may represent the cost of old-age benefits for current workers. 5 Results 5.1 Valuation at the household level Figure 2 plots the average certainty equivalent by age obtained from 50,000 simulations of the baseline scenario as well as the NPV of Social Security wealth computed by discounting expected cash flows at the risk-free rate. 5 Source: The 2013 Annual Report of the Board of Trustees of the Federal Old-Age and Survivors Insurance and Federal Disability Insurance Trust Funds (p.66) 6 Although the Social Security tax rate is 12.4%, 1.8% are dedicated to the disability program. 16

17 [ Insert Figure 2 here ] In the baseline scenario, Social Security has a negative wealth equivalent for workers under 35 years old. This negative wealth equivalent is largely below the net present value of future Social Security cash flows (benefits minus contributions) discounted at the risk free rate. At the time the certainty equivalent reaches zero, the unadjusted present value is already worth nearly two years and a half of average earnings. The dashed lines plot net present values for a variety of discount rates. First, they suggest an internal rate of return around 3%. For the 1949 cohort, the SSA reports internal rates of return of 1.95% for single males with average earnings and 2.46% for single females, who have a longer life expectancy. A simulated IRR above the historical one is consistent with the well-known poor performance of the wage-index over the last decades. The dashed lines also illustrate how the implicit discount rate applied by households evolves with age. Households below 45 years old discount Social Security cash flows at around 4% while households above 60 use the risk-free rate. Close to retirement, the implicit discount rate appears a little bit lower than the risk-free rate, which makes sense if there is little uncertainty about the value of the wage-index at retirement date while the bend point system continues to provide valuable hedging against idiosyncratic labor income risk. Indeed, as noted by Merton (1983), Social Security offers a hedging mechanism against specific risk because individuals who experience negative shocks on their labor income get better returns on their contributions. This insurance feature of Social Security should increase its wealth equivalent at the individual level. However, within-cohort redistribution should play no role for the valuation of Social Security entitlements at the government level and taking it into account may induce a small overvaluation. 5.2 Valuation at the national level To estimate the magnitude of the risk-adjustment at the national scale, I compute, for each age between 20 and 64, a valuation multiple defined as the average certainty equivalent of Social Security divided by the average wage. These multiples are based on simulated data and represent the Social Security value attached to each dollar of current labor income. The risk-adjusted value 17

18 of entitlements of a given cohort can then be deduced by applying the appropriate multiple to its total wages. Similarly, unadjusted valuation multiples are computed by discounting expected future payroll taxes and benefits at the risk-free rate, and dividing the result by the average wage. Figure 3 plots the adjusted and unadjusted multiples obtained in the baseline scenario and shows that most of the risk-adjustment in concentrated among young and middle-aged cohorts. [ Insert Figure 3 here ] Total wages received in 2013 by each cohort born between 1949 and 1992 are computed using the American Community Survey (ACS). Unadjusted and risk-adjusted values of entitlements are estimated by applying the age-dependent valuation multiples to each cohort. Because the model does not take into account survivors benefits, I reintroduce their value on a pro rata basis, assuming that they represent 17.3% of the total present value. For retirees, I value entitlements as an ordinary immediate annuity ending at 85 years old and discounted at the risk-free rate. The coupon received by each cohort is the sum of Social Security income reported in the ACS. Summing over all cohorts gives a valuation of closed-group obligations (i.e. restricted to cohorts that have already participated to the labor force). In the baseline scenario, the unadjusted present value of Social Security entitlements is $31.00 tr. This result cannot be compared to the SSA valuation of closed-group obligations since the SSA assumes a higher risk-free rate. When using the same rate (2.9%), I find a value of $22.29 tr, closer to the SSA estimate of $23.7 tr. The baseline adjusted present value is $19.6 tr, implying a risk adjustment of 36.8%. This riskadjustment is therefore larger than the 19% reported by Geanokoplos and Zeldes (2010). When retirees are excluded, the unadjusted value of entitlements is $24.8 tr, but the adjusted one is nearly twice smaller at $13.4 tr. 18

19 5.3 Portfolio choices Predictions about certainty equivalents cannot be confronted to any dataset unless a survey elicits them from households. However, the key ingredients of the model also provide predictions about the life-cycle pattern of the share of financial wealth that should be invested in the stock market portfolio. These predictions can be confronted to data from the Survey of Consumer Finances. Figure 4 plots the average equity share simulated in the baseline scenario. Overall, the simulated equity share is very low among young households and slowly increasing until age 55. A strong rise of the equity share is predicted when the agent is a few years away from retirement because shocks on the stock markets become less likely to affect the wage-index before she retires. Consequently, Social Security wealth rapidly becomes a bond-like asset after 60 years old, inducing a greater willingness to invest financial wealth in stocks. [ Insert Figure 4 here ] Overall, the prediction of this paper contrasts with many life-cycle models which, starting with Merton (1971), predict that the equity share should be very high among young households and be a decreasing function of age (Viceira (2001), Campbell et al. (2001), Cocco et al. (2005), Chai et al. (2011)). Finance professionals also advice a decreasing equity share. For example, Vanguard Target Retirement Funds have an equity share close to 90% for households below 40 years old. This share then drops to reach 50% at 65 years old. In contrast, data from the SCF show a very low equity share among young households. As explained by Ameriks and Zeldes (2004), the exact shape of the life-cycle pattern is unclear. Indeed, empirical studies on the life-cycle profile of portfolio choices cannot include cohort and year dummies simultaneously because of perfect multicollinearity (age = year cohort). As a consequence, the empirical conclusions depend on the specification. In particular, when year dummies are included, AZ find that the equity share shows little variation throughout the lifecycle. However, it rises rapidly with age when cohort dummies are preferred. In Figure 5, I replicate Ameriks and Zeldes (2004) s analysis using SCF surveys over a longer time period:

20 from In contrast to their findings, I find the predicted equity share to be increasing with age in both models, even though the model with year dummies still suggests a slightly flatter curve. [ Insert Figure 5 here ] An increasing equity share represents a serious investment mistake for research papers predicting the opposite trend. However, as shown by BCG, this pattern can be consistent with a model in which households take into account cointegration between wages and dividends. The model presented in this paper replicates pretty well the life-cycle pattern of the equity share until 55 or 60 years old. However, the data do not support a rise of the equity share around retirement date. The equity share at 65 is the one predicted by the Merton model and results from the safe stream of bond-like cash flows received from Social Security. Portfolio choices after retirement are outside the scope of this paper but possible explanations include new background risks such as health expenditures, a bequest motive, a decline in cognitive abilities, or the inability to adjust labor supply to shocks on financial wealth. Besides, households that should have the highest rise in their equity share when they near retirement are those with high pension to financial wealth ratios. As these households have low financial wealth, fixed investment costs may also explain why they do not have the important equity share at retirement predicted by the Merton model. The idea that an equity share rising with age can be optimal under reasonable assumptions when cointegration between stocks and wages is taken into account is the key result of BCG. With respect to portfolio choices, this paper extends their model by incorporating Social Security benefits, hence providing a more complete analysis of the household optimization problem. The results differ from BCG s conclusions in three ways. First, the optimal equity share is lower for most of the agent s career. This suggests that Social Security risk may explain part of the low participation puzzle. Second, the equity share shows smaller variations between 35 and 55, which seems in line with the data (in particular the year dummies specification). Third, it rises quickly around 60, a prediction that is neither found in the BCG model nor apparent in the data. 7 Their own paper uses surveys from 1989 to

21 6 Sensitivity analysis In this section, I test different calibrations of the model and report the sensitivity of results. Table 3 summarizes the analysis and presents all valuations discussed in this section. [ Insert Table 3 here ] 6.1 Risk aversion Quite surprisingly, tests of different degrees of relative risk aversion ranging from 3 to 6 suggest that smaller coefficients imply greater risk-adjustments. A coefficient of γ = 3 implies a risk adjustment of 52.8%, which is reduced to 45.8% when γ = 6. One possible explanation is that Social Security forces the agent to invest some of her savings at a low rate of return when she would rather invest all her wealth in stocks. This problem would be more severe for low relative risk aversion coefficients. However, in the baseline calibration, less than one worker out of six hits the upper constraint on the equity share (π 1) at some point in their life. Moreover, relaxing the financial constraint and allowing some leverage (π 1.25) does not change the size of the risk-adjustment. Another hypothesis relates to the human capital to financial wealth ratio. For a given wage, a lower risk aversion implies a greater present value of human capital. Therefore, for a given financial wealth, a lower risk aversion means that human capital constitutes a greater share of the representative agent s total portfolio. For a 20 years old worker, the value of human capital is indeed 22% lower when γ = 6 than when γ = 3. Since Social Security has a risk-profile closer to that of the labor market than to that of stocks, then it should have a higher risk premium when human capital represents a larger share of total wealth. Figure 6 presents the life-cycle pattern of the equity share for different levels of risk aversion and suggests that a γ of 5 or 6 is the calibration that fits best the SCF data presented in figure 5. [ Insert Figure 6 here ] 21

22 6.2 Cointegration As expected, the risk-adjustment is a decreasing function of the speed of mean-reversion of the wage-to-dividends ratio. A conservative assumption such as κ = 0.12 has only a 2 percentage points effect on the result. From the point of view of a young worker facing a 45 years long career, whether stock markets shocks are fully transmitted to the market in 5 or 8 years is not that important. In fact, the model shows not difference in the certainty equivalent of Social Security at age 20 when κ is reduced from 0.18 to When κ is set at 0.18, the nationwide risk-adjustment equals -38.6%. This compares to an adjustment of -36.8% and -34.8% when κ equals respectively 0.15 and The risk-adjustment appears to increase by percentage point when κ is increased by Geanokoplos and Zeldes (2010) find a risk-adjustment of 11% for κ =.05 and 23% for κ =.25, which implies a very similar sensitivity of 0.6 percentage point for a 0.01 increase in κ. 6.3 Risk-free rate Figure 7 represents adjusted and unadjusted Social Security average certainty equivalents for different levels of the risk-free rate, but assuming the same equity premium in all scenarios. Adjusted valuations at the beginning of the life-cycle are nearly unaffected by the level of the risk-free rate. As a result, the size of the risk-adjustment is greater when interest rates are low. [ Insert Figure 7 here ] In its 2013 Annual Report, the SSA assumes a long-term real interest rate of 2.9%, far above current levels. In this scenario, the risk-adjustment is only 29.7%. However, if we assume that real interest rates will remain low, the required risk-adjustment is much more important. For a 1% rate, the unadjusted present value raises to $44.01 tr, while the adjusted value is nearly $20 tr lower at $25.1 tr. 22

23 6.4 Growth rate of the wage index Intuitively, the value of Social Security entitlements should be an increasing function of the growth rate of the wage index, which determines the rate of returns on contributions. As a consequence, the unadjusted value is $6.76 tr smaller when wages are expected to grow at g D = than when a higher trend of g D = is assumed. The percentage risk-adjustment also increases with g D. When the pension system offers better yearly returns, a larger share of total Social Security wealth goes to young households for which the risk-adjustment is more important. Overall, the low and high growth scenarios imply riskadjustments of respectively -35.5% and -39.4%. 7 Conclusion The valuation of Social Security entitlements is of prime importance in at least two ongoing debates that rank high on the policy agenda: the sustainability of public debts and likely reforms in pension systems. The life-cycle model presented in this paper suggests that, from the point of view of households, these entitlements may be greatly overvalued when discounted at the risk-free rate. Yet, to the extent that the SSA assumes a real interest rate of 3%, its own valuation of closedgroup obligations in 2013 at $23.7 tr may still be reasonable if real interest rates are in fact lower. Indeed, for a 2% rate, this paper finds a risk-adjusted value of $19.6 tr. However, if the SSA updates its assumption to the current levels of interest rates, large misevaluations could follow. For a 1% rate, the model predicts an error of $19.6 tr. The model presented in this paper may have other uses. One is the evaluation of Social Security reform proposals. A second is the analysis of the consequences of unfunded pension systems on the demand for risky assets, and thus their contribution to important economic puzzles such as the low stock market participation or the equity premium. 23

24 References Ameriks, John and Stephen P Zeldes, How Do Household Portfolio Shares Vary with Age?, Working Paper, Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. Godlstein, Portfolio Choice over the Life-Cycle when the Stock and Labor Markets Are Cointegrated, Journal of Finance, October 2007, 62 (5), Campbell, John Y, Understanding Risk and Return, Journal of Political Economy, April 1996, 104 (2), Campbell, John Y., João F. Cocco, Francisco J. Gomes, and Pascal J. Maenhout, Investing Retirement Wealth: A Life-Cycle Model, in Risk Aspects of Investment-Based Social Security Reform NBER Chapters, National Bureau of Economic Research, Inc, March 2001, pp Chai, Jingjing, Wolfram Horneff, Raimond Maurer, and Olivia S. Mitchell, Optimal Portfolio Choice over the Life Cycle with Flexible Work, Endogenous Retirement, and Lifetime Payouts, Review of Finance, 2011, 15 (4), Cocco, Joao F., Francisco J. Gomes, and Pascal J. Maenhout, Consumption and Portfolio Choice over the Life Cycle, Review of Financial Studies, 2005, 18 (2), Geanokoplos, John and Stephen P. Zeldes, Market Valuation of Accrued Social Security Benefits, in Measuring and Managing Federal Financial Risk NBER Chapters, National Bureau of Economic Research, Inc, March 2010, pp Gomes, Francisco and Alexander Michaelides, Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence, Journal of Finance, , 60 (2), Mehra, Rajnish, The Equity Premium Puzzle: A Review, Foundations and Trends(R) in Finance, September 2007, 2 (1),

25 Merton, Robert, On the Role of Social Security as a Means for Efficient Risk Sharing in an Economy Where Human Capital Is Not Tradable, in Financial Aspects of the United States Pension System, National Bureau of Economic Research, Inc, 1983, pp Merton, Robert C., Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, December 1971, 3 (4), Müller, Christoph, Bernd Raffelhüschen, and Olaf Weddige, Pension obligations of government employer pension schemes and social security pension schemes established in EU countries, Working Paper, Samuelson, Paul A., An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money, Journal of Political Economy, 1958, 66, Viceira, Luis M., Optimal Portfolio Choice for Long-Horizon Investors with Nontradable Labor Income, Journal of Finance, , 56 (2),

26 Tables and Figures Figure 1: Computation of Social Security Benefits Note: This surface represents yearly Social Security benefits (in thousand dollars) as a function of the agent s average indexed yearling earning (AIYE) and the national wage index. The AIYE is the agent s average wage over his career, normalized by the national wage index. Thick lines represent bend points used for the computation of benefits, as detailed in Equation (16) Yearly benefits L 1 (T) AYIE/L 1 (T)

27 Table 1: Augmented Dickey-Fuller Tests Note: I estimate different empirical versions of equation (3) on the period. y(t) = κy(t 1) + b t t + 4 b i y(t i) + c + ɛ(t) 1 For models (1) to (6), the left hand variable is the variation in the log-difference (y) between S&P500 dividends and the average wage. Model (2) tests the existence of a time trend, while models (3) to (6) allow for autocorrelated errors. For models (7) and (8), the left hand variable is the change in the log-difference between cumulated S&P500 stock returns and the average wage (y s ). Both models assume a time trend and model (8) tests the existence of autocorrelated errors.,, indicates p-values below 1%, 5% and 10%. The p-values of ˆκ are from the Dickey-Fuller distributions and are reported in the lower part of the table. Dividends Stock Gains (1) (2) (3) (4) (5) (6) (7) (8) y(t 1).160 ** ***.175 **.196 ***.180 **.200 *.180 (.054) (.054) (.050) (.054) (.055) (.061) (.061) (.065) t *** *** (.000) (.003) (.003) y(t 1).462 ***.577 ***.732 ***.737 ***.018 (.094) (.096) (.104) (.111) (.109) y(t 2) *** *** *** (.106) (.114) (.139) y(t 3).340 ***.382 *** (.108) (.130) y(t 4) (.116) DF p-value RAdj N

28 Table 2: Model Calibration Parameter Description Value Financial markets r risk-free rate 0.02 g D dividends growth rate µ expected stock returns 0.08 σ stock market volatility 0.16 Labor income v 1 Std of permanent labor market shocks v 2 Std of permanent idiosyncratic shocks 0.15 v 3 see text σ κ speed of mean-reversion of y 0.15 α 0 quadratic effect of experience α a see equation (14) 1.14 Preferences γ relative risk aversion 5 ψ discount rate 0.03 T working years (from 20 to 65 years old) 45 R retirement years 20 Initial conditions L 1 (0) average wage index (in thousands) 40 L 2 (0) wage in percentage of L W (0) wealth (in thousands) 50 y(0) abnormal log-diff between L 1 and D see text 28

29 Thousands of US dollars Figure 2: Certainty equivalent and discounted cash flows at different rates Note: This graph plots the average certainty equivalent of Social Security by age. For comparison, I also plot the NPV of future Social Security cash flows (benefits minus payroll taxes) using different discount rates Age DCF at the risk-free rate: Ajdusted r=1% r=2% r=3% r=4% r=5% 29