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Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

The industrialized world has experienced a demographic shift that is straining public pension systems. Employer-sponsored pension plans change from defined benefit to defined contribution. More emphasis is put on individually managed retirement funds. One concern with this movement is the potential negative effect on individual welfare if households' investment behaviour is suboptimal. Using micro-level US data, we compare the optimal utility computed using a lifecycle model with the actual utility as reflected in empirical asset allocation choices. Average estimated welfare costs are below 3% of households' endowment (assets and human capital); yet specific population groups experience higher welfare costs.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

In the USA, and in many other industrialized countries, a demographic shift is taking place that is characterized by low birth rates and increasing longevity (see, for example, United Nations 2007). One consequence of an ageing population is the strain that it puts on public pension systems. Many such systems are pay-as-you-go plans, created under the expectation that current workers would always outnumber retirees. However, the current reality is a shrinking supply of workers relative to retirees. Thus public pension systems can be expected to become less generous. At the same time, employer-sponsored pension plans are relying more on defined contributions instead of defined benefits.

Both trends shift more responsibility for managing retirement investments to households—that is, more emphasis is put on funded, individually managed retirement investments. In light of the evidence that many households have difficulty evaluating simple investment opportunities, including appropriately judging risk and return (see John Hancock Financial Services 2002; Lusardi and Mitchell 2007), shifting the responsibility for old-age provision to households raises a troubling question: as households manage their investments, making savings and asset allocation decisions that affect wealth and consumption over the whole lifecycle, how well are they doing at solving this complex intertemporal allocation problem?

We address this research question by deriving an optimal (benchmark) asset allocation strategy. This strategy is compared with the outcome of households' actual strategies by calculating the welfare costs of not following the benchmark strategy. The benchmark strategy is derived from the solution of a realistically calibrated lifecycle consumption–savings asset allocation model with stochastic labour income, asset returns and lifespan.

In the domain of savings adequacy, this approach is used by Scholz et al. (2006) and Skinner (2007). In the domain of asset allocation, which is the focus of this contribution, the approach has been used by several researchers. Dammon et al. (2004) compare optimal investment choices with commonly observed choices while focusing on the location of assets in taxable versus tax-deferred accounts. Cocco et al. (2005) and Bagliano et al. (2010) compare optimal choices with typical investment recommendations (rules of thumb). Similarly, Bodie and Treussard (2007) and Gomes et al. (2008) evaluate lifecycle (or target date) funds' allocation strategies. Poterba et al. (2006) compare the performance of defined benefit and defined contribution plans. The welfare costs of not following the optimal strategy of renting versus owning a house are calculated by Yao and Zhang (2005).

The work closest to ours is that of Calvet et al. (2007), who measure the welfare costs of underdiversification, or complete non-participation, in risky assets based on Swedish micro-level data. They utilize a classic perfect markets Merton–Samuelson framework (Merton 1969; Samuelson 1969) with lifecycle and endowment-invariant optimal asset allocation policies (‘one-size-fits-all’ for a given degree of risk aversion). We employ a more realistic imperfect markets framework, yielding optimal asset allocation policies that are dependent on the household's age, endowment and preferences. Based on US Survey of Consumer Finances (SCF) data, we calibrate a lifecycle consumption and asset allocation model that accounts for heterogeneity in endowments (income and wealth), risk aversion, time preference and bequest motives when calculating optimal asset allocations—that is, the share of assets invested riskily. Moreover, we include real estate in risky investments. Because the SCF data are not as precise as the Swedish data with respect to the individual location of assets, we abstract from underdiversification issues. Instead, we focus solely on the household-specific share of wealth invested in risky assets for which the SCF provides precise data.

The calibrated lifecycle model employed here gives, in certain calibration variants, a fairly good match to the hump-shaped age profiles of the risky asset share found in the SCF data. Consequently, welfare costs of suboptimal asset allocations are small on average—that is, below 1% of a household's financial resources (including human capital) in model calibrations considering housing as risky asset and below 3% considering only liquid assets. Particular population subgroups with comparatively larger welfare costs are identified. Households that would benefit most from better asset allocations are those in the middle age brackets or having high wealth. Overall, these results suggest that although asset allocation mistakes are present in the data, the incentives for households to correct choices are modest.

In the next section, we define, calibrate and solve the lifecycle model, which yields the optimal asset allocation policies. In Section II, we describe the data and estimate a regression model used for predicting actual asset allocation policies as a function of household characteristics (e.g. age, education, preferences) and endowment (income and wealth). We perform the welfare analysis in Section III and robustness checks in Section IV. We conclude in Section V.

I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

The household's decision problem

For deriving a benchmark asset allocation policy, a standard model for solving intertemporal allocation problems—discounted utility—is employed. The household maximizes the expected utility of consumption C and bequests B (all monetary variables are in nominal terms) over its stochastic lifespan. Its intertemporally separable utility function U(C, B) is defined as

  • display math(1)

where T denotes the maximum remaining lifespan, δ is the subjective discount factor, pt is the age-dependent conditional probability that the household will survive in the period from t − 1 to t, Ct(Bt) is consumption (bequest) at time t and b is the strength of the bequest motive (for example as in Cocco et al. 2005; Inkmann et al. 2011).

The one-period constant relative risk aversion (CRRA) utility function for consumption Ut(Ct), with γ as the coefficient of relative risk aversion, is given by

  • display math(2)

The utility of the bequest, Ut(Bt), is defined by replacing Ct in equation (2) by Bt. Nominal consumption and bequests at time t are adjusted for inflation at rate π.

At each point in time t, the household decides how much to consume (implicitly determining savings) and how to allocate savings. Financial wealth at time t is denoted by Wt. Savings St are allocated to both a risk-free investment and a risky investment. The proportion of savings invested riskily during each period, αt, earns the risky return Rt, whereas the remainder (1 − αt) is compounded at the risk-free return Rf (= 1 + risk-free rate). Conditional on survival, the household receives stochastic labour income Lt until age 64 at the end of each year t. The retirement age is exogenously fixed at age 65. From that age, Lt refers to a deterministic (government) pension income stream. The household cannot short-sell the risky asset but has some borrowing opportunities. In each period, the household may take on debt Dt at interest rate rd, up to a maximum amount inline image , which is equal to the lowest possible labour income received at the end of that period discounted at rd.1

The maximization problem is given by

  • display math(3)

subject to consumption constraints

  • display math(4)

subject to borrowing constraints

  • display math(5)

and subject to no-short-sale constraints

  • display math(6)

In the following subsections, the model is calibrated with a set of baseline parameters. Further calibration alternatives are introduced as robustness checks in Section IV. The technique for solving the optimization problem of equations (3) to (6) is described in the Appendix. Table 1 summarizes the calibration parameters.

Table 1. Parameter Calibration for the Benchmark Model
ParameterSymbolBaseline valuesRobustness checks
Notes
  1. This table summarizes the baseline parameter values chosen to calibrate the benchmark lifecycle model and alternative parameter values used for robustness checks.

Relative risk aversion γ 1, 2, 3, 4Optimally selected
Subjective discount factor δ 0.8, 0.9, 0.95, 0.97, 0.99 
Bequest parameter b 0, 1, 2, 3 
Survival probability p t US Life Tables 2003 
Retirement age 65 
Marginal tax rate 0% 
Log-normal nominal risky asset return R t   
Expected return on equity 1.1151 
S.D. of return on equity 0.1996 
Expected return on housing 1.0111 
S.D. of return on housing 0.1011 
Correlation of equity returns with housing returns 0.2 
Portfolio share of equity in risky assets 46%100%
Expected risky returnE(Rt)1.05901.0490, 1.1151
S.D. of risky returnStd(Rt)0.11580.1996
Risk-free nominal return R f 1.0361 
Nominal borrowing interest rate r d 0.0961 
Maximum borrowing per year inline image 65% × E(Lt)/(1 + rd) 
Inflation π 0.0310 
Log-normal nominal labour income L t   
Expected growth rates during work life Lifecycle profile, Cocco et al. (2005) 
Correlation of growth rates with risky returns 01
Expected growth rates during retirement 3.1% (= inflation rate) 
Replacement factor 35% 
S.D. during work life 0.19 × E(Lt) 
S.D. during retirement 00.19 × E(Lt)
Human capital discount rate 0.03610.0590
(welfare cost measure) (= risk-free rate)(= risky return)

Preference parameters

The household's preferences are described by setting the constant of relative risk aversion γ, the subjective discount factor δ and the bequest parameter b. The range for the values of these parameters is set in accordance with the literature (see, for example, Laibson et al. 1998; Cocco et al. 2005; Inkmann et al. 2011). Allowing for heterogeneity between households' preferences, specific values for each household are assigned based on responses to three survey questions in the SCF. A robustness check for the assignment of risk aversion is provided in Section IV. Responses to such survey questions have been shown to be valid measurements because they are related to household behaviour (Laitner and Juster 1996; DeVaney and Chiremba, 2005; Kimball et al. 2008; Gouskova et al. 2010; Kapteyn and Teppa 2011) and correlate with quantitative survey measures (Guiso et al. 2011).

Risk aversion is assigned based on the question: ‘Which of the statements on this page comes closest to the amount of financial risk that you are willing to take when you save or make investments?’ For the response ‘Take substantial financial risks expecting to earn substantial returns’, γ is set to 1; for ‘Take above average financial risks expecting to earn above average returns’, γ is set to 2; for ‘Take average financial risks expecting to earn average returns’, γ is set to 3; and for ‘Not willing to take any financial risks’, γ is set to 4 (see Tables 2 and 3).

Table 2. Descriptive Statistics
 Full sampleSelected sample
N = 4519N = 2969
MeanMedianS.D.MeanMedianS.D.
Notes
  1. This table presents summary statistics for the 2004 Survey of Consumer Finances (SCF). Statistics are based on the SCF sample weight. ‘Human capital’ is the risk-free rate discounted lifecycle income stream based on the income profile (see Section I). Other variables are defined in Table 3.

Age49.5748.0017.2750.2948.0017.77
No. of children0.810.001.140.770.001.09
Married0.420.000.490.420.000.49
Education—low0.140.000.350.120.000.33
Education—middle0.490.000.500.500.000.50
Education—high0.370.000.480.380.000.49
Occupation—employed0.601.000.490.691.000.46
Occupation—unemployed0.040.000.200.040.000.19
Occupation—retired0.240.000.430.270.000.44
Occupation—self-employed0.120.000.320.000.000.00
Black0.140.000.340.130.000.33
Hispanic0.090.000.290.080.000.27
Risk aversion—high0.420.000.490.410.000.49
Risk aversion—average0.380.000.490.390.000.49
Risk aversion—below average0.160.000.370.160.000.37
Risk aversion—low0.030.000.180.030.000.18
Time preference—very high0.130.000.340.130.000.34
Time preference—high0.260.000.440.260.000.44
Time preference—middle0.280.000.450.290.000.45
Time preference—low0.140.000.350.140.000.35
Time preference—very low0.190.000.390.180.000.38
Bequests—very important0.260.000.440.260.000.44
Bequests—important0.290.000.460.300.000.46
Bequests—somewhat important0.280.000.450.280.000.45
Bequests—not important0.170.000.380.170.000.37
Labour income59,16840,000125,76757,98941,600103,096
Human capital1,728,0351,161,1194,032,9401,739,0321,219,2034,088,312
Retirement income—very satisfactory0.080.000.270.090.000.28
Retirement income—satisfactory0.110.000.320.120.000.32
Retirement income—enough0.340.000.470.350.000.48
Retirement income—inadequate0.190.000.400.200.000.40
Retirement income—totally inadequate0.270.000.450.250.000.43
Owns risky financial0.601.000.490.631.000.48
Risky financial191,29132001,832,832113,9693600921,800
Percent risky financial0.330.250.340.300.210.32
Owns risky total0.811.000.390.851.000.36
Risky total420,453130,0032,260,956316,168135,0001,173,376
Percent risky total0.690.850.350.680.830.35
Owns house0.711.000.450.741.000.44
House value229,161110,000807,526202,200120,000458,300
Percent house0.520.600.380.530.620.38
In debt0.771.000.420.781.000.41
Debt79,07822,272172,04875,96426,000134,358
Percent debt36.460.25772.011.650.276.77
Financial assets277,29223,1002,094,203191,64625,0001,157,734
Total assets506,453154,2002,527,815393,845160,5001,415,306
Labour income/Financial assets185.981.481888.4319.801.7572.47
Labour income/Total assets88.200.261287.159.490.2845.41
Table 3. Variable Definitions
VariableDefinition
AgeAge of head of household respondent
No. of childrenNumber of children in the household
MarriedMarital status, 0 = married or partnership, 1 = otherwise
Education—lowEducation of head of household = no degree, on-the-job training, no high school diploma/GED
Education—middleEducation of head of household = high school diploma, GED, some college (holdout group)
Education—highEducation of head of household = college degree
Occupation—employedOccupation of head of household = work for someone else (holdout group)
Occupation—unemployedOccupation of head of household = other groups not working
Occupation—retiredOccupation of head of household = retired/disabled
Occupation—self-employedOccupation of head of household = self-employed/partnership
BlackRace/ethnicity of household respondent = black/African American
HispanicRace/ethnicity of household respondent = Hispanic
Risk aversion—highWillingness to take financial risks = ‘Not willing to take any financial risks’
Risk aversion—averageWillingness to take financial risks = ‘Take average financial risks expecting to earn average returns’ (holdout group)
Risk aversion—below averageWillingness to take financial risks = ‘Take above average financial risks expecting to earn above average returns’
Risk aversion—lowWillingness to take financial risks = ‘Take substantial financial risks expecting to earn substantial returns’
Time preference—very highSaving and spending planning horizon = ‘Longer than 10 years’
Time preference—highSaving and spending planning horizon = ‘Next 5–10 years’
Time preference—middleSaving and spending planning horizon = ‘Next few years’ (holdout group)
Time preference—lowSaving and spending planning horizon = ‘Next year’
Time preference—very lowSaving and spending planning horizon = ‘Next few months’
Bequests—very importantLeaving an estate or inheritance to heirs = ‘Very important’
Bequests—importantLeaving an estate or inheritance to heirs = ‘Important’
Bequests—somewhat importantLeaving an estate or inheritance to heirs = ‘Somewhat important’ (holdout group)
Bequests—not importantLeaving an estate or inheritance to heirs = ‘Not important’
Labour incomeTotal amount of pretax income of household excluding income from investments (financial assets and real estate)
Retirement income—very satisfactoryRetirement income received or expect to receive = ‘Very satisfactory’
Retirement income—satisfactoryRetirement income received or expect to receive = ‘Satisfactory’
Retirement income—enoughRetirement income received or expect to receive = ‘Enough’ (holdout group)
Retirement income—inadequateRetirement income received or expect to receive = ‘Inadequate’
Retirement income—totally inadequateRetirement income received or expect to receive = ‘Totally inadequate’
Owns risky financialDummy for Risky financial > 0
Risky financialRisky financial assets (directly held stocks; risky share invested in investment funds, trusts, annuities, and managed investment accounts, quasi-liquid retirement accounts; mortgage-backed, corporate, and foreign bonds; other financial assets (e.g. loans to someone else, future proceeds from lawsuits))
Percent risky financialRisky financial divided by Financial assets
Owns risky totalDummy for Risky total > 0
Risky totalRisky financial assets (Risky financial) + House value
Percent risky totalRisky financial divided by Total assets
Owns houseDummy for House > 0
House valueValue of houses and real estate, including land
Percent houseHouse value divided by Total assets
In debtDummy for Debt > 0
DebtValue of debt
Percent debtDebt divided by Total assets
Financial assetsFinancial assets, including risky (Risky financial) and non-risky financial assets (money, checking, savings, and call market accounts; saving bonds; cash value of life insurance; tax-exempt bonds; US government and government agency bonds; non-risky share invested in investment funds, trusts, annuities, and managed investment accounts; quasi-liquid retirement accounts), excluding non-financial assets (e.g. cars, paintings)
Total assetsFinancial assets + House value

Time preference—the subjective discount factor—is assigned based on the question: ‘In planning saving and spending, which of the time periods listed on this page is most important to you?’ For the response ‘Longer than 10 years’, δ is set to 0.99; for ‘Next 5–10 years’, δ is set to 0.97; for ‘Next few years’, δ is set to 0.95; for ‘Next year’, δ is set to 0.90; and for ‘Next few months’, δ is set to 0.80.

The strength of the bequest motive is assigned based on the question: ‘Some people think it is important to leave an estate or inheritance to their surviving heirs, while others don't. Which is closer to your feelings? Would you say it is…?’ For the response ‘Not important’, b is set to 0; for ‘Somewhat important’, b is set to 1; for ‘Important’, b is set to 2; and for ‘Very important’, b is set to 3.

For survival probabilities, the US Life Tables 2003 are used (see Arias 2006), which reflect average population mortality; the maximum age is 100 years, thus the maximum remaining lifespan T is given by 100 minus the current age of the household.

Asset returns

As a proxy for the risky asset, a broadly defined market portfolio consisting of equity and real estate is used. (Section IV contains a robustness check considering only liquid assets.) The financial crisis of 2007–09 made clear that housing wealth can be very risky, and thus should be included in risky assets. Although real estate is not as liquid as stocks, there is evidence that households adjust their house size and consider the possibility of trading in real estate according to their individual situation (see Banks et al. 2010). The proportion of equity and real estate in the portfolio is set equal to the overall share of equity and real estate held by households (and non-profit organizations) in the USA; the portfolio is composed of 46% equity and 54% real estate.2

As a proxy for the equity return, a broad-based US stock market index including data from 1926 to 2006 is used (see Morningstar 2007). After deducting typical transaction costs of an index-investment fund of 0.7% per year, the nominal mean of the equity return is 1.1151 and the standard deviation is 0.1996. Real estate returns have a mean real return of 1.0 and a standard deviation of 0.1. After considering annual maintenance costs of 1.5% and annual transaction costs of 0.4320% of the house market value, as well as the inflation rate reported below, the nominal mean real estate return equals 1.0111 and the standard deviation is 0.1011 per year.3 Finally, as in Cauley et al. (2007), the correlation coefficient for equity and real estate returns is set to 0.2. The nominal mean of the overall portfolio return on the risky asset, consisting of equity and real estate, Rt, is then 1.0590, with a standard deviation of 0.1158. Risky asset returns are modelled to be log-normal and i.i.d.

As a proxy for the risk-free return, the short-term money market is used. Given the same sample periods, the nominal risk-free return Rf is set to 1.0361 per year (see Morningstar 2007), again considering typical transaction costs for an index-based investment of 0.18%. Again using the same sample period, inflation is set at 0.031 (see Morningstar 2007). Borrowing costs are taken from Davis et al. (2005), who use an empirically calibrated borrowing wedge of 6 percentage points—that is, rd is set to 0.0961 in nominal terms per year.

Labour income

Following, for example, Koijen et al. (2010), the labour income process is calibrated to match empirically observed lifecycle household income profiles as estimated by Cocco et al. (2005). The mean real growth rates of income during the lifecycle before retirement (until age 64) are age- and education-specific (low, middle, high).4 Real income profiles are generally hump-shaped in age. During retirement, labour income is exogenously replaced by (government) pension income by multiplying the income at age 64 with an expected future replacement factor of 35% (see Reno and Lavery 2007). Afterwards, nominal pension income grows with the inflation rate of 3.1% per year.

To account for the riskiness of labour income, each period's labour income is modelled log-normally distributed and subject to transitory shocks5 uncorrelated with the risky asset return (as in Li and Yao 2007). The mean of Lt is given by the current income at year t = −1 with real growth rates plus growth due to the inflation rate as specified above. Until age 64, the standard deviation for US households is set to 0.19 × E(Lt) (see Carroll and Samwick 1997). From age 65 (during retirement), there is no labour income uncertainty. In the calculations, it is assumed that there are no taxes.

II Analysis of Actual Asset Allocation Behaviour

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

Data description and sample selection

The data are taken from the 2004 wave of the US Survey of Consumer Finances (SCF). The dataset contains detailed information on 4519 households, including demographics, wealth, income and other characteristics. Data were collected via a dual-frame sample design. About 3000 households are drawn from a representative sample of households to reflect characteristics that are broadly distributed in the population, such as home and vehicle ownership. The other set of 1500 survey cases is drawn from an oversampling of wealthy households.

The following analyses exclude self-employed households—that is, households for which the standard lifecycle model with labour income and a traded risky asset most likely will not be appropriate—and cases equal to zero or smaller in the labour income and wealth (total assets) distributions. With these modifications, the final dataset comprises 2969 cases. Summary statistics on demographic and financial characteristics for the full dataset and the sample selected are shown in Table 2. Variables are defined in Table 3.

Choice of the regression model and variable selection

The literature distinguishes two general hypotheses with respect to the investment decision process (e.g. Bertaut and Starr-McCluer 2002). According to the first hypothesis, the decision to invest in risky assets (participation) is made simultaneously with the decision about how much to invest (the risky share). According to the second hypothesis, there is a two-stage investment process: at the first stage, the household decides whether to invest in the risky asset; the second stage independently involves how much to invest. The benchmark model from Section I implies a simultaneous investment decision process; that is, the participation and share decisions are made jointly. Consequently, a Tobit model is used as opposed to a two-stage probit (participation)/truncated OLS (share) regression.6 The Tobit model, in comparison to an OLS model, takes into account that the dependent variable, the risky share (percent risky total), is censored at 0 and 1.

Regression variables are selected by matching the risky share determinants of the benchmark model of Section I with the SCF data counterparts. Other previously identified empirical determinants of the risky asset share (Curcuru et al. 2010) not modelled in the benchmark model are included as controls in the regression analysis of the welfare costs (Section III).

The dependent variable is the risky asset share of investments (percent risky total), equal to the amount of risky assets divided by wealth (total assets—see Table 3). Following the calibration of the benchmark model, risky assets encompass equity, risky bonds and real estate (see Table 3). The independent variables are age, education, retirement status, total assets, labour income, debt, risk aversion, time preference and the strength of the bequest motive.

Based on the benchmark model, a major determinant for the risky asset share is the ratio of the household's expected labour income to wealth. The risky asset share increases in the labour income-to-wealth ratio because labour income serves partially as a risk-free asset (Viceira 2001; Cocco et al. 2005). This means that with increasing labour income, the household invests its financial wealth more riskily to account for the fact that a larger fraction of its resources is implicitly invested in a risk-free asset. Thus the coefficient for ratio of labour income to total assets should be positive, and the coefficient for ln(total assets) should be insignificant (as for a CRRA household, the ratio of income to assets alone is sufficient to determine the risky share). Having more debt is expected to decrease the risky asset share (Teplá 2000; Davis et al. 2005).

Increasing age generally should have a negative effect on the risky asset share, given some fixed value for the labour income-to-total assets ratio. With increasing age, the number of periods with expected future labour income decreases, leading to a lower risky asset share. Likewise, retired households have a lower expected income stream, thus a negative coefficient of the ‘Occupation—retired’ variable is expected as well, possibly picking up non-linearities in the age effect not yet incorporated by the age and age2 terms.

The effect of education on the risky asset share in the benchmark model is driven by differences in expected future labour income. Higher education is associated with higher growth rates of labour income during the lifecycle until age 52, resulting in higher expected labour income (given identical current income) in the early stage of the lifecycle. This leads to a higher risky asset share in this stage, which is reversed later on. The expected sign of low (high) education is negative (positive), due to expected overall labour income growth rate differentials according to education. However, the opposite effect is expected when education is interacted with age due to the reversal of the gap in expected growth rates between the education levels.

The impact of risk aversion on the risky asset share is straightforward: the risky asset share decreases with increasing values for the risk aversion parameter γ; consequently, the dummy variable for ‘Risk aversion—high’ should have a negative sign, and the other risk aversion dummy variables should have positive signs. The effect of a change in the subjective discount factor δ depends on the household's expected labour income. Both increases and decreases of the risky asset share are possible. In general, increasing δ will result in higher savings because the household places more weight on future utility. By saving more, the household's expected labour income decreases relative to the higher savings-induced increase in wealth, thus resulting in a lower risky asset share. However, putting more weight on future utility also implies that the household's subjective value of future labour income is larger (less discounted), and thus the risky asset share can also increase. The overall effect in general is rather small and ambivalent; consequently, the sign for the ‘Time preference’ dummy variables cannot be predicted. The stronger the bequest motive, the more the household will save to provide resources for bequests. This in turn decreases the labour income-to-total assets ratio, leading to a lower risky asset share (Gomes and Michaelides 2005). Thus for the dummy variables for ‘very important’ and ‘important’ bequest motives, positive coefficients are expected, and for ‘unimportant’ bequest motives, a negative coefficient is expected.

Regression results

Table 4 displays the results of the regression analysis.

Table 4. Determinants of the Risky Asset Share
Dependent variablePredictionPercent risky total
Coeff.S.E.
Notes
  1. This table presents the results from a Tobit regression of the risky share (Percent risky total) on household characteristics given in the SCF 2004 data that have a counterpart in the benchmark model. Variables are defined in Table 3.

  2. *, **, *** denote statistical significance at the 10%, 5%, 1% level, respectively.

Ageimage_n/ecca12036-gra-0001.png0.00900.0021***
Age2−0.00010.0000***
Education—low−0.02560.0628
Education—high+0.02930.0383
Age * Education—low+0.00170.0011
Age * Education—high−0.00190.0007***
Occupation—retired−0.03630.0174**
ln(Total assets)00.07710.0033***
Labour income/Total assets+−0.01170.0009***
Percent debt−0.00090.0016
Risk aversion—high0.02700.0137**
Risk aversion—below average+−0.00830.0151
Risk aversion—low+0.03090.0291
Time preference—very high±−0.01990.0175
Time preference—high±−0.01380.0148
Time preference—low±0.03190.0188*
Time preference—very low±0.03620.0175**
Bequests—very important−0.03670.0147**
Bequests—important−0.02290.0145
Bequests—not important+0.02160.0170
Constant −0.34580.0615***
N  2969
McFadden R2 0.6032

With respect to the coefficients with a clear prediction of the sign, the following results are obtained. The age effect (age, age2) is hump-shaped, peaking at age 44, and the retirement dummy variable is negative and significant. Thus the age effect matches predictions from age 44 onward. The education and age–education interaction coefficients confirm predictions, but they are insignificant except for the ‘Age * Education—high’ interaction term. The effects of labour income and total assets are both in the opposite direction to the prediction of the benchmark model: more assets and less labour income increase the risky assets share. The impact of having debt is positive and confirms the prediction; however, the coefficient is insignificant. The coefficient for the dummy variable for high risk aversion is significant and in the opposite direction to the predictions, while the other risk aversion coefficients are insignificant. With respect to the strength of the bequest motive, only the coefficient for having strong bequest motives is significant and negative, thus confirming the predictions. Overall, the regression results partially confirm the predictions; that is, asset allocation behaviour follows the predictions of the benchmark lifecycle model with respect to only some factors.

III Welfare Analysis

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

Measuring welfare costs

We next develop the measure for analysing welfare costs incurred by households that choose different risky asset shares than those proposed by the benchmark model. In particular, an equivalent wealth variation measure (Merton and Samuelson 1974) is used. We first calculate the expected utility resulting from an optimal asset allocation for each household in the SCF data. This optimal utility is given by the value function (see the Appendix) for t = 0 from the benchmark model inline image —that is, inline image (t ∊ {0, 1, …, T - x}). Next, we calculate the expected utility resulting from the predicted actual asset allocation behaviour inline image (i.e. inline image ).

The prediction of actual behaviour is done based on the Tobit regression model and calculated as follows. First, for each household the residual of the Tobit regression model of Section II is calculated (result A). Result A represents each household's idiosyncratic variation in asset allocation behaviour. Second, the regression equation's prediction is calculated for each household for each time period, incorporating both the household's time-invariant (e.g. education, time preference) and time-variant (e.g. age, assets) characteristics (result B). Result B represents the asset allocation behaviour that is common to all households with the same characteristics. Finally, the constant term A and the time-variant term B are added to yield the household's predicted risky asset share inline image.

To derive the welfare costs, equation (7) is solved for ΔW0. ΔW0 is then normalized by the household's resources, that is, divided by the sum of assets at = 0 and the household's discounted expected labour income according to equation (8):

  • display math(7)
  • display math(8)

This final resulting relative measure, WCO (welfare cost), allows comparisons across households with different endowments (total assets and expected labour income). The economic interpretation of this measure answers the question of how much welfare it will cost the household if it does not follow an optimal lifecycle asset allocation strategy, that is, the strategy proposed by the benchmark model.

WCO relates welfare costs to the discounted total financial resources expected over a lifetime (i.e. the sum of current assets and the discounted value of future labour income). Small values of WCO indicate that moving the asset allocation toward the benchmark model would not enhance expected utility very much, whereas high values of WCO indicate that the household could be considerably better off—that is, it loses a great deal of utility by not following the benchmark.

The specification of inline image includes actual behaviour with respect to asset allocation, as predicted by the regression model. The decision with respect to consumption Ct(Wt) and borrowing Dt(Wt) is optimized conditional on the empirically predicted asset allocation. Thus, not modelling further effects from underdiversification (see Calvet et al. 2007; Campanale 2009), the results can be interpreted as a lower boundary of welfare costs, since, in reality, savings and borrowing could differ from this optimal choice.

Results

Welfare costs, as measured by WCO, in the SCF calculated according to equation (8), range between virtually zero and 9.5%. On average, welfare costs are small, with a mean (median) welfare cost of 0.5% (0.2%), with the 25% (75%) percentile at 0.02% (0.6%). At first glance, the average magnitude of welfare costs seems to be small; however, welfare costs are measured in relation to total lifetime wealth, including a large stock of discounted labour income. In dollar terms, the mean (median) welfare costs are $12,770 ($2131), with the 25% (75%) quartile at $247 ($10,727). These amounts are slightly smaller than the welfare costs of underdiversification found for Swedish investors by Calvet et al. (2007). The variation in mean welfare costs according to major demographic characteristics is shown in Panel A of Figure 1 (age) and Panel A of Table 5 (labour income and total assets). With respect to these household characteristics, welfare costs are hump-shaped in age, increasing in assets and have no clear tendency in labour income.

Table 5. Distribution of Welfare costs
Labour income quintileTotal assets quintile
12345
Notes
  1. This table shows the distribution of welfare costs over the labour income and total assets quintiles. Panel A refers to the original welfare cost measure (equation (8)), while Panel B refers to the adjusted welfare cost measure (equation (9)). Each cell shows the mean welfare costs in percent for households that are located in the respective quintiles for labour income and total assets. Quintiles and mean welfare costs are calculated using the SCF sample weight. Variables are defined in Table 3. Calibration parameters = baseline values according to Table 1.

Panel A: Welfare costs in percent
10.150.360.390.931.14
20.120.180.300.590.81
30.050.220.290.560.78
40.250.210.340.621.02
50.290.200.240.580.98
Panel B: Adjusted welfare costs in percent
10.090.420.451.011.15
20.040.210.380.660.80
30.010.250.390.640.80
40.200.190.450.691.05
50.220.120.330.661.01
image

Figure 1. Mean welfare costs for different household ages. Notes: Means are calculated using the SCF sample weight. Panel A shows the welfare costs for the baseline calibration according to Table 1. Panel B shows the welfare costs for the calibration that includes only the households' liquid assets, i.e. excluding housing wealth from the analysis.

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To understand the relationships between household characteristics and the size of welfare costs, next, welfare costs are regressed on a set of household characteristics. The set of household characteristics includes two subsets. The first subset encompasses the variables already included in the predictive regression of the risky asset share (compare Table 4), which are the variables that have a counterpart in the benchmark lifecycle model. The second subset encompasses variables that neither were included in the risky share regression nor have a counterpart in the lifecycle model, but which have been shown in empirical studies to explain the risky share in household portfolios (see, for example, Curcuru et al. 2010). The purpose of including the second subset of variables is to control for the part of welfare costs that originates from modelling choices rather than from household behaviour. This subset includes the following variables: number of children and marital status (see Love 2010), race/ethnicity (proxying for heterogeneity in life expectancy, see Brown 2003), subjective expectations on the adequacy of retirement income (addressing potential heterogeneity in replacement ratios), and the difference between the household's actual ratio of housing wealth to total risky assets and the ratio used in the calibration of the benchmark model of 54% (conditional on having risky assets). Regression results are shown in Table 6.

Table 6. Determinants of Welfare costs
Dependent variableWelfare costs
Coeff.Bootstrap S.E.
Notes
  1. This table presents the results from an OLS regression of the welfare costs in percent (WCO) on household characteristics. Standard errors are bootstrapped (10,000 replications). Variables are defined in Table 3.

  2. *, **, *** denote statistical significance at the 10%, 5%, 1% level, respectively.

  3. Calibration parameters = baseline values according to Table 1.

Age0.04720.0050***
Age2−0.00040.0001***
Education—low−0.03460.1429
Education—high−0.00660.0855
Age * Education—low0.00130.0024
Age * Education—high0.00020.0017
Occupation—retired0.06800.0596
ln(Total assets)0.09160.0125***
Labour income/Total assets0.00350.0010***
Percent debt−0.00020.0028
Risk aversion—high−0.07870.0382**
Risk aversion—below average0.15850.0537***
Risk aversion—low−0.04360.0717
Time preference—very high0.13380.0597**
Time preference—high0.04830.0435
Time preference—low−0.21480.0417***
Time preference—very low−0.33610.0358***
Bequests—very important0.01300.0455
Bequests—important−0.03200.0386
Bequests—not important0.08960.0509*
No. of children0.00420.0149
Married0.08070.0363**
Black0.04110.0486
Hispanic−0.05180.0482
Retirement income—very satisfactory−0.02060.0590
Retirement income—satisfactory0.07470.0529
Retirement income—inadequate−0.02400.0431
Retirement income—totally inadequate−0.08220.0401**
((House/Risky total) − 0.54) * Owns risky total−0.32600.0541***
Constant−1.74850.1654***
N 2969
Adj. R20.2069

The discussion of the determinants of welfare costs focuses first on the variables with significant coefficients included in first subset of household characteristics, that is, the household characteristics accounted for in both the benchmark model and the risky share regression.

For a comprehensive interpretation of the regression results, one general effect needs to be explained first. In the baseline calibration, most households invest too little in risky assets. The benchmark model reveals that the expectation of receiving labour income in the future increases the optimal risky asset share. For younger households and households with high income (relative to assets), the optimal share is often 100%. Such households should invest as much as possible in risky assets given the constraints in equations (5) and (6). Over the whole sample, there is on average a positive gap between the optimal and the predicted risky asset allocation share. The mean optimal share according to the model is 92%; the actual mean of the risky asset share in the selected sample is 68%. In consequence, whenever the predicted risky asset share rises in a certain variable that lowers the optimal risky share of the benchmark model, the gap to the optimal allocation decreases with resulting lower welfare costs.

Welfare costs are higher for households with a higher endowment of assets and labour income. At first glance this seems puzzling, given that the risky share prediction regression (compare Table 4) implies increasing risky asset shares in assets. Thus higher assets correspond with a smaller gap to the optimal risky asset share. One result of the benchmark model, however, is that high assets in relation to labour income lead to high savings in proportion to assets, since the stock of future labour income (implicit savings) becomes less important. Households with fewer assets but larger expected labour income may save nothing or even borrow to finance consumption. The wealthier a household is, therefore, the more do its (although smaller) deviations from the optimal asset allocation affect a larger savings amount in relation to assets, leading to higher welfare costs. This finding is in line with the results of Calvet et al. (2007) regarding the welfare costs of underdiversification. They find that wealthier households, although financially more sophisticated and better diversified, incur larger welfare costs due to the larger amounts of wealth at stake.

Having below average risk aversion is associated with higher welfare costs, while high risk aversion implies lower welfare costs. With below average risk aversion, the empirically predicted risky asset share decreases slightly (and the coefficient is insignificant), while the benchmark model's investment in the risky asset increases. As a result, the gap between the optimal risky asset share and the empirical asset share increases, with resulting higher welfare costs. Likewise, households with high risk aversion have lower welfare costs, since the ‘wrong’ positive sign of the risky share regression narrows the gap to the optimal risky asset share. Welfare costs are higher with higher time preference—that is, when δ is higher. A higher δ makes the household more future oriented, and thus it increases its savings, which also means that the potential amount invested suboptimally also increases. Furthermore, with a higher δ, all future welfare costs are less heavily discounted. With respect to the third preference parameter, the strength of the bequest motive, a significant positive impact on welfare costs is found for households with no bequest motive. Although the risky share prediction coefficients adjust the risky share in the correct direction, the correction is not sufficient. The increase in the optimal risky share for households having no bequest motive is three times as large as produced by the risky share regression coefficient, thus the gap to the optimal share increases.

With respect to the household's age, regression results confirm the univariate findings depicted in Figure 1: welfare costs are hump-shaped in age, peaking at an age of 57. This shape is produced by a complex interplay of an age-related impact of predicted asset allocation behaviour, an optimal age-related asset allocation part, and an age-related impact of the strength of subjective discounting in the utility function. The pure asset allocation part cannot be precisely disentangled; however, the general shape can be explained. Figure 2 shows age profiles of the individual components that enter the welfare cost measure according to equation (8), and it shows the differences between the benchmark model's optimal risky assets shares and actual SCF risky asset shares.

image

Figure 2. Mean summary statistics for different household ages. Notes: Means are calculated using the SCF sample weight. Welfare costs in dollar terms are derived according to equation (8) without normalizing welfare costs by ‘Total assets’ and ‘Human capital’. Human capital is the risk-free rate discounted lifecycle income stream based on the income profile (see Section I). The optimal risky asset share is the risky asset share according to the benchmark model. Other variables are defined in Table 3. Calibration = baseline values according to Table 1.

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According to Figure 2, welfare costs are comparatively low, also in dollar terms, for young households. For such households, the ratio of labour income to assets is typically very high and savings are often zero (or households even borrow). Consequently, deviations from any optimal asset allocation (although indeed present in Figure 2) do not matter much if the investment is very small in comparison to total lifetime resources (the asset allocation's contribution to welfare costs). In addition, future welfare costs (when savings will increase) are still heavily discounted in the utility function, so whatever future behaviour is predicted by the benchmark model, deviations do not matter much from the perspective of the present. With higher age, savings increase, and asset allocation deviations from the benchmark model increase in relative importance. At the same time, future welfare costs are discounted less heavily, thus overall welfare costs increase. Finally, at very high ages, welfare costs derive from the discounted utilities of just a few—by the survival probability strongly discounted (compare equation (A1) in the Appendix)—periods, with less potential to sum to large amounts. The age profile of welfare costs, therefore, generally is not driven by the choice of the normalization variables in the welfare cost measure.

Some of the coefficients of the variables in the second subset of household characteristics are significant, highlighting that some factors not accounted for in the lifecycle model correlate with the welfare cost measure. To get an insight into the importance of this correlation, an adjusted measure of welfare costs is calculated. In particular, the welfare cost measure is adjusted by subtracting the part that is explained by the non-modelled household characteristics in the welfare cost regression according to equation (9):

  • display math(9)

The vector X contains the household characteristics of the second subset of the welfare cost regressors (number of children, marital status, race/ethnicity, retirement income adequacy, difference between the actual share of housing in risky assets and the share in the benchmark model), and inline image contains the estimated coefficients for these regressors from the welfare cost regression. These adjusted welfare costs are on average higher than the welfare costs calculated originally. The mean (median) increases slightly from 0.48% (0.16%) to 0.51% (0.25%). The labour income and assets gradients of the adjusted welfare costs are given in Panel B of Table 7 (baseline calibration), highlighting further this effect. In consequence, the original welfare cost measure can be interpreted as a rather conservative estimate of the actual welfare cost.

Table 7. Summary Statistics for Welfare costs and Risky Asset Shares—Alternative Calibrations
CalibrationWelfare costs in percentMean risky asset share
Mean25% quartileMedian75% quartileSCFModel
Notes
  1. This table presents summary statistics for welfare costs, adjusted welfare costs, and the SCF actual and model optimal risky asset shares for alternative calibrations. Statistics are based on the SCF sample weight. Panel A refers to the original welfare cost measure (equation (8)), while Panel B refers to the adjusted welfare cost measure (equation (9)). Welfare cost statistics are calculated using the SCF sample weight. Calibrations are defined as follows: Baseline calibration = baseline values according to Table 1; Risky pensions = standard deviation of pension income equals work life standard deviation; Correlated labour income = correlation between risky asset and labour income return equals one; Alternative discount rate = discount rate for calculating the present value of labour income equals the risky asset's expected return; Risky return minus 100 BPS = expected risky return is 100 basis points lower; Optimal risk aversion = lifecycle model relative risk aversion parameter is varied between 1 and 4, chosen as the value minimizing welfare costs; Financial assets = analysis is based on the households' liquid assets, i.e. excluding housing wealth from the analysis.

Panel A: Welfare costs and risky asset share
Baseline calibration0.480.020.160.610.680.92
Risky pensions0.470.020.150.600.680.92
Correlated labour income0.440.020.150.560.680.89
Alternative discount rate0.610.030.200.770.680.92
Risky return minus 100 BPS0.240.010.090.250.680.82
Optimal risk aversion0.310.010.110.380.680.91
Financial assets2.360.010.453.260.300.87
Panel B: Adjusted welfare costs in percent
Baseline calibration0.510.110.250.65  
Risky pensions0.510.110.260.63  
Corrrelated labour income0.480.100.240.61  
Alternative discount rate0.740.300.520.95  
Risky return minus 100 BPS0.220.000.150.32  
Optimal risk aversion0.320.050.140.40  
Financial assets2.880.691.383.74  

In summary, the analysis shows that welfare costs stemming from choosing suboptimal risky asset shares are, on average, small, with a mean cost of 0.5% of a household's total resources (total assets and labour income). Welfare costs are significantly higher for households that are in the middle age brackets and have high assets, high labour income, below average risk aversion and a higher time preference. These results contribute to the literature that tries to reconcile households' low participation in risky assets, by, for example, allowing (like the present model) for heterogeneous investors (e.g. Gomes and Michaelides 2005), non-standard preferences (e.g. Polkovnichenko 2007), and the different risk properties of labour income (e.g. Bodie and Treussard 2007; Benzoni et al. 2007; Storesletten et al. 2007). We find only small welfare costs due to asset allocation mistakes, especially for young households for which standard lifecycle models predict choices rather distant from actual choices. Thus a potential alternative explanation of low participation of young households is that they have only modest incentives to correct their choices.

IV Model Discussion and Robustness Checks

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

The following discussion focuses on potential shortcomings of the benchmark model, and provides numerical results for model calibrations that incorporate alternatives for the labour income process, the risky asset return, the preference parameters and the treatment of housing as a risky asset.

Model discussion

The empirical analysis relies on cross-sectional data on asset allocations. This implies that when making predictions for lifecycle behaviour, the model does not account for time or cohort effects with respect to asset allocation. In general, risky asset ownership increases over time with the birth cohort (e.g. Ameriks and Zeldes 2004), due to cheaper access to financial information and/or lower transaction costs for trading. Furthermore, the introduction of lifecycle funds (also called target date funds) as the default in employer-sponsored pension schemes tends to increase exposure to risky assets (Mitchell et al. 2009). For most households, therefore, the results would show lower welfare costs if one took time and cohort effects into consideration, because the gap to the optimal risky asset share would decrease over time. Tendencies with respect to the covariates used in the welfare cost prediction, however, should be fairly robust with respect to time and cohort effects given that these effects are more or less equal for all households of the same cohort.

With respect to the assumptions on mortality, the life tables that we use reflect survival probabilities that are relevant for the population average. There is empirical evidence that mortality is heterogeneous with respect to the financial endowment, with wealthier people tending to live longer (Brown 2003). Although not specifically modelled, the impact of heterogeneous mortality on welfare costs can be inferred from the results on the impact of the subjective discount factor (time preference). Changing the survival probability is mathematically very similar to changing the subjective discount factor (compare equation (A1) in the Appendix). Thus assuming lower survival probabilities for richer households is expected to increase the welfare costs for them (and vice versa for poorer households).

In the model, households save optimally, conditional on their predicted asset allocation decisions. Because this will not be the case in the real world, overall welfare could be higher for all households due to suboptimal savings. The analysis of savings adequacy for the USA in Scholz et al. (2006) suggests, particularly with respect to the labour income endowment, that there are some systematic deviations. The savings decision comes closer to the optimal solution with increasing labour income. As a result, the positive coefficient for the labour income-to-total assets ratio may become smaller.

Both decisions of whether or not to invest in risky assets, and how much to invest in risky assets are modelled in the benchmark lifecycle model and in the predictive regression to be made simultaneously. Thus the framework used here is focusing on the analysis of the ultimate risky asset share. Participation decisions per se receive considerable interest in the literature (e.g. Calvet et al. 2007; van Rooij et al. 2011), leading to the question of whether the estimated welfare costs are driven by participation decisions. This question cannot be answered straightforwardly. Welfare costs derive from all future periods of the lifecycle, and participation may vary over the lifecycle. Thus comparing households currently participating with non-participating households can deliver only partial information on the direction of effect (assuming that current non-participation is informative for future decisions). Mean welfare costs of non-participating households (15% of the sample) are 0.25% and smaller than those of participating households (0.52%). Comparing major characteristics of the two groups shows that non-participating households have considerably fewer assets (difference in total asset means −$457,481), and that labour income is larger in relation to assets (difference in labour income to total assets ratio means 55.80). Thus, in line with the results on the determinants of the welfare costs of Section III, welfare costs are higher for participating households since they have a higher amount of investable wealth at stake. Given that non-participating households' total endowment is comprised primarily of their exogenously given labour income, non-participation is not driving the results.

The labour income process

Three aspects of the labour income process are considered in the following numerical robustness check. First, a variant of the benchmark model is calculated that accounts for risky (rather than risk-free) pension income by imposing the same volatility of labour income for both working years and retirement years. This check can be viewed as accounting for the uncertainty of a future government pension policy or for the possibility that a part of the pension income may be risky because it is derived from an occupational pension. Second, independence between labour income and the risky asset return was assumed. Bodie and Treussard (2007) argue that labour income may be strongly correlated with risky assets during earlier years of a career—that is, that labour income betas decrease with age, implying lower optimal risky asset shares during the work life. Therefore a model variant where labour income is perfectly correlated with the risky asset return is considered. Third, for normalizing welfare costs by the sum of total assets and the discounted labour income, as in Calvet and Sodini (2009), the risk-free rate was used for discounting future labour income. A higher discount rate may be appropriate, given the riskiness of labour income. Welfare costs are thus normalized using the risky return as the discount rate.

Summary statistics on resulting welfare costs for these three model variants (labelled ‘Risky pensions’, ‘Correlated labour income’ and ‘Alternative discount rate’) are given in Table 7 with the original calibration (Baseline calibration); welfare cost regressions results are given in Table 8, columns 2, 3 and 4.

Table 8. Determinants of Welfare costs—Alternative Calibrations
Dependent variableBaseline calibrationRisky pensionsCorrelated labour incomeAlternative discount rateRisky return minus 100 BPSOptimal risk aversion
(1)(2)(3)(4)(5)(6)
Welfare costsWelfare costsWelfare costsWelfare costsWelfare costsWelfare costs
Coeff.Boot. S.E.Coeff.Boot. S.E.Coeff.Boot. S.E.Coeff.Boot. S.E.Coeff.Boot. S.E.Coeff.Boot. S.E.
Notes
  1. This table presents the results from an OLS regression of the welfare costs in percent (WCO) on household characteristics for alternative calibrations. Standard errors are bootstrapped (10,000 replications). Variables are defined in Table 3.

  2. *, **, *** denote statistical significance at the 10%, 5%, 1% level, respectively.

  3. Calibrations are defined as follows: Baseline calibration = baseline values according to Table 1; Risky pensions = standard deviation of pension income equals work life standard deviation; Correlated labour income = correlation between risky asset and labour income return equals one; Alternative discount rate = discount rate for calculating the present value of labour income equals the risky asset's expected return; Risky return minus 100 BPS = expected risky return is 100 basis points lower; Optimal risk aversion = lifecycle model relative risk aversion parameter is varied between 1 and 4, chosen as the value minimizing welfare costs.

Age0.04720.0050***0.04560.0049***0.03990.0047***0.06060.0062***0.00630.00490.02840.0032***
Age2−0.00040.0001***−0.00040.0001***−0.00030.0000***−0.00050.0001***−0.00010.0001**−0.00020.0000***
Education—low −0.03460.1429−0.06990.1315−0.01970.1297−0.08840.1880−0.05800.1253−0.05610.0772
Education—high−0.00660.0855−0.00520.0843−0.04420.08040.06210.1068−0.17410.0896*0.05210.0585
Age * Education—low0.00130.00240.00190.00230.00110.00220.00230.00290.00310.00230.00030.0015
Age * Education—high0.00020.00170.00020.00170.00090.0016−0.00100.00210.00180.0019−0.00120.0012
Occupation—retired 0.06800.05960.06420.06010.08900.05780.03730.06830.22610.0718***−0.04130.0343
ln(Total assets)0.09160.0125***0.09250.0124***0.08980.0121***0.09500.0152***0.16890.0173***0.03760.0065***
Labour income/ Total assets0.00350.0010***0.00290.0008***0.00300.0008***0.00440.0014***0.00460.0010***0.00020.0004
Percent debt−0.00020.00280.00080.00250.00280.0026−0.00240.00390.01090.0044**−0.00170.0014
Risk aversion—high−0.07870.0382**−0.07500.0386*−0.06430.0360*−0.10800.0477**0.17590.0475***−0.04180.0282
Risk aversion—below average0.15850.0537***0.15870.0525***0.17620.0511***0.18780.0648***−0.13590.0377***−0.03770.0306
Risk aversion—low−0.04360.0717−0.02710.07040.00410.0706−0.03730.0944−0.21230.0580***−0.16790.0365***
Time preference—very high0.13380.0597**0.13230.0591**0.14620.0572**0.17550.0729**0.07880.05740.07180.0390*
Time preference—high0.04830.04350.04740.04370.05770.04110.06780.05380.05510.04210.07780.0322**
Time preference—low−0.21480.0417***−0.22280.0404***−0.20140.0380***−0.28380.0531***−0.01860.0476−0.17000.0271***
Time preference—very low−0.33610.0358***−0.33060.0357***−0.31050.0336***−0.44800.0451***−0.11800.0377***−0.22880.0246***
Bequests—very important0.01300.04550.01410.04560.01440.04280.01280.05660.08670.0404**−0.00500.0302
Bequests—important−0.03200.0386−0.03210.0382−0.02780.0351−0.05010.04710.03310.0329−0.03740.0272
Bequests—not important0.08960.0509*0.08660.0504*0.08060.0473*0.11780.0629*0.06800.04310.03370.0342
No. of children0.00420.01490.00520.01510.00250.01410.01070.01960.00790.01250.00340.0095
Married0.08070.0363**0.08080.0364**0.08280.0342**0.09000.0459**0.15960.0321***0.03560.0255
Black0.04110.04860.05090.04880.04650.04480.06320.06410.01730.03000.04200.0374
Hispanic−0.05180.0482−0.07540.0432*−0.06360.0409−0.06000.0662−0.03120.0353−0.02060.0264
Retirement income—very satisfactory−0.02060.0590−0.02420.0583−0.02000.0569−0.03190.0711−0.07930.0553−0.02200.0383
Retirement income—satisfactory0.07470.05290.06850.05220.05700.04940.09510.0648−0.08670.0366**0.08900.0412**
Retirement income—inadequate−0.02400.0431−0.02830.0435−0.02130.0413−0.03020.05470.05030.0426−0.03500.0282
Retirement income—totally inadequate−0.08220.0401**−0.09560.0396**−0.08630.0366**−0.12170.0487**0.07190.0410*−0.06220.0267**
((House / Risky total) − 0.54) * Owns risky total−0.32600.0541***−0.32300.0534***−0.30000.0513***−0.38720.0667***−0.38040.0602***−0.11890.0386***
Constant−1.74850.1654***−1.71280.1651***−1.60270.1599***−1.94930.2037***−1.83560.2103***−0.82490.0999***
N 296929692969296929692969
Adj. R20.20690.20840.21300.19380.18620.1490

Compared to the baseline calibration, welfare costs in the case of risky pensions are slightly lower. Lower welfare costs result from the slightly lower benchmark share of risky assets in this calibration—that is, the gap to the optimal share becomes slightly smaller. Regression coefficients for the welfare cost regression (see Table 8, column 2) are similar to the baseline calibration.

With perfectly correlated labour income, the benchmark risky asset share decreases more strongly from 92% to 89% (see Table 7), resulting in lower welfare costs (sample mean decreases from 0.48% to 0.44%). Regression coefficients in the welfare cost regression are also similar to the baseline calibration (see Table 8, column 3).

Assuming a higher discount rate for future labour income when normalizing welfare costs leads to higher welfare costs, because the denominator in equation (8) becomes smaller. Mean welfare costs increase from 0.48% to 0.61% (see Table 7). Welfare costs are again hump-shaped in age and peak as well at age 57. The slope of the increase until that age implied by the age and age2 coefficients, however, is now steeper. The steeper slope originates from the fact that with a higher discount rate, an increase in age—that is, a decrease in the number of periods and incomes to discount—has a smaller impact on the change in the discounted value of labour income. The denominator in equation (8) grows slower in age given a higher discount rate, resulting in age having a less moderating impact on the normalized welfare costs. The other regression coefficients in the welfare cost regression are similar to the baseline calibration (see Table 8, column 4).

The risky asset return

In the next robustness check, the calibration of the risky asset return is examined. For the calibration of the return distribution, historical data were used. This approach may overstate the prospective premium over the risk-free rate because economic conditions may have become less risky (see Mehra and Prescott 2003). This would imply taking a too-optimistic view of future capital market conditions. Furthermore, no sophisticated stochastic processes for the risky return, or catastrophic scenarios like a financial crisis, were incorporated. This too could lead to an overly optimistic outlook for capital market conditions. To get an idea of how sensitive the results are to different assumptions about the risky return distribution, a variant of the model is calculated in which the expected return for the risky asset is reduced by 100 basis points (labelled ‘Risky return minus 100 BPS’). Corresponding summary statistics are given in Table 7, and welfare cost regression results are shown in Table 8, column 5. Welfare costs in this calibration decline substantially. The mean welfare costs decrease from 0.48% to 0.24%. This decrease is caused by the now lower optimal risky asset share in the benchmark model, which reduces the gap to the optimal risky asset share for most households. Over the whole sample, the resulting benchmark model mean optimal risky asset share (82%) is now closer to the SCF data (68%) than in the other calibration variants. The match to the data is also closest over the whole age range of households. This is shown in Panel A of Figure 3, where the risky asset shares of the SCF data and the benchmark model risky asset shares for the various calibration variants are plotted for different ages. Thus, extending the results of Gomes and Michaelides (2005), it is found that a lifecycle model accounting for heterogeneity in preferences is able to improve the match with the SCF data, here especially with respect to the risky asset share variation in age.

image

Figure 3. Mean risky asset shares. Notes: Means are calculated using the SCF sample weight. Calibrations are distinguished by including all risky assets (Panel A) or only liquid risky assets (Panel B), and are defined as follows. Baseline calibration = baseline values according to Table 1; Risky pensions = standard deviation of pension income equals work life standard deviation; Correlated labour income = correlation between risky asset and labour income return equals one; Risky return minus 100 BPS = expected risky return is 100 basis points lower; Optimal risk aversion = lifecycle model relative risk aversion parameter is varied between 1 and 4, chosen as the value minimizing welfare costs; Financial assets = analysis is based on the households' liquid assets, i.e. excluding housing wealth from the analysis. The risky asset share in the calibration with a higher discount rate for labour income in the welfare cost calculation equals the baseline calibration and is thus not plotted.

Download figure to PowerPoint

In the calibration variant with a smaller expected risky return, there is a lower tendency for underinvestment in risky assets. Thus some of the regression coefficients, especially the coefficients for risk aversion in the welfare cost regression, change their sign, possibly now indicating welfare costs from overinvestment. Still, however, the positive impact of wealth on the welfare costs remains, and becomes even stronger (compare Table 8, column 5).

Preference parameters

Next, the impact of the choice of the household's preference parameters in the benchmark model on the robustness of results is examined. Initially, preference parameters (relative risk aversion, time preference and strength of the bequest motive) were chosen according to the household's answers to qualitative survey questions. Although there is some evidence that the qualitative survey questions such as the one used in the SCF are related to household behaviour (e.g. Laitner and Juster 1996; DeVaney and Chiremba 2005; Kimball et al. 2008; Gouskova et al. 2010; Kapteyn and Teppa 2011) and correlate with quantitative survey measures (Guiso et al. 2011), the information provided by the survey measures may still be very noisy. In this robustness check, the impact of selecting the wrong parameters is examined. The approach taken is to select preference parameters that minimize the resulting welfare costs. The robustness check focuses on the relative risk aversion parameter γ. This parameter is selected for two reasons. First, among the three preference parameters, this parameter shows the most potential for misspecification in the risky share regression, as estimated coefficients signs were the opposite of predictions (compare Table 4). Second, this parameter shows the strongest impact on the optimal risky asset share in the benchmark model.

The difference between the resulting sample average of the optimal risky asset share for the highest and the lowest parameter value selected is 13.2 percentage points. For the time preference parameter the difference is 6.3 percentage points, and for the bequest parameter it is 2.1 percentage points. Choosing the value for γ in the benchmark model in order to minimize welfare costs first of all improves the fit to the SCF data. While the correlation of the values in the original calibration to the risky asset share in the SCF data was −0.0747 (p-value 0.0001), the correlation of the optimally selected parameter is −0.1099 (p-value 0.0000). In 71% of the cases, the optimal parameter differs from the original choice, with upward revisions in 59% and downward revisions in 41% of the changes. The resulting mean welfare costs in this calibration decrease from 0.48% to 0.31% (compare Table 7). The regression coefficients in the welfare cost regression with respect to the main determinants identified, however, still show similar tendencies: again welfare costs are hump-shaped in age and increasing in assets, thus proving the robustness of the regression results (see Table 8, column 6).

The treatment of housing as risky asset

The final numerical robustness check examines the treatment of housing. Throughout the analysis, it was argued that since housing wealth is risky, it should be included in the risky assets of the household. Housing wealth is different in some aspects from other risky assets. It is less liquid, and it may provide the household non-pecuniary services. To some extent, the welfare costs regressions already control for the housing decision not being explicitly in the benchmark model by including a variable that picks up heterogeneity in the share of housing wealth within the risky asset holdings.

To check robustness of the model further, a variant is calculated where the emphasis is put only on the liquid wealth of the household. In particular, housing wealth is excluded from the household's assets. The wealth variables in the risky share regression and the benchmark model now refer to the household's financial assets (percent risky financial), and the risky return distribution uses the historical parameters from stocks. Housing wealth enters the analysis only at two points. First, the normalization of the welfare costs according to equation (8) is still being done with respect to the household's total assets, thereby making it easier to compare the welfare costs on the same scale used in previous calculations. Second, the welfare cost regression includes a new variable measuring the importance of housing (the share) within the household total assets to control for systematic effects of not modelling the housing wealth component at all. The respective results of the risky share regression and welfare cost regression are given in Table 9, summary statistics on the welfare costs are given in Table 7, and the respective age profile of welfare costs is shown in Panel B of Figure 1.

Table 9. Determinants of the Risky Asset Share and Welfare costs—Only Liquid Assets Considered
Dependent variablePercent risky financialWelfare costs
Coeff.Boot. S.E.Coeff.Boot. S.E.
Notes
  1. This table presents the results from a Tobit regression of the risky share of liquid assets (Percent risky financial) and from an OLS regression of the welfare costs in percent (WCO) on household characteristics in the same model calibration. Standard errors in the welfare cost regression are bootstrapped (10,000 replications). Variables are defined in Table 3.

  2. *, **, *** denote statistical significance at the 10%, 5%, 1% level, respectively.

Age0.00710.0027***0.14800.0246***
Age2−0.00010.0000***−0.00140.0002***
Education—low 0.32550.0871***−1.39190.4705***
Education—high−0.02650.04880.65580.4830
Age * Education—low−0.00720.0016***0.02370.0085***
Age * Education—high0.00080.0009−0.01250.0088
Occupation—retired −0.04700.0229**0.44020.2389*
ln(Financial assets)0.00310.0002***0.01640.0019***
Labour income/Financial assets−0.00580.0006***0.00050.0009
Percent debt−0.00110.0018−0.03200.0104***
Risk aversion—high−0.12260.0179***−0.47190.1702***
Risk aversion—below average0.05710.0188***0.09840.2164
Risk aversion—low0.09740.0358***1.21590.4788**
Time preference—very high−0.01350.02220.53460.2656**
Time preference—high−0.00010.01890.32910.2071
Time preference—low0.01380.0246−1.06230.1947***
Time preference—very low−0.00460.0232−1.69890.1589***
Bequests—very important−0.02680.0189−0.24930.2044
Bequests—important−0.04980.0187***−0.28640.1850
Bequests—not important0.01950.02190.88280.2318***
No. of children  0.05730.0683
Married  0.42160.1597***
Black  0.17060.2315
Hispanic  −0.13510.2178
Retirement income—very satisfactory  −0.32410.2569
Retirement income—satisfactory  0.57370.2620**
Retirement income—inadequate  −0.33420.1930*
Retirement income—totally inadequate  −0.52080.1846***
House/Total assets  −1.08220.1871***
Constant−0.08300.0716−2.11300.7422***
N 29692969
McFadden R2/Adj. R20.40730.2087

The risky share regression coefficients show similar signs as in the original calibration, except that the risk aversion coefficients now have signs in line with the predictions. The mean welfare costs in this calibration are high, increasing from 0.48% to 2.36%. This strong increase is driven by the now larger difference between the optimal risky share in the benchmark model and the actual share (referring now only to liquid risky assets) in the SCF data (compare Table 7). In consequence the gap to the optimal risky assets share is wider, leading to higher welfare costs. In comparison to Cocco et al. (2005), who, with a similar calibration, compare welfare costs of following asset allocation rules of thumb versus optimal behaviour, the welfare costs found here are smaller. The reason is because the current model incorporates more flexibility and heterogeneity, such as with respect to households' preference parameters. The main factors driving welfare costs, however, are robust in this specification; again welfare costs are hump-shaped in age (compare Panel B of Figure 1) and increase in the household's wealth (compare Table 9).

Overall, the numerical robustness checks confirm that welfare costs are small. The average welfare costs depend on the chosen model, and range between 0.24% and 0.61% in models with housing as risky asset, and amount to 2.36% in a model considering only liquid assets. The general tendencies with respect to household-specific covariates, especially the increasing welfare costs for higher household's financial endowment, are robust with respect to the assumptions made.

V Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

Motivated by the increased reliance of pension plans on privately managed retirement funds, we estimate households' welfare costs stemming from investing a suboptimal share of wealth into risky assets. Using micro-level data from the US Survey of Consumer Finances, we calibrate a lifecycle consumption and asset allocation model that incorporates heterogeneity in household endowments (income and wealth), risk aversion, time preference and bequest motives.

The model gives in certain calibration variants a fairly good match to the hump-shaped age profiles of the risky asset share found in the SCF data. Welfare costs of suboptimal asset allocations are small on average. In model calibrations treating housing as a risky asset, they are below 1% of a household's financial resources (including human capital), and in a calibration considering only liquid assets, they are below 3%. Particular population subgroups with comparatively larger welfare costs are identified. Households that would benefit most from better asset allocations are characterized by middle age brackets and high wealth. Overall, these results suggest that although asset allocation mistakes are present in the data, the incentives for households to correct their choices are modest.

A topic for future research is determining which strategy or public policy measure—for example, improvements in financial literacy (see, for example, Lusardi and Mitchell 2007; van Rooij et al. 2011) or use of asset allocation default options in pension plans (see, for example, Beshears et al. 2009)—should be implemented to reduce welfare costs. Combining knowledge about the effectiveness of various improvement strategies with the results on the determinants of welfare costs as estimated in this paper could help considerably in improving asset allocation behaviour.

Appendix: Solving Technique for the Lifecycle Model

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

The optimization problem of equations (3) to (6) is solved backwards via stochastic dynamic programming. The Bellman equation for this problem depends on three state variables: time t, wealth Wt, and labour income Lt. The Bellman equation (with V denoting the value function) is given for t = 0, 1, …, T - 1 by

  • display math(A1)

subject to the constraints of equations (4) to (6).

In the last period, remaining wealth is consumed (or bequeathed), and the value function is given by UT(WT). For CRRA utility, the Lt state variable often can be eliminated by dividing Wt by Lt (see Carroll 2004). The econometric results show that households do not behave exactly according to CRRA. Thus when integrating empirical asset allocations into the model (depending on both state variables), the Lt state variable cannot be eliminated. Technically, equation (A1) is solved by referring only to the Wt state. The Lt state is considered implicitly in that equation (A1) is solved for each household separately, thus referring to each household's expected labour income path.

The Bellman equation cannot be solved analytically, hence a numerical technique is used. First, at each point in time t, the Wt state space is discretized into I points inline image , with i = 1, 2, …, I. The upper and lower bounds of this inline image grid are chosen to be non-binding in all periods. The distributions of the risky return Rt and the labour income Lt are discretized using Gaussian quadrature methods. Since in the last period (i.e. at t = T), the value function VT(WT) is given by UT(WT), the numerical solution algorithm starts at the penultimate period (i.e. at t = T − 1). For each inline image , equation (A1) is solved with the Mathematica® 8.0 implemented non-linear optimizer NMaximize, yielding the optimal decisions inline image , inline image , inline image , and the function value of inline image . Next, a continuous function is fitted to the points inline image , which delivers a continuous approximation of the value function Vt(Wt). The fitting algorithm used guarantees that relative risk aversion displayed by optimal decisions inline image , inline image , inline image is inherited to the approximation of the value function Vt(Wt). Finally, the problem is rolled back to the preceding period.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References

The authors thank the editor, Kevin Sheedy, and the anonymous reviewers for their constructive guidance through the review process. For their comments and suggestions we thank Mike B. Adams, Pierluigi Balduzzi, Marie-Eve Lachance, Alex Michaelides, Hato Schmeiser, Paul Sengmüller and seminar participants at the University of Hannover, Annual Meeting of the American Risk and Insurance Association, Annual Conference of the German Insurance Science Association, Risk Theory Society, Netspar Pension Workshop, Annual Congress of the European Economic Association, Annual Meeting of the Verein für Socialpolitik, University of Amsterdam, Humboldt-Universität zu Berlin, Goethe-Universität Frankfurt, University of Wisconsin-Madison, University of St Gallen, Symposium on Finance, Banking, and Insurance, Annual Meeting of the American Economic Association, Copenhagen Business School, Maastricht University, Chinese University of Hong Kong, University of Waterloo, University of Mannheim, SAVE Conference.

Notes
  1. 1

    This approach follows in principle the model variant considered in Davis et al. (2005) where the realistic borrowing costs assumption is used. Lowest possible labour income is taken from the Gaussian quadrature methods discretized distribution of labour income (see the Appendix) and equals 65% of expected labour income, which is received with a probability of 5%.

  2. 2

    Thus it is assumed that the capital market in general is in equilibrium, and overall shares in the economy for equity and real estate reflect the composition of the market portfolio. Data on equity and real estate shares are taken from Federal Reserve Flow of Funds Accounts of the United States, data from 2003 to 2007, Tables B100 and B100e (see www.federalreserve.gov/releases/z1/20080306, accessed 4 July 2013). Following the SCF-based analyses of Bertaut and Starr-McCluer (2002) and Wachter and Yogo (2010), risky bonds (i.e. corporate and foreign bonds) are included in equity. The 46% share of equity is then given by the sum of: directly and indirectly held corporate equities, and mutual fund shares and corporate and foreign bonds.

  3. 3

    The values for real returns and annual costs fall within the range of estimates reported in Campbell and Cocco (2003), Cocco (2005), Yao and Zhang (2005), Cauley et al. (2007), and Li and Yao (2007). Transaction costs are annualized based on data in Cocco (2005), who reports costs of 8% per trade and an annual probability for trading of 5.44%.

  4. 4

    In particular, the real growth rates depend on a third-order age polynomial. For low education, the coefficients for the age polynomial (constant, age, age2/10, age3/100) are −2.1361, 0.1684, −0.0353, 0.0023; for middle education they are −2.1700, 0.1682, −0.0323, 0.0020; for high education they are −4.3148, 0.3194, −0.0577, 0.0033 (Cocco et al. 2005).

  5. 5

    The literature discusses whether shocks to labour income are permanent, transitory, or both (e.g. as in Vissing-Jorgensen, 2002; Cocco et al. 2005). Recent empirical evidence gives rather mixed results on this question (Guvenen 2007). The modelling choice was made in favour of the computationally less intense income process as standard state reduction techniques are not applicable here when solving the lifecycle model (see the Appendix).

  6. 6

    As a robustness check, a likelihood ratio test was used to compare the performance of the one-stage Tobit specification versus an alternative two-stage probit/truncated OLS regression. The Tobit model was not rejected at the 1% level.

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  1. Top of page
  2. Abstract
  3. Introduction
  4. I Optimizing Asset Allocations and Wealth over the Lifecycle—Setting the Asset Allocation Benchmark
  5. II Analysis of Actual Asset Allocation Behaviour
  6. III Welfare Analysis
  7. IV Model Discussion and Robustness Checks
  8. V Conclusions
  9. Appendix: Solving Technique for the Lifecycle Model
  10. Acknowledgments
  11. References
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