Measuring welfare costs
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 .
To derive the welfare costs, equation (7) is solved for ΔW_{0}. ΔW_{0} is then normalized by the household's resources, that is, divided by the sum of assets at t = 0 and the household's discounted expected labour income according to equation (8):
- (7)
- (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 includes actual behaviour with respect to asset allocation, as predicted by the regression model. The decision with respect to consumption C_{t}(W_{t}) and borrowing D_{t}(W_{t}) 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 costsLabour income quintile | Total assets quintile |
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1 | 2 | 3 | 4 | 5 |
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Notes |
Panel A: Welfare costs in percent |
1 | 0.15 | 0.36 | 0.39 | 0.93 | 1.14 |
2 | 0.12 | 0.18 | 0.30 | 0.59 | 0.81 |
3 | 0.05 | 0.22 | 0.29 | 0.56 | 0.78 |
4 | 0.25 | 0.21 | 0.34 | 0.62 | 1.02 |
5 | 0.29 | 0.20 | 0.24 | 0.58 | 0.98 |
Panel B: Adjusted welfare costs in percent |
1 | 0.09 | 0.42 | 0.45 | 1.01 | 1.15 |
2 | 0.04 | 0.21 | 0.38 | 0.66 | 0.80 |
3 | 0.01 | 0.25 | 0.39 | 0.64 | 0.80 |
4 | 0.20 | 0.19 | 0.45 | 0.69 | 1.05 |
5 | 0.22 | 0.12 | 0.33 | 0.66 | 1.01 |
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 costsDependent variable | Welfare costs |
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Coeff. | Bootstrap S.E. |
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Notes |
Age | 0.0472 | 0.0050*** |
Age^{2} | −0.0004 | 0.0001*** |
Education—low | −0.0346 | 0.1429 |
Education—high | −0.0066 | 0.0855 |
Age * Education—low | 0.0013 | 0.0024 |
Age * Education—high | 0.0002 | 0.0017 |
Occupation—retired | 0.0680 | 0.0596 |
ln(Total assets) | 0.0916 | 0.0125*** |
Labour income/Total assets | 0.0035 | 0.0010*** |
Percent debt | −0.0002 | 0.0028 |
Risk aversion—high | −0.0787 | 0.0382** |
Risk aversion—below average | 0.1585 | 0.0537*** |
Risk aversion—low | −0.0436 | 0.0717 |
Time preference—very high | 0.1338 | 0.0597** |
Time preference—high | 0.0483 | 0.0435 |
Time preference—low | −0.2148 | 0.0417*** |
Time preference—very low | −0.3361 | 0.0358*** |
Bequests—very important | 0.0130 | 0.0455 |
Bequests—important | −0.0320 | 0.0386 |
Bequests—not important | 0.0896 | 0.0509* |
No. of children | 0.0042 | 0.0149 |
Married | 0.0807 | 0.0363** |
Black | 0.0411 | 0.0486 |
Hispanic | −0.0518 | 0.0482 |
Retirement income—very satisfactory | −0.0206 | 0.0590 |
Retirement income—satisfactory | 0.0747 | 0.0529 |
Retirement income—inadequate | −0.0240 | 0.0431 |
Retirement income—totally inadequate | −0.0822 | 0.0401** |
((House/Risky total) − 0.54) * Owns risky total | −0.3260 | 0.0541*** |
Constant | −1.7485 | 0.1654*** |
N | 2969 |
Adj. R^{2} | 0.2069 |
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.
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):
- (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 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 CalibrationsCalibration | Welfare costs in percent | Mean risky asset share |
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Mean | 25% quartile | Median | 75% quartile | SCF | Model |
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Notes |
Panel A: Welfare costs and risky asset share |
Baseline calibration | 0.48 | 0.02 | 0.16 | 0.61 | 0.68 | 0.92 |
Risky pensions | 0.47 | 0.02 | 0.15 | 0.60 | 0.68 | 0.92 |
Correlated labour income | 0.44 | 0.02 | 0.15 | 0.56 | 0.68 | 0.89 |
Alternative discount rate | 0.61 | 0.03 | 0.20 | 0.77 | 0.68 | 0.92 |
Risky return minus 100 BPS | 0.24 | 0.01 | 0.09 | 0.25 | 0.68 | 0.82 |
Optimal risk aversion | 0.31 | 0.01 | 0.11 | 0.38 | 0.68 | 0.91 |
Financial assets | 2.36 | 0.01 | 0.45 | 3.26 | 0.30 | 0.87 |
Panel B: Adjusted welfare costs in percent |
Baseline calibration | 0.51 | 0.11 | 0.25 | 0.65 | | |
Risky pensions | 0.51 | 0.11 | 0.26 | 0.63 | | |
Corrrelated labour income | 0.48 | 0.10 | 0.24 | 0.61 | | |
Alternative discount rate | 0.74 | 0.30 | 0.52 | 0.95 | | |
Risky return minus 100 BPS | 0.22 | 0.00 | 0.15 | 0.32 | | |
Optimal risk aversion | 0.32 | 0.05 | 0.14 | 0.40 | | |
Financial assets | 2.88 | 0.69 | 1.38 | 3.74 | | |
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.