Differential Rates of Return and Racial Wealth Inequality

Using data on household balance sheets from the Survey of Consumer Finances and data on macroeconomic rates of return from Jordà et al. (Q J Econ. 134(3):1225–1298, 2019) we construct two alternative series for household rates of return by race from 1989 to 2016. Our estimates suggest a persistent racial gap in the rate of return on assets between 1 and 4 percentage points. The gap in returns remains even after conditioning on demographic factors, labor market factors, credit history, portfolio composition, household attitudes toward savings, financial literacy, and inheritance—suggestive of a role for discrimination. Recentered influence function (RIF) decompositions indicate between 40 and 53%—1.2 to 1.6 percentage points—of the difference in median returns between Black and White households is unexplained by observable characteristics. A standard Oaxaca-Blinder decomposition suggests that differential rates of return can explain up to 14% of the racial wealth gap at the mean. Finally, our data on differential rates of return allow us to effectively rule out explanations for the racial wealth gap based on myopia or excessive time preference. Given observed series for consumption and rates of return, a standard lifecycle model requires Black households to discount the future less than White households in order to match the data.


Introduction
A large literature documents a significant, persistent gap in median net worth between White and Black households in the USA (Blau and Graham 1990;Altonji and Doraszelski 2005;Hamilton and Darity 2010;Williams 2017;Darity and Mullen 2020). In 2016, the median net worth of White households was 9.5 times the median net worth of Black households and 7.4 times the median net worth of Hispanic households (Authors' Calculations, Survey of Consumer Finances 2016). The racial wealth gap has important welfare implications: a lack of liquid wealth holdings translates to a large loss of consumption for Black and Hispanic households in response to negative income shocks (Ganong et al. 2020 Despite the welfare implications of the racial wealth gap, racial wealth inequality poses a puzzle for economic researchers, because racial differences in median wealth are much larger than those implied by racial differences in median income. In 2016, the median income of White households was only 1.7 times the median income of non-White households. In a standard lifecycle model of consumption and saving, the distribution of wealth should reflect the distribution of earnings (Modigliani and Brumberg 1954;Hendricks 2002;Williams 2017), a prediction inconsistent with a racial wealth gap five times the size of the racial income gap. While racial income inequality may explain part of the racial wealth gap (e.g., see Aliprantis and Carroll 2019, who provide evidence that the racial income gap is a significant factor influencing the racial wealth gap), it is not a sufficient explanation.
Various behavioral explanations for the racial wealth gap have been examined, including differences in the receipt of inheritances (Blau and Graham 1990;Menchik and Jianakoplos 1997;Gittleman and Wolff 2004), permanent income (Altonji and Doraszelski 2005), saving behavior (Altonji and Doraszelski 2005;Gittleman and Wolff 2004), and cultural differences in attitudes toward risk, financial decision making, time preference, or expectations of family support (Chiteji and Hamilton 2002;Scholz and Levine 2004;Boshara et al. 2015). With the exception of inheritances-which Menchik and Jianakoplos (1997) estimate explain 10-20% of the racial wealth gap-and differences in parental and sibling need-which Chiteji and Hamilton (2002) find explains up to 27% of the racial wealth gap-the aforementioned behavioral explanations are only weakly supported by the data. Differences in savings behavior cannot explain the racial wealth gap: conditional on income, evidence suggests Blacks save slightly more than Whites Darity and Mullen 2020). Using quantile decompositions, Dal Borgo (2019) finds that after adjusting for socioeconomic characteristics-including retirement assetsthe gap in savings rates between Black and White households disappears (although importantly, Dal Borgo (2019) suggests the ethnic gap-between White and Hispanic households-remains).
One possible explanation for the racial wealth gap not mentioned above lies in differences in the rate of return on investment across race. Variation in rates of return may arise due to differences in portfolio composition, educational attainment, financial literacy, or overt discrimination. For much of the twentieth century discriminatory treatment in Federal mortgage lending (redlining) and the use of restrictive covenants prevented Black households from accumulating housing wealth in desirable neighborhoods (Katznelson 2005;Oliver and Shapiro 2006;Rothstein 2017), thus forcibly reducing the rate of return those households could achieve. Black assets were often intentionally destroyed by White rioters, as in Wilmington, North Carolina, in 1898;Tulsa, Oklahoma, in 1921;andRosewood, Florida, in 1923 (Darity andFrank 2003;Darity Jr. 2008;Darity and Mullen 2020). Estimates indicate that Black landowners had 24,000 acres of farms and timberland stolen from them in the first three decades of the twentieth century (Darity and Frank 2003). Importantly, racial differences in access to sources of wealth are not limited to the past: Black and Latino mortgage applicants are rejected more frequently than Whites-even when credit history proxies are accounted for (Munnell et al. 1996;Charles and Hurst 2002). Black-owned firms are more than twice as likely to be denied loans as White-owned firms with similar credit scores (Blanchflower et al. 2003). Similarly, recent evidence suggests Black and Hispanic auto loan applicants face higher borrowing costs than White applicants with similar credit scores, despite lower rates of default, consistent with racial bias (Butler et al. 2020). Finally, Dymski et al. (2013) show that racial minorities were "superincluded" in the predatory subprime mortgage lending that preceded the Great Recession, inevitably impacting the rate of return on housing received by minority households. Given these and other considerations, Darity and Mullen (2020) argue that " [W]ealth is the best single indicator of the cumulative impact of White racism over time" (p. 31). Differential access to high rates of return is therefore a potentially important explanatory component of the racial wealth gap.
In this paper, we explore racial differences in rates of return in the USA from 1989 to 2016. Few studies have explicitly assessed the role differential returns play in preserving racial wealth inequality. Menchik and Jianakoplos (1997) argue that racial differences in rates of return have little effect on the racial wealth gap, although their estimate is not based on actual rates of return, instead estimating a rate of return based on household portfolio composition. Similarly, Blau and Graham (1990)-rather than using measured returns-use a simulation to conclude that differential rates of return are unimportant in explaining the racial wealth gap. 1 Using PSID data, Gittleman and Wolff (2004) find no statistically significant difference in the rate of return between Black and White households, and no statistically significant effect of rates of return on racial wealth inequality. However, Gittleman and Wolff's (2004) analysis excludes households reporting non-positive wealth, calculates rates of return as the sum of capital gains divided by initial period wealth (thus ignoring differences in interest and dividend income) and weights regressions by start-ofperiod wealth (possibly understating differences in average or median returns by race). In this paper, we overcome the limitations of previous studies in two ways. First, we measure returns directly, that is inclusive of capital income; second, we include households with non-positive net worth.
Using the Survey of Consumer Finances (SCF) we calculate household rates of return using both a direct approach and a matched balance sheet approach using data on macroeconomic rates of return from Jordà et al. (2019) we document a persistent 1-to-4 percentage point gap in average rates of return between White and Black households. The gap in rates of return remains even after conditioning on educational attainment, portfolio composition, financial background, financial literacy, and income-suggesting that racial differences in rates of return may be driven by discrimination. Recentered influence function (RIF) decompositions indicate between 40 and 53% of the Black-White gap in rates of return (1.2 to 1.6 percentage points) is unexplained by observable characteristics. Oaxaca-Blinder decompositions show that differences in rates of return explain approximately 14% of the racial wealth gap at the mean. For comparison, differences in personal credit history and inheritance explain approximately 10% and 5% of the gap, respectively. Of the factors we control for, only differences in portfolio composition consistently explain a larger portion of the racial wealth gap than differential rates of return. Finally, we use the new series on rates of return to simulate a standard lifecycle model. Given observed rates of return, the model suggests that Black households must discount the future less than White households in order to match observed patterns of consumption across race in the Consumer Expenditure Survey (CEX), effectively ruling out explanations of the racial wealth gap based on myopia or excessive time preference. Moreover, a simple welfare exercise suggests that policies aiming to equalize rates of return-such as Hamilton and Darity's (2010) Baby Bonds proposal-are welfare improving.

The Racial Wealth Gap: Stylized Facts
Fact 1 The racial wealth gap is large and persistent. Figure 1 presents series for median net worth for White, Black, and Hispanic households from 1989 to 2016, in constant 2016 dollars, 2 where net worth is measured as the difference between total assets and total liabilities held by the household. In 1989 the median real net worth of White households was approximately $136,021, the median real net worth of Black households was $37,485, and the 2 As is common practice in the racial wealth gap literature Williams 2017; Darity and Mullen 2020) we use "wealth" and "net worth" interchangeably, with the understanding that the racial wealth gap and racial gap in net worth are synonymous. Any reference to wealth in the paper should be interpreted as referring to net worth. median real net worth of Hispanic households was $18,603. In 2016, the median real net worth of each group was $164, 400, $17,300, and $22,200, respectively. Not only has there been no tendency for the racial wealth gap to decline, but the gap in wealth holdings between Whites and Blacks has increased over the sample period. From 1989 to 2016 median White wealth increased from 4 times median Black wealth to nearly 10 times median Black wealth.
Changes in the racial wealth gap in the latter part of this period are driven by the differential response by race of wealth to the Great Recession. Although White households faced a steeper initial decline in wealth over the 2007-2010 period (as a percentage of total pre-recession household wealth, losses were actually smaller for White households, as we show in "Comparison of Household Balance Sheets by Race"), net worth for White households has since recovered. In contrast, neither the net worth of Blacks nor Hispanics has recovered from the damage done by the recession.
The persistence of the racial wealth gap over time matches the persistence in racial differences in earnings observed in the literature (albeit with racial differences in wealth holdings far exceeding racial differences in income). Bayer and Charles (2018) show that-after narrowing between 1940 and the mid-1970s-the median Black-White earnings gap has since grown as large as it was in 1950.

Fact 2
The racial wealth gap increases over the lifecycle.   . Estimates in constant 2016 dollars, constructed using sample weights. Excludes households reporting zero or negative income. Excludes households reporting zero or negative assets $0 at age 21 to approximately $200,000 by retirement (age 65). The combination of Figs. 1 and 2 indicates that any explanation of the racial wealth gap must be able to account for the fact that the racial wealth gap is both persistent over time and increasing with age.
Fact 3 At observed income levels, small differences in rates of return can explain lifecycle variation in the racial wealth gap-even with equal savings rates.
Consider a basic accumulation equation for household wealth: W t+1 = s(Y t + rW t ) + W t , where W t is wealth in period t, Y t is labor income, r is the rate of return on wealth, and s is the household savings rate. Figure 3 presents simulations of the racial wealth gap over the lifecycle under alternative assumptions about income, rates of return, and savings. The figure is suggestive about the possibility of rates of return to explain the racial wealth gap. Assuming that at age 20 each race earns the average income of individuals less than 30 in the SCF ($47,879 for Whites, $37,454 for Blacks), that all incomes grow at 2% per year, and that each race saves at a constant rate of 10% out of both labor and capital income, a 1 percentage point difference in rates of return is sufficient to generate $118,298 racial wealth gap at retirement. The median income simulation (starting at the SCF values of $26,123 for Blacks and $39,184 for Whites), with the same assumptions about wage growth and differences in rates of return, delivers a wealth gap of $97,334 at retirement. Finally, with identical starting incomes (equal to $100,000) and savings rates (10%), a 3 percentage point difference in rates of return generates a $54,105 racial wealth gap at retirement. Reducing the White saving rate by 1% relative to Blacks, a 3 percentage point difference in rates of return nonetheless delivers a positive wealth gap of $49,526 at retirement age, as the gray line shows. Thus, if the gap in rates of return is persistent, the existence of differential rates of return can potentially explain both variation in the racial wealth gap by age and the endurance of the racial wealth gap over time. The estimates of the racial wealth gap under the above alternative assumptions should be taken as conservative, given the assumption of income growth at 2% per year for all races.

Data Description
Survey of Consumer Finances To obtain information on household wealth, we use the summary extract public data from the Survey of Consumer Finances (SCF) for sample years 1989 to 2016. 3 The SCF is a triennial cross-sectional survey of US families, including information on families' balance sheets, pensions, income, and demographic characteristics. It is well known that high income, high wealth households are over represented in the SCF. Despite this shortcoming, 4 the SCF remains an important source of information on household wealth, as there exists no comparably detailed public data enabling a breakdown of wealth holdings by race over time. Although other public where W t is wealth at time t, Y t is labor income, s is the savings rate, and r is the rate of return. Savings rates are assumed to be equal to 10% in equal savings simulations. We assume labor income grows at 2% per-year in all simulations. Initial income values are set equal to either the mean or median income by race for individuals under thirty. Mean income for Whites under thirty is $47,879. Mean income for Blacks under thirty is $37,454. Median income for Whites under thirty is $39,184. Median income for Blacks under thirty is $26,123. In the simulation with lower White savings, the savings rate of White households is set to 9%. In the "Identical Incomes" simulation, income of both groups is set to $100,000. Rates of return are assumed to be 0.08 for Blacks and 0.09 for Whites in the 1 percentage point gap simulation. In the 3 percentage point gap simulation, the White rate of return is increased to 0.11 surveys-such as the Panel Study of Income Dynamics (PSID)-allow the study of some dimension of wealth holdings by race over time, none are as comprehensive as the SCF. In a direct comparison of the SCF and PSID Pfeffer et al. (2015) write that "[T]he SCF is the gold standard measure of household wealth, providing detailed estimates of all asset components and more accurately representing very wealthy households, which is tremendously important given the concentration of wealth at the top of the distribution" (p. 15). Further, recent work by Saez and Zucman (2016) suggests that despite its expanded coverage of high income, high wealth households, the SCF may nonetheless understate inequality in both wealth and capital income (as measured by the top 0.1% share), relative to estimates obtained from tax data. To the extent that the SCF oversamples wealthy households from all race and ethnicity categories, estimates of the racial wealth gap based on SCF data are unlikely to be overstated. 5 Table 1 presents sample means for several key variables from the SCF. To arrive at our final sample, we drop observations reporting negative assets (for which we cannot 5 Racial differences in net worth may in fact be understated in the SCF. Steinbaum (2019) points out that individuals experiencing housing instability, incarceration, cohabitation without economic dependence, or any number of "non-traditional" living arrangements are excluded from SCF households, and therefore absent from estimates of the racial wealth gap. calculate a rate of return) and observations reporting negative income. 6 Our final sample consists of 44,468 familyyear observations, spanning the 1989-2016 period. We report sample means for the full sample, as well as separately for Black, Hispanic, and White respondents. The SCF separates respondents into four different race and ethnicity categories: White non-Hispanic, Black, Hispanic, and Other. Because it is not possible to disaggregate the otherwise heterogeneous "Other" category in the public-use version of the SCF, we limit our analysis to White, Black, and Hispanic respondents. Net worth is measured as the difference between household assets and liabilities. Throughout the paper, we use the terms "net worth" and "wealth" interchangeably. "Financial Asset Share" measures the share of household assets in the form of financial assets-such as stocks, bonds, mutual funds, retirement accounts, or checking accounts. "Non-Financial Asset Share" measures the share of household assets in the form of non-financial assets, including the household's primary residence (if owned), other real-estate, business assets, and other non-financial assets such as vehicles. The detailed portfolio composition variables reflect controls included in later regressions. In particular, we include controls for the share of the primary residence in total assets, the share of businesses in  Sample means calculated using SCF sample weights. Standard deviations in parentheses total assets, the share of stocks and pooled investment funds in total assets, the share of retirement accounts in total assets, and the share of total assets in low-yield savings accounts (such as CDs, bonds, or liquid savings accounts). The omitted portfolio categories in later regressions are thus "Other Non-Financial Assets" which include vehicles, other real-estate, and other valuables (such as gold or fine art), and "Other Financial Assets," which captures the share of household assets held in the cash value of any whole life insurance policies, the share of household assets in trusts, annuities, and other managed accounts, and other financial assets such as loans from the household to someone else, future proceeds from lawsuits, royalties, etc. The "Financial Literacy" variable-available only in the 2016 wave of the SCF-records the number of correct answers to three common financial literacy questions. 7 The variable 7 The questions are the following: (1) "Do you think that the following statement is true or false: Buying a single company's stock usually provides a safer return than a stock mutual fund?" (2) "Suppose you had $100 in a savings account and the interest rate was 2 percent per year. After five years, how much do you think you would have in the account if you left the money to grow?" and (3) "Imagine that the interest rate on your savings account was 1 percent per year and inflation was 2 percent per year. After one year, would you be able to buy." The first question is true/false, the second and third question are multiple choice with options more than/exactly/less than a given amount.
"Willing to Take Financial Risk" is a dummy variable equal to one if a respondent indicated they would be willing to "'[T]ake substantial financial risks expecting to earn substantial returns." "Saved Money in the Last Year" is equal to one if the respondent indicated they saved any money at all in the last year. "Working in Some Way" is equal to one if the respondent indicated they are in some way employed. The highly aggregated industry categories correspond to the groupings identified in the SCF summary extract public data. A description of how our rate of return estimates are constructed is contained in "Constructing Rates of Return".

Jordà-Schularick-Taylor Macrohistory Database
We obtain data on macroeconomic rates of return for different asset classes from Jordà et al. (2019). The Jordà-Schularick-Taylor Macrohistory Database (JST) provides information on three major classes of return on investment: equity total return, housing total return, and bond total return, on an annual basis for a panel of 17 countries since 1870. JST data on equity returns are constructed from a broad range of sources including articles in economic and financial journals, stock exchange listings, newspapers, yearbooks of statistical offices and central banks, and company reports. For a majority of the JST sample, estimated equity returns rely on indices weighted by market capitalization of individual stocks, selected so as to be representative of the  (2000).
To calculate the bond rate, JST use the total return on US long-term bonds from Barclays (2016). Finally, to calculate housing returns JST make use of data from Knoll et al. (2017) and Knoll (2017). Housing returns are computed as the sum of the rental yield and capital gains (measured as the growth of a country-specific house price index). To avoid large variation in average rates of return caused by aberrational movements in a single year, we take 5-year rolling averages of the equity, bond, and housing rates. As a robustness check, Appendix A2.2 provides additional results using both single-year and 3-year rolling average values for each category of return. Table 2 presents statistics on household wealth ownership by race, following the taxonomy in Dettling et al. (2017). Table 2 further details the extent of the racial disparity in wealth holding. 8 Black and Hispanic families are far less likely to report ownership in every high-value asset category-primary residence, retirement assets, business assets, and stocks and mutual funds-than White families. In the full sample, only 8% of Black families, and 6% of Hispanic families, report owning positive amounts of equity in directly held stocks, stock mutual funds, and combination mutual funds, compared to nearly 28% of White families. Further, Black and Hispanic families are more likely to report having a negative credit experience, including being more than 60 days late on payments, having a payment-to-income ratio in excess of 40%, or requiring a payday loan in the past year. Finally, although housing equity is larger as a share of total assets for both Black and Hispanic families, the average total value of housing wealth among homeowners is much less for these families-approximately $85,384 for Blacks and $108,538 for Hispanics, compared to $177,516 for Whites. Figure 4 further decomposes differences in wealth ownership by race over time by examining two key categories of assets: housing, and equity in directly held stocks, stock mutual funds, and combination mutual funds. Figure 4 illustrates that housing consistently constitutes a larger share of total assets for Black and Hispanic homeowners, although this share has fallen for Black homeowners in recent years. In contrast, corporate equities (both directly held and via mutual fund holdings) consistently make up a much smaller portion of Black and Hispanic portfolios, to the tune of three to four percentage points less than the share in White portfolios.

Comparison of Household Balance Sheets by Race
Finally, an additional dimension on which the portfolios of White, Black, and Hispanic households can be differentiated is the extent to which the value of assets in these portfolios appreciate or depreciate over time. In particulargiven the large share of housing in Black and Hispanic portfolios-a key question concerns the extent to which the value of the housing assets of each group was impacted by the 2007-2010 recession. Figure 5 plots the average change in housing net worth for each group between 2007 and 2010 as a percent of average start-of-period total net worth. The figure is illustrative of differential returns to housing wealth during the Great Recession. The average value of lost housing wealth totals nearly 10% of total pre-recession net worth for Black households, compared to only 2% of pre-recession net worth for White households. Examination of portfolio differences by race is therefore suggestive of at least two proximate channels through which Black, Hispanic, and White households may receive differential rates of return: differential ownership of high-return assets (such as corporate equities), and differential returns within the same asset class (e.g., housing).

Constructing Rates of Return
We construct household rates of return using two alternative methods. First, we calculate rates of return as the weighted average of macroeconomic rates of return from JST, where the weights are given by the shares of various assets in the household's portfolio: where i denotes households, j denotes asset type, and t denotes time period. φ ij t are the weights, given by the share of asset type j in household i's portfolio at time t. We refer to this as the "matched balance sheet approach." The crosswalk between JST rates of return and SCF asset types is presented in Appendix 1. 9 One disadvantage of the matched balance sheet approach is that all householdlevel variation in rates of return is driven by differences in portfolio composition, rather than differences in the return on specific assets. That is, rates of return for different asset classes are assumed to be constant across race (taken from the JST data which are at the country-year level), 9 For a similar application, see Ederer et al. (2020), who use the JST data to construct estimates of rates of return across the wealth distribution for a cross section of European countries. with variation in household-level returns being driven solely by portfolio composition (via the asset shares, φ ij t ). In other words, the matched balance sheet rate of return estimate merely depends on household asset composition and the aggregate ROA for a particular asset class in a particular time period. This is potentially problematic, as "Comparison of Household Balance Sheets by Race" suggests differential returns by race on the same asset are an important source of overall differential returns. Thus, we also implement a "direct" approach for calculating rates of return for each household, based on observed capital income and asset holdings in the SCF 10  Figure 6 presents a time-series plot of average rates of return using both the matched balance sheet approach and the direct approach, from 1989 to 2016. Although there are small differences across the two approaches, both series are close in value and display similar trends. Rates of return were roughly constant from 1989 to 2007, underwent a large decline during the Great Recession, and by 2016 have recovered slightly. Figure 7 presents time-series plots of average rates of return across Black, Hispanic, and White households separately for each of our constructed series for rates of return. In line with the differences in means presented in Table 1, both methods of constructing rates of return are indicative of persistent differences in the return on assets across race. For both series, the gap in returns between White and Black households and White and Hispanic households is roughly constant during the period prior to the Great Recession, falls slightly during the Great Recession, and increases in the post-recession period. 10 The value of assets in the denominator does not include the value of capital income, so the estimated return is a true rate of return, rather than a share. 11 E.g., if the reported value of unrealized capital gains on a respondent's primary residence was $100,000, and the respondent had owned the residence for 10 years, the annualized value of unrealized capital gains would be $10,000. If the respondent also owned other real estate worth $70,000, and they had owned this for 10 years, the annualized value of unrealized capital gains on that asset would be $7,000. The total annualized value of unrealized capital gains for the respondent would be the sum of unrealized gains on the primary residence ($10,000) and other real estate ($7000).

Conditional Differences in the Rate of Return
While the unconditional differences in rates of return presented in Fig. 7 are striking, the policy implications of racial differences in rates of return invariably depend on the source of such differences. Differences in rates of return may arise for a variety of reasons including differences in portfolio composition, financial literacy, credit history, education or, as already suggested above, discrimination. In this section, we run OLS regressions to estimate conditional differences in rates of return across race, controlling for factors other than race which may influence household rates of return. We note that the results of this exercise should be interpreted with caution: because racial differences in variables such as portfolio composition, credit history, financial literacy, and education may also be driven by discrimination-and are thus outcomes of the notional experiment-it would be a mistake to rule out discrimination based on a statistically insignificant regression coefficient on race in the following exercise. On the other hand, the persistence of economically and statistically significant differences in rates of return across race after conditioning on other factors-while not conclusive-would be strongly suggestive of a role for discrimination. Additionally, if the racial wealth gap is influenced by historical differences in rates of return due to discrimination, e.g., due to redlining-an undeniable fact (Small and Pager 2020)-our study of relatively recent data will necessarily miss such differences, thereby understating the impact of historical discrimination. 12 Table 3 presents results using the matched balance sheet series on returns. In each regression the dependent variable is the rate of return for household i in sample year t. We include a dummy variable for both Black and Hispanic SCF respondents, such that White respondents constitute the reference group. Column (1) excludes all controls. Column (2) adds demographic controls including age, sex, marital status, number of kids, and educational attainment. Column (3) adds labor market controls including (log) income, employment status, and industry. Column (4) adds controls for credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, whether the respondent indicates they are willing to take a financial risk, and a variable capturing whether the respondent believes it is important to leave a bequest. Our inclusion of a control capturing the self-reported importance of leaving a bequest follows Williams (2017), who includes controls for "cultural" factors potentially influencing household financial outcomes in a decomposition of the racial wealth gap. In this case, individuals with a stronger bequest motive may invest more aggressively, leading to higher returns. For similar reasons, we include a dummy variable measuring whether an SCF respondent indicated they would be willing to "[T]ake substantial financial risks expecting to earn substantial returns." Column (4) also adds year-fixed effects. Finally, column (5) presents results from only the 2016 sample, including a control for financial literacy. Although it is acknowledged that financial literacy is both conceptually co-mingled with numeracy (Skagerlund et al. 2018) and conditioned on exposure to-or familiarity with-financial services and investing (specifically, measured financial literacy may depend on the way one is socialized, particularly via informal channels such as parental influence, e.g., see, Hudson et al. 2017), we believe it important to demonstrate that differences in measured financial literacy do not explain away racial  Table 3 Conditional differences in rates of return, matched balance sheet series Robust standard errors in parentheses, *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Column (1) excludes all controls. Column (2) adds demographic controls including age, sex, marital status, number of kids, and educational attainment. Column (3) adds labor market controls including (log) income, employment status, and industry. Column (4) adds controls for credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, whether the respondent indicates they are willing to take a financial risk, and a variable capturing whether the respondent believes it is important to leave a bequest. Column (4) also adds year-fixed effects. Finally, column (5) presents results from only the 2016 sample, including a control for financial literacy differences in rates of return, and thus include it as a control in column (5).
In every specification there remain statistically and economically significant racial differences in rates of return. For both Black and Hispanic households the rate of return on assets is one to two percentage points lower than the rate of return earned by White households, even after conditioning on demographics, education, labor market factors, financial history, attitude toward financial risk, attitude toward bequests, and financial literacy. With respect to the regression coefficients estimated for the control variables, there are few surprises. Being denied credit (or indicating one fears being denied credit) is associated with a lower rate of return, indicating a willingness to take financial risk and a strong bequest motive are positively related to the rate of return, and the financial literacy variable is positively related to the rate of return (although the regression coefficients on race remain significant when this variable is included). Among the demographic controls, having a college degree, the number of children in the household, and the age of the respondent are all positively related to the rate of return. Table 4 presents results from repeating the same exercise using the direct series for rates of return. This series has the advantage of allowing controls for portfolio composition, which are absent from Table 3 due to the fact that differences in rates of return across households in the matched balance sheet series are driven solely by differences in portfolio composition, such that portfolio composition would be mechanically correlated with R Matched . In contrast, R Direct -calculated using measurements of capital income accruing to each household-does not suffer from this problem. To save on space, Table 4 suppresses results for the demographic and labor market control variables.
Column (1) once again excludes all controls. Column (2) adds demographic controls including age, sex, marital status, number of kids, and educational attainment. Column (3) adds labor market controls including (log) income, employment status, and industry. Column (4) adds controls for credit history-including whether the individual was turned Robust standard errors in parentheses, *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Column (1) excludes all controls. Column (2) adds demographic controls including age, sex, marital status, number of kids, and educational attainment. Column (3) adds labor market controls including (log) income, employment status, and industry. Column (4) adds controls for credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, a variable capturing whether the respondent believes it is important to leave a bequest, and a variable capturing whether the respondent believes it is important to leave a bequest. Column (4) also adds controls for portfolio composition, which includes an indicator variable for whether the respondent is a homeowner, the share of household assets in the primary residence, businesses, stocks and pooled investment funds, retirement accounts, and low-yield savings accounts. The omitted portfolio composition category is thus a combination of other non-financial and financial assets, including vehicles, other real estate, other valuables (such as gold or fine art), the cash value of whole life insurance policies, the share of household assets in trusts, annuities, and other managed accounts, loans from the household to someone else, future proceeds from lawsuits, and royalties. We assume a zero rate of return for assets in the omitted category. Column (5) adds year-fixed effects. Finally, column (6) presents results from only the 2016 sample, including a control for financial literacy down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, a variable capturing whether the respondent believes it is important to leave a bequest, and a variable capturing whether the respondent believes it is important to leave a bequest. Column (4) also adds controls for portfolio composition, which includes an indicator variable for whether the respondent is a homeowner, the share of household assets in the primary residence, businesses, stocks and pooled investment funds, retirement accounts, and low-yield savings accounts. The omitted portfolio composition category is thus a combination of other non-financial and financial assets, including vehicles, other real estate, other valuables (such as gold or fine art), the cash value of whole life insurance policies, the share of household assets in trusts, annuities, and other managed accounts, loans from the household to someone else, future proceeds from lawsuits, and royalties. We assume a zero rate of return for assets in the omitted category. Column (5) adds year-fixed effects. Finally, column (6) presents results from only the 2016 sample, including a control for financial literacy. Table 4 confirms that the rate of return on assets obtained by Black households is statistically significantly less than the return obtained by White households in every specification except columns (4) and (5) (which nonetheless maintain a negative sign), ranging from -1.76 percentage points in column (3) to − 4.34 percentage points in column (1). The results for Hispanic households are mixed, becoming statistically insignificant and even positive in column (5) when controls for portfolio composition are included. However, because the impact of race (and racial discrimination) on rates of return may operate through portfolio composition, one cannot rule out meaningful differences in rates of return based on insignificant result in Table 4 when portfolio composition is included. In fact, the estimated regression coefficients on the portfolio composition variables may shed light on the mechanism through which households receive differential rates of return. In particular, although being a homeowner is associated with a higher average rate of return in columns (4) and (5), conditional on owning a home the share of the primary residence in household assets is negatively related to the return the household receives. Given that housing wealth makes up a larger share of wealth for Black and Hispanic homeowners-and that the decline in housing wealth during the Great Recession (as a percentage of net worth) was greater for Blacks and Hispanics- Table 4 provides strong evidence that differential returns to housing by race play a role in explaining the gap in rates of return observed in Fig. 7. Other statistically significant elements of portfolio composition include the share of assets in businesses and the share of assets in stocks and pooled investment funds-which positively impact the rate of return, and the share of assets in retirement accounts and low-yield savings accounts-which negatively impact the rate of return.
Taken together, Tables 3 and 4 provide evidence suggestive of a role for discrimination in explaining differential rates of return across race, insofar as significant differences in rates of return by race are not explained away by controlling for a wide variety of observable factors plausibly related to the difference in returns. However, the results in this section cannot prove discrimination, nor do the results indicate what portion of differences in returns by race might be due to discrimination. Thus, in "Decomposition Analysis" we implement a decomposition analysis to asses the contribution of observable factors (such as portfolio composition) to differential rates of return by race, before assessing the contribution of differential returns to the racial wealth gap.

Decomposition Analysis
In this section we adopt a technique common in the literature on racial wealth inequality to decompose both rates of return and the racial wealth gap into their observable and unobservable components: the regression-based Oaxaca-Blinder decomposition (Menchik and Jianakoplos 1997;Altonji and Doraszelski 2005;Williams 2017). Consider the decomposition in the context of the racial wealth gap. If the wealth of Whites and Blacks (W w , W b ) can each be expressed as a function of individual characteristics (X w , X b ) and the returns to those characteristics (β w , β b ), the difference in the observed averages of White and Black wealth can be expressed as either: Depending on whether the difference is evaluated at the β w coefficients-obtained from a regression of wealth on individual characteristics in the White sample-or the β b coefficients-obtained from a regression of wealth on individual characteristics in the Black sample. This method decomposes differences in average wealth into a portion explained by differences in observables-the first term in Eqs. 3 and 4-and an unexplained portion-the second term in Eqs. 3 and 4, due to differences in the estimated regression coefficients. Following Neumark (1988), we use β coefficients from a pooled regression over both groups as an estimate for the return to individual characteristics, such that the decomposition can be alternatively expressed as: where the first term is again the portion of the racial wealth gap explained by observed characteristics, and the second term is the unexplained portion of the racial wealth gap. In each decomposition we measure wealth with the inverse hyperbolic sine transformation of household net worth (commonly used in the literature on wealth inequality, e.g., Thompson and Suarez 2015;Kakar et al. 2019). The inverse hyperbolic sine transformation is similar to a logarithmic transformation, but admits negative values, therefore allowing the inclusion of households reporting negative net worth. 13 We use the Oaxaca-Blinder decomposition in two applications: (1): to assess what fraction of differential rates of return are explained by observable characteristics-and thereby what fraction can plausibly be attributed to unobservable discrimination, and (2): to assess the contribution of differential rates of return to the racial wealth gap.
Alternatively, because decompositions at the mean may be influenced by extreme values in the distribution of either the rate of return or net worth, we follow Firpo et al. (2009) andFirpo et al. (2018) and use the re-centered influence function (RIF) approach to decompose additional moments of the unconditional distribution of net worth and the rate of return. In particular, the method described by Firpo et al. (2009)-focused on unconditional quantile regression-makes it possible to obtain the partial effects of explanatory variables on any unconditional quantile of the dependent variable. This strategy allows for a generalization of the Oaxaca-Blinder decomposition for analyzing differences in the outcome distribution across groups. 14 When decomposing the rate of return, we focus on three different quantiles in addition to the mean: the 15th percentile, the median, and the 85th percentile. When decomposing the racial wealth gap, we focus only on the mean and the median. 13 The assumed functional form is the one associated with the "asinh" command in Stata 16. In particular: Inverse Hyperbolic Sine(x) = ln{x + (x × x + 1) 1 2 }. 14 See Rios-Avila (2020) for a complete description of the RIF decomposition methodology. We implement all RIF decompositions using the "oaxaca rif" command in Stata 16. Following the suggestion of Firpo et al. (2018), standard errors for RIF decompositions are calculated using bootstrap methods. Table 5 presents the results from our decompositions of the rate of return gap between Black and White households. The suite of explanatory variables is identical to those in "Conditional Differences in the Rate of Return". The first block of results excludes controls for portfolio composition. In all decompositions of the rate of return we take the average of the "direct" and "matched balance sheet" estimates of the rate of return as the dependent variable. Full results for each component of the explained portion are presented in Appendix A2.3. Following common practice in the literature (Firpo 2017), we interpret the unexplained portion of the gap in rates of return (or "structural effects") as discrimination.

Rate of Return Decompositions
The results in Table 5 suggest that a statistically and economically significant portion of the difference in returns between White and Black households at both the median and the 15th percentile of the distribution is unexplained by observables, regardless of whether controls for portfolio composition are included as explanatory variables. The unexplained difference in median rates of return ranges from 1.2 percentage points to 1.6 percentage points, or approximately 40-53% of the difference in median returns. At the 15th percentile, nearly the entire difference in returns (73-86%) is unexplained by observables. In contrast, the unexplained portion of the rate of return gap is sensitive to the inclusion of portfolio composition controls at the mean and the 85th percentile. Absent portfolio composition controls, approximately 1 percentage point of the difference in returns at both the mean and 85th percentile is unexplained by observables. With the inclusion of portfolio composition controls, the unexplained difference at the mean becomes statistically insignificant, and the unexplained difference at the 85th percentile becomes positive. We interpret this change as indicating a larger role for portfolio composition in explaining differences in rates of return at the top of the distribution-with the caveat that if differences in portfolio composition are themselves the result of racial discrimination, this result would not rule out discrimination as an explanation for the rate of return gap. Further, the significance of the unexplained fraction of the rate of return gap at the median and 15th percentile-despite the inclusion of controls for portfolio composition-suggests that discrimination operating through some channel other than portfolio composition is responsible for persistent differences in returns at the bottom of the distribution. Table 6 presents the results from our decompositions of the rate of return gap between White and Hispanic households. Similar to the results in Table 5, the unexplained Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Explanatory variables include demographic controls-including age, sex, marital status, number of kids, and educational attainment, labor market controlsincluding (log) income, employment status, and industry, credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, a variable capturing whether the respondent believes it is important to leave a bequest, and a variable capturing whether the respondent believes it is important to leave a bequest, and year-fixed effects portion of the Hispanic-White rate of return gap is both statistically and economically significant at the 15th percentile and the median, and insignificant or positive (when portfolio composition is controlled for) at the mean and 85th percentile. At the median, between 1.3 and 2.03 percentage points of the gap in rates of return between White and Hispanic households remains unexplained, approximately 38-57% of the overall gap in rates of return. At the 15th percentile, between 0.4 and 0.5 percentage points of the gap in rates of return is unexplained by differences in observable characteristics, approximately 67-83% of the total gap at the 15th percentile. The insignificant (or positive) values of the unexplained portion of the gap in returns at the mean and 85th percentile indicates again that if discrimination is to blame for differences in rates of return further up the distribution, it is primarily though its impact on portfolio composition (or other observable characteristics). In contrast, the size and significance of the unexplained portion at the bottom of the distribution of returns is suggestive of a role for discrimination as an explanation for the racial gap in rates of return through channels other than portfolio composition. Table 7 presents the results for decompositions of the Black-White racial wealth gap. In addition to the rate of return, each decomposition of the racial wealth gap controls for a slightly expanded set of variables, including:

Racial Wealth Gap Decompositions
(1) portfolio composition: whether the household is a Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Explanatory variables include demographic controls-including age, sex, marital status, number of kids, and educational attainment, labor market controlsincluding (log) income, employment status, and industry, credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, a variable capturing whether the respondent believes it is important to leave a bequest, and a variable capturing whether the respondent believes it is important to leave a bequest, and year-fixed effects homeowner, the share of stocks and mutual funds in the household's portfolio, the share of retirement account savings in the household's portfolio, the share of lowreturn liquid savings in the household's portfolio, the share of the household's primary residence in its portfolio, the share of business assets in the household's portfolio, the share of debt holdings in credit card debt, the share of debt holdings in mortgage debt, (2) credit history: whether the household was turned down for credit in the past five years, whether the household feared being turned down for credit in the past five years, (3) saving behavior: whether the household saved at all in the last year, 15 whether 15 The savings indicator variable is not available in the 1989 wave of the SCF. The results presented here are therefore for 1992 onward. Dropping the savings variable and including the 1989 wave in the decomposition makes little difference to our results.
the household rates giving a bequest as important, and whether the household reported willingness to take financial risks, (4) inheritances: the amount of any inheritances received, whether the household expects to receive future inheritances, (5) income, (6) labor market: employment status, industry of employment, and (7) demographics: age, number of kids, marital status, educational attainment. In Appendix A2.4, we implement a separate decomposition on only the 2016 SCF wave, to allow for the inclusion of a financial literacy control. Lastly, because single-year rates of return may be over-or under-stated relative to long-run trends when the direct approach to constructing rates of return is applied (thereby potentially overstating the contribution of rates of return to the racial wealth gap as a result of large single-year movements in capital gains) we take a conservative approach and drop observations  Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. The table presents results from decompositions of the Black-White wealth gap using a Oaxaca-Blinder decomposition at the mean and a Re-centered Influence Function decomposition at the median. Estimates obtained using SCF sample weights. Columns (1) and (2) assess the contribution of differential rates of return using the matched balance sheet approach. Columns (3) and (4) assess the contribution of differential rates of return using the direct approach. Columns (5) and (6) include both estimates of household rates of return. Because single-year rates of return may be over-or under-stated relative to long-run trends when the direct approach to constructing rates of return is applied (thereby overstating the contribution of rates of return to the racial wealth gap as a result of large single-year movements in capital gains) we drop observations in the top and bottom 5% of the distribution of returns when R Direct is used. Appendix 2 presents additional results from the full sample reporting rates of return in the top or bottom 5% of the distribution of returns when R Direct is used. Appendix A2.4 presents additional results from the full sample, suggesting our findings are not primarily driven by the presence (or absence) of these potential outliers.
The decomposition results suggest that the racial wealth gap is almost entirely explained by observable household characteristics. A significant portion of the explained variation in racial wealth holdings is due to differences in rates of return. Using just the matched-balance sheet estimates of household rates of return, columns (1) and (2) suggest that differential rates of return explain between 7 and 14% of the racial wealth gap. Using just the direct estimates of household rates of return, columns (3) and (4) suggest a more modest (albeit nonetheless significant) contribution of differential rates of return to the racial wealth gap between 3 and 5%. Finally, using both estimates of the rate of return, columns (5) and (6) suggest rates of return explain between 7 and 15% of the racial wealth gap.
Consistent with earlier literature, our results also suggest a role for differences in the receipt of inheritances (between 2.8 and 5%), demographics-including education, age, and family structure (6 to 10%), and credit history (5.6 to 11.6%, also a function of discrimination). Second, we note that because portfolio composition on the asset side of the balance sheet will be correlated with either measure of rate of return (in a mechanical sense with R Matched , in an indirect sense with R Direct ) it is likely difficult to completely separate the effects of rates of return and portfolio composition in the decomposition analysis. Across the board portfolio composition consistently explains approximately 30% of the racial wealth gap. These results hardly rule out a role for rates of return however, because at least one obvious channel through which differences in portfolio composition contribute to racial wealth inequality is via their impact on the rate of return earned by the household over the long run-a featured missed in our analysis because the SCF is a repeated cross section, presenting only a snapshot of household rates of return at a given point in time, rather than following the rate of return earned by a given household over time. Finally we note that savings behavior and attitudes toward future bequests appear to explain only a tiny fraction of the racial wealth gap (between 1.1 and 3.7%). In "Simulating Euler Equations", we provide further evidence ruling out a major role for differential time preference as an explanation for racial wealth inequality. Table 8 presents results from decompositions of the Hispanic-White racial wealth gap. The results suggest a similar role for rates of return in explaining the wealth gap between Whites and Hispanics to that found in Table 7. In particular, differential rates of return appear to directly explain between 2.5 and 24% of the Hispanic-White racial wealth gap. Other than rates of return-and similar to the Black-White wealth gap decompositionsportfolio composition appears to be the most important explanatory factor in decomposing the difference between White and Hispanic net worth. However, it must be emphasized that-to the extent that one channel through which portfolio composition impacts net worth is via rates of return-the importance of portfolio composition in the decomposition exercise does not suggest differential rates of return are unimportant. To the contrary, the significance of the portfolio composition variables as an indirect proxy for the impact of rates of return in both Tables 7 and 8, in addition to the direct effect of the rate of return, suggests that differential rates of return are one of the primary drivers of the racial wealth gap for both Blacks and Hispanics.

Welfare Considerations in a Simple Lifecycle Model
In light of our empirical findings, the purpose of this and the next Section is twofold. First, we highlight the welfare implications of differential rates of return. As such, this exercise is normative in nature, and is not aimed at providing a positive theory of the economy that explains why differential rates of return persist in equilibrium: the empirical analysis above already provided evidence that discrimination is an important factor. Second, we illustrate an additional empirical application of our series on rates of return at the household level by backing out the implied rate of time preference across households via simulation exercises using a standard lifecycle model of consumption and saving.
To begin, consider a bare-bone canonical lifecycle model. The economy is populated by j = {1, . . . , N} types of households. Households live for two periods: in the first period, household j inelastically supplies labor services and earns a wage w j that can be allocated between consumption and savings for retirement. In the second period, the household retires and consumes its entire firstperiod savings. The total population is constant. With log utility u j (c j,t , c j,t+1 ) = ln c j,t + β ln c j,t+1 for householdtype j , equal discount factors β ∈ (0, 1) and householdspecific rate of return r j , the choice of consumption in the two periods is: Let α j ∈ (0, 1) be the share of type-j households in the population such that j α j = 1, and consider a social planner choosing rates of returns {r j,t+1 } N j =1 so as to maximize the weighted average of the population's indirect utilities = j α j u * j (c * j,t , c * j,t+1 ) = j α j (1 + β) ln w j + β ln(1 + r j,t+1 ) + (where ≡ ln[1/(1 + β)] + β ln[β/(1 + β)]) subject to the adding-up constraint j α j r j,t+1 =r t+1 , the average rate of return in the economy at time t + 1. It is straightforward to show that the welfare-maximizing choice involves equal rates of return r j,t+1 =r t+1 ∀j , with the implication that any difference in rates of returns in the economy is welfare-reducing. 16 Again for normative purposes, suppose next that-as argued by Hamilton and Darity (2017), Dal Borgo (2019), and Darity and Mullen (2020)-Black households tend to save more than White households all else equal. In the simple lifecycle model above, this consideration can be embedded as follows. Take any two household types {i, j } such that 0 < β i < · · · < β j < 1, consistent with 16 In a model explicitly including production with intensive production function y = f (k) with the usual properties, the planner would choose an allocation of capital across households {k j } N j =1 taking into account that r = f (k j ) and w j = f (k j ) − f (k j )k j , which delivers equal amounts of capital for every household k j =k ∀j , equal rates of return, and equal wages. Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. The table presents results from decompositions of the Hispanic-White wealth gap using a Oaxaca-Blinder decomposition at the mean and a Re-centered Influence Function decomposition at the median. Estimates obtained using SCF sample weights. Columns (1) and (2) assess the contribution of differential rates of return using the matched balance sheet approach. Columns (3) and (4) assess the contribution of differential rates of return using the direct approach. Columns (5) and (6) include both estimates of household rates of return. Because single-year rates of return may be over-or under-stated relative to long-run trends when the direct approach to constructing rates of return is applied (thereby overstating the contribution of rates of return to the racial wealth gap as a result of large single-year movements in capital gains) we drop observations in the top and bottom 5% of the distribution of returns when R Direct is used. Appendix 2 presents additional results from the full sample the parameter restrictions implied by "Simulating Euler Equations". In this case, welfare maximization requires which implies a welfare-maximizing ranking of the rate of returns r j,t+1 > r i,t+1 . Thus, normative considerations would require a benevolent planner to reward the most patient individuals with higher rates of return, as is intuitive. The next Section-that takes the presence of differential rates of return as a given and thus has a positive aimlends support to the argument by Hamilton and Darity (2017), Dal Borgo (2019), and Darity and Mullen (2020) that Black households in fact do compensate facing lower rates of return on assets through saving at higher rates. The combination of these two results and the empirical analysis amounts to the argument that the presence of differences in rates of return that cannot be explained by economic fundamentals or observable characteristics-and are largely rooted in structural discrimination-is not only unfair: it is economically inefficient.
(a) 1989 Initial Values Actual Trend in Relative Consumption

Simulating Euler Equations
In a simple log-utility model, the Euler equation with discount factor β j = 1 1+ρ j , where ρ j is household j 's rate of time preference, is sufficient to characterize household j 's utility-maximizing saving path. To examine the implications of our series for household rates of return in an infinite-horizon setting, we use the observed data on rates of return, along with data on the relative consumption of White and Black households from the Consumer Expenditure Survey (CEX), to simulate alternate paths for household consumption. In particular, we are interested in the following question: What rate of time preference would be required to match the actual trend in relative (White/Black) consumption, given the observed series in rates of return? To answer the question, we set initial values for consumption to match the ratio of average White to average Black consumption in an initial period (either 1989 or 1995) and iterate on separate consumption Euler equations for White and Black households using our observed series on rates of return, under various assumptions about the rate of time preference. In this exercise we use the matched balance sheet series for returns. Because the SCF is triennial, we interpolate values for White and Black rates of return in missing years by assuming a linear annual trend in the interval. 17 We obtain the trend series in relative consumption by applying a 17 E.g., for the years 1990 and 1991 we assume a linear trend between the rates of return observed in 1989 and 1992.
Hodrick-Prescott filter to the annual ratio of White to Black consumption in the CEX. 18 Figure 8 presents the results of our simulations. We present two alternative series: a simulation beginning in 1989, and a simulation beginning in 1995. The reason for the two starting points is that the CEX data displays a sizable decline in White/Black relative consumption between 1989 and 1995. Accordingly, the 1995 starting point is more conservative. We assume that the rate of time preference for White households is ρ w = 3%, and simulate several alternative series for relative consumption under differing assumptions about the size of the Black rate of time preference, ρ b , relative to the White rate.
Given our series on rates of return, the average rate of time preference among non-White households must be less than the rate of time preference in White households in order to match observed data on relative consumption. In particular, our results suggest that if the rate of time preference of White households is ρ w = 3%, the Black rate of time preference, ρ b , must be somewhere between 1 and 2% if our simulations are to match the data on relative consumption. This finding effectively rules out explanations of the racial wealth gap rooted in myopia or excessive time preference.
Next, we simulate the wealth gap implied by differential rates of return, taking into account our findings above on relative rates of time preference and the fact that the racial wealth gap at age 20 is basically zero. We start with equal rates of time preference (ρ = .03 for both groups) and identical initial net worth. Since the initial number is arbitrary, we take the average between median Black and Fig. 9 Racial wealth gap simulations -Euler equation. Simulations of the wealth accumulation equation W t+1 = β(1 + r t+1 )W t for the two groups with log utility and identical initial wealth median White net worth in 1989, i.e. $49,268, to start the simulation. In this scenario, a 1% rate of return differential as in Fig. 3 implies a racial wealth gap of $213,754 at retirement. This result is presented in Fig. 9, alongside the actually observed racial wealth gap.
Finally, we look at differences in the rate of time preference across races in the presence of a higher difference (3%) in rates of return. Our consumption simulations illustrate that, with a White rate of time preference of 3%, the corresponding time preference rate for Black households should be lower. We also know from Fig. 2 that the racial wealth gap at age 61 is over $320,000. With values ρ w = 0.03 and ρ b = 0.01, Fig. 9 shows that a 3% gap in rates of return generates a racial wealth gap around $320,000 at age 62. 19 Other scenarios can also be thought of: a lower difference in rates of return and a higher Black rate of time preference (but still lower than Whites) will deliver similar results, for instance. Regardless, the main takeaway is that our analysis appears to rule out a higher time-impatience of Black households-as compared to White households-as a main cause of consumption and wealth differentials across races. 20

Conclusion
Differential rates of return are an important explanatory component of the racial wealth gap. Using data on 19 Note that, while the data plot that features stochastic variation, the plots for the Euler equation are purely deterministic. These plots showcase the highly exponential nature of the solution to the Euler equation, which explains the strongly convex shape of the corresponding curve. 20 We omit simulations with zero gap in rates of return because they either would deliver a zero wealth gap (under identical time preference) or a counterfactually negative wealth gap (under Blacks being more patient than Whites, as implied by the orange line in Fig. 9). Observe further that the simulations are deterministic. household balance sheets from the Survey of Consumer Finances and data on macroeconomic rates of return from Jordà et al. (2019) we construct two alternate series for household rates of return by race from 1989 to 2016. Our estimates suggest a persistent racial gap in the rate of return on assets between 1 and 4 percentage points. The gap in returns remains even after conditioning on demographic factors, labor market factors, credit history, portfolio composition, household attitudes toward savings, financial literacy, and inheritance-suggestive of a role for discrimination. Recentered influence function decompositions indicate that between 40 and 53% (that is, 1.2 to 1.6 percentage points) of the difference in median returns between Black and White households is unexplained by observable characteristics. A standard Oaxaca-Blinder decomposition suggests that differential rates of return can explain up to 14% of the racial wealth gap at the mean. Finally, our data on differential rates of return allow us to effectively rule out explanations for the racial wealth gap based on myopia or excessive time preference. Given observed series for consumption and rates of return, a standard lifecycle model requires Black households to discount the future less than White households in order to match the data.
Two questions remain: What is the source of differential returns? And what-if any-are the policy implications of our findings? To the first, the large unexplained portion of differential returns indicated by the RIF decomposition exercise suggests a role for discrimination. The USA has a long history of discrimination against Blacks and Hispanics: asset markets are no exception. If discrimination prevents Blacks and Hispanics from purchasing homes in desirable neighborhoods, from obtaining funding to start a business, or from being hired by employers offering retirement plans with generous returns, then the rate of return on assets obtained by Black and Hispanic households will inevitably fall. Given the outsize role of housing in the portfolios of Blacks and Hispanics, structural racism in housing markets-via redlining, residential segregation, reduced access to amenities, and exposure to environmental hazards 21 -undoubtedly lowers returns. Additionally, both the SCF data and prior literature make clear that Blacks have higher exposure to asset-stripping credit products like payday loans, subprime mortgages (Dymski et al. 2013), credit cards, and high-interest auto loans (Butler et al. 2020), such that higher relative debt-servicing costs are likely to be a mechanism-outside of differential rates of returnthrough which credit-market discrimination contributes to differential wealth accumulation. Although exploring every microeconomic mechanism through which discrimination leads to lower returns is outside the scope of this paper, we note that the large and growing literature on stratification economics speaks in some detail on these issues (Stewart 2010;Darity et al. 2015;Williams 2017).
To the second question, our exercise in "Welfare Considerations in a Simple Lifecycle Model" suggests that any policy which aims to equalize rates of return will be welfareenhancing. Current policy efforts to address racialized differences in lending, borrowing, and saving suggest mixed progress toward equality. On the one-hand, recent attention has been paid to the possibility of making race an explicit criteria in Community Reinvestment Act (CRA) lending (Adams 2009), and the Federal Reserve has acknowledged racial inequality as a central economic concern (Saphir and Marte 2020). On the other hand, the Treasury Department's 2017 decision to end the myRA savings program-a key feature of which was to give workers without a defined contribution plan through their employers the opportunity to save and invest-is likely to disproportionately impact Black and Hispanic workers who are barred from employer 401(k) plans due to labor market discrimination. In any case, it is clear that more aggressive policy efforts than those currently on the table will be needed to address racial differences in rates of return. Policies such as Hamilton and Darity's (2010) Baby Bonds proposal-a progressive system (conditional on net worth) of trust funds granted upon birth, deposited in federally managed investment accounts with guaranteed returns, and accessible once the child turns 18 years of age-fit this bill. In addition to directly reducing the racial wealth gap, by helping to equalize returns across race Baby Bonds have the added benefit of preventing the reemergence of the racial wealth gap in the future. A policy of pure reparations for African Americans-as advocated by Darity and Mullen (2020)-may also help to equalize rates of return, to the extent that a large one-time transfer 21 E.g., Shertzer et al. (2016) show that when comprehensive zoning was implemented in Chicago, industrial use zoning was disproportionately allocated to neighborhoods populated by racial and ethnic minorities. enables Black households to acquire assets yielding greater returns. Beyond direct transfer policies like Baby Bonds and reparations, policy efforts designed to reverse the negative impact of structural racism on housing markets will help close the gap in differential returns. To the extent that low home values are linked to poor-performing, underfunded neighborhood schools (Bogin and Nguyen-Hoang 2014), targeted investments in public education in majority Black and Hispanic neighborhoods will boost home values, reducing both the racial wealth gap and the gap in rates of return. Alternatively, targeted public investment in environmental goods-such as brownfield clean-up, relocation of hazardous waste facilities, modernization of water-treatment infrastructure, and increased pollution regulation-will have a similar effect, to the extent that Black and Hispanic neighborhoods are disproportionately impacted by negative environmental externalities (a claim for which there is ample empirical evidence, e.g., see Hamilton 1995;Pulido 2000;Shertzer et al. 2016;Banzhaf et al. 2019 and recent events in Flint, MI).
These suggestions come with a caveat. Although a standard welfare analysis has indicated that there may be efficiency reasons to address racial differences in wealth accumulation and rates of return, these should not limit the scope of policy action on racial disparities. Justice, not efficiency, should be the litmus test for economic action on racial inequality. We conclude by echoing Darity and Hamilton's (2010) call for bold policies for economic justice to address structural racial inequality-including the gap in household rates of return. Table 9 presents the crosswalk between the JST and SCF data for the main asset categories in the SCF. For other assets, including vehicles, life insurance, transaction accounts (including checking accounts, savings accounts, and money market mutual funds), gold, silver, jewelry, antiques, future proceeds from lawsuits, mineral investments, and other miscellaneous sources of financial and non-financial wealth we assign a zero rate of return. Table 10 presents additional summary statistics of detailed portfolio composition variables for the full sample and separately for Blacks, Hispanics, and Whites.    Figure 10 also presents a comparison of the 3-year rolling average matched estimate and the direct approach estimate of the rate of return. Table 11 presents results from regressions estimating the conditional difference in returns-identical to the regressions in "Conditional Differences in the Rate of Return"using the matched balance sheet series for returns with both  2019) rates. In every case there remains a statistically significant gap in rates of return between White and Black households and White and Hispanic households close in magnitude to the estimates presented in the body of the paper. These results suggest that neither the use of a rolling average, nor the length of the window selected, are driving our main results.

A2.3 Full rate of return decomposition results
Tables 12 and 13 present the full results for the contribution of control variables to the explained portion of the Black-White gap in rates of return from the RIF decompositions in "Decomposition Analysis". Significant contributors to the gap in returns in Table 12 include the variables capturing credit history, inheritance, and income. Table 13 indicates the importance of portfolio composition in explaining differences in returns: observed differences in homeownership, the ownership of equity in stocks and pooled investment funds, and business wealth explain a significant portion of the observed difference in returns.
Tables 14 and 15 present the full results for the contribution of control variables to the explained portion of the Hispanic-White gap in rates of return from the RIF decompositions in "Decomposition Analysis". The results suggest similar factors-credit history, income, inheritance, home ownership, share of assets in stocks and pooled investment funds, and business wealth-are important contributors to the explained portion of the Hispanic-White gap in rates of return. For both the Black-White gap in rates of return and the Hispanic-White gap in rates of return, differences in portfolio composition appear to have the biggest effect at the top of the distribution of returns-consistent with the discussion in the body of the paper. Additionally, credit history (in the form of either being denied credit or reporting having feared being denied credit) appears to be a   Robust standard errors in parentheses, *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Column (1) excludes all controls. Column (2) adds demographic controls including age, sex, marital status, number of kids, and educational attainment. Column (3) adds labor market controls including (log) income, employment status, and industry. Column (4) adds controls for credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, whether the respondent indicates they are willing to take a financial risk, and a variable capturing whether the respondent believes it is important to leave a bequest. Column (4) also adds year-fixed effects. Finally, column (5) presents results from only the 2016 sample, including a control for financial literacy significant explanatory factor of differences in returns only at or below the mean.

A2.4 Additional estimates of racial wealth gap decomposition
Decomposition estimates using the direct rate of return series Table 16 presents results from additional wealth gap decompositions that do not exclude the top and bottom 5% of the distribution of R Direct . In every specification rates of return continue to explain a statistically significant portion of the racial wealth gap for both Black and Hispanics, albeit with an attenuated magnitude in some cases.

Decomposition estimates from 2016 sample including financial literacy control
Tables 17 and 18 present results from decompositions using only the 2016 SCF sample to allow for the inclusion of a control for financial literacy. The primary results of the paper remain unchanged, with the fraction of the racial wealth gap explained by differential rates of return actually increasing in some cases. In contrast, in no specification does financial literacy explain a statistically significant Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Explanatory variables include demographic controls-including age, sex, marital status, number of kids, and educational attainment, labor market controlsincluding (log) income, employment status, and industry, credit history-including whether the individual was turned down for credit in the last five years and whether the individual has feared being turned down for credit in the last five years-the value of inheritances received, a variable capturing whether the respondent believes it is important to leave a bequest, and a variable capturing whether the respondent believes it is important to leave a bequest  Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Variables the same as Table 12, with the exception of the addition of controls for portfolio composition fraction of the Black-White wealth gap or the Hispanic-White wealth gap.

A2.5 Winsorization
Tables 19 through 21 present results from our main specifications with both the rate of return and net worth winsorized at the 90th and 10th percentile. 22 Intuitively, winsorization should minimize the influence of outliers (in either rates of return or net worth) on the results of decompositions at the mean. While there are small differences in the magnitude of the estimates using winsorized values, the results remain qualitatively similar to those in the body of the paper.

A2.6 Household debt
An important consideration not explicitly addressed in the main text is the role of household debt-both as a factor influencing differential returns, and as a factor contributing to the racial wealth gap. For instance, Fagereng et al. (2020) argue that consideration of debt when measuring the returns to net worth can result in negative measured returns. Further, debt-constrained households may be limited in their investment options (or may have limited liquidity with which to invest), in such as a way as to exacerbate existing differences in access to high-return assets. Table 22 presents summary statistics for the debt-toincome ratio and liability shares for households with debt. The average debt-to-income ratio is largest for Black 22 In this case, winsorization involves replacing values above the 90th percentile or values below the 10th percentile value with the values corresponding to either the 90th or 10th percentile. households and smallest for Hispanic households. In terms of differences across race, mortgage debt accounts for the largest share of total debt for White and Hispanic households (although the share is larger for White households). In contrast, installment debt (student loan debt, auto loan debt, other revolving debt payments) accounts for the largest share of total debt for Black households. Credit card debt makes up a larger share of overall debt for Black and Hispanic households, compared to Whites.
To assess the role of household debt, we repeat estimates for (A) conditional differences in returns, (B) decompositions of the rate of return at the median, and (C) Oaxaca-Blinder decompositions of the racial wealth gap. Tables 23 through 25 present these results. The central results from the main text are unaltered. There remains a substantial gap in rates of return even after conditioning on household debt, a significant portion of differences in rates of return remain unexplained at the median, and rates of return explain a significant portion of the racial wealth gap-even after accounting for household debt explicitly.

A2.7 Additional rate of return decompositions
This section presents results for our rate of return decompositions on each rate of return measure separately, rather than taking the average of the two measures as in the main text. The decompositions are contained in Tables 26 through 29. As in "Conditional Differences in the Rate of Return", we exclude portfolio composition controls from specifications where the matched-balance sheet rate of return is the dependent variable, given the way it is constructed. Although there is variation in the fraction of the difference in returns left unexplained across Black and Hispanic groups and percentiles of the distribution of Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Variables the same as Table 12   Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. Decomposition at the mean follows the standard Oaxaca-Blinder decomposition. Other decompositions follow the RIF approach of Firpo et al. (2018). Variables the same as Table 13 returns, the results remain qualitatively similar to those obtained when using the average of the matched balance sheet and direct returns. 23 In fact, the results presented here 23 It is worth drawing attention to the fact that the largest variation occurs in the decompositions of the Hispanic-White rate of return gap using the direct measure. In particular, the fraction of the gap that remains unexplained is diminished (or even positive) when controls for portfolio composition are included in the decomposition (although the Black-White return differential shows a similar pattern when portfolio composition controls are included). As in the main text, we do not believe that this result rules out differences in returns due to discrimination. Rather, it simply begs the question of what is causing the differences in portfolio composition to arise (these differences themselves possibly the result of discrimination) that are leading to differences in rates of return?
illustrate the usefulness of taking the average of the two measures of the rate of return in previous decompositions, in that taking the average potentially mitigates bias or interpretative error caused by focusing on a single rate of return measure.

A2.8 Wealth gap decompositions without portfolio composition
Tables 30 and 31 contain decompositions of the racial wealth gap excluding portfolio composition controls. In nearly every specification the unexplained portion of the wealth gap is increased, relative to the results presented in the main text, as is the fraction explained by the rate of return.    (1) and (2) assess the contribution of differential rates of return using the matched balance sheet approach. Columns (3) and (4) assess the contribution of differential rates of return using the direct approach. Columns (5) and (6) include both estimates of household rates of return Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. The table presents results from decompositions of the Hispanic-White wealth gap using a Oaxaca-Blinder decomposition at the mean and a Re-centered Influence Function decomposition at the median. Estimates obtained using SCF sample weights. Columns (1) and (2) assess the contribution of differential rates of return using the matched balance sheet approach. Columns (3) and (4) assess the contribution of differential rates of return using the direct approach. Columns (5) and (6) include both estimates of household rates of return 1, **p < 0.05, ***p < 0.01. Presents results using values for the rate of return winsorized at the 90th and 10th percentile. All variables otherwise defined as in the main text  (1) and (2) present results for the Black-White wealth gap. Columns (3) and (4) present results for the Hispanic-White wealth gap. Columns (1) and (3) use the matched-balance sheet series for returns. Columns (2) and (4) use the direct series for returns. All other variables are defined as in the main text  Means calculated using SCF sample weights. Standard errors in parentheses. "Other Residential Real Estate Debt" includes land contracts, loans for residential property other than the principal residence, misc vacation, and installment debt reported for cottage or vacation homes. "Other Secured Debt" consists of lines of credit other than those secured by the primary residence. In the SCF, debt for nonresidential real estate is netted out of the corresponding assets. "Credit Card Debt Share" consists of the amount outstanding on all credit cards and revolving store accounts after the last payment. "Installment Debt Share" consists of loans for vehicles, education loans and other installment type loans (such as loans for durables or hospital bills). Finally, "Other Debt Share" includes loans against pensions, loans against life insurance, margin loans, and miscellaneous loans Robust standard errors in parentheses, *p < 0.1, **p < 0.05, ***p < 0.01. Estimates obtained using SCF sample weights. The omitted debt category is "other debt" which includes loans against pensions, loans against life insurance, margin loans, and miscellaneous loans   Robust standard errors in parentheses. *p < 0.1, **p < 0.05, ***p < 0.01. The table presents results from decompositions of the Hispanic-White wealth gap using a Oaxaca-Blinder decomposition at the mean and a Re-centered Influence Function decomposition at the median. Estimates obtained using SCF sample weights. Columns (1) and (2) assess the contribution of differential rates of return using the matched balance sheet approach. Columns (3) and (4) assess the contribution of differential rates of return using the direct approach. Columns (5) and (6) include both estimates of household rates of return. Because single-year rates of return may be over-or under-stated relative to long-run trends when the direct approach to constructing rates of return is applied (thereby overstating the contribution of rates of return to the racial wealth gap as a result of large single-year movements in capital gains) we drop observations in the top and bottom 5% of the distribution of returns when R Direct is used. Appendix 2 presents additional results from the full sample