Estimation and impact of gender differences in risk tolerance.

Neelakantan, Urvi

I. INTRODUCTION

Are women systematically less risk-tolerant than men? The answer has critical implications for the financial well-being of women (Schubert et al., 1999). For example, corporations may pass women over for promotion if it is believed that they are unable to make risky financial decisions (Johnson and Powell, 1994). Financial advisers, believing that women are less risk tolerant, may recommend conservative portfolios that have lower returns (Wang, 1994).

This paper uses a simple model of individual portfolio choice to estimate the distributions of risk tolerance for women and men. Moments of the model are matched to data on Individual Retirement Accounts (IRAs) from the Health and Retirement Study (HRS) to obtain estimates of the distributions' parameters. Under the assumption of constant relative risk aversion (CRRA) utility, results show that the mean risk tolerance is 0.25 for women and 0.28 for men. Economists generally assume that risk aversion, the reciprocal of risk tolerance, falls in the 0-10 range (Jagannathan and Kocherlakota, 1996). The estimates obtained in this paper are consistent with this assumption.

The impact of the results is gauged in the context of gender differences in wealth accumulation. Less risk-tolerant individuals are more conservative investors. Conservative investments can lead to low levels of wealth accumulation and may account for the gender gap in wealth (Lyons et al., 2008). Data on older Americans from the HRS show that men have 103% more wealth in their IRAs than do women. (1) The results from this paper show that gender differences in risk tolerance alone would lead to men accumulating 10% more wealth than women. Gender differences in risk tolerance can thus account for nearly 10% of the gap in wealth accumulation. By comparison, gender differences in earnings can account for 51% of the wealth gap.

The results have implications for recent economic and policy changes that are making individuals increasingly responsible for their own financial security. Financial security depends greatly on the ability to accumulate adequate retirement wealth (Wolff, 1998). The results suggest that while the difference in risk tolerance does contribute to the gender gap in wealth accumulation, the role of the gender gap in earnings remains a primary concern.

II. LITERATURE REVIEW

Gender differences in risk tolerance have been observed in responses to survey questions and risk-tolerance instruments. The National Longitudinal Survey of Youth 1979 (NLSY79) and the HRS both include a series of questions about choosing between two jobs, one that pays respondents their current income with certainty and the other that has a 50-50 chance of doubling their income or reducing it by a certain fraction. Analyzing the HRS question, Barsky et al. (1997) found that men tended to be somewhat more risk-tolerant than women. Spivey (2008) corroborated this finding using the NLSY79. Grable and Lytton (1998) and Sung and Hanna (1996) studied the responses to a subjective risk-tolerance question in the Survey of Consumer Finances (SCF) and concluded that women were significantly less risk-tolerant than men. Grable (2000) used a used a 20-item instrument to assess risk tolerance among faculty and staff at a university and also found that women were less risk-tolerant than men.

Researchers have also made inferences about women's risk tolerance based on observed wealth accumulation and investment choices. Using SCF data, Jianakoplos and Bernasek (1998) found that as individuals' wealth increased, the proportion of wealth held in risky assets increased for both men and women, but the effect was significantly smaller for single women. They concluded that single women were less risk-tolerant than single men. Dwyer et al. (2002) found that women took less risk than men in their mutual fund investment decisions. However, the impact of gender on risk-taking was reduced significantly when controlling for the investor's financial knowledge.

A related literature has studied gender differences in portfolio allocation within retirement plans without necessarily attributing the difference to risk tolerance. Bernasek and Shwiff (2001) collected data from faculty members employed at five universities in Colorado and found that women invested a significantly lower percentage of their defined contribution plan in stocks. Using data from federal workers in a government-sponsored Thrift Savings Plan, Hinz et al. (1997) found that, compared to men, a smaller fraction of women participated in an equity fund and invested a smaller share of their assets in that fund. Sunden and Surette (1998) used SCF data to show that single women were less likely than single men to invest their defined contribution plan mostly in stocks. In contrast to these papers, Papke (1988) found no effect of gender on investment choices in the National Longitudinal Survey (NLS) of mature women. Rather, people who were able to choose their investments put 14 percentage points more in stocks than people with no choice.

The approach of this paper is to use observed gender differences in portfolio allocation to estimate the distributions of risk tolerance for men and women. The next section describes the theoretical model that is used to derive the estimates.

III. THEORETICAL FRAMEWORK

The theoretical framework that is used to estimate the distribution of risk tolerance follows the exposition of Jagannathan and Kocherlakota (1996) of the classic Merton-Samuelson framework (Merton, 1969; Samuelson, 1969).

In each time period t, individual i has wealth [w.sup.i.sub.t]. Individuals must decide how much of their wealth they wish to allocate to a risk-free asset, [b.sup.i.sub.t], that earns a certain return, [r.sup.m], and a risky asset, [s.sup.i.sub.t], that earns a stochastic return, [[??].sup.s.sub.t] ([theta]), where [theta] denotes the state of nature. The objective of the individual is to maximize expected utility, u([w.sup.i.sub.T]), where [w.sup.i.sub.T] is the wealth accumulated by period T. The objective function can be written in this way--ignoring the individual's consumption-savings decision--because it is assumed that u(*) is a CRRA utility function, that returns [[??].sup.s.sub.t] ([theta]) are independent and identically distributed over time, and that there are no short-sale constraints. Under these assumptions, the portfolio allocation decision is independent of the consumption-savings decision (Samuelson, 1969).

The individual's problem in period t, given initial period wealth, [w.sup.i.sub.0], and interest rates, [[??].sup.s.sub.t] and [r.sup.m], is to choose savings in the risk-tree asset, [b.sup.i.sub.t], and savings in the risky asset, [s.sup.i.sub.t], to solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[gamma].sup.i] is the individual's risk aversion (i.e., 1/[[gamma].sup.i] is risk tolerance).

[FIGURE 1 OMITTED]

Let [[rho].sup.i.sub.t] = [s.sup.i.sub.t]/[w.sup.i.sub.t] be the share of the individual's wealth invested in the risky asset. It can be shown that the individual's problem is the same every period t and that the share of wealth he or she invests in the risky asset is a constant [[rho].sup.i*], that solves (2)

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Though it cannot be solved analytically, Equation (1) shows that [[rho].sup.i*] does not depend on wealth. In other words, given interest rates, the individual's investment in the risky asset depends only on his or her individual risk tolerance. The relationship is illustrated in Figure 1 for values of risk aversion, [[gamma].sup.i], ranging from 1 to 10. It shows that as risk aversion increases, the individual chooses to invest a smaller share of his or her wealth in the risky asset. (3) Equation (1) can thus be used to calculate the unique optimal value of [[rho].sup.i*] for a given [[gamma].sup.i].

IV. DATA

The data used in the estimation come from the 2006 wave of the HRS. The HRS is a longitudinal study that has surveyed older Americans every other year since 1992. (4) Using data on older Americans enables us to look at realized wealth accumulation around retirement age.

The HRS has detailed information on financial assets, though most of this information is collected at the household level. It reports the amount of money in the household's three largest IRAs and also reports to which household member these accounts belong. This paper focuses on IRAs because they are the only component of wealth that can be attributed to an individual rather than to a household, which may consist of more than one person. In 2006 the HRS reported for the first time the percentage of the IRA invested in stocks. In this paper, stocks in IRAs are treated as the risky asset.

The HRS had 18,469 respondents in 2006, of whom 5,265 had an IRA. Those who were missing information about the amount of money in their IRA account or the percentage invested in stocks were dropped, yielding a sample of 3,156 observations. Of these 1,553 were men and 1,603 were women. Table 1 describes the sample. (5)

Table 2 shows the mean and standard deviation of the share of stocks in IRA accounts for men and women. Men had an average of $193,367.30 in their IRAs while women had an average of $95,037.01. Observe that men hold a larger share of their IRA assets in stocks compared to women. The estimation described in the next section will attempt to match these key facts.

V. ESTIMATION AND RESULTS

The aim of this paper is to estimate the distribution of risk tolerance separately for men and women. Let [[gamma].sup.i.sub.m], and [[gamma].sup.i.sub.f] denote the risk aversion of an individual male and female, respectively. Assume that risk tolerance (the reciprocal of risk aversion) follows a lognormal distribution:

log 1/[[gamma].sup.i.sub.m] ~ N([[mu].sub.m], [[sigma].sup.2.sub.m])

log 1/[[gamma].sup.i.sub.f] ~ N([[mu].sub.f], [[sigma].sup.2.sub.f]).

Previous literature has shown that it is reasonable to assume that risk tolerance is lognormally distributed because it restricts risk tolerance to be non-negative and fits the data well (Kimball et al., 2008).

The parameters of the lognormal distributions of 1/[[gamma].sup.i.sub.m] and 1/[[gamma].sup.i.sub.f] are derived using the method of moments simulation. Specifically, the simulation is used to find the values of [[mu].sub.m], [[sigma].sub.m], [[mu].sub.f], and [[sigma].sub.f], to match the data in Table 2. The procedure used to calculate the parameter values for females is described in detail below. The procedure for males is identical.

The simulation begins by drawing values of [[gamma].sup.i.sub.f] from a lognormal distribution with parameters [[mu].sub.f] and [[sigma].sub.f] set to an arbitrary initial value. The value of [[rho].sup.i*]. is calculated for each [[gamma].sup.i.sub.f] by numerically solving Equation (1). (6) The mean and standard deviation of the calculated values of [[rho].sup.i*]] are computed. The process is repeated for a wide variety of values of [[mu].sub.f] and [[sigma].sub.f] until the mean and variance of [[rho].sup.i*] in the simulation match the data in the second row of Table 2.

The results are reported in Table 3. For men, the mean and standard deviation of the underlying normal distribution were found to be [[mu].sub.f] = -1.4661 and [[sigma].sub.f] = 0.5991. For women, the corresponding values were [[mu].sub.f] = -1.5879 and [[sigma].sub.f] = 0.6603. The mean and median of risk tolerance for men and women are also reported. The mean risk tolerance of women is lower than the mean risk tolerance of men.

The importance of these results can be illustrated by estimating their contribution to the gender gap in wealth accumulation. As reported earlier, the average man in the HRS has over twice the amount of wealth in his IRA compared to the average woman. How much of this gap can be attributed to the gender difference in risk tolerance? This question can be answered using sample simulations. First assume that both men and women started out with the same amount of initial wealth, [w.sup.i.sub.0]. Initial wealth can be thought of as the present discounted value of future income at time 0. From Equation (1) we can calculate that the average male, with a risk tolerance of 0.28, would invest 64% of his wealth in stocks each period. We consider 10,000 different paths of stock returns and calculate the amount of wealth accumulated over 38 years for each path. (7) Over a period of 38 years (the average number of years that males in the HRS sample worked), men would accumulate wealth equal to 6.5278 [w.sup.i.sub.0]. (8)

The same exercise is repeated for the average female with a risk tolerance of 0.25, who would invest 59% of her wealth in stocks. Over a period of 38 years, she would accumulate 5.9122 [w.sup.i.sub.0]. Men thus accumulate 10% more wealth than women. Since the actual difference in IRA wealth accumulation is 103%, the difference in risk tolerance accounts for about 10% (10%/103 %) of the gap in wealth accumulation.

To get a sense of the relative magnitude of the role of risk tolerance, the results are compared to the role played by the gender gap in income. In the sample, the average earnings of men is approximately 53% more than the average earnings of women. (9) Now assume that both men and women invest 64% of their initial wealth in stocks but men's initial wealth, [w.sup.i.sub.0], is 53% more than women's wealth, 0.655 [w.sup.i.sub.0]. This is appropriate if we think of initial wealth as the present discounted value of future earnings. From the simulation, men would accumulate 6.5278 [w.sup.i.sub.0] as before, while women would accumulate 4.2705 [w.sup.i.sub.0]. The gap persists, which means men end up with 53% more wealth than women. The gender gap in income thus accounts for 51% (53%/103%) of the gender gap in wealth.

Combining both the gender gap in income and in risk tolerance, we find that women would accumulate 3.8678 [w.sup.i.sub.0]. The exercise thus accounts for two-thirds of the actual gender gap in wealth. Other factors such as differences in the amount of time worked, which are outside the scope of the model, could account for the rest.

VI. CONCLUSION

This paper used a simple theoretical model to infer individuals' risk tolerance from observed investment choices. Parameters of the distribution of risk tolerance for men and women were estimated by matching moments of the model to data on asset allocation in IRAs. On average, women were found to be less risk-tolerant than men. The estimates were used to measure the impact of risk tolerance on the gender gap in wealth accumulation. The gender difference in risk tolerance accounted for 10% of the gap in accumulated wealth. The gender difference in earnings was found to have a more substantial impact, accounting for 51% of the gap in accumulated wealth.

The findings have implications for policies that are making individuals increasingly responsible for managing their own retirement savings. The results suggest that women might optimally choose to invest a smaller fraction of their wealth in risky assets, which might slightly exacerbate the gender gap in wealth. However, the effect of the gender difference in earnings on the gender gap in wealth is likely to remain a primary concern.

It is important to acknowledge a couple of limitations of this study. First, the sample is restricted to older Americans and previous research has shown that risk preferences may change with age. For example, Dohmen et al. (2005) find that risk tolerance decreases with age. Thus, it may not be possible to generalize the estimates from this study to the U.S. population as a whole. Second, the role of factors such as knowledge and education are outside the scope of the theoretical framework used in this paper. As a result, estimates of the preference parameter may be capturing not just gender differences in risk aversion but, more generally, differences in the taste for stocks. A more comprehensive look at the factors that affect earnings, risk aversion, and hence wealth accumulation is left for future research.

ABBREVIATIONS

CRRA: Constant Relative Risk Aversion

HRS: Health and Retirement Study

IRAs: Individual Retirement Accounts

NLS: National Longitudinal Survey

NLSY79: National Longitudinal Survey of Youth 1979

SCF: Survey of Consumer Finances

doi: 10.1111/j.1465-7295.2009.00251.x

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(1.) This is in line with previous estimates. For example, Johnson and Powell (1994) found that even among full-time workers approaching retirement age with pension coverage, men had 76% greater pension wealth than had women.

(2.) See Jagannathan and Kocherlakota (1996).

(3.) Note that it is possible for the share of the risky asset in the portfolio to exceed 100% because the risk-free asset can be used to borrow money to finance this in the first period.

(4.) The HRS is sponsored by the National Institute of Aging (grant number NIA U01AG009740) and is conducted by the University of Michigan.

(5.) It would have been ideal to analyze the portfolio allocation of never-married persons alone to ensure that allocations were not influenced by the spouse's preferences. However, as Table 1 shows, there are insufficient cases in the data to be able to do this.

(6.) For computational simplicity, a stylized distribution of returns is chosen in the solution to Equation (1). It is assumed that the return on the risk-free asset, [r.sup.b] is fixed 1% and the return on risky assets, [r.sup.s.sub.t], is either 27.03, 13, or -15.25% with equal probability. This yields a mean return of 8.26% with a standard deviation of 17.58%, which corresponds to the S&P 500 for 1871-2004. The return data are taken from http://www.econ.yale.edu/shiller/data.htm.

(7.) As before, it is assumed that the realized stock return each period is 27.03, 13, or -15.25% with equal probability.

(8.) This, and the results reported hereafter, is the median of the 10,000 simulated paths.

(9.) The ratio of men's to women's average total income is also the same.

Neelakantan: Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Champaign, Urbana, IL 61801. Phone 1-217-333-0479, Fax 1-217-333-5538, E-mail urvi@illinois.edu

TABLE 1
Descriptive Statistics

 Men Women
 (N = 1553) (N = 1603)

Age 67.2 65.3
Marital Status (%)
 Married/Partnered 85.1 66.4
 Divorced/Separated 7.1 11.5
 Widowed 5.3 19.0
 Never Married/Unknown 2.5 3.1
Race (%n)
 White 94.4 93.1
 Black 3.0 3.8
 Other 2.6 3.1
Education (%)
 < 12 years 7.6 2.0
 12 years 25.8 12.8
 13-15 years 20.5 15.9
 16 years 20.3 20.4
 > 16 years 25.2 21.8
 Unknown 0.6 27.1

TABLE 2 Share of Stocks in IRAs

 Mean Standard Deviation

Men 64.4% 41.8%
Women 59.5% 44.0%

TABLE 3
Estimated Parameter Values

 Risk Tolerance

 [mu] [sigma] Mean Median

Men -1.4661 0.5991 0.2749 0.2308
Women -1.5879 0.6603 0.2541 0.2044