Loss-aversion and household portfolio choice

Journal of Empirical Finance 17 (2010) 441–459

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Journal of Empirical Finance j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j e m p f i n

Loss-aversion and household portfolio choice Stephen G. Dimmock a,⁎, Roy Kouwenberg b,c a b c

Michigan State University, 306 Eppley Center, East Lansing, MI, 48824, United States Mahidol University, Thailand Erasmus University Rotterdam, The Netherlands

a r t i c l e

i n f o

Article history: Received 7 August 2008 Received in revised form 20 November 2009 Accepted 24 November 2009 Available online 30 November 2009 JEL classification: G11 D14 D1

a b s t r a c t In this paper we empirically test if loss-aversion affects household participation in equity markets, household allocations to equity, and household allocations between mutual funds and individual stocks. Using household survey data, we obtain direct measures of each surveyed household's loss-aversion coefficient from questions involving hypothetical payoffs. We find that higher loss-aversion is associated with a lower probability of participation. We also find that higher loss-aversion reduces the probability of direct stockholding by significantly more than the probability of owning mutual funds. After controlling for sample selection we do not find a relationship between loss-aversion and portfolio allocations to equity. © 2009 Elsevier B.V. All rights reserved.

Keywords: Portfolio choice Loss-aversion Stock market participation Limited participation Prospect theory

1. Introduction Calibrated models of household portfolio choice with standard preferences suggest that in frictionless economies almost all households should own equity (e.g., Haliassos and Bertaut, 1995; Heaton and Lucas, 1997). Empirical studies have found equity market participation much lower than is implied by these models (Bertaut, 1998; Haliassos and Bertaut, 1995; Vissing-Jorgenson, 2002). Explanations for this divergence have focused on two explanations: frictions such as background risk and investment costs,1 and non-standard preferences such as loss-aversion. Loss-aversion implies that households frame events as either gains or losses relative to a reference point, and weight losses more heavily than gains. Theoretical papers such as Ang et al. (2004), Barberis et al. (2006), Benartzi and Thaler (1995), Berkelaar et al. (2004), Gomes (2005), and Polkovnichenko (2005) show that loss-averse households will either not participate in equity markets, or they will allocate considerably less of their wealth to equities relative to households with standard preferences. In this paper we empirically examine how loss-aversion affects household portfolio choice. The DNB Household Survey conducted in The Netherlands contains a series of intertemporal choice questions based on experimental work by Thaler (1981) and Loewenstein (1988). Similar to Tu (2004), we use these questions to measure household loss-aversion. We then show that this measure has significant explanatory power for households' equity market participation decisions and the choice between mutual funds and direct stockholding.

⁎ Corresponding author. Tel.: + 1 517 432 7133. E-mail addresses: [email protected] (S.G. Dimmock), [email protected] (R. Kouwenberg). 1 See Cocco et al. (2005), Davis et al. (2006), Viceira (2001), and Vissing-Jorgenson (2002). 0927-5398/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jempfin.2009.11.005

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Following Tu (2004) we measure loss-aversion from 16 survey questions involving the speed-up and delay of payoffs, both gains and losses. With standard utility functions, people will have a single discount rate across gains and losses, and across speed-up and delay. This prediction is strongly violated in the responses to these questions. Most people require much higher compensation to delay receiving a sure gain than the amount they are willing to pay to expedite receipt of the same gain. Loewenstein (1988), Loewenstein and Prelec (1992), and Thaler (1981) show that a loss-averse individual who does not integrate payments with existing consumption plans, but rather frames the payments as gains and losses relative to a reference point, can have multiple discount rates. The predictions of these models are supported by experiments conducted by Benzion et al. (1989), Loewenstein (1988), Shelley (1993), and Thaler (1981) among others. Frederick et al. (2002, p. 370) explain the intuition behind this result as follows: “Shifting consumption in any direction is made less desirable by loss-aversion, since one loses consumption in one period and gains it in another. When delaying consumption, loss-aversion reinforces time discounting, creating a powerful aversion to delay. When expediting consumption, loss-aversion opposes time discounting, reducing the desirability of speed-up…”. Clearly a key issue for this paper is the reliability of our measures of loss-aversion. We test the internal consistency of our measures in two ways. First, as we have multiple questions designed to measure the same set of latent variables, we use Cronbach's alpha — the standard psychometric test for reliability in these cases — and find strong evidence to support the notion that similar questions are consistently measuring the same underlying concepts. Second, there are simple, logical relations which must hold between many of the responses. We find that only a small percent of household responses violate these relations. Further we show that the responses to the DNB Household Survey are both qualitatively and quantitatively similar to a large number of experimental studies. Finally, we discuss other potential explanations for the pattern of responses and show that the data reject alternative explanations. We hypothesize that our empirical measure of loss-aversion is a proxy for the level of loss-aversion that households experience when investing in actual markets. Using our loss-aversion measure we show that households with higher loss-aversion are significantly less likely to participate in equity markets. For an average household if their estimated loss-aversion coefficient increases from the 25th to the 75th percentile, the probability of owning stocks decreases by 7%, relative to the sample mean. These results are robust to controlling for a direct survey measure of risk-aversion and a wide variety of other variables used in previous studies, such as age, education, income, financial wealth, and unsecured debt. Although loss-aversion predicts household equity market participation, after controlling for sample selection we do not find a significant relationship between loss-aversion and portfolio allocations. If investors are loss-averse and exhibit narrow framing, meaning they evaluate gains and losses on securities in isolation rather than after integrating their entire portfolio, then the bundling of returns will affect the relative attractiveness of mutual funds and individual stocks. Consistent with this idea, we find that loss-aversion affects the type of equity that households hold. Households with higher loss-aversion avoid investing in individual stocks to a greater extent than they avoid mutual funds. This paper provides direct empirical evidence on the importance of loss-aversion for household decision making. To our knowledge it is the first paper to empirically measure the heterogeneity in loss-aversion across a representative sample of households and use this information to explain household portfolio choice. This paper is related to a branch of the literature on household portfolio choice that shows how psychological factors measured through survey questions can predict household equity market participation. Barsky et al. (1997) show that hypothetical questions designed to measure risk-aversion are significantly related to household stock market participation. Hong et al. (2004) show that more social households are more likely to participate in the equity market. Guiso et al. (2008) show that households that are more trusting of others have higher participation rates and allocations to equity. Puri and Robinson (2007) show that optimism and equity market participation are related. The remainder of this paper is organized as follows. Section 2 outlines the theories and hypotheses tested in this paper. Section 3 describes the data source and the variables. Section 4 presents and discusses our measure of loss-aversion. Section 5 presents the results. Section 6 concludes. 2. Theory and hypotheses Heaton and Lucas (1997) calibrate a model of a representative household's portfolio choice using standard utility functions and parameter values drawn from the US economy. They find that all households should participate in equity markets and that, in the absence of market frictions, they should allocate all of their financial wealth to equity. However, numerous empirical papers have shown that many households do not participate in the equity market, and many participants own only small amounts of equity.2 There are two broad streams of research attempting to explain these two puzzles: models based on market frictions such as participation costs, credit constraints,3 and background risk such as risky labor income4; and models based on non-standard preferences. Note that these two classes of explanations are not mutually exclusive. While our focus is on non-standard preferences, we acknowledge that market frictions play a role in determining household portfolios. However, as discussed by 2

See for example Bertaut (1998) and Vissing-Jorgenson (2002). Davis et al. (2006) show that if there are frictions in the credit markets it is possible to get non-participation and realistic portfolio allocations. VissingJorgenson (2002) presents evidence that some of the participation puzzle can be explained with fixed entry costs, which could include costs such as learning, fees, taxes etc. 4 Cocco et al. (2005), Heaton and Lucas (1997, 2000) and Viceira (2001) show that if a household's labor income has a high correlation with equity or a high standard deviation it is optimal to hold a safer portfolio. Guiso et al. (1996) and Vissing-Jorgenson (2002) empirically demonstrate that households with high background risk are less likely to participate and hold less equity. 3

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Barberis et al. (2006) market frictions cannot explain non-participation among the wealthy. Haliassos and Bertaut (1995) document substantial non-participation at all income levels, including the top 1%. Thus, while we do not dispute that market frictions are relevant and affect participation, they offer an incomplete explanation. This paper empirically tests the effect of one particular type of non-standard preferences, loss-aversion, on the portfolio choices of households. Prospect theory, introduced in Kahneman and Tversky (1979), differs from the standard expected utility model in four ways: 1) Individuals frame events as gains and losses relative to a reference point, representing the status quo or an aspiration level. 2) Individuals are loss-averse meaning that losses are weighted about twice as heavily as gains.5 3) Individuals are risk averse in the region of gains and risk seeking in the region of losses. 4) Individuals use subjective probability weights that overweight small objective probabilities. This paper is concerned with the first two features of prospect theory — loss-aversion relative to a reference point. Many authors use the term loss-aversion to mean a combination of features one, two and three, including the convex–concave shape of the utility function. In this paper loss-aversion refers only to the sudden increase in the slope of the value function for payoffs that fall below the reference point. Ang et al. (2004), Barberis et al. (2006), Gomes (2005), and Polkovnichenko (2005) show that stock market non-participation can be explained by loss-aversion. If households are loss-averse over stock market fluctuations the potential pain from stock market declines outweighs the pleasure from gains even with a high equity premium.6 These theoretical papers lead to a clear, one-sided hypothesis for our study; greater levels of loss-aversion will result in a lower probability of participating in the equity markets. Ang et al. (2004), Benartzi and Thaler (1995) and Berkelaar et al. (2004) further show that loss-aversion will lead to lower portfolio allocations to equity. The increased sensitivity to losses decreases the benefit of owning equity and as a result lossaverse households allocate less of their wealth to equity. We argue that loss-aversion will also affect the type of equity that households choose to hold. Barberis and Huang (2001) derive a model in which investors are loss-averse and frame narrowly, deriving utility directly from the gains and losses of individual equity securities. With these preferences households with higher loss-aversion will prefer mutual funds over individual stocks, as mutual funds integrate the returns of many securities into a single package. This encourages investors to frame returns at a broader level than does ownership of individual stocks, making mutual fund ownership relatively more attractive for loss-averse investors. Consider the following simple example. An investor with a loss-aversion coefficient of 2.5 has an equally weighted portfolio of three equity securities. Suppose that in a given year two of the stocks increase in value by 10% and one stock decreases in value by 10%. The value of the portfolio has increased by 3.3% and an investor who frames at the level of the portfolio will have a gain from investing in stocks. However, if the investor frames at the level of individual stocks the pain from the single loss outweighs the pleasure of the two gains. Since mutual funds package equities in a way that encourages framing at the portfolio level, loss-averse investors will prefer to own mutual funds rather than individual stocks. Even if investors do manage to integrate all portfolio gains and losses, i.e. if investors frame at the portfolio level, we would still expect loss-averse individuals to prefer mutual funds over portfolios of individuals stocks. The reason is that in practice individual investors typically hold under-diversified portfolios, see for example Ivkovic et al. (2008). The excessive unsystematic risk in a typical under-diversified individual stock portfolio will lead to occasional large losses, which are very painful for loss-averse investors. 3. Data The data source for this paper is the CentERdata DNB Household Survey, a household survey conducted by CentERdata at Tilburg University in The Netherlands.7 We use this dataset because it contains information about household wealth, income, and financial assets, as well as a set of questions we can use to extract a measure of loss-aversion. This paper uses data from the 1997– 2002 waves of the DNB Household Survey as the questions used to measure loss-aversion are unavailable in other years. The DNB Household Survey is conducted entirely online. To avoid the obvious sample selection effect of limiting the survey to households with internet access, CentERdata provides all households with a set-top box, which allows internet access through a television and phone lines. Alessie et al. (2002) and Das and van Soest (1999) provide an excellent introduction to this data. Comparing the DNB Household Survey results to national accounts data and microdata on household wealth published by Statistics Netherlands they find that it is generally representative of Dutch households. Although no household survey can ever be entirely free of potential biases caused by non-response, their findings suggest that this problem is limited in the DNB Household Survey. As Das and van Soest (1999) and Tu (2004) discuss, there is considerable attrition in this dataset. However, there is no clear reason why attrition would be correlated with loss-aversion and so it seems unlikely to bias our results. 8 To be included in this study in a given year, the household must have answered the General Information section of the Household module, the Assets and Liabilities module, the Health and Income module, and the Economic and Psychological 5 For experimental evidence see: Kahneman and Tversky (1979), Loewenstein (1988), and Thaler et al. (1997). Tversky and Kahneman (1991) provide a good review of the experimental evidence. 6 Barberis et al. (2006) show that loss-aversion alone is not sufficient to explain stock market non-participation. Investors must also narrowly frame stock market gains and losses, meaning that they evaluate stock returns in isolation rather than after integrating stocks with their other sources of wealth. 7 For more information see http://www.centerdata.nl/en/TopMenu/Projecten/DNB_household_study/. 8 The number of observations per household is uncorrelated with our measure of loss-aversion. However, as will become clear in the next section, it is possible that the precision of our estimate of loss-aversion is increasing in the number of observations.

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Table 1 Summary statistics. Variable

Full sample

Equity owners

Non-owners

Panel A — control variables Total financial assets Income Age

67,007 50,505 49.9

134,268 60,665 52.5

35,534 45,750 48.7

Panel B — employment status Regular employment Unemployed Retired Disabled Self-employed Other

65.4% 1.4% 15.6% 8.9% 1.9% 6.9%

60.5 1.2 19.9 10.5 1.9 6.0

67.6 1.5 13.6 8.1 1.9 7.4

Panel C — education Low education Intermediate/low education Intermediate/high education Vocational 1 Vocational 2 University education

5.4% 11.5% 10.3% 31.8% 24.5% 16.5%

4.5 10.1 11.8 26.4 27.8 19.5

5.8 12.2 9.6 34.3 22.9 15.1

5.0 56.2% 105.6%

4.8 66.9 21.8

5.1 51.2 144.8

Panel D — other control variables Risk-aversion Home owner Unsecured debt to total financial assets

This table contains summary statistics for variables used in this paper. Means are shown for the full sample as well as for equity owners, and non-owners. The means are pooled across households and time periods. All monetary values are inflation adjusted to 1997 levels. The summary statistics are sample weighted following the method of Alessie et al. (2002).

Concepts module. For all monetary variables, such as income and asset ownership, we aggregate within households and we adjust for inflation to 1997 prices. For non-monetary variables we take the response of the self-identified household head.9 If the household head's answer is not available we use the spouse's response. If neither the head, nor the spouse, responds, we drop the household from the sample. Summary statistics of the control variables used in this paper are presented in Table 1. This table shows results for all respondents, equity owners, and non-owners. Following Alessie et al. (2004) summary statistics are computed using sampling weights based on information from Statistics Netherlands. We refer to Donkers and van Soest (1999) and Alessie et al. (2004) for an analysis of equity ownership in the earlier DNB Household Survey waves 1993–1998, using a similar set of control variables. 3.1. Household wealth and income Numerous authors have shown that wealth is an important determinant of portfolio allocation, e.g. Bertaut (1998), VissingJorgenson (2002). As Panel A of Table 1 shows, equity owners in this sample have greater wealth than non-participants. VissingJorgenson argues that this is consistent with fixed costs of market entry, such as minimum investments or investment advising fees, as it is easier for wealthy households to pay these costs. As in Alessie et al. (2002), total financial assets is defined as the sum of: all savings and checking accounts, bonds, stocks, mutual funds, money market funds, single-premium annuity insurance policies, cash value of life insurance, employer sponsored savings plans, money lent to friends and family, and other savings or investments. To avoid having outliers drive our results we winsorize total financial assets at the 99th percentile by replacing all observations in the top percentile with the value of the 99th percentile of total financial assets. We define income as total income before taxes less dividends and interest income. This variable is highly skewed and is winsorized at the 1st and 99th percentiles. Several studies have shown that equity ownership increases with income (e.g. Bertaut, 1998; Haliassos and Bertaut, 1995; Vissing-Jorgenson, 2002). Alessie et al. (2004) and Donkers and van Soest (1999) find a significant positive relation between equity ownership and wealth, as well as income, in the DNB Household Survey. As can be seen in Panel A of Table 1 equity owners have considerably higher incomes than non-owners. 3.2. Demographic and other control variables The DNB Household Survey contains the standard demographic variables used in studies of household portfolio choice: age, employment status, and education. Summary statistics of these variables are presented in Table 1. 9 When data is missing we use imputed amounts based on the method of Alessie et al. (2002). CentERdata provides this imputed data on its website. Our equity ownership variables are never imputed.

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Table 2 Equity ownership. Year

Proportion of sample holding equity

Equity/TFA

Pooled 1997 1998 1999 2000 2001 2002

33.1% 30.7% 30.8% 34.5% 31.0% 35.9% 35.7%

37.1% 35.3% 38.0% 38.4% 41.6% 39.1% 31.3%

This table shows the proportion of the sample population holding equity pooled across all years and by each year in the first column. In the second column it shows the average allocation of total financial assets (TFA) to equity among households that hold equity, both pooled across all years and by year.

We include the measure of risk-aversion used in Alessie et al. (2002). This variable is the response to a seven point Likert scale question asking to what extent the household head agrees with the statement “I think it is more important to have safe investments and guaranteed returns, than to take a risk to have a chance to get the highest possible returns”. A high value indicates greater risk-aversion. Bertaut (1998), Haliassos and Bertaut (1995), and Puri and Robinson (2007) use a similar measure from the survey of consumer finances. We include indicator variables to control for the household head's employment status. Employment is measured with a series of indicator variables where the default is regular paid employment. The definitions that we use follow Alessie et al. (2002): Regular Employment, Unemployed, Retired, Disabled, Self-Employed, and Other. In studies using American data, the number of years of education is typically used as a control variable. This is not appropriate for this data set as The Netherlands has a Germanic education system in which the education stream is more important than the length of education. At a young age children are divided into different education streams depending on the results of standardized tests and input from parents. Lower education streams are designed for individuals who will not seek higher education. Accordingly, we use indicator variables for different education streams where the default is college education. Table 1 Panel C shows that university educated households and vocational education level 2 households (white collar vocational education such as accounting and actuarial science) are more likely to own equities. Cocco (2005) and Yao and Zhang (2005) argue that home ownership has an important effect on equity ownership. Home ownership can crowd out investment in equities, particularity for younger, credit constrained households. Panel D of Table 1 shows that there is extensive homeownership in the sample and equity owners and wealthier households are more likely to own homes. Davis et al. (2006) and Cocco et al. (2005) argue that credit constraints are an important factor in household portfolio choice. To control for credit constraints we use the debt to total financial assets ratio.10 This ratio varies widely across households and is markedly higher for non-equity owners.

3.3. Equity ownership The key dependent variable in this paper is publicly traded equity, which includes publicly traded stocks and mutual funds. Mutual fund ownership includes balanced funds, so this variable will include fixed income ownership for some households.11 To study household portfolio allocations we use the ratio of publicly traded equity to total financial assets. The first column of Table 2 shows the proportion of the population holding equity over time. Column two shows portfolio allocations to equity, conditional on equity market participation. Allocations generally follow the market's rise and fall during this time period.

4. Measuring loss-aversion In addition to demographic information, income, and wealth, the DNB Household Survey also contains an “Economic and Psychological Concepts” module. This module includes a series of questions based on work by Thaler (1981) and Loewenstein (1988), showing that loss-aversion affects intertemporal choice. Thaler shows that individuals discount gains and losses at different rates. Loewenstein shows a related result; individuals will demand more to defer receipt of a payment than they will pay to expedite receipt. This pattern of responses implies intransitivity and is thus incompatible with standard preferences, but these authors show it is consistent with a loss-averse value function. Donkers and van Soest (1999) provide an excellent discussion of these questions in the DNB Household Survey. 10 Debt is the sum of: private loans, extended lines of credit, debt with mail-order firms, loans from family and friends, student loans, credit card debt, and other debt not reported elsewhere. We have tried a version of this where debt is defined simply as credit card debt. This version is never significant in the regressions and the results on loss-aversion are unaffected. 11 The Netherlands has a system of employer pensions that cover the vast majority of employees. For legal reasons over 99% of these pensions are defined benefit plans, and thus tax deferred equity investment in retirement accounts is not a significant issue during this time period.

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4.1. The questions The DNB Household Survey includes 16 questions about intertemporal choice which vary across four dimensions: delaying versus speeding up a payment, gains versus losses, the timing of the decision, and the size of the payment. For example: 1. Imagine that you win a prize of ƒ 1000 (€454)12 in the National Lottery. The prize is to be paid out today. Imagine however, that the lottery asks if you are prepared to wait A YEAR before you get the prize. There is no risk involved in this wait. How much extra money would you ask to receive AT LEAST to compensate for the waiting term of a year? If you agree on the waiting term without the need to receive extra money for that, please type 0 (zero). 2. Imagine that you receive notice from the National Lottery that you have won a prize worth ƒ 1000 (€454). The money will be paid out after A YEAR. The money can be paid out at once, but in that case you receive less than ƒ 1000 (€454). How much LESS money would you be prepared to receive AT MOST if you would get the money at once instead of after a year? If you are not interested in receiving the money earlier or if you are not prepared to receive less for getting the money earlier, please type 0 (zero). The first question frames the decision as the delay of a gain while the second question frames the decision as the speed-up of a gain. The survey asks similar questions for a three-month decision period and for 100,000 guilders, for a total of eight questions involving a gain. These eight questions are then repeated asking for compensation and payments required to delay or speedup a loss, where the loss comes from a tax assessment.13 The questions are repeated in each year of the survey from 1997 to 2002.14 4.2. Loss-aversion and intertemporal choice In this section, we derive equations that link the answers to the 16 survey questions about intertemporal choice to the parameters of the value function of prospect theory. The approach for deriving the equations follows Loewenstein (1988) and Tu (2004). Based on the available empirical and experimental evidence in Loewenstein (1988), Loewenstein and Prelec (1992), and Thaler (1981), we assume that individuals do not integrate the payoffs mentioned in the 16 questions with existing consumption plans, but rather evaluate them as gains and losses relative to a reference point. The value of a payoff sequence offering X0 at time 0 and XT at time T is expressed as: VðX0 ; XT ; RÞ = vðX0 –RÞ + δðTÞvðXT –RÞ

ð1Þ

where R designates the reference point, δ(T) denotes the individual's discount factor for a period of length T and v(·) is the value function used to evaluate payoffs. For convenience, we assume v(0) = 0. We first consider the case where an individual will receive a gain of amount X, in the present at time 0. The individual is willing to delay the receipt of X to time T, if the payment is increased by the amount PDG.15 This implies that the individual is indifferent between receiving (X, 0) and (0, X + PDG), and hence VðX; 0; RÞ = Vð0; X + PDG ; RÞ

ð2Þ

vðX–RÞ + δðTÞvð–RÞ = vð0–RÞ + δðTÞvðX + PDG –RÞ

ð3Þ

where R denotes the individual's reference point for payments at time 0 and at time T, subject to 0 b R ≤ X.16 We use the value function of prospect theory for v(·), but to simplify the analysis, we follow Barberis and Huang (2001) and Barberis et al. (2001) and set the curvature parameter of the value function equal to one: 

x; if x ≥ 0 λx; if x b 0 where λ N 1 implies loss-aversion. Using this specification of the value function, Eq. (3) can be written as: vðxÞ =

ð4Þ

X–R–δðTÞλR = –λR + δðTÞðX + PDG –RÞ

ð5Þ

PDG = ½ð1–δðTÞÞðX–RÞ + ð1–δðTÞÞλR = δðTÞ

ð6Þ

12 In 1997–2001 the question asks for a response based on ƒ (Dutch guilders) 1000. This is approximately $500 US dollars through most of the sample period. In 2002, after The Netherlands switched to the euro, the question is asked as given above, including both guilders and the equivalent amount in euros. The capitalization of certain words follows the original questionnaire. 13 In all cases the counterparty is the government of The Netherlands. This is done to eliminate counterparty risk from the decision. 14 Donkers and van Soest (1999) test the relationship between the implied discount rates from similar questions in the 1993 and 1995 waves of the survey and risky asset holdings. They find a significant positive relationship in one year and an insignificant relationship in another. 15 Throughout this paper the subscript DG refers to delay of gain, DL refers to delay of loss, SG refers to speed-up of gain, and SL refers to speed-up of loss. 16 The individual has either completely (R = X), or partially (0 b R b X), adjusted to receiving the amount X.

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To simplify the exposition, we consider the special case of complete reference point adjustment (R = X). Let pDG = PDG / X and r = R / X = 1, then the following equation expresses the relative premium pDG demanded in return for delaying the gain as a function of the loss-aversion parameter and the discount rate: pDG = ð1–δðTÞÞ½ð1–rÞ + λr = δðTÞ = λð1–δðTÞÞ = δðTÞ

ð7Þ

Given λ N 1 and 0 b δ(T) ≤ 1, the premium is positive and bounded. Following similar steps, we derive expressions for the three other types of questions (SG, DL and SL): pSG = ð1–δðTÞÞ

ð8Þ

pDL = ð1 = λÞð1–δðTÞÞ = δðTÞ

ð9Þ

pSL = ð1–δðTÞÞ

ð10Þ

We refer to Appendix A for derivations and more general expressions with a variable reference point r (with 0 b r ≤ 1). As a small numerical example, consider λ = 2.25, T = 1 and δ(T) = 0.90. The equations above then predict pDG = 0.250, pDL = 0.049, pSG = 0.100 and pSL = 0.100, quite close to the actual mean survey response of pDG = 0.246, pDL = 0.039, pSG = 0.038 and pSL = 0.100. The mean absolute deviation between the four actual means and the predictions is 0.019, and the model only has difficulty replicating the mean response to the speed-up of a gain (pSG) questions. As an alternative approach, suppose that individuals use the single constant rate p to answer all four questions, as is the case in a standard utility framework with unlimited borrowing and lending at the same risk-free rate (p). The rate p = 0.077 gives the lowest possible mean absolute deviation of 0.050. When we use different borrowing and lending rates, pB (for the DG and SL questions) and pL (for the DL and SG questions), the best fit is found at pB = 0.181 and pL = 0.038, with a mean absolute deviation of 0.037, which is still about twice as large as the approximation error of the loss-aversion model (7)–(10). Above we follow Loewenstein (1988) in assuming that the individual uses a single reference point R for payoffs at time 0 and time T, when evaluating whether to delay the gain. Implicitly, we assume that the person feels a loss in the period without a payoff (for example, when not choosing to delay, a loss at time T). Loewenstein (1988, p. 204) points out that this does not mean that when time T actually arrives, the person will really experience a loss; only that when evaluating the choice between receiving the gain at time 0 or at time T, the period without a payoff is assigned a loss for sake of comparison.17 Analyzing the same data, Tu (2004) assumes that the respondent's reference point is equal to zero at the time when no payment was initially expected. This implies a zero reference point at time T when considering delay of a gain (or loss) and a zero reference point at time 0 when considering speed-up. We derive alternative estimates of loss-aversion under this reference point assumption and let the data reveal whether the Loewenstein (1988) or Tu (2004) reference point specification fits better. Following Tu (2004), indifference between receiving (X, 0) and (0, X + PDG) implies: vðX–RÞ + δðTÞvð0Þ = vð0–RÞ + δðTÞvðX + PDG Þ

ð11Þ

Substituting the specification of the value function given in Eq. (4) and using pDG = PDG / X and r = R / X, Eq. (11) can be written as (see Appendix B): pDG = ½ð1–δðTÞÞ + ðλ–1Þr = δðTÞ

ð12Þ

Given λ N 1 and 0 b δ(T) ≤ 1, the delay premium is positive and bounded. Following similar steps, we can derive an equation for speed-up of gains: pSG = ð1–λÞδðTÞr + ð1–δðTÞÞ

ð13Þ

Note that pSG can easily become negative, for example when λ N 1 and δ(T) = 1, because loss-averse individuals with a reference point equal to X will experience a considerable loss at time T in case of speed-up and will actually demand compensation (i.e. PSG b 0). The speed-up question in the DNB Household Survey explicitly asks respondents to type zero when they are not willing to pay for speed-up.18 In line with the framing of the question, we therefore define the relative speed-up payment as in Eq. (13) as long as it is positive, and zero otherwise: pSG = maxfð1–λÞδðTÞr + ð1–δðTÞÞ; 0g

ð14Þ

17 The assumption that people have a single reference point to compare payoffs at different points in time is analogous to the assumption in prospect theory that people use a single reference point to evaluate the different possible payoffs from a gamble. People will tend to update their reference point once one particular outcome actually occurs. Similarly, in the current setting as time passes the individual may eventually update his or her reference point. 18 “If you are not interested in receiving the money earlier or if you are not prepared to receive less for getting the money earlier, please type 0 (zero).”

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Following similar steps, we derive the following expressions for pDL and pSL: pDL = maxf½ð1–λÞð1 = λÞr + ð1–δðTÞÞ = δðTÞ; 0g

ð15Þ

pSL = ðλ–1Þð1 = λÞδðTÞr + ð1–δðTÞÞ

ð16Þ

As a small numerical example, consider r = R / X = 1, λ = 2.25, T = 1 and δ(T) = 0.90. The equations above then predict pDG = 1.5, pDL = 0, pSG = 0 and pSL = 0.6, quite far from the actual mean survey response of pDG = 0.246, pDL = 0.039, pSG = 0.038 and pSL = 0.100, with a mean absolute deviation of 0.537. It turns out that with the reference point specification of Tu (2004) the data can be fitted much closer when the reference point does not adjust completely to the promised payments (R b X) and lossaversion is less than two (λ b 2). For example, with r = R / X = 0.25, λ = 1.50, T = 1 and δ(T) = 0.90, the predicted values are pDG = 0.25, pDL = 0.019, pSG = 0 and pSL = 0.175, and the mean absolute deviation is 0.0345. Later on we will inspect the estimation results to see which model specification best fits the data, using various levels of reference point adjustment. 4.3. Descriptive statistics and reliability 4.3.1. Descriptive statistics Before beginning estimation we check the answers to the 16 questions for two types of errors. First, some respondents answer zero to all questions in a particular year. In total 345 of the 7590 (4.5%) responses are all-zero. We discard these observations, as they probably indicate that the respondent did not want to spend time and thought on answering the questions.19 Second, we check the data for implausibly high responses. Fortunately, the frequency of implausibly large answers is relatively small. We winsorize the data at X for the delay of gain category, and at 50% of X for the other categories, as this amounts to approximately 2% of the responses for each category.20 Table 3 Panel A shows descriptive statistics of the winsorized responses. Generally, respondents require high compensation to delay gains. On average respondents require 252 guilders additional compensation to delay a gain of 1000 guilders for one year. Only 4.7% of the respondents agree to delay the payment without any compensation. On the other hand, respondents are willing to pay only 43 guilders to speed-up a 1000 guilders gain due in one year. Further, about 63.3% of the respondents are not willing to pay to speed-up a gain of 1000 guilders. The large divergence in answers to the questions about delay and speed-up of gains indicates intransitivity and is a clear violation of the traditional discounted expected utility framework, which predicts that the discount rates for all 16 questions are either equal (assuming borrowing and lending at the same rate) or very similar (without borrowing). In general respondents are unwilling to pay much to delay losses. For example respondents will pay 35 guilders on average to delay a loss of 1000 guilders for one year, with 56.7% of the respondents not willing to pay at all. On the other hand, respondents require a loss reduction of 99 guilders on average to accept speeding up a loss of 1000 guilders by one year, while 23.7% of the respondents do not require a reduction. The results in Table 3 Panels B and C are based on households' average discount rate within each category of questions. Below the diagonal in Panel B are the correlations between the average discount rates. If respondents have standard utility we would expect all correlations to equal one. Although all of the correlations are significantly positive, they are considerably lower than one. The percentages above the diagonal in Panel B show how often the discount rate identified in the row is greater than the discount rate in the column. Panel C shows p-values of significance tests of the differences in medians between different categories of questions. 4.3.2. Reliability The validity of our measures of loss-aversion and reference points is crucial for this study. Fortunately, the DNB Household Survey consists of four sets of closely related questions and we can check the consistency of answers to these sets of related questions. For example, there are four questions about delay of gains, with a gain of either 1000 or 100,000 guilder, and with a horizon of either three months or one year. Although we expect some variation in the relative compensation pDG as a function of the amount at stake and the horizon, the four answers should satisfy certain basic constraints. For example, if a respondent asks for no compensation to delay a payment of 1000 guilders for one year, but demands 500 guilders to delay the same amount for three months, this would indicate a judgment error or lack of concentration. Table 4 Panel A displays the frequency of responses where the answer to a question with a horizon of three months is larger than the answer to the same question with a horizon of 12 months (related questions are compared in pairs of two, and frequencies averaged). Table 4 also displays the frequency of responses where the answer to a question with a payoff of 1000 guilder is larger than the answer to the same question with a payoff of 100,000 guilder. The overall frequency of both types of errors is 2.9% and 2.4%, respectively, indicating that the majority of respondents avoid these kinds of judgment errors. To reduce measurement error we delete these observations. 19

Within our preference framework an all-zero response is only possible when the individual is not loss-averse (λ = 1) and has a discount rate of zero (δ = 1). The payments asked for in the speed-up questions reduce the payoff X (gain or loss), and therefore theoretically the answer should be less than or equal to X. Only 0.1% of the respondents violate this constraint in their answers to speed-up questions. We decided to set the cutoff point for winsorizing the answers at the level 0.5X, as this roughly coincides with the 98th percentile for the two speed-up questions and the delay of losses question. However, the answers for delay of gains questions are typically much larger compared to the other questions, and there the 98th percentile lies roughly at 1 × X. 20

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Table 3 Descriptive statistics of the payments for delay/speed-up of gains/losses. Question

Horizon

Amount

Type of payment

Mean

Median

Std. dev.

% Zero answers

Panel A: Descriptive statistics DG 3m DG 12 m DG 3m DG 12 m DL 3m DL 12 m DL 3m DL 12 m SG 3m SG 12 m SG 3m SG 12 m SL 3m SL 12 m SL 3m SL 12 m

1000 1000 100,000 100,000 1000 1000 100,000 100,000 1000 1000 100,000 100,000 1000 1000 100,000 100,000

Receive extra Receive extra Receive extra Receive extra Lose more Lose more Lose more Lose more Receive less Receive less Receive less Receive less Loss reduction Loss reduction Loss reduction Loss reduction

10.2% 25.2% 7.8% 17.0% 1.2% 3.5% 0.6% 2.2% 1.3% 4.3% 1.4% 3.8% 4.6% 9.9% 2.7% 6.4%

3.5% 10.0% 2.0% 10.0% 0.0% 0.0% 0.0% 0.5% 0.0% 0.0% 0.0% 0.0% 2.5% 7.5% 1.0% 5.0%

20.6% 30.6% 20.1% 26.1% 3.6% 6.9% 1.2% 3.6% 5.4% 13.6% 6.5% 12.9% 5.6% 11.3% 4.5% 7.9%

18.3% 4.7% 6.4% 1.6% 69.5% 56.7% 48.9% 42.7% 76.8% 63.3% 59.3% 48.8% 27.8% 23.7% 21.6% 20.8%

DL

SG

SL

Panel B: Correlations and proportion of discount rates in Column N Rates in row DG – 97.2% DL 0.203 – SG 0.102 0.192 SL 0.242 0.181

DG

89.0% 69.5% – 0.165

63.0% 28.5% 37.9% –

Panel C: Significance tests of differences in medians DG – DL 0.000 SG 0.000 SL 0.000

– 0.000

–

– 0.000 0.000

Panel A shows summary statistics of the responses to the 16 questions about the speed-up and delay of gains and losses, after winsorizing 2% of the extreme observations in the right tail of the distribution. “DG” refers to questions about the delay of a gain (lottery prize), “DL” to questions about the delay of a loss (tax assessment), while “SG” and “SL” refer to speed-up of a gain and speed-up of a loss, respectively. The time-period mentioned in the questions — i.e. for delay and speed-up — is either three months or one year, as indicated in the 2nd column. The size of the payoff mentioned in the questions is either 1000 Dutch guilders or 100,000 guilders, as indicated in the 3rd column. In the DNB Household Survey households are asked to indicate the minimum amount they want to receive to accept delay of a gain (DG) and speed-up of a loss (SL), and instructed to write “0” if they do not require compensation to accept DG or SL. Households are asked to indicate the maximum amount they are willing to pay to speed-up gains (SG) and delay losses (DL), and instructed to write “0” if they do not want to consider SG or DL at any price. The last four columns of Panel A show descriptive statistics of the winsorized household answers to the 16 questions, including the proportion of households that give “0” as their answer to a particular question. Panel B shows correlations between average discount rates below the diagonal. The numbers above the diagonal show the percentage of observations for which the average discount rate for the type of question indicated in the row is larger than the average discount rate for the question type in the column. Panel C shows p-values from tests of the equality of discount rates. The p-values are based on a Wilcoxon matched pairs signed rank test. The null hypothesis is that across households the median difference between the rates is zero.

In survey-based research, Cronbach's alpha (Cronbach, 1959) is frequently used to measure the reliability of a set of questions intended to measure the same underlying latent variable. Cronbach's alpha is a function of the average cross-sectional correlation of the respondents' answers. Most researchers consider empirical values of alpha above 0.70 as an acceptable level of reliability, see, e.g. Nunnally (1978). We expect an individual's answers to the four questions about speed-up of gains to reflect the individual's underlying latent preference for speeding up gains, apart from some small variations caused by changes in time horizon and the payoff amount. Table 4 Panel B reports Cronbach's alpha for the four sets of related questions (DG, DL, SG and SL) for each survey year (1997–2002), as well as the average over six years. All alpha estimates are above 0.70 and the average alpha for each set of questions is above 0.80. Hence, it is clear that the responses to related questions are not random, but closely related with acceptable levels of reliability.21 4.3.3. Discussion of alternative explanations It is important to establish that not only are the data consistent with preferences based on loss-aversion, but that other potential explanations fail. If households have standard preferences and there is a single market interest rate for both borrowing and lending, then all households should report a rate equal to the market interest rate for each of the 16 questions. Equality of discount rates within households is overwhelmingly rejected by pairwise signed rank tests. If there is a wedge between borrowing 21 Interestingly, if we group variables into groups by time or the amount of money the Cronbach's alphas are lower. For example, the Cronbach's alphas of questions with a time period of three months and twelve months are 0.701 and 0.706 respectively. The Cronbach's alphas of the 1000 guilder questions and 100,000 guilder questions are both 0.753. These results suggest that grouping the questions by the question frame produces grater response consistency than grouping by the time period or amount of money.

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Table 4 Frequency of suspect responses and reliability estimates. Panel A: Average frequency of suspect responses

DG DL SG SL

P (1000) N P (100,000)

P (3 m) N P (12 m)

0.015 0.035 0.021 0.045

0.011 0.016 0.032 0.037

Panel B: Cronbach's alpha

DG DL SG SL

Average

1997

1998

1999

2000

2001

2002

0.919 0.894 0.885 0.821

0.888 0.909 0.885 0.798

0.896 0.936 0.832 0.763

0.921 0.847 0.871 0.839

0.967 0.891 0.878 0.781

0.948 0.914 0.925 0.875

0.893 0.868 0.916 0.870

This table shows the reliability of the responses to the survey questions used to extract our measure of loss-aversion. Panel A shows the frequency that answers to 1000 guilder questions are larger than for 100,000 guilder questions and the frequency of answers for delay/speed-up questions with a three month horizon exceeding the answers for equivalent questions with a 12 month horizon. Panel B shows Cronbach's alpha for related questions.

and lending rates then all households should report a single rate for lending transactions (as all households can lend at the same risk-free interest rate), but possibly households will report heterogeneous borrowing rates. However, with standard preferences each household should be internally consistent and report a single borrowing rate. Again, this is overwhelmingly rejected by pairwise t-tests. The average within household difference in borrowing rates is significantly different from zero,22 implying that on average the households' responses are intransitive. If there are borrowing constraints that prevent households from accessing credit markets it is possible to get some variation across rates for a given individual due to the concavity of the utility function. Concavity predicts that households will report higher rates for losses than gains, as the utility function is steeper for losses than gains. However, the largest rates are reported for the delay of gain questions, not loss questions, in direct violation of this prediction. The effect of concavity should be stronger for the 100,000 guilder questions than the 1000 guilder questions, as most households will be close to risk neutral once the 1000 guilders is integrated with current wealth. There is no evidence to support this prediction. Concavity of the utility function also fails to explain the magnitude of the intra household range of reported rates. For a calibrated power utility function a household with median wealth, no access to credit markets, and a risk-aversion coefficient of 3 would have only a 0.47% annualized difference in rates across responses to the 1000 guilder questions. Even assuming a riskaversion coefficient of 50, this implies at most a difference across rates of 2% for a given household. In the data there is an average difference across rates of over 20%, implying that the variation in responses predicted by a standard power utility function is too small by many orders of magnitude.23 Benzion et al. (1989) discuss another potential explanation they call the implicit risk approach. Perhaps respondents believe that there is some counterparty risk and thus wish to accelerate gains while delaying losses. However, implicit risk should affect delay and speed-up of gains questions symmetrically and delay of losses and speed-up of losses symmetrically. This is flatly rejected by the data. There are two additional issues sometimes raised in response to survey questions: that households respond at random to survey questions, or that households respond to hypothetical questions differently than they would respond to real situations. The questions asked in the DNB Household Survey have been asked in numerous prior studies, such as Benzion et al. (1989), Loewenstein (1988), Shelley (1993) and Thaler (1981) all of whom find similar results to the DNB Household Survey. This suggests responses are not random but measure some systematic characteristic of preferences. Loewenstein (1988) asks questions similar to those in the DNB using both real and hypothetical rewards, and finds similar results in both cases. Finally we note that the relevance of hypothetical behavior for real behavior is jointly tested along with the hypothesis that loss-aversion matters. Since our loss-aversion measures come from hypothetical questions, our results can be viewed as tests of the joint statement: 1. Loss-aversion matters and, 2. Loss-aversion can be reliably measured from hypothetical questions. If either part is wrong we will find coefficients indistinguishable from zero. Since we find significant coefficients, with the signs as predicted, we can reasonably interpret this as evidence that household responses to hypothetical questions do contain relevant information about preferences.

22 Even assuming households face different borrowing rates for 1000 guilders and 100,000 guilders, there is strong evidence that households report different intra household borrowing rates across 1000 guilder questions, and across 100,000 guilder questions, depending on the framing of the question. 23 As a further check that our results are inconsistent with standard preferences we derive a system of equations similar to Eqs. (7)–(10), but using a power utility function that integrates the payoffs with household wealth. We estimate risk-aversion for each household based on their responses to the 16 survey questions and their wealth. The sum of squared errors is, on average, two times as large for these estimates as for estimates based on a loss-averse utility function.

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4.4. Estimation 4.4.1. Notation and assumptions To simplify the discussion in this section we begin by introducing the following extended notation to describe the survey data: PDG,i,t(X, T) denotes the payment that individual i wants to receive for the delay of a gain of size X (1000 or 100,000 guilders) over a period of time T (three months or one year), measured in survey year t (1997–2002). We define the relative amount as pDG,i,t(X ,T) = PDG,i,t(X, T) / X. Our aim is to keep the specification of the empirical preference model parsimonious, as for each survey participant we have a limited number of data points. We assume that the preference parameters vary from individual to individual, but do not depend on the survey year t. For the discount rate δi(X, T) we use the specification δi(X, T) = δTi . For the reference point we will initially assume that it fully adjusts to the gain or loss initially expected (r = R / X = 1). We will then re-estimate the model with reference point adjustment of 50% and 25% (r = 0.50 and r = 0.25), respectively, and compare model fit. Much of the experimental work in the area of intertemporal preferences was designed not just to show that loss-aversion affects intertemporal choice, but also that individuals' time-preferences are dynamically inconsistent. Because of this, to avoid any possibility that dynamic inconsistency is driving our measure, we report results using only the eight questions where the time choice is over one year. Both the loss-aversion estimates and all of the results in this paper are very similar if we derive a measure using all 16 questions, or if we use only the three-month questions. As the panel is not balanced, not every household has answered the questions in each of the six years from 1997 through 2002. Further, we eliminate all pairs of suspect answers reported in Panel A of Table 4. We use nDG,i to denote the number of years that individual i provided answers to the delay of gain question (for T = 1) and the answer did not violate the conditions in Panel A of Table 4. From here onwards summation over time t, denoted by Σt, is presumed to include only years with non-suspect data available for individual i for that particular question. 4.4.2. Estimation Our aim is to estimate the loss-aversion parameter (λi), and discount rate (δi) of individual i from his or her indicated speed-up and delay payments (PSG,i,t(X,T), PDG,i,t(X,T), PSL,i,t(X,T) and PDL,i,t(X,T)). The preference parameters should satisfy the following feasibility conditions: λi N 0, 0 b δi ≤ 1. This leaves us with two parameters to estimate from eight equations, one for each survey question (T = 1 only). As an example, below we show the equations for the Loewenstein (1988) specification with full reference point adjustment, i.e. r = 1: Σt pDG;i;t ðX; TÞ = nDG;i –½λi ð1–δTi Þδ−T i 

= 0;

Σt pSG;i;t ðX; TÞ = nSG;i –½ð1–δTi Þ

= 0;

T −T Σt pDL;i;t ðX; TÞ = nDL;i –½ð1 = λi Þð1–δi Þδi 

= 0;

Σt pSL;i;t ðX; TÞ = nSL;i –½ð1–δTi Þ

= 0;

ð17Þ

with X a {1000; 100,000}, T = 1 and t {1997, 1998, …., 2002} and subject to λi N 0, 0 b δi ≤ 1. We estimate system (17) for each individual i separately using the Generalized Methods of Moments (GMM).24 Let uSG,i(X,T), uDG,i(X,T), uSL,i(X,T) and uDL,i(X,T) denote the errors in the eight moment equations, for a given set of preference parameters, for individual i. We minimize the sum of the squared errors to estimate the parameters25,26: 2

2

2

2

ΣX ΣT ½uSG;i ðX; TÞ + uDG;i ðX; TÞ + uSL;i ðX; TÞ + uDL;i ðX; TÞ :

We derive and estimate two similar systems as in Eq. (17) for the case of partial reference point adjustment with r = 0.50 and r = 0.25, respectively. Further, we also derive a system to estimate λi and δi under the reference point assumptions of Tu (2004), with r = 1, r = 0.5 and r = 0.25. Overall, six different specifications are estimated under alternative assumptions the about reference points. Our aim is to compare model fit and to use the estimates from the model with the lowest sum of squared errors. 24 The number of unknown parameters in system (17) is two, while the number of moment conditions is eight. The mean of each moment condition is estimated with n = 1 up to n = 6 yearly observations. On average the two parameters are estimated with 21 answers to speed-up and delay questions, as 2.6 survey years are on average available per household. The number of observations ranges from eight, for households with only one year of survey data available, to 48 for households with six years of data. Due to the small number of observations and the non-linearity of the system, consistency and unbiasedness of the estimates cannot be guaranteed. 25 We set the weighing matrix for the errors equal to an identity matrix. We do not attempt to estimate the covariance matrix of the errors, e.g. as part of a 2stage GMM estimation procedure, as this implicitly involves estimating (8 × 9) / 2 = 36 additional unknown parameters in the covariance matrix. 26 As constrained optimization in practice is limited to equality constraints and inequality constraints (≤ and ≥), we model the strict inequality constraint λi N 0 as λi ≥ 0.1, and δi N 0 as δi ≥ 0.1. The lower bound for λi and δi is set at 0.1 — away from zero — because the inverse of these parameters (1 / λi and 1 / δi) occur in the moment equations, and near-zero values might lead to error propagation and numerical instability.

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4.4.3. Estimation results Table 5 displays the estimation results, using each household's mean response to the questions, averaged over the survey years 1997 through 2002, as the dependent variable. Panel A displays descriptive statistics of the loss-aversion estimates λi, for six different reference point specifications: following Loewenstein (1988) and Tu (2004), respectively, with r = 1, r = 0.5 and r = 0.25. Panel B shows statistics of the estimated discount factor δi and panel C displays model fit (sum of the squared estimation errors). Under the assumption that households use a single reference point when making a trade-off between payoffs at time 0 and time T, as in Loewenstein (1988), the loss-aversion estimates are relatively high (median between 1.7 and 2.6), while the median discount rate is about 5% annually. Under the assumption that households have a zero reference point at times when no payment was expected initially, as in Tu (2004), the loss-aversion estimates are relatively small (median between 1.1 and 1.4), while the estimated discount rates are close to zero (median of 0%). Hence, the two different reference point setups lead to considerably different estimates, both qualitatively and quantitatively. We assess model fit by inspecting the average sum of squared errors in Panel C. The Loewenstein specification fits best when the reference point adjusts completely to the payoff (r = 1, with mean squared error of 0.043), while fit deteriorates considerably in case of partial adjustment (r = 0.5 and r = 0.25). The Tu specification achieves best fit with low reference point adjustment, at r = 0.25 (the mean squared error is 0.045). Overall, the best fit is achieved by the Loewenstein setup with full reference point adjustment (r = 1). For this specification the average discount factor δi is equal to 0.95, implying a discount rate of 5% per year. The median loss-aversion estimate λi across the 2526 households is 2.47, which is close to the loss-aversion estimate of 2.25 reported by Tversky and Kahneman (1992). However, the distribution of loss-aversion estimates is skewed to the right and the average is considerably higher at 5.61, driven by a number of households with great reluctance to intertemporal trade-offs. In the subsequent sections we will use two sets of loss-aversion estimates while analyzing equity ownership: estimates derived from the Loewenstein specification with r = 1 and estimates from the Tu setup with r = 0.25. We choose these two specifications because they have the best fit based on mean squared error. Further, by using two sets of loss-aversion estimates that are substantially different (correlation = 0.658), we aim to establish the robustness of our results with respect to the reference point assumptions.

5. Results In this section we estimate the relation of loss-aversion and estimated reference points with equity ownership. We show that households with higher reported loss-aversion are less likely to participate in the equity market and avoid direct stockholding to a greater extent than mutual funds. After controlling for sample selection, we do not find a significant relation between lossaversion and allocations to equity. Table 5 Household loss-aversion and time-preference estimates. Mean

Median

Std. dev.

5th%

95th%

Panel A: Loss-aversion estimates Loewenstein, r = 1 Loewenstein, r = 0.5 Loewenstein, r = 0.25 Tu, r = 1 Tu, r = 0.5 Tu, r = 0.25

5.61 3.50 4.96 1.17 1.33 1.64

2.47 1.68 2.61 1.11 1.17 1.36

8.22 6.50 6.57 0.19 0.41 0.76

1.00 0.47 0.25 0.99 1.01 1.03

25.11 10.76 17.92 1.60 2.26 3.23

Panel B: Discount factor estimates Loewenstein, r = 1 Loewenstein, r = 0.5 Loewenstein, r = 0.25 Tu, r = 1 Tu, r = 0.5 Tu, r = 0.25

0.95 0.93 0.93 1.00 0.98 0.97

0.96 0.95 0.95 1.00 1.00 1.00

0.043 0.050 0.053 0.006 0.040 0.054

0.86 0.83 0.82 0.98 0.90 0.85

0.99 0.99 0.99 1.00 1.00 1.00

Panel C: Estimation error Loewenstein, r = 1 Loewenstein, r = 0.5 Loewenstein, r = 0.25 Tu, r = 1 Tu, r = 0.5 Tu, r = 0.25

0.043 0.098 0.106 0.071 0.055 0.045

0.011 0.026 0.026 0.018 0.015 0.013

0.084 0.161 0.186 0.124 0.099 0.086

0.0005 0.0012 0.0012 0.0012 0.0005 0.0004

0.201 0.525 0.592 0.343 0.231 0.198

This table shows descriptive statistics of the estimated preference parameters and estimation error. All parameters are estimated subject to the constraints: 0.1 ≤ λi ≤ 40 and 0.1 ≤ δi ≤ 1. Panel A shows summary statistics of the estimates of households' loss-aversion parameters. Panel B shows summary statistics of the estimates of households' discount factors. Panel C summarizes estimation error.

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Table 6 Random-effects probit model of the participation decision. Variable

Loss-aversion Discount rate

1

Lagged participation TFA / 1000 TFA / 1,000,000 Squared Homeowner Debt to TFA Income / 1000 Income/1,000,000 squared Age Age squared Employment effects Education effects Constant Log likelihood Akaike I.C.

3

4

Loewenstein

Loewenstein

Tu

Tu

− 0.014** [− 2.38] − 0.677 [− 0.65]

− 0.015** [− 2.55] − 0.564 [− 0.55] − 0.110*** [− 4.71] 0.246*** [9.54] 1.154*** [12.60] 0.017*** [17.06] − 0.019*** [− 12.38] 0.245*** [3.09] − 0.053*** [− 3.23] 0.003** [2.16] − 0.009 [− 1.01] − 0.039* [− 1.95] 0.346* [1.74] Yes Yes Yes − 1986.9 4033.8

− 0.093* [− 1.66] − 0.124 [− 0.16]

− 0.103* [− 1.84] − 0.12 [− 0.15] − 0.110*** [− 4.70] 0.247*** [9.60] 1.159*** [12.66] 0.017*** [17.07] − 0.019*** [− 12.37] 0.247*** [3.12] − 0.053*** [− 3.22] 0.004** [2.28] − 0.009 [− 1.07] − 0.040** [− 1.99] 0.345* [1.74] Yes Yes Yes − 1989.6 4039.1

Risk-aversion Initial participation

2

0.249*** [9.68] 1.152*** [12.60] 0.017*** [17.14] − 0.019*** [− 12.37] 0.242*** [3.05] − 0.052*** [− 3.22] 0.003** [2.13] − 0.008 [− 0.93] − 0.038* [− 1.92] 0.328 [1.64] Yes Yes Yes − 1997.9 4053.9

0.251*** [9.73] 1.157*** [12.65] 0.017*** [17.15] − 0.019*** [− 12.35] 0.245*** [3.08] − 0.052*** [− 3.22] 0.004** [2.24] − 0.009 [− 1.00] − 0.039* [− 1.95] 0.326 [1.63] Yes Yes Yes − 2000.5 4059.1

*,**,*** Significant at the 10%, 5%, and 1% level respectively. N = 5810. This table shows random-effects probit estimates of the participation decision. Marginal effects are reported rather than coefficients. The dependent variable equals one if the household owns equity. Financial variables are measured across all members of the household. Other variables, such as age, employment, riskaversion, and loss-aversion, are measured as the value given by the household head. Z-scores are shown in parentheses below the marginal effect estimates. The last two rows show the log likelihood and the Akaike information criterion.

5.1. Participation To test if loss-aversion affects households' equity market participation decisions we estimate a random-effects panel probit model on the full unbalanced panel. Table 6 shows the estimation results where the dependent variable equals one if the household owns equity. In the first two columns, loss-aversion is measured using the assumptions of Loewenstein (1988) and full reference point adjustment (r = 1). In the third and fourth columns, loss-aversion is measured using the assumptions of Tu (2004) and partial reference point adjustment (r = 0.25). In all columns loss-aversion has a significant negative relation with equity market participation, however the loss-aversion measure based on Loewenstein (1988) is considerably more significant, and the Akaike information criterion suggests this measure provides a better fit. In all specifications we include the estimated discount rates, but they are not significantly related to equity participation and their inclusion has little effect on the loss-aversion coefficients. The estimates also indicate that loss-aversion is economically significant. After setting all other variables equal to their mean, if loss-aversion changes from the 25th percentile to the 75th percentile this results in a change in the probability of owning stocks of 1.7% points for the Loewenstein loss-aversion measure and 2.0% points for the Tu loss-aversion measure. While this may seem low, this represents approximately a 10% to 15% increase relative to the mean probability of owning equity.27 This is economically comparable to changing wealth by 5% points. In columns two and four we include our measure of risk-aversion. It is highly significant, but its inclusion has little effect on the magnitude of the loss-aversion estimates, and the significance of loss-aversion actually increases. Lagged participation is highly significant, indicating considerable persistence in equity market participation. Coefficients on the other control variables are generally consistent with existing findings in the household portfolio literature and results reported for the 1993–1998 waves of the DNB Household Survey in Donkers and van Soest (1999) and Alessie et al. (2004). We also include equity market participation in 1996 to handle the initial conditions problem, as suggested in Wooldridge (2005). Wealthier, home owning, higher income 27 While the unconditional probability of owning equity is around 25%–30%, the probability of owning equity is quite low for households with total financial assets near the mean value. This is caused by the fact that equity ownership is concentrated among wealthier households.

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households are more likely to participate in equity markets. Unlike in many other empirical studies, age is not very significant. This may be because age is highly correlated with several other variables in the employment effects category, notably the retirement indicator variable. The estimated coefficient of the ratio of unsecured debt to total financial assets is negative and significant. This is intuitively reasonable as for indebted households paying off consumer debt is their best investment opportunity. Overall, the results in this section support the hypothesis that loss-aversion has a significant impact on household participation. Households with a higher level of loss-aversion are unwilling to risk the pain that comes from declines in equity prices and so avoid equity. 5.2. Portfolio allocations Having examined the effect of loss-aversion and reference points on equity market participation, our next step is to test their effect on portfolio allocation. Miniaci and Weber (2002) and Vissing-Jorgenson (2002) argue some households that desire to hold equity are unable to, because of fixed costs of participating in equity markets, and as a result it is important to control for sample selection effects when examining household portfolio allocations. Because of the possibility of fixed participation costs we estimate portfolio allocations using the panel sample selection model of Wooldridge (1995). We estimate the inverse Mill's ratio for each household using the panel probit regressions presented in the previous subsection. As independent variables we include each households' average wealth, income, and age, and the squares of each of these averages, with the averages taken across all years the household is in the panel. These averages function as quasi-fixed effects and remove any unobserved household level effects which are correlated with these variables. Table 7 shows the sample selection model results. In columns one and two loss-aversion is measured based on Loewenstein (1988) and in columns three and four loss-aversion is based on Tu (2004). In all cases loss-aversion is not significant. It appears that loss-aversion affects households' participation decision, but conditional upon participation loss-aversion does not affect allocations. If we estimate these regressions on the sample of participants without including the Mill's ratio we find qualitatively similar results. The combination of a significant effect on participation but no effect on allocation is surprising. However, participation represents the result of a clear decision by a household. It is less clear that allocations are the result of conscious decisions. For example, Agnew et al. (2003) find that in an average year only 12.5% of investors rebalance their equity holdings. Table 7 Sample selection model estimates of portfolio allocations. Variable

Loss-aversion Discount rate

1

2

3

4

Loewenstein

Loewenstein

Tu

Tu

0.002 [1.31] − 0.110 [− 0.52]

0.002 [1.26] − 0.113 [− 0.53] − 0.007* [− 1.79] 0.000 [− 1.00] 0.000 [− 0.45] − 0.022 [− 1.60] 0.009** [2.08] 0.000 [0.23] 0.000 [− 0.14] 0.016 [1.07] 0.001 [0.01] − 0.055*** [− 3.99] Yes Yes Yes Yes Yes

0.019 [1.54] − 0.023 [− 0.14]

0.017 [1.45] − 0.032 [− 0.20] − 0.007* [− 1.76] 0.000 [− 1.01] 0.000 [− 0.45] − 0.022 [− 1.59] 0.009** [2.09] 0.000 [0.22] 0.000 [− 0.14] 0.016 [1.07] 0.002 [0.02] − 0.055*** [− 4.01] Yes Yes Yes Yes Yes

Risk-aversion TFA / 1000 TFA/1,000,000 squared Homeowner Debt to TFA Income / 1000 Income/1,000,000 squared Age Age squared Mill's ratio Employment effects Year effects Education effects Quasi-fixed effects Constant

0.000 [− 1.13] 0.000 [− 0.33] − 0.022* [− 1.65] 0.009** [2.08] 0.000 [0.18] 0.000 [− 0.12] 0.016 [1.07] − 0.001 [− 0.01] − 0.058*** [− 4.20] Yes Yes Yes Yes Yes

0.000 [− 1.14] 0.000 [− 0.33] − 0.022 [− 1.64] 0.009** [2.09] 0.000 [0.16] 0.000 [− 0.13] 0.016 [1.07] 0.000 [0.00] − 0.058*** [− 4.24] Yes Yes Yes Yes Yes

*,**,*** Significant at the 10%, 5%, and 1% level respectively. N = 1921. This table shows sample selection model estimates of the asset allocation decision. The dependent variable is the proportion of total financial assets allocated to equity. Financial variables are measured across all members of the household. Other variables, such as age, employment, risk-aversion, and loss-aversion, are measured as the value given by the household head. T-statistics shown in parentheses below parameter estimates.

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Table 8 Ordered probit model of the equity type decision. Variable

Loss-aversion Discount rate

1

TFA/1,000,000 squared Homeowner Debt to TFA Income / 1000 Income/1,000,000 squared Age Age squared Employment effects Year effects Education effects Constant

3

4

Loewenstein

Loewenstein

Tu

Tu

− 0.011 (3.19)*** − 0.483 (0.84)

− 0.012 (3.42)*** − 0.391 (0.68) − 0.092 (6.68)*** 0.008 (17.63)*** − 0.009 (12.86)*** 0.123 (2.63)*** − 0.038 (2.96)*** 0.001 (1.09) 0 (0.07) 1.308 (26.56)*** − 0.024 (2.01)** Yes Yes Yes Yes

− 0.059 (1.82)* − 0.663 (1.44)

− 0.068 (2.08)** − 0.676 (1.48) − 0.092 (6.67)*** 0.008 (17.77)*** − 0.009 (12.99)*** 0.123 (2.62)*** − 0.038 (2.97)*** 0.001 (1.29) − 0.001 (0.22) 1.319 (26.74)*** − 0.024 (1.99)** Yes Yes Yes Yes

Risk-aversion TFA / 1000

2

0.008 (17.68)*** − 0.008 (12.93)*** 0.114 (2.44)** − 0.037 (3.04)*** 0.001 (1.04) 0.001 (0.11) 1.326 (27.03)*** − 0.025 (2.03)** Yes Yes Yes Yes

0.008 (17.80)*** − 0.009 (13.02)*** 0.114 (2.44)** − 0.037 (3.07)*** 0.001 (1.24) 0 (0.04) 1.337 (27.20)*** − 0.024 (2.01)** Yes Yes Yes Yes

*,**,*** Significant at the 10%, 5%, and 1% level respectively. N = 5810. This table shows results from an ordered probit model. The dependent variable equals zero if the household does not own equity, one if the household owns mutual funds but not individual stocks, two if the household owns mutual funds and individual stocks, and three if the household owns only individual stocks. Standard errors are clustered by household.

5.3. Equity type In this subsection we examine the effect of loss-aversion and reference points on households' investment choices between mutual funds and direct stock holdings. If a household is loss-averse and frames at the level of each individual stock, as in Barberis and Huang (2001), the pain from losses on individual stocks may outweigh the pleasure from gains, even if the overall portfolio return is positive. A mutual fund effectively integrates the gains and losses from individual stocks into a single reported return. Thus loss-averse households subject to narrow framing and loss-aversion will place a higher value on owning a mutual fund than on directly owning the component securities. In Table 8 we show the results of an ordered probit regression28 where households are divided into four categories: no equity, mutual funds only, mutual funds and individual stocks, and individual stocks only. No equity ownership is the default category and the other categories are coded as one, two, and three respectively. Standard errors are clustered by household. Consistent with our hypothesis, the results in Table 8 show that loss-aversion has a significant negative coefficient. These results indicate that not only does loss-aversion reduce the probability of a household owning equity: loss-aversion affects the type of equity participants select.29 Loss-aversion increases households' preference for having individual stock returns integrated into a single mutual fund return. In columns two and four we include risk-aversion as a control variable. Risk-aversion is highly significant, and its inclusion increases the significance of loss-aversion.

6. Conclusion Despite the high equity premium many households choose not to participate in the equity markets, and across participating households there is great heterogeneity in allocations to equity. These empirical facts are difficult to reconcile with normative results obtained from frictionless models using standard utility functions. One proposed explanation for these facts is that

28

The results are similar if we use a multinomial probit model. We have also estimated a probit model making a direct comparison between households owning mutual funds only and households owning individual stocks only (i.e. excluding households holding no equity and households holding both mutual funds and stocks). The direct comparison reduces the sample size from 5810 to 1364, so statistical power is probably lower. The coefficient of the Loewenstein loss-aversion measure is significantly negative at the 5% level, as predicted, while controlling for risk-aversion and other household characteristics. For the Tu measure the coefficient is negative as well, but insignificant. 29

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households do not in fact have standard utility functions but are loss-averse. In this paper we empirically test how loss-aversion affects household portfolio choice. We use a unique dataset from The Netherlands which contains data on household portfolios and as well as a series of questions which allow us to directly estimate each household's loss-aversion coefficient. Following the methods in Tu (2004), we derive our behavioral measures from a series of questions asking for rates of time-preference across gains versus losses and speeding up versus delaying transactions. These questions are based on experimental work by Loewenstein (1988) and Thaler (1981) who show that loss-averse individual's discount rates will vary depending on the framing of intertemporal choices. We then use this measure to test how loss-aversion affects household portfolio choice. We find that households with higher loss-aversion are less likely to participate in equity markets and avoid direct stockholding to a greater extent than mutual funds. However, after controlling for sample selection we do not find a significant relation between loss-aversion and household portfolio allocations to equity. Overall, the results indicate that loss-aversion is an important feature of households' investment decision making process with the ability to explain puzzling features of empirical household financial behavior. Acknowledgement We would like to thank Jeffrey Brown, Louis K.C. Chan, Bing Han, Terry Odean, Joshua Pollet, Thierry Post, Allen Poteshman, Joshua White, William Ziemba, three anonymous referees, and seminar participants at Case Western Reserve University, Michigan State University, Tulane University, University of Alberta, University of Illinois at Chicago, University of Illinois at Urbana-Champaign, and the People and Money Conference at DePaul University for helpful comments. This paper uses data from the DNB Household Survey. We are grateful to CentERdata at Tilburg University for providing this data. The usual disclaimer applies. Appendix A A.1. Delay of losses, Loewenstein specification Individuals select the delay payment such that they are indifferent between both alternatives: vð–ðX–RÞÞ + δðTÞvð–ð–RÞÞ = vð–ð–RÞÞ + δðTÞvð–ðX + PDL –RÞÞ

ðA1Þ

Using the specification of the value function, Eq. (A1) can be written as: –λðX–RÞ + δðTÞR = R–δðTÞλðX + PDL –RÞ

ðA2Þ

δðTÞλPDL = ð1–δðTÞÞλðX–RÞ + ð1–δðTÞÞR

ðA3Þ

Let pDL = PDL / X and r = R / X, then we find: pDL = ð1–δðTÞÞ½ð1–rÞ + ð1 = λÞr = δðTÞ

ðA4Þ

Given λ N 1 and 0 b δ(T) ≤ 1, the payment for delay of losses is non-negative. In the special case r = 1, we find: pDL = ð1 = λÞð1–δðTÞÞ = δðTÞ

ðA5Þ

A.2. Speed-up of gains, Loewenstein specification Individuals have chosen the speed-up payment such that they are indifferent between both alternatives: vð0–RÞ + δðTÞvðX–RÞ = vðX–PSG –RÞ + δðTÞvð0–RÞ

ðA6Þ

Please note that (X − PSG − R) is positive when R ≤ X − PSG, and negative when R N X − PSG. We substitute the piece-wise linear value function of prospect theory v(·): –λR + δðTÞðX–RÞ = ðX–PSG –RÞ–δðTÞλR; –λR + δðTÞðX–RÞ = λðX–PSG –RÞ–δðTÞλR;

PSG = ð1–δðTÞÞðX–RÞ + ð1–δðTÞÞλR; PSG = ð1–ð1 = λÞδðTÞÞðX–RÞ + ð1–δðTÞÞR;

if R≤ X–PSG if R N X–PSG

if R≤X–PSG if R N X–PSG

ðA7Þ

ðA8Þ

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457

Let pSG = PSG / X and r = R / X, then we find: if r≤1–pSG pSG = ð1–δðTÞÞ½ð1–rÞ + λr; pSG = ð1–ð1 = λÞδðTÞÞð1–rÞ + ð1–δðTÞÞr; if r N 1–pSG

ðA9Þ

Further we can prove that r ≤ 1 − pSG is equivalent to λ ≤ δ(T)(1 − r) / [(1 − δ(T))r], so that the expression for pSG in Eq. (A9) becomes fully exogenous.30 Given λ N 1 and 0 b δ(T) ≤ 1, the payment for speed-up of gains is non-negative. In the two special cases r = 1 and r = 0, Eq. (A9) reduces to: pSG = ð1–δðTÞÞ

ðA10Þ

A.3. Speed-up of losses, Loewenstein specification Individuals will select the speed-up premium such that they are indifferent between both alternatives: vðRÞ + δðTÞvð–ðX–RÞÞ = vð–ðX–PSL –RÞÞ + δðTÞvðRÞ

ðA11Þ

Please note that −(X − PSL − R) can be positive (e.g. when R = X) or negative (when R b X − PSL), depending on the premium PSL and the value of the reference point. Using the specification of the value function, Eq. (A11) can be written as: λPSL = ð1–δðTÞÞλðX–RÞ + ð1–δðTÞÞR; if PSL b X–R; PSL = ð1–δðTÞλÞðX–RÞ + ð1–δðTÞÞR; if PSL ≥ X–R:

ðA12Þ

Let pSL = PSL / X and r = R / X, then we find31: pSL = ð1–δðTÞÞ½ð1–rÞ + ð1 = λÞr; pSL = 1–λδðTÞ + ðλ–1ÞδðTÞr;

if rbλδðTÞ = ð1 + ðλ–1ÞδðTÞÞ if r≥λδðTÞ = ð1 + ðλ–1ÞδðTÞÞ

ðA13Þ

Given λ N 1 and 0 b δ(T) ≤ 1, the premium demanded for speed-up of losses is non-negative. In the two special cases r = 1 and r = 0, Eq. (A13) reduces to: pSL = ð1–δðTÞÞ

ðA14Þ

Appendix B B.1. Delay of losses, Tu specification Individuals select the delay payment such that they are indifferent between both alternatives: vð–ðX–RÞÞ + δðTÞvð0Þ = vð–ð0–RÞÞ + δðTÞvð–ðX + PDL ÞÞ

ðB1Þ

Using the specification of the value function, Eq. (B1) can be written as: –λðX–RÞ = R–δðTÞλðX + PDL Þ

ðB2Þ

δðTÞλPDL = ð1–λÞR + λXð1–δðTÞÞ

ðB3Þ

Let pDL = PDL / X and r = R / X, then we find: pDL = ½ð1–λÞð1 = λÞr + ð1–δðTÞÞ = δðTÞ

ðB4Þ

Note that the “indifference” payment pDL for the delay of losses can become negative, for example when λ N 1 and δ(T) = 1. A negative solution means that the individual does not want to delay the loss and needs to be compensated to do so. The survey question about delay of a tax assessment explicitly instructs respondents to answer zero in this case: “If you are not interested in getting an extension of payment or if you are not prepared to pay more for the extension of payment, please type 0 (zero).” In line with the framing of the question, we therefore define the relative delay payment pDL as Eq. (B4) when it is positive, and zero otherwise: pDL = maxf½ð1–λÞð1 = λÞr + ð1–δðTÞÞ = δðTÞ; 0g

ðB5Þ

30 Given λ N 0, it is easy to show that the condition pSG = (1 − δ(T))[(1 − r) + λr] ≤ 1 − r is equivalent to (1 − δ(T))λr ≤ δ(T)(1 − r). Similarly, pSG = (1 − (1 / λ) δ(T))(1 − r) + (1 − δ(T))r N 1 − r is equivalent to (1 − δ(T))λr N δ(T)(1 − r). 31 Given λ N 0, it is easy to show that the condition pSL = (1 − λδ(T))(1 − r) + (1 − δ(T))r ≥ 1 − r is equivalent to r ≥ λδ(T) / (1 + (λ − 1)δ(T)). Similarly, pSL = (1 − δ(T))[(1 − r) + (1 / λ)r] b 1 − r is equivalent to r b λδ(T) / (1 + (λ − 1)δ(T)).

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B.2. Speed-up of gains, Tu specification Individuals have chosen the speed-up payment such that they are indifferent between both alternatives: vð0Þ + δðTÞvðX–RÞ = vðX–PSG Þ + δðTÞvð0–RÞ

ðB6Þ

We substitute the piece-wise linear value function of prospect theory v(·): δðTÞðX–RÞ = ðX–PSG Þ–λδðTÞR

ðB7Þ

PSG = ð1–λÞδðTÞR + Xð1–δðTÞÞ

ðB8Þ

As mentioned in the main text, PSG can become negative: in that case respondents are not willing to speed-up the gain and instructed to write down zero. Hence: PSG = maxfð1–λÞδðTÞR + Xð1–δðTÞÞ; 0g pSG = maxfð1–λÞδðTÞr + ð1–δðTÞÞ; 0g

ðB9Þ ðB10Þ

B.3. Speed-up of losses, Tu specification Individuals will select the speed-up premium such that they are indifferent between both alternatives: vð0Þ + δðTÞvð–ðX–RÞÞ = vð–ðX–PSL ÞÞ + δðTÞvð–ð0–RÞÞ

ðB11Þ

Using the specification of the value function, Eq. (B11) can be written as: –λδðTÞðX–RÞ = –λðX–PSL Þ + δðTÞR

ðB12Þ

λPSL = ðλ–1ÞδðTÞR + λXð1–δðTÞÞ

ðB13Þ

Let pSL = PSL / X and r = R / X, then we find: pSL = ðλ–1Þð1 = λÞδðTÞr + ð1–δðTÞÞ

ðB14Þ

Given λ N 1 and 0 b δ(T) ≤ 1, the premium demanded for speed-up of losses is non-negative. References Agnew, J., Balduzzi, P., Sundén, A., 2003. Portfolio choice and trading in a large 401(k) plan. American Economic Review 93, 193–215. Alessie, R.J.M., Hochguertel, S., van Soest, A., 2002. Household portfolios in The Netherlands. In: Guiso, L., Haliassos, M., Jappelli, T. (Eds.), Household Portfolios. The MIT Press, Cambridge, MA, pp. 341–388. Alessie, R.J.M., Hochguertel, S., van Soest, A., 2004. Ownership of stocks and mutual funds: a panel data analysis. Review of Economics and Statistics 86, 783–796. Ang, A., Bekaert, G., Liu, J., 2004. Why stocks may disappoint. Journal of Financial Economics 76, 471–508. Barberis, N., Huang, M., 2001. Mental accounting, loss aversion, and individual stock returns. Journal of Finance 56, 1247–1292. Barberis, N., Huang, M., Santos, T., 2001. Prospect theory and asset prices. Quarterly Journal of Economics 116, 1–53. Barberis, N., Huang, M., Thaler, R.M., 2006. Individual preferences, monetary gambles and stock market participation: a case for narrow framing. American Economic Review 96, 1069–1090. Barsky, R.B., Juster, F.T., Kimball, M.S., Shapiro, M.D., 1997. Preference parameters and behavioral heterogeneity: an experimental approach in the health and retirement survey. Quarterly Journal of Economics 112, 537–579. Benartzi, S., Thaler, R.H., 1995. Myopic loss-aversion and the equity premium puzzle. Quarterly Journal of Economics 110, 73–92. Benzion, U., Rapoport, A., Yagil, J., 1989. Discount rates inferred from decisions: an experimental study. Management Science 35, 270–284. Berkelaar, A., Kouwenberg, R., Post, T., 2004. Optimal portfolio choice under loss aversion. Review of Economics and Statistics 86, 973–987. Bertaut, C.C., 1998. Stockholding behavior of U.S. households: evidence from the 1983–1989 survey of consumer finances. Review of Economics and Statistics 80, 263–275. Cocco, J., 2005. Portfolio choice in the presence of housing. Review of Financial Studies 18, 535–567. Cocco, J.F., Gomes, F.J., Maenhout, P.J., 2005. Consumption and portfolio choice over the life cycle. Review of Financial Studies 18, 491–533. Cronbach, L.J., 1959. Coefficient alpha and the internal structure of tests. Psychometrika 16, 297–334. Das, M., van Soest, A., 1999. A panel data model for subjective information on household income growth. Journal of Economic Behavior and Organization 40, 409–426. Davis, S.J., Kubler, F., Willen, R., 2006. Borrowing costs and the demand for equity over the life cycle. Review of Economics and Statistics 88, 348–362. Donkers, B., van Soest, A., 1999. Subjective measures of household preferences and financial decisions. Journal of Economic Psychology 20, 613–642. Frederick, S., Loewenstein, G., O'Donoghue, T., 2002. Time discounting and time-preference: a critical review. Journal of Economic Literature 40, 351–401. Gomes, F.J., 2005. Portfolio choice and trading volume with loss-averse investors. Journal of Business 78, 675–706. Guiso, L., Jappelli, T., Terlizzese, D., 1996. Income risk, borrowing constraints, and portfolio choice. American Economic Review 86, 158–172. Guiso, L., Sapienza, P., Zingales, L., 2008. Trusting the stock market. Journal of Finance 63, 2557–2600. Haliassos, M., Bertaut, C.C., 1995. Why do so few hold stocks? Economic Journal 105, 1110–1129. Heaton, J., Lucas, D., 1997. Market frictions, savings behavior, and portfolio choice. Macroeconomic Dynamics 1, 76–101. Heaton, J., Lucas, D., 2000. Portfolio choice in the presence of background risk. Economic Journal 110, 1–26. Hong, H., Kubik, J.D., Stein, J.C., 2004. Social interaction and stock market participation. Journal of Finance 59, 137–163.

S.G. Dimmock, R. Kouwenberg / Journal of Empirical Finance 17 (2010) 441–459

459

Ivkovic, Z., Sialm, C., Weisbenner, S., 2008. Portfolio concentration and the performance of individual investors. Journal of Financial and Quantitative Analysis 43, 613–656. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decision under risk. Econometrica 47, 363–391. Loewenstein, G., 1988. Frames of mind in intertemporal choice. Management Science 34, 200–214. Loewenstein, G., Prelec, D., 1992. Anomalies in intertemporal choice: evidence and an interpretation. Quarterly Journal of Economics 107, 573–597. Miniaci, R., Weber, G., 2002. Econometric issues in the estimation of household portfolio models. In: Guiso, L., Haliassos, M., Jappelli, T. (Eds.), Household Portfolios. The MIT Press, Cambridge, MA. Nunnally, J.C., 1978. Psychometric Theory, Second Edition. McGraw-Hill Inc., New York, NY. Polkovnichenko, V., 2005. Household portfolio diversification: a case for rank-dependent preferences. Review of Financial Studies 18, 1467–1501. Puri, M., Robinson, D.T., 2007. Optimism and economic choice. Journal of Financial Economics 86, 71–99. Shelley, M.K., 1993. Outcome signs, question frames and discount rates. Management Science 39, 806–815. Thaler, R.H., 1981. Some empirical evidence on dynamic inconsistency. Economics Letters 8, 201–207. Thaler, R.H., Tversky, A., Kahneman, D., Schwartz, A., 1997. The effect of myopia and loss-aversion on risk taking: an experimental test. Quarterly Journal of Economics 112, 1937–1996. Tu, Q., 2004. Reference points and loss aversion in intertemporal choice, Unpublished working paper, Tilburg University. Tversky, A., Kahneman, D., 1991. Loss-aversion in riskless choice: a reference dependent model. Quarterly Journal of Economics 106, 1039–1061. Tversky, A., Kahneman, D., 1992. Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty 5, 297–323. Viceira, L.M., 2001. Optimal portfolio choice for long-horizon investors with nontradable labor income. Journal of Finance 56, 433–470. Vissing-Jorgenson, A., 2002. Towards an explanation of household portfolio choice heterogeneity: nonfinancial income and participation cost structures, Unpublished working paper, University of Chicago. Wooldridge, J.M., 1995. Selection corrections for panel data models under conditional mean independence assumptions. Journal of Econometrics 68, 115–132. Wooldridge, J.M., 2005. Simple solutions to the initial conditions problems in dynamic, nonlinear panel data models with unobserved heterogeneity. Journal of Applied Econometrics 20, 39–54. Yao, R., Zhang, H.H., 2005. Optimal consumption and portfolio choices with risky housing and borrowing constraints. Review of Financial Studies 18, 197–239.