Loss-aversion and household portfolio choice

Loss-aversion and household portfolio choice
Stephen G. Dimmock a,
⁎, Roy Kouwenberg b,c
a
Michigan State University, 306 Eppley Center, East Lansing, MI, 48824, United States
b
Mahidol University, Thailand
c
Erasmus University Rotterdam, The Netherlands
a r t i c l e i n f o a b s t r a c t
Article history:
Received 7 August 2008
Received in revised form 20 November 2009
Accepted 24 November 2009
Available online 30 November 2009
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.
JEL classification:
G11
D14
D1
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.
Journal of Empirical Finance 17 (2010) 441–459
⁎ Corresponding author. Tel.: +1 517 432 7133.
E-mail addresses: dimmock@msu.edu (S.G. Dimmock), cmroy@mahidol.ac.th (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
Contents lists available at ScienceDirect
Journal of Empirical Finance
journal homepage: www.elsevier.com/locate/jempfin

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).
3
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. Vissing-
Jorgenson (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.
442 S.G. Dimmock, R. Kouwenberg / Journal of Empirical Finance 17 (2010) 441–459

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 loss-
averse 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.
443
S.G. Dimmock, R. Kouwenberg / Journal of Empirical Finance 17 (2010) 441–459

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), Vissing-
Jorgenson (2002). As Panel A of Table 1 shows, equity owners in this sample have greater wealth than non-participants. Vissing-
Jorgenson 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.
Table 1
Summary statistics.
Variable Full sample Equity owners Non-owners
Panel A — control variables
Total financial assets 67,007 134,268 35,534
Income 50,505 60,665 45,750
Age 49.9 52.5 48.7
Panel B — employment status
Regular employment 65.4% 60.5 67.6
Unemployed 1.4% 1.2 1.5
Retired 15.6% 19.9 13.6
Disabled 8.9% 10.5 8.1
Self-employed 1.9% 1.9 1.9
Other 6.9% 6.0 7.4
Panel C — education
Low education 5.4% 4.5 5.8
Intermediate/low education 11.5% 10.1 12.2
Intermediate/high education 10.3% 11.8 9.6
Vocational 1 31.8% 26.4 34.3
Vocational 2 24.5% 27.8 22.9
University education 16.5% 19.5 15.1
Panel D — other control variables
Risk-aversion 5.0 4.8 5.1
Home owner 56.2% 66.9 51.2
Unsecured debt to total financial assets 105.6% 21.8 144.8
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).

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.
Table 2
Equity ownership.
Year Proportion of sample holding equity Equity/TFA
Pooled 33.1% 37.1%
1997 30.7% 35.3%
1998 30.8% 38.0%
1999 34.5% 38.4%
2000 31.0% 41.6%
2001 35.9% 39.1%
2002 35.7% 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.
445

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 speed-
up 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 0bR≤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:
vðxÞ =
x; if x≥ 0
λx; if x b 0

ð4Þ
where λN1 implies loss-aversion. Using this specification of the value function, Eq. (3) can be written as:
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 (0bRbX), adjusted to receiving the amount X.

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 λN1 and 0bδ(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 0br≤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 + PDGÞ 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 λN1 and 0bδ(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 λN1 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 b0). 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).”
447

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 (RbX) and loss-
aversion is less than two (λb2). 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).
20
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.

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.
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 3 m 1000 Receive extra 10.2% 3.5% 20.6% 18.3%
DG 12 m 1000 Receive extra 25.2% 10.0% 30.6% 4.7%
DG 3 m 100,000 Receive extra 7.8% 2.0% 20.1% 6.4%
DG 12 m 100,000 Receive extra 17.0% 10.0% 26.1% 1.6%
DL 3 m 1000 Lose more 1.2% 0.0% 3.6% 69.5%
DL 12 m 1000 Lose more 3.5% 0.0% 6.9% 56.7%
DL 3 m 100,000 Lose more 0.6% 0.0% 1.2% 48.9%
DL 12 m 100,000 Lose more 2.2% 0.5% 3.6% 42.7%
SG 3 m 1000 Receive less 1.3% 0.0% 5.4% 76.8%
SG 12 m 1000 Receive less 4.3% 0.0% 13.6% 63.3%
SG 3 m 100,000 Receive less 1.4% 0.0% 6.5% 59.3%
SG 12 m 100,000 Receive less 3.8% 0.0% 12.9% 48.8%
SL 3 m 1000 Loss reduction 4.6% 2.5% 5.6% 27.8%
SL 12 m 1000 Loss reduction 9.9% 7.5% 11.3% 23.7%
SL 3 m 100,000 Loss reduction 2.7% 1.0% 4.5% 21.6%
SL 12 m 100,000 Loss reduction 6.4% 5.0% 7.9% 20.8%
DG DL SG SL
Panel B: Correlations and proportion of discount rates in ColumnNRates in row
DG – 97.2% 89.0% 63.0%
DL 0.203 – 69.5% 28.5%
SG 0.102 0.192 – 37.9%
SL 0.242 0.181 0.165 –
Panel C: Signiﬁcance tests of differences in medians
DG –
DL 0.000 –
SG 0.000 0.000 –
SL 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.
449

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 risk-
aversion 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.
Table 4
Frequency of suspect responses and reliability estimates.
Panel A: Average frequency of suspect responses
P (1000)NP (100,000) P (3 m)NP (12 m)
DG 0.015 0.011
DL 0.035 0.016
SG 0.021 0.032
SL 0.045 0.037
Panel B: Cronbach's alpha
Average 1997 1998 1999 2000 2001 2002
DG 0.919 0.888 0.896 0.921 0.967 0.948 0.893
DL 0.894 0.909 0.936 0.847 0.891 0.914 0.868
SG 0.885 0.885 0.832 0.871 0.878 0.925 0.916
SL 0.821 0.798 0.763 0.839 0.781 0.875 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.
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.

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)=δi
T
. 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 N0, 0bδi ≤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:
ΣtpDG;i;tðX; TÞ= nDG;i–½λið1–δT
i Þδ−T
i = 0;
ΣtpSG;i;tðX; TÞ= nSG;i–½ð1–δT
i Þ = 0;
ΣtpDL;i;tðX; TÞ= nDL;i–½ð1 = λiÞð1–δ
T
i Þδ
−T
i = 0;
ΣtpSL;i;tðX; TÞ= nSL;i–½ð1–δT
i Þ = 0;
ð17Þ
with Xa{1000; 100,000}, T=1 and t {1997, 1998, …., 2002} and subject to λi N0, 0bδi ≤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
:
ΣXΣT ½uSG;iðX; TÞ
2
+ uDG;iðX; TÞ
2
+ uSL;iðX; TÞ
2
+ uDL;iðX; TÞ
2
:
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 2-
stage 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 N0
as λi ≥0.1, and δi N0 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.
451

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 loss-
aversion 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 5.61 2.47 8.22 1.00 25.11
Loewenstein, r=0.5 3.50 1.68 6.50 0.47 10.76
Loewenstein, r=0.25 4.96 2.61 6.57 0.25 17.92
Tu, r=1 1.17 1.11 0.19 0.99 1.60
Tu, r=0.5 1.33 1.17 0.41 1.01 2.26
Tu, r=0.25 1.64 1.36 0.76 1.03 3.23
Panel B: Discount factor estimates
Loewenstein, r=1 0.95 0.96 0.043 0.86 0.99
Loewenstein, r=0.5 0.93 0.95 0.050 0.83 0.99
Loewenstein, r=0.25 0.93 0.95 0.053 0.82 0.99
Tu, r=1 1.00 1.00 0.006 0.98 1.00
Tu, r=0.5 0.98 1.00 0.040 0.90 1.00
Tu, r=0.25 0.97 1.00 0.054 0.85 1.00
Panel C: Estimation error
Loewenstein, r=1 0.043 0.011 0.084 0.0005 0.201
Loewenstein, r=0.5 0.098 0.026 0.161 0.0012 0.525
Loewenstein, r=0.25 0.106 0.026 0.186 0.0012 0.592
Tu, r=1 0.071 0.018 0.124 0.0012 0.343
Tu, r=0.5 0.055 0.015 0.099 0.0005 0.231
Tu, r=0.25 0.045 0.013 0.086 0.0004 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.

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
Table 6
Random-effects probit model of the participation decision.
Variable 1 2 3 4
Loewenstein Loewenstein Tu Tu
Loss-aversion −0.014** −0.015** −0.093* −0.103*
[−2.38] [−2.55] [−1.66] [−1.84]
Discount rate −0.677 −0.564 −0.124 −0.12
[−0.65] [−0.55] [−0.16] [−0.15]
Risk-aversion −0.110*** −0.110***
[−4.71] [−4.70]
Initial participation 0.249*** 0.246*** 0.251*** 0.247***
[9.68] [9.54] [9.73] [9.60]
Lagged participation 1.152*** 1.154*** 1.157*** 1.159***
[12.60] [12.60] [12.65] [12.66]
TFA/1000 0.017*** 0.017*** 0.017*** 0.017***
[17.14] [17.06] [17.15] [17.07]
TFA/1,000,000 Squared −0.019*** −0.019*** −0.019*** −0.019***
[−12.37] [−12.38] [−12.35] [−12.37]
Homeowner 0.242*** 0.245*** 0.245*** 0.247***
[3.05] [3.09] [3.08] [3.12]
Debt to TFA −0.052*** −0.053*** −0.052*** −0.053***
[−3.22] [−3.23] [−3.22] [−3.22]
Income/1000 0.003** 0.003** 0.004** 0.004**
[2.13] [2.16] [2.24] [2.28]
Income/1,000,000 squared −0.008 −0.009 −0.009 −0.009
[−0.93] [−1.01] [−1.00] [−1.07]
Age −0.038* −0.039* −0.039* −0.040**
[−1.92] [−1.95] [−1.95] [−1.99]
Age squared 0.328 0.346* 0.326 0.345*
[1.64] [1.74] [1.63] [1.74]
Employment effects Yes Yes Yes Yes
Education effects Yes Yes Yes Yes
Constant Yes Yes Yes Yes
Log likelihood −1997.9 −1986.9 −2000.5 −1989.6
Akaike I.C. 4053.9 4033.8 4059.1 4039.1
*,**,*** 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, risk-
aversion, 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.
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.
453

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 1 2 3 4
Loss-aversion 0.002 0.002 0.019 0.017
[1.31] [1.26] [1.54] [1.45]
Discount rate −0.110 −0.113 −0.023 −0.032
[−0.52] [−0.53] [−0.14] [−0.20]
Risk-aversion −0.007* −0.007*
[−1.79] [−1.76]
TFA/1000 0.000 0.000 0.000 0.000
[−1.13] [−1.00] [−1.14] [−1.01]
TFA/1,000,000 squared 0.000 0.000 0.000 0.000
[−0.33] [−0.45] [−0.33] [−0.45]
Homeowner −0.022* −0.022 −0.022 −0.022
[−1.65] [−1.60] [−1.64] [−1.59]
Debt to TFA 0.009** 0.009** 0.009** 0.009**
[2.08] [2.08] [2.09] [2.09]
Income/1000 0.000 0.000 0.000 0.000
[0.18] [0.23] [0.16] [0.22]
Income/1,000,000 squared 0.000 0.000 0.000 0.000
[−0.12] [−0.14] [−0.13] [−0.14]
Age 0.016 0.016 0.016 0.016
[1.07] [1.07] [1.07] [1.07]
Age squared −0.001 0.001 0.000 0.002
[−0.01] [0.01] [0.00] [0.02]
Mill's ratio −0.058*** −0.055*** −0.058*** −0.055***
[−4.20] [−3.99] [−4.24] [−4.01]
Year effects Yes Yes Yes Yes
Quasi-fixed effects Yes Yes Yes Yes
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.

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
Table 8
Ordered probit model of the equity type decision.
Variable 1 2 3 4
Loss-aversion −0.011 −0.012 −0.059 −0.068
(3.19)*** (3.42)*** (1.82)* (2.08)**
Discount rate −0.483 −0.391 −0.663 −0.676
(0.84) (0.68) (1.44) (1.48)
Risk-aversion −0.092 −0.092
(6.68)*** (6.67)***
TFA/1000 0.008 0.008 0.008 0.008
(17.68)*** (17.63)*** (17.80)*** (17.77)***
TFA/1,000,000 squared −0.008 −0.009 −0.009 −0.009
(12.93)*** (12.86)*** (13.02)*** (12.99)***
Homeowner 0.114 0.123 0.114 0.123
(2.44)** (2.63)*** (2.44)** (2.62)***
Debt to TFA −0.037 −0.038 −0.037 −0.038
(3.04)*** (2.96)*** (3.07)*** (2.97)***
Income/1000 0.001 0.001 0.001 0.001
(1.04) (1.09) (1.24) (1.29)
Income/1,000,000 squared 0.001 0 0 −0.001
(0.11) (0.07) (0.04) (0.22)
Age 1.326 1.308 1.337 1.319
(27.03)*** (26.56)*** (27.20)*** (26.74)***
Age squared −0.025 −0.024 −0.024 −0.024
(2.03)** (2.01)** (2.01)** (1.99)**
Year effects Yes Yes Yes Yes
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.
28
The results are similar if we use a multinomial probit model.
29
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.
455

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 λN1 and 0bδ(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 RNX−PSG. We substitute the piece-wise linear
value function of prospect theory v(·):
–λR + δðTÞðX–RÞ = ðX–PSG–RÞ–δðTÞλR; if R≤ X–PSG
–λR + δðTÞðX–RÞ = λðX–PSG–RÞ–δðTÞλR; if R N X–PSG
ðA7Þ
PSG = ð1–δðTÞÞðX–RÞ + ð1–δðTÞÞλR; if R≤X–PSG
PSG = ð1–ð1 = λÞδðTÞÞðX–RÞ + ð1–δðTÞÞR; if R N X–PSG
ðA8Þ

Let pSG =PSG /X and r=R/X, then we find:
pSG = ð1–δðTÞÞ½ð1–rÞ + λr; if r≤1–pSG
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 λN1 and 0bδ(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 RbX−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; if rbλδðTÞ = ð1 + ðλ–1ÞδðTÞÞ
pSL = 1–λδðTÞ + ðλ–1ÞδðTÞr; if r≥λδðTÞ= ð1 + ðλ–1ÞδðTÞÞ
ðA13Þ
Given λN1 and 0bδ(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 + PDLÞÞ ðB1Þ
Using the specification of the value function, Eq. (B1) can be written as:
–λðX–RÞ = R–δðTÞλðX + PDLÞ 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 λN1 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 λN0, 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))rN1−r is equivalent to (1−δ(T))λrNδ(T)(1−r).
31
Given λN0, 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]b1− r is equivalent to rbλδ(T)/(1+(λ−1)δ(T)).
457

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–PSGÞ + δðTÞvð0–RÞ ðB6Þ
We substitute the piece-wise linear value function of prospect theory v(·):
δðTÞðX–RÞ = ðX–PSGÞ–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 ðB9Þ
pSG = maxfð1–λÞδðTÞr + ð1–δðTÞÞ; 0g 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–PSLÞÞ + δðTÞvð–ð0–RÞÞ ðB11Þ
Using the specification of the value function, Eq. (B11) can be written as:
–λδðTÞðX–RÞ = –λðX–PSLÞ + 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 λN1 and 0bδ(T)≤1, the premium demanded for speed-up of losses is non-negative.
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