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Prospect Theory and Its Applications in
Bing Han and Jason Hsu
Current Version: December 2004
Han is with the Fisher College of Business at the Ohio State University. He can be reached at
firstname.lastname@example.org, or (614) 292-1875. Hsu is with the Research Aﬃliates, LLC. He can be reached
at email@example.com or (626) 584-2145.
Prospect theory is an important theory for decision making under uncertainty. It
departs from the traditional expected utility framework in important ways. It pro-
vides psychological underpinnings for the behavioral approaches to portfolio selection
that are quite diﬀerent from the traditional approaches such as the mean-variance
framework. Prospect theory was developed by two psychologist, Daniel Kahneman
and Amos Tversky, and published in the Econometrica in 1979. Kahneman won the
2002 Nobel Prize for Economics “for having integrated insights from psychological
research into economic science, especially concerning human judgment and decision-
making under uncertainty.”
The 1979 paper on prospect theory was singled out for
praise by the Royal Swedish Academy of Science.
Since its ﬁrst app earance, prospect theory has been revised and improved in many
ways (e.g., Tversky and Kahneman, 1992, Wakker and Amos Tversky, 1993, Wakker
and Zank, 2002). It has also been widely applied to many areas of social science.
This paper reviews the prosp ect theory and its applications in ﬁnance.
1 What is Prospect Theory?
The traditional ﬁnance theory assumes that investors make decision under uncertainty
by maximizing expected utility of wealth or consumption. The expected utility theory
is mathematically elegant and is a rational-based framework built upon axioms. How-
ever, the underlying assumptions have been shown by many studies to be inaccurate
description of how people actually behave when choosing among risky alternatives.
Kahneman and Tversky (1979) propose prospect theory as a descriptive model of
decision making under uncertainty. The prospect theory is NOT a normative theory,
but a descriptive approach to explain real world behavior. Kahneman and Tversky
relied on a series of small experiments to identify the manner in which people make
choices in the face of risk.
Like its mean-variance theory counterpart from the traditional approach, prospect
theory focuses on the way people choose among alternatives. But the theories are
Tversky passed away in 1996.
For example, the famous Allias paradox show that the independence axiom of expected utility
theory is routinely violated in the real life.
diﬀerent. People who conform to prospect theory tend to violate the principles that
underlie mean-variance theory.
1.1 Key Elements of Prospect Theory
There are four ingredients in prospect theory that distinguishes it from mean-variance
theory. First, people in mean-variance theory choose among alternatives based on the
eﬀect of the outcomes on the levels of their wealth. In contrast, people in prospect
theory choose based on the eﬀect of outcomes on changes in their wealth, relative to
their reference point. In other words, prospect theory agents evaluate outcomes in
terms of gains and losses relative to a reference point.
Second, people in mean-variance theory are risk averse in all of their choices. In
contrast, prospect theory agents are risk-averse in the domain of gains but risk-seeking
when all changes in wealth are perceived as losses. Consider the following experiment
that illuminates the features of prospect theory. Imagine that you face a concurrent
choice within two pairs (A vs. B and C vs. D), where:
• A = a sure gain of $24,000
• B = a 25% chance to gain $100,000 and a 75% chance to gain nothing.
• C = a sure loss of $75,000
• D = 75% chance to lose $100,000 and 25% chance to lose nothing.
Kahneman and Tversky found that more people chose A than B and more people
chose D than C. This common choice is a puzzle if agents are always risk-averse and
never risk-seeking. While the choice of A over B is consistent with risk aversion, the
choice of D over C is not. Instead it is consistent with risk-seeking. Note that the
$25,000 expected gain of B (25% of a $100,000 gain), is greater than the sure $24,000
gain of A, so the common choice of A over B is consistent with risk-aversion. However,
the common choice of D over C indicates that most people make some choices as if
This aspect of prospect theory is similar to habit-formation or catching-up-with-Jones utility
function where agents care about their consumption relative to some benchmark levels (e.g., their
neighbor’s consumption, or a certain level of consumption they have been used to have). Other
aspects of prospect theory clearly are diﬀerent from the habit-formation utility.
they are risk-seeking. Note that the expected $75,000 loss of D (75% of a $100,000
loss) is equal to the sure $75,000 loss of C, but D is riskier than C since it can imp ose
a $100,000 loss. Kahneman and Tversky refers to the choice of D over C as “aversion
to a sure loss,” since C imposes a sure loss while D does not. An individual who has
not made peace with his losses is likely to accept gambles that would be unacceptable
The third feature of prospect theory is often called “loss aversion.” An individual
is loss averse if she or he dislikes symmetric 50-50 bets and, moreover, the aversiveness
to such bets increases with the absolute size of the stakes. In other words, there is an
asymmetry in how prospect agent perceive gains and losses of equal amounts. Loss
aversion says that the disutility of giving up a valued good is much higher than the
utility gain associated with receiving the same good.
Loss aversion applies when one
is avoiding a loss even if it means accepting a higher risk. Some argue that investors
and traders show no risk aversion, but an aversion against losses.
The concept of loss aversion can be illustrated by an example in Samuelson (1963).
Samuelson once oﬀered a colleague the following bet: ﬂip a coin, heads you win $200
and tails you lose $100. Samuelson reports that his colleague turned this bet down: “I
won’t bet because I would feel the $100 loss more than the $200 gain.” This sentiment
is the intuition behind the concept of loss aversion.
Finally, people in mean-variance theory treat risk objectively, by its probabilities.
In contrast, the utility of prospect theory agent depends not on the original probability
but rather on the transformed probability. These transformed probabilities can be
viewed as decision weights, or subjective probabilities. They do not just measure the
perceived likelihood of an event. Instead, they measure the impact of events on the
desirability of prospects. This features of the prosect theory can explain several key
violations of expected utility theory, including the famous Allais’ parodox. People
in prospect theory overweight small probabilities. Overweighting small probabilities
explains people’s demand for lotteries oﬀering a small chance of large gain, and for
insurance protecting against a small chance of a large loss.
To summarize, under prospect theory, people evaluate risk using a value function
This is sometimes called endowment eﬀect: the value appears to change when a good is incor-
porated into one’s endowment. For example, there is a large diﬀerence in willingness to pay and
willingness to accept, even if the the sellers where just equipped with the good.
that is deﬁned over gains and losses, is concave over gains and convex over losses, and
is kinked at the origin; and using transformed rather than objective probabilities by
applying a weighting function.
1.2 Prospect Theory Value Function
Let us illustrate how prospect theory agent evaluates risk, and how the four elements
of the prospect theory can be reﬂected in the value or utility function. Consider a
simple gamble that with probability p, you get x and with probability q, you receive
y, where x < 0 < y, and p + q = 1. In the expected utility framework, the agent
evaluate this risk by computing
p U (W + x) + q U (W + y)
where W is agent’s current wealth. Under the prospect theory of Kahneman and
Tversky (1979), the agent assigns the gamble the value
π(p)v(x) + π(q)v(y)
where π is a probability transformation, and v is an S-shaped prospect theory value
function like that shown in Figure 1.
Loss aversion implies a kink in the prospect theory value function around the
reference point (the “origin”), with the slope being steeper for losses than for gains.
The decline in utility for a loss (measured relative to a reference point) exceeds the
increase in utility for an equal-sized gain (relative to the same reference point). Kah-
neman and Tversky (1979) infer the kink in utility from the widespread aversion to
bet that payoﬀs $110 when a head comes up with a coin ﬂip and -$100 when a tail
come up. Such aversion is hard to explain with diﬀerentiable utility function, because
the very high local risk aversion required to do so typically predicts implausibly high
aversion to large-scale gambles, such as 50:50 bet to win $20 million or lose $10,000,
which clearly is not reasonable (e.g., Epstein and Zin, 1990, Rabin, 2000, Barberis,
Huang and Thaler, 2003). Extensive experiments reveal that people are generally
about twice as unhappy about a given loss as the joy brought by a gain of the same
size. For example, the disutility of lossing $100 is roughly twice the utility of gaining
A typical prospect theory value function is given by the following:
v(W ) =
(W − R)
1 − γ
, if W ≥ R;
v(W ) = −λ
(R − W )
1 − γ
, if W < R
where R be a reference level, γ is a positive constant, and λ > 1 is another constant.
In ﬁgure 1, γ = 0.5 and λ = 2.25. Function v(W ) is continuous and diﬀerentiable
(except at W = R). It is S-shaped (see Figure 1): it is monotonic increasing; it is
concave in the region W > R and convex in the region W < R. The S-shaped value
function can also be generated by the following function:
v(W ) = 1 − e
, if W ≥ R;
v(W ) = λ(e
− 1), if W < R
1.3 Cumulative Prospect Theory
Perhaps the most important change to the original prospect theory is that of Tver-
sky and Kahneman (1992) about how probabilities are transformed. The original
speciﬁcation in Kahneman and Tversky (1979) applies only to binomial gambles,
and violates the ﬁrst-order stochastic dominance property. The essence of change in
Tversky and Kahneman (1992) is that the transformation is ﬁrst applied to the cu-
mulative density function rather than directly to the probabilities. Thus, the Tversky
and Kahneman (1992) version is usually called the cumulative prospect theory. The
cumulative prospect theory applies to the most general gambles. It is also consistent
with ﬁrst-order stochastic dominance.
More precisely, cumulative prospect theory says that the agent evaluates a gamble
< · · · < x
= 0 < x
< · · · < x
with corresponding probabilities p
, ·, p
, · · · , p
by assigning it a value
where v is a S-shaped value function as above, and
+ . . . + p
) − w
+ . . . + p
) if 0 ≤ i ≤ n;
+ . . . + p
) − w
+ . . . + p
), if −m ≤ i ≤ 0
being the probability weighting function for gains and losses respec-
tively. The most common probability weighting function is given by
(p) = w
+ (1 − p)
, 0 < α ≤ 1
1.4 Prospect Theory and Framing
Before we turn to the applications of prospect theory in ﬁnance, it is worth emphasiz-
ing that the framing of alternatives aﬀect choices in the prospect theory, and people
are assumed to be passive in accepting the frames or problem descriptions oﬀered
to them. In contrast, the traditional theory implicitly assumes that frames does not
aﬀect choice. For example, in the example described before (see Page 2), the common
choice of the A & D combination is stochastically dominated by the less frequently
chosen B & C combination. Note that A & D oﬀers a 25% chance to win $24,000 and
a 75% chance to lose $76,000, while B & C oﬀers a 25% chance to win $1,000 more
and a 75% chance to lose $1,000 less. Even though the Kahneman and Tversky’s
experiment instructions indicate that the choice among A, B, C and D is concurrent,
people tend to frame the choice into one from A and B and one from C or D, over-
looking the link between the two choices and its relationship to the ﬁnal level of their
wealth. This has important implication for investors’ portfolio choices, which we will
discuss in detail later.
Prospect theory also implies a unique relationship of risk taking to positive and
negative framing. Negatively framed problems decrease risk bearing and encourage
risk seeking. Since losses loom larger than gains, it appears that humans follow
conservative strategies when presented with a positively-framed dilemma, and risky
strategies when presented with negatively-framed ones. To illustrate, consider the ex-
periment in Kahneman and Tversky (1984) where they asked a representative sample
of physicians the following question.
Imagine that the U.S. is preparing for the outbreak of an unusual Asian
disease, which is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Assume that the exact scientiﬁc
estimates of the consequences of the programs are as follows: If program
A is adopted, 200 people will be saved. If program B is adopted, there
is a one-third probability that 600 people will be saved and a two-thirds
probability that no people will be saved. Which of the two programs
would you favor?
Notice that the preceding dilemma is positively framed. It views the dilemma
in terms of “lives saved.” When the question was framed in this manner, 72% of
physicians chose A, the safe-and-sure strategy, but only 28% chose program B, the
risky strategy. An equivalent set of physicians considered the same dilemma, but
with the question framed negatively:
Imagine that the U.S. is preparing for the outbreak of an unusual Asian
disease, which is expected to kill 600 people. Two alternative programs to
combat the disease have been proposed. Assume that the exact scientiﬁc
estimates of the consequences of the programs are as follows: If program
C is adopted, 400 people will die. If program D is adopted, there is a
one-third probability that nobody will die and a two-thirds probability
that 600 people will die. Which of the two programs would you favor?
The two questions examine an identical dilemma. Two hundred of 600 people
saved is the same as 400 of 600 lost. However, when the question was framed nega-
tively, and physicians were concentrating on losses rather than gains, they voted in a
dramatically diﬀerent fashion. When framed negatively, 22% of the physicians voted
for the conservative strategy and 72% of them opted for the risky strategy! Clearly,
framing can powerfully inﬂuence the way a problem is perceived, which in turn can
lead to the favoring of radically diﬀerent solutions.
Related to framing, psychologists have long known of the existence of the “positiv-
ity bias,” which states that humans overwhelmingly expect good things (as opposed
to neutral or bad things) to occur. If perceivers construct a world in which primar-
ily positive elements are expected, then negative information becomes perceptually
salient. We also know that people stop to examine disconﬁrmations to a much higher
degree than conﬁrmations.
Negative information is often highly informative and
thus may be assigned extra weight in the decision-making process. This may have
important implications for investors’ reaction for good v.s. bad earnings or analyst
2 Applications of Prospect Theory
Two types of applications are emphasized. Prospect theory can help us understand
the portfolio choices and trading b ehavior of both individual investors and money
managers in the ﬁnancial market, such as why many investors tend to hold on to
their losers and why they hold very undiversiﬁed portfolios. Prospect theory has also
been applied to explain almost all well-known asset pricing “anomalies” including
the equity premium puzzle, the proﬁtability of value and momentum strategy, ex-
cess volatility, IPO underpricing and long-term performance of IPOs. Each of these
applications will be discussed in this section.
2.1 Portfolio Choice
Expected utility theory implies that investors hold well diversiﬁed portfolios, vary
their risk exposure by selecting the right mix of the risk free security and a risky
fund that is itself well diversiﬁed. By way of contrast, prospect theory implies that
investors do not choose well-diversiﬁed portfolios. In particular, people ignore covari-
ance among security returns and therefore, choose stochastically dominated portfolios
that lie below the eﬃcient frontier. They also combine very safe and very risky choices
in their portfolios (insurance and lottery tickets).
By not looking, you won’t ﬁnd anything that may cause you to worry. If you look, you may ﬁnd
something you don’t want to ﬁnd!
Shefrin and Statman (2000) explain that behavioral portfolios of securities are
structured as separate layers of a pyramid, e.g., a “downside protection” layer and
an “upside potential” layer. For example, in the case of pension fund portfolios, the
line between the downside protection and upside potential layers is the full funding
line. The downside protection layer contains assets needed for full funding of pension
obligations. The upside potential layer contains assets beyond those necessary for
full funding. For individual investors, they may have a low aspiration layer that is
designed to avoid poverty as well as a high aspiration layer for a shot at riches.
Prospect theory aﬀects the contents of layers in the pyramid of behavioral portfo-
lios. The parameters that are relevant to asset allocation in the behavioral portfolio
theory are the relative importance of the upside potential goal relative to the down-
side protection goal and the reference points of the upside and downside goals. The
greater importance that the investor attaches to the upside potential goal, the higher
is his allocation of his wealth to the upside potential layer. At the same time, a higher
reference point for the upside potential layer will be accompanied by the selection of
securities that are more “speculative.” The portfolios of prospect theory investors are
sensitive to the location of the reference point. For low reference points, prospect the-
ory investors choose traditional portfolios. Higher reference points induce risk-seeking
behavior, or the reluctance to engage in trade.
The S-shape of the utility function also matters for behavioral p ortfolio choice.
Higher concavity in the domain of gains reﬂects earlier satiation with a given security,
and early satiation leads to an increase in the number of securities in a layer. Another
determining factor is the degree of aversion to realization of losses. Investors who are
aware of their aversion to the realization of losses hold more cash so as to avoid the
need to satisfy liquidity needs by realization of losses. Moreover, portfolios of such
investors contain securities held solely because selling them entails the realization of
losses. These portfolios might seem well diversiﬁed, but the large number of securities
they contain is designed for avoiding the realization of losses, not the beneﬁt of
2.2 Disposition Eﬀect
This subsection discusses investors’ reluctance to realize losses in more detail. Shefrin
and Statman (1985) coin the term “the disposition eﬀect” to describe the tendency
of investors to hold onto their losing stocks to a greater extent than they hold onto
their winners. The disposition eﬀect has been documented for both individual and
professional investors in many markets and for many countries.
For example, Odean (1998) analyzes accounts at a large brokerage house and
found that there was a greater tendency to sell stocks with paper capital gains than
those with paper losses. Grinblatt and Keloharju (2001) ﬁnd a similar eﬀect among all
types of investors in Finland, even after controlling for a variety of variables that may
determine trading. Heath, Huddart and Lang (1999) uncover disposition behavior
relative to a reference price of a prior high for the stock price by studying the option
exercise behavior of over 50,000 employees at seven corporations. Shapira and Venezia
(2001) show that both professional and independent investors in Israel exhibit the
disposition eﬀect, although the eﬀect is stronger for independent investors. Garvey
and Murphy (2004) provide evidence that even professional traders are also subject to
the disposition eﬀect. Lo cke and Mann (1999) present evidence for the existence of a
disposition eﬀect within a sample of professional futures traders. Frazzini (2004) and
Wermers (2003) ﬁnd evidence for disposition eﬀect among mutual fund managers.
The disposition eﬀect also inﬂuences agents in the IPO and housing markets (e.g.,
Case and Shiller, 1988, Genesove and Mayer, 2001, Kaustia, 2004)
The disposition eﬀect can be explained by the prosp ect theory, combined with
the concept of mental accounting (e.g., Thaler, 1985). Mental accounting provides
a foundation for the way that decision makers set reference points for the accounts
that determine gains and losses. In the context of ﬁnancial transactions, the key
mental accounting issues concern aggregation: how transactions are grouped both
cross-sectionally (e.g., are securities evaluated one at a time or as portfolios) and
intertemporally (how often are portfolios evaluated). The main idea of mental ac-
counting is that decision makers tend to segregate diﬀerent types of gambles into
separate accounts, and then apply prospect theory to each account by ignoring pos-
The key feature of prospect theory required to explain disposition eﬀect is con-
vexity over losses (the S-shaped value function). Intuitively, when an investor ﬁrst
bought a stock, he op ened a mental account and keeps a running score on the gain
or loss for his position. Suppose the investor suﬀers a paper loss. If he sells the stock
and realizes the loss, the mental account is closed and the loss becomes certain. This
is very painful for a prospect theory agent because of loss aversion and will inﬂict him
trying to avoid such a loss. In fact, because he is risk-seeking in a losing situation, he
will take even greater position (“throw good money after bad”) in hope that prices
will recover so he can break even sooner. On the other hand, in a winning situation
the circumstances are reversed. Investors will become risk averse and quickly take
proﬁts, not letting proﬁts run.
To demonstrate disposition eﬀect using prospect theory utility function, consider
Figure 1, which plots the S-shaped value function for outcomes in a particular stock.
Let us analyze how this S-shape alters traditional investment behavior. The curve
above the point labelled “reference point” has the shape of power utility. For true
power utility, the fraction of wealth invested in the stock is increasing in the stock’s
expected return, but is unaﬀected by the (initial wealth) starting point. How is this
demand function shifted by the substitution of a convex utility function to the left of
the inﬂection point? Comparing a starting position at Point D with Point C in Figure
1, one can infer that demand is increased more at Point C. If we start from Point
D, gambles rarely end up in the convex portion of the curve. Indeed, for any given
positive mean return, demand increases as the starting position moves left of point
D because gambles experience an increasing likelihood of outcomes in the convex
portion of the value function. This pattern of larger demand (for a given mean) as
the starting position moves left continues as our starting position crosses the inﬂection
point and moves into the convex region. Clearly, the critical determinant of demand
is the starting position in the value function.
When the relevant mental accounts employ the cost basis in a stock as the reference
point, the starting positions are dictated by the unrealized capital gain or loss in the
stock. Stocks that are extreme winners start the investor at Point D. Stocks that are
extreme losers start the investor at Point A, and so forth. It follows that a PT/MA
demand function diﬀers from that of a standard utility investor not just because
winners are less desirable than losers, other things equal. One also concludes that
there is a greater appetite for large losers (point A) than for small losers (point B).
Moreover, there is a lesser desire to shun small winners (point C) than large winners
(point D) because of the greater degree to which realizations in the convex region
enter the expected value calculation.
2.3 Home Bias
Under the mean-variance framework, one of the key alleged beneﬁts of international
diversiﬁcation is the minimization of risk for a given exp ected return. This contra-
dicts the so-called “home bias”, documented by French and Poterba (1991) and many
others. The home bias refers to the ﬁnding that American investors hold more U.S.
stocks and fewer foreign stocks than the amounts predicted by mean-variance opti-
mization. It is a puzzle within the mean-variance framework but is is consistent with
behavioral portfolio theory.
Stracca (2002) argues that if prospect theory is an accurate description of human
attitudes towards risk, the beneﬁts of international diversiﬁcation would be reduced
to a signiﬁcant extent. He shows that risk concentration (“do put your eggs in the
same basket”) may be optimal for a cumulative prospect theory agent, provided that
the subjective probability of obtaining a perfect hedge is negligible, and the agent
sees the allocation of risks as a self-contained decision problem. The intuitive reason
behind this result is that a prospect theory agent is risk-seeking over losses, with
the consequence that the property of diversiﬁcation of averaging downside risks is
welfare-reducing rather than welfare-improving. In other words, risk diversiﬁcation
does not lead to risk (loss) minimization for cumulative prospect theory agent, in
contrast with a standard expected utility agent.
While prospect theory can explain the tendency to concentrate risks on a single
asset rather than to hold a well diversiﬁed portfolio, it can not explain why the
single asset the investor chooses is a domestic one. One reason for this could be a
greater familiarity with domestic assets. Consider a foreign stock and a domestic
stock with an identical distribution of payoﬀs. Since foreign stocks seem less familiar
than domestic stocks, investors may perceive it as having higher variance of payoﬀs.
That perception leads to a low allocation to foreign stocks. A direct implication is a
behavioral portfolio theory prediction that the home bias would decline as investors
became more familiar with foreign stocks. There is no such prediction in mean-
variance portfolio theory.
2.4 Equity Premium
Mehra and Prescott (1985) discovered the equity premium puzzle. The puzzle is that
the historic equity premium has been very large. Over the time period Mehra and
Prescott studied (1889-1978) the annual real return on the S&P 500 was about 7
percent while the return on T-bills was less than 1 percent. Since 1978, stocks have
done even better. When these large return diﬀerences are cumulated, the diﬀerentials
become staggering. For example, a dollar invested in the S&P 500 on January 1, 1926,
was worth over $1100 by the end of 1995, while a dollar invested in T-bills was worth
only $12.87. Mehra and Prescott show that it is diﬃcult to explain the combination of
a high equity premium and a low risk-free rate within a standard neoclassical model.
The implicit coeﬃcient of relative risk aversion needed to produce such numbers was
over 30, while most estimates put it close to 1.
Benartzi and Thaler (1995) oﬀer an explanation of the equity premium based on
the prospect theory. The key feature of prosp ect theory used in their explanation
is loss aversion. They also need two closely related behavioral concepts: mental
accounting and narrow framing. A ﬁnancial investor can be modelled as making a
series of decisions about the allocation of his assets. Mental accounting determines
both the framing of decisions and the experience of the outcomes of these decisions.
An investor who frames decisions narrowly will tend to make short-term choices rather
than adopt long-term policies. An investor who frames past outcomes narrowly will
evaluate his gains and losses frequently. In general, narrow framing of decisions
and narrow framing of outcomes tend to go together, and the combination of both
tendencies deﬁnes a myopic investor.
Benartzi and Thaler (1995) examine the eﬀect of myopic loss aversion on risk
attitudes and equity premium. They argue that the attractiveness of the risky asset
depends on the time horizon of the investor, and valuation depends on investors’ time
horizon. Since the stock price is generally the frame of reference, the probability of
loss or gain is important. Note that the more frequently one evaluate his portfolio,
the more likely he sees losses and hence suﬀer from loss aversion. The probability
of a gain or a loss for a risky asset that pays an expected 7 percent per year with a
standard deviation of 20 percent (like stocks) in the very short term is close to 50-
50. To a loss-averse investor who evaluate his portfolio frequently, the stock market
appears very risky. On the other hand, an investor who is prepared to wait a long
time before evaluating the outcome of the investment as a gain or a loss will ﬁnd
the risky asset more attractive, since the longer the holding period, the higher the
probability that he ends up with a positive total return.
Thus, if losses cause more mental anguish than equivalent gains cause pleasure,
the experienced utility associated with owning stocks is lower for the more myopic
investor. Long-term investors (individuals who evaluate their portfolios infrequently)
are willing to pay more for an identical risky asset than short-term investors (frequent
evaluation). The more often that a loss-averse investor evaluates his portfolio, the
higher return he would demand in order to hold risky stocks. Benartzi and Thaler
(1995) ﬁnd that investors with myopic loss aversion would be indiﬀerent between the
historical returns of stocks and T-bills if they evaluate their portfolios about once per
2.5 Aggregate Stock Market Return Puzzles
Besides the equity premium puzzle, prospect theory can help explain other puzzling
features of the aggregate stock market. Here we discuss two such examples: the
volatility puzzle, and the predictability of price-earnings ratio.
The volatility puzzle is that stock market levels appear to move around too much.
For example, ratios of price to earnings in the U.S. stock market have often been
very high. The standard rationalization of this is that investors must be expecting
high cashﬂows and earnings in the future, and are therefore happy to pay high prices
today. However, historical data shows that high levels of price-earnings ratios are
not, on average, followed by higher earnings. In this sense, it is a puzzle why prices
were so high to begin with.
Historical data also shows that the price-earnings ratio can predict future returns
on the stock market. High levels of the price-earnings ratio have generally led to lower
subsequent returns, and low levels of the ratio to higher returns.
Barberis, Huang and Santos (2001) show that prospect theory help explain both
puzzles above. They need two features from prospect theory: agents receive direct
utility from changes in the value of their ﬁnancial wealth and they are loss averse.
Barberis, Huang and Santos (2001) do not need the convexity over losses or the
probability weighting features of the prospect theory.
However, they need to combine prosp ect theory, originally developed for one-shot
game, with the “house money” eﬀect of Thaler and Johnson (1990) which describe
how people integrate gains and losses in dynamic setting. The house money eﬀect
simply says that people are more likely to bet recklessly in casinos with money they
have recently won.
The key assumption in Barberis, Huang and Santos (2001) is that prospect theory
agent’s loss aversion changes over time depending on their previous investment results.
If they have recently made a lot of money in the stock market, they may be less
nervous, or less loss averse, because any loss they incur will be cushioned by their
prior gains. However, if they have recently been burnt by painful losses in the stock
market, they may be more nervous about any additional setbacks, in other words,
more loss averse.
Barberis, Huang and Santos (2001) conﬁrm Benartzi and Thaler’s ﬁnding (derived
in a single-period setup) in a dynamic model. They show that their model predicts
large equity premia, in line with those observed in the data. The reason is that the
investors in our model are loss averse: they are much more sensitive to losses than
to gains, and therefore they are uncomfortable with the frequent ﬂuctuations of the
stock market and demand a large average premium to compensate them for this risk.
To understand how Barberis, Huang and Santos (2001) resolve the volatility puz-
zle, suppose that the stock market receives some good news about earnings. This
will push the stock market up, generating substantial gains for investors. Now that
they have gains, investors will be less loss averse, because these gains will cushion
any subsequent losses. Since they are less risk averse than before, they are prepared
to pay even more for stocks, and push stock market prices even higher. Therefore, a
changing degree of loss aversion may explain why prices appear to move more than
is justiﬁed by news about earnings.
The resolution of the predictability of price-earnings ratio is similar. After a good
piece of earnings news, the stock market goes up, generating gains for investors, who
become less loss averse and push the stock market even further up. Since their prior
gains make them feel more comfortable, investors demand a lower average return as
compensation for staying in the stock market. Therefore, high prices are on average
followed by lower returns, in line with the ﬁndings of predictability in the data.
2.6 Cross-sectional Return Predictability
Prospect theory has also been used to explain two most important cross-sectional
stock return patterns: the predictability of book-to-market ratio (i.e., the proﬁtabil-
ity of value strategy) and the predictability of past returns (i.e., the proﬁtability
Barberis, Huang and Santos (2001) applies to the puzzles for the aggregate market.
Barberis and Huang (2001) extend Barberis, Huang and Santos (2001) to multiple
assets. They ﬁnd that prospect theory combined with the concept of individual mental
accounting works the best in explaining the cross-sectional expected return patterns,
such as the value premium. The intuition is the same as that for the predictability
of price-earnings ratio for the aggregate market.
Grinblatt and Han (2004) show that prospect theory and mental accounting can
explain the proﬁtability momentum strategy, or the persistence in the returns of stocks
over horizons between three months and one year documented ﬁrst by Jegadeesh and
Titman (1993). They ﬁrst show that demand for a stock by a prospect theory agents
deviate from that of a fully rational investor, with the distortions being inversely
related to the unrealized proﬁt they have experienced on the stock. A stock that
has been privy to prior go od news has excess selling pressure relative to a stock that
has been privy to adverse information. Such demand perturbation tends to generate
price underreaction to public information. This distorts equilibrium prices relative to
those predicted by standard utility theory. In equilibrium, past winners tend to be
undervalued and past losers tend to be overvalued.
As the above mispricing gets corrected, return predictability arises: past winners,
which tend to be undervalued, will continue to go up; past losers, which tend to
be overvalued, will continue to go down. This leads to momentum. Interestingly,
Grinblatt and Han (2004) ﬁnd that the dynamic updating of the aggregate reference
prices, which occurs as shares change hands through the trading process, naturally
brings a convergence of the market prices and the fundamental values of the stocks.
2.7 IPO Pricing Puzzles
There is a large literature on the pricing of initial public oﬀerings. Two puzzles on
IPOs stand out: one is the initial underpricing of IPOs; the other is the poor abnormal
long-run performance of IPOs. Prospect theory has been applied to explain both.
2.7.1 IPO Underpricing
During 1990-1998, companies going public in the U.S. left over $27 billion of money
on the table, where the money left on the table is deﬁned as the ﬁrst day price gain
multiplied by the number of shares sold. This number is approximately twice as large
as the fees paid to investment bankers, and represents a substantial indirect cost to
the issuing ﬁrm. If the shares had been sold at the closing market price rather than
the oﬀer price, the proceeds of the oﬀering would have been higher by an amount
equal to the money left on the table. The investors’ proﬁts come out of the pocket of
the issuing company and its pre-issue shareholders.
Many explanations have been advanced for why IPOs are underpriced (for a re-
view, see, e.g., Ritter and Welch, 2002). Loughran and Ritter (2002) oﬀer a new
explanation based on prospect theory. Prospect theory assumes that issuers care
about the change in their wealth, rather than the level of wealth. The other key
aspect of the prospect theory used in Loughran and Ritter’s explanation for IPO
underpricing is loss-aversion (of the issuers). The issuers anchor their reference point
at the IPO ﬁlling date.
Loughran and Ritter’s model focuses on the covariance of the money left on the
table and issuers’ unanticipated wealth changes. Issuers do not get upset about the
severe underpricing, especially when the issue takes place at a price above the ﬁlling
range, because of their loss-averse preferences: they have gained a lot on their shares,
and the underpricing is a relatively small “losses,” so they irrationally aggregate the
two and are still relatively happy. Issuers treat the opportunity cost of leaving money
on the table as less important than direct fees. They sum the wealth loss from leaving
money on the table with the larger wealth gain from a price jump, producing a net
increase in wealth for pre-issue shareholders.
Loughran and Ritter’s prospect theory explanation for IPO underpricing can be
recast in terms of a bargaining model in which underwriters want a lower oﬀer price
and issuing ﬁrms desire a higher oﬀer price. When unexpectedly strong demand
becomes apparent during the pre-selling period, issuing ﬁrms acquiesce in leaving
more money on the table. When demand is unexpectedly weak, issuing ﬁrms negotiate
more aggressively, leaving little money on the table.
Empirically, Loughran and Ritter (2002) show that most of the money left on the
table comes from a minority of IPOs, i.e., those where the oﬀer price is revised upwards
from what had been anticipated at the time of distributing the preliminary prospectus.
The oﬀer price is increased in response to indications of strong demand, but it could
have been increased even further. Thus, at the same time that underpricing is diluting
the pre-issue shareholders of these ﬁrms, these shareholders are receiving the good
news that their wealth is much higher than they had anticipated.
Loughran and Ritter’s argument can be illustrated with the example of Netscapes’
IPO. James Clark, a company co-founder, held 9.34 million shares. Approximately
one month before going public, Netscape ﬁled a preliminary prospectus with the
Securities and Exchange Commission. This prospectus contained a projected number
of shares to be issued and an anticipated price range for the oﬀering. Based up on the
midpoint of the ﬁle price range of $12-14, the expected value of his Netscape holdings
equaled $121 million at the time that the preliminary prospectus was ﬁled. At the
closing market price on the ﬁrst day of trading, his shares were worth $544 million, a
350% increase in this component of his pre-tax wealth in the course of a few weeks.
So at the same time that he discovered that he had been diluted more than necessary
due to the large amount of money left on the table, he discovered that his wealth had
increased by hundreds of millions of dollars. Most people would not be upset if they
found themselves in this situation.
Based on Loughran and Ritter (2002), Ljungqvist and Wilhelm (2004) derive a
behavioral measure of the IPO decision-maker’s satisfaction with the underwriter’s
performance. They ﬁnd that prospect theory has explanatory power for IPO market
behavior. For example, IPO ﬁrms are less likely to switch underwriters for their ﬁrst
seasoned equity oﬀering when their behavioral measure indicates they were satisﬁed
with the IPO underwriter’s performance.
The prospect theory explanation also predicts a positive correlation between the
ﬁlling price revision (equal to the diﬀerence between the midpoint of the ﬁlling price
range and the oﬀer price) and the ﬁrst-day return. Following upward revisions, issuers
are willing to tolerate higher levels of underpricing than following downward revisions.
This explains the ﬁndings of Hanley (1993) that IPOs where the oﬀer price is revised
upwards see much higher ﬁrst-day price jumps, on average, than those where the oﬀer
price is revised downwards.
Another unique implication of the prospect theory explanation is that market
return between the ﬁlling date and the issue date is related to the IPO underpricing:
much more money is left on the table following recent market rises than after market
falls. Thus, the prospect theory explanation of IPO underpricing also leads to a
theory of hot issue markets.
2.7.2 Long-run Under-performance of IPO
Ritter (1991) reports that the average holding period return for a sample of 1526
U.S. IPOs between 1975 and 1984 underperformed control ﬁrms of similar size and
industry by nearly 29% after three years. Loughran and Ritter (1995) reported that
from 1970 to 1990 the companies going public produced an average return of just
5% for the next ﬁve years, whereas a control group of nonissuing ﬁrms produced an
average return of 12%. The long-run underperformance of the IPOs seems to be a
consensus of the researchers, and is not restricted to U.S. markets (see Ritter and
Welch, 2002, for review and discussions). Controlling for risk using the CAPM or
Fama and French’s three-factor model cannot eliminate the underperformance either.
Several studies argue theoretically and/or empirically that the cumulative prospect
theory can explain the long-run underperformance of IPOs (see, e.g., Brav, and
Heaton, 1996, Ma and Shen, 2003, and Barberis and Huang, 2004). For example,
Barberis and Huang (2004) show that probability weighting can have unusual pricing
eﬀects in an economy with a positively skewed security. The key observation is that
if the new security is suﬃciently skewed, some investors may choose to hold large
undiversiﬁed positions in it, thereby making the distribution of their overall wealth
lottery-like. Since a cumulative prospect theory investor overweights the tails of a
probability distribution, he loves lottery-like wealth distributions, and is therefore
willing to pay a very high price for the skewed security. Thus, the skewed security
can become overpriced, relative to the price that would be set by investors who do
not weight probabilities, and thus earn a very low average return.
Under cumulative prospect theory, the relationship between a securitys skewness
and its expected return is nonlinear: a highly skewed security can be overpriced
and earn a low average return, but a security that is merely moderately skewed is
priced fairly. The security needs to be suﬃciently positively skewed before prospect
theory investors are willing to take undiversiﬁed position in it, since there is a trade-
oﬀ between diversiﬁcation and skewness preference. A suﬃciently p ositively skewed
security can be overpriced even if there are many skewed securities in the economy,
again because investors may prefer a large undiversiﬁed position in just one skewed
security to a diversiﬁed portfolio.
IPOs have positively skewed return distribution, probably because, being young
ﬁrms, a large fraction of their value is in the form of growth options. Barberis and
Huang (2004) show that based on the historical data, IPOs have suﬃcient skewness so
that investors with cumulative prospect theory preferences calibrated to experimental
data would require an average return that is several percentage points below the
market return. Under cumulative prospect theory, then, the historical performance
of IPOs may not be so puzzling.
Similar idea can be applied to shed light on so-called “equity stub anomalies
whereby parent companies appear undervalued relative to their publicly traded sub-
sidiaries (Mitchell and Staﬀord, 2002, Lamont and Thaler, 2003). If a subsidiary is
valued mainly for its growth options, its returns may be p ositively skewed, leading
investors to overprice it relative to its parent, and thereby generating a low stub
value. Thus, in the presence of cumulative prospect theory investors, a ﬁrm can cre-
ate value by spinning oﬀ subsidiaries that are valued mainly for their growth options.
If a subsidiary of this kind is traded as a separate entity, its stock is likely to have a
positively skewed return distribution. If investors overprice such securities, the ﬁrm
will be more valuable when its parts are traded separately, than when they are traded
as a single bundle.
Prospect theory is a descriptive model of decision making under uncertainty. Under
prospect theory, people evaluate risk using a value function that is deﬁned over gains
and losses, is concave over gains and convex over losses, and is kinked at the origin;
and using transformed rather than objective probabilities by applying a weighting
This paper only reviews a portion of of topics that prospect theory can help us
understand. Other areas where prospect theory has interesting applications includes
commercial banking (e.g., Johnson, 1994, Godlewski, 2004), investment banking (e.g,
Willman et al, 2002), hedge funds (e.g., Siegmann and Lucas, 2002, Kouwenberg
and Ziemba, 2004) venture capital ﬁrms (Dubil and Maretno, 2003), and analyst
behavior (e.g., Ding, Charoenwong, and Seetoh, 2004). This list is still growing fast.
Prospect theory, together with other behavioral concepts such as mental accounting
and narrowing framing, may oﬀer a new paradigm for understanding behavior of
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Figure 1: Prospect Theory Value Function
This ﬁgure plots an example of the S-shaped prospect theory value function, generated by the
U(W ) =
(W − R )
1 − γ
, if W ≥ R;
U(W ) = −λ
(R − W )
1 − γ
, if W < R
where R is a reference level, γ = 0.5 and λ = 2.25.
* Reference Point