Journal of Financial Economics 104 (2012) 251–271

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Journal of Financial Economics

journal homepage: www.elsevier.com/locate/jfec

Realization utility$
Nicholas Barberis a,n, Wei Xiong b
a
Yale School of Management, New Haven, CT 06511, United States
b
Princeton University, Princeton, NJ 08544, United States

a r t i c l e i n f o abstract

Article history: A number of authors have suggested that investors derive utility from realizing gains
Received 20 January 2010 and losses on assets that they own. We present a model of this ‘‘realization utility,’’
Received in revised form analyze its predictions, and show that it can shed light on a number of puzzling facts.
10 February 2011
These include the disposition effect, the poor trading performance of individual
Accepted 14 March 2011
investors, the higher volume of trade in rising markets, the effect of historical highs
Available online 25 October 2011
on the propensity to sell, the individual investor preference for volatile stocks, the low
JEL classification: average return of volatile stocks, and the heavy trading associated with highly valued
D03 assets.
G11
& 2011 Elsevier B.V. All rights reserved.
G12

Keywords:
Behavioral finance
Disposition effect
Trading
Individual investors

1. Introduction gains and losses on assets that they own. Suppose, for
example, that an investor buys shares of a stock and then,
When economists model the behavior of individual a few months later, sells them. We consider a model in
investors, they typically assume that these investors which he receives a burst of utility right then, at the
derive utility only from consumption or from total wealth. moment of sale. The amount of utility depends on the size
In this paper, we study the possibility that investors also of the gain or loss realized—on the difference between the
derive utility from another source, namely from realized sale price and the purchase price—and is positive if the
investor realizes a gain, and negative otherwise. This
source of utility, which we label ‘‘realization utility,’’ is
$
We thank Daniel Benjamin, Patrick Bolton, John Campbell, Lauren
not new to our paper: other authors also discuss it. Our
Cohen, Erik Eyster, Nicolae Garleanu, Simon Gervais, Bing Han, Vicky contribution is to offer a comprehensive analysis of its
Henderson, Bige Kahraman, Peter Kelly, Antonio Mele, Matthew Rabin, implications for trading behavior and for asset prices.
Chris Rogers, Paul Tetlock, Jeffrey Wurgler, the referees, and seminar Why might an investor derive utility from realizing a
participants at Arizona State University, Brown University, Cornell
gain or loss? We think that realization utility is a
University, Harvard University, the LSE, New York University, Notre
Dame University, Oxford University, Princeton University, the University consequence of two underlying cognitive processes. The
of California at Berkeley, the University of Texas at Austin, Yale first has to do with how people think about their investing
University, the Gerzensee Summer Symposium, and the NBER for helpful history. Under this view, people do not think about their
comments. We are especially grateful to Xuedong He, Jonathan Ingersoll, investing history purely in terms of the return they have
and Lawrence Jin for many discussions about this project.
n
Corresponding author.
earned on their portfolio. Rather, they often think about it
E-mail addresses: nick.barberis@yale.edu (N. Barberis), as a series of investing episodes, each one defined by
wxiong@princeton.edu (W. Xiong). three things: the name of the investment, the purchase

0304-405X/$ - see front matter & 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jfineco.2011.10.005
252 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

price, and the sale price. ‘‘I bought IBM at $80 and sold it investor makes his allocation decision by maximizing the
at $100’’ might be one such episode. ‘‘We bought our discounted sum of expected future utility flows. In our
house for $260,000 and sold it for $320,000’’ might be baseline model, we assume a linear functional form for
another. realization utility. Later, we also consider a piecewise-
The second cognitive process that, in our view, under- linear specification.
lies realization utility has to do with how people evaluate We find that, under the optimal strategy, an investor
their investing episodes. We suspect that many investors who is holding a position in a stock will voluntarily sell
use a simple heuristic to guide their trading, one that this position only if the stock price rises sufficiently far
says: ‘‘Selling a stock at a gain relative to purchase price is above the purchase price. We look at how this ‘‘liquida-
a good thing—it is what successful investors do.’’ After all, tion point’’ at which the investor sells depends on the
an investor who buys a number of stocks in sequence and expected stock return, the standard deviation of the stock
manages to realize a gain on all of them does end up with return, the time discount rate, the transaction cost, and
more money than he had at the start. The flip side of the the likelihood of a liquidity shock.
same heuristic says: ‘‘Selling a stock at a loss is a bad The model has a number of interesting implications.
thing—it is what unsuccessful investors do.’’ Indeed, an One of the more striking is that, even if realization utility
investor who buys a number of stocks in sequence and has a linear or concave functional form, the investor can
realizes a loss on all of them does end up with less money be risk seeking: all else equal, his initial value function can
than he had at the start. be an increasing function of the standard deviation of
In summary, an investor feels good when he sells a stock returns. The intuition is straightforward. A highly
stock at a gain because, by selling, he is creating what he volatile stock offers the chance of a large gain which the
views as a positive investing episode. Conversely, he feels investor can enjoy realizing. Of course, it may also drop a
bad when he sells a stock at a loss because, by selling, he lot in value; but in that case, the investor will simply
is creating what he views as a negative investing episode. postpone selling the stock until he is forced to sell by a
We do not expect realization utility to be important for liquidity shock. Any realized loss therefore lies in the
all investors or in all circumstances. For example, we distant, discounted future and does not scare the investor
expect it to matter more for individual investors than for very much at the time of purchase. Overall, then, the
institutional investors who, as trained professionals, are investor may prefer more volatility to less.
more likely to think about their investing history in terms We use our model to link realization utility to a number
of overall portfolio return than as a series of investing of financial phenomena. Among the applications we discuss
episodes. Also, since realization utility depends on the are the disposition effect (Shefrin and Statman, 1985;
difference between sale price and purchase price, it is Odean, 1998), the subpar trading performance of individual
likely to play a larger role when the purchase price is investors (Barber and Odean, 2000; Barber, Lee, Liu, and
more salient. It may therefore be more relevant to the Odean, 2009), the higher volume of trade in bull markets
trading of individual stocks or to the sale of real estate than in bear markets (Stein, 1995; Statman, Thorley, and
than to the trading of mutual funds: the purchase price of Vorkink, 2006; Griffin, Nardari, and Stulz, 2007), the effect of
a stock or of a house is typically more salient than that of historical highs on the propensity to sell (Grinblatt and
a fund. Keloharju, 2001), the individual investor preference for
In our view, the idea that some investors derive utility volatile stocks (Kumar, 2009), the low average return of
directly from realizing gains and losses is a plausible one. volatile stocks (Ang, Hodrick, Xing, and Zhang, 2006), and
But in order to claim that realization utility is a significant the heavy trading associated with highly valued assets—as,
driver of investor behavior, we cannot appeal to mere for example, in the case of U.S. technology stocks in the late
plausibility. To make a more convincing case, we need to 1990s (Hong and Stein, 2007).
build a model of realization utility and then see if the Of these applications of realization utility, the most
model explains a range of facts and leads to new predic- obvious is the disposition effect, the greater propensity of
tions that can be tested and confirmed. individual investors to sell stocks that have risen in value,
In this paper, we take up this challenge. We construct a rather than fallen in value, since purchase. In combination
model of realization utility, discuss its predictions, and with a sufficiently positive time discount rate, realization
show that it can shed light on a number of empirical facts. utility generates a strong disposition effect: the investor
We start with a partial equilibrium framework but also in our model voluntarily sells a stock only if it is trading at
show how realization utility can be embedded in a full a gain relative to purchase price.
equilibrium model. This allows us to make predictions not While the link between realization utility and the
only about trading behavior but also about prices. disposition effect is clear, we emphasize that realization
Our partial equilibrium model is an infinite horizon utility is not a ‘‘relabeling’’ of the disposition effect. On the
model in which, at each moment, an investor allocates his contrary, it is just one of a number of possible theories of
wealth either to a risk-free asset or to one of a number of the disposition effect and can be distinguished from other
stocks. If the investor sells his holdings of a stock, he theories through carefully constructed tests. For example,
receives a burst of utility based on the size of the gain or another theory of the disposition effect, one that has
loss realized and pays a proportional transaction cost. He nothing to do with realization utility, is that investors
also faces the possibility of a random liquidity shock: if have an irrational belief in mean-reversion. Later in the
such a shock occurs, he must immediately sell his asset paper, we discuss an experiment that can distinguish this
holdings and exit the asset markets. At each moment, the view from the realization utility view.
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 253

Our other applications are more subtle. For example, More recently, Barberis and Xiong (2009) use a two-
our model predicts that individual investors—the investor period model to study the trading behavior of an investor
group that is more likely to think in terms of realization who derives utility from realized gains and losses with a
utility—will have a much greater propensity to sell a utility function that is concave over gains and convex over
stock once its price moves above its historical high. losses. They observe that, consistent with Shefrin and
Imagine a stock that rises to a high of $45, falls, and then Statman (1985), the investor often exhibits a disposition
rises again, passing its previous high of $45 and continu- effect. They do not study any other implications of realiza-
ing upwards. Our model predicts that there will be tion utility, nor do they link it to any other applications.2
relatively little selling as the stock approaches $45 for In this paper, we offer a more comprehensive analysis
the second time—any realization utility investors with of realization utility. We construct a richer model—an
liquidation points of $45 or lower will have sold already infinite horizon model that allows for transaction costs
when the stock first approached $45—but once the stock and a stochastic liquidity shock. We derive an analytical
moves above the historical high of $45, realization utility solution for the investor’s optimal trading strategy. We
investors with liquidation points higher than $45 will show how realization utility can be incorporated into
start to sell. In line with the evidence of Grinblatt and both a model of trading behavior and a model of asset
Keloharju (2001), then, our model predicts that historical pricing. We document several basic implications of reali-
highs will have a sharp effect on individual investors’ zation utility. And we discuss many potential applica-
propensity to sell. tions, rather than just one.
The idea that people derive utility from gains and In Section 2, we present a partial equilibrium model of
losses rather than from final wealth levels was first realization utility, one that also assumes a linear func-
proposed by Markowitz (1952), but is particularly asso- tional form for the realization utility term. In Section 3,
ciated with Kahneman and Tversky (1979): it is a central we use a piecewise-linear functional form. In Section 4,
element of their prospect theory model of decision-mak- we show how realization utility can be embedded in a
ing. Finance researchers have typically taken Kahneman model of asset prices. Section 5 discusses a range of
and Tversky’s message to be that they should study applications and testable predictions, while Section 6
models in which investors derive utility from paper gains concludes.
and losses. Benartzi and Thaler (1995), for example,
assume that investors derive utility from fluctuations in 2. A model of realization utility
their financial wealth, while Barberis, Huang, and Santos
(2001) and Barberis and Huang (2001) assume that they Before presenting our model, we briefly note two of
derive utility from fluctuations in the value of their stock our assumptions. First, we assume that realization utility
market holdings or in the value of specific stocks that is defined at the level of an individual asset—a stock, a
they own. house, or a mutual fund, say. Realization utility is trig-
The idea that people might derive utility from realized gered by the act of selling. But when an investor makes a
gains and losses has received much less attention. The sale, he is selling a specific asset. It is therefore natural to
concept first appears in Shefrin and Statman (1985). define realization utility at the level of this asset. This
Among several other contributions, these authors point assumption has little bite in our baseline model because,
out, with the help of a numerical example, that if an in this model, the investor holds at most one risky asset at
investor derives utility from realized gains and losses and any time. However, it becomes more important when we
has a utility function that, as in prospect theory, is discuss an extension of our model in which the investor
concave over gains and convex over losses, then he will can hold several risky assets simultaneously.
exhibit a disposition effect. A second assumption concerns the functional form for
Shefrin and Statman (1985) justify their emphasis on realization utility. In this section, we use a linear func-
realized gains and losses by reference to ‘‘mental account- tional form so as to show that we do not need elaborate
ing,’’ a term used to describe how people think about, specifications in order to draw interesting implications
organize, and evaluate their financial transactions. In their out of realization utility. In Section 3, we also consider a
view, when an investor sells a stock, he is closing a mental piecewise-linear functional form.
account that was opened when he first bought the stock. We work in an infinite horizon, continuous time
The moment of sale is therefore a natural time at which to framework. An investor starts at time 0 with wealth W0.
evaluate the transaction: a realized gain is seen as a good At each time t Z0, he has the following investment
outcome and a realized loss as a poor outcome. Realized options: a risk-free asset, which offers a constant con-
gains and losses thereby become carriers of utility in their tinuously compounded return of r; and N risky assets
own right. Although described using different language, indexed by i 2 f1, . . . ,Ng. The most natural application of
this motivation for realization utility is similar to our our model is to understanding how individual investors
own.1 trade stocks in their brokerage accounts. We therefore
often refer to the risky assets as stocks.

1 2
Other authors also discuss realization utility. For example, Thaler Barberis and Xiong (2009) do not say very much about realization
(1999) writes that ‘‘one clear intuition is that a realized loss is more utility because it is not their main focus. Their paper is primarily about
painful than a paper loss. When a stock is sold, the gain or loss has to be the trading behavior of an investor who derives prospect theory utility
‘declared’ both to the tax authorities and to the investor (and spouse).’’ from paper gains and losses.
254 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

The price of stock i, Si,t , follows: positive one if the capital gain exceeds what he could
dSi,t have earned by investing in the risk-free asset.
¼ ðr þ mÞ dt þ s dZ i,t , ð1Þ The key feature of our model is that the investor
Si,t
derives utility from realizing a gain or loss. If, at time t,
where Z i,t is a Brownian motion and where, for iaj, dZ i,t he moves his wealth from a stock into the risk-free
and dZ j,t may be correlated. In the interval between t and asset or into another stock, he receives a burst of utility
t þ dt, stock i also pays a dividend flow of given by
Di,t dt ¼ aSi,t dt: ð2Þ
uðð1kÞW t Bt Þ: ð6Þ
The stock’s expected excess return—throughout the
paper, ‘‘excess’’ means over and above the risk-free The argument of the utility term is the realized gain or
rate—is therefore a þ m: the dividend yield a plus the loss: the investor’s wealth at the moment of sale net of
expected excess capital gain m. For now, we assume that the transaction cost, ð1kÞW t , minus the cost basis of the
each of a, m, and s is the same for all stocks. stock investment Bt. Throughout this section, we use the
The dividends Di,t do not play a significant role in the linear functional form
partial equilibrium analysis in Sections 2 and 3. The only uðxÞ ¼ x: ð7Þ
reason we introduce them is because, as we will see in
Section 4, they make it easier to embed realization utility We emphasize that the investor only receives the burst
in a full equilibrium framework. To prevent the dividends of utility in (6) if he moves his wealth from a stock into
from unnecessarily complicating the analysis, we make the risk-free asset or into another stock. If he sells a stock
the following assumptions about them: that the investor and then immediately puts the proceeds back into the
consumes them; and that he receives linear consumption same stock, he derives no realization utility from the sale,
utility nor is the cost basis affected. Realization utility is asso-
vðcÞ ¼ bc ð3Þ ciated with the end of an investing episode. It is hard to
argue that the sale of a stock represents the end of an
from doing so, where b determines the importance of episode if, after selling the stock, the investor immedi-
consumption utility relative to the second source of utility ately buys it back.
that we introduce below. We assume that the investor does not incur a transac-
We assume that, at each time t, the investor either tion cost if he sells the risk-free asset. If we measure the
allocates all of his wealth to the risk-free asset or all of his cost basis for this asset in the same way as for a stock, it
wealth to one of the stocks; for simplicity, no other follows that the realized gain or loss from selling the risk-
allocations are allowed. Therefore, over any interval of free asset is always zero. The investor therefore receives
time during which the investor maintains a position in realization utility only when he sells a stock, not when he
one particular asset, his wealth Wt evolves according to sells the risk-free asset.
dW t XN The investor also faces the possibility of a random
¼ r dt þ ðm dt þ s dZ i,t Þyi,t , ð4Þ liquidity shock whose arrival is governed by a Poisson
Wt i¼1
process with parameter r. If a shock occurs, the investor
where yi,t takes the value one if he is holding stock i at immediately sells his holdings, exits the asset markets,
time t, and zero otherwise. Note that, if yi,t ¼ 1 for some i and, if he was holding a stock at the time of the shock,
and t, then yj,t ¼ 0 for all jai. We also suppose that, if the receives the burst of utility in (6). We think of this shock
investor sells his position in a stock at time t, he pays a as capturing a sudden consumption need that forces the
proportional transaction cost kWt, 0 r k o1. investor to draw on the funds in his brokerage account.
An important variable in our model is Bt. This variable, We include it because it ensures, as is reasonable, that the
which is formally defined only if the investor is holding a investor cares not only about realized gains and losses but
stock at time t, measures the cost basis of the stock also about paper gains and losses. It also gives us a way of
position, in other words, the reference point relative to varying the investor’s horizon: when r is high, the
which the investor computes his realized gain or loss. One investor effectively has a short horizon; when it is low,
possible definition of the cost basis is the amount of he has a long horizon.
money the investor put into the time t stock position at At each moment, the investor makes his allocation
the time he bought it. This is the definition we use, with decision by maximizing the discounted sum of expected
one adjustment. We take the cost basis to be the amount future utility flows. Suppose that, at time t, his wealth is
of money the investor put into the stock position at the allocated to a stock. His value function then depends on
time he bought it, scaled up by the risk-free return two things: on the current value of his position, Wt, and
between the time of purchase and time t, so that on the cost basis of the position, Bt. We therefore denote it
as VðW t ,Bt Þ. Since the utility functions in (3) and (7) are
Bt ¼ W s erðtsÞ , ð5Þ
homogeneous of degree one, and since the prices of the
where sr t is the moment at which the time t stock risky assets all follow geometric Brownian motions, the
position was purchased. This definition is tractable and value function must also be homogeneous of degree one,
may be more realistic than the alternative that sets the so that, for z 40,
cost basis equal to the original purchase price: the
investor may only think of an investing episode as a VðzW t , zBt Þ ¼ zVðW t ,Bt Þ: ð8Þ
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 255

Now suppose that, for some positive W, price. The variable

VðW,WÞ Z 0: ð9Þ Wt
gt ¼ ð12Þ
Bt
Note that VðW,WÞ is the value function that corresponds
in words, the value of the stock position the investor is
to investing wealth W in a stock now, so that current
holding at time t relative to its cost basis—plays an
wealth and the cost basis are both equal to W. Since
important role in the solution. To simplify the statement
VðW t ,Bt Þ is homogeneous of degree one, if (9) holds for
of the proposition, we define
some positive W, then it holds for all positive W. Later, we
will compute the range of parameter values for which (9) d0  dr: ð13Þ
holds. For now, we note that, so long as the time discount
As we will see, the investor’s behavior does not depend on
rate d exceeds the risk-free return r, condition (9) implies
d and r separately, but only on the difference between
two things. First, it implies that, at time 0, the investor 0
them. We sometimes refer to d as the ‘‘effective’’ discount
allocates his wealth to one of the N stocks: since the risk- 0
rate and assume throughout that d 40. The proof of the
free asset generates no utility flows, he allocates to a stock
proposition is in the Appendix.
as early as possible. Second, and using the same logic,
condition (9) implies that, if, at any time t 4 0, the Proposition 1. Unless forced to exit the stock market by a
investor sells his holdings of a stock, he will then liquidity shock, an investor with the decision problem in (10) will
immediately use the proceeds to buy another stock. sell his holdings of a stock if the gain g t ¼ W t =Bt reaches a
We can now formulate the investor’s decision pro- liquidation point g t ¼ g n Z 1. If the transaction cost k is positive,
blem. Suppose that, at time t, the investor is holding stock then g n 41. The value function is VðW t ,Bt Þ ¼ Bt Uðg t Þ, where3
i. Let t0 be the random future time at which a liquidity 8
shock occurs. Then, at time t, the investor solves < ag g1 þ ab þ rð1kÞ g  r
>
if g t 2 ð0,g n Þ,
Uðg t Þ ¼
t
r þ d0 m t r þ d0
(Z >
: ð1kÞð1 þ Uð1ÞÞg 1
minft, t g
0
if g t 2 ½g n ,1Þ,
t
VðW t ,Bt Þ ¼ max Et edðstÞ vðDi,s Þ ds
tZt t ð14Þ

þ edðttÞ ½uðð1kÞW t Bt Þ where

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
   
þ Vðð1kÞW t ,ð1kÞW t ÞIft o t0 g 1 4 1 2 2 0 1 2 5
) g1 ¼ 2 m s þ 2ðr þ d Þs  m s 2 40
s 2 2
þ edðt tÞ uðð1kÞW t0 Bt0 ÞIft Z t0 g
0
ð10Þ
ð15Þ
and
subject to (3), (4), (5), and (7). Ifg is an indicator function
that takes the value one if the condition in the curly d0
a ¼ g1 0 : ð16Þ
brackets is met, and zero otherwise. To ensure that the g n ðg1 1Þðr þ d Þ
investor does not hold his time 0 stock position forever,
The liquidation point g n is the unique root, in the range ½1,1Þ, of
without selling it, we impose the following parameter 0  
restriction, which, in words, requires that the expected 0 ab 1
kðr þ d Þ r þ
B 1k C
excess capital gain is not too high: ðg1 1ÞB Cg gn1  g1 g gn1 1 þ 1 ¼ 0: ð17Þ
@1 d0 ðr þ d0 mÞ A 1k
  
k ab
m oðr þ drÞ 1 rþ : ð11Þ
dr 1k

Note that this implies m o r þ dr, a simpler condition In summary, the optimal strategy takes one of two
that we will sometimes also use. forms. If the model parameters are such that Uð1Þ Z 0,
To understand the formulation in (10), note that the where Uð1Þ is the value function per unit wealth from
investor’s problem is to choose the optimal time t, a buying a stock at time 0—equivalently, if condition (9)
random time in the future, at which to realize the gain or holds—the investor buys a stock at time 0 and voluntarily
loss in his stock holdings. Suppose first that t o t0 , so that sells it only if it reaches a sufficiently high liquidation
the investor voluntarily sells the stock before a liquidity point, at which time he immediately invests the proceeds
shock arrives. In this case, the investor receives a burst of in another stock, and so on. In particular, the investor
utility uðð1kÞW t Bt Þ when he sells at time t; and a cash never voluntarily sells a stock at a loss. If, on the other
balance of ð1kÞW t which he immediately invests in hand, Uð1Þ o 0, the investor allocates his wealth to the
another stock. If t Z t0 , however, the investor is forced risk-free asset at time 0 and keeps it there until a liquidity
out of the stock market by a liquidity shock and receives shock arrives.4
realization utility uðð1kÞW t0 Bt0 Þ from the gain or loss at
the moment of exit. Finally, while holding the stock, the
3
investor receives a continuous stream of dividends. Since g n Z 1, the term Uð1Þ which appears in the second row of
The proposition below presents the solution to the Eq. (14) can be obtained from the first row of the equation. It equals
0 0
a þ ðab þ rð1kÞÞ=ðr þ d mÞr=ðr þ d Þ.
decision problem in (10). It states that if the investor buys 4
To be clear, if g n ¼ 1:05, say, the investor sells his holdings of a
a stock, his optimal strategy is to sell it voluntarily only if stock once the value of the position is 5% higher than the cost basis.
its price rises a sufficient amount above the purchase Given the definition of the cost basis in (5), this means that the value of
256 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

0
For expositional simplicity, we have assumed that the choose an effective discount rate of d ¼ 0:08 because, as
investor holds at most one stock at any time. However, we will see later, this generates a trading frequency
Proposition 1 can also tell us how the investor trades in a similar to that observed in actual brokerage accounts.
setting where he holds several stocks simultaneously. The graph illustrates an interesting implication of
Suppose that, at time 0, he spreads his wealth across a realization utility, namely that the investor is willing to
number of stocks. Suppose also, as is natural in the case of buy a stock with a negative expected excess return, so
realization utility, that he derives utility separately from long as its standard deviation s is sufficiently high. The
the realized gain or loss on each stock. Finally, suppose intuition is straightforward. So long as s is sufficiently
that if a liquidity shock occurs, the investor sells all of his high, even a negative expected excess return stock has a
holdings and exits the asset markets. Under these non-negligible chance of reaching the liquidation point g n ,
assumptions, the investor’s decision problem is ‘‘separ- at which time the investor can enjoy realizing a gain. Of
able’’ across the different stocks he is holding and the course, more likely than not, the stock will perform
solution to (10) in Proposition 1 describes how he trades poorly. However, since the investor does not voluntarily
each one of his stocks. realize losses, this will only bring him disutility in the
A corollary to Proposition 1—one that also holds for event of a liquidity shock. Any realized loss therefore lies
the piecewise-linear specification we consider in Section in the distant, discounted future and does not scare the
3—is that, in this multiple-concurrent-stock extension of investor very much at the time of purchase. Overall, then,
our basic model, the investor is indifferent to diversifica- investing in a stock with a low expected return can
tion. For example, he is indifferent between investing sometimes be better than investing in the risk-free asset.
W0 in just one stock at time 0 as compared to investing Figs. 2 and 3 show how the liquidation point g n and
W 0 =2 in each of two stocks at time 0. The time 0 value initial utility per unit wealth Uð1Þ depend on the para-
0
function for the first strategy, W 0 Uð1Þ, is the same as the meters m, s, d , k, and r. The graphs on the left side of each
time 0 value function for the second strategy, namely figure correspond to the liquidation point, and those on
W 0 Uð1Þ=2þ W 0 Uð1Þ=2. the right side, to initial utility. For now, we focus on the
solid lines; we discuss the dashed lines in Section 3.
To construct the graphs, we start with a set of benchmark
2.1. Results
parameter values. We use the same benchmark values
throughout the paper. Consider first the asset-level para-
In this section, and again in Section 3, we draw out the
meters a, m, s, and k. We assume a dividend yield a of 0.015,
implications of realization utility through two kinds of
an expected excess capital gain on stocks of m ¼ 0:015—note
analysis. First, we compute the range of parameter values
that this implies an expected excess stock return of
for which condition (9) holds, so that the investor is
a þ m ¼ 0:03—a standard deviation of stock returns of
willing to buy a stock at time 0. Second, we look at how
s ¼ 0:5, and a transaction cost of k¼0.005. As for the
the liquidation point g n and initial utility per unit wealth 0
investor-level parameters d , r, and b, we use an effective
Uð1Þ depend on each of the model parameters. The first 0
time discount rate of d ¼ 0:08, a liquidity shock intensity of
analysis therefore concerns the investor’s buying beha-
r ¼ 0:1, and a consumption utility weight of b ¼ 1. The
vior, and the second, his selling behavior. When assigning 0
graphs in Figs. 2 and 3 vary each of m, s, d , k, and r in turn,
parameter values, we have in mind our model’s most
keeping the other parameters fixed at their benchmark
natural application, namely, the trading of stocks by
values.
individual investors.
The top-right graph in Fig. 2 shows that, as is natural,
The shaded area in the top graph in Fig. 1 shows the
initial utility is increasing in the expected excess capital
range of values of the expected excess stock return a þ m
gain m. The top-left graph shows that the liquidation point
and standard deviation of stock returns s that satisfy
is also increasing in m: if a stock that is trading at a gain
Uð1Þ Z0—in other words, condition (9)—so that the
has a high expected return, the investor is tempted to
investor is willing to buy a stock at time 0, but also the
hold on to it rather than to sell it and incur a
restriction in (11), so that he sells the stock at a finite
transaction cost.
liquidation point.5
0 The middle-right graph illustrates an important impli-
To create the graph, we assign values to d , k, r, a, and
cation of realization utility: that, as stock return volatility
b, and then search for values of m and s such that both
goes up, initial utility also goes up. Put differently, even
Uð1Þ Z0 and condition (11) hold. We set the transaction
though realization utility has a linear functional form, the
cost to k¼ 0.005 and the liquidity shock intensity r to 0.1,
investor is risk seeking. The intuition for this parallels the
so that the probability of a shock over the course of a year
intuition for why the investor is sometimes willing to buy
is 1e0:1  0:1. We also set the dividend yield a to 0.015
a stock with a low expected return. The more volatile a
and the consumption utility weight b to 1. Finally, we
stock is, the more likely it is to reach its liquidation point,
at which time the investor can enjoy realizing a gain. Of
(footnote continued) course, a volatile stock may also decline a lot in value. But
the position at the time of sale is more than 5% higher than it was at the the investor does not voluntarily realize losses and so will
time of purchase. only experience disutility in the event of a liquidity shock.
5
The unshaded area in the bottom-left of the graph corresponds to
parameter values for which Uð1Þo 0, so that the investor does not buy a
Any realized loss therefore lies in the distant, discounted
stock at time 0. The unshaded area in the right of the graph corresponds future and does not scare the investor very much at the
to parameter values that violate restriction (11). time of purchase. Overall, then, the investor prefers more
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 257

Linear realization utility

1

Standard deviation
0.8

0.6

0.4

0.2

0
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Expected excess return

Piecewise−linear realization utility

1
Standard deviation

0.8

0.6

0.4

0.2

0
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Expected excess return

Fig. 1. Range of values of a stock’s expected excess return and standard deviation for which an investor who derives utility from realized gains and losses
is willing both to buy the stock and to sell it once its price reaches a sufficiently high liquidation point. The top graph corresponds to the case in which
realization utility has a linear functional form. The bottom graph corresponds to the case in which realization utility has a piecewise-linear functional
form, so that the investor is 1.5 times as sensitive to realized losses as to realized gains.

volatility to less.6 A similar intuition explains why, in the point falls. An investor with a high discount rate is impatient
middle-left graph, the liquidation point is increasing in and therefore wants to realize gains sooner rather than later.
volatility. The top graphs in Fig. 3 show how the liquidation point
The trading patterns we have just described—the and initial utility depend on the transaction cost k. As
buying of low expected return stocks and the preference expected, a higher transaction cost lowers time 0 utility. It
for volatile stocks—are not behaviors that we associate also increases the liquidation point: if it is costly to sell a
with sophisticated investors. We emphasize, however, stock, the investor waits longer before doing so.
that our model is not a model of sophisticated investors. What happens when there is no transaction cost? The
It is a model of unsophisticated investors—specifically, of top-left graph in Fig. 3 suggests that, in this case, the
investors who use a simple heuristic to guide their liquidation point is g n ¼ 1. It is straightforward to check
trading, one that says that selling an asset at a gain is a that when k¼0, (17) is indeed satisfied by g n ¼ 1, so that the
good thing and that selling an asset at a loss is a bad thing. investor realizes all gains immediately. In other words, in
What Figs. 1 and 2 demonstrate is that an investor who our model, it is the transaction cost that stops the investor
thinks in these terms can be drawn into stocks with low from realizing all gains as soon as they appear.
expected returns and high volatility. We discuss some The bottom graphs in Fig. 3 show how the liquidation
evidence consistent with this prediction in Section 5.7 point and initial utility depend on r, the intensity of the
The bottom-left graph in Fig. 2 shows that when the liquidity shock. The liquidation point depends on r in a
investor discounts the future more heavily, the liquidation non-monotonic way. There are two forces at work here.
As the liquidity shock intensity r goes up, the liquidation
point initially falls. One reason the investor delays realiz-
6
ing a gain is the transaction cost that a sale entails. For
In mathematical terms, this prediction is related to the fact that,
r 40, however, the investor knows that he will be forced
while instantaneous utility is linear, the value function Uðg t Þ in (14) is
0
convex: since, from (11), m o r þ d , we have g1 41 and a 40, which, in out of the stock market at some point. The present value
turn, imply the convexity of UðÞ. of the transaction costs he expects to pay is therefore
7
For the case of linear realization utility, the predictions that the lower than in the absence of liquidity shocks. As a result,
investor will be willing to buy stocks with low expected returns and that he is willing to realize gains sooner.
he will be risk seeking are robust to changes in the model parameters. In
the next section, however, we will see that when the investor is more
At higher levels of r, however, another factor makes
sensitive to realized losses than to realized gains, these predictions do the investor more patient. If he is holding a stock with a
not always hold. gain, he is reluctant to exit the position because he will
258 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

Liquidation point Initial utility

1.5
1.4
1

1.2
0.5

1 0
−0.04 −0.02 0 0.02 0.04 −0.04 −0.02 0 0.02 0.04
Expected excess capital gain μ Expected excess capital gain μ

1.5
1.4
1

1.2
0.5

1 0
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Standard deviation σ Standard deviation σ

3 1.5
2.5
1
2
0.5
1.5
1 0
0.04 0.06 0.08 0.1 0.04 0.06 0.08 0.1
Effective time discount rate δ′ Effective time discount rate δ′

Fig. 2. Sensitivity of the liquidation point at which an investor sells a stock, and of the initial utility from buying it, to the stock’s expected excess capital
0
gain m, its standard deviation s, and the effective time discount rate d . The investor derives utility from realized gains and losses. The solid lines
correspond to the case where realization utility has a linear functional form. The dashed lines correspond to the case where realization utility has a
piecewise-linear functional form, so that the investor is 1.5 times as sensitive to realized losses as to realized gains.

Liquidation point Initial utility

3 1.5
2.5
1
2
0.5
1.5
1 0
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
Transaction cost k Transaction cost k

3
1.4
2

1.2
1

1 0
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Liquidity shock arrival rate ρ Liquidity shock arrival rate ρ

Fig. 3. Sensitivity of the liquidation point at which an investor sells a stock, and of the initial utility from buying it, to the transaction cost k and the
arrival rate r of an exogeneous liquidity shock. The investor derives utility from realized gains and losses. Realization utility has a linear functional form.

then have to invest the proceeds in another stock, which The bottom-right graph shows that as the liquidity
might do poorly and which he might be forced to sell at a shock intensity rises, initial utility falls. A high intensity r
loss by a liquidity shock. This factor pushes the liquida- makes it more likely that in the near future, the investor
tion point back up. will be forced to exit the stock market with a painful loss.
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 259

Several of the implications of realization utility that we The investor’s decision problem is now
have described can also be obtained in a two-period (Z 0 minft, t g
version of our model. However, our infinite horizon VðW t ,Bt Þ ¼ max Et edðstÞ vðDi,s Þ ds
framework has at least one advantage. In an infinite tZt t
horizon model, the structure of the optimal trading
þ edðttÞ ½uðð1kÞW t Bt Þ
strategy is simpler than in a two-period model: the
investor either holds the risk-free asset or else buys a þ Vðð1kÞW t ,ð1kÞW t ÞIft o t0 g
)
series of stocks in sequence, selling each one whenever it
þ edðt tÞ uðð1kÞW t0 Bt0 ÞIft Z t0 g
0

reaches a fixed liquidation point. The reason for this ð19Þ

simple structure is that in the infinite horizon model,
the environment is stationary: the value function does not subject to (3), (4), (5), and (18). This is the same as
depend explicitly on time, t. In a two-period model, the decision problem (10) in Section 2 except that uðÞ is no
environment is non-stationary and so the optimal trading longer linear but instead takes the form in (18).
strategy, while similar to that in our model, has a more In the Appendix, we prove:
complex structure.
We have also studied an extension of our model in Proposition 2. Unless forced to exit the stock market by a
which the value of the dividend yield a, the expected liquidity shock, an investor with the decision problem in (19)
excess capital gain m, and the standard deviation of will sell his holdings of a stock if the gain g t ¼ W t =Bt reaches a
returns s differ across stocks. In this case, the investor liquidation point g t ¼ g n Z 1. If the transaction cost k is positive,
follows a strategy that is similar to the one described then g n 4 1. The value function is VðW t ,Bt Þ ¼ Bt Uðg t Þ, where
8  
above, but that is restricted to a subset of the available >
> g ab þ rlð1kÞ rl 1
>
> bg t 1 þ gt  if g t 2 0, ,
stocks. Specifically, for each stock i, the investor computes >
>
< r þ d0 m r þ d0 1k
 
V i ðW,WÞ, the value function from investing wealth W in Uðg t Þ ¼ g g ab þ rð1kÞ r 1
>
> c1 g t 1 þ c2 g t 2 þ g if g t 2 ,g ,
stock i today. Suppose that stock j, with parameter values >
>
>
r þ d0 m t r þ d0 1k n
>
: ð1kÞg ð1þ Uð1ÞÞ1
aj , mj , and sj , maximizes V i ðW,WÞ across all stocks; and t if g t 2 ½g n ,1Þ,
suppose also that there are several stocks, which together ð20Þ
comprise a set M, that have the same parameter values
as stock j. Then, so long as V j ðW,WÞ Z0, the investor where g1 is defined in (15), where
allocates his wealth to a stock drawn from M at time 0, 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
   
sells it when it reaches the liquidation point specified 1 4 1 2 2 0 1 2 5
g2 ¼  2 m s þ 2ðr þ d Þs þ m s 2 o0
in Proposition 1, and then immediately reinvests the s 2 2
proceeds in another stock drawn from M, and so on.
ð21Þ
Fig. 2 tells us something about the characteristics of
the stocks in the agent’s preferred set M: a stock is more
and where b, c1, c2, and g n are determined from
likely to be in M, the higher its expected excess capital
ðl1Þrð1kÞg2 ðmg1 rd Þ
0
gain m and the higher its standard deviation s. Realization
c2 ¼ 0 0 , ð22Þ
utility therefore has implications not only for an investor’s ðg1 g2 Þðr þ d mÞðr þ d Þ
selling behavior, but also for his buying behavior.
g g d0
ðg1 1Þc1 g n1 þ ðg2 1Þc2 g n2 ¼ , ð23Þ
r þ d0
3. The case of piecewise-linear utility
     
1 g1 1 g2 1 g1 ðl1Þmr
c1 þc2 ¼b þ
In Section 2, we took the functional form for realiza- 1k 1k 1k 0 0
ðr þ d mÞðr þ d Þ
tion utility uðÞ to be linear. However, in reality, investors ð24Þ
may be more sensitive to realized losses than to realized
gains. We therefore now look at what happens when uðÞ g g kab þ ð1kÞðmd Þ d 0 0

is piecewise-linear rather than linear: c1 g n1 þ c2 g n2 þ gn þ

r þ d0 m r þ d0
(  
x if x Z0, rlðmkrkd0 Þ
uðxÞ ¼ l 41, ð18Þ ¼ ð1kÞg n b þ 0 0 : ð25Þ
lx if x o0, ðr þ d Þðr þ d mÞ

where l determines the relative sensitivity to realized

losses as opposed to realized gains.8 Specifically, given values for the asset-level parameters
a, m, s, and k, and for the investor-level parameters d0 , r,
8 l, and b, we first use (22) to find c2; we then obtain c1
It is not clear whether a piecewise-linear form is more reasonable
than a linear one. There is, of course, the well-known concept of ‘‘loss from (23); we then use (24) to find b; finally, (25) allows
aversion,’’ but this is the idea that people are more sensitive to wealth us to solve for the liquidation point g n .
losses than to wealth gains, in other words, more sensitive to paper
losses than to paper gains. It is the premise of this paper that utility from
realized gains and losses is distinct from utility from paper gains and (footnote continued)
losses and that it may have different psychological roots. Even if people necessarily follow that they are also more sensitive to realized losses
are more sensitive to paper losses than to paper gains, it does not than to realized gains.
260 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

3.1. Results a higher l means that the investor is more reluctant to sell
a stock at a gain, because if he does, he will have to invest
The shaded area in the lower graph in Fig. 1 shows the the proceeds in a new stock, which might go down and
range of values of the expected excess stock return a þ m which he might be forced to sell at a loss by a liquidity
and standard deviation of stock returns s for which the shock. The right graph shows that, as the sensitivity to
investor is willing to buy a stock at time 0—in other losses goes up, initial utility falls: a high l means that the
words, condition (9) is satisfied—but also to sell the stock investor may be forced, by a liquidity shock, to make an
at a finite liquidation point. We set the asset-level para- especially painful exit from a losing position.
meters a and k to their benchmark values from before, The dashed lines in Fig. 2 show how the liquidation
0
namely 0.015 and 0.005, respectively; and we set the point g n and initial utility Uð1Þ depend on m, s, and d
0
investor-level parameters d , r, and b to their benchmark when the investor is more sensitive to losses than to
0
values of 0.08, 0.1, and 1, respectively. Finally, we assign l gains. Here, we vary each of m, s, and d in turn, keeping
the benchmark value of 1.5. the other parameters fixed at their benchmark values
Relative to the upper graph—the graph for the Section
ða, m, s,kÞ ¼ ð0:015, 0:015, 0:5, 0:005Þ,
2 model with linear realization utility—we see that the
investor is now more reluctant to invest in a stock with a 0
ðd , r, l, bÞ ¼ ð0:08, 0:1, 1:5, 1Þ: ð27Þ
negative expected excess return. For a realization utility
investor, the problem with investing in such a stock is By comparing the dashed lines to the solid lines—the lines
that it raises the chance that he will be forced, by a that correspond to linear realization utility—we see that,
liquidity shock, to make a painful exit from a losing for our benchmark parameter values, allowing for greater
position. A high sensitivity to losses makes this prospect sensitivity to losses preserves the qualitative relationship
0
all the more unappealing. The investor therefore only between g n and Uð1Þ on the one hand, and m, s, and d on
invests in a negative expected excess return stock if it is the other.
highly volatile, so that it at least offers a non-negligible The dashed line in the middle-right graph of Fig. 2
chance of a sizeable gain that he can enjoy realizing. deserves particular attention. It shows that, for the bench-
When l 4 1, the prediction that the investor will be mark values in (27), initial utility Uð1Þ is still increasing in
willing to invest in a stock with a negative expected stock return volatility s. Put differently, even though
excess return depends heavily on the parameters r, l, the functional form for realization utility is now concave,
0
and d . If the liquidity shock intensity or the sensitivity to the investor is still risk seeking. However, when l 41, this
0
losses rise significantly above their benchmark values, or prediction is sensitive to the values of r, l, and d . If the
if the discount rate falls significantly below its benchmark sensitivity to losses or the liquidity shock intensity rise
value, the investor will no longer be willing to buy a significantly, or if the discount rate falls significantly, the
negative expected excess return stock, whatever its prediction is reversed: initial utility becomes a decreasing
volatility. function of s and the investor is risk averse, not risk
The graphs in Fig. 4 show how the liquidation point g n seeking.
and initial utility per unit wealth Uð1Þ depend on the It is worth emphasizing the crucial role that the
0
sensitivity to losses l. These graphs vary l while main- discount rate d plays in determining whether the investor
taining is risk seeking or risk averse, and whether he is willing to
ða, m, s,kÞ ¼ ð0:015, 0:015, 0:5, 0:005Þ, buy stocks with low expected returns. Roughly speaking,
buying a stock offers the investor either a short-term
0 realized gain, should the stock perform well, or a long-
ðd , r, bÞ ¼ ð0:08, 0:1, 1Þ: ð26Þ
term realized loss, should the stock perform poorly. The
In the left graph, we see that the more sensitive the more impatient the investor is, the more he focuses on
investor is to losses, the higher the liquidation point: the short-term gain as opposed to the long-term loss.

Liquidation point Initial utility

1.4 1

0.8
1.3

0.6
1.2
0.4

1.1
0.2

1 0
1 1.1 1.2 1.3 1.4 1.5 1 1.1 1.2 1.3 1.4 1.5
Sensitivity to losses λ Sensitivity to losses λ

Fig. 4. Sensitivity of the liquidation point at which an investor sells a stock, and of the initial utility from buying it, to l, his relative sensitivity to realized
losses as opposed to realized gains.
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 261

As a result, he is more likely to be risk seeking and to In this economy, the equilibrium conditions are
invest in stocks with low expected returns.9
V i ðW,WÞ ¼ 0, i ¼ 1, . . . ,N, ð31Þ

4. An asset pricing model where V i ðW t ,Bt Þ is the value function for an investor
whose wealth Wt is allocated to stock i and whose cost
In Sections 2 and 3, we studied realization utility in a basis is Bt. In words, these conditions mean that an
partial equilibrium model of trading behavior. In this investor who is buying a stock is indifferent between
section, we show how it can be embedded in an asset allocating his wealth to that stock or to the risk-free asset.
pricing model. We do not necessarily expect realization Why are Eqs. (31) the appropriate equilibrium condi-
utility to have an impact on the prices of all stocks; it may, tions? Note that, under the conditions in (31), we can
at most, affect the prices of stocks held and traded clear markets at time 0 by assigning some investors to
primarily by individual investors. Of course, the only each stock and the rest to the risk-free asset. If, at any
way to know for sure is to derive the pricing implications point in the future, some investors sell their holdings of
of realization utility and to compare these predictions to stock i because of a liquidity shock, they immediately
the available facts. withdraw from the asset markets. If some investors sell
Embedding non-standard preferences in a full equili- their holdings of stock i because, for these investors, the
brium can be challenging. To make headway, we study stock has reached its liquidation point, the conditions in
the simplest possible model, one with homogeneous reali- (31) mean that they are happy to then be assigned to the
zation utility investors. Consider an economy with a risk- risk-free asset. Finally, the conditions in (31) mean that, if
free asset and N risky stocks indexed by i 2 f1, . . . ,Ng. The some investors do sell their holdings of stock i, whether
risk-free asset is in perfectly elastic supply and earns a because of a liquidity shock or because the stock reaches
continuously compounded return of r. The risky stocks are its liquidation point, we can reassign other investors from
in limited supply. The dividend process for stock i is the risk-free asset to stock i, thereby again clearing the
dDi,t market in this stock.10
¼ ðr þ mi Þ dt þ si dZ i,t , ð28Þ Formally, the decision problem for an investor holding
Di,t
stock i at time t is
where Z i,t is a Brownian motion and where, for iaj, dZ i,t (Z
minft, t0 g
and dZ j,t may be correlated. The parameters mi and si are
V i ðW t ,Bt Þ ¼ max Et edðstÞ vðDi,s Þ ds
constant over time but can vary across stocks. tZt t
The price of stock i at time t, Si,t , is set in equilibrium.
We hypothesize that
þ edðttÞ uðð1ki ÞW t Bt ÞIft o t0 g
1
Si,t ¼ Di,t , ð29Þ )
ai
þ edðt tÞ uðð1ki ÞW t0 Bt0 ÞIft Z t0 g ,
0
ð32Þ

where ai will be determined later. By investing in stock i,

an investor therefore receives the dividend stream Di,t , subject to (3), (5), (18), and
which he consumes, and also the price fluctuation given by
dW s
dSi,t ¼ ðr þ mi Þ ds þ si dZ i,s , t r s o minft, t0 g, ð33Þ
¼ ðr þ mi Þ dt þ si dZi,t : ð30Þ Ws
Si,t
where t0 is the random future time at which a liquidity
The expected excess return of stock i is therefore ai þ mi . shock arrives. This differs from the decision problem in
The economy contains a continuum of realization (19) in that it imposes the market clearing condition (31):
utility investors. At each time t Z 0, each investor must after selling his stock holdings at time t, the investor’s
either allocate all of his wealth to the risk-free asset or all future value function is zero. We summarize the solution
of his wealth to one of the stocks. We allow for transac- to the decision problem in (32) in the following proposi-
tion costs, liquidity shocks, and piecewise-linear utility. tion. The proof is in the Appendix.
0
As noted above, the investors are homogeneous, so that d ,
r, l, and b are the same for all of them. Transaction costs,
Proposition 3. Unless forced to exit the stock market by a
however, can differ across stocks. The transaction cost for
liquidity shock, an investor with the decision problem in (32)
stock i is ki.
will sell his holdings of a stock if the gain g t ¼ W t =Bt reaches
a liquidation point g t ¼ g n Z1. If the transaction cost ki is
9
We have also studied another extension of the model in Section 2, positive, then g n 4 1. The value function when holding stock i
one that assumes hyperbolic, rather than exponential, discounting. We
find that hyperbolic discounting has a significant effect on the trading
10
behavior of an investor who is guided by realization utility. The more We assume here that whenever we need to reassign investors
present-biased the investor is, the lower the liquidation point: a from the risk-free asset to one of the stocks, there are always enough
present-biased investor is impatient to realize gains. More generally, investors holding the risk-free asset to make this possible. This can
hyperbolic discounting is one way of thinking about the high discount happen if, for example, investors who leave the asset markets because of
rate d required by condition (11). a liquidity shock later re-enter.
262 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

at time t is V i ðW t ,Bt Þ ¼ Bt U i ðg t Þ, where the same realization utility preferences. They do not
8   describe an equilibrium when investors have heteroge-
>
> g a b þ rlð1ki Þ rl 1
>
> bg t 1 þ i gt  if g t 2 0, neous realization utility preferences, nor when some
>
>
< r þ d0 mi r þ d0 1ki
 
U i ðg t Þ ¼ g g a b þ rð1ki Þ r 1 , investors have expected utility preferences defined only
>
> c1 g t 1 þ c2 g t 2 þ i gt  if g t 2 ,g n
>
> r 0
þ d mi r þd
0
1ki over consumption. We conjecture that in an economy
>
>
: ð1k Þg 1 if g t 2 ½g n ,1Þ with both expected utility and realization utility inves-
i t

ð34Þ tors, the expected utility investors will partially—but only

partially—attenuate any pricing effects caused by the
where g1 and g2 are given by realization utility investors. The predictions of the model
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 in this section should therefore hold more strongly among
   
1 4 1 2 2 0 1 2 5
g1 ¼ 2 mi  si þ2ðr þ d Þsi  mi  si 2 4 0, stocks traded by investors whose thinking is especially
si 2 2 influenced by realization utility.
ð35Þ
5. Applications
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
   
1 4 1 2 2 1
g2 ¼  2 mi  si þ 2ðr þ d Þs2i þ mi  s2i 5 o 0 0
Our model may be able to shed light on a number of
si 2 2
financial phenomena. We now discuss some of these
ð36Þ potential applications. We divide the applications into
those that relate to trading behavior (Section 5.1) and
and where b, c1, c2, and g n are determined from
those that relate to asset prices (Section 5.2). In Section
ðl1Þrð1ki Þg2 ðmi g1 rd Þ
0
5.3, we briefly discuss a few of the testable predictions
c2 ¼ 0 0 , ð37Þ
ðg1 g2 Þðr þ d mi Þðr þ d Þ that emerge from our framework.

g g d0 5.1. Trading behavior

ðg1 1Þc1 g n1 þðg2 1Þc2 g n2 ¼ , ð38Þ
r þ d0
5.1.1. The disposition effect
 g1  g2  g1
1 1 1 ðl1Þmi r The disposition effect is the finding that individual
c1 þc2 ¼b þ 0 0
1ki 1ki 1ki ðr þ d mi Þðr þ d Þ investors have a greater propensity to sell stocks that
ð39Þ have gone up in value since purchase, rather than stocks
that have gone down in value (Shefrin and Statman, 1985;
 0 
g g ð1ki Þðmi d rlÞ rl d0 Odean, 1998). This fact has turned out to be something of
c1 g n1 þ c2 g n2 þ 0 þ 0 b gn ¼  :
r þ d mi rþd r þ d0 a puzzle, in that the most obvious potential explanations
ð40Þ fail to capture important features of the data. Consider, for
example, the most obvious potential explanation of all,
the ‘‘informed trading’’ hypothesis. Under this view,
The equilibrium expected excess return of stock i is investors sell stocks that have gone up in value because
ai þ mi . The parameter mi is the expected excess dividend they have private information that these stocks will
growth rate and is exogenously given. To determine ai , we subsequently fall, and they hold on to stocks that have
require that the value function satisfies the condition in gone down in value because they have private informa-
(31), namely V i ðW,WÞ ¼ 0, or equivalently, U i ð1Þ ¼ 0. The tion that these stocks will rebound. The difficulty with
parameter ai is therefore given by this view, as Odean (1998) points out, is that the prior
winners people sell subsequently do better, on average,
ai b þ rlð1ki Þ rl than the prior losers they hold on to. Odean (1998) also
bþ  ¼ 0: ð41Þ
r þ d0 mi r þ d0 considers other potential explanations based on taxes,
0
Since the parameters d , r, l, and b are constant across rebalancing, and transaction costs, but argues that none of
investors, ai is constant over time, as assumed earlier.11 them is fully satisfactory.
In Section 5.2, we use the model described in this Our analysis shows that a model that combines reali-
section to illustrate the effect of realization utility on asset zation utility with a sufficiently positive time discount
prices. We emphasize that conditions (31) only describe rate predicts a strong disposition effect. Unless forced to
an equilibrium when all investors in the economy have sell at a loss by a liquidity shock, the investor in our
model only sells stocks trading at a price higher than the
original purchase price.
11
In our model, it is the buyers of the risky assets, not the sellers, In simple two-period settings, Shefrin and Statman
who set prices. In other words, the condition V i ðW,WÞ ¼ 0 is determined
(1985) and Barberis and Xiong (2009) show that realiza-
by buyer behavior, not seller behavior. To see this, suppose that an
investor is trying to sell stock i. If V i ðW,WÞ 40, then all investors holding tion utility, with no time discounting but with a func-
the risk-free asset will want to switch to the stock and the market will tional form for utility that, as in prospect theory, is
fail to clear. On the other hand, if V i ðW,WÞ o 0, there will be no one for concave over gains and convex over losses, can predict a
the seller to trade with: no one holding the risk-free asset will want to disposition effect. This paper proposes a related but
switch to the stock. Only if V i ðW,WÞ ¼ 0 can we clear the market. The
fact that prices are set by buyers has an important corollary: it means
distinct view of the disposition effect, namely that it
that the price of a stock does not depend on the average cost basis of the arises from realization utility with a linear functional form
investors holding it. for utility and a positive time discount rate.
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 263

We emphasize that realization utility does not, on its 5.1.2. Excessive trading
own, predict a disposition effect. In other words, to generate Using a database of trading activity at a large discount
a disposition effect, it is not enough to assume that the brokerage firm, Barber and Odean (2000) show that, after
investor derives pleasure from realizing a gain and pain transaction costs, the average return of the individual
from realizing a loss. We need an extra ingredient in order investors in their sample falls below the returns on a
to explain why the investor would want to realize a gain range of benchmarks. This is puzzling: why do people
today, rather than hold out for the chance of realizing an trade so much if their trading hurts their performance?
even bigger gain tomorrow. Shefrin and Statman (1985) and Barber and Odean (2000) consider a number of potential
Barberis and Xiong (2009) point out one possible extra explanations, including taxes, rebalancing, and liquidity
ingredient: a prospect theory functional form for utility. needs, but conclude that none of them can fully explain
Such a functional form indeed explains why the investor the patterns they observe.
would expedite realizing a gain and postpone realizing a Our model offers an explanation for this post-transac-
loss. Here, we propose an alternative extra ingredient: a tion-cost underperformance. Under this view, the inves-
sufficiently positive time discount rate. tors in Barber and Odean’s (2000) sample are guided by
Our model is also well-suited for thinking about the realization utility. This leads them to trade: specifically, to
disposition-type effects that have been uncovered in other sell stocks that have risen in value since purchase so that
settings. Genesove and Mayer (2001), for example, find that they can enjoy bursts of positive utility, and to then invest
homeowners are reluctant to sell their houses at prices the proceeds in new stocks. However, by trading, they
below the original purchase price. Our analysis shows that a incur transaction costs that cause them to underperform
model that combines linear realization utility with a posi- the benchmarks.
tive time discount rate can capture this evidence. It is possible to compute the probability that the
Of all the applications we discuss in Section 5, the investor in our model sells a stock within any given
disposition effect is the most obvious, in the sense that it interval of time after the initial purchase. Doing so will
is very clear how the effect follows from our initial help us compare the trading frequency predicted by our
assumptions. However, as we noted in the Introduction, model with that observed in actual brokerage accounts.
realization utility is in no sense a relabeling of the When the investor first establishes a position in a stock, at
disposition effect. On the contrary, it is just one of a time 0, say, we have g 0 ¼ 1. When gt reaches an upper
number of possible theories of the disposition effect, and barrier g n 4 1 or when a liquidity shock arrives, he sells
can be distinguished from other theories through care- the stock. To compute the probability that the investor
fully constructed tests. sells the stock within s periods after establishing the
An example of a test that distinguishes various the- position, we therefore need to compute the probability
ories of the disposition effect can be found in Weber and that gt reaches g n in the interval ð0,sÞ or that there is a
Camerer (1998). These authors test the realization utility liquidity shock during the same interval. The next propo-
view of the disposition effect against the alternative view sition, which we prove in the Appendix, reports the result
that it stems from an irrational belief in mean-reversion. of this calculation.
In a laboratory setting, they ask subjects to trade six
stocks over a number of periods. In each period, each Proposition 4. The probability that the investor sells a stock
stock can either go up or down. The six stocks have within s periods of the date of purchase is
2 0  
different probabilities of going up in any period, ranging s2 1
from 0.35 to 0.65, but subjects are not told which stock is ln g n þ m s
6 B 2 C
associated with each possible up-move probability. GðsÞ ¼ 1ers þers 6 B
4N @ pffiffi C
A
s s
Weber and Camerer (1998) find that, just as in field
data, their subjects exhibit a disposition effect. To try to 0  
understand the source of the effect, the authors consider s2 13
B ln g n  m s 7
an additional experimental condition in which the experi- 2 2 C
þ eðð2m=s Þ1Þln g n NB
@ p ffiffi C7:
A5
menter liquidates subjects’ holdings and then tells them s s
that they are free to reinvest the proceeds in any way they
like. If subjects were holding on to their losing stocks ð42Þ
because they thought that these stocks would rebound,
we would expect them to re-establish their positions in
these losing stocks. In fact, subjects do not re-establish The expression in the square parentheses in (42) is the
these positions. This casts doubt on the mean-reversion probability that gt reaches g n in the interval ð0,sÞ. With
view of the disposition effect and lends support to the this information in hand, it is easy to interpret the
realization utility view, namely that subjects were refus- equation. The investor trades during the interval ð0,sÞ if
ing to sell their losers simply because it would have been one of two mutually exclusive events occurs: if there is a
painful to do so. Under this view, subjects were relieved liquidity shock in ð0,sÞ; or if there is no liquidity shock in
when the experimenter intervened and did it for them.12 ð0,sÞ but gt reaches g n in ð0,sÞ. The probability of a trade in
ð0,sÞ is therefore the probability of a liquidity shock in
ð0,sÞ, namely 1ers , plus the probability of no liquidity
12
See Kaustia (2010) for additional evidence against the mean- shock, namely ers , multiplied by the probability that gt
reversion view of the disposition effect. reaches g n .
264 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

Trade probability Trade probability

1 1

0.5 0.5

0 0
−0.04 −0.02 0 0.02 0.04 0.2 0.4 0.6 0.8
Expected excess capital gain μ Standard deviation σ

1 1

0.5 0.5

0 0
0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
Effective time discount rate δ′ Transaction cost k

0.5

0
1 1.1 1.2 1.3 1.4 1.5
Sensitivity to losses λ

Fig. 5. Probability that an investor who derives utility from realized gains and losses will sell a specific stock within a year of buying it. The graphs show
0
how this probability varies with the stock’s expected excess capital gain m, its standard deviation s, the effective time discount rate d , the transaction
cost k, and the relative sensitivity to realized losses as opposed to realized gains l.

Fig. 5 shows how the probability of selling a stock The top graphs in Fig. 5 show that, interestingly, a
within a year of purchase, Gð1Þ, depends on the model different factor dominates in each case. As m rises, the
parameters. To construct the graphs, we use the model of probability of a trade falls. Roughly speaking, as m rises,
Section 3 which allows for a transaction cost, a liquidity the liquidation point rises more quickly than the stock’s
shock, and piecewise-linear utility. For any given para- ability to reach it. As s rises, however, the probability of a
meter values, we compute the liquidation point g n from trade goes up: in this case, the liquidation point rises less
(22)–(25) and substitute the result into the expression for quickly than the stock’s ability to reach it.
0
Gð1Þ in Proposition 4. The graphs vary each of m, s, d , k, The bottom-left graph, which varies l, shows that the
and l in turn, keeping the remaining parameters fixed at probability of a trade declines as the sensitivity to losses
their benchmark values rises. If l is high, the investor is reluctant to sell a stock
ða, m, s,kÞ ¼ ð0:015, 0:015, 0:5, 0:005Þ, trading at a gain because if he does, he will have to buy a
new stock, which might go down and which he might be
0 forced to sell at a loss by a liquidity shock.
ðd , r, l, bÞ ¼ ð0:08, 0:1, 1:5, 1Þ: ð43Þ
Barber and Odean (2000) find that in their sample of
Some of the results in Fig. 5 are not very surprising. households with brokerage accounts, the mean and med-
The middle-left graph shows that as the investor becomes ian annual turnover rates are 75% and 30%, respectively.
more impatient, the probability of a trade rises. And the Fig. 5 shows that for the benchmark parameter values, our
middle-right graph shows that as the transaction cost model predicts a trading frequency that is of a similar
falls, the probability of a trade again rises. order of magnitude. When s ¼ 0:5, for example, the
The top-left and top-right graphs, which vary m and s probability that an investor trades a specific stock in his
respectively, are less predictable. In both cases, there are portfolio within a year of purchase is approximately 0.6.
two factors at work. On the one hand, for any fixed Of course, the fact that the trading frequency predicted by
liquidation point g n , a higher m or s raises the likelihood our model is similar to that observed in actual brokerage
that g n will be reached within the year-long interval. accounts is not an accident: we chose the benchmark
0
However, as we saw in Fig. 2, the liquidation point g n value of d to ensure that this would be the case.
itself goes up as m and s go up, thereby lowering the When we say that realization utility can help us
chance that g n will be reached. Without computing Gð1Þ understand ‘‘excessive trading,’’ we do not mean that it
explicitly, it is hard to know which factor will dominate. can explain the high overall volume of trading in financial
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 265

markets. Rather, we mean something narrower: that it Wurgler, 2007). If some investors have very positive
can help us understand why individual investors trade as sentiment and push stock prices up as a result, realization
much as they do in their brokerage accounts, given that utility investors will start trading heavily. This creates a
they would earn higher returns, on average, if they traded link between turnover and sentiment.
less. While realization utility investors are keen to trade a
stock that has risen in value, they are not keen to trade a
stock that has fallen in value. It is therefore an open
question as to whether an increase in the fraction of 5.1.5. The effect of historical highs on the propensity to sell
investors in the economy who are guided by realization Our model implies that there will be more trading in
utility would lead to an increase in the overall volume of rising markets, but it can also make more precise predic-
trading. tions as to how trading activity will vary over time. For
example, it predicts that individual investors—the inves-
5.1.3. Underperformance before transaction costs tor group that is more likely to think in terms of realiza-
Some studies find that the average individual investor tion utility—will have a much higher propensity to sell a
underperforms benchmarks even before transaction costs stock once its price moves above its historical high.
(Barber, Lee, Liu, and Odean, 2009). Our model may be To see this, consider a stock that, on January 1st, is trading
able to shed light on this by way of one of the predictions at $30. Suppose that it then rises through January and
we discussed in Sections 2 and 3: that an investor who February, reaching a high of $45 by February 28th. It then
thinks in terms of realization utility is often willing to buy declines significantly through most of March but, towards
a stock with a low expected return, so long as the stock’s the end of March, starts rising again, passing through the
volatility is sufficiently high. previous high of $45 on March 31st and continuing upwards.
Suppose that the investing population consists of two Our model predicts that after the stock passes $45 on
groups: individuals, who think in terms of realization March 31st, there will be a sharp increase in selling by
utility; and institutions, who do not. Since individuals are individual investors. To see why, note that there will be very
guided by realization utility, they may be more willing little selling between February 28th and March 31st. During
than institutions to buy stocks with low expected returns. this time, the stock is trading below its high of $45. The only
Moreover, since the average portfolio return before trans- investors who would want to sell in this interval are those
action costs across all investors must equal the market targeting liquidation points below $45. But the majority of
return, we should observe the average individual under- these investors will have sold the stock already, before
performing market benchmarks before transaction costs February 28th, when the stock first reached $45. Once the
and the average institution outperforming the bench- stock moves above $45 on March 31st, however, investors
marks, again before transaction costs. This prediction is targeting liquidation points higher than $45 will start sell-
broadly consistent with the available evidence.13 ing. As claimed above, then, individual investors’ propensity
to sell a stock will increase sharply as the stock price moves
5.1.4. Trading volume in rising and falling markets above its historical high.
Researchers have found that in many different asset Our prediction is consistent with the available evi-
classes, trading volume is higher in rising markets than in dence. Grinblatt and Keloharju (2001) find that house-
falling markets (Stein, 1995; Statman, Thorley, and holds’ propensity to sell a stock does increase strongly
Vorkink, 2006; Griffin, Nardari, and Stulz, 2007). Robust once the stock price moves above its historical high for
though this finding is, there are few explanations for it. that month. Similarly, albeit in a different context, Heath,
The equilibrium model of Section 4 offers a way of Huddart, and Lang (1999) find that executives are much
understanding it. In that model, there is indeed more more likely to exercise stock options when the underlying
trading in rising markets. In a rising market, the stocks stock price exceeds its historical high. Finally, Baker, Pan,
held by realization utility investors start hitting their and Wurgler (2009) show that, when a firm makes a
liquidation points. When this happens, these investors takeover bid for another firm, the offer price is more likely
sell their stocks to other realization utility investors. As a to slightly exceed the target’s 52-week historical high
result, trading volume goes up. than to be slightly below it; and that there is a discontin-
The same line of reasoning can motivate the use of uous increase in deal success as the offer price rises
turnover as a measure of investor sentiment (Baker and through the 52-week high. This is consistent with the
idea that, as a consequence of realization utility, investors
are more likely to sell their shares in the target company
13
So far, our model has pointed to two ways in which realization at a price that exceeds the historical high.14
utility can lower an investor’s Sharpe ratio: it leads him to buy stocks
with low expected returns and high volatility; and by encouraging him
to trade, it leads him to incur transaction costs. There is one more
14
channel through which realization utility can harm the investor’s It is tempting to interpret Grinblatt and Keloharju’s (2001)
performance—a channel that, while important, lies outside our model. finding as evidence that investors use the historical high as an explicit
A strategy that sells winners but holds on to losers will lower the reference point: for example, that they derive utility from the difference
investor’s average return if his typical holding period coincides with the between the price at which they sell a stock and its historical high. Our
horizon at which stocks exhibit momentum. At least for some investors, analysis shows, however, that Grinblatt and Keloharju’s (2001) result
this does appear to be the case: the investors in Barber and Odean’s can arise in a model in which the only explicit reference point is the
(2000) sample hold stocks for a few months, on average—a horizon at purchase price. The historical high emerges as a reference point
which stock returns exhibit significant momentum. endogenously because of the nature of the investor’s optimal strategy.
266 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

0 1

Expected excess return

−0.005 0.8

Trade probability
−0.01
0.6
−0.015
0.4
−0.02
0.2
−0.025
0
0.1 0.2 0.3 0.4 0.5 −0.025 −0.02 −0.015 −0.01 −0.005
Standard deviation Expected excess return

Fig. 6. Expected return, standard deviation, and probability of sale in an economy populated by investors who derive utility from realized gains and
losses. The top-left graph shows the equilibrium relationship between expected excess return and standard deviation in a cross-section of stocks. The
top-right graph shows, for the same cross-section, the equilibrium relationship between a stock’s expected excess return and the probability that, after
buying the stock, an investor sells it within a year of purchase. Realization utility has a piecewise-linear functional form, so that investors are 1.5 times as
sensitive to realized losses as to realized gains.

5.1.6. The individual investor preference for volatile stocks and assume that the excess dividend growth rate and the
Kumar (2009) analyzes the trades of approximately transaction cost are the same for all stocks, namely
60,000 households with accounts at a large discount m ¼ 0:03 and k¼ 0.005, respectively. For values of s
brokerage firm. He finds that, as a group, the individual ranging from 0.01 to 0.5, we use equilibrium condition
investors in his sample overweight highly volatile stocks: (41) to compute the dividend yield a and hence the
these stocks make up a larger fraction of the value of the expected excess return a þ m that a stock with any given
aggregate individual investor portfolio, constructed using standard deviation must earn in order for its market to
these data, than they do of the aggregate market portfolio. clear.15
Realization utility offers a way of understanding this. The top-left graph in Fig. 6 plots the resulting relation-
As we saw in Sections 2 and 3, investors who are guided ship between standard deviation and expected excess
by realization utility often have a strong preference for return. The graph confirms our prediction: more volatile
volatile stocks. Moreover, these investors are more likely stocks earn lower average returns; in this sense, they are
to be individuals than institutions. overpriced.16
The top-left graph also shows that for the parameter
5.2. Asset pricing values in (44), stocks earn negative average excess
returns, which is inconsistent with the positive historical
Our model may also be helpful for understanding equity premium. A negative equity premium is not a
certain asset pricing patterns. We now discuss three generic prediction of our model: for values of r and l
applications of this type. that are somewhat higher than those in (44), and for
0
values of d that are somewhat lower, the investors
5.2.1. The low average return of volatile stocks become risk averse rather than risk seeking and the equity
Ang, Hodrick, Xing, and Zhang (2006) show that, in the premium turns positive. It is difficult, however, for the
cross-section, and after controlling for previously known homogeneous agent economy we are analyzing to gen-
predictor variables, a stock’s daily return volatility over erate both a positive equity premium and a negative
the previous month negatively predicts its return in the relationship between volatility and average return in the
following month. This finding, which holds not only in the cross-section. We conjecture that it may be possible to
U.S. stock market but in many international stock markets generate both of these facts in an economy with
as well, is puzzling. Even if we allow ourselves to think of
a stock’s own volatility as risk, the result is the opposite of
what we would expect: it says that ‘‘riskier’’ stocks have 15
Since m is the excess dividend growth rate, a negative value of m
lower average returns. does not necessarily mean that the dividend growth rate is negative, just
Our model offers a novel explanation for this finding. that it is below the risk-free rate. Since, for the parameter values in (44),
We noted earlier—see the middle-right graph in the investors in our economy are risk seeking, the dividend growth rate
Fig. 2—that for some parameter values, realization utility must be below the risk-free rate to prevent prices from exploding, just
as, in a standard Gordon growth model with risk-neutral investors, the
investors are risk seeking. As a result, they will exert
dividend growth rate has to be below the risk-free rate. Note that a
heavy buying pressure on stocks that are highly volatile. negative excess dividend growth rate m does not necessarily imply a
These stocks may then become overpriced. If so, their negative expected excess return. The expected excess return is a þ m.
subsequent average return will indeed be low. This can be positive even if m is negative.
16
We now check this intuition using the equilibrium In our model, the risky assets are infinitely lived. We have studied
a variant of the model in which the risky assets stochastically ‘‘expire’’
model of Section 4. We assign all investors the same based on the arrival of Poisson-distributed liquidation shocks. We find
benchmark parameter values that in an economy with realization utility investors, a short-horizon
0 asset—one with a higher liquidation shock intensity—can earn a higher
ðd , r, l, bÞ ¼ ð0:08, 0:1, 1:5, 1Þ ð44Þ Sharpe ratio than a long-horizon asset.
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 267

heterogeneous realization utility investors, some of whom expected returns—stocks that are more ‘‘overpriced’’—do
are risk seeking and some of whom are risk averse. indeed experience more turnover.
Another way of reconciling the top-left graph with the
positive historical equity premium is to say that the result in 5.2.3. Momentum
the graph only applies to stocks that are primarily held by Grinblatt and Han (2005) study an economy in which
investors who think in terms of realization utility—most some investors’ demand for a stock depends, negatively,
likely, individual investors. Since these stocks constitute a on the difference between the current stock price and the
small fraction of the total stock market capitalization, they price they paid for the stock. They show that in this
play only a minor role in determining the aggregate equity economy, as in actual data, stock returns exhibit momen-
premium. One prediction of this view is that the cross- tum. The authors suggest one possible foundation for the
sectional relationship between volatility and average return demand function they propose, namely, a combination of
documented by Ang, Hodrick, Xing, and Zhang (2006) should prospect theory and mental accounting. Our model sug-
be stronger among stocks traded by individual investors. This gests a different, albeit related foundation: linear realiza-
is exactly the finding of Han and Kumar (2011). tion utility. In combination with a sufficiently positive
time discount rate, linear realization utility also leads to a
5.2.2. The heavy trading of highly valued assets demand function for a stock that depends, negatively, on
A robust empirical finding is that assets that are highly the difference between the current stock price and the
valued, and possibly overvalued, are also heavily traded purchase price. This, in turn, suggests that momentum
(Hong and Stein, 2007). Growth stocks, for example, are may ultimately stem, at least in part, from realization
more heavily traded than value stocks; the highly priced utility.
technology stocks of the late 1990s changed hands at a rapid A limitation of the pricing model in Section 4 is that it
pace; and shares at the center of famous bubble episodes, does not allow us to illustrate the link between realization
such as those of the East India Company at the time of the utility and momentum: in that model, stock returns are
South Sea bubble, also experienced heavy trading. not predictable. To see why the link breaks down, recall
Our model may be able to explain this coincidence of the original intuition for it. The idea is that if a stock rises
high prices and heavy trading. Specifically, it predicts that in value, realization utility investors will start selling it in
this phenomenon will occur for assets whose value is order to realize a gain. This selling pressure causes the
especially uncertain. stock to become undervalued. Subsequently, the stock
Suppose that the uncertainty about an asset’s value price moves higher, on average, as it corrects from this
goes up, thereby increasing s, the standard deviation of undervalued point to a more reasonable valuation. An
returns. As noted earlier, investors who think in terms of upward price move is therefore followed by another
realization utility will now find the asset more attractive. upward price move, on average. This generates a momen-
If there are many such investors in the economy, the tum effect in the cross-section of stock returns.
asset’s price will be pushed up. In our model, realization utility investors do indeed
At the same time, the top-right graph in Fig. 5 shows start selling when a stock rises in value. However, this
that as s goes up, the probability that an investor will does not depress the stock price because of the perfectly
trade the asset also goes up: simply put, a more volatile elastic demand for the stock from other realization utility
asset tends to reach its liquidation point more rapidly. In investors. As a result, there is no momentum. We suspect
this sense, the overvaluation will coincide with higher that the link between realization utility and momentum
turnover, and this will occur when uncertainty about the can be formalized in an economy with both realization
asset’s value is especially high. Under this view, the late utility investors and expected utility investors. In such an
1990s were years when realization utility investors, economy, when realization utility investors sell a stock
attracted by the high uncertainty of technology stocks, that is rising in value, their selling will depress the stock
bought these stocks, pushing their prices up; as (some of) price because the demand from expected utility investors
these stocks rapidly reached their liquidation points, the will not be perfectly elastic.
realization utility investors sold them and then immedi-
ately bought new ones. 5.3. Testable predictions
We now check this intuition using the equilibrium frame-
work of Section 4. As in our discussion of the low average In Sections 5.1 and 5.2, we argue that realization utility
return of volatile stocks, we assign all investors the bench- offers a simple way of understanding a range of financial
mark parameter values in (44) and assume that the excess phenomena. In this section, we briefly note a few of the
dividend growth rate and the transaction cost are the same new predictions that emerge from our framework.
for all stocks, namely m ¼ 0:03 and k¼0.005, respectively. One set of predictions is based on the graphs in Fig. 5,
For values of s ranging from 0.01 to 0.5, we again use which show how the probability of trade depends on
condition (41) to compute the corresponding equilibrium various parameters. One of these predictions, that the
expected excess return; but this time, as a guide to the investor is more likely to trade a stock within a year of
intensity of trading, we also use (42) to compute Gð1Þ, the purchase when transaction costs are lower, is not unique
probability of a trade within a year of purchase. to our model. However, the figure also suggests some
The top-right graph in Fig. 6 plots the resulting other, more novel predictions: that the probability that
relationship between the expected excess return and the the investor trades a stock within a year of purchase is an
trade probability. It confirms that stocks with lower increasing function of his impatience and of the stock’s
268 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

volatility, and a decreasing function of his sensitivity to subjects while they trade stocks in an experimental
losses. market. The authors use the neural data to test some
The prediction relating the probability of trade to a theories of investor behavior, including the one presented
stock’s volatility is straightforward to test empirically. To in this paper.
test the predicted link between trade probability and Finally, it would be useful to think about other appli-
impatience and between trade probability and sensitivity cations of realization utility. These applications may again
to losses, we need estimates of impatience and loss concern the trading and pricing of financial securities, or
sensitivity, which may be difficult to obtain. In recent they may be drawn from quite different areas of study.
years, however, researchers have pioneered clever tech- After all, the core idea that, in our view, underlies
niques for extracting information about investors’ psy- realization utility—that people break their experiences
chological profiles. Grinblatt and Keloharju (2009), for down into episodes and receive a burst of utility when an
example, use military test scores from Finland to estimate episode comes to an end—strikes us as one that may be
overconfidence. This success makes us more optimistic relevant in many contexts, not just the financial market
that a test of the link between trade probability on the context that we have focused on in this paper.
one hand, and impatience and loss sensitivity on the
other, can also be implemented.
If we are indeed able to measure investor impatience, Appendix A
there are other predictions that can be tested. As noted
earlier, two of the more striking implications of realiza-
tion utility—that investors will be willing to buy stocks Proof of Proposition 1. At time t, the investor can either
that are highly volatile and that have low expected liquidate his position or hold it for an infinitesimal period
returns—depend crucially on the discount rate d. Roughly dt. We therefore have
speaking, a stock with a low expected return or with high
volatility offers the investor the prospect of realizing
either a short-term gain or a long-term loss. The higher VðW t ,Bt Þ ¼ maxfuðð1kÞW t Bt Þ þVðð1kÞW t ,ð1kÞW t Þ,
the discount rate d, the more attractive this tradeoff
vðDi,t Þ dt þ ð1r dtÞEt ½eddt VðW t þ dt ,Bt þ dt Þ
becomes. In short, then, if we are able to measure investor
impatience, we should find that more impatient investors þ r dt½uðð1kÞW t Bt Þg: ð45Þ
allocate more to stocks with low expected returns,
thereby earning low portfolio returns even before taking The first argument of the ‘‘max’’ function corresponds to
transaction costs into account; and also that they tilt their the case where the investor liquidates his position at time
portfolios more heavily towards volatile stocks. t: he receives realization utility of uðð1kÞW t Bt Þ and
cash proceeds of ð1kÞW t which he immediately invests
in another stock. The second argument of the ‘‘max’’
6. Conclusion function corresponds to the case where the investor
instead holds his position for an infinitesimal period dt:
A number of authors have suggested that investors he receives utility vðDi,t Þ dt from the flow of dividends;
derive utility from realizing gains and losses. We present with probability er dt  1r dt, there is no liquidity
a model of this ‘‘realization utility,’’ study its predictions, shock during the interval and his value function is the
and show that it can shed light on a number of expected future value function discounted back; and with
puzzling facts. probability 1er dt  r dt, there is a liquidity shock, in
There are several possible directions for future research. which case he sells his holdings, exits the asset markets,
First, while many of our model’s implications match the and receives realization utility of uðð1kÞW t Bt Þ.
observed facts, some do not. For example, our model predicts Given the homogeneity property in (8), we can write
too strong a disposition effect: in our framework, investors the value function as
never voluntarily sell stocks at a loss, while, in reality, they
clearly do. It would be useful to see whether an extension of
VðW t ,Bt Þ ¼ Bt Uðg t Þ:
our model—one that modifies our preference specification in
some way, or that allows for richer beliefs about expected
stock returns—can make more accurate predictions.17
Substituting this into (45), canceling Bt from both sides,
Another natural research direction involves testing the
and applying Ito’s lemma gives
implications of realization utility. To do this, we can use
field data on investor trading behavior; or experimental
data, as in Weber and Camerer (1998). Another type of Uðg t Þ ¼ maxfuðð1kÞg t 1Þ þ ð1kÞg t Uð1Þ,Uðg t Þ
data that has recently become available is neural data. For  0
þ abg t þ 12s2 g 2t U 00 ðg t Þ þ mg t U 0 ðg t Þðr þ d ÞUðg t Þ
example, Frydman, Barberis, Camerer, Bossaerts, and

Rangel (2011) use functional magnetic resonance imaging þ ruðð1kÞg t 1Þ dtg: ð46Þ
(fMRI) technology to monitor the brain activity of 28
Eq. (46) implies that any solution to (10) must satisfy
17
Two recent studies that take up this question are Ingersoll and Jin
(2011) and Henderson (forthcoming). Uðg t Þ Z uðð1kÞg t 1Þ þ ð1kÞg t Uð1Þ ð47Þ
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 269

and We now verify that the function Uðg t Þ summarized in

abg t þ 12 2 g 2t U 00 ðg t Þ þ g t U 0 ðg t Þ
s m Eq. (14) satisfies conditions (47) and (48). Define

r 0
ð þ d ÞUðg t Þ þ uðð1kÞg t 1Þ r 0:
r ð48Þ f ðgÞ  ð1kÞð1 þ Uð1ÞÞg1:
By construction, f ðgÞ is a straight line that coincides with
0
Formally speaking, the decision problem in (10) is an UðgÞ for g Z g n . Since g1 4 1—this follows from m o r þ d
optimal stopping problem. To solve it, we first construct a which, in turn, follows from restriction (11)—U(g) in
function Uðg t Þ that satisfies conditions (47) and (48) and Eq. (14) is a convex function. It must therefore lie above
that is both continuous and continuously differentiable— the straight line f(g) for all g og n . Condition (47) is
this last condition is sometimes known as the ‘‘smooth therefore satisfied.
pasting’’ condition. If we are able to do this, then, given We now check that condition (48) holds. Define
0
that certain technical conditions are satisfied, the con- HðgÞ  12s2 g 2 U 00 ðgÞ þ mgU 0 ðgÞðr þ d ÞUðgÞ þ ðab þ rð1kÞÞgr:
structed function Uðg t Þ will indeed be a solution to
problem (10). For g o g n , HðgÞ ¼ 0 by construction. For g Z g n , UðgÞ ¼ f ðgÞ,
We construct Uðg t Þ in the following way. If gt is low, so that
  
specifically, if g t 2 ð0,g n Þ, we suppose that the investor 0 ab 0
HðgÞ ¼ ð1kÞg ðr þ d mÞð1 þUð1ÞÞ r þ þd :
continues to hold his current position. In this ‘‘continua- 1k
tion’’ region, condition (48) holds with equality. If gt is
Substituting (51) and (16) into this expression, we obtain
sufficiently high, specifically, if g t 2 ðg n ,1Þ, we suppose ( " #
that the investor liquidates his position. In this ‘‘liquida- d0 ðr þ d0 mÞ 1
HðgÞ ¼ ð1kÞg 1þ g
tion’’ region, condition (47) holds with equality. As in the r þ d0 ðg1 1Þg n1
statement of the proposition, we refer to g n as the
k d0
liquidation point.  ðab þ rð1kÞÞ
1k ð1kÞg
Since uðÞ is linear, the value function UðÞ in the ( " #
continuation region satisfies d0 ðr þ d0 mÞ 1
rð1kÞg 1 þ g
1 2 2 00
s g t U ðg t Þ þ mg t U 0 ðg t Þðr þ d0 ÞUðg t Þ þ ðab þ rð1kÞÞg t r ¼ 0: r þ d0 ðg1 1Þg n1
2
k d0
The solution to this equation is  ðab þ rð1kÞÞ
1k ð1kÞg n
g ab þ rð1kÞ r g d 0
Uðg t Þ ¼ ag t 1 þ g ¼
0
ðr þ d mg1 Þ:
r þ d0 m t r þ d0 g n ðr þ d0 Þðg1 1Þ
for g t 2 ð0,g n Þ, ð49Þ
The last equality follows by applying (17). Using (15), it is
0
where g1 is given in (15) and where a is determined straightforward to show that if m o r þ d , as assumed in
0
below. restriction (11), then r þ d mg1 4 0. Therefore, HðgÞ o 0
In the liquidation region, we have for g Z g n , thereby confirming that condition (48) holds
Uðg t Þ ¼ ð1kÞg t ð1 þUð1ÞÞ1: ð50Þ for all g t 2 ð0,1Þ.
To formally complete the derivation of Proposition 1,
Note that the liquidation point g n satisfies g n Z 1. For if we have proved a verification theorem. This theorem uses
g n o1, then g t ¼ 1 would fall into the liquidation region, the fact that conditions (47) and (48) hold everywhere to
which, from (50), would imply confirm that the stopping strategy proposed above is
Uð1Þ ¼ ð1kÞUð1Þk: indeed the optimal one. For space reasons, we do not
present the details of this step here. &
For k4 0 and Uð1Þ Z0, this is a contradiction. Since g n Z 1,
then, we infer from (49) that Proof of Proposition 2. The proof is very similar in struc-
ture to the proof of Proposition 1. We therefore present
ab þ rð1kÞ r
Uð1Þ ¼ a þ  : ð51Þ only the key steps. From (8), the value function takes the
r þ d0 m r þ d0
form
VðW t ,Bt Þ ¼ Bt Uðg t Þ:
The value function must be continuous and continu-
ously differentiable at the liquidation point g n . This Following the same reasoning as in the proof of Proposi-
implies tion 1, we find that UðÞ again satisfies Eq. (46) and
inequalities (47) and (48). The only difference is that uðÞ
g ab þ rð1kÞ r
ag n1 þ g  ¼ ð1kÞg n ð1þ Uð1ÞÞ1 now has the piecewise-linear form in (18).
r þ d0 m n r þ d0 As before, we conjecture two regions: a continuation
region, g t 2 ð0,g n Þ, and a liquidation region, g t 2 ðg n ,1Þ.
g 1 ab þ rð1kÞ
ag1 g n1 þ ¼ ð1kÞð1 þUð1ÞÞ: In the continuation region, UðÞ satisfies
r þ d0 m
1 2 2 00
2s g t U ðg t Þ þ mg t U 0 ðg t Þðr þ d0 ÞUðg t Þ
Solving these two equations, we obtain the expression for
þ abg t þ ruðð1kÞg t 1Þ ¼ 0: ð52Þ
a in (16) and the nonlinear equation for g n in (17). It is
straightforward to check that if restriction (11) holds, The form of the uðÞ term depends on whether its argu-
Eq. (17) has a unique solution in the range ð1,1Þ. ment, ð1kÞg t 1, is greater or less than zero. Note that the
270 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271

cross-over point, g t ¼ 1=ð1kÞ, lies below g n , so that Since, from (53),

g n Z1=ð1kÞ. For if g n o 1=ð1kÞ, then g t ¼ 1=ð1kÞ would
ab rlðmkrkd0 Þ
be in the liquidation region, which, from (20), would imply Uð1Þ ¼ bþ 0 þ 0 0 ,
  r þ d m ðr þ d Þðr þ d mÞ
1
U ¼ Uð1Þ, we obtain
1k

contradicting the reasonable restriction that Uðg t Þ be g g ab þ ð1kÞðmd0 Þ d0

c1 g n1 þ c2 g n2 þ 0 gn þ
strictly increasing in gt. Since g n Z 1=ð1kÞ, we further r þ d m r þ d0
 
subdivide the continuation region ð0,g n Þ into two subre- ab rlðmkrkd0 Þ
gions, ð0; 1=ð1kÞÞ and ð1=ð1kÞ,g n Þ. ¼ ð1kÞg n bþ 0 þ 0 0 ,
r þ d m ðr þ d Þðr þ d mÞ
For g t 2 ð0; 1=ð1kÞÞ, (52) becomes
which reduces to Eq. (25); and also Eq. (23). &
1 2 2 00
2s g t U ðg t Þ þ mg t U 0 ðg t Þðr þ d0 ÞUðg t Þ þ ðab þ rlð1kÞÞg t rl ¼ 0:
The solution to this equation is Proof of Proposition 3. We solve the decision problem in
(32) using the same technique as the one employed in the
g ab þ rlð1kÞ rl proofs of Propositions 1 and 2. In particular, we replace a,
Uðg t Þ ¼ bg t 1 þ gt 
r þ d0 m r þ d0 m, s, and k in (46) with ai , mi , si , and ki—the dividend
 
1 yield, expected excess capital gain, standard deviation,
for g t 2 0, , ð53Þ
1k and transaction cost of stock i, respectively. We also note
that U i ð1Þ ¼ 0 in equilibrium. It is then straightforward to
where g1 is defined in (15) and where b is determined obtain the results in Proposition 3. &
below.

For g t 2 1=ð1kÞ,g n , (52) becomes Proof of Proposition 4. Define xt  ln g t and xn  ln g n .
1 2 2 00
s g t U ðg t Þ þ mg t U 0 ðg t Þðr þ d0 ÞUðg t Þ þ ðab þ rð1kÞÞg t r ¼ 0: Then,
2

The solution to this equation is

s2
dxt ¼ mx dt þ s dZ t , mx ¼ m :
2
g g ab þ rð1kÞ r
Uðg t Þ ¼ c1 g t 1 þ c2 g t 2 þ g
r þ d0 m t r þ d0
  If the investor has not yet traded, what is the prob-
1 ability that he trades at least once in the following s
for g t 2 ,g n ,
1k periods? Note that he will trade if the stock price rises
where sufficiently high so that the process xt hits the barrier xn ;
2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 or if there is a liquidity shock. The probability is therefore
   
1 4 1 2 2
0 2 1 2 5 a function of xt and of the length of the period s. We
g2 ¼  2 m s þ 2 r þ d s þ m s o0
s 2 2 denote it by pðx,sÞ.
Since a probability process is a martingale, its drift is
and where c1 and c2 are determined below. zero, so that
The value function must be continuous and continu-
ps þ mx px þ 12s2 pxx þ rð1pÞ ¼ 0:
ously differentiable at g t ¼ 1=ð1kÞ. We therefore have
      The last term on the left-hand side is generated by the
1 g1 1 g1 1 g2 ðl1Þmr
b ¼ c1 þc2  0 0 , liquidity shock: if a liquidity shock arrives, the probability
1k 1k 1k ðr þ d mÞðr þ d Þ
of a trade jumps from p to 1. The probability function
which is (24), and must also satisfy two boundary conditions. First, if the
  process xt is already at the barrier xn , there is a trade for
1 g1 1
bg1 sure:
1k
    pðxn ,sÞ ¼ 1, 8s Z 0:
1 g1 1 1 g2 1 ðl1Þð1kÞr
¼ c 1 g1 þ c2 g2  :
1k 1k r þ d0 m Second, if the length of the remaining time period is zero
and the price level is such that x o xn , there will be no
Together, these equations imply Eq. (22). trade:
In the liquidation region, g t 2 ðg n ,1Þ, using the fact that
g n Z1, we have pðx,0Þ ¼ 0, 8x oxn :

Uðg t Þ ¼ ð1kÞg t ð1 þ Uð1ÞÞ1:

The solution to the differential equation, subject to the
The value function must be continuous and continuously boundary conditions, is
differentiable at the liquidation point, so that
pðx,sÞ ¼ 1ers
g1 ab þ rð1kÞ
g2 d0     
c 1 g n þ c2 g n þ 0 g n ¼ ð1kÞg n ð1þ Uð1ÞÞ xxn þ mx s xxn mx s
r þ d m r þ d0 þers N pffiffi
2
þ eð2mx =s Þðxxn Þ N pffiffi :
s s s s
g 1 g 1 ab þ rð1kÞ Substituting x ¼ 0, xn ¼ ln g n , and mx ¼ ms2 =2 into this
c1 g1 g n1 þ c2 g2 g n2 þ ¼ ð1kÞð1 þ Uð1ÞÞ:
r þ d0 m expression, we obtain the result in Proposition 4. &
 N. Barberis, W. Xiong / Journal of Financial Economics 104 (2012) 251–271 271

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