Prospect Theory Applications in Finance Nicholas Barberis Yale

Prospect Theory Applications in Finance

Nicholas BarberisYale University

September 2011

1

Overview

• in behavioral finance, we investigate whether certainfinancial phenomena are the result of less than fullyrational thinking

Preferences

• prospect theory

• ambiguity aversion

Beliefs

• representativeness, law of small numbers

• non-belief in the law of large numbers

• conservatism, belief perseverance, confirmation bias

• overconfidence

• in this note, we look at prospect theory applicationsin finance

2

Overview, ctd.

• almost all models of financial markets assume thatinvestors evaluate risk according to expected utility

– but this framework has had trouble matching manyempirical facts

• can we make progress by replacing expected utilitywith a psychologically more realistic preference spec-ification?

– e.g. with prospect theory

3

Prospect Theory, ctd.

Four key features:

• the carriers of value are gains and losses, not finalwealth levels

– compare v(x) vs. U (W + x)

• v(·) has a kink at the origin

– captures a greater sensitivity to losses (even smalllosses) than to gains of the same magnitude

– “loss aversion”

– inferred from aversion to (110, 12;−100, 1

2)

• v(·) is concave over gains, convex over losses

– inferred from (500, 1) � (1000, 12) and (−500, 1) ≺

(−1000, 12)

5

Prospect Theory, ctd.

• transform probabilities with a weighting function π(·)that overweights low probabilities

– inferred from our simultaneous liking of lotteriesand insurance, e.g. (5, 1) ≺ (5000, 0.001) and(−5, 1) � (−5000, 0.001)

Note:

• transformed probabilities should not be thought of asbeliefs, but as decision weights

6

Cumulative Prospect Theory

• proposed by Tversky and Kahneman (1992)

• applies the probability weighting function to the cu-mulative distribution function:

(x−m, p−m; . . . ; x−1, p−1; x0, p0; x1, p1; . . . ; xn, pn),

where xi < xj for i < j and x0 = 0, is assigned the value

n∑i=−m

πiv(xi)

πi =

⎧⎪⎪⎨⎪⎪⎩

π(pi + . . . + pn) − π(pi+1 + . . . + pn)π(p−m + . . . + pi) − π(p−m + . . . + pi−1)

for0 ≤ i ≤ n−m ≤ i <

• the agent now overweights the tails of a probabilitydistribution

– this preserves a preference for lottery-like gambles

7

Cumulative Prospect Theory, ctd.

• Tversky and Kahneman (1992) also suggest functionalforms for v(·) and π(·) and calibrate them to experi-mental evidence:

v(x) =

⎧⎪⎪⎨⎪⎪⎩

xα

−λ(−x)αfor

x ≥ 0x < 0

π(P ) =Pδ

(Pδ + (1 − P )δ)1/δ

with

α = 0.88, λ = 2.25, δ = 0.65

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P

w(P

)

Figure 2. The figure shows the form of the probability weighting function proposed byTversky and Kahneman (1992), namely w(P ) = P δ/(P δ +(1−P )δ)1/δ. The dashed linecorresponds to δ = 0.65, the dash-dot line to δ = 0.4, and the solid line to δ = 1.

24

Narrow framing

• in traditional models, an agent evaluates a new gam-ble by merging it with his pre-existing risks and check-ing if the combination is attractive

• but in experimental settings, people often seem toevaluate a new gamble in isolation (Tversky andKahneman, 1981)

– this is “narrow framing”

– get utility directly from the outcome of the gamble,not just indirectly from its contribution to totalwealth

• e.g. the rejection of a 50:50 bet to win $110 or lose$100 is probably evidence not only of loss aversion,but also of narrow framing

– Barberis, Huang, Thaler (2006); Koszegi and Ra-bin (2007)

9

Narrow framing, ctd.

• in what follows, we sometimes take prospect theory’s“gains” and “losses” to be gains and losses in specificcomponents of wealth

– e.g. gains and losses in stock market wealth, or ina specific stock

• we lack a full theory of narrow framing, but possibleelements are:

– anchoring on the first thing you think of

– accessibility (Kahneman, 2003)

– regret

• we interpret utility from gains and losses in (compo-nents of) wealth as utility from news about futureconsumption

– but narrow framing strains this interpretation slightly

10

Prospect theory applications

[1]

• the cross-section of stock returns

– one-period models

– new prediction: the pricing of skewness

– probability weighting plays the most critical role

[2]

• the aggregate stock market

– intertemporal representative agent models

– try to address the equity premium, volatility, pre-dictability, and non-participation puzzles

– loss aversion plays a key role; but probability weight-ing also matters

[3]

• trading behavior

– multi-period models

– try to address the disposition effect and other trad-ing phenomena

– all aspects of prospect theory play a role

11

Prospect theory applications, ctd.

Note:

• in all applications, we have to decide on the degree ofnarrow framing

– not easy, given the lack of a full theory of framing

• also need to decide on the reference point relative towhich gains and losses are computed

– typically take the reference point to be initial wealthscaled up by the risk-free rate

• the risk-free rate is, in some ways, a more plausiblereference point than “expectations”

– it is much more salient; and is a natural defaultreturn

• preliminary quantitative calculations (see later) alsocast some doubt on “expectations” as the unique ref-erence point

• a conjecture is that the reference point lies somewherebetween the risk-free rate and the expected return

12

Themes

• prospect theory is helpful for thinking about financialphenomena

– particularly a model that applies prospect theoryto gains and losses in financial wealth

• for finance applications, probability weighting maybe the most important and useful element of prospecttheory

• we should investigate reference points other than therisk-free rate

– and need a better theory of narrow framing

• we also need a better understanding of “dynamics”

– e.g. of how past gains and losses affect risk atti-tudes

13

The cross-section

Barberis and Huang (2008)

• single period model; a risk-free asset and J risky assetswith multivariate Normal payoffs

• agents have identical expectations about security pay-offs

• agents have identical CPT preferences

– defined over gains/losses in wealth (i.e. no narrowframing)

– reference point is initial wealth scaled up by therisk-free rate, so utility defined over W = W1 −W0Rf

– full specification is:

V (W ) =∫ 0−∞ v(W ) dπ(P (W ))−∫ ∞

0 v(W ) dπ(1−P (W ))

(continuous distribution version of Tversky andKahneman, 1992)

14

The cross-section, ctd.

In this economy, the CAPM holds!

• CPT preferences satisfy first-order stochastic domi-nance (FOSD):

W1 FOSD W2 ⇒ V (W1) > V (W2)

• under multivariate Normality, the investor’s utility isF (μW, σ2

W ), which, for fixed σ2W, is increasing in μW

⇒ investors choose portfolios on the mean-varianceefficient frontier

• market clearing ⇒ tangency portfolio is the marketportfolio ⇒ CAPM

• first proved by De Giorgi, Hens, and Levy (2003)

15


• now introduce a small, independent, positively skewedsecurity into the economy, return Rn

• in a representative agent economy with concave EUpreferences, the security would earn an average excessreturn of zero

• we find that, in an economy with CPT investors, thesecurity can earn a negative average excess return

– skewness itself is priced, in contrast to EU models,where only coskewness matters

• equilibrium involves heterogeneous holdings

(for now, assume short-sale constraints)

– some investors hold the old market portfolio and alarge, undiversified position in the new security

– others hold the old market portfolio and no posi-tion at all in the new security

– heterogeneous holdings arise from non-unique globaloptima, not from heterogeneous preferences

16


• from before, the market return, excluding the newsecurity, is Normally distributed:

RM ∼ N(μM, σ2M)

• give the new security a “lottery-like” binomial distri-bution:

payoff ∼ (L, q; 0, 1 − q)

Rn ∼ (L

pn, q; 0, 1 − q)

– think of L as large, q as small

• set values for

– preference parameters: α = 0.88, λ = 2.25, δ =0.65

– the risk-free rate, Rf = 1.02, and the market stan-dard deviation, σM = 0.15

– the skewed security payoff L = 10, q = 0.09

• search for a market risk premium μM and a price of theskewed security pn for which there is a heterogeneousholdings equilibrium

17


• equilibrium condition V (RM) = 0 ⇒ μM = 0.075

– i.e. a high equity premium

• equilibrium condition sup x>0V (RM + xRn) = 0 ⇒pn = 0.925, so that the skewed security earns a neg-ative excess return

E(Rn) − Rf =(0.09)(10)

0.925− 1.02

= −0.047

Intuition:

• since it contributes skewness to the portfolios of someinvestors, it is valuable, and so earns a low averagereturn

• not surprising that a CPT investor likes a skewedportfolio

– more surprising that he likes a skewed security,even if it is small

18

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16−10

−8

−6

−4

−2

0

2

4

6

8x 10

−3

x

utili

ty

FIGURE 3. A HETEROGENEOUS HOLDINGS EQUILIBRIUM. Notes: The figureshows the utility that an investor with cumulative prospect theory preferences derives fromadding a position in a positively skewed security to his current holdings of a Normallydistributed market portfolio. The skewed security is highly skewed. The variable x isthe fraction of wealth allocated to the skewed security relative to the fraction of wealthallocated to the market portfolio. The two lines correspond to different mean returns onthe skewed security.

56


• the skewed security only earns a negative excess returnif it is highly skewed

– e.g. for q = 0.15, there is no heterogeneous hold-ings equilibrium

– security can’t contribute enough skewness to over-come lack of diversification

• for moderately skewed securities, there is a homoge-neous agent equilibrium

– here, the expected excess return is zero

• the result that a (highly) positively skewed securityearns a negative average excess return holds:

– even if there are many skewed securities

– even if short sales are allowed

– qualitatively, even if expected utility agents arepresent

19

0 0.05 0.1 0.15 0.2 0.25−0.2

−0.1

0

0.1

x

utili

ty

FIGURE 4. A HOMOGENEOUS HOLDINGS EQUILIBRIUM. Notes: The figureshows the utility that an investor with cumulative prospect theory preferences derivesfrom adding a position in a positively skewed security to his current holdings of a Nor-mally distributed market portfolio. The skewed security is only moderately skewed. Thevariable x is the fraction of wealth allocated to the skewed security relative to the fractionof wealth allocated to the market portfolio. The three lines correspond to different meanreturns on the skewed security.

57

0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14−20

−15

−10

−5

0

5

10

15

20

q

expe

cted

exc

ess

retu

rn (

perc

ent)

FIGURE 5. SKEWNESS AND EXPECTED RETURN. Notes: The figure shows theexpected return in excess of the risk-free rate earned by a small, independent, positivelyskewed security in an economy populated by cumulative prospect theory investors, plottedagainst a parameter of the security’s return distribution, q, which determines the security’sskewness. A low value of q corresponds to a high degree of skewness.

58


Empirical evidence and applications

• several papers test the model’s basic prediction thatskewness is priced in the cross-section

– Zhang (2006) uses returns on stocks similar to stockX to predict the skewness of stock X

– Boyer, Mitton, Vorkink (2010) use a regressionmodel to predict future skewness

– Conrad, Dittmar, Ghysels (2010) use option pricesto infer the perceived (risk-neutral) distribution ofthe underlying stock

• all three studies find supportive evidence

20


Empirical evidence and applications, ctd.

• low average return on IPOs

– Green and Hwang (2011) show that IPOs predictedto be more positively skewed have lower long-termreturns

• “overpricing” of out-of-the-money options

– Boyer and Vorkink (2011) find that stock optionspredicted to be more positively skewed have lowerreturns

• low average return on stocks with high idiosyncraticvolatility (Ang et al., 2006; Boyer, Mitton, Vorkink,2010)

• low average return on OTC stocks (Eraker and Ready,2011)

• diversification discount (Mitton and Vorkink, 2008)

• under-diversification

– Mitton and Vorkink (2010) find that undiversifiedindividuals hold stocks that are more positivelyskewed than the average stock

21


Remarks:

• an example of how psychology can lead us to usefulnew predictions

• the model manages to capture both the high and lowrisk premia we observe

– skewness, or the lack of it, is key

• alternative framing assumptions?

– the pricing of skewness should follow even moredirectly under stock-level narrow framing

• alternative reference points?

– not clear that the reference point matters verymuch here

22


[1]





[2]





[3]





23

The aggregate stock market

• can prospect theory help us understand the propertiesof, and attitudes to, the aggregate stock market?

– e.g. equity premium, volatility, predictability, andnon-participation puzzles

• Benartzi and Thaler (1995) note that a model in whichinvestors are loss averse over annual changes in theirfinancial wealth predicts a large equity premium

• three elements:

– loss aversion

– annual evaluation

– narrow framing

• Benartzi and Thaler (1995) emphasize the first twoelements

– “myopic loss aversion”

• the idea seems to be gaining acceptance

24

The aggregate stock market, ctd.

Subsequent developments:

• formalizing the argument

• emphasizing the role of narrow framing

• studying the role of probability weighting

• trying to address the volatility puzzle as well

25


Formalizing the argument

• to fill out the argument, we need to embed it in thesetting where the equity premium is usually studied

– an intertemporal, representative agent model whereconsumption plays a non-trivial role

– e.g. where preferences include a utility of consump-tion term alongside the prospect theory term

• two ways of doing this:

– Barberis, Huang, and Santos (2001)

– Barberis and Huang (2009)

• Barberis and Huang (2008) reviews both methods

26


Formalizing the argument, ctd.

Method I: Barberis, Huang, and Santos (2001)

• intertemporal model; three assets: risk-free (Rf,t),stock market (RS,t+1), non-financial asset (RN,t+1)

• representative agent maximizes:

E0

∞∑t=0

⎡⎢⎢⎢⎣ρ

t C1−γt

1 − γ+ b0ρ

t+1C−γt v(GS,t+1)

⎤⎥⎥⎥⎦

GS,t+1 = θS,t(Wt − Ct)(RS,t+1 − Rf,t)

v(x) =

⎧⎪⎪⎨⎪⎪⎩

xλx

forx ≥ 0x < 0

, λ > 1

– this assumes narrow framing of the stock market

– and that the reference point is the risk-free rate

– v(·) captures loss aversion

– we ignore concavity/convexity and probability weight-ing for now

• for “reasonable” parameters, get a substantial equitypremium, although not as large as in Benartzi andThaler (1995)

27



Method II: Barberis and Huang (2009)

• start from the standard recursive utility specification

Vt = H(Ct, μ(Vt+1))

W (C, x) = ((1 − β)Cρ + βxρ)1ρ, 0 < β < 1, 0 = ρ < 1

μ(x) = (E(xζ))1ζ

• can adjust this to incorporate narrow framing

Vt = H⎛⎜⎝Ct, μ(Vt+1) + bi,0

∑iEt(v(Gi,t+1))

⎞⎟⎠

• in the three asset context from before:

Vt = H (Ct, μ(Vt+1) + b0Et(v(GS,t+1)))

GS,t+1 = θS,t(Wt − Ct)(RS,t+1 − Rf,t)

v(x) =

⎧⎪⎪⎨⎪⎪⎩

xλx

forx ≥ 0x < 0

, λ > 1

ζ = ρ

28



• this specification is better than Method I

– it is tractable in partial equilibrium

– it admits an explicit value function⇒ easy to checkattitudes to monetary gambles

– it does not require aggregate consumption scalingC

• can now show that for parameter values that predictreasonable attitudes to large and small-scale monetarygambles, get substantial equity premium

29


The role of narrow framing

• Barberis, Huang, Santos (2001) find that narrow fram-ing is a critical ingredient

– loss aversion over total wealth fluctuations doesn’tproduce a significant equity premium

– the “loss aversion / narrow framing” approach?

– how do we justify the narrow framing?

The role of probability weighting

• De Giorgi and Legg (2010) bring probability weightingand concavity/convexity into the Barberis and Huang(2009) framework

– they show that probability weighting can signifi-cantly increase the equity premium

– because the aggregate market is negatively skewed

30


The volatility puzzle

• Barberis, Huang, and Santos (2001) also build in dy-namic aspects of loss aversion

– based on evidence in Thaler and Johnson (1990),assume that loss aversion decreases (increases) af-ter past gains (losses)

– can be interpreted in terms of “capacity for dealingwith good or bad news”

– generates excess volatility in addition to a high eq-uity premium

• more work is needed on how past gains and lossesaffect risk attitudes

– amplification seems very important in financial mar-kets, but the mechanisms are unclear

31



– broad, wealth-level framing does not produce a sig-nificant equity premium

– very narrow stock-level framing produces a largerequity premium than stock market-level framing(Barberis and Huang, 2001)

– stock market-level framing seems natural here

32


• alternative reference point assumptions?

– both “Method I” and “Method II” allow for refer-ence points other than the risk-free rate

e.g. Method I

E0

∞∑t=0

⎡⎢⎢⎢⎣ρ

t C1−γt

1 − γ+ b0ρ

t+1C−γt v(GS,t+1)

⎤⎥⎥⎥⎦

GS,t+1 = θS,t(Wt − Ct)(RS,t+1 − Rz)

v(x) =

⎧⎪⎪⎨⎪⎪⎩

xλx

forx ≥ 0x < 0

, λ > 1

• we can compute the equilibrium equity premium forRz = Rf and Rz = E(RS)

• in preliminary calculations, we find that, when thereference point is the expected stock return

– the equilibrium equity premium is typically toohigh (intuition?)

– and is extremely sensitive to the value of b0

– in particular, the model only matches the historicalpremium for a very narrow range of b0

33


[1]





[2]





[3]





35

Trading behavior

• can prospect theory help us understand how peopletrade stocks over time?

• a particular target of interest is the “disposition effect”

– individual investors’ greater propensity to sell stockstrading at a gain relative to purchase price, ratherthan at a loss

• at first sight, prospect theory, in combination withstock-level narrow framing, appears to be a promisingapproach

• but it turns out that we need to be careful how weimplement prospect theory

– prospect theory defined over annual stock-level trad-ing profits does not generate a disposition effectvery reliably

– Barberis and Xiong (2009), “What Drives the Dis-position Effect?...”

36

Trading behavior, ctd.

• consider a simple portfolio choice setting

– T + 1 dates: t = 0, 1, . . . , T

– a risk-free asset, gross return Rf each period

– a risky asset with an i.i.d binomial distributionacross periods:

Rt,t+1 =

⎧⎪⎪⎨⎪⎪⎩

Ru > Rf with probability 12

Rd < Rf with probability 12

, i.i.d.

• the investor has prospect theory preferences definedover his “gain/loss”

– simplest definition of gain/loss is trading profit be-tween 0 and T, i.e. WT − W0

– we use WT − W0RTf

– i.e. again, reference point is initial wealth scaledup by the risk-free rate

37


The investor therefore solves

maxx0,x1,...,xT−1

E[v(ΔWT )] = E[v(WT − W0RTf )]

where

v(x) =

⎧⎪⎪⎨⎪⎪⎩

xα

−λ(−x)αfor

x ≥ 0x < 0

,

subject to

Wt = (Wt−1 − xt−1Pt−1)Rf + xt−1Pt−1Rt−1,t

WT ≥ 0

• we are assuming stock-level narrow framing

– and are ignoring probability weighting

• we can now derive an analytical solution for any num-ber of trading periods

38


Results

• the investor usually exhibits the opposite of the dis-position effect

– only when T is high and the expected stock returnis low does he exhibit a disposition effect

• for T = 2 and for the Tversky and Kahneman (1992)parameterization, he always exhibits the opposite ofthe disposition effect

39


Why does the disposition effect always fail for the TKparameters in the two-period case? (t = 0, 1, 2)

• for the investor to buy the stock at time 0, in spite ofhis loss aversion, it must have a high expected return

– this implies that the time 1 gain is larger than thetime 1 loss in magnitude

– it also implies that, after a time 1 gain, the investorgambles to the edge of the concave region (v(·) isonly mildly concave over gains)

– after a time 1 loss, the investor gambles to the edgeof the convex region

• but it takes a larger position to gamble to the edge ofthe concave region after the time 1 gain than it doesto gamble to the edge of the convex region after thetime 1 loss

⇒ the investor takes more risk after a gain than aftera loss, contrary to the disposition effect

40

−100 −80 −60 −40 −20 0 20 40 60 80 100 120−100

−80

−60

−40

−20

0

20

40

60

80

100

gain/loss

utils

Time 1 and time 2 gains/losses plotted on the value function

D

C

D’

B

A

B’

45



– stock-level framing seems reasonable

– the predictions of broader framing have not beenworked out


– a very promising direction

– e.g. Meng (2011) suggests that a reference pointhigher than the risk-free rate may make it easierto predict a disposition effect

41


Other approaches to explaining the disposition effect?

• one idea is to apply prospect theory to realized gainsand losses (Shefrin and Statman, 1985)

– i.e. to assume “realization utility”

• e.g. if you buy a stock at $40 and sell it at $60

– you get a burst of positive utility at the momentof sale, based on the size of the realized gain

• prospect theory applied to realized gains and lossesdoes predict a disposition effect more reliably (Bar-beris and Xiong, 2009)

• what is the source of realization utility?

– people often think about their investing history asa series of investing episodes

– and they think of selling a stock at a gain (loss) asa “good” (“bad”) episode

⇒ when an investor sells an asset at a gain, hefeels a burst of pleasure because he is creating apositive new investing episode

42


• Barberis and Xiong (2010), “Realization Utility,” studylinear realization utility, coupled with a positive timediscount factor

– the investor derives utility from the sale price ofan asset minus the purchase price scaled up by therisk-free rate

• look at both portfolio choice and asset pricing

• if the expected stock return is high enough, the in-vestor’s optimal strategy is to buy a stock at time0

– and to sell it only if its value rises a certain percent-age amount above the (scaled up) purchase price

– he then immediately invests the proceeds in an-other stock, and so on

• for some parameter values, the investor is risk-seeking

43


• applications:

– the disposition effect

but also:

– “excessive trading”

– the underperformance of individual investors evenbefore transaction costs

– the greater turnover in bull markets

– the greater selling propensity above historical highs

– the individual investor preference for volatile stocks

– the negative premium to volatility in the cross-section

– the fact that overpriced assets are also heavily traded

– momentum

44



– realization utility fits most naturally with asset-level framing


– can be studied in our framework

Note:

• neural data offer some support for realization utility

– Frydman, Barberis, Camerer, Bossaerts, Rangel(2011)

45


Summary

• a model in which the investor derives prospect theoryutility from annual trading profits does not deliver adisposition effect very reliably

• a model in which the investor derives prospect the-ory utility from realized gains and losses delivers adisposition effect more reliably

– but the disposition effect follows even from linearrealization utility, coupled with a positive time dis-count factor

Note:

• the annual trading profit model may not be dead(Meng, 2011)

• the trading models we have seen ignore probabilityweighting

– in dynamic settings, probability weighting leads toa time inconsistency that may be important insome contexts

– e.g. in casinos (Barberis, 2011)

46

Summary





– no narrow framing needed



– try to address the equity premium, volatility puz-zles


– typically assume stock market-level narrow fram-ing





– typically assume stock-level narrow framing

47

Themes

• prospect theory seems to be quite helpful for thinkingabout financial phenomena

– particularly a model that applies prospect theoryto gains and losses in financial wealth

• for finance applications, probability weighting maybe the most important and useful element of prospecttheory

• we need to investigate reference points other than therisk-free rate

– and to think harder about narrow framing

• finally, we need a better understanding of dynamics

– how past gains and losses affect risk attitudes

48

Documents

Prospect Theory Applications in Finance Nicholas Barberis Yale