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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 small

losses) than to gains of the same magnitude

loss aversion

inferred from aversion to (110, 12;100, 12) v() is concave over gains, convex over losses

inferred from (500, 1) (1000, 12) and (500, 1) (1000, 1

2)

5

Prospect Theory, ctd.

transform probabilities with a weighting function ()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:

(xm, pm; . . . ; x1, p1; x0, p0; x1, p1; . . . ; xn, pn),

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

ni=m

iv(xi)

i =

(pi + . . . + pn) (pi+1 + . . . + pn)(pm + . . . + pi) (pm + . . . + pi1) for

0 i nm i V (W2) under multivariate Normality, the investors utility is

F (W, 2W ), which, for fixed

2W, is increasing in W

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

The cross-section, ctd.

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

The cross-section, ctd.

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

RM N(M, 2M) 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: = 0.88, = 2.25, =

0.65

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

the skewed security payoff L = 10, q = 0.09

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

17

The cross-section, ctd.

equilibrium condition V (RM) = 0 M = 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 0.1610

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 cross-section, ctd.

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 cant 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 0.250.2

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 0.1420

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 securitys return distribution, q, which determines the securitysskewness. A low value of q corresponds to a high degree of skewness.

58

The cross-section, ctd.

Empirical evidence and applications

several papers test the models 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

The cross-section, ctd.

Empirical evidence and applications, ctd.

low average return on IPOs Green and Hwang (2011) show that IPOs predicted

to be more positively skewed have lower long-termreturns

overpricing of out-of-the-money options Boyer and Vorkink (2011) find that stock options

predicted 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

The cross-section, ctd.

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 more

directly under stock-level narrow framing

alternative reference points? not clear that the reference point matters very

much here

22

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

23

The aggregate stock market

can pr