Can Probability Weighting Help Prospect Theory Explain the - - PowerPoint PPT Presentation

Can Probability Weighting Help Prospect Theory Explain the Disposition Effect?

Vicky Henderson University of Warwick and Oxford-Man Institute & David Hobson (Warwick) & Alex S.L. Tse (Warwick → Cambridge)

9th Bachelier World Congress, New York, July 15-19, 2016

Overview

Psychologists have uncovered a wealth of behavioral biases in the way we make decisions under uncertainty → behavioral economics

◮ Thinking Fast and Slow, D Kahneman (Nobel Prize 2002) ˆ

• ◮ US and UK governments have Behavioral Insights

Teams/Units to provide policy recommendations Prospect theory is most well known and well supported by experimental evidence of many non-EU theories. (Tversky and Kahneman (1979, 1992))

Experimental & empirical evidence suggests.....

Tend to prefer a certain $500 to a 50% chance of $1000 risk averse over gains But prefer a 50% chance of losing $1000 to a certain loss of $500 risk seeking over losses Averse to gambles such as ($110, 50%; −$100, 50% ) loss averse Tend to prefer a

1 1000 chance of $5000 to a certain $5

But prefer a certain loss of $5 to a

1 1000 chance of losing $5000

• verweight tail events

Delay realization of losses (relative to gains) - disposition effect

Our Aims

We study dynamic stopping models with investors who have prospect theory (PT) preferences. Goal is to show inclusion of probability weighting component of PT can:

◮ Generate more realistic trading behavior than existing PT

models

◮ Assist PT in matching the magnitude of the disposition effect

The Disposition Effect

◮ The disposition effect is the stylized fact that investors have a

higher propensity to sell a winner (gain relative to purchase price) than a loser (lost relative to purchase price). (Shefrin and Statman (1985)).

◮ Documented for individual investors by Odean (1998),

institutional investors, in real estate and options markets. Experimental evidence (Weber and Camerer (1998) and Magnani (2015)).

◮ Shefrin and Statman (1985) propose the disposition effect can

be explained by Prospect Theory

Proportion of days with sales producing a given magnitude of return. Using large US retail brokerage dataset. (Thanks to Matthew Burgess (Warwick))

Prospect Theory (Tversky & Kahneman (1979, 1992))

◮ U a continuous, increasing S-shaped value function defined

• ver gains and losses relative to a reference level R with

U(0) = 0.

◮ utility from gains and losses rather than final wealth ◮ risk averse (concave) over gains and risk seeking (convex) over

losses

◮ steeper for losses than for gains to capture loss aversion,

U(−x) + U(x) < 0 or U′(0+) < U′(0−)

◮ w± : [0, 1] → [0, 1] a pair of increasing functions with

w±(0) = 0 and w±(1) = 1. Typically w± concave on [0, q±] and convex on [q±, 1] for some q±.

◮ probability weighting function applied to CDF to overweight

tails of distribution

Tversky and Kahneman (1992) suggest functional forms and calibrate to experimental data:

◮ Value function:

U(y) = yα+ y ≥ 0 −k(−y)α− y < 0 where α± ∈ (0, 1). k > 1 governs loss aversion. TK estimate α+ = α− = 0.88 and k = 2.25. Wu and Gonzales (1996) estimate α+ = 0.5.

◮ Probability weighting function:

w±(p) = pδ± (pδ± + (1 − p)δ±)1/δ± for 0.28 < δ± ≤ 1. TK estimated δ+ = 0.61, δ− = 0.69. Wu and Gonzales (1996) estimate δ+ = 0.71.

−3 −2 −1 1 2 3 −3 −2 −1 1 2 3 x v(x)

(a) TK (1992) value function with α± = 0.5, k = 1.5

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w(p) p w+ w−

(b) TK (1992) weighting functions with δ = 1, δ+ = 0.61 and δ− = 0.69

Prospect Theory

Prospect theory value of cont. random variable Y (reference level centred) is given by (Kothiyal, Spinu and Wakker (2011)) EU(Y ) =

• R+

w+(P(U(Y ) > y))dy −

• R−

w−(P(U(Y ) < y))dy If w±(p) = p then EU(Y ) = EU(Y ). Denoting by G(.) the quantile function of Y , a change of variable & IBP gives EU(Y ) = 1

1−P(Y >0)

U(G(q))w′

+(1−q)dq+

P(Y <0) U(G(q))w′

−(q)dq

Formulation of the Investor’s Asset Sale Problem

Investor can choose to sell asset with price Pt at time t at any time in the future. Suppose asset price P = (Pt)t≥0 is a (regular, time homogeneous) diffusion on (0, ∞) with initial value P0. Goal of the investor is to choose the best time τ to sell the asset to maximize: sup

τ EU(Pτ − R)

where U is value function and R reference level. Note if w±(p) = p, studied by Henderson (2012) (see also Kyle, Ou-Yang and Xiong (2006)).

Formulation of the Investor’s Asset Sale Problem

Let s be scale function of P with s(R) = 0 such that Xt = s(Pt) is a (local) martingale. Then Xt is transformed gains/losses relative to reference and EU(Pτ − R) = Ev(Xτ) where v(x) := U

• s−1(x) − R
• .

Suppose L := s(0) > −∞. Let M = s(∞) ≤ ∞. We can take Xτ rather than τ as the decision variable giving: sup

ν∈A

Ev(X) where A = {ν :

• yν(dy) = X0 = s(P0)}

Skorokhod embedding tells us for every law ν with support on [s(0), s(∞)] = [L, M] and such that

• yν(dy) = X0 there is a

stopping rule τ such that Xτ ∼ ν.

Some Related Work

◮ Xu and Zhou (2013) study a similar problem just for gains or

just for losses.

◮ We will study the combined gains/losses problem and give

additional conditions under which a single point mass on losses is optimal.

◮ Probability weighting introduces time inconsistency (Barberis

(2012)).

◮ We focus here on the behavior of a sophisticated investor who

can precommit.

◮ Ebert and Strack (2015) study behavior of naive PT agents -

with pure strategies they never stop.

◮ Henderson, Hobson and Tse (2015) show this is no longer the

• nly behavior when naive agents can follow randomized
• strategies. (Saturday 2:50pm)

Two Assumptions

w± are inverse-S shaped: concave on [0, q±] & convex on [q±, 1].

Assumption (S-shaped on v)

v is concave on [0, M) and convex on [L, 0] and v′(0+) = ∞ Define E(x; f , c) = (x − c)f ′(x) f (x) − f (c) = f ′(x)

f (x)−f (c) x−c

as the elasticity of a function f relative to a reference point c.

Assumption (Elasticity)

• a. E(x; v, L) is increasing in x for x ∈ [L, 0]
• b. E(p; w−, r) is decreasing in p for 0 ≤ p ≤ min{r, q−} for any r

in [0, 1]. Elasticity Assumption b. holds for all the popular weighting functions - TK (1992), Prelec (1998), Goldstein and Einhorn (1987)

Solving for the Optimal quantile function

◮ Adopt a sequential optimization approach (Xu and Zhou

(2013), Jin and Zhou (2008)) by decomposition of X = X + − X −

◮ Optimal X +:

◮ solving a concave maximization problem via a Lagrangian

approach

◮ Optimal X −:

◮ solving a convex maximization problem where solution is a

3-point distribution with atoms at L, 0 and some I ∈ [L, 0] in general

◮ Under assumptions on v ′(0+) and elasticity of v and w−, the

• ptimal rv only puts one atom on losses.

◮ Combine by finding optimal allocation of probability mass over

gains and losses subject to mean constraint

Theorem

Under the Assumptions, optimal prospect has a distribution consisting of:

◮ a point mass in the loss regime and ◮ a point mass at some level a in the gains regime, together with

a continuous distribution on the unbounded interval (a, ∞). The optimizer has quantile function GX of the form

GX(u) =          −

1 1−φ

φ

0 dy(v ′)−1 λ w ′

+(ψ∧y)

• − X0
• u ≤ 1 − φ

(v ′)−1

λ w ′

+(ψ)

• 1 − φ < u ≤ 1 − ψ

(v ′)−1

λ w ′

+(1−u)

• 1 − ψ < u ≤ 1

(1) for some λ > 0, φ ∈ [0, 1], ψ ≤ q+ ∧ φ such that X +

0 ≤

1 1 − φ φ dy(v ′)−1

• λ

w ′

+(ψ ∧ y)

• ≤ X0 − L.

Base Model - TK value & weighting functions

◮ Pt follows a GBM

dPt = Pt(ρdt + σdBt) (2)

◮ Define β := 1 − 2ρ

σ2 . For ρ < σ2/2 or β > 0, have

s(p) = pβ − Rβ and L > −∞.

◮ TK value and weighting functions with (α+, α−, k) and

(γ+, γ−).

◮ v(x) = U(s−1(x) − P0) is S-shaped if and only if β ≤ 1,

α+ < β.

◮ Assumptions on v ′(0+) and elasticity functions of v and w−

all satisfied.

Proposition

Under the above specification, the problem is well posed if α+/δ+ < β. We will work with α+/δ+ < β ≤ 1.

Optimal sale strategy involves:

◮ a fixed stop-loss threshold ◮ a random gain level supported on [(v′)−1 λ w′

+(ψ)

• , ∞)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Quantile function of Pτ ∗ = s−1(X ∗). Parameters are α+ = 0.5, α− = 0.9, δ+ = δ− = 0.7, k = 1.25, β = 0.9, R = 1 and P0 = 1.

Impact of Probability Weighting

δ+=δ- 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.5 1 1.5 2 2.5 3 3.5

Loss Threshold Gain Distribution (lower bound and 99th percentile)

Distributional properties of Pτ ∗ = s−1(X ∗). Vary δ± on y-axis. Parameters are α+ = 0.5, α− = 0.9, δ+ = δ− = 0.7, k = 1.25, β = 0.9, R = 1 = P0.

Impact of Probability Weighting

When δ± → 1 (no weighting):

◮ Recover two-sided threshold strategy of Henderson (2012). ◮ One sided threshold strategies have been by derived by Kyle,

Ou Yang and Xiong (2006), Barberis and Xiong (2012).

◮ Typically, gain threshold close to reference level and loss

threshold (if non-zero!) much further away. Why? Marginal utility decreases in size so small gains, large losses desirable.

◮ Investor’s return distribution tends to be left or negatively

skewed. When δ± = 1 (weighting):

◮ Investor puts some mass on extreme gain levels because these

are overweighted - long right tail.

◮ When pw is sufficiently strong (but not violating earlier

Proposition) there is a non-zero stop-loss threshold.

◮ Investor’s return distribution now tends to be right or

positively skewed.

Empirical and experimental evidence

◮ Fundamental difference in behavior regarding gains and losses

in our model mirrors what we see in markets & in the lab.

◮ Stop-loss strategies are in widespread use in financial markets

but not that easily justified by financial theory. Stop or take gain strategies are much less common.

◮ EU settings predict stop-gain level at which investor should sell

but tend to put any lower threshold at −∞.

◮ If models do predict a stop-loss, they tend to also predict a

stop-gain. Eg. PT models of Henderson (2012), Ingersoll and Jin (2013).

◮ Experimental evidence (Strack and Viefers (2015)) shows

players do not play cut-off or threshold strategies on gains.

The Disposition Effect - Odean’s measure

◮ Frequency with which they sell winners and losers relative to

their opportunities to sell each.

◮ Odean (1998) defines PGR (PLR) as the number of times a

gain (loss) is realized as a fraction of the total number of times a gain could have been realized.

◮ Odean (1998) finds PGR = 0.148 and PLR = 0.098 in his

dataset of 10000 investors at a large US discount brokerage

• firm. Disposition measure is 1.51, investors realize gains at a

50% higher rate than losses.

Linking PT to the disposition effect I

S shaped value function - Static intution of Shefrin and Statman (1985) -

◮ risk aversion on gains encourages stopping ◮ risk seeking on losses encourages waiting/gambling over losses

Rigorous models show it is a challenge - Kyle, Ou Yang and Xiong (2006), Barberis and Xiong (2009, 2012), Henderson (2012). Tend to obtain disposition that is much too extreme (or even infinite, if lower sale threshold at zero!). Ingersoll and Jin (2013) improve - by reinvestment - need alterations to the standard TK preferences, very extreme parameters or mixing with Poisson traders None use the probability weighting of PT!

Linking PT to the disposition effect II

Inverse S probability weighting -

◮ Intution tells us that -

◮ overweighting extreme gains encourages waiting ◮ overweighting extreme losses encourages stopping

This works against the disposition effect!

◮ We want to see if the combined effect of probability weighting

and S shape value function can bring us closer to reality

A Model-based Disposition Measure I

Henderson (2012) proposes a model-based measure of the rate of selling at gains (losses) to proxy Odean’s statistic. RG = P(Pτ > P0) E( τ

0 1(Pu>P0)du),

RL = P(Pτ < P0) E( τ

0 1(Pu<P0)du)

Then the disposition measure can be defined as: D = RG RL = P(Pτ > P0) E( τ

0 1(Pu>P0)du)

E( τ

0 1(Pu<P0)du)

P(Pτ < P0) . (3) We say the disposition effect occurs if D > 1.

A Model-based Disposition Measure II

Proposition

Suppose the scaled price process satisfies dXt = σ(Xt)dBt with X0 = 0. Let ν∗ be the probability law of the target scaled prospect, Xτ. Then D depends on the optimal prospect, but not on the stopping rule τ used to generate that prospect. Moreover D = ∞

0 ν∗(dx)

M

1 σ2(x) (uν∗(x) − x) dx L 1 σ2(x) (uν∗(x) + x) dx L ν∗(dx)

(4) where for any measure ν, uν(x) = EX∼ν[|X − x|] is the potential

• f the law ν.

Matching Odean’s Disposition Effect

δ+=δ- 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.2 0.4 0.6 0.8 1 1.2

(a) δ+ = δ− Plot of log10D against δ+ = δ−. Parameters used are α+ = 0.5, α− = 0.9, β = 0.9, k = 1.25, R = 1, P0 = 1. The horizontal dashed lines mark Odean’s disposition estimate of log10 1.51 ≈ 0.18.

Model based Sales Intensity

◮ We also derive a level-dependent selling rate per unit time

consistent with the investor achieving the optimal distribution.

◮ Generalize to many heterogeneous investors with different

preference parameters.

◮ Matches qualitative features of empirical data - disposition

• effect. Quite close on magnitudes including daily probabilities
• f sale in the data.

Concluding Remarks

◮ We fully characterize the optimal prospect for a PT investor

under some mild elasticity conditions on v and w− and show it consists of a single loss threshold, and a point mass on gains at a together with a continuous distribution on (a, ∞).

◮ PT can indeed predict realistic levels of the disposition effect

• nce we include probability weighting.

Thank you!

Barberis N., 2012, “A Model of Casino Gambling”, Management Science, 58, 35-51. Barberis N. and W. Xiong, 2009, “What Drives the Disposition Effect? An Analysis of a Long-standing Preference-based Explanation”, Journal

• f Finance, 64, 751-784.

Barberis N. and W. Xiong, 2012, “Realization Utility”, Journal of Financial Economics, 104, 2, 251-271. Ebert S. and Strack P., 2015, “Until the Bitter End: On Prospect Theory in a Dynamic Context‘’, American Economic Review, 105(4), 1618-1633. Henderson V., 2012, “Prospect theory, Liquidation and the disposition effect”, Management Science, 58, 445-460. Henderson V., Hobson D. and A.S.L.Tse, 2014, Randomized Strategies and Prospect Theory in a Dynamic Context, Under review. Henderson V., Hobson D. and A.S.L.Tse, 2016, “Can Probability Weighting Help Prospect Theory Explain the Disposition Effect?”, Available soon, Slides on my website.

Ingersoll J.E. and L.J. Jin, 2013, “Realization Utility and Reference-Dependent Preferences, Review of Financial Studies, 26, 3, 723-767. Kyle, A.S., Ou-Yang, H. and W. Xiong, 2006, “Prospect Theory and Liquidation Decisions,” Journal of Economic Theory, 129, 273-288. Odean, T., 1998, “Are Investors Reluctant to Realize Their Losses?”, Journal of Finance, 53, 5, 1775-1798. Shefrin, H. and M. Statman, 1985, “The Disposition to Sell Winners too Early and Ride Losers too long: Theory and Evidence,” Journal of Finance, 40, 777-790. Strack P. and P. Viefers, 2015, “Too Proud to Stop: Regret in Dynamic Decisions”, Working paper. Tversky, A. and D. Kahneman, 1992, “Advances in Prospect Theory: Cumulative Representation of Uncertainty”, Journal of Risk and Uncertainty, 5, 297-323. Xu Z., and X. Y. Zhou, 2013, “Optimal Stopping under Probability Distortion”, Annals of Applied Probability, 23, 251-282.