Prospect theory and fat tails

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1 Risk ad Decisio Aalysis 1 (2009) DOI /RDA IOS Press Prospect theory ad fat tails Philip Maymi Polytechic Istitute of New York Uiversity, New York, NY, USA phil@maymi.com Abstract. A behavioral represetative ivestor who evaluates a sigle risky asset based o cumulative prospect theory will ofte iduce high kurtosis, egative skewess, ad persistet autocorrelatio ito the distributio of market returs eve if the asset payoffs are merely a sequece of idepedet coi tosses. hese fidigs cotiue to hold eve whe the ivestor is simply loss averse. Keywords: Loss aversio, kurtosis, prospect theory, fat tails, behavioral What causes fat tails ad extreme evets i market returs? Oe possibility is that the market prices accurately reflect the uderlyig busiess risk, ad that busiess risk itself has rare but extreme possibilities. his possibility is the implicit assumptio i statistical models of market returs. he busiess risks are presumed to be reflected i the market ad so we study the market process to deduce the distributio of the uderlyig busiess risks. he alterate possibility, ad the oe I follow here, is that the market process itself may augmet the possibility of extreme risks, eve whe the uderlyig busiess risk has o rare but extreme evets. How do we kow which possibility is the right oe? he secod does makes a specific but hard to test predictio: if we could observe two markets o the same asset, oe by huma traders subject to stadard behavioral tedecies ad psychological biases, ad oe by risk eutral robots, the behavioral market would have more extreme evets tha the risk eutral oe. Experimetal results do suggest that bubbles ad crashes are a product of huma tradig ad ca dissipate as experiece ad group familiarity grows, cf. [3] for a review of 72 such experimets. he aim of this paper is to see if applyig stadard behavioral models of ivestor psychology ad decisio makig to the repeated evaluatio of a sequece of biomial gambles geerates ew extreme evets i the market prices that do ot occur i the uderlyig busiess risk. Suppose there is a sigle represetative ivestor tradig a sigle market asset whose fudametal risk is as beig as a coi toss, with o extreme evets, ad kow probabilities. If the ivestor is risk eutral, the asset will always be worth its expected value, ad because the expected value will ot chage i ay extreme way over time, either will the returs of the market asset. Similarly, if the ivestor maximizes the expected utility of his total wealth, for stadard utility fuctios, o ew extreme evets are itroduced. However, research over the past few decades has show that actual ivestors appear to be either risk eutral or expected utility maximizers. A few stylized facts have emerged: people ted to be loss averse, feelig about twice as much pai from losses as they feel pleasure from gais; people evaluate opportuities based o chages to their wealth, ot o the overall levels of their wealth; ad people ted to be risk-seekig i the domai of losses, willig to overpay for gambles that might reduce their loss, ad risk-averse i the domai of gais, as they are scared of losig what they have eared so far. hese three observatios form the basis of the cumulative prospect theory of [5]. Aother cosistet psychological observatio of huma behavior is metal accoutig [4]. Metal accoutig recogizes that people ted to view their assets i separate accouts, evaluatig salary icome differetly from bous icome, savigs moey from vacatio moey, ad so forth. I additio, people do ot igore suk costs: as [2] has show, idividual ivestors display a dispositio effect, a tedecy to sell wiig stocks but hold o to losig stock i the hope that they recover their prior losses, eve if they would t re-ivest i those losig stocks if they were forced to liquidate their positios ad realize their losses /09/$ IOS Press ad the authors. All rights reserved

2 188 P. Maymi / Prospect theory ad fat tails It could be the case that eve such a behavioral represetative ivestor would still ot geerate extreme evets; after all, he is merely evaluatig risky assets a little differetly, ad the theories ad models of behavioral fiace have bee put forth ad tested to match the psychological realities faced by huma traders, ot explicitly to model fat tails or extreme evets. But it turs out that with reasoable parameter assumptios, stable fudametal risk ideed gets trasformed ito market prices with high kurtosis, egative skewess, ad persistet autocorrelatios, all of the troublig features of real markets. his approach is offered as a proof of cocept that we eed ot merely pick statistical models that fit the data from the top dow but that we ca explore huma psychology to geerate price paths from the bottom up. I build the model with examples ad ituitio i Sectio 1, explore its implicatios i detail i Sectio 2, discuss ad cosider simple loss aversio istead of the etirety of cumulative prospect theory i Sectio 3, ad coclude with directios for future research i Sectio Model here is a sigle risky asset i the market that exists for periods ad pays off a coi toss each period. Each coi toss, g,givesu with probability π ad d with probability 1 π. Deote by g the distributio resultig from idepedet coi tosses. Ay gamble X is a list of payoffs ad associated probabilities. Sort these payoffs to express the gamble as: X = { (x m, q m );...;(x 1, q 1 ); (x 0, q 0 ); (x 1, q 1 );...;(x, q ) }, where x i <x j for i<j, x 0 = 0, ad the q i are the probabilities of havig the associated payoff x i. here is a sigle behavioral represetative ivestor who evaluates gambles based o the cumulative prospect theory of Kahema ad versky [5]. Specifically, his evaluatio of gamble G is v[g] where v[ ] is the cumulative prospect theory valuatio fuctio: v[x] q i v (x i ), where { v x α for x 0, (x) = λ( x) α for x<0, w(q i + + q ) w(q i q ) for 0 i<, qi = w(q m + + q i ) w(q m + + q i 1 ) for m i<0, ad w(q) = q δ (q δ + (1 q) δ ) 1/δ. he parameters estimated by [5] from experimetal data are α = 0.88, λ = 2.25, ad δ = he cumulative prospect theory value of a gamble takes three steps: first, all of the payoffs are evaluated based o v, which icorporates loss aversio through the fact that λ > 1 ad cocavity over gais ad covexity over losses through the fact that α>0; secodly, the probabilities are adjusted to reflect the propesity of ivestors to overweigh extreme outcomes; fially, the sum of the product of the traslated payoffs ad the traslated probabilities computes the value of the gamble to a behavioral ivestor. Like utility, the prospect theory value of a gamble is used to compare two gambles: the oe with the higher value is the presumed choice of the behavioral ivestor. A sure amout P is a gamble that pays off P with probability oe. At each time t = 0,..., 1, the represetative ivestor holds oe uit of the risky asset ad determies the market price that makes him idifferet betwee holdig the risky asset or holdig cash. he certaity equivalet C of the asset at time t = 0isthesolutio C 0 to the followig equatio: v[g] = v[c 0 ]. Cosider a umerical example where = 10, π = 0.5, u = 300, ad d = 100. he C 0 = By compariso, the expected value of te such coi tosses is E[g(10)] = Figure 1 shows the ratio of the certaity equivalet to the expected value for ragig from 1 to 100. he more coi flips, the closer the price gets to the expected value. Appedix A proves that uder loss aversio the limit of the ratio ap-

3 P. Maymi / Prospect theory ad fat tails 189 proaches oe as the umber of coi flips icreases to ifiity. At time t = 1, suppose that the results of the first coi toss are such that the asset retured A 1 {u, d}, givig the ivestor a urealized gai or loss. He evaluates the asset relative to his origial etry poit so that the ew certaity equivalet C 1 is foud from: v[g( 1) + A 1 ] = v[c 1 ], ad i geeral the certaity equivalet C t at time t is foud from: [ ] t v g( t) + A i = v[c t ], i=1 where A k is the result of the kth coi toss. It is more coveiet to deal with scaled umbers. Defie: Fig. 1. Coi toss prospect theory valuatio. Defie the price of a sequece of fair coi flips payig off 300 o heads ad 100 o tails as the certaity equivalet uder cumulative prospect theory valuatio. his figure shows the ratio of the price to the expected value of the gamble for ragig from 1 to 100. As the umber of coi flips icreases, the prospect theory price approaches but ever reaches the expected value. p t C t t i=1 A i E[g( t)] = C t t i=1 A i ( t)(πu + (1 π)d). he p t is the price of the gamble, because it is the excess of the certaity equivalet relative to the ivestor s actual gais ad losses to date, expressed as a portio of the expected value of the remaiig gamble. he ituitio for the umerator is that the ivestor could i priciple choose betwee cotiuig to ivest i the risky asset or realizig his gais ad losses to date ad holdig cash. he certaity equivalet C t expressed how much the ivestor was willig to pay relative to zero to stay i the risky asset; the scaled excess price p t represets the more iterestig umber of how much the ivestor is willig to pay relative to what he has already gaied or lost so far, scaled as a portio of the remaiig expected value to make the umbers more comparable across differet times t. Note that the iitial certaity equivalet equals the iitial price, C 0 = p 0, so a alterate iterpretatio of Fig. 1 is that the price is always below the expected value. Our particular umerical example is useful for two reasos: oe, the expected value of a sigle coi toss is exactly 100, makig scalig easy, ad two, the prospect theory value of ay coi tosses plus ay amout A of accumulated urealized gais ad losses will always exceed the prospect theory value of holdig A i cash, as we ca see i Fig. 2 ad as we ca prove for the special case of loss aversio i Appedix B, meaig that Fig. 2. he graph above is the differece betwee (a) the prospect theory value of a fair coi toss payig either 300 or 100 plus accumulate profits or losses ragig from 1000 to 1000, ad (b) the prospect theory value of the accumulated profits or losses by themselves. Specifically, it is a plot of v[g + x] v[x], where v is the cumulative prospect theory valuatio fuctio, g is the fair coi toss, ad x varies alog the x-axis from 1000 to the ivestor will always choose the gamble over holdig cash, thus assurig positive prices p t. However, the particular prices at which he is idifferet do chage, ad it is the distributio of these prices that we wish to explore. We ca solve for the distributio of possible prices p t () where t is the umber of heads that have occurred to date. Hece the distributio of p t is: { p t p t () with probability } t π (1 π) t,

4 190 P. Maymi / Prospect theory ad fat tails where p t () is such that: v[g( t) + u + (t )d] = v[c t ()] ad as above: p t () = C t() (u + (t )d) ( t)(πu + (1 π)d). For the same umerical example, p 0 = 0.75 ad the distributio of p 1 is: p 1 = { 0.76 if A1 = u, with probability p, 0.72 if A 1 = d, with probability 1 p. I other words, whe the first coi toss is up, the price of the asset rises, ad whe it is dow, the price falls, eve though the value of the remaiig ie coi tosses is idepedet of that first toss, ad eve though ivestors ought to igore suk costs by traditioal ecoomic reasoig. his property of behavioral ivestors to icorporate prior gais ad losses ito evaluatios of future prospects may be part of the explaatio for the excess volatility puzzle, or the fidig that the stock market teds to move aroud too much, relative to the volatility of the uderlyig earigs. As compaies report relatively radom earigs, ivestors appear to overreact ad cause a eve greater price drop, but as Barberis et al. [1] poit out, the reaso may be that behavioral ivestors have chagig levels of loss aversio resultig from the gais or losses geerated by previous market moves. he same effect happes i our model here. 2. Results Cotiuig the umerical example from the model, Fig. 3 plots the histogram of the prices across time. Each white label is the digit correspodig to the time t for which the histogram is plotted. Figure 4 plots the idividual price histograms for t = 6,...,9. As t icreases, the histogram spreads out, ad at least visually is far from ormal. Figure 5 plots all of the 2 = 1024 possible price paths. All of the paths start from p 0 = 0.75 ad may Fig. 3. Histograms of prices. For = 10, π = 0.5, u = 300 ad d = 100, the graph above plots the histograms of the implied prices of the behavioral represetative ivestor after t = 1,...,9 coi tosses. he bars are labelled with t, so for example the highest possible price of 1.25 occurs with 1.8% probability whe t = 9. Fig. 4. Fial histograms of prices. For = 10, π = 0.5, u = 300 ad d = 100, the graph above plots the histograms of the implied prices of the behavioral represetative ivestor after t = 6,..., 9 coi tosses.

P. Maymi / Prospect theory ad fat tails 191 Fig. 5. All price paths. Each lie i the graph below represets oe possible path of prices implied by the behavioral represetative ivestor followig t = 0,.

5 P. Maymi / Prospect theory ad fat tails 191 Fig. 5. All price paths. Each lie i the graph below represets oe possible path of prices implied by the behavioral represetative ivestor followig t = 0,...,9ofthe = 10 coi tosses that retur u = 300 or d = 100 with equal probability π = 0.5. Fig. 6. Kurtosis ad skewess. he top lie shows the kurtosis of the implied price distributios of the behavioral represetative ivestor after t out of = 10 coi tosses payig out u = 300 or d = 100 with equal probability π = 0.5, ad the bottom lie shows the correspodig skewess. Fig. 7. Autocorrelatios. he six graphs above show the histogram of autocorrelatios for give lags of returs calculated from the price paths implied by a behavioral represetative ivestor evaluatig = 10 coi flips that result i u = 300 or d = 100 with equal probability π = 0.5. of them follow a smooth arc, but several extremes paths are also geerated. We ca compute the skewess ad kurtosis of the implied distributios as a fuctio of the time t. hese are show i Fig. 6. he kurtosis exceeds three for all t > 5, reachig a maximum ear 40 at t = 9, ad the skewess is early always egative, except for t = 9. We ca also compute the autocorrelatios of each path: give a particular price path, we calculate overlappig returs of lag l ad compute the correlatio betwee successive such returs. Figure 7 shows the histogram of these autocorrelatios across all possible paths. Virtually ay autocorrelatio is possible, though as the lag icreases, a correlatio ear oe emerges as the mode.

6 192 P. Maymi / Prospect theory ad fat tails 3. Discussio Which of the assumptios of cumulative prospect theory are ecessary to geerate these results? We ca reproduce the results for differet values of the cumulative prospect theory parameters. I particular, if the probability weightig parameter of cumulative prospect theory, δ, is set equal to oe, the the probabilities are uadjusted, ad if the curvature parameter α is also set equal to oe, the the prospect theory valuatio of a gamble reduces to a straightforward expected value where losses are multiplied by λ = I this limited model, without risk aversio over gais or risk seekig over losses, ad without overweightig the likelihood of extreme evets, the same results cotiue to hold. I other words, it is just the loss aversio ad the metal accoutig that create extreme evets. Figure 8 plots all of the possible price paths implied by a loss averse ivestor. As before, the possible prices spread out widely. Figure 9 shows the skewess ad kurtosis of the resultig price distributios. he effects are eve more proouced. he kurtosis exceeds three for all t>2, ad the skewess is cosistetly egative. Figure 10 shows the lagged autocorrelatios of the resultig price series. As before, the possible correlatio ca be quite high with sigificat probability. 4. Coclusio We have see how simple loss aversio ca result i extreme distributios eve whe the uderlyig busiess risk has o extremes. I geeral, the results hold uder cumulative prospect theory, though the miimum required assumptios seem to be oly loss aversio, experiecig losses as about twice as paiful as gais are pleasat, ad metal accoutig, icorporatig the previous gais ad losses o a asset with its future values whe evaluatig it. hese two assumptios loss aversio ad metal accoutig are amog the most well-documeted i the behavioral fiace literature ad the most stable across both idividual ad istitutioal ivestors. he fact that they also geerate extreme market price distributios may suggest that it is the activity of the ivestors that is causig the extreme evets, ad ot the uderlyig busiess risk. Future research could replace the discrete biomial distributio with a cotiuous ormal or other distributio. We could cosider multiple risky assets, or allow for other ivestors, icludig the possibility of arbitrageurs ad of overlappig geeratios where ew ivestors eter the market with o accumulated profits or losses. Appedix: Proofs Assumig oly loss aversio, so that the probability weightig parameter δ ad the curvature parameter α of cumulative prospect theory are set equal to oe, the the ivestor with loss aversio parameter λ = 2.25 evaluates gambles X = { (x m, q m );...;(x 1, q 1 ); (x 0, q 0 ); (x 1, q 1 );...;(x, q ) } Fig. 8. All price paths for loss aversio. Each lie i the graph above represets oe possible path of prices implied by the loss averse represetative ivestor followig t = 0,..., 9 of the = 10 coi tosses that retur u = 300 or d = 100 with equal probability π = 0.5. Fig. 9. Kurtosis ad skewess uder loss aversio. he top lie shows the kurtosis of the implied price distributios of the loss averse represetative ivestor after t out of = 10 coi tosses payig out u = 300 or d = 100 with equal probability π = 0.5, ad the bottom lie shows the correspodig skewess.

7 P. Maymi / Prospect theory ad fat tails 193 Fig. 10. Autocorrelatios uder loss aversio. he six graphs above show the histogram of autocorrelatios for give lags of returs calculated from the price paths implied by a loss averse represetative ivestor evaluatig = 10 coi flips that result i u = 300 or d = 100 with equal probability π = 0.5. with the simpler fuctio: v[x] = = 1 λx i q i + = E[X] x i q i i=0 x i q i + (λ 1) 1 1 x i q i. x i q i A sequece of fair coi flips payig off u with probability π ad d with probability 1 π is the gamble g: g = {( ku + ( k)d, ) π k (1 π) k k } for k = 0,...,. A. Proof of covergece I this special case of loss aversio ad for our umerical example where u = 300, d = 100, ad π = 0.5, we ca prove that the certaity equivalet of the loss aversio value of the gamble approaches the gamble s expected value i the limit. heorem 1. If α = δ = 1, u = 300, d = 100, ad π = 0.5, the lim t C = E[g] where C is the certaity equivalet give by v[g] = v[c ]. Proof. For coi tosses, we ca solve for the miimum umber of heads k that guaratee a positive outcome: ku + ( k)d >0, which implies: k> 4 for our particular u ad d. he the loss aversio value equals the expected value plus 1.25 times the

8 194 P. Maymi / Prospect theory ad fat tails probability-weighted sum of the egative payoffs, or rearragig terms: E[g] v[g] = /4 (300k ) ( k)100 k k=0 < /4 < 125 2( /4 ) 2 ad therefore the risk premium approaches zero as approaches ifiity because: 2( ) /4 lim 2 = 0. hus we have show that v[g] approaches E[g] as teds to ifiity. he certaity equivalet C is defied as v[g] = v(c ) = C because C is always positive. herefore the certaity equivalet C approaches the expected value E[g] as approaches ifiity. B. Proof of positivity I this special case ad for our umerical example, we ca prove that a loss averse ivestor will always choose the gamble relative to ay startig poit. as: π (1 π) ( u + ( )d + x ) = + λ π (1 π) =0 (u + ( )d + x) x ( = = + λ π (1 π) =0 ) π (1 π) ( u + ( )d ) (u + ( )d) [ + x 1 + π (1 π) =0 + (λ 1) =0 = v[g] + x(λ 1) =0 ] π (1 π) π (1 π). We have see from the earlier proof that v[g] is positive for our umerical example. We have assumed x is positive. We kow that λ 1 = 1.25 is positive. Ad the fial term is just a sum of positive probabilities. herefore the etire sum is positive, ad therefore v[g + x] v[x] > 0forx>0 ad for ay,i particular for = 1. Now cosider the case x<0. Let y = x be its absolute value so that y>0. he v[x] = v[ y] = λy ad, defiig as above, we ca evaluate heorem 2. If α = δ = 1, u = 300, d = 100 ad π = 0.5, the v[g(1) + x] >v[x] for all x. Proof. Cosider first the case x>0. he v[x] = x. Call the critical value of the umber of heads such that the payoff from the gamble for > always exceeds or equals x ad the payoff from the gamble for is always less tha x. he we ca evaluate: v[g + x] x as: v[g + x] v[x] = v[g y] ( λy) = v[g y] + λy = π (1 π) ( u + ( )d y ) + λ =0 π (1 π)

9 P. Maymi / Prospect theory ad fat tails 195 (u + ( )d y) + λy = π (1 π) ( u + ( )d ) = + λ π (1 π) =0 (u + ( )d) [ + y λ π (1 π) =0 (λ 1) =0 = v[g] [ + y(λ 1) 1 + =0 ] π (1 π) ] π (1 π). As before, we have see from the earlier proof that v[g] is positive for our umerical example. We have assumed x = y is egative, so y is positive. We kow that λ 1 = 1.25 is positive. Ad the fial term is just oe plus a sum of positive probabilities. herefore the etire sum is positive, ad therefore v[g + x] v[x] > 0forx<0. herefore we have show that a loss averse ivestor will always choose the gamble of coi tosses per our umerical example for ay startig value ad ay umber. Refereces [1] N. Barberis, M. Huag ad. Satos, Prospect theory ad asset prices, Quarterly Joural of Ecoomics 116 (2001), [2]. Odea, Are ivestors reluctat to realize their losses?, he Joural of Fiace 53 (1998), [3] D.P. Porter ad V.L. Smith, Stock market bubbles i the laboratory, Joural of Behavioral Fiace 4 (2003), [4] R.H. haler, Metal accoutig matters, Joural of Behavioral Decisio Makig 12 (1999), [5] A. versky ad D. Kahema, Advaces i prospect theory: Cumulative represetatio of ucertaity, Joural of Risk ad Ucertaity 5 (1992),

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