Fnance 400 A. Penat - G. Pennacch Prospect Theory and Asset Prces These notes consder the asset prcng mplcatons of nvestor behavor that ncorporates Prospect Theory. It summarzes an artcle by N. Barbers, M. Huang, and T. Santos (2000) Prospect Theory and Asset Prces, Quarterly Journal of Economcs (forthcomng). Prospect Theory devates from von Neumann-Morgenstern expected utlty maxmzaton because nvestor utlty s a functon of recent changes n, rather than smply the current level of, fnancal wealth. In partcular, nvestor utlty characterzed by Prospect Theory may be more senstve to recent losses than recent gans n fnancal wealth. Ths effect s referred to as loss averson. Moreover, losses followng prevous losses create more dsutlty than losses followng prevous gans. After a run-up n asset prces, the nvestor s less rsk-averse because subsequent losses would be cushoned by the prevous gans. Ths s the so-called house money effect. An mplcaton of ths ntertemporal varaton n rsk-averson s that after a substantal rse n asset prces, lower nvestor rsk averson can drve prces even hgher. Hence, asset prces dsplay volatlty that s greater than that predcted by observed changes n fundamentals, such as changes n dvdends. Ths also generates predctablty n asset returns. A substantal recent fall (rse) n asset prces ncreases (decreases) rsk averson and expected asset returns. It can also mply a hgh equty rsk premum because the excess volatlty n stock prces leads loss-averse nvestors to demand a relatvely hgh average rate of return on stocks. Prospect theory assumes that nvestors are overly concerned wth changes n fnancal wealth, that s, they care about wealth changes more than would be justfed by how these changes affect consumpton. The dea was advanced by D. Kahneman and A. Tversky (1979) Prospect Theory: An Analyss of Decson Under Rsk, Econometrca 47, p. 263-291. Ths psychologcal noton s based on expermental evdence. For example, R. Thaler and E. Johnson (1990) Gamblng wth the House Money and Tryng to Break Even, Management Scence 36, p.643-660 fnd that ndvduals faced wth a sequence of gambles are more wllng to take rsk f they have made gans from prevous gambles. The Barbers, Huang, and Santos model assumptons are as follows. 1
Assumptons: A.1 Technology: A dscrete-tme endowment economy s assumed. The rsky asset (or portfolo of all rsky assets) pays a dvdend of pershable output of D t at date t. The paper presents an Economy I model, where aggregate consumpton equals dvdends. Ths s the standard Lucas (1978) economy assumpton. However, the paper focuses on ts Economy II model whch allows the rsky asset s dvdends to be dstnct from aggregate consumpton because there s assumed to be addtonal (non-traded) non-fnancal assets, such as labor ncome. In equlbrum, aggregate consumpton, C t, then equals dvdends, D t,plusnonfnancal ncome, Y t,becauseboth dvdends and nonfnancal ncome are assumed to be pershable. Aggregate consumpton and dvdends are assumed to follow the jont lognormal process ln ³C t+1 /C t = g C + σ C η t+1 (1) ln (D t+1 /D t ) = g D + σ D ε t+1 where the error terms are serally uncorrelated and dstrbuted η t ε t N 0 0, 1 ω ω 1 The return on the rsky asset from date t to date t + 1 s denoted R t+1. A one-perod rsk-free nvestment s assumed to be n zero-net supply, and ts return from date t to date t + 1 s denoted R f,t. 1 A.2 Preferences: Representatve, nfntely-lved ndvduals maxmze lfetme utlty of the form " Ã!# X E 0 ρ t C1 γ t 1 γ + b tρ t+1 v (X t+1,s t,z t ) t=0 (2) where C t s the ndvdual s consumpton at date t, γ > 0, and ρ s a tme dscount factor. X t+1 1 Snce the rsk-free asset s n zero net supply, the representatve ndvdual s equlbrum holdng of ths asset s zero. R f,t s nterpreted as the shadow rskless return. 2
s the gan n (change n value of) the ndvdual s rsky asset poston between date t and date t + 1. S t s the date t value of the ndvdual s rsky asset holdngs, and z t s a measure of the ndvdual s pror gans as a fracton of S t. z t < (>) 1 denotes a stuaton n whch the nvestor has earned pror gans (losses) on the rsky asset. Rsky asset gans are assumed to be measured relatve to the alternatve of holdng wealth n the rsk-free asset: The pror gan factor, z t, s assumed to follow the process X t+1 = S t (R t+1 R f,t ) (3) z t = 1 + η Ã! R z t 1 1 R t (4) where 0 η 1. Ifη =0,z t = 1 for all t. However, f η = 1, z t s smaller (larger) than z t 1 when rsky asset returns were relatvely hgh last perod, R t > R. In ths case, the benchmark rate, z t adjusts slowly to pror asset returns. In general, the greater s η, the longer s the nvestor s memory n measurng pror gans from the rsky asset. v ( ) s a functon characterzng the prospect theory effect of rsky asset gans on utlty. 2 For the case of z t = 1 (noprorgansorlosses),thsfunctondsplayspurelossaverson: X t+1 f X t+1 0 v (X t+1,s t, 1) = λx t+1 f X t+1 < 0 where λ > 1. Hence, ceters parbus, losses have a dsproportonately bgger mpact on utlty. When z t 6= 1, the functon v ( ) reflects Propect Theory s house money effect. In the case of pror gans (z t 1), the functon takes the form X t+1 f R t+1 z t R f,t v (X t+1,s t,z t )= (5) X t+1 +(λ 1) S t (R t+1 z t R f,t )fr t+1 <z t R f,t The nterpretaton of ths functon s that when a return exceeds the cushon bult by pror 2 Snce v ( ) depends only on the rsky asset s returns, t s assumed that the ndvdual s not subject to loss averson on nonfnancal assets. 3
gans, that s, R t+1 z t R f,t,taffects utlty one-for-one. However, when the gan s less than the amount of pror gans, R t+1 <z t R f,t, t has a greater than one-for-one mpact on dsutlty. Inthecaseofprorlosses(z t > 1), the functon becomes X t+1 f X t+1 0 v (X t+1,s t,z t )= λ (z t ) X t+1 f X t+1 < 0 (6) where λ (z t )=λ + k (z t 1), k>0. Here we see that losses that follow prevous losses are penalzed at the rate, λ (z t ), whch exceeds λ and grows larger as pror losses become larger (z t exceeds unty). Fnally, the prospect theory term n the utlty functon s scaled to make the rsky asset prce-dvdend rato and the rsky asset rsk premum be statonary varables as aggregate wealth ncreases over tme. 3 The form of ths scalng factor s chosen to be where b 0 > 0andC t s aggregate consumpton at date t. 4 Soluton to the Model: b t = b 0 C γ t (7) The state varables for the ndvdual s consumpton - portfolo choce problem are wealth, W t,andz t. Intutvely, snce the aggregate consumpton - dvdend growth process n (1) s an ndependent, dentcal dstrbuton, the dvdend level s not a state varable. We start by assumng that the rato of the rsky asset prce to ts dvdend s a functon of only the state varable z t,thatsf t P t /D t = f t (z t ), and then show that an equlbrum exsts n whch ths s true. 5 Gven ths assumpton, the return on the rsky asset can be wrtten as R t+1 = P t+1 + D t+1 P t = 1 + f (z t+1) D t+1 (8) f (z t ) D t 3 Wthout the scalng factor, as wealth (output) grows at rate g D, the prospect theory term would domnate the conventonal constant relatve rsk averson term. 4 Because C t s assumed to be aggregate consumpton, the ndvdual vews b t as an exogeneous varable. 5 Ths s plausble because the standard part of the utlty functon dsplays constant relatve rsk averson. Wth ths type of utlty, optmal portfolo proportons would not be a functon of wealth. 4
= 1 + f (z t+1) e g D+σ D ε t+1 f (z t ) It s also assumed that an equlbrum exsts n whch the rsk-free return s constant, that s, R f,t = R f. Ths wll be verfed by the soluton to the agent s frst order condtons. Makng ths assumpton smplfes the form of the functon v. From (5) and (6) t can be verfed that v s proportonal to S t.hence,v (X t+1,s t,z t ) can be wrtten as v (X t+1,s t,z t )=S t bv (R t+1,z t ) where for z t < 1 R t+1 R f f R t+1 R f bv (R t+1,z t )= (9) R t+1 R f +(λ 1)(R t+1 z t R f )fr t+1 <R f and for z t > 1 R t+1 R f f R t+1 R f bv (R t+1,z t )= (10) λ (z t )(R t+1 R f ) f R t+1 <R f The ndvdual s maxmzaton problem s then max {C t,s t} E 0 subject to the budget constrant " X Ã t=0 ρ t C1 γ t 1 γ + b 0ρ t+1 C γ t S t bv (R t+1,z t )!# (11) W t+1 =(W t + Y t C t ) R f + S t (R t+1 R f ) (12) and the dynamcs for z t gvenn(4). Defne ρ t J (W t,z t ) as the derved utlty of wealth functon. Then the Bellman equaton for ths problem s J (W t,z t )=max {C t,s t } Ct 1 γ h 1 γ + E t b 0 ρc γ t S t bv (R t+1,z t )+ρj (W t+1,z t+1 ) (13) Takng the frst order condtons wth respect to C t and S t one obtans 0=C γ t ρr f E t [J W (W t+1,z t+1 )] (14) 5
h 0 = E t b 0 C γ t bv (R t+1,z t )+J W (W t+1,z t+1 )(R t+1 R f ) = b 0 C γ t E t [bv (R t+1,z t )] + E t [J W (W t+1,z t+1 ) R t+1 ] R f E t [J W (W t+1,z t+1 )] (15) It s straghtforward to show that (14) and (15) mply the standard envelope condton Substtutng ths nto (14), one obtans the Euler equaton C γ t = J W (W t,z t ) (16) " µct+1 # γ 1 = ρr f E t C t (17) Usng (16) and (17) n (15) mples 0 = b 0 C γ h t E t [bv (R t+1,z t )] + E t C γ t+1 R t+1 = b 0 C γ h t E t [bv (R t+1,z t )] + E t C γ t+1 R t+1 R f E t h C γ t /ρ C γ t+1 (18) or " µ # γ Ct+1 1 = b 0 ρe t [bv (R t+1,z t )] + ρe t R t+1 C t (19) In equlbrum, condtons (17) and (19) hold wth the representatve agent s consumpton, C t, replaced wth aggregate consumpton, C t. Usng the assumpton n (1) thataggregate consumpton s lognormally dstrbuted, we can compute the expectaton n (17) to solve for the rsk-free nterest rate: Usng (1) and (8), condton (19) can also be smplfed: R f = e γg C 1 2 γ2 σ 2 C /ρ (20) 1 + f (zt+1 ) 1 = b 0 ρe t [bv (R t+1,z t )] + ρe t e g D+σ D ε t+1 e g C +σ C η t+1 γ f (z t ) (21) or 6
1 = µ 1 + f (zt+1 ) b 0 ρe t bv e g D+σ D ε t+1,z t f (z t ) +ρe g D γg C + 1 2 γ2 σc(1 ω 2 2 ) 1 + f (zt+1 ) Et e (σ D γωσ C )ε t+1 f (z t ) (22) The prce - dvdend rato, P t /D t = f t (z t ), can be computed numercally from (22). However, because z t+1 = 1 + η z t R t+1 ³ R 1 and R t+1 = 1+f(z t+1) f(z t ) e g D+σ D ε t+1, z t+1 depends upon z t, f (z t ), f (z t+1 ), and ε t+1,thats z t+1 = 1 + η Ã z t Rf (z t ) e g D σ D ε t+1 1 + f (z t+1 ) 1! (23) Therefore, (22) and (23) need to be solved jontly. Barbers, Huang, and Santos descrbe an teratve numercal technque for fndng the functon f ( ). Gven all other parameters, they guess an ntal functon, f (0), and then use t to solve for z t+1 n (23) for gven z t and ε t+1. Then, they fnd a new canddate soluton, f (1), usng the followng recurson that s based on (22): f (+1) (z t ) = ρe g D γg C + 1 2 γ2 σc(1 ω 2 2 ) Et hh1 + f () (z t+1 ) e (σ D γωσ C )ε t+1 Ã!# 1 + f +f () (z t ) b 0 ρe t "bv (z t+1 ) f () e g D+σ D ε t+1,z t, z t (z t ) (24) where the expectatons are computed usng a Monte Carlo smulaton of the ε t+1. Gven the new canddate functon, f (1), z t+1 s agan found from (23). The procedure s repeated untl the functon f () converges. For reasonable parameter values, Barbers, Huang, and Santos fnd that P t /D t = f t (z t )s a decreasng functon of z t. The ntuton was descrbed earler: f there were pror gans from holdng the rsky asset (z t s low), then nvestors become less rsk averse and bd up the prce of the rsky asset. Usng ther estmate of f ( ), the uncondtonal dstrbuton of stock returns s smulated from a randomly generated sequence of ε t s. Because dvdends and consumpton follow separate 7
processes and stock prces have volatlty exceedng that of dvdend fundamentals, the volatlty of stock prces can be made substantally hgher than that of consumpton. Moreover, because of loss averson, the model can generate a sgnfcant equty rsk premum for reasonable values of the consumpton rsk averson parameter γ. Because the nvestor cares about stock volatlty, per se, a large premum can exst even though stocks may not have a hgh correlaton wth consumpton. 6 The model also generates predctablty n stock returns: returns tend to be hgher followng crashes and smaller followng expansons. An mplcaton of ths s that stock returns are negatvely correlated at long horzons, a feature documented by recent emprcal research. 6 Recall that n standard consumpton asset prcng models, an asset s rsk premum depends only on ts return s covarance wth consumpton. 8