9. Mood Impacts on Probability Weighting Functions%3a “Large-gamble” Evidence

The Journal of Socio-Economics 37 (2008) 13971411

Mood impacts on probability weighting functions:Large-gamble evidence

Doron Kliger a,, Ori Levy b,1a The University of Haifa, Haifa, Israel

b Coral Capital Management, Minneapolis,MN 55402, United States

Abstract

This paper integrates considerations of mood into non-expected utility theories and extends the existing literature on how moodinfluences peoples decisions and choices. An important element in many non-expected utility theories is the probability weightingfunction (PWF), that nonlinearly weights physical probabilities. Using US market price data, we attempt to establish an empiricalrelation between investors mood and these PWFs. To proxy investors mood, we rely on an established medical phenomenon,seasonal affective disorder, a source of depression caused by the scarcity of daylight time during fall and winter, as well as ona measure of cloudiness. We find statistical evidence indicating that bad mood causes investors to systematically distort theirPWFs. 2007 Elsevier Inc. All rights reserved.

JEL classication: D81Keywords: Market data; Mood; Non-expected utility; Probability weighting functions; SAD; Weather

1. Introduction

In this paper we integrate mood effects into non-expected utility theory, extending thereby the existing literature onthe influence of mood on peoples decision-making processes.2 We do so by exploring peoples probability weightingfunctions (PWFs), an important building block of prominent non-expected utility theories such as prospect theory(Kahneman and Tversky, 1979), rank-dependent expected utility (Quiggin, 1982; Yaari, 1987), and cumulative prospecttheory (Tversky and Kahneman, 1992). In essence, PWFs relax the linearity-in-probability property inherent in expectedutility theory by allowing the physical probabilities to be nonlinearly transformed into decision weights.3 Using anew approach that permits the recovering of subjective decision weights from a panel data of US market prices, weestablish an empirical relation between investors moods and their PWFs.

Corresponding author. Tel: +972 4 8249587; fax: +972 4 8240059.E-mail addresses: [email protected] (D. Kliger), [email protected] (O. Levy).

1 The research was conducted while Levy was at the University of Haifa, Israel.2 Loewenstein (2000) discusses the importance of incorporating visceral factors in economic modelling. Loewenstein et al. (2001) discuss the

role of affect experienced in the moment decisions are taken. Hirshleifer (2001) provides a comprehensive survey on psychological considerationsin asset pricing.

3 Camerer (1995) and Starmer (2000) are two excellent surveys.

1053-5357/$ see front matter 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.socec.2007.08.010

1398 D. Kliger, O. Levy / The Journal of Socio-Economics 37 (2008) 13971411

A sizeable literature in psychology considers how emotions and moods influence individuals decision making.The DSM-IV (American Psychiatric Association, 1994) states that individuals experiencing major depressive episodesoften have difficulty making decisions. Austin et al. (1992), Brown et al. (1994), Beats et al. (1996), and Elliott et al.(1996) document extensive neuro-psychological deficits on tests of attention, memory and executive functioning indepressed patients. Psychological stress is demonstrated to detract decision-makers from rational choice paradigmsand affects the quality of decision-making in Sieber (1974), Levi and Tetlock (1980), and Janis (1982). Bad mood isshown to be associated with non-rational, self-defeating behavior (e.g., Keinan, 1987; Wegener and Petty, 1994; Leithand Baumeister, 1996). Finally, Pacini et al. (1998) show that nondepressed individuals behave more optimally withincreased incentives and more meaningful contexts, while depressed individuals, less optimally. Whereas a subclinicallydepressed group made more optimal decisions than a nondepressed control group under trivial conditions (the so-calleddepressive realism effect), the depressed group responded less and the control group more optimally under moreconsequential conditions. Furthermore, the depressed group reported engaging in less rational processing and in moremaladaptive experiential processing in everyday life than did the control group.4 Overall, this literature suggests thatadverse mood/depression may be responsible for distorting individuals thinking and behavior. This paper empiricallyquestions whether bad mood is associated with deviations from linearity-in-probability (expected utility), over andabove the deviations in neutral or good mood.5

To proxy investors mood, we rely on two psychological/medical phenomena: seasonal affective disorder (SAD),and the effect of weather conditions on mood. SAD is a type of fall and winter depression that affects numerous peopleeach year between September and April. It is caused by a biochemical imbalance in the hypothalamus due to theshortening of daylight hours and the lack of sunlight in the fall-winter season (e.g., Rosenthal et al., 1984).6Kamstraet al. (2003) document a seasonal pattern in stock market returns and attribute it to SAD. Garrett et al. (in press) findthat a conditional CAPM that allows the price of risk to vary in relation to seasonal variation in the length of day isable to capture the SAD effect.7

Accumulated psychological evidence suggests that weather conditions (humidity levels, number of sunshine hours,precipitation volume) affect peoples moods (e.g., Cunningham, 1979; Sanders and Brizzolara, 1982; Howarth andHoffman, 1984; Eagles, 1994). Saunders (1993) and Hirshleifer and Shumway (2003) examine the relationship betweenweather and market index returns. Using a measure of cloud coverage in cities where exchanges are located as a proxyfor investors mood, they find strong support for their weather-market return hypothesis.

Using experimental (laboratory) data, Tversky and Kahneman (1992), Camerer and Ho (1994), Tversky and Fox(1995), Wu and Gonzalez (1996), Gonzalez and Wu (1999), and Donkers et al. (2001) find the PWF to be regres-sive, overweighting (underweighting) small (large) probabilities, and inverse S-shaped, first concave then convex,capturing individuals high sensitivity to probability changes near the distributions endpoints. Our theoretical setupuses the observation that marginal utilities and stochastic discount factors (SDFs) are proportional cross-sectionally.8We specify the utility function and the subjective SDF in accordance with the cumulative prospect theory. Conse-quently, physical probabilities are (possibly) nonlinearly weighted by a regressive, inverse S-shaped PWF. A paneldata of European call options written on the market index assists us in the empirical phase. Option prices are knownto be particularly useful in an analysis such as ours that wishes to recover information on a state-by-state basis.After decision weights are recovered, we will be in a position to shed light on a possible relationship betweeninvestors mood and their degree of rationality in a real market setting where unrealistic behavior is incontestablycostly.

The rest of the paper is organized as follows. The proposed theoretical framework is described in Section 2. Section3 implements the devised framework on market data, and Section 4 concludes.

4 Pacini et al. cite recent literature on an inverse relation between the demonstration of the depressive realism effect and the realism of theexperimental conditions/the emotional involvement of the participants in the outcomes of their efforts.

5 Rottenstreich and Hsee (2001) find that the probability weighting function is flatter for vivid outcomes that evoke emotions than for pallidoutcomes. Compared with affect-poor prizes, affect-rich prizes yielded more pronounced overweighting of small probabilities, more pronouncedunderweighting of large probabilities, and less sensitivity to intermediate probability variations.

6 A less debilitating form is known as the subsyndromal SAD or winter blues.7 Kamstra et al. (2000) show that the FridayMonday return is significantly lower on daylight-saving weekends than other weekends.8 The SDF is a ratio of state prices (i.e., prices of ArrowDebreu contingent claims) and decision weights assigned by investors to physical

probabilities. See Section 2 below.

D. Kliger, O. Levy / The Journal of Socio-Economics 37 (2008) 13971411 1399

2. Theory

Our study exploits a cross-sectional relation between investors marginal utilities and subjective SDFs9:

U t (s, ) [

Qt(s, )DW(s;Pt())

], s = 1, . . . , S, (2.1)

where U t (s, ) is the time t + , state s, marginal utility, Qt(s, ) the state price (the price of a claim that guar-anties one $US in state s and nothing otherwise), DW(s;Pt()) the decision weight for state s, where Pt() (Pt(1, ), . . . , Pt(S, )) is the probability distribution function available to decision-makers, Pt(s, ) being the state-sprobability, and S is the total number of states of nature.10

The application of PWFs may be viewed as a generalization of the treatment of probabilities by expected utilitytheory, where the decisions weights equal their corresponding state probabilities. Under this view, the analysis in ourstudy also provides an indirect test of expected utilitys linearity-in-probabilities property.

In the sequel, we elaborate on the specification of the utility (value) function and the decision weights. Then, weoutline the testable hypotheses resulting from our analysis.

2.1. Utility function

We adopt the following specification of the utility (value) function, due to Tversky and Kahneman (1992):

U[xs;, ] ={

xs , xs 0 (xs ), xs < 0

. (2.2)

In (2.2), > 1 is the loss aversion parameter losses are allowed (but not restricted) to be more painful thanequal-size gains are pleasurable; (0, 1) measures the curvature of the value function investors may be risk aversein the gain-domain and risk seekers in the loss-domain, with a diminishing marginal sensitivity to the lotterys outcomewhen moving away from the reference point; xs is the state-s relative (to current wealth) payoff.11 The marginal utilitycommensurate with (2.2) is

U [xs;, ] =

{ (x1s ), xs < 0 (x1s ), xs > 0

. (2.3)

2.2. Decision weights

Under expected utility theory, a state-s decision weight equals its corresponding state probability, thusDW(s;Pt()) = Pt(s, ). Non-expected utility theories, however, employ PWFs to introduce variant sensitivity tothe state probabilities, resulting thereby in DW(s;Pt()) = Pt(s, ).

The decision weights may be assigned differently in the loss and gain domains (Tversky and Kahneman, 1992):

DW(s;P()) w(

si=1

P(i, ))

w(

s1i=1

P(i, ))

, (2.4)

9 For an intuitive interpretation of (2.1), consider the state-specific discount factors Mt(s, ), s = 1, . . . , S, each of which defined by Qt(s, ) =Mt(s, ) DW(s;Pt()). The idea in this definition is as follows: the decision weight, DW(s;Pt()), may be regarded as the (subjectively) expectedpayoff generated by the state-s ArrowDebreu security. Therefore, discounting by Mt(s, ) yields the ArrowDebreu securitys present value whichis its state price, Qt(s, ). Having established that Mt(s, ) = Qt(s, )/DW(s;Pt()), note that the time t + , state-s, marginal utility, U t (s, ), isproportional to the marginal rate of substitution (the rate at which the investor is willing to substitute consumption at time t + for consumption attime t). Therefore, U t (s, ), is also proportional to Mt(s, ).10 Qt() (Qt(1, ), . . . ,Qt(S, )) is known as the state price density (SPD). Qt(s, )/DW(s;Pt()) is the state-s stochastic discount factor.11 Consistent with the laboratory findings of Tversky and Kahneman, the value function is assumed to be reflective, and hence homogeneous

in the payoff xs. Since investors preferences are recovered from their valuation of index returns, and as their investment level is unobserved, onlyreflective value functions are empirically identified.


DW+(s;P()) w+(

Si=s

P(i, ))

w+(

Si=s+1

P(i, ))

, (2.5)

where DW() and DW+() correspond to the decision weights for losses and gains, respectively, and w() and w+()are the respective PWFs. An intuitive interpretation of the decision weights represented by Eqs. (2.4) and (2.5) maybe found in Gonzalez and Wu (1999).

We use the following specification of the PWF, due to Tversky and Kahneman (1992) (see also Camerer and Ho,1994; Wu and Gonzalez, 1996):

w(P ; ) = P

(P + (1 P) )1/ , (2.6)

w+(P ; +) = P+

(P+ + (1 P)+ )1/+ , (2.7)

where (0, 1] and + (0, 1].12 The loss- and gain-domain decision weights may therefore be treated as functionsof and +, respectively, that is, DW(s;w(P ; )) and DW+(s;w+(P ; +)).

2.3. Testable hypotheses

Henceforth, we explore the effect of investors mood on the decision weights they assign to possible outcomes. Werelate mood to the regressiveness of the PWFs, i.e., we allow and + to be mood-dependent. The non-comprehensivepsychological evidence cited above suggests that individuals in a downbeat mood hold more unrealistic views thanothers and would engage in a less rational, distorted thinking and behavior in meaningful contexts with significantconsequences. If indeed bad mood dissociates investors from rational behavior by intensifying deviations from linearity-in-probability, then it would make w() and w+() more regressive. Given that the larger , the less regressive thePWF, the following testable hypothesis and its alternative result:

H0: and + are unrelated to investors mood,HA:

and + are larger the better investors mood is.

Moreover, we expect the increased regressiveness to be more pronounced in the loss domain, reflecting investorsincreased subjective probability to large losses when in bad mood. To test these hypotheses, we compare marginalutilities summarized in (2.3) with SDFs, as in (2.1). The decision weights required for the SDFs evaluation are calculatedin agreement with (2.4) and (2.5). Then, we test whether the parameters characterizing the loss- and gain-domain PWFs, and +, are mood-dependent.

3. Empirical analysis

3.1. The data

The option data used in this analysis were provided to us by Jens Jackwerth and Mark Rubinstein.13 The datasetextends from December 1987, immediately after the October 1987 crash, through December 1995, and includes 74independent cross-sections of option prices for which there were no arbitrage violations.14 To back out time-dependentstate prices, (Qt(s, ))s S,t T , we build on Breeden and Litzenberger (1978) and use prices of European call optionswritten on the S&P500 index, and traded on the Chicago Board Options Exchange. These prices are sampled on amonthly basis and are averages of bid and ask quotes adjusted by a penalty parameter.

12 The special case = + = 1 implies linearity-in-probabilities, i.e., DW(s;P()) = Pt(s, ).13 A comprehensive description of the data can be found in Jackwerth and Rubinstein (1996) and Jackwerth (2000).14 Rubinstein (1994), Jackwerth and Rubinstein (1996), and Christensen and Prabhala (1998) document a regime shift in option prices following

the October 1987 crash.


Table 1Data description

No. of options Time to expiration (day) Volatility (%)Mean 15.41 31.61 15.73Median 15.00 30.00 14.26Maximum 26.00 43.00 35.72Minimum 6.00 23.00 9.17S.D. 4.58 4.06 5.42

No. of observations 74

The table describes the 74 cross-sections of options in our dataset. Volatility (the rightmost column) is the annual, at-the-money volatility impliedby the Black and Scholes (1973) option pricing formula. The sample extends from December 1987 through December 1995.

Table 1 describes the selected sample. Overall, the dataset consists of 1140 options, with 626 options in a cross-section, and an average of 15.4. The cross-sectional time-to-expiration ranges from 23 to 43 days, with an average of31.6. At-the-money volatility, implied by the Black and Scholes (1973) pricing equation, ranges from 9.2% to 35.7%(in December 1987), averaging 15.7%.

We recover time-dependent physical state probabilities, (Pt(s, ))s S,t T , using daily S&P500 index returns takenfrom the University of Chicagos Center for Research on Security Prices files. These physical probabilities are estimatedby a Monte-Carlo simulation with an underlying GARCH model for the S&P500 return process (Bollerslev, 1986).All technical details are collected in Appendix A.

A measure of cloudiness serves as one of our mood proxies (Saunders, 1993; Hirshleifer and Shumway, 2003). Skycoverage from sunrise to sunset (SCSS) records collected at LaGuardia Field, New York City, were obtained fromthe National Climatic Data Center (NCDC). The cloud coverage variable is reported by the NCDC on a scale from 0to 10, where 10 indicates total cloud coverage. The cut-off point for the cloud coverage variable we will use in ourempirical analysis (Section 3.2 below) is commensurate with the observation made by Saunders (p. 1339) that 100percent cloud cover days are quite different from other days, including 90 percent cloud cover days. Approximately 85percent of all rain occurs on 100 percent cloud cover days. Partially cloudy days, most of which have no rain, are notparticularly depressing, even when a little rain falls. Therefore, one would not reasonably expect significant differencesin weather-induced mood influences between days with 30 percent cloud cover versus days with 80 or 90 percent cloudcover.

Fig. 1 depicts the sky coverage data. Notice that the sky was completely covered with clouds (SCSS = 10) in abouta third of the days.

3.2. Mood-affected probability weighting functionsstatistical results

To deal with our testable hypotheses, we estimate the parameters underlying the cross-sectional relation betweenmarginal utilities and SDFs given by Eq. (2.1), where marginal utilities and decision weights required for the SDF areas in Sections 2.1 and 2.2, respectively:

(x1s ) = At [

Qt(s, )DW(s;w(P ; ))

], xs < 0;

(x1s ) = At [

Qt(s, )DW+(s;w+(P ; +))

], xs > 0. (3.1)

We allow for idiosyncratic time effects by the cross-sectional estimation of the scaling parameters At .As we have mentioned, we proxy investors mood by SAD and a measure of cloudiness. In the remaining of this

subsection, we conjecture that the SDF in (2.1) is mood-related, and present results using both mood proxies. Werestate that results are based on repeated cross-sections of option prices endowed with a unique qualitythe ability torecover state-wise preference information.


Fig. 1. The figure describes the distribution of sky coverage from sunrise to sunset (SCSS). SCSS is reported by the National Climatic Data Centeron a scale from 0 to 10, where 10 indicates total cloud coverage. Data are collected at LaGuardia Field, New York City. The number of includedobservations is 74.

3.2.1. SAD as a mood proxyTo assess the role of mood, captured by the SAD proxy, in shaping investors PWFs, we partition and + in

(2.6) and (2.7) in the following way: = F F + W W + S S, (3.2)

and

+ = +F F + +W W + +S S, (3.3)where F = 1 for cross-sections sampled in September through November (fall), W = 1 for cross-sections sam-pled in December through February (winter), and S = 1 for cross-sections sampled in March through August(spring/summer).15

Fig. 2 depicts the daylight duration within the year cycle. As in Kamstra et al. (2003) who build on accumulatedpsychological and medical evidence, we hypothesize that: (a) the depressive effects of SAD are asymmetric aboutwinter solstice. The asymmetric influence in fall relative to winter is due to the fact that the daylight time decreasesover the fall, shifting thereby investors into bad mood, and increases over the winter, shifting investors into goodmood16; (b) the SAD is a fall- and winter-related phenomenon, i.e., variations in the duration of daylight throughspring and summer months have no systematic effect on investors mood.

To test the effect of mood on investors PWFs, we define the following null hypothesis:

H0(SAD) :

{F = W = S (Loss domain)+F = +W = +S (Gain domain)

. (3.4)

If the PWFs are independent of investors mood are season-independent then our null hypothesis should not berejected. If, however, mood affects investors PWFs, then it should be rejected in favor of several alternatives. First,

15 Since options expire in the US on the third Friday of each month, option prices are sampled on the months third part. Hence the choice of thecut-off points.16 Indeed, letting trading days in autumn include those between September 21 through December 20, Kamstra et al. (2003) find that returns are

relatively low prior to winter solstice, and relatively high following winter solstice.


Fig. 2. The figure depicts the within-year cycle of daylight duration.

comparing the fall season with spring and summer time, we expect:

HA1(SAD) :

{F <

S (Loss domain)

+F < +S (Gain domain)

. (3.5)

The rational behind this one-sided alternative hypothesis is that the decrease in daylight time throughout the fallbrings about depression which is manifested in inferior probability judgments and larger deviations from linearity-in-probability.17

The second alternative hypothesis considers fall and winter mood:

HA2(SAD) :

{F <

W (Loss domain)

+F < +W (Gain domain)

, (3.6)

i.e., as daylight time lengthens during the winter season and shortens during fall, investors winter mood should bebetter than their fall-mood, and their PWF in winter should be less regressive than their PWF in fall.18

Lastly, we compare winter and spring/summer times:

HA3(SAD) :

{S <

W (Loss domain)

+S < +W (Gain domain)

. (3.7)

That is, as daylight time lengthens throughout the winter, while lengthens and then shortens over the spring/summerperiod, investors mood in winter should be better than their spring/summer mood.19

Nonlinear least squares estimation results of Eq. (3.1), with a partition of according to (3.2) and (3.3), are reportedin Table 2. The corresponding PWFs are depicted in Figs. 3 and 4. Consistent with the accumulated experimentalevidence, our market-based parameter estimates indicate that the PWFs are regressive and inverse S-shaped, as all the estimates are significantly smaller than unity, ranging from 0.500 to 0.859 (standard errors not exceeding 0.056).Moreover, the utility (value) function exhibits loss aversion, with estimated at 1.262, significantly larger than one(standard error of 0.094), and diminishing marginal sensitivity, with estimated at 0.845, significantly smaller thanone (standard error of 0.018).

17 Note that daylight time is shorter in the fall than in the spring/summer period. Thus, this alternative hypothesis strengthens in case daylightduration, in addition to changes in daylight time, affects investors mood.18 Clearly, the average number of daylight hours is the same in both seasons, so the possibility that daylight duration affects mood is inconsequential.19 Note, however, that as daylight time is shorter in winter than in the spring/summer period, the possibility that daylight duration has an effect on

investors mood weakens this alternative hypothesis.


Fig. 3. The figure depicts the estimated gain-domain mood-dependent PWFs based on the results reported in Table 2. Investors mood is proxied byan established medical phenomenon, seasonal affective disorder (SAD). Fall, spring/summer, and winter PWFs are denoted by the letter F, S, andW, respectively.

Fig. 4. The figure depicts the estimated loss-domain mood-dependent PWFs based on the results reported in Table 2. Investors mood is proxied byan established medical phenomenon, seasonal affective disorder (SAD). Fall, spring/summer, and winter PWFs are denoted by the letter F, S, andW, respectively.


Table 2The SAD effectparameter estimates

Coefficient Estimated value

F 0.500 (0.029)S

0.568 (0.031)W 0.609 (0.024)+F 0.762 (0.051)+S

0.788 (0.047)+W 0.859 (0.056) 1.262 (0.094) 0.845 (0.018)The table presents nonlinear least squares estimation results of the value and probability weighting functions. Investors mood is proxied by anestablished medical phenomenon, seasonal affective disorder (SAD). The different parameters are delineated in Section 3.2. The sample extendsfrom December 1987 through December 1995. White heteroskedasticity-consistent standard errors appear in parentheses.

Table 3The SAD effecthypotheses testing

Null hypothesis F-statistic Prob (%)Panel (a): Joint hypotheses

F = S = W +F = +S = +W 4.92 0.06F = S +F = +S 4.12 1.66F = W +F = +W 6.65 0.14S

= W +S = +W 0.77 46.26

Coefficient difference Estimated value F-statistic (%)Panel (b): Simple hypotheses

S

F 0.068 3.026 (4.11%)W F 0.109 9.996 (0.08%)W S 0.041 1.289 (12.83%)+S

+F 0.026 0.114 (36.79%)+W +F 0.097 1.594 (10.36%)+W +S 0.071 1.378 (12.04%)

The table details hypotheses testing results for the case where seasonal affective disorder (SAD) serves as the mood proxy. Panel (a) shows theresults of testing joint hypotheses on the seasons PWF coefficients. Prob is the F-statistics associated p-value. Panel (b) shows the results of testingsimple (i.e., one-difference) hypotheses on the seasons PWF coefficients. One-tailed p-values appear in parentheses.

We turn now to deal with our conjecture that bad mood induces more regressive probability distortions. Indeed, ourestimates unequivocally support the conjecture that mood affects the formation of the PWFs, as all the estimates areordered as predicted: F <

S <

W and

+F <

+S <

+W .

The results of joint hypotheses comparing the seasons PWF coefficients are reported in panel (a) of Table 3.Indeed, the joint null hypothesis, F = S = W +F = +S = +W , is firmly rejected with an F-value of 4.92 (p-value of 0.06%). The joint pair-wise nulls, F = S +F = +S , F = W +F = +W , are also soundly rejected withrespective F-values of 4.12 and 6.65 (respective p-values of 1.66% and 0.14%). The pair-wise null S = W +S =+W , however, has an F-value of only 0.77 (p-value of 46.26%).20

Panel (b) of Table 3 reports the results of simple (i.e., one-difference) hypotheses on the PWF coefficients. Theone-tail p-values of the pair-wise seasonal differences in the estimates range from 0.08% to 36.79% and are generallymore significant in the loss domain. As expected, the strongest alternative hypothesis is the one comparing the PWFsof the fall and winter seasons. The larger, more significant, differences in the estimates observed in the loss-domain

20 Recall that the low F-value of the comparison between summer and winter is consistent with the presence of an effect of daylight duration (inaddition to daylight time changes) on investors mood. To test for this effect, we re-estimate Eq. (3.1), restricting F = W and +F = +W , to yieldtwo periods which differ in their daylight duration, but contain identical daylight time changes. Consistent with our expectations, the fall-winterPWFs are found significantly more regressive than the springsummer PWFs.


Table 4The cloud coverage effectparameter estimates

Coefficient Estimated value

L 0.502 (0.030)M 0.559 (0.024)H 0.621 (0.036)+L 0.740 (0.054)+M 0.801 (0.041)+H 0.878 (0.074) 1.278 (0.094) 0.839 (0.018)The table presents nonlinear least squares estimation results of the value and probability weighting functions. Investors mood is proxied by ameasure of cloudiness. The different parameters are delineated in Section 3.2. The sample extends from December 1987 through December 1995.White heteroskedasticity-consistent standard errors appear in parentheses.

relatively to the differences in the gain-domain PWFs corroborate our expectations regarding investors behavior. Theyprovide evidence that the investors increase the subjective probabilities of large losses when they are in bad mood, anddo so more profoundly than on the subjective probabilities of possible gain outcomes.

3.2.2. Sky coverage as a mood proxyWe turn now to the case where the sky coverage variable (SCSS) serves as a mood proxy. We partition and +

as follows:

= L L + M M + H H, (3.8)+ = +L L + +M M + +H H, (3.9)

where L = 1 indicates total cloud coverage (i.e., days where SCSS = 10), M = 1 indicates days where SCSS =3, . . . , 9, and H = 1 indicates days where SCSS = 0, 1, 2.

If PWFs are independent of investors mood, and + should not depend on the degree of sky coverage, and thus

H0(SCSS) :

{L = H = M (Loss domain)+L = +H = +M (Gain domain)

. (3.10)

If, alternatively, investors mood as proxied by SCSS affects investors PWFs, then

HA4(SCSS) :

{L <

M <

H (Loss domain)

+L < +M <

+H (Gain domain)

. (3.11)

Nonlinear least squares estimation results of Eq. (3.1) with the partition as in (3.8) and (3.9) are reported inTable 4. The corresponding PWFs are depicted in Figs. 5 and 6.

The main message communicated by the results clearly confirms the analysis with the SAD variable. In particular, thecloud cover-dependent PWFs are regressive and inverse S-shaped (all estimates are below 0.878), and the estimatedutility function is concave for gains, convex for losses (estimated of 0.839) and manifests loss aversion ( estimatedat 1.278).

Similar to the picture emerging from the SAD variable analysis, the estimates are perfectly ordered as predictedby the mood-affecting-PWFs hypothesis. The gain and loss PWFs are more regressive, the more cloud-covered the skyis, i.e., L <

M <

H and

+L <

+M <

+H .

The joint hypotheses comparing the estimated PWFs are reported in Panel (a) of Table 5. Similar to the case ofSAD as a mood proxy, the joint null hypothesis, L = M = H+L = +M = +H , is rejected (F-value of 2.47, p-value of 4.37%). The joint pair-wise nulls, L = H +L = +H , L = M +L = +M , and M = H +M = +H ,have respective F-values of 3.76, 2.05, and 1.12 (prob. = 2.37%, 12.91%, and 32.61%). As expected, the strongestalternative hypothesis is the one comparing the PWFs in the two extreme mood states (i.e., L and H).


Fig. 5. The figure depicts the estimated loss-domain mood-dependent PWFs based on the results reported in Table 4. Investors mood is proxied by ameasure of cloudiness (SCSS). The PWFs are denoted by the letters L, M, and H. L = 1 indicates total cloud coverage (i.e., days where SCSS = 10),M = 1 indicates days where SCSS = 3, . . . , 9, and H = 1 indicates days where SCSS = 0, 1, 2.

Fig. 6. The figure depicts the estimated gain-domain mood-dependent PWFs based on the results reported in Table 4. Investors mood is proxiedby a measure of cloudiness (SCSS). The PWFs are denoted by the letters L, M, and H. L = 1 indicates total cloud coverage (i.e., days whereSCSS = 10), M = 1 indicates days where SCSS = 3, . . . , 9, and H = 1 indicates days where SCSS = 0, 1, 2.


Table 5The cloud coverage effecthypotheses testing

Null hypothesis F-statistic Prob (%)Panel (a): Joint hypotheses

L = M = H +L = +M = +H 2.47 4.37L = M +L = +M 2.05 12.91L = H +L = +H 3.76 2.37M = H +M = +H 1.12 32.61

Coefficient difference Estimated value F-statistic

Panel (b): Simple hypothesesM L 0.057 2.605 (5.34%)H L 0.119 7.157 (0.38%)H M 0.062 2.232 (6.78%)+M +L 0.061 0.710 (19.98%)+H +L 0.138 2.342 (6.32%)+H +M 0.077 1.082 (14.92%)

The table details hypotheses testing results for the case where cloud coverage serves as the mood proxy. Panel (a) shows the results of testingjoint hypotheses on the estimated PWF coefficients. Prob is the F-statistics associated p-value. Panel (b) shows the results of testing simple (i.e.,one-difference) hypotheses on the estimated PWF coefficients. One-tailed p-values appear in parentheses.

Panel (b) of Table 5 reports the results of the simple hypotheses on the estimated coefficients. The one-tail p-values of the differences in estimates between each two SCSS classes range from 0.38% to 19.98%, providing morepronounced significance levels than those in the SAD analysis. As in the SAD case, the differences in the loss domainare more significant than those in the gain domain. The increased significance of the differences in the loss domainimply, as expected, that the investors increase the subjective probabilities of large losses when they are in bad mood.

4. Conclusions

This paper questioned whether mood influences investors choices. In particular, in a setup where physical proba-bilities are distorted by way of a probability weighting function, characteristic of many non-expected utility theories,we attempted to show how emotional states influence these PWFs. For that purpose a market approach utilizing realUS market price data, was developed and implemented.

Psychological evidence suggests that bad mood/depression distorts individuals conscious thinking and behaviorand is associated with a more unrealistic, less rational behavior (at least in meaningful contexts with significant con-sequences). In a cumulative prospect theory setup, using information on marginal utilities and subjective probabilitiesembedded in option prices, we tested whether adverse mood states are associated with more pronounced deviationsfrom expected utility. To proxy investors mood, we relied on two proxy measures, an established medical phenomenonknown as the seasonal affective disorder (SAD), and cloud coverage.

Using repeated cross-sections of European call options written on the market index, we were able to recoverpreference information on a state-by-state basis to find, as conjectured, statistical evidence indicating that bad moodcauses investors to systematically distort their PWFs, over and above the distortion caused when in good mood.

We believe the novelty in the approach presented in this paper is twofold. First, on the theoretical aspect, we link therole of mood in the decision-making process to the building blocks of prospect theory and cumulative prospect theory.Specifically, the setup we provide enables detecting the effect of mood on investors PWFs, one of the key features innon-expected utility theories. Second, on the empirical aspect, we employ a new approach that enables estimating thePWFs using real market prices.

In a recent paper, Loewenstein (2000) concludes with the following remarks (p. 431), Visceral factors have impor-tant, but often underappreciated, consequences for behavior. . . To predict or make sense of viscerally driven behavior,it is necessary to incorporate visceral factors into models of economic behavior. Based on our large-gamble evidencefor interrelations between investors emotions and the way the physical environment is perceived when decisions aretaken, this studys conclusion in complete agreement with Loewenstein is that integrating considerations of moodinto the economic literature is particularly warranted.


Acknowledgements

We wish to thank Jens Jackwerth and Mark Rubinstein for the option data used in the empirical analysis of thispaper. We would like to thank the Editor, Morris Altman, for the helpful remarks.

Appendix A. SPDs and physical PDFs

A.1. State prices

Consider the option pricing interpretation of Cox and Ross (1976):

Ct = exp(RFt ) Et [max(0,u

Pt+,s

K)] = exp(RFt )

Pt,s max(0,

u

Pt+,s

K) dPut+,s, (A.1)

where Ct is the current price of a European call option with strike price K, RFt the risk-free rate till the optionsexpiration, the time-to-expiration, Et denotes the conditional expectation operator according to the risk-neutral PDF,Put+,s the s th state expiration-date underlying assets price, and Pt,s is the expiration-date risk-neutral probability ofthe s th state. Differentiating (A.1) with respect to K yields

Ct

K= exp(RFt )

K

Pt,s dPut+,s. (A.2)

Differentiating with respect to K once more yields

Qt(s, ) = exp(RFt ) Pt,s =2CtK2

K=Put+,s

, (A.3)

where (Breeden and Litzenberger, 1978)2CtK2

K=Put+,s

= lim0

Ct(Put+,s + ) Ct(Put+,s)2

Ct(Put+,s) Ct(Put+,s )

2. (A.4)

Practically, the s th state price is approximated by pricing a butterfly spread centered around K = Put+,sashort position of two European call options with strike prices K = Put+,s at a price of Ct(K = Put+,s) each, andlong two European call options with strike prices K = (Put+,s + ) and K = (Put+,s ) at respective prices ofCt(K = Put+,s + ) and Ct(K = Put+,s ). This position promises an expiration-date payoff of conditional onstate Put+,s occurring. Dividing by to normalize the payoff to unity, the s th state price is obtained:

[Ct(Put+,s + ) Ct(Put+,s)]

[Ct(Put+,s) Ct(Put+,s )]

. (A.5)

A.2. Time-varying physical state probabilitiesa GARCH model for the S&P500 return process

Let the return process be characterized by the following generalized auto-regressive conditional heteroskedasticity(GARCH) model (Bollerslev, 1986):

rt = 1 + 2 rt1 + t, (A.6)2t = 1 + 2 2t1 + 3 2t1, (A.7)

where rt logRt is the daily logarithmic rate of return, and 2t is the one-period ahead forecast variance based on pastinformation (the conditional variance). Having daily return observations ranging from July 1962 through December1995, the model is estimated using maximum likelihood, to yield (1, 2, 1, 2, 3).

After we estimate the model, we use Monte-Carlo simulation to determine the simple return, as follows. We let r0be the daily return on the day the cross-section of option prices is sampled, and let 0 and 20 be the forecasted valueof the error term (innovation) and the conditional variance, respectively. ( 1, 2, 3) along with (0, 20 ) determine, inaccordance with (A.7), 21 . Drawing the first return innovation 1, along with (1, 2) and r0 determines, in accordance


with (A.6), rs1. Updating the conditional variance and drawing the second return innovation, yields rs2. The procedurecontinues through the N th innovation (N < is the number of actual trading days). The N-day simulated log return isthen

Rt,i nN

rsi , (A.8)

and the N-day simulated simple return is

Rt,i = exp( Rt,i) 1. (A.9)We repeat this procedure to produce 100,000 simulated return realizations. The physical state-s probability, Pt(s, ),

is estimated as follows:

Pt(s, ) = 1100, 000 100,000

i=1I(Rt,i s), (A.10)

where I() is an indicator function that receives the value 1 whenever Rt,i falls within state s and zero otherwise.

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Mood impacts on probability weighting functions: "Large-gamble" evidenceIntroductionTheoryUtility functionDecision weightsTestable hypotheses

Empirical analysisThe dataMood-affected probability weighting functions-statistical resultsSAD as a mood proxySky coverage as a mood proxy

ConclusionsAcknowledgementsSPDs and physical PDFsState pricesTime-varying physical state probabilities-a GARCH model for the S&P500 return process

References

Documents

9. Mood Impacts on Probability Weighting Functions%3a “Large-gamble” Evidence