Investor Mood and Financial Markets

Journal of Economic Behavior & Organization 76 (2010) 267–282

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Journal of Economic Behavior & Organization

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Investor mood and financial markets

Hui-Chu Shu ∗

Department of International Business, China University of Science and Technology, No.245, Sec. 3, Academia Rd., Nangang Dist., Taipei City 115, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 4 December 2009Received in revised form 3 June 2010Accepted 10 June 2010Available online 25 June 2010

JEL classification:G11G12D81E44

Keywords:Investor moodAsset pricingBehavioral financeTime preferenceRisk attitude

a b s t r a c t

Numerous studies in recent decades have linked investor mood and financial market behav-ior, but most works have been empirical investigations. This paper bridges the gap betweenempirical findings and financial theory. By slightly modifying the Lucas (Lucas, R.E., 1978.Asset prices in an exchange economy. Econometrica 46, 1429–1445.) model, this studyshows how investor mood variations affect equilibrium asset prices and expected returns.

Analysis results indicate that both equity and bill prices correlate positively with investormood, with higher asset prices associated with better mood. Conversely, expected assetreturns correlate negatively with investor mood. Further, the mood effect on asset pricesincreases when investors are in a good mood, and mood variations exhibit a greater influ-ence on equity markets than on bill markets. Results of this study suggest that investor moodis a vital factor in equilibrium asset prices and returns, and integrating investor mood intoasset-pricing models helps to interpret the growing body of seemingly anomalous evidenceregarding investor behavior.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

This study attempts to link asset-pricing theory, empirical evidence, and psychological research to enhance knowledgeof the role of investor mood in financial markets. In conventional economic analysis, the significance of mood has long beenneglected, whether because its influence is perceived as transient and unimportant, or because investors are assumed tobe entirely rational. However, the traditional perspective is now under challenge. Ample evidence suggests that mood doessignificantly influence decision-making, especially when the decision involves risk and uncertainty.

The extent to which investor psychology influences economic behavior has been broadly studied in recent decades(see, e.g. the reviews of Hirshleifer, 2001; Daniel et al., 2002; Nofsinger, 2005; and Lucey and Dowling, 2005). Researcherssuggest that mood markedly affects judgment and decision-making, subsequently altering investor behavior. The mood orpsychological state of investors when making decisions can affect their preferences, risk assessments and rational cogitationsand, ultimately, their investment decisions. Therefore, financial decisions should vary with investor mood.

A strand of empirical research in behavioral finance has accumulated persuasive evidence that stock returns are relatedto mood proxy variables, such as weather (e.g. Saunders, 1993; Hirshleifer and Shumway, 2003; Krivelyova and Robotti,2003; Cao and Wei, 2005; Chang et al., 2006; Keef and Roush, 2007; and Shu and Hung, 2009), biorhythms (e.g. Kamstra etal., 2000; Kamstra et al., 2003; and Yuan et al., 2006), and beliefs (Dowling and Lucey, 2005). These studies argue that certainvariables affect the mood or emotions of investors and thus influence their decisions. Consequently, asset prices and returnsfluctuate with investor mood.

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268 H.-C. Shu / Journal of Economic Behavior & Organization 76 (2010) 267–282

However, compared to the extensive empirical evidence that mood affects financial markets, its effects on equilibriumasset prices are relatively unsubstantiated. Although some studies in the past decade have modeled investor psychology, mosthave focused on psychological bias (e.g. narrow-framing, overconfidence, representativeness heuristic, over- and under-reaction, ambiguity aversion and familiarity) or on concern about future feelings (e.g. loss- and disappointment-aversion)rather than on mood variations. Thus, the influence of shifting investor mood on equilibrium asset prices and returns remainsan open challenge.

Thus, this study attempts to fill the above gap in the literature by investigating the influence of investor mood variations onfinancial markets using a simple general equilibrium asset-pricing model. By slightly modifying the Lucas (1978) model, thisstudy demonstrates how slight mood variations can induce financial market fluctuation. Drawing on psychological literature,time preference and risk attitude are employed as mood factors. This analytical work shows that the model satisfactorilyexplains many financial market phenomena.

This study offers a general equilibrium perspective of the claims that better investor mood is associated with higher assetprices, that mood variations have a greater effect on asset prices when most investors are in a good mood than when theyare in a bad mood, and that mood influences equity markets more than it affects bill markets since investing in the formerentails increased complexity and uncertainty. Above analytical results are consistent with psychological concepts.

This preliminary study attempts to link traditional asset-pricing models, behavior finance, and psychology. Traditionalasset-pricing models have been argued to fail to account for historically observed asset returns as they ignore investorpsychology aspects, and empirical research in behavior finance have documented the impact of investor mood on financialmarkets (see the review of Hirshleifer, 2001; Lucey and Dowling, 2005) while lacking explicit economic theoretical expla-nation. If only traditional models, or behavioral finance, or psychology could provide a single but fragmentary knowledgefor the complexity of investor behavior. Standing along, each has its limitations. Merged together, their knowledge andinsights become more powerful, meaningful, and applicable to the reality. This work links the psychology literature withmood factors and shows that introducing mood factors into a traditional asset-pricing model adequately explains severalempirical findings in behavior finance research and also improves understanding of mood in financial markets.

Briefly, this study contributes to the literature by identifying the economic effects of investor mood variations on equi-librium asset prices and returns. In addition to connecting psychology research and asset-pricing theory, this study alsobridges the gap between theory and practical evidence. By associating theory, empirical findings, and psychology, this studyimproves understanding of the role of mood in financial markets. Above all, this study contributes to the growing literatureon mood and investor behavior (e.g. Loewenstein, 2000; Mehra and Sah, 2002; Lo and Repin, 2002; Falato, 2009).

The rest of this paper is organized as follows. Section 2 reviews pertinent psychological literature on how mood affectsjudgment, decision-making, time preferences and risk attitude. Section 3 then summarizes the empirical findings that thisstudy attempts to explain. Next, Section 4 introduces the proposed model and derives the closed-form expressions for theprices and expected returns of equities and bills. Section 5 analyzes the influence of mood variations on asset prices andreturns. Conclusions are finally drawn in Section 6, along with recommendations for future research.

2. Psychology literature review

2.1. Effect of mood on judgment and decision-making

Traditional economic theory assumes that people are always rational. However, the traditional perspective is arguablyunrealistic as it overlooks the influence of mood. Psychological research has amply documented the effects of mood onjudgment and decision-making and suggests that mood is an influential factor in preferences (Loewenstein, 1996; Mehraand Sah, 2002), cognitive processes (Isen, 2001), and in the integration of information (Estrada et al., 1997). Mood is arguablyan important focusing mechanism in economic decision-making (Etzioni, 1988), and good mood is associated with fast andefficient decision-making (Forgas, 1998).

Mood may cause decision-making to deviate from the optimum or from rationality. Loewenstein et al. (2001) developeda “risk-as-feelings” model that incorporated the influence of mood on decision-making. The model assumed that emotionalreactions can influence, and even override, rational cogitations on decisions involving risk and uncertainty, and anticipatedemotions influence the cognitive evaluation process and ultimately the decisions.

Of the numerous psychological theories of how mood affects perception, the one quoted most frequently by financialeconomists is misattribution: people tend to attribute their feelings to the wrong sources, which causes incorrect judgments(Frijda, 1988; Schwarz and Clore, 1983). For example, people in a good mood that is induced by good weather may uncon-sciously attribute this feeling to favorable life prospects. Schwarz and Clore (1983) found that people tend to rate their lifesatisfactions much higher on sunny days than on cloudy or rainy days, even though their well being does not change on adaily basis. Similarly, Wright and Bower (1992) showed that happy people are more optimistic and assign higher probabili-ties to positive events. Forgas and Ciarrochi (2001) asserted that people in a good mood assign a higher value to both actualand potential wealth. Accordingly, Nofsinger (2005) suggested that people in a good mood are more willing to invest in riskyassets than those in a bad mood, and vice versa.

Notably, the effect of mood on judgment and decision-making depends on the information environment and the com-plexity of the decision. According to the affect heuristic theory, using affective impression to make decisions is much easierthan judging probability when the decisions are complex or full of uncertainty (MacGregor et al., 2000). Finucane et al. (2000)

H.-C. Shu / Journal of Economic Behavior & Organization 76 (2010) 267–282 269

found that, people with abundant information tend to rely on simplified rules or heuristics that either use partial informationor deal with information in incomplete ways. The higher complexity of judgment usually impels people to weigh affectivecues more heavily than technical indicators. Forgas (1995) argued that decision characteristics like risk and uncertaintydetermine the importance of feelings. The higher the complexity and uncertainty of a situation, the greater the influenceof emotions on decision making is. Similarly, Conlisk (1996) claimed that cognitive constraints and excessive informationusually hinder people from making fully rational decisions under complex states. Hence, people are prone to make satisfyingrather than optimal decisions. Kaufman (1999) and Hanoch (2002) suggested that people rely on their emotions to makesatisfying decisions under bounded rationality.

Mood effects depend not only on decision characteristics, but also on mood status. People in a good mood are reportedlyeasily influenced by emotional factors. According to the “mood-as-information” theory (Schwarz, 1990), people tend tomake decisions that are congruent with their moods. Specifically, people in a good mood are prone to react to irrelevantinformation, whereas people in a bad mood tend to process information more carefully and to react more strongly totruly relevant news. Similarly, Schwarz and Bless (1991) claimed that good mood is usually associated with optimisticjudgments and tend to cause heuristic styles of information processing. Thus, good mood is associated with greater moodeffects.

Thus, psychological research suggests that mood significantly affects judgment and decision-making, even though peopleare usually unconscious of it. Especially, mood effect on judgment and decision-making are related to the uncertaintyand complexity of decisions, with higher uncertainty and complexity associated with a greater mood effect. Interestingly,compared to people in a bad mood, people in a good mood are more optimistic about their the future prospects and relymore on heuristic styles of information processing; they are more willing to invest in risky assets and are easily affected bymood factors.

2.2. Influence of mood on time preference and risk attitude

Time preference measures the marginal rate of substitution between current and future consumption and hence playsan important role in theories of investment and asset pricing. Since Samuelson’s (1937) discounted utility model, timepreference has conventionally taken as given or as constant.

However, psychological studies suggest that time preference varies with circumstance. Postponing consumption involvesself-control and is therefore related to mood and feelings. Loewenstein and Prelec (1992) demonstrated that time preferencediffers greatly in different decision domains. Becker and Mulligan (1997) developed a model to show that time preferenceis determined endogenously. In their framework, time preference is not a fixed parameter and can be affected by numerousvariables such as wealth, mortality, and uncertainty. As some of these variables are random by nature, individual timepreferences may fluctuate over time.

Loewenstein (2000) further argued that visceral factors such as mood play a critical role in intertemporal choice. Assurveys of economic behavior generally report very low correlations between the different intertemporal trade-offs madeby the same individual, Loewenstein suggested that including the effect of visceral factors may help to explain inconsistenciesin concern for the future over time. In particular, understanding the mood people experience at the time of consuming iscritical for understanding and predicting the intertemporal trade-offs that they make.

Notably, preferences vary with time, but people tend to inaccurately predict the future sequence of their preferencesand systematically overstate the degree to which their future preferences resemble present preferences (Loewenstein et al.,2003). Given intertemporal choices, people tend to overestimate the duration and intensity of their present tastes. Thus,although time preference varies with time, people tend to discount future cash flow by the subjective discount rate of thedecision moment.

Additionally, psychological research also suggests that mood can affect risk assessments and risk attitudes. Johnson andTversky (1983) found that misattribution of mood can influence perceived risk. That is, people may not realize that theirrisky decisions are influenced by mood.

Furthermore, Mann (1992) and Nygren et al. (1996) argued that mood affects individual perception and judgment ofrisk, while Isen and Geva (1987) claimed that mood alters risk preference. Loewenstein (2000) suggested that cognitiveevaluations of risks often diverge from emotional reactions to those risks. Therefore, considering the effect of immediateemotions on risky behavior can elucidate many otherwise anomalous risk-taking phenomena. Meanwhile, economic studiessuggest that people are more risk averse after an initial loss (Thaler and Johnson, 1990) or when they are depressed (Kamstraet al., 2003).

Specifically, risk attitude depends on mood status. People in a good mood reportedly underestimate risk and overestimatebenefit (Finucane et al., 2000), and are more willing than people in a bad mood to invest in risky assets (Nofsinger, 2005). Auet al. (2003) found that foreign exchange traders in a pleasant mood environment tend to be overconfident, take superfluousrisks, make worse decisions, and perform poorly. Conversely, traders in a bad mood environment are more conservative andmake better decisions.

Briefly, previous studies suggest that time preference fluctuates over time. However, when faced with intertemporalchoices, people often assume present preference do not change. Further, time preference and risk attitude are affected bymood and are thus good mood proxy variables. Finally, as people in a good mood tend to take more risks or to undervaluerisk, they are assumed to be less risk averse than people in a bad mood, and vice versa.


3. Evidence

As behavioral finance emerges in mainstream financial research, a growing number of financial studies have attemptedto link investor mood with stock prices. Such studies have investigated the possible relationship between stock prices andmood proxy variables such as weather and biorhythms. Drawing on psychological evidence, these empirical studies arguethat certain variables can cause broadly uniform fluctuations in the mood of large groups of people and might thereforeaffect their financial decisions and, ultimately, stock prices.

Weather, with the documented influence on moods (e.g. Goldstein, 1972; Cunningham, 1979; Sanders and Brizzolara,1982; Schwarz and Clore, 1983; Howarth and Hoffman, 1984; Parrott and Sabini, 1990; Watson, 2000) and behavior (Baronand Ransberger, 1978; Allen and Fisher, 1978; Cunningham, 1979; Schneider et al., 1980; Rotton and Cohn, 2000; Anderson,2001; Pilcher et al., 2002), has attracted much interest. It is well recognized that pleasant weather causes a good mood andnasty weather causes a bad mood (e.g. Schwarz and Clore, 1983; Keller et al., 2005). Weather variables known to correlatewith stock prices include sunshine, temperature, wind, and geomagnetic storms. The research in this area indicates thatpleasant weather triggers a good mood and hence induces a mood misattribution that guides investors to price stocksoptimistically, and vice versa.

Saunders (1993) as well as Hirshleifer and Shumway (2003) found that sunshine is highly correlated with stock returns.They claimed that sunny weather is associated with upbeat and optimistic investor mood, which makes investors more likelyto buy stocks. Substantial psychological evidence supports this argument. Sunshine is a significant weather-based influenceon mood and behavior. As hours of sunshine increase, depression (Eagles, 1994) and skepticism (Howarth and Hoffman,1984) decrease whereas optimism (Howarth and Hoffman, 1984) and general good mood (Persinger, 1975) increase.

Krivelyova and Robotti (2003) further identified a negative correlation between geomagnetic storms and stock returns.According to that study, individuals tend to sell stocks on days with geomagnetic storms, which they contended is driven byinvestors incorrectly attributing their bad mood to negative economic prospects rather than to adverse weather conditions.Consequently, high geomagnetic activity is followed by negative stock returns while stock returns increase during periodsof quiet geomagnetic activity. Similarly, temperature (Cao and Wei, 2005; Chang et al., 2006; and Keef and Roush, 2007) andwind (Keef and Roush, 2007; Shu and Hung, 2009) reportedly have a negative influence on stock prices.

In addition to weather effects, another branch of research indicates that biorhythms (the body’s natural biological cycles)also correlate with stock returns. As in weather-effect research, this body of literature examines whether a widely fluctuatingmood induced by biorhythms is misattributed by investors and allowed to inform the stock investment decision. For instance,Kamstra et al. (2000) found that stock returns on Mondays following daylight savings time changes are significantly lowerthan those on other Mondays. Their results suggested that the anxiety induced by interrupted sleep patterns may temporarilyincrease risk aversion and trigger stock sales, thereby causing falling prices.

A further study by Kamstra et al. (2003) found that the seasonal affective disorder (SAD) explains seasonal variationsin stock returns. Kamstra et al. asserted that depression induced by longer winter nights makes investors more risk-averseand unwilling to hold stocks. Thus, autumn to winter is associated with increasingly negative returns as the length of nightincreases while winter to spring is associated with increasingly positive returns as the length of night decreases. Similarly,Yuan et al. (2006) found that stock returns are significantly lower on the days around a full moon than on the days arounda new moon, and argued that the depressed mood associated with a full moon makes investors value stocks less and thusinduces lower returns during full moon periods. Additionally, Ariel (1990) claimed that high pre-holiday returns can beinterpreted by good investor mood due to the expectation of having a holiday.

Notably, the strength of the mood effect on stock markets may vary with investor mood status. Dowling and Lucey (2005)investigated the relationship between eight mood proxy variables and daily stock returns and found that the relationshipbetween mood proxy variables and equity returns is more pronounced in times of positive recent market performance. Theysuggested that people in a good mood are more likely to allow irrelevant mood factors to influence their decisions.

The strength of mood effects also depends on the complexity of the decision. For instance, small stocks and closed-endfunds exhibit larger prices anomalies, probably due to the greater reliance on emotional decision-making of individualinvestors, who have a large influence on the pricing of these assets (Lee et al., 1991; Chopra et al., 1993). As investmentdecisions are more complex and risky for individual investors than for professional investors, and since greater uncertaintyis linked with greater reliance on mood and feelings in the decision-making process (Forgas, 1995), the assets mainly ownedby individual investors are more likely to be influenced by investor mood than those mainly owned by institutional investors.

Thus, empirical studies indicate that investor mood significantly influences stock prices and that better mood is associatedwith higher prices, and vice versa. Additionally, people in a good mood are more easily affected by irrelevant mood factors,and the effect of mood increases when the decision is complex and uncertain. However, although these empirical findingscoincide with the psychological argument, a economic model is still needed to explain the phenomena. Theoretical supportis needed in order for research into the mood effect on investor behavior to progress and to become paradigmatic, which isthe motivation for this study.

4. Proposed model

This analysis is based on a simple general equilibrium model. The simplicity of the model not only facilitates expositionand derives explicit conclusions, it also suits the questions posed. This analysis employs a variation of the Lucas (1978) pure


exchange model, while assumes that the growth rate of consumption, rather than consumption level, follows a Markovprocess. Consider a closed economy with a representative agent. In period t, the agent wishes to maximize the expectedutility

Ut = Et

⎡⎣

∞∑j=0

ˇju(ct+j)]

⎤⎦ (1)

where ct is the consumption in period t, u( ) is the period utility function, Et[ ] is an expectations operator conditional upon theinformation available at time t, and ˇ ∈ (0, 1) is the subjective discount factor representing time preference. The subjectivediscount factor is used to discount the individual flow of future expected utilities and can thus be considered the marginalrate of substitution between two successive flows of expected utilities. Restated, ˇ represents the patience of the individualfor future consumption, with higher ˇ meaning more patient.

To ensure a stationary equilibrium return process, the period utility function is specified as the constant relative riskaversion (CRRA) class:

u(ct) = c1−˛t

1 − ˛, 0 < ˛ < ∞ (2)

where ˛ represents the parameter of relative risk aversion. When ˛ equals one, the utility function is defined as logarithmic.Thus, this utility function is the standard time-separable iso-elastic utility function, which satisfies the following strictlyincreasing and concave condition: ∂u/∂c > 0;∂2

u/∂2c < 0.

At an optimum, the equilibrium equity return Rt+1 must satisfy the Euler equation:

Et

[ˇ

u′(ct+1)u′(ct)

Rt+1

]= 1 (3)

where Rt+1 = (pt+1 + dt+1)/pt, pt is the ex-dividend share price in period t, dt is the dividend payout in that period, and u′( )denotes the derivative of u with respect to its argument. This definition ensures that the economy is arbitrage-free and thelaw of one price holds.

For risk-free bills, the Euler equation becomes

Et

[ˇ

u′(ct+1)u′(ct)

RBt+1

]= 1 (4)

with

RBt+1 = 1/pB

t (5)

where pBt is the time t price of a one-period risk-free bill that pays one unit of consumption good in period t + 1. Eqs. (3) and

(4) follow from an extension of Lucas (1978) by Mehra (2003).Assume that only one productive unit is producing the consumption good, that the good is perishable, and that one equity

share is competitively traded. As only one productive unit is considered, the return on this share equals the market returns.In equilibrium, all output is consumed in the same period it is produced, no other source of the consumed good is available,and dividend payout in that period equals output in period t. Hence, ct = dt, and it follows that ct+1/ct = dt+1/dt = xt+1, wherext+1 represents the consumption (dividend) growth rate which is assumed to be identically and independently distributed.So that

u′(ct+1)/u′(ct) = x−˛t+1 (6)

4.1. Closed-form solutions for bill prices and returns

Substituting Eqs. (5) and (6) into Eq. (4) yields

PBt = ˇEt

[u′(ct+1)u′(ct)

]= ˇEt(x−˛

t+1) (7)

So that

RBt+1 = 1

ˇEt(x−˛t+1)

(8)

Let the consumption growth rate x follow the geometric Brownian process:

dx

x= udt + �dW (9)

where W is the unit Weiner process. Thus, x follows the lognormal distribution with expected value u − (1/2)�2 and variance�2. The time interval is set to be 1 year.


Then, using the properties of the lognormal distribution, the price and return of the bill can be expressed in a closed-formmanner as follows:

PBt = ˇe−˛u+ ˛�2

2 + ˛2�22 (10)

RBt+1 = 1

ˇe˛u− ˛�2

2 − ˛2�22 (11)

The derivation of (10) and (11) is listed in Appendix A.

4.2. Closed-form solutions for equity prices and returns

Let wt denote the price-dividend ratio, therefore, wt ≡ pt/dt . Since pt is homogeneous of degree 1 in dt, it follows thatpt = wtdt and pt+1 = wt+1dt+1. As Rt+1 = (pt+1 + dt+1)/pt, which can be rewritten as

Rt+1 = (1 + wt+1)xt+1

wt(12)

Thus, substituting (12) into (4) yields:

wt =ˇE(x1−˛

t+1 )

1 − ˇE(x1−˛t+1 )

= ˇyt+1

1 − ˇyt+1(13)

where yt+1 ≡ E(x1−˛t+1 ).

Substituting the definition of wt, that is, wt ≡ pt/dt into (13) yields the closed-form solution of equity prices:

pt = dtˇek

1 − ˇek(14)

where k ≡ (1 − ˛) (u − ˛�2

2 )Algebraic operation yields:

E(Rt+1) = 1ˇ

e˛u+ ˛�22 − ˛2�2

2 − �22 (15)

The derivation of (15) is listed in Appendix A.

5. The economic meaning of variations in mood factors

Since psychological experiments have documented the effects of mood on risk attitude and time preference, this analysisuses these two variables as mood factors to explore the effect of mood. To elucidate the effects of small mood variations onequilibrium asset prices and expected returns, both bill and equity prices and expected returns are partially differentiatedto mood factors to examine the effects of mood.

5.1. Effects of varying mood factors on asset prices

This section investigates how slight variations in mood factors affect asset prices. Based on (10) and (14), the followingis derived:

∂PBt

∂ˇ= e−˛u+ ˛�2

2 + ˛2�22 (16)

∂PBt

∂˛= −(u − �2

2− ˛�2)ˇe−˛u+ ˛�2

2 + ˛2�22 (17)

∂Pt

∂ˇ= dt

e(1−˛) (u− ˛�22 )

(1 − ˇe(1−˛) (u− ˛�22 ))

2(18)

∂Pt

∂˛= −dtˇ(u + �2

2− ˛�2)

ek

(1 − ˇek)2(19)


Table 1Variations in bill prices induced by time preference.

˛

1.01 2 3 4 5 6 7 8 9 10

∂PBt

∂ˇ0.983 0.968 0.954 0.942 0.931 0.921 0.912 0.905 0.899 0.893

This table displays the variations in bill prices that are caused by variations in the time preference. The values in this table are obtained by partiallydifferentiating bill prices to the subjective discount factor as expressed in Eq. (16):

∂PBt

∂ˇ= e−˛u+(˛�2/2)+(˛2�2/2)

Table 2Variations in equity prices induced by the time preference.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 0.025 0.024 0.024 0.023 0.023 0.023 0.022 0.022 0.022 0.0220.2 0.031 0.030 0.030 0.029 0.029 0.028 0.028 0.027 0.027 0.0270.3 0.041 0.040 0.038 0.037 0.037 0.036 0.035 0.035 0.034 0.0340.4 0.056 0.053 0.052 0.050 0.049 0.047 0.046 0.045 0.044 0.0440.5 0.080 0.076 0.073 0.070 0.067 0.065 0.063 0.062 0.060 0.0590.6 0.125 0.117 0.110 0.105 0.100 0.096 0.092 0.089 0.087 0.0850.7 0.222 0.203 0.186 0.173 0.162 0.153 0.146 0.140 0.135 0.1310.8 0.499 0.432 0.381 0.341 0.310 0.286 0.266 0.250 0.237 0.2270.9 1.993 0.475 1.170 0.960 0.813 0.708 0.629 0.570 0.526 0.4910.92 3.112 2.165 1.622 1.284 1.059 0.903 0.790 0.707 0.645 0.5990.94 5.524 3.438 2.396 1.805 1.437 1.192 1.022 0.900 0.811 0.7450.96 12.394 6.278 3.892 2.721 2.058 1.646 1.373 1.185 1.050 0.9530.98 49.151 14.924 7.395 4.565 3.191 2.419 1.942 1.628 1.413 1.2620.99 193.26 28.072 11.249 6.282 4.145 3.027 2.368 1.949 1.669 1.476

This table displays the variations in equity prices that are caused by variations in the time preference. The values in this table are obtained by partiallydifferentiating equity prices to the time preference as expressed in Eq. (18):

∂Pt

∂ˇ= dt

e(1−˛) (u− ˛�22 )

(1 − ˇe(1−˛) (u− ˛�22 ))

2

Tables 1–4 and the corresponding Figs. 1–4 summarize the results of comparative static analysis. This analysis uses theparameter values u = 0.018 and � = 0.035 obtained by Mehra and Sah (2002),1 and the dividend is set at 0.02.2

In Tables 1 and 2, the partial derivative of asset prices with respect to ˇ is positive, meaning that both bill and equityprices increase as the agent becomes more patient. Thus, a higher ˇ is associated with higher asset prices. Fig. 1 shows thedecreasing and nearly linear relationship between ∂PB

t /∂ˇ and ˛. Accordingly, increment in bill prices induced by ˇ increaseswhen the agent is less risk averse.

Meanwhile, the influence of time preference on equity prices is substantial for generally proposed ˇ, namely ˇ > 0.9. Forexample, in the case of ˇ = 0.98, when ˛ = 2, the increment in equity prices induced by ˇ is 14.9 times the increment in ˇ, andthe multiplier rises to 28 when ˇ = 0.99, indicating that a slight shift in time preference can significantly change equity prices.

Tables 3 and 4 show that the partial derivative of bill and equity prices with respect to ˛ is negative. As expected, theprices of both assets fall as the agent becomes more risk averse.

Notably, mood factors exert more influence on equity prices than on bill prices for ˇ > 0.9. For example, when ˛ = 2 andˇ = 0.98, the variation in equity prices induced by ˛ is about seventeen times that in bill prices. Similarly, in the same case,when ˇ changes, the induced increment in equity prices is about fifteen times that in bill prices. These mathematical resultssuggest that equity prices are more easily influenced by mood variations than are bill prices.

Finally, all of the figures reveal an interesting phenomenon. The effects of both mood factors are enhanced in situationsof low ˛ and high ˇ, indicating that an agent with high time preference and low risk aversion is easily affected by moodfactors when pricing assets.

Thus, the above analytical results reveal patterns in line with the empirical findings. First, better investor mood is asso-ciated with higher asset price, since an increased time preference and a decreased risk-aversion coefficient induce higherprices. Psychological research indicates that people in a good mood tend to underestimate risk (Finucane et al., 2000) or totake more risk (Au et al., 2003). Thus, a reasonable inference is that investors in a good mood are less risk averse, and vice

1 Mehra and Sah (2002) obtained the parameter values from data on U.S. per capita real consumption of non-durables and services.2 Changing the dividend rate would induce the partial derivative of asset prices to mood factors to shift the same percentage as the variation in the

dividend rate.


Table 3Variations in bill prices induced by the relative risk-aversion coefficient.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 −0.0016 −0.0014 −0.0013 −0.0012 −0.0010 −0.0009 −0.0008 −0.0007 −0.0006 −0.00050.2 −0.0032 −0.0029 −0.0026 −0.0024 −0.0021 −0.0018 −0.0016 −0.0014 −0.0011 −0.00090.3 −0.0048 −0.0043 −0.0039 −0.0035 −0.0031 −0.0028 −0.0024 −0.0021 −0.0017 −0.00140.4 −0.0064 −0.0058 −0.0052 −0.0047 −0.0042 −0.0037 −0.0032 −0.0027 −0.0023 −0.00180.5 −0.0079 −0.0072 −0.0065 −0.0059 −0.0052 −0.0046 −0.0040 −0.0034 −0.0029 −0.00230.6 −0.0095 −0.0087 −0.0079 −0.0071 −0.0063 −0.0055 −0.0048 −0.0041 −0.0034 −0.00280.7 −0.0111 −0.0101 −0.0092 −0.0082 −0.0073 −0.0065 −0.0056 −0.0048 −0.0040 −0.00320.8 −0.0127 −0.0116 −0.0105 −0.0094 −0.0084 −0.0074 −0.0064 −0.0055 −0.0046 −0.00370.9 −0.0143 −0.0130 −0.0118 −0.0106 −0.0094 −0.0083 −0.0072 −0.0062 −0.0051 −0.00410.92 −0.0146 −0.0133 −0.0120 −0.0108 −0.0096 −0.0085 −0.0074 −0.0063 −0.0053 −0.00420.94 −0.0149 −0.0136 −0.0123 −0.0111 −0.0099 −0.0087 −0.0076 −0.0065 −0.0054 −0.00430.96 −0.0152 −0.0139 −0.0126 −0.0113 −0.0101 −0.0089 −0.0077 −0.0066 −0.0055 −0.00440.98 −0.0156 −0.0142 −0.0128 −0.0115 −0.0103 −0.0091 −0.0079 −0.0067 −0.0056 −0.0045

This table displays the variations in bill prices that are caused by variations in the relative risk-aversion coefficient. The values in this table are obtained bypartially differentiating bill prices to the relative risk-aversion coefficient as expressed in Eq. (17):

∂PBt

∂˛= −(u − �2

2− ˛�2)ˇe−˛u+ ˛�2

2 + ˛2�22

Table 4Variations in equity prices induced by the relative risk-aversion coefficient.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 −4.3E−05 −3.9E−05 −3.5E−05 −3.2E−05 −2.9E−05 −2.6E−05 −2.2E−05 −1.9E−05 −1.7E−05 −1.4E−050.2 −0.00011 −0.00010 −0.00009 −0.00008 −0.00007 −0.00006 −0.00006 −0.00005 −0.00004 −0.000030.3 −0.0002 −0.0002 −0.0002 −0.0002 −0.0001 −0.0001 −0.0001 −0.0001 −0.0001 −0.00010.4 −0.0004 −0.0003 −0.0003 −0.0003 −0.0002 −0.0002 −0.0002 −0.0002 −0.0001 −0.00010.5 −0.0007 −0.0006 −0.0005 −0.0005 −0.0004 −0.0004 −0.0003 −0.0003 −0.0002 −0.00020.6 −0.0013 −0.0011 −0.0010 −0.0009 −0.0007 −0.0006 −0.0006 −0.0005 −0.0004 −0.00030.7 −0.0027 −0.0023 −0.0019 −0.0017 −0.0014 −0.0012 −0.0010 −0.0009 −0.0007 −0.00060.8 −0.0069 −0.0056 −0.0046 −0.0037 −0.0031 −0.0026 −0.0021 −0.0018 −0.0014 −0.00120.9 −0.0312 −0.0062 −0.0157 −0.0118 −0.0091 −0.0072 −0.0057 −0.0045 −0.0036 −0.00280.92 −0.0497 −0.0322 −0.0223 −0.0162 −0.0122 −0.0094 −0.0073 −0.0057 −0.0045 −0.00350.94 −0.0902 −0.0522 −0.0336 −0.0233 −0.0169 −0.0126 −0.0096 −0.0075 −0.0058 −0.00450.96 −0.2067 −0.0974 −0.0558 −0.0358 −0.0247 −0.0178 −0.0132 −0.0100 −0.0077 −0.00580.98 −0.8369 −0.2364 −0.1083 −0.0613 −0.0390 −0.0267 −0.0191 −0.0141 −0.0105 −0.0079

This table displays the variations in equity prices that are caused by variations in the relative risk-aversion coefficient. The values in this table are obtainedby partially differentiating equity prices to the relative risk-aversion coefficient as expressed in Eq. (19):

∂Pt

∂˛= −dtˇ(u + �2

2− ˛�2)

ek

(1 − ˇek)2

Fig. 1. Variations in bill prices induced by the time preference.


Fig. 2. Variations in equity prices induced by the time preference.

Fig. 3. Variations in bill prices induced by the relative risk-aversion coefficient.

Fig. 4. Variations in equity prices induced by the relative risk-aversion coefficient.


Table 5Variations in bill returns induced by the time preference.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 −101.708 −103.285 −104.776 −106.157 −107.425 −108.575 −109.604 −110.506 −111.280 −111.9210.2 −25.427 −25.821 −26.194 −26.539 −26.856 −27.144 −27.401 −27.627 −27.820 −27.9800.3 −11.301 −11.476 −11.642 −11.795 −11.936 −12.064 −12.178 −12.278 −12.364 −12.4360.4 −6.357 −6.455 −6.548 −6.635 −6.714 −6.786 −6.850 −6.907 −6.955 −6.9950.5 −4.068 −4.131 −4.191 −4.246 −4.297 −4.343 −4.384 −4.420 −4.451 −4.4770.6 −2.825 −2.869 −2.910 −2.949 −2.984 −3.016 −3.045 −3.070 −3.091 −3.1090.7 −2.076 −2.108 −2.138 −2.166 −2.192 −2.216 −2.237 −2.255 −2.271 −2.2840.8 −1.589 −1.614 −1.637 −1.659 −1.679 −1.696 −1.713 −1.727 −1.739 −1.7490.9 −1.256 −1.275 −1.294 −1.311 −1.326 −1.340 −1.353 −1.364 −1.374 −1.3820.92 −1.202 −1.220 −1.238 −1.254 −1.269 −1.283 −1.295 −1.306 −1.315 −1.3220.94 −1.151 −1.169 −1.186 −1.201 −1.216 −1.229 −1.240 −1.251 −1.259 −1.2670.96 −1.104 −1.121 −1.137 −1.152 −1.166 −1.178 −1.189 −1.199 −1.207 −1.2140.98 −1.059 −1.075 −1.091 −1.105 −1.119 −1.131 −1.141 −1.151 −1.159 −1.165

This table displays the variations in bill returns that are caused by variations in the time preference. The values in this table are obtained by partiallydifferentiating bill returns to the time preference as expressed in Eq. (15):

∂RBt+1

∂ˇ= − 1

ˇ2e˛u− ˛�2

2 − ˛2�22

versa. Further, since investors in a good mood are usually more optimistic about future prospects (Wright and Bower, 1992;Schwarz and Bless, 1991) and are more willing to invest (Nofsinger, 2005) than are investors in bad moods, a reasonableinference is that good mood is associated with high time preference. Thus, this result provides theoretical support for theargument of empirical studies that improved investor mood increases asset prices (e.g. Saunders, 1993; Hirshleifer andShumway, 2003; Krivelyova and Robotti, 2003; Kamstra et al., 2003; Yuan et al., 2006; Shu and Hung, 2009).

Second, mood factors, especially time preference, significantly influence asset prices. A slight change in time preferencecan cause great variation in equity prices. The variation in risk attitude has a smaller but still unignorable influence. Thus, ifcertain variables alter the time preference of investors even slightly, equity prices may shift significantly.

Third, mood variations have a greater effect on asset prices when investor mood is good as the variation in asset pricescaused by changing mood increases when the agent has lower risk aversion and higher time preference. As investors ina good mood are less risk averse and more patient, these results coincide with psychological argument that people in agood mood are more likely to allow mood-related factors to influence their decision-making process, and also support theempirical findings of Dowling and Lucey (2005).

Finally, mood variations have a stronger association with equity prices than with bill prices, which suggests that equityinvestments are more likely to be influenced by mood fluctuations than are bill investments. The psychological literaturesuggests that mood plays a more important role in decision-making when uncertainty and complexity is greater (Finucaneet al., 2000; Kaufman, 1999; and Hanoch, 2002). Compared to bill investing, equity investing involves much more risk anduncertainty and requires more information. Therefore, mood factors should exert a greater effect on equity investment

Table 6Variations in equity returns induced by the time preference.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 −101.772 −103.475 −105.097 −106.613 −108.019 −109.309 −110.480 −111.526 −112.444 −113.2310.2 −25.443 −25.869 −26.274 −26.653 −27.005 −27.327 −27.620 −27.881 −28.111 −28.3080.3 −11.308 −11.497 −11.677 −11.846 −12.002 −12.145 −12.276 −12.392 −12.494 −12.5810.4 −6.361 −6.467 −6.569 −6.663 −6.751 −6.832 −6.905 −6.970 −7.028 −7.0770.5 −4.071 −4.139 −4.204 −4.265 −4.321 −4.372 −4.419 −4.461 −4.498 −4.5290.6 −2.827 −2.874 −2.919 −2.961 −3.001 −3.036 −3.069 −3.098 −3.123 −3.1450.7 −2.077 −2.112 −2.145 −2.176 −2.204 −2.231 −2.255 −2.276 −2.295 −2.3110.8 −1.590 −1.617 −1.642 −1.666 −1.688 −1.708 −1.726 −1.743 −1.757 −1.7690.9 −1.256 −1.277 −1.297 −1.316 −1.334 −1.349 −1.364 −1.377 −1.388 −1.3980.92 −1.202 −1.223 −1.242 −1.260 −1.276 −1.291 −1.305 −1.318 −1.329 −1.3380.94 −1.152 −1.171 −1.189 −1.207 −1.222 −1.237 −1.250 −1.262 −1.273 −1.2810.96 −1.104 −1.123 −1.140 −1.157 −1.172 −1.186 −1.199 −1.210 −1.220 −1.2290.98 −1.060 −1.077 −1.094 −1.110 −1.125 −1.138 −1.150 −1.161 −1.171 −1.179

This table displays the variations in equity returns that are caused by variations in the time preference. The values in this table are obtained by partiallydifferentiating equity returns to the time preference as expressed in Eq. (22):

∂E(Rt+1)∂ˇ

= − 1ˇ2

e˛u+ ˛�22 − ˛2�2

2 − �22


Table 7Variations in bill returns induced by the relative risk-aversion coefficient.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 0.1643 0.1543 0.1437 0.1326 0.1210 0.1090 0.0966 0.0838 0.0708 0.05750.2 0.0821 0.0771 0.0718 0.0663 0.0605 0.0545 0.0483 0.0419 0.0354 0.02880.3 0.0548 0.0514 0.0479 0.0442 0.0403 0.0363 0.0322 0.0279 0.0236 0.01920.4 0.0411 0.0386 0.0359 0.0331 0.0302 0.0272 0.0241 0.0210 0.0177 0.01440.5 0.0329 0.0309 0.0287 0.0265 0.0242 0.0218 0.0193 0.0168 0.0142 0.01150.6 0.0274 0.0257 0.0239 0.0221 0.0202 0.0182 0.0161 0.0140 0.0118 0.00960.7 0.0235 0.0220 0.0205 0.0189 0.0173 0.0156 0.0138 0.0120 0.0101 0.00820.8 0.0205 0.0193 0.0180 0.0166 0.0151 0.0136 0.0121 0.0105 0.0089 0.00720.9 0.0183 0.0171 0.0160 0.0147 0.0134 0.0121 0.0107 0.0093 0.0079 0.00640.92 0.0179 0.0168 0.0156 0.0144 0.0132 0.0118 0.0105 0.0091 0.0077 0.00630.94 0.0175 0.0164 0.0153 0.0141 0.0129 0.0116 0.0103 0.0089 0.0075 0.00610.96 0.0171 0.0161 0.0150 0.0138 0.0126 0.0114 0.0101 0.0087 0.0074 0.00600.98 0.0168 0.0157 0.0147 0.0135 0.0123 0.0111 0.0099 0.0086 0.0072 0.0059

This table displays the variations in bill returns that are caused by variations in the relative risk-aversion coefficient. The values in this table are obtainedby partially differentiating bill returns to the relative risk-aversion coefficient as expressed in Eq. (21):

∂RBt+1

∂˛= 1

ˇ(u − �2

2− ˛�2)e˛u− ˛�2

2 − ˛2�22

according to the psychological argument. The above finding suggests that, when investor mood changes, investor moodinduces greater variations in the equity market than in the bill market.

5.2. Effects of varying mood factors on asset returns

Varying mood factors that affect equity and bill prices may also impact the expected returns of both assets. The magnitudeand direction of this influence is an interesting inquiry, as is discussed below.

Analogously, partially differentiating the asset returns to mood factors yields the following:

∂RBt+1

∂ˇ= − 1

ˇ2e˛u− ˛�2

2 − ˛2�22 (20)

∂RBt+1

∂˛= 1

ˇ(u − �2

2− ˛�2)e˛u− ˛�2

2 − ˛2�22 (21)

∂E(Rt+1)∂ˇ

= − 1ˇ2

e˛u+ ˛�22 − ˛2�2

2 − �22 (22)

Table 8Variations in equity returns induced by the relative risk-aversion coefficient.

˛

ˇ 1.01 2 3 4 5 6 7 8 9 10

0.1 0.1768 0.1672 0.1570 0.1462 0.1349 0.1231 0.1109 0.0983 0.0853 0.07200.2 0.0884 0.0836 0.0785 0.0731 0.0674 0.0616 0.0554 0.0491 0.0427 0.03600.3 0.0589 0.0557 0.0523 0.0487 0.0450 0.0410 0.0370 0.0328 0.0284 0.02400.4 0.0442 0.0418 0.0392 0.0365 0.0337 0.0308 0.0277 0.0246 0.0213 0.01800.5 0.0354 0.0334 0.0314 0.0292 0.0270 0.0246 0.0222 0.0197 0.0171 0.01440.6 0.0295 0.0279 0.0262 0.0244 0.0225 0.0205 0.0185 0.0164 0.0142 0.01200.7 0.0253 0.0239 0.0224 0.0209 0.0193 0.0176 0.0158 0.0140 0.0122 0.01030.8 0.0221 0.0209 0.0196 0.0183 0.0169 0.0154 0.0139 0.0123 0.0107 0.00900.9 0.0196 0.0186 0.0174 0.0162 0.0150 0.0137 0.0123 0.0109 0.0095 0.00800.92 0.0192 0.0182 0.0171 0.0159 0.0147 0.0134 0.0121 0.0107 0.0093 0.00780.94 0.0188 0.0178 0.0167 0.0156 0.0144 0.0131 0.0118 0.0105 0.0091 0.00770.96 0.0184 0.0174 0.0164 0.0152 0.0141 0.0128 0.0116 0.0102 0.0089 0.00750.98 0.0180 0.0171 0.0160 0.0149 0.0138 0.0126 0.0113 0.0100 0.0087 0.0074

This table displays the variations in equity returns that are caused by variations in the relative risk-aversion coefficient. The values in this table are obtainedby partially differentiating equity returns to the relative risk-aversion coefficient as expressed in Eq. (23):

∂E(Rt+1)∂˛

= 1ˇ

(u + �2

2− ˛�2)e˛u+ ˛�2

2 − ˛2�22 − �2

2


Fig. 5. Variations in bill returns induced by the time preference.

∂E(Rt+1)∂˛

= 1ˇ

(u + �2

2− ˛�2)e˛u+ ˛�2

2 − ˛2�22 − �2

2 (23)

Tables 5–8 and Figs. 5–8 present the comparative static results. Tables 5 and 6 show that, the partial derivatives of expectedasset returns with respect to ˇ are negative, which indicate a negative relationship between patience and expected returns.Thus, a higher ˇ induces a lower expected return. Intuitively, the more patient the agent is, the more willing the agentwould be to sacrifice current consumption to buy assets with lower expected returns. Additionally, as noted in Section 5.1,asset prices in period t rise as ˇ increases. Because asset prices follow a mean-reverting process, this increased period pricereduces the expected returns of the next period. Hence, these results are consistent with those in Section 5.1.

Notably, the negative influence of ˇ on expected asset returns exceeds one in all cases of ˇ considered in Tables 5 and 6,which implies that the decrement in asset returns induced by ˇ exceeds the increment in ˇ. Thus, a slight variation in timepreference induces a significant variation in expected returns.

Tables 7 and 8 reveal that ˛ is positively related to the expected returns for both assets. Hence, a higher risk-aversioncoefficient is associated with a higher expected return. The explanation for this result is that a high expected return inducesa risk averse agent to sacrifice consumption during this period to buy assets. Because increased ˛ reduces asset prices intime t, expected returns for the next period also increase.

In comparison, the influence of ˇ on expected returns is significantly larger than that of ˛. For example, when ˛ = 2 andˇ = 0.98, the variation in expected equity returns induced by ˇ is 1.077 times that of ˇ. However, in the same case, the

Fig. 6. Variations in equity returns induced by the time preference.


Fig. 7. Variations in bill returns induced by the relative risk-aversion coefficient.

Fig. 8. Variations in equity returns induced by the relative risk-aversion coefficient.

variation in ˛ induces just 0.017 times the variation in returns. Both mood factors also exert a greater influence on expectedequity returns than on bill returns, which reflects the greater susceptibility of equity returns to fluctuating investor mood.

In sum, the above analytical results indicate some notable patterns. First, investor mood is inversely related to expectedreturn as increased risk aversion and decreased time preference raise expected asset returns. This finding is economicallyintuitive and is consistent with the psychology literature. As psychological studies demonstrate that individuals in a badmood are pessimistic and unwilling to invest, high expected returns are required to attract them to invest.

Furthermore, time preference substantially influences expected returns. A change in time preference as small as 0.1 caninduce a 10 percent or larger variation in expected equity returns. Meanwhile, expected returns of both assets are much moresensitive to varying time preference than to varying risk attitude, and, compared to bill returns, expected equity returns aremore susceptible to mood factors.

6. Conclusion

Wisdom is cultivated by studying the past and by intelligently applying the lessons learned to the present. Importantly,integrating knowledge from other domains into financial and economic research can help to understand otherwise puzzlingaspects of financial markets. This study adopts a conventional asset-pricing model to show that mild investor mood variationshave rich implications for financial markets. Mood states, a well recognized factor in human perception and behavior, hasan impact on risk attitude and time preference and significantly affects the equilibrium of asset prices and returns. Thus,understanding mood factors can help to resolve some of the most prominent documented market anomalies.


To gain further insight into investor behavior, financial economists have growingly accepted psychological explanations.However, embedding investor mood in theoretical analyses is still at the early stage of research. This analysis provides ageneral equilibrium perspective over a broad set of important empirical findings regarding how investor mood fluctuationsaffect financial markets. Although the causal association between market anomalies and investor mood is well documentedin the behavioral finance literature, most reported data are empirical. This study fills the gap between practical findings andasset-pricing theory.

Specifically, this study offers a novel explanation for financial market volatility. The cause of over-volatility in equitymarkets has attracted much academic interest, but is still unsolved. This study suggests that considering varying investormood can help to explain violent fluctuations in equity markets. This study also suggests that varying time preference sub-stantially affects equity markets. Increased time preferences significantly increase equity prices and reduce expected equityreturns. Thus, if investor time preferences are even slightly altered by certain factors, the equity markets may accordinglyswing violently. The effect of varying risk attitude is smaller but still non-negligible.

Overall, this analysis provides a theoretical interpretation for how mood fluctuations influence asset pricing, and itsuggests that considering investor mood in asset-pricing models can help interpret the growing body of seemingly anomalousevidence in financial markets. As human behavior is motivated by both thought and feelings, neither should be ignored infinancial decision-making. The results of this study suggest that equally weighting thoughts and feelings goes a long waytoward interpreting investor behavior. However, for an enhanced understanding of how investor mood affects financial assetprices, further study is needed to determine what influences investor mood and emotion, how mood fluctuations spread,and why mood influences investor attention to certain groups of stocks. Further, if investor mood can predict asset pricemovements, it may be possible to measure overall investor mood in advance to predict market prices. If so, investors canbenefit from exploiting the prediction. Thus, designing and developing a measurement method is a promising area of futureresearch.

Appendix A.

A.1. Derivation of expressions (10) and (11)

If Y follows the lognormal distribution with expected value � and variance h2, then it is stated as: Y∼LN (�,h) for brevity.For a fixed value of parameter �, the expected value of Y� is:

E[Y�] = e��+ �2h22 (A1)

As x ∼ LN(u − (1/2)�2, �), then use (A1), it is obtained that

E(x�t+1) = e�(u− �2

2 )+ �2�22 = e�u− ��2

2 + �2�22 (A2)

Thus, substituting � = − ˛ into (A2) yields

E(x−˛t+1) = e−˛u+ ˛�2

2 + ˛2�22 (A3)

Then substituting (A3) into (7) yields the desired expression (10), and substituting (A3) into (8) yields the desiredexpression (11).

A.2. Derivation of expression (15)

From (12), we have

E(Rt+1) = Et

[(1 + wt+1)xt+1

wt

](A4)

Because price-dividend ratio is i.i.d., then substitute (13) into (A4) yields

E(Rt+1) = 1ˇ

× E(xt+1)yt+1

(A5)

Substituting � = 1 − ˛ into (A2) yields:

yt+1 = E(x1−˛t+1 ) = e(1−˛) (u− ˛�2

2 ) (A6)

As

E(xt+1) = eu− �22 (A7)

then, substituting (A6) and (A7) into (A5) yields expression (15).


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