Prospect theory for stock markets: Empirical evidence with time-series data

Journal of Economic Behavior & Organization 72 (2009) 835–849

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Journal of Economic Behavior & Organization journal homepage: www.elsevier.com/locate/jebo

Prospect theory for stock markets: Empirical evidence with time-series data夽 Wenlang Zhang a,∗ , Willi Semmler b a b

Research Department, Hong Kong Monetary Authority, 55th Floor, II International Finance Centre, 8 Finance Street, Central, Hong Kong Dept of Economics, Graduate Faculty, New School University, 65 Fifth Ave, New York, NY 10003, United States

a r t i c l e

i n f o

Article history: Received 28 July 2008 Received in revised form 2 August 2009 Accepted 4 August 2009 Available online 13 August 2009 JEL classification: C60 C61 D90

a b s t r a c t Based on the loss aversion model of asset pricing, this paper explores empirical evidence on the prospect theory for stock markets with time-series data. The analysis, using a state-space model, shows that previous gains and losses may have asymmetric effects on investment behavior, pointing to the possibility of break-even effects ignored by assetpricing models using prospect theory. © 2009 Elsevier B.V. All rights reserved.

Keywords: Prospect theory Loss aversion State-space model House-money effect Break-even effect

1. Introduction Consumption-based asset pricing models with time separable preferences have been shown to have serious difficulty matching financial market characteristics, such as risk-free interest rate, equity premium and the Sharpe ratio, to time series data. Therefore, recent development of asset pricing studies has turned to extensions of intertemporal models conjecturing that the difficulties to match real and financial time series characteristics may be related to the simplified structure of the basic model. In order to better match asset price characteristics of the model to data, economic research has extended the baseline stochastic growth model to include different utility functions, such as non-separable preferences of habit formation. Moreover, adjustment costs of investment have also been built into the model. Yet, models with habit formation or adjustment costs have been shown to improve equity premium and Sharpe ratio weakly, see Grüne and Semmler (2007) and Lettau and Uhlig (2000, 2002) for example. They do not generate enough co-variance of consumption growth with asset returns so as to match the data. Therefore, research in new directions is in great demand, and the work by Barberis et al. (2001) seems to be a notable progress in this field.

夽 The views in this paper are only those of the authors and should not be interpreted as the views of their institutions. ∗ Corresponding author. Tel.: +852 2878 1830; fax: +852 2878 1897. E-mail addresses: [email protected] (W. Zhang), [email protected] (W. Semmler). 0167-2681/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jebo.2009.08.003

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The work by Barberis et al. (2001) is based on the so-called “prospect theory” developed by Kahneman and Tversky (1979) to explore decisions under risk. According to this theory, economic agents are concerned with the changes in wealth rather than with its final state. Moreover, economic agents are more sensitive to losses than to gains. That is, an economic agent is more painful with a loss than he feels happy with a gain of the same size. It is further found that economic agents’ utility function is concave with respect to gains and convex with respect to losses, implying that economic agents are risk-averse with respect to gains and risk-seeking with respect to losses. This phenomenon has been called “loss aversion”. Another important concept used by Barberis et al. (2001) is the “house-money” effect explored by Thaler and Johnson (1990) in detail. The house-money effect implies that economic agents’ decision under risk can be affected by their gains and losses in the previous period. In case they have obtained some gains in the previous period, they would be more ready to accept risk in the current period. That is, they become (more) risk-seeking tomorrow in case they obtain some gains today. The effect of losses in the previous period, however, can be more complicated than that of gains. Although the experiments in Kahneman and Tversky (1979) show that people may become (more) risk-averse in the next period if they have some losses in the current period, this is not always true. That is, in some cases people may also become (more) risk-seeking in the next period even if they have some losses in the current period. This is described as the “break-even” effect. In other words, even if they have some losses in the previous period, people may be ready to accept further risks and accept gambles which offer them a chance to break even. Therefore, the effect of losses in the current period on the decision in the next period may not be so clear as that of gains. Barberis et al. (2001) do not consider the break-even effect and just assume that agents become (more) risk-seeking after gains and (more) risk-averse in the aftermath of losses because the break-even effect would make their model much more complicated. An important question is how to measure losses and gains. Barberis et al. (2001) measure losses or gains by the difference of returns from risky and risk-free assets. Barberis et al. (2001) show that loss aversion alone can not justify the high equity premium, large stock volatility and weak correlation between asset returns and consumption, and that these puzzles can be better explained by the house-money effect. House-money effect leads to time-varying risk-aversion which can affect economic agents’ investment behavior on stock markets each period. Employing an advanced dynamic programming algorithm, Grüne and Semmler (2008) study the model of Barberis et al. (2001) numerically and confirm their findings. While Barberis et al. (2001) and Grüne and Semmler (2008) study this issue at the theoretical level, we would like to shed some light on prospect theory and asset pricing at the empirical level.1 Although there exists some empirical research on prospect theory in the literature, most of them has been undertaken with micro-panel data. As previous gains and losses can affect the decision in the current period, the persistence of investment behavior on stock markets may be affected by returns in the past. Against this background, we will explore some empirical evidence on the persistence of investment behavior on stock markets. In particular, we will explore whether there is any break-even effect in investment behavior on stock markets using a state-space model. To be precise, we will estimate the parameters in an extended Euler equation considering asymmetry in investment behavior in booms and recessions. As stated below, the coefficient of the lagged return representing past gains and losses will be more significant in booms than in recessions if there exist break-even effects. The remainder of this paper is organized as follows. In Section 2 we replicate some traditional studies. Section 3 sketches the asset pricing model of Barberis et al. (2001). Section 4 shows empirical evidence on break-even effects with time-series data of the US and Japan. Section 5 concludes the paper. 2. Replication of traditional studies We start with a traditional dynamic stochastic general equilibrium (DSGE) model, exploring its implications for asset pricing. The representative agent maximizes the following objective function:



max E

(Ct ,Kts )

∞ t=0

∞ 1−ı  −1 t Ct



t=0



1−ı

s.t. s Ct + Kts = Wt Nts + Rt Kt−1 Nts = 1

where Ct denotes real consumption, 0 <  < 1 is the discount factor, and ı> 0 is the coefficient of relative risk aversion. Nts is labor supply and Kts the supply of capital. Rt is the gross return on capital which can be stochastic and Wt the real wage of labor. The representative firm is assumed to maximize the following objective function 

d max Zt (Kt−1 ) (Ntd )

(K d

t−1

,Ntd )

1−

d d + (1 − )Kt−1 − Wt Ntd − Rt Kt−1

1 For a recent consumption-based asset pricing model that improves on the equity premium assuming heterogeneous agents, see Gust and Lopez-Salido (2009).

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

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Table 1 Estimates of the parameters in Eqs. (4) and (5). Country

TSLS

US UK Italy Germany France Japan Country



ı



ϕ

Sample

0.046 (2.887) 0.021 (1.285) 0.018 (0.729) 0.059 (2.394) 0.045 (3.733) 0.011 (1.132)

−0.941 (1.825) 0.059 (0.161) 0.098 (0.242) −1.749 (1.788) −0.646 (2.083) 0.119 (0.433)

0.032 (22.281) 0.040 (13.423) 0.052 (16.262) 0.027 (8.956) 0.041 (11.797) 0.027 (12.010)

−0.137 (2.291) 0.037 (0.426) 0.028 (0.475) −0.206 (1.963) −0.249 (3.020) 0.062 (0.929)

1962.4–2004.1 1962.4–1999.1 1970.4–1998.4 1970.4–1998.4 1970.3–1998.4 1962.4–2004.1

GMM

US UK Italy Germany France Japan



ı



ϕ

Sample

J-statistic

0.067 (5.332) 0.028 (1.893) 0.003 (0.106) 0.059 (6.944) 0.094 (7.768) 0.009 (0.836)

−1.537 (4.090) 0.052 (0.141) 0.256 (0.470) −3.307 (7.313) −1.965 (6.334) 0.240 (0.879)

0.031 (19.500) 0.035 (8.163) 0.052 (9.091) 0.028 (14.266) 0.042 (12.258) 0.028 (6.821)

−0.163 (5.844) 0.056 (0.654) 0.029 (0.496) −0.251 (6.823) −0.334 (4.042) 0.090 (1.651)

1962.4–2004.1 1962.4–1999.1 1970.4–1998.4 1970.4–1998.4 1970.3–1998.4 1962.4–2004.1

0.097 0.086 0.095 0.251 0.515 0.088

where log Zt = (1 − ω) log Z¯ + ω log Zt−1 + t ,

t ∼N(0, 2 ), 0 < ω < 1

with Zt denoting the technology shock, Ktd the demand for capital, Ntd the demand for labor and  the depreciation rate of capital. We do not consider leisure in the utility function and just assume Nts to be 1 for simplicity. Assuming general equilibrium, the first-order conditions read2  Ct = Zt Kt−1 + (1 − )Kt−1 − Kt

(1)

−1 Rt = Zt Kt−1 + (1 − )

(2)

   Ct+1 −ı

1 = Et 

Ct

 Rt+1

(3)

¯ − 1 and Let rt and ct denote the deviations of Rt and Ct from their steady states and define ct+1 ≡ ct+1 − ct ,  ≡ (1/R) t+1 ≡ rt+1 − Et [rt+1 ] − ı(ct+1 − Et [ct+1 ]), we know the log-linearization of Eq. (3) reads rt+1 =  + ıct+1 + t+1 .

(4)

Because the firm is assumed to be owned by the household, the return on capital can be considered as the return from equity and bonds issued by the firm. Therefore, one can take Rt as the return on assets. The relation between asset returns and consumption is shown in Eq. (4). Assuming rational expectation, one knows that t+1 has zero mean. Because t+1 is correlated with ct+1 , the ordinary least squares is not an appropriate approach to estimating the parameters in the above equation. Therefore, Campbell et al. (1997) estimate the parameters with the two-stage least squares (TSLS) approach. A problem with the instrument variables (IV) estimation here is that the instruments used in the estimation are only weakly correlated with the independent variables and the standard error of the coefficient tends to be too small for testing the overidentifying restrictions of the model. In order to see whether the orthogonality conditions hold, one can then reverse Eq. (4) and obtains: ct+1 =  + ϕrt+1 + t+1 .

(5)

In case the orthogonality conditions hold, ϕ will asymptotically be the reciprocal of ı. Campbell et al. (1997) estimate the parameters in the above two equations with the annual US data of 1889–1994. The instrument variables they use include the lags of real commercial paper rate, real consumption growth rate and the log of dividend-price ratio. Their estimation result is inconsistent with theories as both ı and ϕ have negative signs. Next, we would like to undertake some similar estimation for several countries. We will use both the TSLS and the generalized methods of moments (GMM) to estimate the parameters in the above two equations as a system. The instruments we use here include one- and two-period lags of stock return (the log difference of share price index),3 real (aggregate private) consumption growth rate, short-term interest rate or treasury bill rate and a constant. The estimation results are shown in

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Fig. 1. Growth rates of total consumption and the consumption of nondurables and services (US, 1962.1—1998.4).

Table 1 with t-statistics in parentheses.4 The determinant residual covariances in all cases are almost zero. It is clear that the estimation with the TSLS is not essentially different from that with the GMM. ı and ϕ in some cases have negative signs and in other cases, they are positive but have relatively insignificant t-statistics. These results are largely consistent with the findings of Campbell et al. (1997), that is, no significant relationship between stock return and the growth rate of real consumption has been found. Note that in the production-based as well as the consumption-based asset pricing models, consumption refers to total (private) consumption. In empirical tests, however, economists usually employ the data of consumption of nondurables and services (N.&S.), as in Campbell et al. (1997). Here we employ quarterly data of total private consumption for estimation as the time series of these two types of consumption are highly correlated. The growth rates of total consumption and the consumption of N.&S. in the US are shown in Fig. 1, with the correlation coefficient being 0.860 (sample period: 1962.1–1998.4). The estimation result with the consumption of N.&S. is found quite similar to that with total consumption. Therefore, in the research below we keep using the data of total consumption since the data of N.&S. consumption has a shorter sample period. 3. Asset pricing model with prospect theory In this section we would like to make a brief sketch of the loss aversion model in Barberis et al. (2001). The representative agent maximizes the following objective function:



E

∞  t=0



C 1−ı + bt t+1 (Xt+1 , At , zt ) t t 1−ı

,

(6)

where Ct is real consumption,  the discount factor and ı the parameter of relative risk aversion. The second term in parentheses stands for the effect of the change in wealth on the agent’s welfare. Xt+1 is the change in wealth and At the value of the agent’s risky assets. zt is a variable measuring the agent’s gains or losses prior to time t as a fraction of At . Barberis et al. (2001) assume that Xt+1 = At Rt+1 − At Rf,t , with Rt+1 being the gross return of risky assets between time t and t + 1, and Rf,t the gross return of risk-free assets between time t and t + 1. They further assume that in case zt ≤ 1



(Xt+1 , At , 1) =

At Rt+1 − At Rf,t At (zt Rf,t − Rf,t ) + At (Rt+1 − zt Rf,t )

for Rt+1 ≥ zt Rf,t for Rt+1 < zt Rf,t ,

(7)

2

See Uhlig (1999) for the derivation of these equations. Because dividend data are unavailable, we just take the growth rate of share price approximately as stock return. 4 J-Statistics are also presented to see the validity of the over-identifying restrictions, since the number of orthogonality conditions is larger than the number of parameters to estimate. The J-statistic reported here is the minimized value of the objective function in the GMM estimation. Hansen (1982) 3

L

claims that N · J →2 (m − q), with N being the sample size, m the number of moment conditions and q the number of parameters to be estimated. Data source: CEIC.

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

with  > 1 and



Xt+1 (zt )Xt+1

(Xt+1 , At , 1) =

for Xt+1 ≥ 0 . for Xt+1 < 0

839

(8)

In case zt > 1, (zt ) is then defined as (zt ) =  + k(zt − 1), k > 0,

(9)

while zt+1 = zt

R¯ Rt+1

+1−

(10)

with ∈ [0, 1] and R¯ a fixed parameter which is chosen to be the long-term risk-free interest rate. Moreover, it is presumed that bt = b0 C˜ t−ı ,

(11)

with b0 (a scaling factor) and C˜ t (aggregate consumption) to be specified so that the price-dividend ratio and the risky asset premium remain stationary. Hereby b0 is an important parameter indicating the relevance that financial wealth has in utility gains or losses relative to consumption. In case b0 = 0, we recover the consumption-based asset pricing model with power utility. Barberis et al. (2001) apply such a model of loss aversion and asset pricing to two stochastic variants of an endowment economy without production. In the first model variant there is only one stochastic pay-off for the asset holder, a stochastic dividend, whereby dividend pay-offs are always equal to consumption. In the other model variant dividends and consumption follow different stochastic processes. The Euler equation for the risky asset reads:





1 = Et Rt+1 with

C˜ t+1 C˜ t

−ı 

+ b0 Et [ˆ(Rt+1 , zt )]

⎧ ⎪ ⎨ Rt+1 − Rf,t ,

(zt − 1)Rf,t + (Rt+1 − zt Rf,t ), ˆ (Rt+1 , zt ) = Rt+1 − Rf,t , ⎪ ⎩ (zt )(Rt+1 − Rf,t ),

(12)

Rt+1 Rt+1 Rt+1 Rt+1

≥ zt Rf,t and zt ≤ 1 < zt Rf,t and zt ≤ 1 ≥ Rf,t and zt > 1 < Rf,t and zt > 1

(13)

We can now compare the two Euler Eqs. (12) and (3). The first term of Eq. (12) is the same as Eq. (3). The difference is that Eq. (12) has an additional term b0 Et [ˆ(Rt+1 , zt )] which shows the effects of the changes in financial assets on the agent’s welfare. As can be seen from Eq. (12), the Euler equation has now six variables, Rt+1 , Ct+1 , Ct , Rf,t , C˜ t , and zt which is a function of Rt−1 and zt−1 . Considering the cases in Eq. 13 separately, one sees that for each single case the right hand side of Eq. (12) is affinely linear in Rt+1 . More precisely, one can rewrite Eq. (12) as



1 + b0 ˛2 Rf,t = Et



C˜ t+1 C˜ t

−



+ b0 ˛1



Rt+1

(14)

with ˛1 and ˛2 given by ˛1 ˛1 ˛1 ˛1

= 1, = , = 1, = (zt ),

˛2 ˛2 ˛2 ˛2

=1 = ( − 1)zt + 1 =1 = (zt )

for for for for

Rt+1 Rt+1 Rt+1 Rt+1

≥ zt Rf,t and zt ≤ 1 < zt Rf,t and zt ≤ 1 ≥ Rf,t and zt > 1 < Rf,t and zt > 1

(15)

Using the equation Rt+1 =

Pt+1 + Dt+1 Pt

for the risky return with Pt denoting the asset price and Dt the dividend, which is chosen equal to C˜ t in Grüne and Semmler (2008), plugging this equation into Eq. (14) and using Eq. (13) we obtain

⎡  ⎤ −  C˜ t+1 /C˜ t + b0 ˛1 (C˜ t+1 + Pt+1 )⎦ Pt = Et ⎣ 1 + b0 ˛2 Rf,t

(16)

=mt+1

Taking C˜ t as the aggregate consumption from the growth model of Brock and Mirman (1972), Grüne and Semmler (2008) solve the above problem numerically with an advanced dynamic programming algorithm and find that the loss aversion

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Table 2 Numerical result of Grüne and Semmler (2008). E(Rf )

E(R)

E(R) − E(Rf )

SD(R)

SR

Cov(m, R)

1.012

1.029

6.944

0.103

0.173

−0.008

model can explain the equity-premium puzzle better than models with habit formation and adjustment costs studied by Boldrin et al. (2001) and Jerman (1998), for example. To be precise, They obtain the C˜ t from the following model MaxE C˜ t

 ∞  t=0

C˜ 1−ı  t 1−ı



t

s.t. Kt+1 = yt AKt˛ − C˜ t ln yt+1 = ln yt + ςt with Kt denoting capital stock, yt a stochastic shock to output and ςt an i.i.d. random variable. Taking A = 5, ˛ = 0.34, = 0.5,  = 0.99, ı = 1,  = 7.5, = 0.9, b0 = 1, K = 3 and setting the standard deviation of ςt at 0.072, they have the results shown in Table 2 where SR stands for the Sharpe-ratio and E(R) − E(Rf ) is the annual equity premium. Grüne and Semmler (2008) also undertake the numerical computation by varying the values of the parameters mentioned above and obtain similar results. Setting = 0.9 and leaving other parameters unchanged, for example, they get E(Rf ) = 1.014 and SR = 0.216. While Grüne and Semmler (2008) solve the loss aversion model numerically with a dynamic programming algorithm, below we would like to shed more light on this issue with actual data. To be precise, we would like to estimate the parameters in the Euler Eq. (12). Although Eq. (12) is hard to log-linearize because of the existence of kinks, one knows Rt+1 is a function of Ct , Ct+1 , Rt , Rf,t , C˜ t and zt . In other words, Et F(Rt+1 , Ct , Ct+1 , Rt , Rf,t , C˜ t , zt ) = 0,

(17)

with F(.) being a measurable function. A simplified assumption is that rt+1 is a linear function of the variables in Eq. (17). In the research below, we just follow this simplification and assume that5 rt = ˛ + ˇct + rt−1 + εt ,

(18)

where εt is an i.i.d. noise with zero mean and constant variance, while rt and ct have the same meanings as in Eq. (4). Note that εt may contain expectation errors which may be correlated with ct , as discussed in Section 2. 4. Empirical evidence of the prospect theory Above we have made a brief sketch of the loss aversion model in Barberis et al. (2001) and explored the difference between the first-order conditions of the traditional model which considers only consumption in the utility function and the loss aversion model which also considers the effects of wealth changes on welfare. An important point of the loss aversion model with house-money effect is that the return on stock in the previous period may affect the agent’s risk aversion in the current period and, therefore, affects the investment behavior on the stock market, which in turn influences return on stock. Barberis et al. (2001) assume that people become (more) risk-seeking in the aftermath of gains and (more) risk-averse with respect to previous losses. Applying this assumption to stock market, the following seems to hold: if the share price increases in the previous period, people may enlarge their investment in stocks and reinforce the booms and, as a result, stock prices may increase further. If the share price decreases, however, people become (more) risk-averse and will probably reduce their investment in stocks. This implies that the demand for stocks will decrease in the aftermath of losses, and as a result, stock prices may decline further because of the withdrawal in stock investment. This kind of investment behavior will then enlarge volatility in asset prices and returns. Moreover, there should be a strong correlation between returns in the current period and in the next period. In the simplified Eq. (18) this may be illustrated by the coefficient . To be precise, one would expect  to be relatively significant. Next, we will estimate  for several countries.

5 The equation below without the rt−1 term is the log linearization of the first term of Eq. (12) and we try to approximate the second term of Eq. (12) by rt−1 . Though we have modeled the influence of the asymmetry of current as well as past gains and losses, we try here first to capture this effect by a delayed rt−1 (past gains and losses). In the later sections we will consider the asymmetry in the influence of past gains and losses by considering a state-dependent coefficient for rt−1 . In future research a more precise linearized form of Eq. 12 may be considered. Theoretically, the risk-free rate is also an independent variable in this equation. We tried the estimation with the GMM and found that the risk-free rate has a relatively insignificant t-statistic, albeit with a right sign. It is −0.088 with the t-statistic being only 0.472 in the case of the US, for example. Therefore, in the estimation below we will not consider the effect of the risk-free rate.

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

841

Table 3 Estimates of the parameters in Eq. (18). Country

ˇ



J-statistic

Cov(rt , ct )

Sample

US UK France Germany Italy Japan

−1.492 (2.422) −0.438 (1.188) −0.472 (1.279) −1.988 (2.156) 0.616 (1.527) 0.034 (0.139)

0.275 (3.439) 0.258 (3.412) 0.169 (3.886) 0.105 (1.816) 0.444 (8.382) 0.343 (7.356)

0.031 0.033 0.063 0.027 0.046 0.040

3.88 ×10−5 1.62 ×10−4 −1.06 ×10−3 −1.57 ×10−4 −3.40 ×10−5 1.37 ×10−4

62.3–04.1 62.3–99.1 65.2–98.4 70.3–98.4 70.2–98.4 62.3–04.1

4.1. Some preliminary evidence In this section we estimate the parameters in Eq. (18) with the GMM using the same instruments as for Eqs. (4) and (5). The estimation results are shown in Table 3 with t-statistics in parentheses. The covariance of rt and ct is also shown there. It is clear that ˇ has a negative sign in most cases or a positive sign with insignificant t-statistics in others. The covariance of rt and ct is relatively low in all cases. On the contrary,  always has significant t-statistics with a positive sign. This supports the argument of Barberis et al. (2001) mentioned above. That is, investment in stocks may be greatly affected by stock returns in the past and as a result, returns from stocks may be greatly affected by their lags rather than by consumption.

4.2. Markov-regime changes in the coefficient: break-even effect? As mentioned above, prospect theory argues that people’s investment behavior can be affected by their previous gains and losses. In case they have obtained some gains in the previous period, they would be more ready to accept risks in the current period. The effect of losses in the previous period, however, can be more complicated than that of gains. Kahneman and Tversky (1979) show that although in many cases people become (more) risk-averse in the next period if they have some losses in the current period, this is not always true. That is, they find that in some other cases people may become (more) risk-seeking in the next period even if they have some losses in the current period because of the break-even effect. That is, previous losses may induce risk-seeking if they fail to adapt to losses or believe an expected gain can be attained. This is also clearly stated by Thaler and Johnson (1990): Thus, while an initial loss may induce risk aversion for some gambles, other gambles, which offer the opportunity to break even, will be found acceptable. Thaler and Johnson (1990) further mention the implication of the break-even effect in investment behavior as follows Also, when options present the opportunity to “break even”, tendencies toward risk-seeking in the domain of losses might be enhanced. Thus, we would expect investments in failing enterprises to be particularly prevalent when there is a hope, however dim, that one might eradicate existing losses. . . . Another potential domain for applying this research is in the study of investment behavior. The break-even effect suggests that individuals are averse to closing an account that shows a loss. This aversion can produce a reluctance to sell securities that have declined in value. Therefore, previous losses in stocks may have two kinds of effects in opposite directions on current investment behavior on the stock market. On the one hand, previous losses may induce risk aversion and make people reduce their demand for stocks and, as a result, reinforce the drops in stock prices. On the other hand, in case there is a break-even effect, previous losses may induce risk seeking. In this case, people will not withdraw their investment in stocks and may even enlarge their demand for stocks. This break-even effect will obviously prevent stock prices from falling further or slow down the falls in stock prices. Therefore, how previous losses affect the investment behavior in stocks depends on which of the two effects dominates. In case neither effect dominates, the effect of previous losses on the investment behavior may be unclear. Barberis et al. (2001), however, assume that people become (more) risk-seeking after gains and (more) risk-averse in the aftermath of losses, and do not consider the break-even effect. In their analysis, the effects of previous gains and losses on investment behavior are symmetric. The break-even effect would, of course, make their model much more complicated. In the estimation above, we have followed the assumption of Barberis et al. (2001) and did not separate the effects of previous losses from those of previous gains, and found that  is relatively significant in the general case. Next, we will explore some empirical evidence on the break-even effect to see whether the effects of previous gains are different from those of previous losses. To be precise, we will explore whether  in Eq. (18) experiences regime changes in the cases of positive and negative returns of stocks. We may modify Eq. (18) as rt = ˛ + ˇct + St rt−1 + εt ,

εt ∼N(0, S2t ),

(19)

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Table 4 Estimates of the parameters in Eq. (19). Parameter

Estimate (t-statistic)

˛

ˇ

0

1

0

1

p

q

0.025 (2.210)

−0.444 (1.233)

−0.061 (0.156)

0.375 (5.613)

0.096 (3.856)

0.043 (7.332)

0.750

0.0003

with St denoting the state of the stock markets. Here for simplicity we just assume two states (booms and recessions), namely, St = 0 and St = 1.6 Moreover, we assume St has a Markov-switching process with the following transitional probabilities: Pr[St = 1|St−1 = 1] = p

(20)

Pr[St = 0|St−1 = 0] = q.

(21)

One can employ the maximum likelihood method to estimate the parameters. In the model above we assume that St is not observed and therefore have to figure out the log likelihood function. Kim and Nelson (1999) show that the log likelihood function can then be explored as follows. Let t denote the information up to t and f (.) the density function, the log likelihood function reads lnL =

T 

ln

 1 



f (rt |St ,

t−1 )Pr[St |

t−1 ]

St =0

t=1

with 1 

f (rt |St ,

t−1 )Pr[St |

t−1 ]

=



St =0

+



1 2 02



exp

−

(rt − ˛ − ˇct − 0 rt−1 )

2 12

exp

Pr[St = 0|

2 02



1

2

−

(rt − ˛ − ˇct − 1 rt−1 )

2

t−1 ]



2 12

Pr[St = 1|

t−1 ]

(22)

where Pr[St = j|

t−1 ] =

1 

Pr[St = j, St−1 = i|

i=0

In order to obtain Pr[St−1 = i| Pr[St = j|

t]

= Pr[St = j|

t−1 ] =

1 

Pr[St = j|St−1 = i]Pr[St−1 = i|

i, j = 0, 1

t−1 ],

i=0 t−1 ],

one should compute the following probability

t−1 , rt ]

=

f (St = j, rt | t−1 ) = f (rt | t−1 )

f (rt |St = j,



t−1 )Pr[St

= j|

t−1 ]

1

f (rt |St = j,

t−1 )Pr[St

= j|

,

t−1 ]

j=0

where t = { t−1 , rt }. Running the above iteration from period to period, one then obtains the time-varying probability of being in each state. In order to start the iteration, one needs starting values for Pr[St = j| t ](j = 0, 1). The steadystate probabilities of St are taken as the starting values, namely, Pr[S0 = 0| 0 ] = (1 − p)/(2 − p − q) and Pr[S0 = 1| 0 ] = (1 − q)/(2 − p − q). The estimation result with quarterly US data of 1962–2004 is shown in Table 4 and Fig. 2.7 It is clear that ˇ has a wrong sign and an insignificant t-statistic, consistent with the estimate using the GMM.  is much lower in state 0 than in state 1 and, moreover, it has an insignificant t-statistic in state 0 (0.156). In state 1 it has a highly significant t-statistic (5.612). What does state 0 or state 1 stand for? We may get some hints from Fig. 2. Fig. 2A is the return of share and Fig. 2B is the probability of being in state 0, namely Pr[St = 0| t ]. We find that if rt is positive (or in boom), Pr[St = 0| t ] is usually low, and if rt is negative (or in recession), Pr[St = 0| t ] is usually high. This indicates that boom is consistent with state 1 and recession is consistent with state 0. Therefore, 1 should be interpreted as the coefficient in a bull market and 0 the coefficient in a bear market. This evidence supports the above statement that the effects of previous

6 We do not preset which state represents booms or recessions, but instead let data speak by identifying the two states with the estimation results. In particular, we will compare the return on stocks with the probability of being in each state. If the probability of being in state 0, for example, rises when the return on stocks is positive, then state 0 stands for booms. Therefore, it is possible that state 1 may stand for booms in this section and represents recessions in the next section in which we use a state-space model with time-varying coefficients. 7 One may argue that the maximum-likelihood estimation method might not be the best way to estimate the parameters in the equation above, since the noise εt may be correlated with ct , and the estimate of ˇ may not be consistent in this case. But as mentioned above, the most interesting parameter here is , not ˇ, therefore, this would not be a major problem in interpreting the results.

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

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Fig. 2. Estimation with Markov-switching coefficient. (A) Return of share and (B) probability of being in state 0.

gains and losses may be asymmetric. Previous gains will enlarge investment in stocks, while the effects of previous losses may be unclear possibly due to the break-even effect. Barberis et al. (2001) measure losses or gains in wealth by the difference from stock return and bond return (or the equity premium). A higher equity premium in the previous period may induce people to enlarge their investment in stocks and vice versa if no break-even effect exists. In the above estimation we have just assumed the transition probabilities to be constant for simplicity, namely, Pr[St = 1|St−1 = 1] = p and Pr[St = 0|St−1 = 0] = q. Next, we would like to explore the effects of the equity premium on the transition probabilities. To be precise, we assume that they are affected by the gap between the stock return and bond return in the previous period, gt−1 .8 Formally we have Pr[St = 1|St−1 = 1, gt−1 ] = p(t) =

exp(1 + 2 gt−1 ) 1 + exp(1 + 2 gt−1 )

(23)

Pr[St = 0|St−1 = 0, gt−1 ] = q(t) =

exp(3 + 4 gt−1 ) . 1 + exp(3 + 4 gt−1 )

(24)

The effect of gt−1 on p(t) can be seen below dp(t) 2 exp(1 + 2 gt−1 ) = 2 dgt−1 [1 + exp(1 + 2 gt−1 )]



> 0, ≤ 0,

if 2 > 0; if 2 ≤ 0.

(25)

If 2 is positive, a higher gt−1 may increase the probability of being in state 1 in the next period, given that the stock market is in state 1 in the current period, and vice versa. The effect of gt−1 on q(t) can be explored similarly. The estimates of parameters in equation 19 assuming time-varying transition probabilities with the US data are presented in Fig. 3 and Table 5. Fig. 3 looks quite similar to Fig. 2. That is, we also find that Pr[St = 0| t ] is low when rt is positive (boom) and vice versa, implying that state 0 is recession and state 1 is boom. ˇ still has a wrong sign with the t-statistic being 1.554. 0 is −0.095 with the t-statistic being as low as 0.229 and 1 is 0.362 with the t-statistic being as significant as 5.948. The estimation result is consistent with that of constant transition probabilities: the effects of previous gains and losses on the investment behavior are asymmetric, pointing to the possibility of break-even effects on stock markets. From Table 5 we see that 2 is positive with the t-statistic being 2.234, implying that gt−1 has significant positive effects on p(t). This indicates that the higher the return on shares relative to that on bonds, the higher the probability for the stock

8 Below we prefer the gap between returns from stocks and from bonds to equity premium because we do not consider dividends from stocks when computing return from stocks.

844

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

Fig. 3. Estimation with time-varying transition probabilities. (A) Return of share and (B) probability of being in state 0.

Table 5 Estimates of the parameters in Eq. (19) with time-varying transition probabilities. Parameter

Estimate (t-statistic)

˛

ˇ

0

1

0

0.027 (2.595)

−0.524 (1.554)

−0.095 (0.229)

0.363 (5.948)

0.102 (6.038)

Parameter

Estimate (t-statistic)

1

1

2

3

4

0.042 (12.376)

2.377 (2.742)

21.332 (2.234)

−38.028 (0.038)

−18.938 (0.018)

market to be in boom in the next period, given that it is also in boom in the current period. This also supports the idea that people may become (more) risk-seeking after gains. In contrast to the case of positive returns, 4 is negative with a relatively insignificant t-statistic, implying that gt−1 might not have much effect on q(t). This seems to support our earlier finding that previous losses may not always induce risk-aversion possibly due to the break-even effect. We present the time-varying transition probabilities in Fig. 4. Fig. 4A is the gap between the return on shares and that on bonds. Fig. 4B shows the path of p(t) (Pr[St = 1|St−1 = 1, gt−1 ]) and Fig. 4C presents the path of q(t) (Pr[St = 0|St−1 = 0, gt−1 ]). It is clear that gt−1 shows significant effects on p(t), which increases as gt−1 rises and decreases when gt−1 declines. q(t) is almost constant and has not been affected significantly by gt−1 .

4.3. Time-varying coefficient in a state-space model with Markov-switching In the previous section we have estimated the coefficient  with Markov-regime changes and found that the effects of previous gains and losses on the investment behavior are asymmetric. In this section we would like to explore this problem further and estimate time-varying paths of . We will employ a state-space model with Markov-switching as follows: rt = ˛ + ˇct + St rt−1 + εt , St = ¯ St + St−1 + t ,

εt ∼N(0, S2t )

t ∼N(0, 2 )

(26) (27)

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

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Fig. 4. Time-varying transition probabilities. (A) Gap between returns from share and returns from bond, (B) (Pr[St = 1|St−1 = 1, gt−1 ]) and (C) (Pr[St = 0|St−1 = 0, gt−1 ]).

with E(εt t ) = 0. As in the previous section  may differ across states, and at the same time,  may also change over time and follows a path proposed by Eq. (27).9 We also assume time-varying transition probabilities as shown in Eqs. (23) and (24) and estimate the time-varying  using the Kalman filter.10 Let t−1 denote the vector of observations available as of time t − 1. In the usual derivation of the Kalman filter in a state-space model without Markov-switching, the forecast of t based on t−1 can be denoted by t|t−1 . Similarly, the matrix denoting the mean squared error of the forecast can be written as Pt|t−1 = E[(t − t|t−1 )(t − t|t−1 ) |

t−1 ],

where E is the expectation operator. In the state-space model with Markov-switching, the goal is to form a forecast of t based not only on t−1 but also conditional on the random variable St taking the value j and on St−1 taking the value i

9 Here we assume the  follows a mean-reverting path. An alternative assumption is that it follows a random-walk path if the estimate of the parameter  is larger than unity in absolute terms. As shown in Table 6, the estimate of the  is much smaller than one, suggesting that the assumption of a mean-reverting path for  is reasonable. 10 Theoretically,  and  may also have Markov property. We will not consider this possibility because there are already 12 parameters to estimate and the efficiency of estimation will be eroded if more parameters need to be estimated.

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Table 6 Estimates of the parameters in Eqs. 26-27. Parameter

Estimate (t-statistic)

˛

ˇ

0

1





0.044 (4.255)

−0.964 (3.142)

0.099 (6.691)

0.041 (12.679)

0.00004 (0.002)

0.466 (1.064)

Parameter

Estimate (t-statistic)

¯ 0

¯ 1

1

2

3

4

0.299 (1.849)

0.093 (0.758)

−0.016 (12.060)

0.429 (0.035)

2.449 (2.833)

26.500 (2.435)

(i, j = 0, 1): (i,j)

t|t−1 = E[t |

t−1 , St

= j, St−1 = i],

and accordingly, the mean squared error of the forecast is Pt|t−1 = E[(t − t|t−1 )(t − t|t−1 ) | (i,j)

t−1 , St

= j, St−1 = i].

Conditional on St−1 = i and St = j (i, j = 0, 1), the Kalman filter algorithm for our model is as follows: (i,j)

(28)

(i,j)

(29)

i , t|t−1 = ¯ j + t−1|t−1 i + 2 , Pt|t−1 = 2 Pt−1|t−1 (i,j)

(i,j)

t|t−1 = rt − ˛ − ˇct − rt−1 t|t−1 , (i,j)

(30)

(i,j)

2 Pt|t−1 + j2 , t|t−1 = rt−1 (i,j)

(i,j)

(31)

(i,j)

(i,j)

t|t = t|t−1 + Pt|t−1 rt−1 [t|t−1 ] (i,j)

(i,j)

(i,j)

2 [t|t−1 ] Pt|t = (1 − Pt|t−1 rt−1

−1

−1 (i,j)

t|t−1 ,

(32)

(i,j)

)Pt|t−1 ,

(33)

(i,j)

(i,j)

where t|t−1 is the conditional forecast error of rt based on information up to time t − 1 and t|t−1 is the conditional variance (i,j)

(i,j)

(i,j)

2 P of the forecast error t|t−1 . It is clear that t|t−1 consists of two parts rt−1 and j2 . When there is no Markov-switching t|t−1

property in the shock variance,

j2

is constant. In order to make the above Kalman filter algorithm operable, Kim and Nelson (i,j)

(i,j)

j

j

(1999) develop some approximations and manage to collapse t|t and Pt|t into t|t and Pt|t respectively.11 In the estimation below, however, we assume time-varying transition probabilities to explore whether the gap between the return on stocks and that on bonds has any effect on the investment behavior on stock markets. The estimation results with the quarterly US data are shown in Table 6, with the likelihood function being 252.630. The difference between the ¯ 0 and ¯ 1 is obvious. It is about 0.3 with the t-statistic being 1.849 in state 0 and only 0.093 with the t-statistic being 0.758 in state 1. The 2 has a relatively insignificant t-statistic, indicating that gt−1 has little effect on the transition probability p(t). The 4 , however, has a positive sign and a relatively significant t-statistic of 2.435. This indicates that the transition probability q(t) changes proportionally to the gap between the stock return and bond return. That is, in case the stock market is in state 0 in the current period, a higher gt indicates a higher probability for the stock market to be also in state 0 in the next period. This is consistent with the finding in the previous section as long as state 0 stands for boom and state 1 stands for recession. We can explore briefly what state 1 and state 0 stand for from Fig. 5. Fig. 5A is the time-varying  in the two states of 0 and 1 over time. It is clear that  is largely stationary and experiences relatively small changes in state 1 and experiences some significant changes in state 0. Fig. 5B is the probability of being in state 0 given the information up to time t, namely Pr[St = 0| t ], and Fig. 5C is the stock return. Comparing Fig. 5B and Fig. 5C, we find that most of the time Pr[St = 0| t ] is high when rt is positive and vice versa. This seems to indicate that state 0 stands for a bull market and state 1 for a bear market. As seen in Fig. 5A, 0 is higher than 1 , implying that previous gains may have induced investment enlargement in stocks, while previous losses may not have induced significant withdrawal of investment in stocks. This can also be seen from the mean of 0 and 1 , with the former being 0.560 (=0.299/(1-0.466)) and the latter being only 0.174 (=0.093/(1-0.466)). This finding is consistent with that in the previous sections.

11 As for the details of the state-space model with Markov-switching, the reader is referred to Kim and Nelson (1999). The procedure applied below is modified from the Gauss procedures developed by Kim and Nelson (1999).

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

Fig. 5. Time-varying  across states. (A) Time-varying t , (B) Pr[St = 0|

t]

847

and (C) stock return.

4.4. Empirical evidence of Japan Next, we will show some empirical evidence with the quarterly data of Japan (1972.1–2004.2). The estimates of the parameters in Eq. (19) are shown in Table 7 and Fig. 6, with the likelihood function being 158.464. It is clear that ˇ has a wrong sign with an insignificant t-statistic, while 0 is not essentially different from 1 . This is different from the case of the US in which 0 is quite different from 1 (−0.061 vs. 0.375). From Fig. 6 we find that the probability of being in state 0 is very low when rt is negative and vice versa, suggesting that state 0 should be boom and state 1 recession. Moreover, there seems to be a clear regime change around 1990 from boom to recession. We have also tried the estimation with time-varying transition probabilities and found that 2 and 4 have relatively insignificant t-statistics. In short, the empirical evidence in Japan looks somewhat different from that of the US. As shown in the previous section, the effects of previous gains and losses on the investment behavior in the current period appear to be relatively asymmetric Table 7 Estimates of the parameters in Eq. (19) for Japan. Parameter

Estimate (t-statistic)

˛

ˇ

0

1

0

1

p

q

0.015 (1.747)

−0.055 (0.228)

0.242 (1.356)

0.356 (3.738)

0.036 (5.835)

0.083 (12.436)

0.989

0.958

848

W. Zhang, W. Semmler / Journal of Economic Behavior & Organization 72 (2009) 835–849

Fig. 6. Empirical evidence of Japan. (A) Return of share and (B) probability of being in state 0.

partly due to the break-even effect. In the case of Japan, however, we find that the effects of previous gains and losses on investment are relatively symmetric, indicating a smaller break-even effect. This may have two implications: (1) Japanese are more risk-averse than Americans. They would be more reluctant to accept break-even opportunities in case they have experienced some losses in the past; or (2) Japanese are not more risk-averse than Americans, and the reason that they would not like to stay in the stock market after losses is that there has been little chance to break even. The second possibility seems to be consistent with the situation of the stock market in Japan, with the return on stocks having been negative most of the time since the beginning of the 1990s. The stock market has been in such a long recession that people did not see any chance in sight to break even, and as a result, would rather withdraw their investment from stocks after having had losses in the past. The situation in the US is, however, quite different. The stock return has been positive most of the time, particularly after the 1990s.

5. Conclusion This paper explores empirical evidence on the prospect theory in asset pricing with time-series data. The asset pricing model with the prospect theory and house-money effect has been shown to be more successful in explaining some puzzles in financial markets such as the equity premium. Based on the loss aversion model, we have first studied an extended Euler equation with Markov-switching coefficients with constant, as well as time-varying transition probabilities and found that the effects of previous gains and losses on investment in stocks have been likely asymmetric, pointing to the possibility of break-even effects.12 Moreover, the gap between returns on stocks and on bonds also seems to be an important factor affecting the persistence in investment behavior. We have then studied a state-space model of time-varying coefficients, with the empirical evidence confirming that the persistence in investment behavior depends on the state of the stock market (recessions or booms). Future research might need to take into account the stock market meltdown of 2008–2009 and to test whether the break-even effect hypothesis will hold. The main message is that, although the asset-pricing model developed by Barberis et al. (2001) with the prospect theory can explain the equity-premium puzzle better than the habit-formation and adjustment-cost models, it has ignored the break-even possibility, which may have some non-neglectable effects on investment behavior, particularly in recessions.

12 Following the framework of prospect theory we have mainly focused on the break-even effects in explaining what is observed from the estimation. There might be alternative factors leading to the asymmetry in the investment behavior. One possible explanation, for example, is that sophisticated financial institutions attempt to look into the future and undertake a contrarian strategy: whenever the stock market is down, and the returns low, they pursue a buying strategy. This will stop the price from falling or staying down.

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