Reference price distribution and stock returns: an analysis based on the disposition effect

Transcription

1 Reference price distribution and stock returns: an analysis based on the disposition effect Submission to EFM symposium Asian Financial Management, and for publication in the EFM special issue March, 2011, Beijing 1

2 Reference price distribution and stock returns: an analysis based on the disposition effect Abstract This paper provides evidence that reference price distribution can predict expected stock return. We develop a model based on the disposition effect by considering the trading activities of shareholders with different capital profit/loss ratios. The model suggests that both investors incentive to selling their shares and their paper capital gain/loss ratios are crucial to stock performance in the next period. It also indicates that the summary statistics of relative capital gains/losses are associated with future returns as they reflect the shape of a distribution in different aspects. By applying four proxy variables the paper finds that the mean and skewness of relative capital gains/losses play important roles in explaining cross-section of stock returns in the Chinese stock markets. Additionally, the empirical results also show that the mean is the key explanatory variable for variations in cross-sectional returns for stocks with positive average capital gains. Meanwhile the skewness works as the key variable for stocks with negative average capital gains. The effects of the proxy variables survive even when the factors identified in the extant literature are taken into consideration. JEL classification: G1; G12 Keywords: Prospect theory; Disposition effect; Reference price distribution; Cross sections in stock return Introduction Shefrin and Statman (1986) find that investors have tendency to sell winners too early and hold on to losers too long. They refer to the phenomenon as disposition effect. It is a widely held belief that Kahneman and Tversky s (1979) prospect theory and Thaler s (1980) mental 2

3 accounting constitute the basic explanation for this effect 1. This paper contributes to literature on the disposition effect with theoretical modeling and empirical investigations from a market-wide angle. We examine whether or not reference price distribution can predict stock return based on the disposition effect. Prospect theory describes how one evaluates choices under uncertainty, and provides possible explanations for phenomena not in accordance with classical expected utility theory. The principle of prospect theory is based on the different risk attitudes people have when confronted with various situations. That is, an individual is more likely to be risk seeking when facing a loss, and risk averse when facing a gain. If an investor is subject to prospect theory and mental accounting (referred to as a PT-MA investor in the paper), her utility function is concave on the positive domain and convex on the negative domain of investment profit as shown in Figure 1. For example, there are four PT-MA investors G1, G2, G3 and G4 whose current positions are indicated in the figure. Investor G1 should be the most risk seeking and investor G4 the most risk averse. Correspondingly, investor G1 has more demand for the stock than investor G4. (Insert Figure 1 here) Critical to the utility value of the PT-MA investor is her reference price, by which she is determined to be in a position of loss or gain. As in Figure 1, given the market price, G1 s reference price is the highest and G4 s is the lowest. The reference prices of G1 and G2 are higher and those of G3 and G2 are lower than market price. When the stock experiences an appreciation, all investors positions on the utility curve will move to the right if their reference prices remain unchanged. Reference prices, however, usually update from time to time. These adjustments may be due to variations in trading activities. For instance, G4 may sell her shares to a new investor; e.g., G4, and the position of G4 will be at a point near the inflection point. The update may also be due to the investor s expectation of future return. After an appreciation of the stock price, if investor G4 has a higher expectation for future return, then the reference price will also increase. Thus, she will move to the right with a shorter distance than under a fixed reference price. As in Odean (1998) and Grinblatt and Han (2005), this paper assumes that the reference prices are the 1 Recently literature of this area contributes to integrating psychological evidence on risk preferences into models of equilibrium prices, for example, Barberis, Huang, and Santos (2002), Barberis, Huang, and Thaler (2003), and Grinblatt and Han (2005). 3

4 volume-weighted averages of their purchase prices. For a preliminary understanding of the effect of PT-MA investors behaviors on stock price, we can imagine a situation as follows: A market consists of two types of investors. One is a PT-MA investor whose risk preference is defined by prospect theory. The other is a usual rational investor. The decision making should be markedly different for the two types of investors. We assume that, however, an investor, regardless of her type, will always make decisions like a rational one before holding any share of a certain stock. When the stock price converges to its fundamental value from below, PT-MA investors gradually short their shares to new stockholders when they have paper gains, while rational ones hold on to them if they expect a further price rise in the future. If it turns out that the stock price rises, more and more transactions are initiated by PT-MA investors during the good period. This delays the market price quick approach to the fundamental value, resulting in a high probability of a sustaining rise in price during the following period. On the other hand, if the stock price falls to the fundamental, rational investors will sell their shares while PT-MA ones will keep holding them if they are having unrealized. As the market price drops, more and more investors will be facing a loss status, and sell-initiated transactions will happen with less frequency during the bad period. In this case, trading activities of PT-MA investors also defer the stock price s approach to the fundamental value. Therefore, a higher average capital loss more possibly forecasts a further decrease in the stock price in the future. Similarly we can speculate that a stock with more winning shares may have higher returns in the next period. The paper models the demand of a PT-MA investor by adding an extra term due to the disposition effect to the rational linear demand as in Grinblatt and Han (2005). But we differentiate each share s reference price. If a share is sold, its reference price changes to the new market price. By this way we find that the selling probability of each share partially determines the expected return. The aggregate effect of PT-MA investors trading behaviors is associated with how their paper gain/loss ratios are distributed. Therefore the model requires us to take into account the shape of a relative-capital-gain distribution when studying the variations in stock return caused by PT-MA investors. The theoretical model and empirical examination by Grinblatt and Han (2005) only 4

5 documents the role of capital gains overhang in explaining the continuation of stock return. The paper goes forward to reassess the issue by two steps. The first step is to investigate how stock price is affected by an individual investor with a certain profit/loss ratio. The second step is to aggregate the effects from all shareholders. This process indicates that the shape of a reference price distribution should be taken into account. Therefore we expect that the summary statistics, such as the variance, skewness, and kurtosis, should have implications to stock s performance in the future, because they characterize a distribution in different aspects. Besides the average capital gains as in Grinblatt and Han (2005), the roles of higher moment statistics in predicting future return can be speculated as follows. Consider two stocks with the same average capital gains; and one stock has higher variance than the other. The first stock more possibly has more shares with extremely high profits and high losses as well. If the shareholders trade as the disposition effect suggests, then the first stock may not have reacted to the exogenous shock sufficiently. Therefore the first stock will face higher return relative to the second one. Now suppose that both stocks have the same average and variance on their relative capital gains. If the first one has larger skewness, then it may have higher return in the next period. It is because that the first stock has more extremely winning shares. The argument is similar to that for variance. Along this way, we hope to reveal the relation between the summary statistics with cross-section in stock returns. To obtain the summary statistics of relative capital gains for a stock, we have to collect the transaction records with all trades. However it is not available for most stock markets around the world 2. The account-based data have much advantage in the investigation of investors decisions, since they provide precise purchase prices, sell prices, holding periods, etc. For example, Shefrin and Statman (1985) make use of trading records from individual investors and aggregate data on mutual fund trades. Odean (1998) randomly selects 10,000 accounts at a brokerage house. These empirical studies based on partial trading records can provide evidence on the existence of the disposition effect in corresponding financial markets. But if we are interested in the responses of stock return to trading behaviors, these data from a segment of a market may not sufficient. The 2 The data applied by Grinblatt and Keloharju (2000) (2001) consist of all accounted-based transaction records in Finland stocks. 5

6 main concern on these data is that they don t comprehensively reflect the behaviors of common investors, since they do not consist of the whole market participants. In the empirical investigation, the paper instead uses stock-based transaction records 3. Our theoretical model seems to ensure the applicability of market data. The paper constructs four proxy variables using daily average prices and trading volumes. They correspond to the four moment summary statistics mentioned above. The main concern about our methodology is that the distribution of relative capital gain/loss may not be able to represent the true distribution. It is possible that these shares that were held by investors who had not traded them for a long time are not included. We believe, however, those investors trading incentives usually don t obey the disposition effect. So it may not a big issue when we care only for the effect of PT-MA investors. Recently in an empirical investigation to the market-wide disposition effect, Kaustia (2004) also uses the market data, but only focus on the variations in trading volume when stock prices cross different thresholds after their IPOs. The empirical results in the paper seem to provide evidence on the theoretical model based on the disposition effect. By applying the Fama-MacBeth methodology, we find evidence that some of the proxy variables can predict cross-sectional variations in stock returns. The mean (ARC) and skewness (SRC) of relative capital gains are positively related to stock returns in the next month. We further examine the associations by dividing the stock into two groups according to the sign of a stock s ARC in each month. Fama-MacBeth regressions reveal that different proxy variables stand out for stocks in different groups. For example, for stocks in G-group (with positive ARC, winning group), ARC plays the key role in explaining the cross-section of stock returns: a stock with higher ARC has a higher return over the next month. For stocks in L-group (with negative ARC, losing group), the key proxy variable is SRC: a higher SRC implies a higher future return. Finally, we include into our regressions the factors identified in extant literature being able to explain the cross-section of stock returns and re-examine the effects of the four proxy variables. We find that the explanatory abilities of the proxy variables remain significant after these inclusions. 3 Webber and Camerer (1998) mention that control variables such as investors expectations and individual decisions are not observable in market data. 6

7 The structure of the paper is as follows. Section 2 sets up the model, and specifies regression equation for empirical examinations. Section 3 details the methodology to calculate the proxy variables by using common daily market data. Section 4 provides descriptive statistics of the proxy variables and explores our preliminary regression results. The possible explanations for the connection between our empirical results and theoretical model are also offered in this section. Section 5 examines the results obtained in Section 4 by taking into accounts other factors that potentially affect our findings, and provides extra evidence. Section 6 concludes the paper. 1. The model 2.1 Model setup and stock return Assume that the number of a stock s shares outstanding is K. By mental accounting, we suppose that the K shares are held by K investors. Those investors who currently don t have any share are rational. The fundamental of the stock evolves as V V (1) t1 t t1 where V t is the fundamental value at time t, and t follows a normal distribution with a mean of zero. Equation (1) assumes that the fundamental of the stock is normally distributed with the mean of the stock s historical values. Investors in the market are divided into two categories: One is the rational investor, and the other is subject to prospect theory and mental accounting, whom we refer to as the PT-MA investor. The trading behaviors of rational investors drive the market price to revert to its fundamental, but those of PT-MA investors may cause the market price to deviate from it. The direction of deviation depends on the aggregate trading strength of all investors with different ratio of capital gains/losses. As proposed by Grinblatt and Han (2005), the demand functions are Rational Dk, t 1 bt ( Vt Pt ), if k is rational ;. (2) 7

8 PT MA k, t t t t k, t t D 1 b ( V P) C P, if k is PT-MA. (3) D Rational and D PT-MA denote the demands of rational investors and PT-MA investors respectively; P t is the market price of the stock at date t; C k,t is the reference price (mental cost) of the k th PT-MA investor if she is PT-MA investor, known to her prior to time t; b t represents the slope of the rational investors demand function. The positive parameter λ measures the relative strength of the PT-MA investors to induce extra demand. Equation (3) implies that the relative strength is the same for PT-MA investors if their reference prices are equal. We further assume that each shareholder is PT-MA investor with probability. Thus the total market demand D t is the summation of all individuals demands, K Rational PT / MA 1,. D D D t t k t k1 By substituting (2) and (3) into the market demand function, we rearrange the right-hand side and obtain t t t t t k, t t k1 K D K Kb ( V P) b C P. (4) The market clearing condition implies that the equilibrium price is K 1 P V C. (5) t t k, t 1 K 1 k1 The equilibrium price is the weighted average of the fundamental value and reference prices of all PT-MA investors. In comparison with the equilibrium price in Grinblatt and Han (2005), this formula takes into account the possible different reference prices, and the possibility of being PT-MA investors. Our model is arguably more realistic, since the trading activities of investors at different positions of utility curve implied by prospect theory should be distinguished explicitly. For each share, its holder s reference evolves as follows, Ck, t1 Pt 1{ sold } k, t 1 1 { sold } k, t Ck, t, (6) where indicator function 1 sold k, t is one if share k is sold at time t, zero otherwise. This specification is reasonable, because whenever one share is traded, its holder s reference price 8

9 should be current market price. Therefore by virtue of the results obtained above, the change in price from date t to date t 1 can be represented as K 1 P P V V C C. t 1 t t 1 t k, t 1 k, t 1 K 1 k1 By substituting expression of C k,t+1 in equation (6), and taking expectation on both sides, we obtain the expected change in price, which is K EP P Pr 1 1 P C, t1 t { sold} k, t t k, t K 1 k1 since the expected changes in fundamental are zero by assumption. Pr 1{ sold} k, t 1 represents the probability that share k is sold at time t. Therefore the expected stock return is K P P 1 t C t P t k, t E Pr 1{ sold} k, t 1 Pt K 1 k1 P. (7) t Equation (7) demonstrates that the expected change in stock price depends on two critical factors Pt C besides and. One is the relative profit/loss ratio, i.e., Pt probability that a share is sold, i.e., Pr 1{ sold} k, t 1 k, t. The other is the. Both are positively correlated to expected return. The more important implication of equation (7) is that the distribution of relative capital gains (RC) determines the expected stock return. To discuss the effect of the relative profit/loss distribution on expected return, we have to make assumption on the selling probability Pr 1{ sold} k, t 1. In terms of the disposition effect, investors tend to sell their winning shares and keep holding losing ones. We suppose that Pr 1 1 { sold} k, t is an increasing function of Pt C Pt k, t. Although it still holds for a rational shareholder, the equation (7) mainly consider the trading behaviors of PT-MA investors. The expected return is determined by the distribution of relative capital gains/losses. It can be identified from the summation notation in the right-hand side of equation (7). Essentially the expected return reflects the aggregate effect of investors with different positions along the profit 9

10 axis. Based on the expression, we should calculate the relative capital gain/loss and the selling probability for each shareholder in order to forecast the stock return. Our expected return formula also predict the result by Grinblatt and Han (2005), which documents that the expected return is an increasing function of unrealized capital gains. However it may due to two reasons implied by our model: the higher probability to sell a share and the higher profit ratio. Our model apparently shows that higher moment variables are also meaningful. For example, two stocks have four shareholders with equal number of shares respectively. The relative capital gains/losses for stock one s holders are -0.8, -0.2, 0.9, and 1.1; and -1.2, -0.3, 1.0, and 1.5 for stock two. Both stocks average capital gains are Grinblatt and Han s (2005) model predicts the same expected return for the two stocks. But they may be different, because their holders incentives to trade their shares are not always the same. If we assume the Pr 1 sold 1 0 if stockholders with negative capital always holder their shares, that is, { } P C t P t k, t 0, and that higher positive profit ratio suggests higher selling probability, then the second stock may have higher expected return in the future. Our main interest in this paper is in investigating how reference price distribution predicts expected return. It is well-known that a distribution can be represented by its moment polynomial. Thus we may instead use the summary statistics to characterize a distribution as discussed in the introduction, such as mean, variance, skewness, and kurtosis of relative capital gains/losses. By exploring the associations between these variables and stock return, we are able to know how the shape of a reference price distribution is related to future return. 2.2 Return approximation and explanations According to the discussion in the last paragraph in the previous subsection, we approximate the expected stock return by the following polynomial function, P P t1 t E h0 h1 MEAN1 t h2var t h3sknew t h4kurt t Pt (8) where MEAN,VAR, SKEW, and KURT represent the mean, variance, skewness, and kurtosis 10

11 of relative capital gains for a stock at time t ; h0 captures the proportion of the expected return which is related to higher polynomial terms and other factors. Coefficients h i, i 1,...,4, denote the loadings of expected return on the corresponding variables. The four variables in equation (8) capture different aspects of a relative-capital-gain distribution. MEAN measures whether a stock is a winner or a loser on average. It is comparable to capital gain overhang in Grinblatt and Han (2005). A positive MEAN indicates the average cost of the stock s holders is lower than market price. If MEAN 0, the selling motives of wining holders are more possibly stronger than the buying motives of losing ones. For a stock with MEAN 0, the opposite relation holds. The total effect on the price is reflected by a slower convergence to the expected value. Therefore it predict a continuation in return. If the disposition effect has a market-wide effect, we expect that h1 0. The implication of VAR to the winning and losing proportion is unclear. But conditional on VAR, i.e., giving the profit ratio on average, VAR measures how widely shareholders purchasing costs are distributed. Taking a normal distribution as an example, when its mean is positive, a largervar implies that a higher proportion of negative value. When its mean is negative, we have opposite observation. Thus the sign of h2 is not unambiguous. Financial theory except the disposition effect has connection to the variance in investors profit/loss ratios. For example, VAR is possibly related to the differences in evaluations to a stock across investors. A largervar may reflect a higher dispersion in investors opinions. SKEW measures the asymmetry of a capital gain distribution. A positive SKEW implies that there are more shares with capital gains/losses on the near left side of mean, and more shares distributed to extreme high winning positions. A negative SKEW indicates the reversal direction. The fourth variable KURT measures the proportion of shares with extreme profits and extreme losses, and, at the same time, the proportion of shares with gain/loss very close to the mean is higher. Thus for stock with positive or negative KURT, the effect of KURT on stock return should be different. 11

12 2. Variables and data 3.1 Variable calculations As mentioned in the introduction, we apply the transaction records at individual stock level to calculate the moment variables. The procedure is outlined as follows. For a stock, we start with the last trading dayt. Step 1: accumulate daily turnover ratios backwards until the cumulative turnover ratio reaches 100%. The latest day satisfying the criterion of 100% cumulative turnover is marked as S 4. T Step 2: on the time interval[ S, T ], applying the formulas, which are introduced later, to calculate the variables values at dayt. T Step 3: for dayt 1, repeat Step 1 and 2 to obtain variables values. Step 4: continue the process up to the day when the backward cumulative turnover cannot reach 100%. It seems natural to select the time interval on which the turnover ratios accumulate to 100%. We can think all shares of the stock change hands 5 during the period. Therefore, the stock prices at which investors purchase their corresponding shares form the distribution of reference prices. Given the length of time interval N, we calculate proxy variables using trading volume and average prices over the period. Before that, we need the relative capital gain/loss, RC 6, for a stock s shares purchased at date n, 1n N, which is RC n AC AC AC N n. (9) N where AC n is the average price of the stock at date n. If a stockholder purchases a certain number 4 It is worth noting that if another end date is selected, the starting date changes and the length of the time interval also changes, so the letter ST represents different numbers for different stocks and different end dates. 5 We don t preclude other alternatives, since some investors may hold their shares for a longer time than the interval. For PT-MA investors, if the stock price experiences a long depreciation, in terms of the disposition effect, they may hold their losing shares longer than relevant N; otherwise a 100% accumulative turnover interval should be reasonable enough. 6 The so-called relative capital gains are not the return rates of an investment, but the absolute capital gains normalized by the current price, as explained in the introduction. Moreover our methodology does not deny the legitimacy of other plausible choices. 12

13 of shares at date n at an average price lower than that market price, i.e., ACn ACN, then RC 0, and the holder possesses paper capital gains. n Now we can compute proxy variables for reference price distribution. The formulas are listed as following. N N 2 VOL RC N VOL RC ARC n n n n N ARC n1, VRC n1, N N N N VOL ( N 1) VOL n n n1 n1 N ( ) 3 N N VOL RC ARC N VOL ( RC ARC ) 4 n n N n n N SRC n1, KRC n1. N N 3 / 2 N 2 N N 1 VRC VOL N 1 VRC VOL N n N n n1 n1 (10) where ARC N, VRC N, SRC N, and KRC N represent the mean, variance, skewness, and kurtosis of reference price distribution at date N; VOL n is the number of shares traded at date n. ARC N is a volume-weighted average of relative capital gains/losses, RC n, over the target time interval. These four measures proxy for the variables in equation (8). Denote ARC it, VRC it, SRC it, and KRC it the corresponding variables values for stock i at date t. In our empirical investigation, the regression model becomes r i, t 1 h 0 h1 ARC it h2vrc it h3src it h4 KRC it t. (11) This regression model can capture the association between the reference price distribution and stock return in cross sections. 3.2 The data The data used in this paper are obtained from Shanghai Wind Information Co., Ltd (Wind). As a financial services company, Wind collects and sorts various financial data from the Chinese financial markets. Their products have been used by large securities companies, fund management companies, and investment institutions. We select daily trading records of all A-share stocks publicly traded either on the Shanghai Stock Exchange or on the Shenzhen Stock Exchange. The data consist not only of trading information such as opening price, closing price, average price, 13

14 volume (in share number or RMB), and turnover ratio, but also of several financial statement variables such as the earning-price ratio and market-to-book ratio. The raw sample period, from April 1991 to March 2010, consists of 228 months. We first calculate the four proxy variables defined in the last section using daily data, and then transfer them into monthly data, according to a certain rule proper to each specific variable. For example, the monthly return r m is calculated by P, P 1, P m L m L m1, L 100%, where P ml, is the closing price of the last trading day in month m. The monthly turnover ratio is the sum of daily turnover ratios in that month; and the book-to-market ratio is that of the last trading day in that month. When transferring from daily data to monthly data, we delete the first month s observations after IPOs for all individual stocks, since there are dramatic jumps in stock prices during the IPO months 7. We also exclude stocks from those months in which book values were negative, and those whose trading days were less than 15 days in the month. We exclude data before January 1996 from the final monthly data, because the number of stocks traded in that period is too small. Thus the final data, used in our empirical analysis, consist of 171 months, with 309 individual stocks in January 1996, and 1,426 in March Empirical results 4.1 Summary statistics Figure 2 plots the time series of the 10 th, 50 th, and 90 th percentiles for the cross-section of ARCs from January 1996 to March 2010, for Chinese firms listed either on the Shanghai Stock Exchange or on the Shenzhen Stock Exchange. The bottommost dotted line denotes the 10 th percentile; the solid line in the middle denotes the 50 th percentile; and the dashed line on the top denotes the 90 th percentile. These lines indicate that the ARCs display wide monthly variations over that period. And for most of the period, more than 50 percent of stocks had a negative ARC. The most ARCs dropped dramatically over 2008 and 2009 when the stock markets experienced a 7 For discussion on this issue, the reader can refer to Wang and Xu (2004) and others. 14

15 depression due to the world-wide financial crisis. (Insert Figure 2 here.) Similar to Figure 2, Figure 3 plots the time series of the 10 th, 50 th, and 90 th percentiles of the cross-section of SRCs. The SRCs across stocks also exhibit wide dispersions during the period. Most stocks have positive SRCs in most months in contrast to ARCs. (Insert Figure 3 here) Panel A of Table 1 reports the summary statistics on the returns and four proxy variables constructed in the paper. Since Kahneman and Tversky s (1979) prospect theory suggests different shapes of utility functions for investors with negative and positive RCs, in each month, we divide the stocks with positive ARCs and negative ARCs into two groups: G-group (the winning group, ARC > 0) and L-group (the losing group, ARC < 0). As this process is done on a monthly basis, a stock does not necessarily remain in the same group in different months. After that, within each group, we further divide them into four ARC quartiles and calculate the equal-weighted average monthly return and the simple averages of the ARCs, VRCs, SRCs and KRCs in each quartile. This process provides us with time-series averages of these five variables for each subgroup. The time series mean of the average monthly return of stocks in G-group is 3.11%, which is much greater than in L-group (0.85%). The average returns of stocks in the two groups exhibit different patterns. In L-group, it declines from the lowest ARC quartile, 1.43%, through the highest ARC quartile, 0.56%. While in G-group, it declines from the first quartile, 3.09%, through the third quartile, 2.86%, it reaches the highest value in the fourth quartile, 3.46%. Different from the average monthly return, the means of VRCs and SRCs of stocks in G-group are significantly smaller than those in L-group. The patterns of VRCs and SRCs are also different between the two groups. The means of VRCs and SRCs of stocks in L-group decline with ARC quartiles, whereas those in G-group exhibit a nonlinear pattern. (Insert Table 1 here) In Panel B of Table 1, stocks are first divided into two groups in the same way as in Panel A, i.e., G-group and L-group based on the sign of stocks ARCs. However the quartiles are formed in terms of SRCs, from low to high, of individual stocks in each group. Again the time series 15

16 averages of equal-weighted monthly stock returns and the four proxy variables are reported. The average monthly return increases with the SRC quartiles in L-group, while there is no obvious pattern in G-group. The time series averages of the quartile ARCs decrease with SRC in both groups. This has also been observed in Panel A. The average VRC of L-group is greater than that of G-group; and there is no obvious pattern for KRC in both groups. Table 2 reports the summary statistics of several variables identified by the extant literature, including market capitalization, book-to-market ratio, average turnover over the past 12 months, and three types of past returns. The two panels of Table 2 apply the same grouping procedure as the corresponding panels of Table 1. The market capitalization in Table 2 is measured by the logarithm of the market value, in RMB, of tradable A-shares, denoted by ln(me). Panel A shows that the average size of stocks in G-group is greater than that in L-group. The average size increases slightly with the ARC quartiles in G-group. The book-to-market ratio in Table 2 is the logarithm of book value over market value per share, i.e. ln(be/me). As exhibited in Panel A, the simple average ln(be/me) displays consistent patterns along with ARC, within and between L-group and G-group. The quartile average ln(be/me) descends monotonically from low ARC in L-group, at -1.24, to high ARC in G-group, at These findings suggest that when stocks experience appreciations, ARC increases and BE/ME decreases. In table 2, the turnover ratio corresponding to month t is the average monthly turnover ratio over the previous 12 months, namely during the period from month t-1 to month t-12. Turnover is the trading share volume divided by the number of tradable A-shares in each month. Panel A presents interesting results for average turnover over the previous 12 months. First the average turnover ratio of stocks in L-group, 43.42%, is much less than that of the stocks in G-group, which has a rate of 49.32%. Second, quartile average turnover goes up with ARC in L-group, but falls in G-group. These results imply that trading activities of investors are truly related to their unrealized capital gains/losses. We also consider the three types of past returns, which are short-term, medium-term, and 16

17 long-term historical returns. At month t, the short-term past return r -1, is the stock return in the past month, i.e. in month t-1; the medium-term past return, r -4,-6, is the average monthly return during the three-month period from month t-4 to month t-6; and the long-term past return, r -13,-24, is the average monthly return during the twelve-month period one year ago, i.e. from month t-13 to month t-24. The reason to choose the three types of returns is based on relevant papers in this area. Jegadeesh (1990) and Lehmann (1990) show that contrarian investment strategies based on short-term return reversals, one week or one month, can bring abnormal returns. Jegadeesh and Titman (1993) document significant profits for relative strength trading strategies over 3- to 12- month horizons. In addition, long-term return reversals also have been identified by the related literature such as De Bondt and Thaler (1985). The three types of previous returns in the paper aim to capture the three types of stock return patterns over different horizons. Our empirical investigation shows that the long-term return reversals occur over relatively shorter horizons in Chinese stock markets than their counterparts in developed markets. This seems to be in accordance with the viewpoint that a higher proportion of investors are speculative in Chinese stock markets. It may be supported by the average annual turnover ratio as high as over 500 percent. The high-frequency trading activities in Chinese stock markets lead to relatively faster return reversals. Panel A suggests that the return in the previous month, r -1, and average return over the period from month t-4 to month t-6, r -4,-6, are highly related to ARC. The time series average r -1 of stocks in L-group is -4.38% much less than that of stocks in G-group, which have an average of 13.72%. Moreover quartile average r -1 and r -4,-6 exhibit monotonically increasing patterns with ARC in both groups. This means that stocks experiencing more appreciation during the previous half year have higher ARCs. On the other hand r -13,-24 does not display obvious patterns with ARC quartiles. However, the average r -13,-24 of stocks in L-group, at the 1.35 percent, is significantly higher than in G-group, at the 0.72 percent. It indicates to some extent that long-term return reversals exist in Chinese stock markets. (Insert table 2 here.) Panel B in Table 2 shows that both ln(me) and ln(be/me) are related to SRC, but with opposite directions. The quartile with a higher SRC has a smaller ln(me), but a higher ln(be/me) 17

18 in both L-group and G-group. In comparison with the results in Panel A, this suggests that ARC and SRC indeed capture different aspects of the impact of reference price distribution on cross-sectional returns. From the low SRC column in L-group to the high SRC column in G-group, we observe an increasing pattern of average turnover with SRC quartiles. As for the associations between SRC and the three types of past returns, we find that r -1 decreases from percent in the low SRC quartile to percent in the high SRC quartile in L-group, and decreases from percent in the low SRC quartile to percent in the high SRC quartile in G-group. Whereas the average r -4,-6 exhibits different pattern along with SRC in the groups. In L-group, r -4,-6 is increasing in quartile SRC, while it displays a nonlinear association with SRC in G-group. As shown in Panel A of Table 2, SRC seems independent of r -13,-24. These summary statistics reveal that variations of cross-sectional returns are related to the proxy variables we construct in this paper, specifically the ARC and SRC of stockholders relative capital gains. Meanwhile, our variables seem not to be disentangled from some factors identified in extant literature, indicating a need for further investigation. 4.2 The preliminary results We use the Fama-MacBeth 8 (1973) method to analyze the relationship between stock returns and the proxy variables. As in equation (11), we use r it, the monthly return of stock i in month t, as dependent variable; and the four proxy variables on the last trading day of month t-1 as explanatory variables, i.e., mean, variance, skewness, and kurtosis of relative capital gains/losses of stock i. t 1 is the error term containing the cross-sectional variations across stocks in month t, which is not captured by the four variables. This regression preliminarily explores the implications of our model. If prospect theory and mental accounting determine the utility valuation of some investors in the Chinese stock markets, the distribution of investors reference prices should play a role in the stock returns in the next month. Thus cross-sectional variation across stocks returns should be explained by such proxy 8 Thanks to Dayong Huang s reminding us about the Fama-MacBeth regression. As Mitchell A. Perterson suggests, we use the Stata ado file xtfmb.ado by Daniel Hoechle for Fama-MacBeth regressions throughout the empirical investigation in this paper. 18

19 variables as these we construct. As described before, we divide the stocks into G-group and L-group based on the signs of their ARCs for the asymmetry property of risk attitudes. We run cross-sectional OLS regressions of individual stock returns on ARC, VRC, SRC, and KRC of relative capital gains/losses for each of the 171 months from January 1996 to March 2010, and then get the estimates and t-statistics of the 171 coefficients for each variable. The time series average of the estimates and their standard deviations are reported in Table 3. The three panels of Table 3 present the results from the estimations with the full sample, L-group sample and G-group sample respectively. (Insert Table 3 here) Panel A of Table 3 reports estimation results with the full data. It suggests that ARC and SRC play significant roles in explaining the cross-section of stock returns, but VRC and KRC do not. The estimated coefficient for ARC is positive and significant at the 5% level. This result is consistent with the representation of expected return derived from our theoretical model. It suggests that a stock currently with more winning shares will perform better in next period than a stock with fewer winning shares. A stock with a higher ARC most likely experienced more fractions of sell-initiated transactions recently than another stock with a lower ARC. This is because PT-MA investors are prone to selling their winning stocks, and keep holding their losing ones. It implies that the stock had slower convergence to its expected price no matter the target value is high or low. A higher ARC for a stock implies that price s under-reaction will be more prominent during the next period, which leads to a higher return for the stock in the next month. So the results of ARC are consistent with Grinblatt and Han (2005), and can be explained by the disposition effect. Panel A of Table 3 shows that the coefficient estimate of SRC is also positive and even more significant (at the 1% level), demonstrating that a stock with a distribution having a longer right tail has a higher return in the next month when other variables are fixed. This empirical examination illustrates that in addition to ARC, SRC is also important to predict stock returns. When ARC, VRC, and KRC are equal, a higher positive SRC means that more stockholders hold very high profit ratios, and at the same time, more stockholders profit ratios cluster at a level less than the mean. There is a similar argument to the explanation of ARC: if a stock has a higher SRC, 19

20 there are a relatively smaller proportion of PT-MA stockholders who are holding small winning shares but more high-ratio wining ones. So the probability of under-reaction to shocks is higher for the stock, forecasting a higher return in the near future. It thus makes sense that the coefficient estimate for SRC is significantly positive. This result also suggests that besides the average capital gains of a stock s holders, the skewness plays an important role in explaining the differences in stock returns. This result also confirms the assumption in the model where Pr(1 1) is high sold when a share has high profit ratio. The empirical examination provides evidence on our conjecture that the reference price distribution is critical to a stock s expected return. However our empirical examination does not find that the proxy variables VRC and KRC hold explanatory ability in the regressions 9. In order to differentiate the effects of reference price distribution on winner and loser stocks, we repeat the process on stocks in G-group and in L-group separately. We run cross-sectional regressions of stock returns on the four proxy variables for each group in every month. The results are reported in panels B and C of Table 3. As expected from the disposition effect, ARC plays a key role in determining the return of stocks in G-group, while SRC does so for stocks in L-group. For stocks in L-group, the estimated coefficient of ARC is insignificant, whereas for stocks in G-group, SRC does not have any explanatory ability. The insignificance of ARC for stocks in L-group shows that stockholders reluctance to short their losing shares is not correlated with their loss ratios in their investments. Prospect theory forecasts that stockholders with a higher loss ratio should have a higher demand, leading to their purchases of the stock. However, the extra demand by these stockholders does not seem to be reflected in the market. The relevant literature provides evidence that investors tend to hold on to losers too long, but it seldom touches upon purchasing incentives. Nevertheless, neither our results in the paper, nor Shefrin and Statman (1985) suggests that the average extra demand of PT-MA investors is equal, regardless of the absolute value of ARC as long as it is negative. The insignificance of ARC for stocks in L-group may imply that investors with different paper losses are all equally reluctant to realize their losses. Another possible explanation is the investors in Chinese stock markets are shortage of cash, which prevent 9 However our examination does not exclude that VRC and KRC are able to predict a stock s return over spans of other than a one-month period. As a matter of fact, examinations not reported in the paper indicate that VRC and KRC can predict a stock s return over a one week period. These regression results are available upon request. 20

21 them from satisfy their demands. For stocks with positive ARCs, in panel C of Table 3, ARC is the only significant explanatory variable among the four proxies. Note that the expected return formula represented by equation (7) indicates the positive association between relative capital gains and stock returns in the next period. The empirical result is consistent with this prediction, and has a larger magnitude and higher significance level than that obtained in the full sample. As the theoretical framework implies, a stock with a higher positive ARC has more winning shares, which result in more stockholders decline in demand, causing more delay in the stock s price approaching the value implied by the fundamental. The insignificance of SRC in G-group shows that the effect of ARC on the returns of stocks with positive ARC dominates SRC. One possible way to verify this is to examine the trading frequencies of the stocks in L-group and G-group during the following month. This is due to two considerations: One is an implication of the disposition effect, which says that stockholders with paper gains are more likely to sell their shares. The other is that a stock with a positive ARC is more likely to have a higher proportion of winning shares than a stock with a negative ARC. If the trading activities deduced by ARC are much higher than those by SRC, then ARC s effect may dominate that of SRC. We can compare the difference between the average turnover of stocks in L-group and in G-group. By applying a t-test, we find that the t-statistic is , and the average turnover in the following month is 39.75% for L-group and 65.86% for G-group, showing that stocks with a positive ARC are indeed more active than those with a negative ARC 10. The lack of significance of ARC, and significance of SRC for stocks in L-group and the opposite for those in G-group, provide evidence for the asymmetry property of prospect theory. It suggests that we should not only pay attention to the gain/loss ratios of stockholders investments, but also to the shape of the distribution of their relative capital gains. 4. Controlling for other factors The literature on investigating the cross-section of stock returns has identified a number of 10 A deeper inspection is to check the distribution of turnover ratio with respect to different stock ARCs. The highest turnover is close to where ARC is equal to zero, but with a very long tail for negative ARCs, where turnover is very low. 21

22 explanatory variables. Fama and French (1992) show that size and book-to-market equity capture the cross-section of expected stock returns. Jegadeesh and Titman (1993) document that strategies of buying past well-performing stocks and selling poorly-performing stocks can bring investors significant positive abnormal returns. In addition, some studies have postulated that cross-sectional stock returns decrease with stock turnover and/or volume (Datar et al., 1998; Hu, 1997; Rouwenhorst, 1998; and Chordia et al., 2001). In this section, we aim to reassess the relationships explored in the previous section, by controlling for factors that potentially have an impact on the cross-sectional stock returns. 5.1 Controlling for size and book-to-market ratio Since size and book-to-market equity are two well-known factors in explaining cross-section of stock returns, we add them into the regression model (11) to inspect the obtained results. As in the preliminary investigation, we run Fama-MacBeth regressions for full data, then the two groups with negative and positive ARCs separately. For each sample, we run three regressions: The first includes only the size (ln(me)), the second includes only the book-to-market ratio (ln(be/me)), and the third includes both as control variables. In order to keep consistent with the update of our four proxy variables 11, we use tradable market equities and book-to-market ratio values at the beginning of each month in each individual monthly regression. In the three panels of Table 4, the results of regressions for the three groups are reported respectively. (Insert Table 4 here) Panel A in Table 4 demonstrates that our proxy variables hold their ability to explain the cross-sectional variations in stock returns when we control for size effect. For the full sample, similar to the results without any control in the previous section, both ARC and SRC are positively associated with stock returns over the next period when market equity, ln(me), is controlled. Both coefficient estimates are significant at the 10% level. For L-group and G-group, the results in Panel A show that market capitalization cannot eliminate the effects we have found in preliminary regressions. Moreover, ln(me) loses its explanatory ability in G-group. As for L-group, the 11 We also test our model using ln(me) in June of each year, and ln(be/me) in December of the previous year, which are used in Fama and MacBeth (1973). The regression results are similar to those we report in this paper. 22

23 coefficient estimate for SRC is 0.49 and significant at the 1% level; while the estimate for ARC is 8.20 and significant at the 5% level in G-group. As the results reported in Table 3, ARC is insignificant for L-group, and SRC is insignificant for G-group. The insignificance of the size variable for stocks in G-group suggests that relative strength of trading activities of PT-MA investors, who hold shares of large firms, are stronger than those of small firms. When a high enough proportion of the shares is owned by winning stockholders, i.e., high positive ARC, the strength of investors behaviors with high risk aversion can offset the effect of the firm s market equity. To confirm this, we examine the association between firm size and ARC. Panel A of Table 2 indicates that the larger firms in G-group indeed have higher ARCs on average. But the association between SRC and firm size for stocks in L-group is negative as shown in Panel B of Table 2. This results in higher significance levels for both SRC and ln(me) in L-group, are comparable to those for full data. When controlling for only the book-to-market ratio, as illustrated in panel B, the variables, which can explain the differences in stock returns, are still ARC and SRC. In the full sample, when book-to-market ratio is controlled, the coefficient estimate of ARC is 4.62 and significant at the 5% level; that of SRC is 0.35 and significant at the 5% level as well. The regression results for stocks in the L-group show that SRC is the key variable in explaining the cross-sectional stock returns. For stocks in G-group, the effect of ARC is still prominent. These results justify our findings in the preliminary regressions. In addition, unlike size, the book-to-market ratio plays a significant role in determining the returns on stocks in all three groups, indicating that its explanatory power is independent of the proxy variables constructed in the paper. Panel C of Table 4 reports the regression results when both market equity and book-to-market ratio are controlled. It provides further evidence on the findings in the last section. Both ARC and SRC are significant at the 5% level in the full sample; the coefficient estimate of SRC is 0.40 with a 5% significance level for L-group; and ARC s coefficient estimate is 9.77 with a 1% significant level for G-group. An interesting finding in the full sample is that the coefficient estimate of VRC is positive and significant at the 5% level. If VRC contains information on trading activities of PT-MA investors, it may be able to forecast stock returns. Given other conditions, a stock with a higher ARC means that a certain amount of shares are distributed into deeper losses and wins. As 23