Prospect theory and risk-return trade-off: An international study

Transcription

1 Prospect theory and risk-return trade-off: An international study Dazhi Zheng West Chester University Huimin Li* West Chester University Thomas C. Chiang Drexel University This version: January 16, 2018 Abstract This paper examines the risk-return relation under the impact of prospect theory for international markets. Following Grinblatt and Han (2005) and Wang, Yan and Yu (2017), we calculate the capital gain overhang (CGO) to measure the psychological evaluation of past returns. Using the double sorting methodology, we find that a negative risk-return trade-off generally exists in international markets when CGO value is low, and the further GLM regression results confirm our findings. In the portfolio analysis, for stocks that have smaller size, higher price-to-book ratio, lower trading volume turnover, and higher momentum returns, the negative risk-return relationship is found to be more prominent when CGO is negative. The evidence also indicates that the prospect theory effect is more pronounced when market is in crisis condition. The above findings are less significant for all emerging markets group. JEL Classification: G14, G15, G40 Keywords: Prospect theory, capital gains overhang, risk-return trade-off, international markets, stock portfolios *corresponding author; Tel.:

2 1. Introduction In recent years, prospect theory has been used to explain how investors may allocate money to a stock based on their mental evaluation of the stock s past return distribution. The theory was first proposed in Kahneman and Tversky (1979) and extended in Tversky and Kahneman (1992). The theory, based on its S-shaped utility function, differs from traditional utility theory in three useful implications: (1) investors evaluate outcomes, not based on the wealth levels, but based on their perception of gains and losses relative to a reference point (reference-dependent preferences); (2) Investors are more sensitive to losses than to the same-magnitude gains (loss aversion); (3) Due to concavity of the utility function when there was a gain and convexity of the function when there was a loss, investors tend to be risk-averse in times of gains and risk-seeking in times of losses. Based on this theory, researchers have explored to explain many market phenomena, such as disposition effect (investors tend to sell winners sooner than they should have and hold on to losers longer than they should have, e.g. in Li and Yang, 2013; Dacey and Zielonka, 2008; Grinblatt and Han, 2005), negative-feedback trading strategy (investors buy stocks when prices declined and sell stocks when prices rose, e.g. in Yao and Li, 2013), equity premium puzzle (stocks generally earn a much higher risk premium than what the traditional risk measures can explain, e.g. in Benartzi and Thaler, 1995; Barberis and Huang, 2006), and insignificant or even negative risk-return trade-off relationship (the risk-return relationship is not significantly positive as traditional CAPM theory predicts, e.g. in Ang et al, 2006, 2009; Frazzini and Pedersen, 2014; Baker, Bradley and Wurgler, 2011). In this paper, we extend the research by Wang, Yan and Yu (2017) to explore more in-depth the risk-return trade-off relationship in an international context. In the past few years, studies have used different measures of prospect theory reference point to investigate the impact of prospect theory value on stock returns after controlling for various risk

3 measures in a cross-sectional setting. For example, Barberis, Mukherjee and Wang (2016) find that stocks with higher prospect theory value tend to earn a lower subsequent return as investors buy into these stocks while stocks with lower prospect theory value tend to earn a higher subsequent return as investors have less demand for these stocks and these stocks tend to be underpriced. The prospect theory value used is from Tversky and Kahneman (1992) and depends on parameter estimates from experimental data in this same paper. However, the psychological evaluation of the past return distribution could simply be based on a measure called Capital Gain Overhang (CGO). This measure was first introduced by Grinblatt and Han (2005) and later used in Wang, Yan and Yu (2017) to study risk-return trade-off relationship. Intuitively, this measure shows that how much a stock price is above the reference price formed based on weighted historical prices. The CGO measure is easily constructed (more details in section 2) and does impose different weights on prices that are more recent vs. prices that are father away in the past. It is also intuitive as explained above, so we use CGO measure in this study as a proxy for prospect theory value. Wang, Yan and Yu (2017) argue that for stocks that have negative CGO meaning capital loss relative to the reference price, investors may become more risk seeking and hold the stocks longer than they should have (disposition effect). This could result in a negative risk-return relationship as subsequent returns would be low for this high-risk investment. On the other hand, for stocks that have positive CGO meaning capital gain, investors may be risk averse and tend to sell the stocks quicker, therefore resulting in a positive risk-return relationship. This study follows Wang, Yan and Yu (2017) and further investigates the negative risk-return tradeoff relationship in the following aspects. First, we extend the study to twenty two advanced markets and twenty emerging markets. The findings of Wang, Yan and Yu (2007) are found to hold in many of these international markets, which is consistent with Barberis, Mukherjee and Wang (2016). We

4 also report the results for market groups such as advanced markets, emerging markets, G7 countries, Asian markets and Latin American markets. By looking deeper into these different market groups, we are able to see which markets will see more significant risk-return relationship anomaly. Second, this study uses a generalized linear model (GLM) to examine the relationship between beta risk and excess returns for an individual market or a market group. The Fama-Macbeth regressions used in other research papers report time-series average coefficients for cross-sectional regressions. When the time-series estimates for the coefficients vary greatly over the sample period, these averages may not represent the best estimate for the risk-return relationship. The GLM, however, does not assume a normal distribution for the dependent variable (the excess returns in this study) and therefore fits our data better. It also allows us to run a panel regression so that we do not have to get time-series averages for the coefficients. Third, this study further explores this risk-return relationship anomaly for stocks with different fundamental values such as size, price-to-book ratio, short term momentum, long term reversal and trading volume. We hypothesize that for stocks that have smaller market capitalization, higher price-to-book ratio, larger momentum and lower trading volume turnover, investors could be more risk-seeking at times of negative CGO to get out of potential loss region, therefore resulting in a larger impact from the prospect theory on this riskreturn relationship. This study forms different portfolios based on these stock fundamental variables and perform GLM regressions. Lastly, to further examine the CGO s impact on the risk-return relationship, we take out two subsamples: one termed crisis period representing the global financial crisis period (from July 2007 to July 2009) and the pre-crisis tranquil period (from March 2002 to June 2007) representing the period between the dotcom bubble burst and the crisis period. During the global financial crisis period, investors became more risk-seeking as they tried to recover from their unrealized capital losses at the beginning of the crisis. This behavior would magnify the

5 impact of CGO on the risk-return trade-off, meaning that when CGO is negative, the risk-return trade-off has a more negative relationship. None of the previous studies in prospect theory has looked in such depth at the stock fundamentals and crisis sub-periods and this study, to our knowledge, is the first in the literature. The main findings from this study are as follows. The double sorts give us very consistent results across different regional market groups. When CGO is high, the higher the Beta risk, the higher the excess returns. When CGO is lower, this risk-return relation becomes weaker and even turns negative in a few cases. For example, in all advanced markets, when CGO is high (i.e. prior capital gains), the excess returns for high-beta stocks are 16 basis points higher than those for low-beta stocks per week; when CGO is low (i.e. prior capital losses), the difference is only 10 basis points. So the positive risk-return relationship becomes weaker as CGO becomes lower. For all emerging markets, when CGO is high, the high-beta stocks have weekly excess returns about 10 basis points higher; when CGO is low, the low-beta stocks have weekly excess returns almost 2 basis points higher. So the positive risk-return relationship turns into negative when CGO is much lower. We call this risk-return trade-off anomaly. This finding also holds for some, but not all, individual countries. Countries like the U.S., the U.K., Germany, Italy, and Korea follow the same pattern. When using GLM method, we find that for all the regional market groups except for all emerging markets, the risk-return relation is positive when CGO is positive while it is negative when CGO is negative, even after controlling for size, market-to-book ratio, momentum and long term reversal four factors. This finding is consistent with the double sort results and with findings from Wang, Yan and Yu (2017). In the more advanced markets, the U.S. has the strongest dependence on CGO for its risk-return relationship followed by Italy. In the emerging markets, Argentina has the strongest dependence on CGO for its risk-return relationship followed by China and Brazil.

6 When dividing the whole sample into large and small size stock portfolios, the results indicate that the small-size stock portfolios have stronger prospect theory impact on the risk-return relation as the interaction terms between CGO and Beta are all positive other than Asian markets, which might lead to the negative coefficient for all emerging markets. This impact is also of larger magnitude for small size portfolios. Intuitively, for investors who hold small stocks, they tend to be more risk seeking when they have negative unrealized capital loss as the small stocks could give them more of a chance to win it back. The similar finding seems to hold for high price-to-book ratio stock portfolios. All market groups have positive coefficients for the interaction term except for European countries with insignificant but still positive coefficient, suggesting prospect theory could be a good explanation for negative risk-return relationship when CGO is negative. For low price-to-book ratio stock portfolios, these coefficients turn negative and significant except for Latin America group. Again, the high price-tobook ratio stocks probably give investors a sense of possibility that their unrealized loss could be recovered as these stocks tend to be more volatile as well. When trading volume turnover is used to divide the stocks into portfolios, we see that the low turnover portfolios have stronger prospect theory effect compared to the high turnover portfolios. Barberis, Mukherjee and Wang (2016) show that the predictive power of prospect theory value for subsequent stock returns is stronger among illiquid stocks as these stocks are less subject to arbitrage. Similarly, it is possible that the impact of CGO on the risk-return relation is also stronger for illiquid stock portfolios in international markets due to limits to arbitrage. For stock portfolios that have high momentum, the risk-return relation depends more on CGO levels in all market groups. In other words, there is a more significant negative risk-return relationship

7 when CGO is negative due to more risk seeking behavior by investors for past winners. This is not true for almost all market groups for momentum loser portfolios. In summary, for stocks that have smaller size, higher price-to-book ratio, higher momentum and lower trading volume turnover, the negative risk-return relationship is more prominent when CGO is negative. When we divide the sample into tranquil and crisis sub-periods, the risk-return relation anomaly is stronger during the crisis period. This is intuitive as investors try to recover their unrealized capital losses during a crisis and tend to be more risk taking when CGO is negative. When we compare the prospect theory effect among different market groups, while most market groups share similar risk-return relation pattern under CGO moderating, the emerging market group is usually the only group (or among the only few groups) that has the opposite sign of coefficient for the interaction term between Beta and CGO: i.e. in the full model regression analysis, size sorted portfolio analysis, trading volume turnover sorted portfolio analysis, and the different market condition analysis. It suggests that the risk-return relation is less likely to be affected by investors prospect theory behavior in emerging markets. An explanation is that for emerging markets, there are more information asymmetry and therefore investors could be difficult to get accurate reference price for stocks. The rest of the paper is organized as follows. In Section 2, we discuss the model and statistical methods used. Section 3 explains the data sample and summary statistics. Section 4 presents the empirical results and explanations for the results. Section 5 concludes the paper.

8 2. Estimation models and research methodology Following Grinblatt and Han (2005) and Wang, Yan and Yu (2017), we adopt the turnover-based measure to calculate the reference price. Specifically, for each week t, the reference price for each individual stock is defined as the following: T n 1 RR t = 1 (V k n=1 t n τ=1(1 V t n+τ ))P t n (1) where V t is the stock s trading volume turnover in week t, T is 104 weeks, 1 the number of weeks in the previous two years, and k is a constant that makes the weights on past prices sum to one. The trading volume turnover is calculated as weekly trading volume divided by the total number of shares outstanding for the stock. According to Grinblatt and Han (2005), the weight on stock price at time t-n is the probability that the share purchased at week t-n has not been traded. The capital overhang (CGO) at week t is computed as the percentage difference between the market price and the reference price: CCC t = P t RR t P t (2) To avoid the micro-structure issue from daily data and the less prominent behavioral effect from low frequency data, we use twenty years daily data to form the weekly CGO values for each stock. Following Barberis, Mukherjee, and Wang (2016), we write the regression equation below to test investors risk-return relation under prospect theory/disposition effect in each individual international market or market group: R i,t+1 = b 1 CCO i,t + b 2 RRRR i,t + b 3 RRRR i,t CCC i,t + b 4 SSS t + b 5 HHH t + b 6 MMM t + b 7 RRR t + ε i,t+1 (3) 1 Wang, Yan and Yu (2017) use 260 weeks (5 years) data to compute the reference price, but for international markets, especially many emerging markets, the data are much shorter than those in the U.S. market, so we use 104 weeks (2 years) instead. However, the results are similar when we test some markets using a longer period of data (260 weeks).

9 The above Equation 3 is estimated by the generalized linear model (GLM) in our empirical analysis, and according to our hypotheses a positive coefficient (b 3 ) is indicative of existence of prospect theory/mental accounting investors in the market. Stock and stock index returns are calculated as R t+1 = 100 (log(p t+1 ) log (P t )), where P t denotes either the individual stock price or the stock market index at time t, and since we calculate weekly returns, the prices are all Wednesday closing stock prices. All returns in our estimations are excess returns over the short term (1-month or 3-month) domestic interest rate. We use Beta as the risk measure, and it is the coefficient of the weekly CAPM regression in the past 104 weeks with a minimum of two years of data. SMB and HML represent the size factor and the value factor that are calculated as the return difference between the small size stock portfolio and the large size stock portfolio, and between the high bookto-market (B/M) equity stock portfolio and the low B/M stock portfolio, respectively. MOM is the return difference between winner and loser stock portfolios for month t-12 to t-1, and REV is the return difference between winner and loser stock portfolios for month t-36 to t-12. We form SMB, HML, MOM, and REV portfolios according to Fama and French (1993, 2012, and 2015). 2 At the end of every month, we sort stocks in each market based on stock fundamentals including size, book-to-market ratio, momentum, and long term reversal to form portfolios. Stocks are sorted independently to form two groups based on size, and three groups based on book-tomarket ratio, momentum, and long term reversal. The portfolio breakpoint is 50% when two size portfolios are formed, and the breakpoints are 30% and 70% when three portfolios are formed based on the other fundamentals. As shown in Appendix Table 1, for each market the size factor SMB t is the average return on the three small stock portfolios minus the average return on the three large stock portfolios (2 X 3 sorts). The value factors HML t (same for the momentum factor MOM t and 2 For detailed information on data and Fama-French portfolio formation, please refer to Appendix Table 1.

10 the long term reversal factor REV t ) are calculated as the average return on the two highest B/M portfolios minus the average return on the two lowest B/M portfolios (2 X 3 sorts). These four control variables are used as common pricing factors for cross-sectional stock returns. [Appendix Table 1] 3. Data description Stock data for all advanced markets and emerging markets are collected from Thomson Datastream. The data consist of pricing information and fundamental variables for individual stocks and stock market indexes. At individual stock level, the following variables are collected for this study: stock price, trading volume turnover, market capitalization, and price-to-book ratio. At market level, we collect the market price index and domestic short term interest rate to serve as the risk-free rate. Domestic pricing factors such as size, book-to-market ratio, and momentum, etc. are constructed according to Fama and French (1993, 2012, and 2015). 3 Regional and global fundamental factors are collected from Kenneth R. French s data library. 4 As noted by Ince and Porter (2006), there are issues regarding data coverage, classification, and integrity for international markets in the Datastream International data. In addition, according to Brennan, Huh, Subrahmanyam (2011), extreme values in returns/trading volumes may cause illiquidity issue and affect the validity of the model. Therefore, to compile the data, we set a firm s observations to be missing if its stock returns and trading volumes on Wednesdays are in the extreme bottom 1% of the cross-section in each market. To fix the massive stale data problem, we follow Ince and Porter (2006) and drop observations with security prices and trading volumes that have zero variance for more than one week during the periods. 5 Moreover, 20 or more stocks are required for each market in each month 3 See Appendix Table We use weekly data for empirical analysis and daily data for some of the cleaning rules.

11 to ensure meaningful analysis, and therefore even we collect data from July 1997 to July 2017 for all markets, the actual starting date for each company varies in our sample and also emerging/smaller markets tend to have shorter sample periods. The whole dataset includes 22 advanced markets: Australia (AU), Belgium (BG), Canada (CA), Denmark (DK), Finland (FN), France (FR), Germany (BD), Greece (GR), Hong Kong (HK), Israel (IS), Italy (IT), Japan (JP), Netherlands (NL), New Zealand (NZ), Norway (NW), Portugal (PT), Singapore (SG), Spain (ES), Sweden (SD), Switzerland (SW), United Kingdom (UK), and the U.S. (US); and 20 Emerging markets: Argentina (AR), Brazil (BR), Bulgaria (BL), China (CN), Egypt (EG), Hungary (HN), India (IN), Indonesia (ID), Korea (KO), Malaysia (MY), Mexico (MX), Morocco (MX), Philippines (PH), Poland (PO), Romania (RM), Russia (RS), Saudi Arabia (SR), South Africa (SA), Taiwan (TA), and Turkey (TK). Furthermore, we constructed six market groups: All advanced markets (22 advanced markets), all emerging markets (20 emerging markets), G7 excluding the U.S. (CA, FR, BD, IT, JP, and UK), 6 advanced European markets (BG, DK, FN, FR, BD, GR, IT, NL, NW, PT, ES, SD, SW, and UK), Eastern Asian markets (HK, JP, SG, CN, ID, KO, MY, PH, and TW), and Latin American markets (AR, BR, and MX). [Table 1] Table 1 provides summary statistics of stock return and main pricing variables for each market group and all major individual markets. 7 The results in Table 1 show that from year 1997 to 2017 emerging markets group has the highest average weekly return (0.43%) among all the market groups, but it also comes with the highest return standard deviation (4.10%). On the other hand, Latin American market has the lowest average weekly return (0.25%), and G7 market (excluding 6 The U.S. market is excluded from G7 market group because it is a much larger market than the rest and the results could be dominated by the U.S. market performance. 7 The definition for each variable can be found in Section 2.

12 the U.S.) has the lowest return standard deviation (2.62%). Among major individual markets, Germany has the highest average weekly return (0.45%) and Taiwan s average return is the lowest (0.06%). China s stock market is the most volatile with standard deviation of 4.93% and Mexico has the lowest standard deviation of 1.72%. In general, advanced markets are less risky and the average returns are higher than emerging markets. The average capital gain overhang is negative for most markets except for the U.S. and Argentina, indicating that for most international markets the weekly stock prices are on average lower than the reference prices. Average Betas are around 1 for most markets, while the standard deviation of Betas is the highest in Latin American markets and lowest in G7 markets. The signs are mixed for Fama-French pricing factors (SMB and HML), the momentum and the long term reversal variables. 4. Empirical analysis We conduct a series of empirical analyses, and the results are presented as follows: First, following Wang, Yan and Yu (2017), we report the excess stock returns of double sorted stock portfolios by their lagged CGO and Beta values. Second, we adopt the generalized linear model (GLM) to run regression of excess stock returns on CGO, Beta, and the interaction term between CGO and Beta for each market and market group in our sample. The portfolio analysis follows next where we sort common stocks in each market group into different portfolios based on some important fundamental variables: size, price-to-book ratio, trading volume turnover and the short term momentum, and compare the regression results between different portfolios. Lastly, we test the risk-return relation with CGO as the moderator under different market conditions and compare investors behavior during tranquil period and crisis period Double sorted portfolios

13 Double sorting is a simple way to compare the portfolio returns based on two variables of our interest. In this section, we are interested in comparing portfolio returns that are formed based on stocks Beta risk and CGO. At the beginning of each month, we sort all common stocks based on their lagged CGO for each market group and put them into five CGO groups with CGO1 being the group that has the highest CGO and CGO5 being the group that has the lowest CGO. Then within each CGO group, stocks are sorted based on their lagged Beta and further put into five Beta groups with Beta1 with highest Beta and Beta5 with lowest Beta. The portfolios are then held for one month and the equally-weighted weekly excess returns and return differences between Beta1 and Beta5 groups within each CGO group are reported in Table 2. [Table 2] Panel A reports the results for all market groups: all advanced markets, all emerging markets, G7 markets (excluding the U.S.), advanced European markets, Asian markets and Latin American markets. Generally speaking, within each CGO group, the higher the beta risk, the higher the excess returns for the portfolios except for the results in Asian and Latin American markets for CGO3 and CGO5 groups and in all emerging markets for CGO5 group. This suggests that positive risk-return trade-off holds for most advanced markets, but does not hold for some emerging markets especially when the CGO is low. When comparing across different CGO groups, we can see the lower the CGO, the weaker the risk-return relationship and sometimes negative risk-return relationship. For example, in all advanced markets, when CGO is high (i.e. prior capital gains), the excess returns for high-beta stocks are 16 basis points higher than those for low-beta stocks per week; when CGO is low (i.e. prior capital losses), the difference is only 10 basis points. So the positive risk-return relationship becomes weaker as CGO becomes lower. For all emerging markets, when CGO is high, the high-beta stocks have weekly excess returns about 10 basis points higher; when CGO is low, the

14 low-beta stocks have weekly excess returns almost 2 basis points higher. The negative risk-return trade-off seems to hold for emerging markets. This finding is consistent with Wang, Yan and Yu (2017) and suggests the U.S. evidence could be generalized to international markets. However, when we look more specifically into individual markets and use the same double-sort procedure, the results are somewhat mixed for individual markets. Panel B reports the results for G7 countries while Panel C reports for the major Asian and Latin American markets. Among the G7 countries, Germany, Italy, the U.K. and the U.S. have the same result as the overall G7 markets. Canada and Japan have all negative risk-return relationship for different CGO groups, but for lower CGO, they also have a more negative (larger absolute value for coefficient) risk-return relationship. France has a reversed trend, i.e. the lower the CGO, the stronger the positive risk-return relationship. Among the countries in Panel C, Eastern Asian markets (China, HK, Korea, and Singapore) have similar trend as the U.S./G7 markets result, suggesting that East Asian markets share more similar investor behavior with the U.S. market than with the Latin American markets. Even though double sorting method is a simple way to see the relationship, it also has some drawbacks. It cannot go beyond two factors. In later sections, we also want to explore stock fundamentals influence on risk-return relationship and we are looking at three factors of interest, double sorting can no longer help. It also cannot report the specific coefficients for certain independent variables of interest so that we do not know the magnitude of the impact. So the methods we use in the following sections are mostly based on generalized linear regression models GLM regression on stock returns We adopt the generalized linear model (GLM) as the main technique in our empirical analysis. GLM is a more flexible form of the ordinary least square (OLS) as it allows for response variables

15 that have error distribution models other than a normal distribution, and it is particularly suitable for our panel stock return data analysis Reduced GLM model for each market group According to the prospect theory and the findings from Grinblatt and Han (2005) and Wang, Yan and Yu (2017), prospect theory (PT)/ Mental accounting (MA) investors tend to be more risk averse when the capital over hang (CGO) is positive and more risk taking when the CGO is negative, so the risk-return relation under a positive CGO would be positive and under a negative CGO would be negative. Therefore, we should observe a positive and significant coefficient for the interaction term between CGO and Beta if the PT/MA investors widely exist in the market. The first model we test is a reduced GLM model derived from Equation 3: R i,t+1 = b 1 CCC i,t + b 2 BBBB i,t + b 3 BBBB i,t CCC i,t + ε i,t+1 (3 ) In this model, no other control variables are included in the regression except for the main variables: CGO, Beta, and the interaction term between CGO and Beta. 8 The regression results for market groups are reported in Table 3. 9 [Table 3] The coefficients for CGO have mixed signs, specifically, they are negative and significant for G7 (except U.S.) and advanced European markets, but are positive and significant for Asian and Latin American markets. The coefficients for Beta are positive and significant for all market groups, indicating that at the aggregate level, stock excess returns and risk are positive and significant for international financial markets. The coefficients for the interaction term between CGO and Beta, are 8 See section 2 for definition of the variables. 9 Most individual market results are consistent with their corresponding market group. To save space, we shall only report the results from market groups. Some of the individual market regression results can be found in section 4.2.3, and the rest are available upon requests.

16 also mostly positive and significant for the market groups except for the emerging markets group, which is negative and significant. The results confirm our hypotheses that investors that have prospect theory behavior generally exist in international markets, especially in advanced and Asian markets, where the risk-return relation is dependent on the stock s CGO value, i.e. under a positive CGO, the risk-return relation is positive while under a negative CGO, the risk-return relation is negative. They are also consistent with the findings of Wang, Yan and Yu (2017) in the U.S. market Full GLM model for each market group The reduced model provides evidence on how the risk-return relation is affected by CGO values in international markets. However, stock excess returns are affected by other pricing factors as well. Fama and French (1992, 1993) propose a three-factor model to explain the variation of crosssectional stock returns and they argue that small size stocks and high book-to-market stocks have return premiums over large and low book-to-market stocks. The model has been widely used in empirical asset pricing studies. Jegadeesh and Titman (1993) and Carhart (1997) find that buying past winning and selling past losing stocks generate significant positive returns, so the momentum factor is often used together with the Fama-French factors. As stock market momentum might be caused by investors overreaction on short term information (De Bondt and Thaler, 1985, 1987), stock returns can also experience a long term reversal. Therefore, we include the above pricing variables and form the full model to test the risk-return relation with the CGO as the moderator: R i,t+1 = b 1 CCC i,t + b 2 BBBB i,t + b 3 BBBB i,t CCC i,t + b 4 SSS t + b 5 HHH t + b 6 MMM t + b 7 RRR t + ε i,t+1 (3)

17 where SMB, HML, MOM, and REV are the corresponding pricing factors small minus big, high minus low, short term momentum winner minus loser, and long term reversal winner minus loser, 10 and the estimation results of Equation 3 for all market groups are reported in Table 4. [Table 4] The results in Table 4 are consistent with those in Table 3: the signs for coefficients of CGO are mixed, specifically, positive and significant for all emerging markets, Asian markets, and Latin American markets, while negative and significant for G7 (excluding the U.S.) and advanced European markets. The coefficients for the interaction term between Beta and CGO are mostly positive and significant except for all emerging market group. The pricing factors, SMB, HML, MOM, and REV, are significant for most market groups, and the signs of the coefficients are consistent with previous studies as well. 11 The results indicate that even if the widely used pricing factors are significant in explaining stock returns, it seems that they don t have much impact on the risk-return relation Full GLM model for major individual markets Since our data include a large number of markets and many of the smaller markets have much less stocks compared to those large markets, and especially in some small emerging markets very few stocks have enough data points to form the variables, 12 our later analyses are all based on market groups instead of individual markets. However, in this section, we present the estimation results of Equation 3 for some major individual markets that form those market groups. [Table 5] 10 See section 2 for definition of the variables. 11 For example, the signs for the coefficients of SMB and HML are positive and significant for most market groups, indicating that there are small and high book-to-market stock returns premiums. 12 For example, to compute Beta and CGO, the stock needs at least two years data without more than a month missing value.

18 Panel A of Table 5 shows the regression results for G7 markets: CA, FR, BD, IT, JP, UK, and U.S.; Panel B of Table 5 shows the regression results for some major Asian and Latin American markets: CN, HK, KO, SG, TW, AR, BR, and MX. The results of individual markets are in general consistency with those in their corresponding market group: the coefficients of CGO are positive (some not significant) for most individual markets except only for Italy and Argentina. The coefficients of the interaction term between Beta and CGO are mostly positive and significant for Asian and Latin American markets but have mixed signs for G7 markets (positive and significant in Italy, the U.K. and the U.S., negative and significant in Japan and Germany, insignificant in Canada and France). The results imply that investors in the same region tend to have similar behavior so it might be more meaningful to investigate markets in groups Portfolio analysis In this section, we present the empirical results for risk-return relation under CGO s impact for different portfolios in international markets. The stocks are sorted by the values of their market value (size), price-to-book ratio (P/B ratio), trading volume turnover (VO), and past 12 to past 1 month return at the beginning of every month (MOM). We then divide all stocks into five groups and for each market to form the portfolios. For market groups, we simply combine stocks of the same portfolio in all markets of the market group to form the market group portfolio Size sorted portfolios The estimation results for size sorted portfolios are reported in Table 6. In Panel A, Equation 3 is applied only to the largest 20% stocks in each market of the market groups and in Panel B, Equation 3 is applied only to the smallest 20% stocks in each market of the market groups To save space, the estimation results for middle-sized portfolios and for individual markets are not reported in the paper, but they are available upon request.

19 [Table 6] The coefficients for CGO are positive and significant for both large size and small size portfolios for most of the market groups, indicating that excess stock returns are positively affected by capital gain overhang, and it is consistent with Wang, Yan and Yu (2017). The coefficients for Beta are all positive and significant for small size portfolios, but they are mixed for large size portfolios, showing that the inconsistent risk-return relation are mostly for larger stocks. The coefficients for the interaction term between CGO and Beta have mixed signs as well, but they are more significant for smaller stock portfolios and have higher absolute value than those for larger stock portfolios, especially for all advanced market group, G7 (excluding the U.S.) group, advanced European market group and Latin American market group. The results suggest that the investors that hold smaller stocks tend to have more prospect theory behavior. It is likely that those investors who hold smaller stocks are more risk taking when they have negative unrealized capital loss than when they hold larger stocks Price-to-Book ratio sorted portfolios The estimation results for price-to-book ratio (P/B) sorted portfolios are reported in Table 7. In Panel A, Equation 3 is applied only to the highest 20% P/B ratio stocks in each market of the market groups and in Panel B, Equation 3 is applied only to the lowest 20% P/B ratio stocks in each market of the market groups. 14 [Table 7] Similar to size sorted portfolios, the coefficients for CGO are mostly positive and significant for both high P/B ratio and low P/B ratio stock portfolios for all market groups (except for Latin 14 To save space, the estimation results for middle-ranged P/B ratio portfolios and for individual markets are not reported in the paper, but they are available upon request.

20 American markets). The coefficients for Beta are all positive and significant for low P/B ratio stock portfolios, but they are less significant for high P/B ratio portfolios, showing that the inconsistent risk-return relation are mostly for high P/B ratio stocks. The coefficients for the interaction term between CGO and Beta have a sharp contrast between high P/B and low P/B stock portfolios: they are all positive and mostly significant (except for advanced European markets) for high P/B stock portfolios, but are mostly negative and significant (except for Latin American markets) for low P/B stock portfolios. The results suggest that investors holding high P/B ratio stocks tend to have more prospect theory behavior than those holding low P/B stocks, as investors who hold high P/B stocks are more risk taking when they have negative unrealized capital loss than those who hold low P/B stocks Trading volume turnover sorted portfolios The estimation results for trading volume turnover (VO) sorted portfolios are reported in Table 8. In Panel A, Equation 3 is applied only to the highest 20% VO stocks in each market of the market groups and in Panel B, Equation 3 is applied only to the lowest 20% VO stocks in each market of the market groups. 15 [Table 8] The coefficients for CGO are consistent with previous results that almost all coefficients are positive and significant for both high and low VO stock portfolios for all market groups (except for Latin American markets). The coefficients for Beta are also mostly positive and significant for both high and low VO stock portfolios (except for all emerging markets), indicating that the positive risk-return relation is pronounced for both heavily traded and thinly traded stocks in international 15 To save space, the estimation results for middle-ranged trading volume turnover portfolios and for individual markets are not reported in the paper, but they are available upon request.

21 markets. The coefficients for the interaction term between CGO and Beta are more positive and significant for low VO stock portfolios than for high VO stock portfolios (except for all emerging market group, which is the opposite). The results suggest that when investors hold thinly traded stocks, they are more likely to hold them even longer when they have negative unrealized capital loss. It might be because the thinly traded stocks have higher trading costs and investors will incur an even higher loss when selling those stocks Momentum sorted portfolios The estimation results for short term momentum (MOM) sorted portfolios are reported in Table 9. In Panel A, Equation 3 is applied only to the top 20% past winner stocks in each market of the market groups and in Panel B, Equation 3 is applied only to the bottom 20% past loser stocks in each market of the market groups. 16 [Table 9] The coefficients for CGO are mostly positive and significant for loser stock portfolios in all market groups (except for Latin American markets), but they are insignificant or negative for winner stock portfolios, suggesting that capital gain overhang affects stock excess returns more for past losers than for past winners. The coefficients for Beta are generally positive and significant especially for past winner stocks (except for Latin American markets), but they are mixed for past loser stocks, showing that the inconsistent risk-return relation mostly exist in past loser stocks. The coefficients for the interaction term between CGO and Beta vary differently between past winner and past loser stock portfolios: they are all positive and significant for past winner stock portfolios, but are either negative or not significant for past loser stock portfolios. The results imply that when past winner 16 To save space, the estimation results for middle-ranged P/B ratio portfolios and for individual markets are not reported in the paper, but they are available upon request.

22 stocks have negative unrealized capital losses, investors tend to be more risk taking and hold the stocks longer than those past loser stocks. Intuitively, the investors might believe that the past winner stocks have a higher chance to bounce back when its price is lower than the reference price Stock returns under different market conditions In this section, we present the empirical results for the same GLM estimation models under tranquil vs. crisis period. During a crisis period, investors usually face large losses and they tend to have more gambling mentality, i.e. risk-seeking behavior. 17 The crisis period is defined as from July 2007 to July 2009, when the subprime mortgage crisis spread into a global financial crisis. During this period, the global stock markets suffered greatly and S&P500 lost about one third over this period of time. The tranquil period is from March 2002 (the end of the dotcom bubble burst period) until June 2007 right before the start of the crisis. The results for the full GLM regression model under these two sub-periods for all regional markets are reported in Panel A and Panel B of Table 10 respectively. [Table 10] This table provides support for our prediction. During the crisis period, the interaction term between CGO and Beta is mostly significantly positive, except for Latin American group and all emerging market group. This is consistent with our hypothesis above that investors tend to be more riskloving during crisis and we will see negative risk-return relationship when CGO is negative. In Panel B, none of the markets shows significantly positive coefficient for the interaction term except for the Asian market. However, the magnitude for the Asian market during tranquil period (.0115) is much lower than that during the crisis period (.1098). The weaker moderating effect of CGO 17 Investors tend to have more behavioral anomalies during crisis periods. For example, herding activity is found more prominent under crisis and down market conditions (Chiang and Zheng 2010).

23 during tranquil period suggests prospect theory is at work to a larger degree in the Asian market during crisis period. 5. Conclusion This study examines the risk-return relationship anomaly from a behavioral finance perspective. The traditional CAPM theory states that investors earn higher returns by taking higher risks. However, empirically many studies have found opposite results (for example, Fama and French, 1992; Ang et al, 2006, 2009; Frazzini and Pedersen, 2014; Baker, Bradley and Wurgler, 2011). To investigate this anomaly, many have given different explanation such as wrong measures of risk, benchmarking, etc. This study explores this anomaly again in the international context from the prospect theory point of view following Wang, Yan and Yu (2017). Prospect theory argues that investors become more risk seeking when the stock price is below a certain reference price and become risk averse when the stock price is above the reference point (Tversky and Kahneman, 1992). As investors become more risk seeking, they tend to hold the risky stocks for too long, therefore resulting in lower subsequent returns. As investors become risk averse, the positive riskreturn relationship remains. To measure the prospect theory value, we follow Grinblatt and Han (2005) and Wang, Yan and Yu (2017) and use capital gain overhang (CGO), which is defined as how much the stock price is over or below the reference price point (calculated as weighted average historical prices). We first use firm-level data in 42 markets to form portfolios based on Beta and CGO (double sorting) to get preliminary results in different markets and find that in higher CGO groups, the positive risk-return relationship remains in all market groups including all advanced markets, all emerging markets, G7, advanced European markets, Asian markets and Latin American markets; in lower CGO groups, the relationship is much weaker and sometimes becomes negative.

24 We then use the generalized linear regression models (GLM) to investigate the moderator role CGO plays in the risk-return trade-off relationship. This method brings similar results as the double sorting method even after controlling for Fama-French, momentum, and long term reversal pricing factors. The next question arises: do stock fundamental variables matter? In other words, do investors have different behavior in terms of trading risky stocks for different types of stocks? By performing the same GLM method, we find that stocks that have smaller market capitalization, higher price-to-book ratio, lower trading volume turnover and higher momentum tend to see stronger impact from CGO on the risk-return relationship. This corroborates the prospect theory in that investors tend to more risk-taking with these types of stocks and therefore causing a more negative risk-return relationship when CGO is low. The different investor behavior during crisis period is also examined. During the global financial crisis from July 2007 to July 2009, CGO has a consistent and strong positive impact on the riskreturn relationship across all market groups while in the tranquil period before the crisis there is no significant impact from CGO except in the Asian market. This evidence again supports the prospect theory as investors possibly became more risk taking to try to recover their losses from the beginning of the crisis. When we compare the prospect theory effect among different market groups, it is evident that the risk-return relation is least affected by investors behavior in all emerging markets group.

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