Applications of Regret Theory to Asset Pricing

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1 Alications of Regret Theory to Asset Pricing Anna Dodonova * Henry B. Tiie College of Business, University of Iowa Iowa City, Iowa Tel.: address: anna-dodonova@uiowa.edu This aer resents a theoretical model of asset ricing that analyses how the behavior of stock returns is affected by the resence of regret averse investors on the market. Regret aversion is a well established sychological theory that suggests that some eole have regrets when they see that their decisions turn out to be wrong even if they aeared correct with information available ex-ante. The idea of regret extends naturally to finance by assuming that investors comare their returns with exogenous benchmarks. Using this assumtion, the model redicts that the market will over-react on good or bad news, so that there will be an excess volatility of stock returns. It also hels to exlain such wellestablished emirical uzzles as the ositive short-run and negative long-run autocorrelation of stock returns and it redicts a ositive correlation between future trading volume and the disersion of the realized stock returns. In addition to that, the framework develoed in this aer hels to analyze how an imrovement of stock market accessibility for non-rofessional traders affects the redictability of stock returns. Using recent stock market data the second art of the aer rovides an emirical analysis of some of the model s imlications. * I am esecially geful to Roger Gordon for his advice and numerous detailed comments. I also thank Sugato Bhattacharyya, Michael Brennan, Colin Camerer, Joshua Coval, James Hines, David Hirshleifer, George Loewenstein, Tyler Shumway, Richard Thaler, and articiants of the SIRIF Conference in Behavioral Finance, Edinburgh 2001 and Summer Institute on Behavioral Economics, Berkeley 2002 for helful discussions. 1

2 1 Introduction Until recently the main assumtions economists made about eole s references have had their roots in the rinciles ostulated by von Neumann and Morgenstern (1947). These axioms are believed to reresent the basis for ional choice under uncertainty. However, there are a number of economic, financial, and exerimental data that are not consistent with the ional agent hyothesis. The high mean and volatility of stock returns (Mehra and Prescott (1995)), the ositive short-run correlation of stock returns (Jegadesh and Titman (1993), Conrad and Kaul (1988, 1989)), and the negative long-run correlation of stock returns (De Bond and Thaler (1985, 1987), Fama and French (1988), Poterba and Summers (1988)) are among the uzzles that can not be exlained by any existing asset ricing model with fully ional investors. In order to exlain these uzzles a number of researchers try to incoroe ideas from sychology into asset ricing models. The majority of these models assume some kind of irionality in eole s behavior (e.g. overconfidence). Another aroach that assumes that eole are ional but care not only about their consumtion lies in between the two extreme aroaches of traditional consumtion-based asset ricing models and models involving some irionality in eole s behavior. Prosect theory, roosed by Kahneman and Tversky (1979) is robably the best-known reresentative of this aroach. Regret theory, indeendently roosed by Bell (1982) and Loomes and Sugden (1982), is another sychological model that considers ional agents whose concerns are not limited to their ayoffs. Both rosect and regret theories were used to exlain numerous evidences of violations of the exected utility theory axioms documented by 2

3 Kahneman and Tversky (1979). Even though these two theories describe two different well documented behavioral biases they both assume that a erson comares his wellbeing (consumtion, wealth, ortfolio return, etc.) with some benchmark. Prosect theory assumes that this benchmark is defined by the ast 1 while in regret theory this benchmark is not fixed ex-ante but her deends on the future state of the world. So, the main assumtion of regret theory is that eole after making their decisions under uncertainty may have regrets if their decisions turn out to be wrong even if they aeared correct with information available ex ante. This very intuitive assumtion imlies that erson s utility function among other things should deend on the realization of not chosen, and in this sense irrelevant, alternatives. That is why I believe that regret theory is very aroriate to the analysis of investors behavior. Indeed, in reality, an investor may observe not only his own ortfolio erformance but also returns on other stocks or ortfolios in which he was able to invest but decided not to. So, it seems very natural to assume that the investor may feel joy/disaointment if his own ortfolio outerformed/undererformed some benchmark ortfolio or ortfolios. This aer is aimed to analyze how ossible regrets affect investors ortfolio decisions and, thus, the behavior of stock returns. In the rest of the introduction I make a comarison between rosect and regret theories, review the existing aers that aly the results of rosect theory to asset ricing, and rovide the basic intuition for my model. In Part 2 of this aer I resent a theoretical model of asset ricing based on the assumtion that some investors may be regret averse and discuss its emirical imlications. To test whether the model is indeed a good model 1 Prosect theory assumes that a erson s utility deends on his gains/losses in comarison with his wealth 3

4 of investors behavior, in Part 3 of the aer I rovide an emirical analysis of some of the model s imlications. In Part 4 I conclude. Before I come to the exlaining of the underlying intuition for my model it is imortant to show how regret theory differs from rosect theory and how these differences may hel us to say something new about the stock market behavior. In contrast to the exected utility theory, rosect theory assumes that eole s utility is defined over their gains or losses in comarison with some reference oint and not over the value of their final assets. It also assumes that eole s utility from gain w is lower than their disutility from the same loss w and that eole are risk-averse over gains and risk-loving over losses. In addition to these loss aversion assumtions, rosect theory assumes that eole tend to overweight low robabilities and underweight high robabilities. In its turn, regret theory assumes that eole s utility deends on their wealth (in the same way as conventional exected utility theory does). However, it also takes into account the fact that eole may have regrets when their decisions turn out to be the wrong even if they aeared correct at the time they were made. So, as I have already mentioned, this assumtion imlies that erson s utility function among other things should deend on the realization of not chosen, and in this sense irrelevant, alternatives. in the revious eriod. 4

5 Decision regret can be illusted by a simle examle. Assume that a erson has a choice between lotteries L 1 and L 2. Lottery L 1 gives him $2 if the result of a fair coin toss is heads and $0 if it is tails. Lottery L 2 gives a certain ayoff of $1. Regret theory redicts that if the erson chooses lottery L 2 (a certain ayoff of $1), he receives higher utility when the result of the coin toss is tails than when it is heads. Prosect theory, however, does not distinguish these two alternatives since the reference oint in rosect theory does not deend on the future realization of irrelevant (not chosen) alternatives. Similar examle deals with the stock market. Assume that a regret averse investor chooses to invest in S&P 500 and gets 9% return. In this case he feels better about his investment decision if he knows that during the same eriod NASDAQ (that was another investment ossibility) gave 4% than if he knows that NASDAQ gave 15%. There are several aers that try to exlain the uzzling behavior of asset rices using the results of rosect theory. Benartzi and Thaler (1995) use the ideas of rosect theory to exlain the equity remium uzzle. They consider long-term investors who myoically evaluate their ortfolios and make myoic decisions from time to time. Using simulations, they find that if investors evaluate their ortfolios annually then the size of the equity remium is consistent with estimated arameters of rosect theory. Barberis, Huang, and Santos in their 1999 aer Prosect Theory and Asset Prices continue to ursue this rosect theory aroach. Their theory is based on the standard Lucas asset-ricing model (1978) and incoroes the ideas of rosect theory and the evidence of Thaler and Johnson (1990) on the influence of rior gains or losses on risky choice. They construct a model with only one risky asset that hels to exlain the high 5

6 mean, volatility, and long-run negative correlation of aggregate market returns. This aroach, however, fails to say anything about the behavior of individual stock returns. This haens because rosect theory does not distinguish sources of gains and losses. Thus, a change in an agent s risk aversion has the same effect on the rice of any articular security. So, for examle, if the whole market erformed well then the model of Barberis, Huang, and Santos (1999) redicts a decrease in an agent s risk aversion and an increase in rices of all securities (even of those that erformed oorly). To be able to say anything about individual stock returns rosect theory should be augmented with a theory of mental accounting. An attemt to build such a theory was made by Barberis and Huang (2000). They roose a model that tries to exlain the high mean, volatility, and negative long-run correlation of returns on an individual asset. In addition to the assumtions about investors behavior made by Barberis, Huang, and Santos (1999), they adot the idea of the seae mental accounting (Thaler [1985]). Barberis and Huang (2000) assume that an investor values each stock return seaely. In articular, they allow the ossibility for the investor to have different risk aversion while he invests in different stocks at the same time. This aer makes an imortant ste in the right direction. It recognizes that eole imlicitly kee track of different transactions in different mental accounts and that any theory that does not take this fact into account is incomlete 2. In reality by the time investors evaluate their ortfolio erformances they already have information about the erformance of every stock traded on the market and are able to 6

7 make a comarison between their ortfolio erformance and the erformance of any other hyothetical ortfolio. So, the benchmark with which they comare their returns is not fixed ex ante and there is a lot of room for regret since it is reasonable to assume that an investor feels the worse the larger the difference between his ortfolio return and the best ex ost return he could have received if he made a different investment decision. These effects of the cross sectional distribution of returns on the eole s utility cannot be catured by rosect theory 3 but are taken into account by regret theory. This aer resents a new aroach to the roblem of exlaining the high mean, volatility, and time series correlation of stock returns based on the ideas of regret theory. The intuition behind my model is the following: Assume that some asset erforms well at t=1 and that an investor initially did not invest much in it. The situation that delivers him maximum regrets is the one in which at t=2 the asset erforms well again since in this case he missed the right stock two times in a row. To insure himself against this situation, the investor has an incentive to urchase more of this first-eriod-good asset right after he gets news about is. This incentive leads to the uward shift of the aggregate demand function for that asset and, therefore, to the higher rice for it at t=1. Thus, ossible regrets lead to more volatile rices in comarison with the results roduced by the standard CAPM model. If combined with eole s risk aversion, higher rice volatility leads to the larger equity remium. 2 See Dodonova (2002) who also uses the ideas of rosect theory and mental accounting in the context of ortfolio choice, ortfolio management and asset ricing. 3 Prosect theory assumes that a erson s utility deends only on changes of his wealth, i.e., it deends only on his ortfolio return and not on the erformance of each articular asset (so, in this sense rosect theory does not distinguish the sources of gains or loses) 7

8 Moreover, such an overricing of the asset that erformed well leads to the negative long-run correlation of stock returns. The ositive short-run correlation comes from the fact that initially the market does not know how regret-averse investors are or how many of regret averse investors are on the market. This uncertainty roduces an additional risk, and, therefore, risk-averse investors who does not suffer from regret aversion bias demand an additional risk remium in the short run 4. 2 Model Assume that investors live during 3 eriods only. Denote these eriods as t=0, t=1, and t=2. There are 2 risky securities. Each risky security is a share of a firm whose value of the assets in lace is $1. Without loss of generality, we can assume that the suly of each security is exogenous and equal to 1. Each firm in eriods t=1 and t=2 genees rofits of D or D with equal robabilities. Each firm is liquidated at time t=2 with liquidation value of $1. The rofits are erfectly negatively correlated across firms at each time eriod 5 and indeendent across time. All rofits and the liquidation value are aid at t=2. The risk-free interest e is zero. There are 2 tyes of investors: active investors and assive investors. Active investors invest in stocks to earn money, they trade frequently and they are comletely ional with an increasing concave utility function u a (c) of their consumtion at t=2. Passive investors consider stocks as an instrument of saving and trade only if they feel that they 4 This intuition will become more clear after the model it stated. 8

9 made a wrong ortfolio allocation decision in the ast. In addition assive investors care not only about their consumtion but also about the fact that they may feel regrets if their investment decisions turn out to be the wrong ones ex ost. Their utility function is given by u ( 1 α ) ν ( max{ 1+ π + π, + π + } c) ( c) = α ua ( c) 1,1 1,2 1 2,1 π 2, 2 where π is the rofit of firm i at time t = j ; ν is an increasing convex function that i, j measures investor s regrets (so that assive investors are risk-averse in regrets 6 ); and α is a random variable between zero and one 7. Here, I comare a erson s ayoff c with the maximum ayoff he could have received if at t=0 he invested all his wealth in the best ex ost security 8 and did not change his ortfolio at t=1. This assumtion can be justified in the following way: The investor comares his ortfolio return with the maximum ossible return he could have had if at t=0 he made the best ex ost investment decision. Thus, the investor considers the ossibility of trade at t=1 as a ossibility to correct his investment mistakes made at t=0 and not as a ossibility to reallocate his ortfolio in order to receive additional gains. Since he comares his return with the best return (return that he could have received if at t=0 he made the best ex ost decision), he does not consider the ossibility of ortfolio reallocation at t=1 as the ossibility to beat the 5 The qualitative results of the model will be the same for any correlation between rofits. I assume erfectly negative correlation just to simlify the algebra because this assumtion allows me to get rid of the wealth effect and allow interest e to be equal to 0 (exogenous). 6 See Bell (1982) and Bell (1983) for arguments in favor of risk-aversion in regrets. 7 The results of the model will be the same if we assume that α is constant (i.e. the degree of regret aversion is known) but the number of regret-averse investors who decide to trade on the stock market is random. Furthermore, the long-run results of the model do not require such an assumtion at all. 8 The qualitative results of the model will be the same for any endogenous benchmark ortfolio that gives different weights to the best and the worst ex ost ortfolios. 9

10 best return. Since assive investors consider stocks only as a method of savings, I do not allow them to make short sales and to buy or sell otions. I also assume that when assive investors make their investment decisions at t=0 they do not realize that they are regret averse 9. Assume that both active and assive investors have an equal initial wealth of $1 and let the number of investors of each tye (active and assive) be normalized to 1. Consider the following sequence of events: 1) At t=0 trade takes lace at the market. At this time the rice of each asset is 1 and all investors ut ½ of their wealth in asset 1 and ½ of their wealth in i, 0 = the second asset 10. 2) At t=1 each firm i genees rofit π i, 1 Investors observe these rofits and trade takes lace. However, only active investors trade at the market at this time. 3) At t = 1+ regret aversion arameter α of assive investors is realized. Trade takes lace. Both active and assive investors trade at the market. 4) At t=2 each firm i genees rofit π i, 2. Then both firms are liquidated and the entire value of the firm i, equal to ( 1+ π i,1 + π i, 2 ), is aid to its shareholders. Consider how stock rices at t=1 and at t = 1+ react on the realized rofits π = D 1,1 and π = D (i.e., good news about firm #1 and bad news about firm #2). 2,1 9 This assumtion is needed only to simlify algebra and to have that initially all investors hold identical ortfolios. This assumtion does not affect the qualitative results. 10 This is so since there is no systematic risk in the model and the exected terminal ayoff is 1. 10

11 If all investors are ional, then at t=1 and at can be found from the following three equations: Investors maximize their objective function: t = 1+ rices are the same. These rices γw max E ua γ 1,1 + s. t.: w = 11, + + (1 γ ) w ( 1 + D + π ) + ( 1 D + π ) 2,1+ 1,2 2,1+ 2,2 (1) equilibrium condition for investors ortfolio allocation (share of investors wealth invested in the first asset): γ = 1,1+ 1,1+ + 2,1 + ; (2) and equilibrium (no arbitrage) condition: + 2, (3) 1,1+ = The first order condition of investors maximization roblem (1) together with the distribution of π i, 2 and no-arbitrage condition (3) is given by: D D γ u a 1,1 + 2,1 + 1, , ,1+ γ u a 1,1+ 1 γ ( 1+ 2D) + ( 1 2D) 1 γ + = 0 2,1+ 2,1+ + (4) Substitution of (2) and (3) into (4) allows me to receive: D D u a 1,1 + 2,1 + 1, ,1+ = 0 (5) 11

12 and in combination with 1,1+ + 2,1 + = 2 equation (5) imlies that stock rices under the assumtion of ional behavior are given by (6) and (7): 1,1+ = 1+ D (6) 2,1+ = 1 D (7) Now, consider the case in which there are regret-averse investors on the market. In this case stock rices are going to be different from (6) and (7). I want to find qualitative characteristics of stock rices at t = 1+. To do that, assume that rices at t = 1+ are equal to the rices derived under the assumtion that all investors are ional. That is, assume that 1,1+ = 1,1 + = 1+ D and = 2,1 + = D and find how regret-averse and ional investors would behave if 2,1+ 1 they observe these rices. As it was shown above, if 1,1+ = 1+ D and 2,1+ = 1 D, then at time t = 1+ ional 1+ D investors invest $ 2 1 D in asset #1 and $ 2 in asset #2. In an equilibrium the total investment in firm i should be equal to i,1+, regret-averse investors should also invest 1+ D $ in asset #1 and 2 1 D $ 2 in asset #2. The exected utility of assive (regret-averse) investor is given by 12

13 U { αu ( c) ( 1 α ) ν ( max{ 1+ D + π, D + } c) } ( γ ) = max E a 1,2 1 π 2, 2 γ (8) γ 1 γ 1 D where c = ( 1 + D + π ) + ( D + π ) 1 + D 1,2 1 2,2 Using the distribution of π i, 2, assumtions that 1,1+ = 1+ D and,1+ = 1 D 1, equilibrium condition γ 1+ D =, and the fact that 2 1+ D γ = maximizes u a (c), one 2 can receive: U ( γ ) 1 α 1 1 = γ 2 1 D 1 + D ( v (2D) v (0)) > 0 (9) Therefore, at these rices, regret-averse investors have incentives to invest more in asset #1 and the total investment in asset #1 is larger than 1,1+. Since investment in asset #1 deends negatively on and ositively on and in the equilibrium the total 1,1+ 2,1+ investment in asset #1 should be equal to 1,1+ one may conclude that it should be the case that: 1,1+ > 1,1+ 1 and = + D (10) 2,1+ < 2,1+ 1 = D. (11) 13

14 Note, that rices and are random and deend on the realization of the 1,1+ 2,1+ arameter of regret aversion α. Moreover, in the equilibrium regret-averse (assive) investors allocate more of their wealth in the asset that erformed well in the ast than ional (active) investors do. Now, consider time eriod t=1. At t=1 only active (ional) investors trade on the market. They know, however, that at t = 1+ rices are going to be = + D and = D. Therefore, at t=1 rices differ from 1,1+ > 1,1+ 1 2,1+ < 2,1+ 1 ional rices and satisfy equations (12) and (13): 1,1 > 1,1 = 1+ D (12) 2,1 < 2,1 = 1 D. (13) So, the model redicts that the market will overreact on good or bad news and, as a result, the model redicts excess rice volatility (in articular, asset rices are more volatile than the resent value of the underlying firm s future rofits). In combination with investors risk aversion, excess rice volatility leads to the larger exected returns. Thus, we have the following result: Proosition 1: In the resence of regret-averse investors on the market the volatility of stock rices is higher than the volatility of the underlying business characteristics of the firms (such as 14

15 future exected rofits) and exected stock returns are higher 11 than they would be in a market oulated only by risk-averse ional investors. While trading at t=1, active investors take rofit out of these rice differences. However, rices at t = 1+ rices into account and try to make a t = 1+ are random, so, riskaverse investors demand a risk remium. More recisely, active investors want to maximize their exected utility at t=2. Thus, the more stock rices at t = 1+ differ from the ional rices, the higher utility they have. Since rices at t = 1+ are random, ional investors want to insure themselves against the situation in which stock rices are close to the ional rices. As a result, they would tend to buy more of asset #2 (the one that erformed oorly) and less of asset #1 (the one that erformed well) even if it may give them negative exected rofits between eriods t=1 and t = 1+. This is so because such a stegy will give them more wealth in the bad state of the world (i.e. when stock rices are close to the ional rices). Therefore, rices at t=1 should satisfy (14) and (15): 1,1 < E1, 1+ (14) 2,1 > E2, 1+ (15) Note, that ast erformance gives no information about future erformance, therefore, exected rices at t=2 (given that π = D and π = D ) should satisfy (16) and (17): 1,1 2,1 11 Although in my model the exected return is zero (because all risk are diversifiable), a slight modification of the model that will allow for non-zero systematic risk will result in high exected return. 15

16 E 1,2 = 1+ D (16) E2,2 = 1 D (17) Equations (10), (14), and (16) imly that if asset #1 erformed well in the ast then 1,1 < E1, 1+ and 1,1 E1, 2 >. Therefore, the exected short-run return on asset #1 is ositive and its exected long-run return is negative. Equations (11), (15) and (17) imly that if asset #2 erformed oorly in the ast then 2 E and 2,1 < E2, 2, i.e.,,1 > 2, 1+ exected short-run return on asset #2 is negative and its exected long-run return is ositive. Thus, we have the following result: Proosition 2: In the resence of regret-averse investors on the market stock rices should exerience a negative long-run correlation. If, in addition, the degree of investors regret-aversion or the number of regret-averse investors on the market is not known until trade takes lace, then stock rices should exerience ositive short-run correlation. Another imortant imlication of the model can be found if we look at the difference in the trading atterns of ional and regret-averse investors. Regret-averse investors try to insure themselves against ossible risk associated with regrets. As a result, they tend to buy assets that erformed well in the ast and to sell those that erformed oorly. Since total asset suly is fixed, ional investors should behave in the oosite way and the trade takes lace between regret-averse and ional investors. Therefore, if the 16

17 disersion of the realized asset returns in the ast was large, then one can exect high trading volume today. Thus, we may state the following result: Proosition 3: In the resence of regret-averse investors on the market there exists a ositive correlation between future market trading volume and the disersion of the realized stock returns. One more way to think about stock market articiants is to divide them into two grous: rofessional traders (who derive the main art of their income out of the stock market trading) and non-rofessional traders (who consider stock market as an instrument of saving and whose wealth is rimarily derived from their labor income). In the framework of my model I may assume that rofessional traders are ional while non-rofessional traders are regret-averse. It is also seems lausible that rofessional traders trade more frequently. This framework allows me to analyze how recent changes of the market trading accessibility (like on-line brokerage) affect stock rices. There are two ways in which an increase of the market trading accessibility affects the trading by nonrofessionals. First, it increases the number of non-rofessionals on the market and the caital that they invest. Second, it allows non-rofessionals to trade more frequently. In the framework of my model one can redict, that an increase in the amount of the caital invested by non-rofessionals leads to an even more severe overreaction on good/bad news, and, thus, to an even higher volatility and mean of returns. Such an overreaction will increase the long-run negative correlation of stock returns even more. The easier and faster market access rovided by on-line trading allows some of the regret-averse investors to trade immediately after the arrival of news. In the framework of my model 17

18 this can be catured by allowing some of the regret-averse investors to trade at time t=1 instead of t = 1+. Such a modification will decrease the difference between stock rices at t=1 and t = 1+, i.e. it will decrease the ositive short-run correlation of stock returns. Since on-line trading has a much greater effect on the same-day access to the market than it has on the next-day access, the model redicts the larger effect of the better on-line trading accessibility on the autocorrelation of the daily returns than on the autocorrelation of weekly or monthly returns. The discussion above can be summarized in the following result: Proosition 4: In the resence of regret-averse investors on the market an introduction of on-line trading will lead to a higher mean and volatility of stock returns, higher long-run negative correlation of stock returns and higher deendence of future market trading volume on the disersion of realized stock returns. If, in addition, the degree of investors regretaversion or the number of regret-averse investors on the market is not known until trade takes lace, then an introduction of on-line trading will also decrease the ositive shortrun correlation of stock returns. The model described above considers an economy with only two risky assets, but it can be easily generalized to the case of many assets. Though this generalization lies beyond the scoe of this aer, I want to discuss several ossible ways of doing that. The key issue in the multi-asset model is the mechanism that investors use to determine the benchmark ortfolio. It is no longer lausible to exect an investor to comare his return with the best ex-ost return among all the assets because it is not feasible for him to 18

19 rocess all the information available on the market and it is highly unlikely that he will feel regret if some asset he never heard of (e.g., shares in a small firm in Uruguay) erforms extremely well. In the sirit of regret theory, the return on the benchmark ortfolio must be easily observable and an investor should consider the benchmark ortfolio as an alternative that was available to him in the ast but which he did not choose. In other words, there is a small subset of assets that investors use to construct the best ex-ost benchmark ortfolio. One ossible way to construct this subset is to restrict investors attention only to the well-known market indices (e.g. NASDAQ,, S&P 500, and DOW). Another ossibility is to restrict investors attention to the stocks of well-known comanies (e.g. the ones with the highest market values or with the highest historical growth level). One more ossibility is to allow the investor to form his benchmark ortfolio only out of the assets that he investigated in the ast. This aroach is even more aealing because it allows different investors to have different benchmark ortfolios. The alication of regret theory to investors behavior in the multi-asset setting will allow one to draw a number of additional emirical imlications about the stock rice behavior and ortfolio allocation. For examle, the behavior of stock returns (mean and volatility) may deend on whether or not this stock belongs to the set of stocks that can be used to construct a benchmark ortfolio. In articular, one may draw some conclusions about the effect of firm s market value on the behavior of its stock returns. Multi-asset setu of the model may also allow one to exlain why uninformed investors do not diversify enough, and, in articular, it may hel to exlain the home bias effect. 19

20 3 Emirical test of the model The fact that stock returns exhibit high mean and volatility as well as negative long-run and ositive short-run correlation is documented by a number of studies, and there are a number of theoretical models (both with fully ional and bounded ional investors) that try to exlain these henomena. The model resented in this aer is differentiated from the other models by its redictions about the effect of the on-line trading on the time-correlation of stock returns and by its rediction about the deendence of the trading volume on the disersion of the lagged realized stock returns. The emirical analysis resented below is aimed to test these new redictions. For this analysis I have chosen the CRSP data on the value-weighted, equal-weighted, NASDAQ and S&P500 market indices during the eriod. Since the year 1995 might be taken as a benchmark year when on-line trading became oular, my dataset consists of 6-year data eriods with and without significant on-line trading. In addition to these four indices I also considered the fifty stocks which were most traded as of January 1995 and that were traded during the whole eriod. Unfortunately, the size of the data set does not allow me to analyze the effects of on-line trading on the long-run autocorrelation of stock returns, which, I believe, are of higher magnitude than the effects of on-line trading on the short-run autocorrelation of returns. In order to check the effect of an introduction of on-line trading on the short-run correlation of stock return, I have estimated the following model using OLS estimation: 20

21 R t = α 0 + α1 Rt 1 + α 2 indt + α 3 Rit + t (18) where R t is the return on chosen index (or ortfolio) at time t, t 1 R is the return at time t- 1, ind t is an indicator function which is set to be equal to 1 for the time eriod and equal to 0 for the time eriod, and Ri t = R 1 ind. Table 1 t t resents the estimation results for daily, weekly and monthly data for the value-weighted, equal-weighted, NASDAQ and S&P500 indices and for the ortfolio of the fifty most traded (as of January 1995) stocks. In order to check the effect of on-line trading on the autocorrelation of individual stock returns, I have examined the return on buy winners and sell losers stegy. The stegy I used in my analysis is the following. Among the fifty stocks that I have chosen for my analysis, each eriod I have located five stocks that had the highest return and five stocks that had the lowest return in the revious eriod. The stegy is to buy the shares (equally-weighted) of these five winners and short-sell the shares (equally-weighted) of these five losers (so that each eriod the ortfolio worth is zero) and hold such a ortfolio for 1 eriod. Using daily, weekly, and monthly data, I estimated the following regression: Rt ind t = β 0 + β1 + t (19) 21

22 where R t is the return on this investment stegy at time t and ind t is an indicator function which is set to be equal to 1 for the time eriod and to 0 for the time eriod. The model redicts that an introduction of on-line trading should decrease the short-run correlation of stock returns, i.e. it redicts negative estimations of α 3 (in regression (18)) and β 1 (in regression (19)). Indeed, for the daily returns, all of the estimations of α 3 (Table 1A) and the estimation of β 1 (Table 2A) are negative and significant at the 8% level, and 4 out of 6 of these estimations are significant at the 2% level. All the weekly (Tables 1B and 2B) and monthly (tables 1C and 2C) estimation results, however, have insignificant estimations of both α 3 and β 1. This fact, again, is consistent with the model because the model redicts a much larger effect of an introduction of on-line trading on the daily data than its effect on the weekly or monthly data 12. Another rediction that makes my model different from the other theoretical models that deal with autocorrelation of stock returns is the rediction about the effect of the disersion of realized stock returns on the future trading volume. To test this rediction I have chosen the same 50 stocks (that were the most traded stocks as of January 1995 and 12 One may come u with another, non-behavioral reasoning for why short-run correlation decreased over the ast several years. In articular, since on-line trading leads to more frequent trading, the effect of nonsynchronous trading (one of the factors contributing to the ositiveness of short-run correlation of stock returns) became less significant in the ast several years. The imortant question of finding which factors (and at what degree) contributed to the observed decrease in the short-run correlation of stock returns lies beyond the scoe of this aer. However, I do not think that changes in the non-synchronous trading can exlain such a decrease alone. In articular, the return on the ortfolio of most traded (as of January 1995) stocks exerienced a significant reduction in the short-run autocorrelation desite the fact that the stocks belonging to this ortfolio were actively traded during the whole samle eriod (only three of the stocks had one non-trading day during the whole samle eriod and two out of these three days were in the suberiod). 22

23 existed during the whole eriod) as I did for the revious hyotheses. I estimated 2 regression models. First, I divided the data into 2 sub-eriods: the time eriod (eriod with small number of on-line investors) and the time eriod (eriod with large number of on-line investors) and estimated the regression: ln( Trade ) γ + (20) t = 0 + γ 1 timet + γ 2 RVt 1 t where Trade t is the number of shares (total for the 50 chosen stocks) traded at eriod t, time t = t is the time elased from the beginning of the samle (in number of eriods), RV t 1 is the disersion of the lagged realized stock returns defined as t 1 50 ( ri, t 1 rt 1 ) RV =, where r i t 1 is the return on asset i at time t-1 and i= 1 50 t 1 = r i, t 1 50 i 2, 1 r is the equal-weighted return on these fifty stocks at time t-1. Second, I ooled the data together and estimated the following regression: ln( Trade ) γ RVi + (21) t = 0 + γ 1 timet + γ 2 RVt 1 + γ 3 indt + γ 4 t t 23

24 where Trade t, time t and t 1 RV are the same as defined above, ind t is an indicator function which is set to be equal to 1 for the time eriod and to 0 for the time eriod, and RVi = RV 1 ind. t t Table 3 resents estimation results for regressions (20) and (21) using daily, weekly and monthly data. Consistent with the model s rediction, the majority of estimations of γ 2 are ositive and significant at the 5% level with the excetion of weekly data results for the time eriod (which is also ositive but significant only at the 10% level) and monthly data for and ooled eriods (which are insignificant). The ositive (at the 1% significance level) estimation of γ 4 imlies that the effect of the disersion of the lagged realized stock returns on the trading volume becomes larger during time eriod. This result is also consistent with the model because the larger number of non-rofessional traders (resulted from the better access to the market) results in the larger reaction of the trading activity on the disersion of the lagged realized returns. 4 Conclusion This aer resents a model of asset ricing under the assumtion that some investors may have regrets. It considers 2 tyes of investors: ional active investors whose goal is to make money by trading on the market, and regret-averse assive investors who consider the stock market as an instrument of saving. The model does not assume any information asymmetry among investors, nor it assumes the existence of a signal of any 24

25 kind about future returns or any consistent attern in earnings and rofitability of underlying firms. The model redicts that the market will over-react on good or bad news and exlains the high mean and volatility of stock returns as well as ositive short-run and negative long-run correlation of stock returns. It also redicts the ositive correlation between future market trading volume and the disersion of the realized stock returns. My model considers how stock market accessibility affects stock rice behavior and redicts that an imrovement in the stock market accessibility for non-rofessional traders leads to an even higher mean and volatility of stock returns and to the larger negative long-run correlation and smaller ositive short-run correlation of returns. The emirical results resented in the aer are consistent with the redictions from the theory that better stock market accessibility gained by non-rofessional traders through on-line trading decreases the short-run correlation of stock returns, and that future trading volume deends ositively on the disersion of the realized stock returns. 25

26 Table 1 Short-run correlation of stock returns Value-weighted return (0.070) Equal-weighted return NASDAQ (0.300) S&P stocks ortfolio Const R t-1 Ind t Ri t A: Daily returns (0.330) (0.128) (0.001) Value-weighted return (0.031) Equal-weighted return NASDAQ (0.196) S&P stocks ortfolio (0.077) Value-weighted return (0.050) Equal-weighted return (0.304) NASDAQ (0.351) S&P stocks ortfolio (0.096) (0.002) (0.582) B: Weekly returns (0.292) (0.523) (0.050) (0.481) C: Monthly returns (0.609) (0.003) (0.197) (0.215) (0.577) (0.920) (0.255) (0.229) (0.966) (0.313) (0.986) (0.261) (0.210) (0.726) (0.322) (0.660) (0.327) (0.154) (0.903) (0.079) (0.015) (0.061) (0.002) (0.928) (0.366) (0.200) (0.622) (0.878) (0.930) (0.372) (0.376) (0.873) (0.500) This table resents the estimation results of the OLS regression of ortfolios returns on the lagged return (R1), indicator (ind =1 for the time eriod and ind=0 for the time eriod), and indicator times lagged return (R1i=ind*R1). The significance levels for each of the estimations are given in the brackets. 26

27 Table 2 Return to buying winners and selling losers Const A: Daily return (0.320) B: Weekly return C: Monthly return (0.018) Ind t (0.001) (0.346) (0.151) This table resents the estimation results of the OLS regression of returns on the buying winners and selling losers investment stegy on indicator (ind =1 for the time eriod and ind=0 for the time eriod). The significance levels for each of the estimations are given in the brackets. 27

28 Table 3 Trading volume and returns volatility time eriod time eriod time eriod time eriod time eriod time eriod time eriod time eriod time eriod Const Time t RV t-1 Ind t Rvi t A: Daily data B: Weekly data (0.094) (0.021) C: Monthly data (0.675) (0.792) (0.436) (0.012) (0.001) (0.001) This table resents the estimation results of the OLS regression of the logarithm of the number of shares traded at the market on time elased from the beginning of the samle eriod (time, in number of eriods), lagged variance of returns (RV1), indicator (ind =1 for the time eriod and ind=0 for the time eriod), and indicator times lagged variance in returns (RV1i=ind*RV1). The significance levels for each of the estimations are given in the brackets. 28

29 References 1) Ang, Andrew, Geert Bekaert, and Jun Liu (2001), Why Stocks May Disaoint, working aer, Columbia University and UCLA 2) Barberis, N., M. Huang, and J. Santos (1999), Prosect Theory and Asset Prices, Quarterly Journal of Economics 116, Barberis, Nicholas and Huang, Ming (2000), Mental Accounting, Loss Aversion, and Individual Stock Returns, Journal of Finance, 56, ) Bell, David (1982), Regret in Decision Making under Uncertainty, Oeions Reserch 30, ) Bell, David (1983), Risk Premium for Decision Regret, Management Science 29, ) Benartzi, Shlomo and Richard Thaler (1995), Myoic Loss Aversion and the Equity Premium Puzzle, Quarterly Journal of Economics, CX, ) Conrad, C., and G. Kaul (1988), Time-Variation in Exected Returns, Journal of Business, 61, ) Conrad, C., and G. Kaul (1989), Mean-Reversion in Short-Horizon Exected Returns, Review of Financial Studies, 2, ) Chora, Lakonishok, and Ritter (1992), Measuring abnormal erformance. Do Stock overreact? Journal of Financial Economics, 31 9) De Bond, Werner and Thaler, Richard (1985), Does the Stock Market Overreact? Journal of Finance, ) De Bond, Werner and Thaler, Richard (1987), Further Evidence on Investors Overreaction and Stock Market Seasonality, Journal of Finance,

30 11) Dodonova, Anna (2002), Portfolio Allocation and the Choice of Benchmarks: Why Investors do not Diversify Enough? working aer, University of Michigan 12) Fama, Eugene and French, Kenneth (1988), Permanent and Temorary Comonent of Stock Prices, Lournal of Political Economics, 31 13) Grinblatt, Mark and Han, Bin (2001), The Disosition effect and Momentum, working aer, Anderson Graduate School of Management, UCLA 14) Hirshleifer, David (2001), Investor Psychology and Asset Pricing, working aer. 15) Jegadeesh, Narasimhan and Titman, Sheridan (1993), Returns to Buying Winners and Selling Losers: Imlication for stock Market Efficiency, Journal of Finance, ) Kahneman, Daniel and Tversky, Amos (1979), Prosect Theory: an Analysis of decision under Risk, Econometrica 47, ) Loomes, Graham and Sugden, Robert (1982), Regret Theory: an Alternative Theory of Rational Choice under Uncertainty, The Economic Journal 92, ) Lucas, Robert (1978), Asset Prices in an Exchange Economy, Econometrica 46, ) Mehra, Rajnish and Edward Prescott (1985), The Equity Premium Puzzle, Journal of Monetary Economics, XV, ) Odean, T. (1998), Are investors reluctant to realize their losses, Journal of Finance, 53, ) Poterba, James M. and Lawrence H. Summers (1988), Mean Reversion in Stock Prices, Journal of Financial Economics 22,

31 22) Shumway, T. (1998), Exlaining returns with loss aversion, University of Michigan Business School Mimeo. 23) Thaler, R. H. (1985), Mental Accounting and consumer choice, Marketing Science, 4, ) Thaler, Richard H. and Eric J. Johnson (1990), Gambling with the House Money and Trying to Break Even: The Effect of Prior Outcomes on Risky Choice, Management Science 36, ) Von Neumann, J and Morgenstern, O (1947), Theory of Games and Economic Behavior, Ed.2. Princeton University Press, Princeton, N.J. 31

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