Regret Theory and Equilibrium Asset Prices

Regret theory is a behavioral approach to decision making under uncertainty. In this paper we assume that there are two representative investors in a frictionless market, a representative active investor who selects his optimal portfolio based on regret theory and a representative passive investor who invests only in the benchmark portfolio. In a partial equilibrium setting, the objective of the representative active investor is modeled as minimization of the regret about final wealth relative to the benchmark portfolio. In equilibrium this optimal strategy gives rise to a behavioral asset priciting model. We show that the market beta and the benchmark beta that is related to the investor’s regret are the determinants of equilibrium asset prices. We also extend our model to a market with multibenchmark portfolios. Empirical tests using stock price data from Shanghai Stock Exchange show strong support to the asset pricing model based on regret theory.


Introduction
The traditional asset pricing models that assume investors are homogeneous cannot explain many anomalies in the financial markets such as the equity premium puzzle [1] and the risk-free rate puzzle [2].They cannot depict the complex behaviors and ignore the diversification of psychology of different investors.
Behavioral asset pricing theories have emerged and grown during the past two decades in part as a reaction to the phenomena described above.Based on the behavioral theories such as Tversky and Kahneman [3] and assuming investors have heterogeneous beliefs, several behavioral asset pricing models have been proposed that revise the investor's utility from different perspectives.For example, Bakshi and Chen [4] care about the investor's relative social status, Constantinides [5], Abel [6] and Campbell and Cochrane [7] consider the habit formation of investors, Barberis et al. [8] focus on investors' loss aversion, while Abel [9] and Gollier [10] explore the envy between investors.Shefrin and Statman [11] derive a behavioral model based on the noise trading theory.In their model, there are two kinds of traders, information traders and noise traders, who interact and affect asset prices.Many of these studies assume that all investors are the same and do not consider the actual investing process of different types of investors.In this study we assume that investors are heterogeneous: there are two kinds of representative investors in a frictionless market, a representative passive investor and a representative active investor.The representative passive investor invests only in the benchmark and the representative active investor selects his own optimal portfolio based on the regret theory.
Regret theory is developed by Bell [12] and Loomes and Sugden [13].Regret aversion is a well-established psychological theory suggesting that people often have regrets when they see that their decisions turn out to be wrong even if they appeared correct with information available exante.The idea of regret extends naturally to finance by assuming that investors compare their returns with exogenous benchmarks.Clarke et al. [14] argue that investors optimize the tracking error due to regret aversion.Wagner [15] develops an asset selection model assuming the investor's utility is based on the regret theory.The model is labeled as the mean-variancecovariance (EVC) criterion.Dodonova and Khoroshilov [16] present a theoretical model of asset pricing that analyses how the behavior of stock returns is affected by the presence of 2 Mathematical Problems in Engineering regret-averse investors in the market.Gollier and Salanié [17] assume that agents are subject to regret and show that regret reduces the equity premium when the macrorisk is positively skewed.In this study we examine the consequences of investors' regret aversion on the optimal decisions under risk, the allocation of risk in the economy, and equilibrium asset prices.
Our paper complements and extends the extant literature of Brennan [18], Gómez and Zapatero [19], Cornell and Roll [20], Cuoco and Kaniel [21], and Brennan et al. [22].In this paper the objective of the representative active investor is modeled as minimization of regret about final wealth relative to the benchmark portfolio in a partial equilibrium setting.Our research differs from the study of Dodonova and Khoroshilov [16] who focus on the impact on volatility and autocorrelation of stock returns and trading volumes.
The rest of the paper is organized as follows.In Section 2 we present our portfolio selection model based on the regret theory.Equilibrium asset prices are analyzed in Section 3. In Section 4 we extend the pricing model to a market with a riskless asset.We also extend the model to multibenchmark in Section 5. Section 6 offers some empirical tests of the asset pricing model.Concluding remarks and possible future research are collected in Section 7.

Portfolio Selection Based on Regret Theory
We assume that  and  are the portfolios of the investor and the benchmark, respectively.The utility of investor is (  ,   ), where   and   are the final wealth after one period.(  ,   ) is twice continuously differentiable.Generally one can measure regret by the change in utility (  ,   ) with respect to a change in the hypothetical final wealth   .According to Bell [12] and Loomes and Sugden [13], we define regret as According to the classical setting, utility is assumed to be a strictly increasing and concave function of final wealth   ; that is, With respect to regret , we assume that the utility function also obeys the following restriction: We assume that there are  risky assets, no riskless asset (which will be introduced in Sections 4 and 5), and the return of the risky assets is r ( = 1, 2, . . ., ).The representative investor selects his optimal portfolio using the regret theory.We assume that the utility function of the investor is quadratic.Expanding the utility function (  ,   ) as per Taylor series, we get where (  ) and Var(  ) are constants because the benchmark portfolio  is given exogenously.According to Wagner [15], to maximize the utility, the problem that the investor needs to solve is where  is the weight vector of risky assets in the portfolio of investor,  is the vector of expected returns,  = ( 1 ,  2 , . . .  ),   is the weight vector of risky assets in benchmark portfolio, and  1 (> 0) and  2 (≥ 0) are the coefficients of absolute risk aversion and regret aversion of the representative investor, respectively.The Lagrange function of ( 5) is where  is the Lagrange multiplier.According to the first order condition, we get Substituting for  in the constraint condition    = 1 from (7), we get where (   −1 /   −1 ) and    −1  are the expected return and variance of the global minimum variance portfolio of risky assets.We use the notation  to represent the global minimum variance portfolio, and   and Var(r  ) are its expected return and variance.From ( 7) and ( 8), we get Equation (9) shows that in the investor's optimal allocation the weight on the benchmark portfolio increases as his regret aversion increases.

Equilibrium Asset Prices
From now on we consider an economy with heterogeneous investors.There are two representative investors in a frictionless market, a representative passive investor and a representative active investor.We assume  is the market portfolio, and   is the weight vector of risky assets in the market portfolio. (0 ≤  ≤ 1) is the fraction of the representative passive investor who invests only in the benchmark, and 1 −  is the fraction of representative active investor who selects his optimal portfolio based on the regret theory.When the market clears, we have Multiplying both sides by the covariance matrix  and substituting ( 9) into (10) yield Premultiplying (11) by   gives the variance of the market portfolio: Based on Lemma A.1 as in the Appendix, we have where  / and  / are the vectors of individual asset betas with respect to the market portfolio and the benchmark portfolio, respectively, and Var(r  ) is the variance of the benchmark return.Substituting Var(r  ) into (13) gives the vector equation that describes the cross-sectional relationship between betas and expected returns: where  / = cov(r  , r )/var(r  ).The th entry in the system of ( 14) is where  0 = ( 2 −  1 )/ Var(r  ),  = ((( 1 −  2 ) +  2 )/(1 − )) Var(r  ), and  / is the beta for asset  computed against portfolio .Equation ( 15) is the asset pricing model based on regret theory that determines the equilibrium asset prices.From (15) we have the following observations.
(1) In equilibrium two types of risk are priced in the market, the market risk and the benchmark risk. /  / −  / is the beta for the benchmark risk.By construction it is orthogonal to the market risk, which deals with the fact that the market portfolio and the benchmark portfolio are likely correlated.
In this way the two risk factors are independent from each other.The benchmark beta can be positive or negative.Generally speaking  /  / −  / is positive for assets that are more correlated with the market portfolio than with the benchmark.(2) When the benchmark is the same as the market portfolio,  / = 1 and ( /  / −  / ) = 0, then ( 15) is similar to the traditional CAPM.(3) Everything else being equal and assuming  1 >  2 , the more passive investors in the market (i.e., the greater ), the greater the impact of the benchmark risk (i.e., the greater ).This is because passive investors invest in the benchmark portfolio only.

Asset Pricing Model with Riskless Asset
We do not consider a riskless asset in Section 3.With the introduction of a riskless asset in the frictionless market as Section 2, the optimal portfolio selection problem of the investor is where   is the return rate of the riskless asset.The first order condition now becomes When the market clears we obtain Similar to (14), we have The th entry in the system of ( 19) is Equation ( 20) is the asset pricing model based on regret theory when there is a riskless asset in the market.If the market portfolio is same as the benchmark, ( 20) is the traditional CAPM.

Pricing Model with Multi-Benchmark
In Sections 3 and 4 there is only one benchmark in the market.
In this section we investigate the situation when the investor's wealth of portfolio is measured against two benchmarks  1 and  2 .This is common in the investment industry, for instance, when an investment manager is assessed against a market portfolio as well as an internal benchmark (Wang [23]).The utility function of investor (  ,   1 ,   2 ) is assumed to be a strictly increasing concave function of final wealth   : We define investor's regret to benchmark  1 and  2 as As the situation with just one benchmark in the market, with respect to regret  1 and  2 , we assume that the utility function obeys the following restrictions: Assuming the utility function is quadratic and expanding it in Taylor series, we obtain Benchmarks  1 and  2 are exogenously given, so (   ), Var(   ) ( = 1, 2) and Cov(  1 ,   2 ) are constants to investors in their portfolio selection problem.

Model without Riskless
Asset.Similar to the method in Section 2, to maximize his utility, the problem that the active investor needs to solve is where  is the weight vector of risky assets in the investor's portfolio,  is the vector of expect return,   1 and   2 are the weight vectors of risky assets in benchmark portfolios  1 and  2 , respectively,  1 (> 0) is the coefficient of absolute risk aversion, and  2 (≥ 0) and  3 (≥ 0) are the coefficients of regret aversion of the representative investor to benchmark portfolio  1 and  2 , respectively.The optimal portfolio selection of the investor is We assume that  1 (0 ≤  1 ≤ 1) and  2 (0 ≤  2 ≤ 1) are the factions of representative passive investors who only invest in benchmark portfolios  1 and  2 , respectively.Then 1− 1 − 2 is the faction of the representative active investor who selects his own optimal portfolio based on the regret theory.When the market clears, we get Similar to Section 3, the th entry in the system of ( 27) is where The structure of ( 28) is similar to that of (15), but it has one more risk factor.Besides the market risk, in equilibrium the expected return also depends on two benchmark risks, which, by design, are independent of the market risk.

Model with Riskless
Asset.In Section 5.1 we do not consider a riskless asset, but now we assume there is a riskless asset in the frictionless market, and   is its return.The optimal portfolio selection problem is The optimal portfolio of the investor is When the market clears we obtain Similar to Section 5.
The th entry in the system of (33) is The structure of (34) is similar to that of (20) but with the inclusion of two benchmark risk factors that are independent of the market risk.

Empirical Tests
6.1.Data.We take the Shanghai Stock Exchange (SSE) 180 index as the benchmark portfolio.The SSE Stock Composite Index (CI) weekly return series is taken as the market portfolio return.For the locally risk-free asset, the weekly return series of the three-month Treasury Bill is used.Our sample begins in January 2006 and extends through December 2012.
As for risky assets, we select the 150 stocks in the SSE 180 index without interruption from January 2006 to December 2012.We select this subsample of SSE 180 to avoid the possible price effect associated with changes in the composition of the index.Thus any abnormal return captured in our test cannot be explained by the assets being added or deleted from the benchmark index.

Methodology.
To test the asset pricing model (20), we take a three-step approach as in Brennan et al. [22] and Gómez and Zapatero [19] in the spirit of Fama and MacBeth [24].
First, in order to eliminate the linear dependence of the market portfolio and the benchmark portfolio, we obtain the residual by means of where r and r denote the weekly return of the market portfolio and benchmark portfolio, respectively.The residual from regression (35), ê, represents the component of the benchmark that, by construction, is independent of the market portfolio.Second, we estimate the betas of the market and benchmark portfolio according to  (r  ) −   =  0 +  / ( (r  ) −   ) +  / ê +   , (36) where  / represents the benchmark beta that, by construction, is orthogonal to the market beta  / .
Finally, we run a cross-sectional regression of stocks' expected returns on the estimated betas as According to the regret theory, the benchmark risk should be priced in equilibrium stock prices.So we expect  2 to be significant.Following the methodology in Gómez and Zapatero [19], the 150 stocks in SSE 180 index are sorted into 10 portfolios according to their estimated market index beta.We summarize the empirical results of (37) in Table 1. 1,  2 is highly significant for all the portfolios, which supports the prediction of the asset pricing models based on regret theory.When the benchmark risk is taken into account, the market risk, as measured by  1 , is no longer significant (with the exception of panel 1).This result is consistent with the findings in Chen et al. [25], Wen and Yang [26], Wu [27], and Morelli [28], who report that in Chinese stock markets the market risk is often not priced when other risks (e.g., size, value, liquidity, and skewness) are also considered.

Empirical Results. As we can see in Table
Shanghai stock market displays some unique characteristics compared to stock markets in many developed countries.Among the over 900 stocks listed at SSE, the SSE 180 Index includes the top companies ranked by market capitalization and trading volume in all ten major industries.As industry leaders, the SSE 180 companies have experienced tremendous growth since the index was established in July 2002.According a report from SSE (http://www.sse.com.cn/market/sseindex/bluechips/introduction/),during the period 2006-2010 SSE 180 index experienced an average annual growth rate of 24.64%, higher than the annual return of 19.32% for the SSE Composite Index.At the year end of 2010, SSE 180 has an average P/E ratio of 18.23, compared to 21.61 for SSE CI.This risk and return profile of SSE 180 index is in contrast to so cov(r  , r ) =  / Var(r  ), that is,   is the vector of covariance between the portfolio  and individual assets.From the assumptions in Lemma A.1, we have  / Var(r  ) =   .