An analysis of momentum and contrarian strategies using an optimal orthogonal portfolio approach

An analysis of momentum and contrarian strategies using an optimal orthogonal portfolio approach Hossein Asgharian and Björn Hansson Department of Economics, Lund University Box 7082 S-22007 Lund, Sweden Phone: +46462228668. Fax, +46462224118, Hossein.Asgharian@nek.lu.se Björn.Hansson@nek.lu.se. Abstract We use a latent factor approach to investigate if the momentum and contrarian profits, observed in the US stock market, should be considered as risk premiums or have non-riskbased explanations. The model is also employed as a benchmark to assess the explanatory power of the traditional asset pricing models in this contest. Our findings show that the profits of the long-run contrarian strategy are related to some other background risk factors, while the momentum and the short-run contrarian profits are mostly non-risk-based. The latter finding mainly supports the behavioural finance explanation of these anomalies. JEL: G11, G12. Keywords: Orthogonal portfolio, asset pricing, CAPM-anomalies, contrarian, momentum. 1

1. Introduction Momentum and contrarian profits are two well-known anomalies in stock markets. Jegadeesh and Titman (1993) sort stocks listed on NYSE based on their prior six month return and show that investing in a group of prior winner stocks results in a higher average return than investing in a group of prior losers. On the other hand, De Bondt and Thaler (1985) sort firms based on three years historical returns and find that the average annual return of the prior loser portfolio is higher than the average return of the prior winner portfolio. There are different motivations for the existence of the momentum and contrarian profits. The traditional asset-pricing literature tries to explain these CAPM-related anomalies by connecting them to some background risk factors not captured by the market portfolio or by changing the assumptions of the CAPM. Fama and French (1996) use two additional factor mimicking portfolios to the market portfolio and Ang et al. (2001) explain the profitability of investing in momentum strategies as a compensation for bearing high exposure to downside risk. On the other hand, the literature in behavioural finance tries to explain the momentum and contrarian effects as a result of investors systematic errors or psychological characteristics. According to Barberis and Shleifer (2003) investors have propensities to buy assets that have experienced price rise in past. The increase in demand for these assets will raise their prices even more in the short-run. The subsequent price correction toward the fundamental value will result in a reversal, which motivates a long-run contrarian strategy. Jegadeesh and Titman (1993) and Hong and Stein (1999) argue that investors underreaction to news release results in gradual convergence of the asset prices toward the fundamental value, causing momentum in the price movements. On the other hand Jegadeesh and Titman (1995) argue that the short-term contrarian profits can be explained by investors overreactions to firm-specific information. 2

As pointed by Moskowitz and Mark Grinblatt (1999), determining whether these anomalies are rooted in behavior that can be exploited by more rational investors at low risk has profound implications for our view of market efficiency and optimal investment policy. The purpose of this paper is to assess to what extent the average return of these strategies can be explained as compensation towards risk versus a result of behavioral irrationality. We employ a latent factor model that can identify the pricing impact of unknown factors without trying to specify the background factors. The model is based on the optimal orthogonal portfolio approach introduced by MacKinlay and Pastor (2000) and extended to asset pricing tests by Asgharian and Hansson (2005). This approach enables us to estimate the maximum amount of the expected returns that can be related to the risk factors. Thereby we can measure the amount of the realized average return that is due to non-risk-based components. This model can also be used as a benchmark for assessing the performance of conventional asset pricing models in explaining these anomalies. These models rely entirely on observed factors and might therefore fail to capture the total risk exposure of the contrarian or momentum portfolio strategies. We use two well-known asset-pricing models for this comparison, i.e. CAPM and the three-factor model of Fama and French (1993). Most of the previous research on the US data indicates that the contrarian strategy is beneficial over short and long horizons, while the momentum strategy works well over medium horizons (between one month and one year). We build or analysis on these findings and use three different portfolio sets based on these three horizons. Overall, our suggested model is more successful in explaining the mean excess returns comparing to the two traditional factor models. We find that the profit from the long-run contrarian strategy is a compensation for risk, while the profits from the short-run contrarian and the momentum strategies are mostly non-risk-based. The latter results mostly support the behavioural finance explanation of these anomalies. 3

The remainder of the paper is organized as follows: section 2 presents the empirical model and the estimation methods; the empirical results and the analyses are in section 3; and section 4 concludes the study. 2. The model and estimation method Consider the following model for N assets: R α + =, [ ε ] N( 0,Σ) t ε t E t ~, (1) where R t is an (N 1) vector of excess returns on N assets. The mean vector, α, in equation (1) consists of two parts, a risk-based component and a non-risk-based component. MacKinlay and Pastor (2000) define a factor model, in which they assume that, α, in equation (1) is 2 entirely due to some latent risk factor h, with mean μ h and variance σ h. In this case, the factor pricing model can be written as: where =, [ u ] N( 0,Φ) R + t βhrht ut E t ~ (2) β h is an (N 1) vector of the sensitivities of N assets to the factor h. Combining equations (1) and (2) gives: α + ε + (3) t = β hrht ut Taking expectation and the variance of both sides gives α = βhμh. (4) 2 Σ = β β σ + Φ h h By substituting for β h we can relate the vector, α, to the residual covariance matrix Σ. This link provides a joint estimate of α and Σ. h To estimate the parameters of the model we maximise the following restricted likelihood function: 4

T T 1 1 ln L( αβ,, θh, Φ) det ( αα θh + Φ) ( t t) R α βf ( αα θh + Φ) ( Rt α βf t) (5) 2 2 t= 1 where θ h is the inverse square Sharpe ratio of the orthogonal portfolio, which is an unknown parameter and should be estimated. It can be shown that the higher the θ h, the more important the contribution of the observed factor in explaining expected returns. MacKinlay and Pastor (2000) presume that this parameter is known. Thus, their approach determines a priori the contribution of the latent factors to expected returns. In contrast, we identify the optimal orthogonal portfolio h endogenously. Since h can be considered as a linear combination of all missing risk factors it should be able to account for entire risk premium included in the mean return (see Asgharian and Hansson (2005)). Therefore the difference between the sample mean excess return and the estimated α, which stands for ( β h μ h ) is a measure of the non-riskbased components of the realised mean excess returns. A bootstrap procedure is used to test for significance of the non-risk-based components of the test portfolios. We resample with replacement the original data 200 times and re-estimate the model using the new samples. The empirical confidence intervals are used for inferences. 1 3. Analysis The analysis is divided into two parts; first we see if our three candidate factor models can explain the mean excess return of the sorted portfolios. Next, we form the zero investment strategies from our sorted portfolios based on the different expected anomalies: the momentum strategy, short-run and long-run contrarian strategies. We then decompose the mean returns of these strategies into risk-based and non-risk-based components. We use three different formation periods to form the portfolios: 1 For detail description of the model and the estimation method see Asgharian and Hansson (2005). 5

Prior(-1) is the monthly return immediately before the month under consideration. This formation period is supposed to capture the short-run reversal. Prior(-12,-2) is based on the cumulative return over an eleven-month period, which begins 12 months and ending two month before the start of the month under consideration. This formation period is supposed to capture the momentum anomaly. Prior(-60,-13) is based on the cumulative return over a four-year period (48 months), which begins five years and ending one year before the start of the month under consideration. This formation period is supposed to capture the long-run reversal. In addition to our proposed model in equation (2), we use two commonly used asset pricing models: i) CAPM: a model with only the value weighted market index, ii) 3FF: Fama and French three-factor model, i.e. a model that, in addition to the market index, includes two mimicking portfolios for size and book-to-market factors. The data consist of monthly returns on ten value-weighted portfolios for each sorting category and three factor portfolios: value weighted market index, HML and SMB portfolios. Data cover the period 193101-200512 and are collected from Kenneth French s homepage. Table 1 shows the sample statistics of the excess returns and Figure 1 plots the mean excess returns and the corresponding 95% confidence intervals of the 10 portfolios sorted based on the firms passed performance. The patterns of mean returns exhibit short run reversal Prior(- 1), momentum Prior(-12,-2) as well as long run reversal Prior(-60,-13). The sample statistics of the strategies: short-run contrarian, momentum and long-run contrarian are shown in Figure 2. The strategies are represented by zero investment portfolios that are constructed by going long in the portfolios in the three highest deciles and going short in the portfolios in the three lowest deciles for the momentum strategy, and vice versa for the contrarian strategies. All three strategies generate significant mean profits: around 0.5% per month. Thus, the historical 6

data supports these strategies as being profitable. The question is whether these profits can be explained by risk-based scenarios or are due to non-risk based components? We now look at the sorted portfolios that latter are used to constract the investment strategies. The mispricing of the different portfolios are estimated according to the two commonly used asset pricing models, CAPM and 3FF and compare the results of these estimations with our benchmark model, i.e. the restricted factor model in equation (2). Since the estimated risk premiums from the restricted model are supposed to capture the entire risk loading of the portfolios against all the latent factors; any difference between the estimated mispricing from the restricted model and the other models may indicate that the latter models do not contain a sufficient number of factors. Table 2 illustrates the result of this comparison. It shows that for the portfolios sorted on Prior(-1), short run reversal, none of the models can explain the mean excess returns for all deciles. In particular all the models show negative and significant mispricing for the highest decile portfolios, deciles 9 and 10. This indicates that the mean returns of these portfolios are not sufficient to cover their risk loadings. In other words, the market must have initially over priced these portfolios. However, for the lower deciles, which were strong evidence for short run reversal just looking at the sample means, our restricted model is successful, while the other two models show significant mispricing. Consequently, the mean excess returns of the portfolios with the worst historical returns can be explained by latent risk factors captured by our restricted factor model. The result for the portfolios sorted based on Prior(-12,-2) is very similar to that of the previous sorting. However, the worst performing portfolio has significant mispricing for all models, and, quite interesting, the 3FF model results in more significant mispricing than CAPM. Thus, the market is underpricing the successful portfolios and overpricing the least successful one. 7

The excess mean returns of the portfolios sorted based on Prior(-60,-13) can be explained by the 3FF model, which is corroborated by our benchmark model, while CAPM fails in explaining the mean returns of the deciles 1 and 3. Therefore, the long-run reversal effects are related to the risk factors. All in all, our suggested model has fewer mispricing than the other two models. The significant non-risk-based components, revealed by our restricted model, point toward the existence of a short-run reversal and a momentum effect for the historically high performing portfolios (higher decile portfolios). To analyse the momentum and contrarian strategies we use the estimates of the three different models for the risk premium and the non-risk-based components from the ten portfolios sorted on the firms passed returns for the three alternative sortings. The corresponding strategies are constructed as zero investment portfolios: for the contrarian (momentum) strategies by going short (long) in the portfolios in the three highest deciles and going long (short) in the portfolios in the three lowest deciles for each sorting. 2 We then compute the sample mean return, the risk premiums and the non-risk-based components of the strategies by deducting the average values of these items for the portfolios in the three largest deciles from the average values for the portfolios in the three smallest deciles. Figure 2 shows the components of sample mean returns and the related confidence intervals for three alternative sortings. The short-run contrarian strategy has a positive and significant non-risk based component (around 0.5%) in all models and the benchmark also shows a small risk-based premium. The risk based scenario can therefore not explain the significant positive profit from this strategy. The momentum strategy has also a significant and large positive non-risk based component 2 As an alternative we define HmL portfolios from six double sorted portfolios as in Fama and French (1993). The result of this approach is almost identical to the result from the HmL portfolios using 10 portfolios. 8

(around 1%). But all models also exhibit a small and significantly negative risk risk premium. Thus, the sample mean underestimates the importance of the non-risk based components. The long-run contrarian strategy has a risk based explanation for all models on the 5% level but not for CAPM on the 10% level (see the 90% confidence interval). CAPM underestimates the risk based contribution and the market portfolio is therefore not sufficient for capturing the factors relevant for this strategy. To summarize: there is a basis for a behavioural explanation of the short-run contrarian and the momentum strategies while the long-run contrarian strategy has a risk based explanation. These results are already evident from using the three-factor model of Fama and French and almost by just relying on the simple CAPM. Furthermore, the validity of the conclusions from these two models are corroborated by our benchmark. 4. Conclusion We use a latent factor model, based on the optimal orthogonal portfolio approach, to analyze the observed contrarian and momentum anomalies in the US stock returns. Our purpose is to investigate if the realized profit from these strategies is a compensation for risk or it is related investors irrational behavior. In addition, we aim to assess the ability of two traditional assetpricing models, i.e. CAPM and the three-factor model of Fama and French, in explaining the anomalies. This is motivated since these models may lack sufficient number of factors and thereby they may overstate the non-risk-based components of the realized profits. Following previous empirical findings for the US data, we construct three different portfolio sets to capture short and long run contrarian strategies as well as a momentum strategy. We start by testing for the existence of non-risk-based components in the mean excess returns of portfolios sorted based on their historical performance. We find a negative and significant non-risk-based component for the deciles nine and ten of portfolios sorted based on a one 9

period lag returns, while the mean excess return of the other deciles can be explained by our proposed model. The portfolios sorted based on a medium length past returns (-12 to -2 months) show very similar results. However, the mean excess returns of the portfolios sorted based on a long-run historical horizon can be entirely explained by factor models. In general, our suggested model is more successful in explaining the mean excess returns comparing to the other two models. To investigate the components of the profits resulted from the three different investment strategies we construct zero investment strategies based on our decile portfolios. We find that the profits from the short-run contrarian and the momentum strategies are mostly non-riskbased, which is in line with the behavioural explanations of these anomalies. On the other hand, the long-run contrarian profit is found to be a compensation for risk. This is in agreement with the traditional asset-pricing viewpoint. Thus, there certainly seems to be anomalies in the historical data and future will tell whether they also can be exploited by more rational investors at low risk (Moskowitz and Mark Grinblatt (1999)). 10

References Ang, A., J. Chen and Y. Xing, 2001, Downside risk and the momentum effect, NBER Working Paper No 8643. Asgharian, H., and B. Hansson, 2005, Evaluating the importance of missing risk factors using the optimal orthogonal portfolio approach, Journal of Empirical Finance 12, 556-575. Barberis, N., and A. Shleifer, 2003, Style investing, Journal of Financial Economics 68, 161 199. De Bondt, W.F.M., and R. Thaler, 1985, Does the stock market overreact? Journal of Finance 40, 793-805. Fama, E.F., and K.R. French, 1993, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, 3-56. Fama, E.F., and K.R. French, 1996, Multifactor explanations of asset pricing anomalies, Journal of Finance 51, 55-84. Hong, H., and J. Stein, 1999, A unified theory of underreaction, momentum trading, and overreaction in asset markets, Journal of Finance 54, 2143 2184. Jegadeesh, N., and S. Titman, 1993, Returns to buying winners and selling losers: implications for stock market efficiency, Journal of Finance 48, 65 91. Jegadeesh, N., and S. Titman, 1995, Overreaction, delayed reaction, and contrarian profits, Review of Financial Studies 8, 973-993. MacKinlay, A.C., and L. Pastor, 2000, Asset pricing models: Implications for expected returns and portfolio selection, Review of Financial Studies 13, 883-916. Moskowitz, T. J., and M. Grinblatt, 1999, Do industries explain momentum? The Journal Of Finance 54, 1999. 11

Table 1. Sample statistics The table shows the descriptive statistics of the excess returns on the decile portfolios sorted based on three different prior returns. Prior(-1) is the monthly return immediately before the month under consideration, Prior(-12,-2) is return over an eleven-month period, which begins 12 months and ending two month before the start of the month under consideration. Prior(-60,-13) is the return over a four-year period beginning five years and ending one year before the start of the month under consideration. Dec.1 Dec.2 Dec.3 Dec.4 Dec.5 Dec.6 Dec.7 Dec.8 Dec.9 Dec.10 Prior(-1) Mean 1.64 1.36 1.26 1.08 0.88 1.03 0.98 0.95 0.79 0.51 St.dev. 8.43 6.77 6.07 5.72 6.46 5.47 5.68 5.95 6.34 7.05 Prior(-12,-2) Mean 0.45 0.84 0.83 0.94 0.89 0.99 1.04 1.17 1.27 1.61 St.dev. 9.62 8.16 7.00 6.47 6.04 5.92 5.64 5.42 5.68 6.45 Prior(-60,-13) Mean 1.51 1.32 1.28 1.09 1.15 1.04 1.06 1.05 0.94 0.90 St.dev. 8.83 7.97 6.99 6.20 6.24 5.78 5.87 5.80 5.85 6.39 Count 900 900 900 900 900 900 900 900 900 900 12

Table 2. Mispricing according to the different factor models The table shows the estimated mispricing of the portfolios sorted based on different prior returns. Mispricings are based on three alternative factor models, i.e. the restricted factor model based on the orthogonal portfolio approach, CAPM and the Fama and French threefactor model (3FF). Prior(-1) is the monthly return immediately before the month under consideration, Prior(-12,- 2) is return over an eleven-month period, which begins 12 months and ending two month before the start of the month under consideration. Prior(-60,-13) is the return over a four-year period beginning five years and ending one year before the start of the month under consideration. Dec.1 Dec.2 Dec.3 Dec.4 Dec.5 Dec.6 Dec.7 Dec.8 Dec.9 Dec.10 Restricted 0.31 0.22 0.20 0.05-0.03 0.04-0.04-0.08-0.27* -0.59* Prior(-1) CAPM 0.43* 0.30* 0.26* 0.10 0.03 0.08 0.00-0.04-0.24* -0.56* 3 FF 0.33* 0.25* 0.25* 0.09 0.09 0.07-0.04-0.07-0.29* -0.65* Restricted -0.82* -0.31-0.20-0.04-0.05 0.06 0.16 0.33* 0.42* 0.76* Prior(-12,-2) CAPM -0.88* -0.36* -0.26* -0.10-0.11-0.01 0.07 0.24* 0.32* 0.63* 3 FF -1.09* -0.51* -0.36* -0.18* -0.18* -0.05 0.07 0.25* 0.33* 0.68* Restricted 0.09-0.04 0.04-0.04 0.01-0.03-0.01-0.01-0.11-0.16 Prior(-60,-13) CAPM 0.31* 0.15 0.20* 0.08 0.13 0.07 0.07 0.06-0.05-0.13 3 FF -0.06-0.13-0.01-0.05 0.02 0.00 0.04 0.05-0.02-0.02 13

Figure 1. Sample means of the 10 sorted portfolios The figure plots sample mean excess returns of the decile portfolios sorted based on three different prior returns and the related 95% confidence interval. Prior(-1) is the monthly return immediately before the month under consideration, Prior(-12,- 2) is return over an eleven-month period, which begins 12 months and ending two month before the start of the month under consideration. Prior(-60,-13) is the return over a four-year period beginning five years and ending one year before the start of the month under consideration. 2.5% Prior(-1) Sample means 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% Dec. 1 Dec. 2 Dec. 3 Dec. 4 Dec. 5 Dec. 6 Dec. 7 Dec. 8 Dec. 9 Dec. 10 2.5% Prior(-12,-2) Sample Means 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% Dec. 1 Dec. 2 Dec. 3 Dec. 4 Dec. 5 Dec. 6 Dec. 7 Dec. 8 Dec. 9 Dec. 10 2.5% Prior(-60,-13) Sample means 2.0% 1.5% 1.0% 0.5% 0.0% -0.5% Dec. 1 Dec. 2 Dec. 3 Dec. 4 Dec. 5 Dec. 6 Dec. 7 Dec. 8 Dec. 9 Dec. 10 14

Figure 2. Components of strategies The figure plots the sample means and the estimated risk-based and non-risk-based components of the zero investment portfolios constructed based on different prior returns and the related 95% (90%) bootstrap confidence interval. The components are estimated with three alternative factor models, i.e. the restricted factor model based on the orthogonal portfolio approach, CAPM and the Fama and French three-factor model (3FF). Short-run contrarian is based on the monthly return immediately before the month under consideration Prior(-1). Momentum is based on the return over an eleven-month period, which begins 12 months and ending two month before the start of the month under consideration Prior(-12,-2). Long-run contrarian is based on the return over a four-year period beginning five years and ending one year before the start of the month under consideration Prior(-60,-13). 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% 95% bootstrap confidence interval Short-run contrarian -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF Momentum 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF Long-run contrarian 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF 15

Figure 2. Components of the strategies (continued) 90% bootstrap confidence interval Short-run contrarian 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF Momentum 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF Long-run contrarian 1,5% 1,0% 0,5% 0,0% -0,5% -1,0% -1,5% RB NRB RB NRB RB NRB Sample NFR CAPM 3FF 16