Four-State Model vs. Market Model: Part I November 2002 Octave Jokung EDHEC Business School Jean-Christophe Meyfredi EDHEC Business School
Abstract The present paper conducts an empirical study by examining the Market Model and the three versions of the 4-State Model (translated, rotated and un-rotated) in a mean-beta framework. Using daily returns from the CAC 40 Index's assets, we find that the explanatory power of the 4-State Model is greater than the one of the Market Model and this effect is improved by rotation. A reduction in the non-systematic risk is also observed when switching from Market Model to 4-State Models. Surprisingly, the betas are more stable when using any version of the 4-State Model. EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship with professionals that the school has cultivated since its establishment in 1906. EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals. 2 EDHEC pursues an active research policy in the field of finance. EDHEC-Risk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright 2015 EDHEC
1. Introduction Testing of asset pricing models has been developing steadily over the past three decades. These tests relate to the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) either in conditional or unconditional contexts. The restrictive assumptions underlying the CAPM, multivariate normal returns or quadratic preferences, together with Roll s (1977) critique (the market portfolio is unobservable), seriously challenge these studies (see Shanken (1982)). The APT allows for more than one factor but unfortunately the specification of factors that affect an asset s returns is another challenge. Nevertheless, Chen, Roll and Ross (1986) find that the residual market factor has an insignificant risk premium in the case of U.S. securities. They used the Fama-Mac Beth (1973) procedure. Black (1972) derives a more general version of the CAPM when the risk-free asset is not available. Recall that the CAPM can be viewed as a linear relation between the expected returns on one hand and the beta on the other hand. This formulation raises the problem of the quality of the test and the stability of the betas. Fama and French (1992) suggest that returns on U.S securities cannot be explained by betas, and then in 1993 they show that market factor, size factor and book to market ratio explain U.S. stock returns from 1963 up to 1990. In fact, the coefficient on size is significant and negative (higher expected returns are associated with small firm s stocks) and the coefficient on book-to-market ratio is also significant and positive (higher returns are expected with firms with large book to market ratios). Harvey and Siddique (2000) reconsider the asset pricing model by incorporating conditional skewness. They find that expected returns should include a component attributed to conditional co-skewness. Their approach outperforms the usual CAPM. In another vein, Chan and Lakonishok (1993) point out the difficulty of making unambiguous inferences from the ever-changing environment generating stock returns. They used the Fama-MacBeth (1973) procedure to test the CAPM with CRSP data from 1926 to 1991 and their estimated SML is not flat. Black (1993) documents several reasons which prove that Fama and French (1992) mis-interpret their own data and the former results related to empirical studies of the CAPM. For instance, Black says that Fama and French do not give any theoretical framework to support their results and Black also criticises the use of the book-to-market ratio. Finally, Black (1993) highlights the fact that some so-called anomalies can arise due to data mining (Shanken (1992)). The aim of our study is to compare two models based on betas, one with a continuous riskreward relation and the other with a discontinuous relation. The fact that usual CAPM fails to explain the relation between risk and return suggests this relation is not continuous. Therefore, in our framework, we expect that the use of a partition, which authorises this relation to be discontinuous, will lead to a higher explanatory power. The theoretical attractiveness of the betas lies in their professional application. Thus, the empirical study of the betas are necessary and this paper adds a new perspective to this important subject by highlighting the advantage of the proposed partition. The paper written by Norsworthy et al. (2001), proposes an alternative to the market model by examining the relationships between the return and the risk through a particular segmentation of the assets. The assets are distributed among four categories according to their relative performance and to the market return. The same authors show the superiority of the explanatory power of the model suggested, namely the 4-State Model, compared to the Market Model by using daily returns of hundred individual assets over the period 1984-1998: all the assets composing the industrial DJ 30, a sample of 30 assets of the S&P mid-cap and 40 assets of the S&P smallcap. Norsworthy et al. then apply their results to the prospect theory. However, with the CAPM framework, they show that the 4-State Model outperforms the Market Model and this result is improved by rotation. They address the explanatory capability of the 4-State Model through time and find strong evidence that the aforementioned model outperforms the classical Market Model. They also show that the 4-State Model implies that more than 60 per cent of asset risk is undiversifiable within their data. 3
Our France based study exhibits similar conclusions to those obtained by Norsworthy et al. We introduce a third version of the 4-state model by using a translation operation rather than a rotation as per. Northworthy et al. Furthermore, we analyse the diversification effect by studying the standard deviation of the standard error of estimate and its variability. The advantage of our approach is that it allows us to work directly with the four state dependent betas in relation to the co-movement of the asset considered and the market, rather than requiring that these betas be aggregated into one indicator which is the beta of the market model. Our paper is organised as follows. Section 2 presents the framework by discussing the Market Model and the three versions of the 4-State Model. The objectives of Section 3 are two-fold. It is devoted to the description of the existing tests that are briefly revisited. Then we apply these tests to examine the improvement of the explanatory power, the reduction of the non-systematic risk, and the increase in the beta s stability when moving from the Market Model to the 4-State Model. Section 4 places emphasis on our main results. The last section presents our conclusion. 2. 4-State Model: Unrotated, Rotated and Translated Most tests of the CAPM start from the following equation: where r At denotes the return on any asset A at time t, r Mt denotes the return on the market at time t, and ε At represents the residual term. α is the intercept for asset A, and β is the beta of that asset. The estimated equation above is a linear relationship between the asset s returns and the market s returns for any individual asset. This equation is known as the Market Model. Let us now present the 4-State Model. Following Norsworthy et al. we partition the observations in four sets according to the signs of the asset and the market returns. We thus define four states as follows: - State 1 corresponds to observations when both the sign of the asset return and the sign of the market return are non-negative - State 2 is related to a positive sign of the market return and a negative for the asset return - State 3 corresponds to observations where both the sign of the asset return and the sign of the market return are negative - State 4 groups all the other observations. We summarise the partition in the following table: State Sign of Asset Return Sign of Market Return 1 r A 0 r M 0 2 r A < 0 r M 0 3 r A < 0 r M < 0 4 r A 0 r M < 0 (1) Given an asset, for each state, we have a single market model described as follows: (2) The 4-State Model itself is obtained by taking into account the last four equations: (3) 4
"The 4-State Model thus has six additional terms that capture the effects of partitioning based on the asset and market returns" (Norsworthy et al. p.11). The rotated version of the 4-State Model is introduced like in Norsworthy et al. by considering the expected asset return and the expected return of the market, and by rotating the abscises axis in order to pass through the expected point given by the two expected values. This rotation leads to a new partition with the same observations but the observations are now in the expectation quadrants. In fact, the observations in this case are partitioned according to the sign of their distance towards the expected value. We expect the rotation to improve the knowledge of the relationship between the risk and the return. Figure 1 presents the rotation. Figure 1: Rotation of the Coordinate Axes The rotation is caracterised by its centre and its angle given by the following equation: From the original observations, we can determine the new coordinates with respect to the new axis. They are given by: (5) (4) (6) At this stage, from their respective expected values and according to the signs of the deviation of the asset returns and the market returns, we define four new states. The conditions become the following: State Sign of Asset Return Sign of Market Return 1 ρ A 0 ρ M 0 2 ρ A < 0 ρ M 0 3 ρ A < 0 ρ M < 0 4 ρ A 0 ρ M < 0 With the initial observations and according to the new states, we use the 4-State Model procedure and we obtain the rotated version of the 4-State Model. The rotated 4-State is expected to be more efficient than the unrotated version. We can also define a third version of the 4-State Model by relocation of the observations. This version is obtained by changing the origin: the coordinates of the new origin are exactly the 5
expected values of the returns. Thus the new coordinates of each observation is given by: (7) (8) We define the new four states in the following manner: State Sign of Asset Return Sign of Market Return 1 r ' A 0 r ' M 0 2 r ' A < 0 r ' M 0 3 r ' A < 0 r ' M < 0 4 r ' A 0 r ' M < 0 We then use the 4-State procedure and obtain the relocated 4-State Model (Translated). Figure 2: Translation of the coordinates axes. 3. Data and Methodology Our study is based on the totality of the assets constituting the French CAC40 Index on 17 July 2002. All data, consisting of closing daily returns of these stocks adjusted for dividends and equity offerings, were calculated from the Fininfo database since 18 February 1997. We also use the CAC40 Index as a proxy for the market portfolio. In total, we have used quotes from the previous five and a half years. In order to work with homogenous data, the six stocks quoted after the beginning of our study were dismissed from our sample. Finally we use 34 stocks for which we indicate the main statistics in the annex. As displayed in Figure 3, we identify two periods. The French Index has shown an increase of 280% from the beginning of our sample until September Y2K (other than the three months between July and October 1998). After this date, the CAC40 decreased by 50%. In order to obtain the different parameter estimates, we use the Ordinary Least Square (OLS). We deal with heteroscedasticity by constructing a consistent estimate of the variance matrix as per White (1980). This well-known approach constitutes a good alternative to the Weighted Least Square when the form of heteroscedasticity is unknown and when we want to avoid imposing any arbitrary form. Thus, we replace the variance of errors by the square of the last. So we have for the variance of any partial regression coefficient the following formula: (9) 6
Figure 3: Evolution of the CAC40 index between 18 February 1997 and 17 July 2002. where are the residuals taken from the regression 3 and represents the residuals obtained from the regression of the regressors r Mt on the remaining regressors in 3. This result converges to So we can verify that we do not need to know the real form of heteroscedasticity. Our first goal is to check the explanatory power of the four models. This will be done by studying the adjusted coefficient of determination and simultaneously considering an increase in the number of regressors. For each model we also estimate the diversification effect which is given by the standard error of estimates. In fact, the model with the lowest standard error will be associated with the best level of diversification. Finally we test for parameters stability using an alternative to the classical Chow test on two arbitrary chosen sub-periods. We divide our sample into two equal sub-samples, each containing 706 observations (from 18 February 1997 to 2 November 1999 for the first one and from 3 November 1999 to 17 July 2002 for the second). In fact, the main drawback of the Chow test is that we cannot determine whether the difference between the two regressions is due to change in intercept terms or in slope coefficients. Indeed we construct a dummy approach. Each dummy variable is set at 1 if the observation corresponds to the considered state and 0 otherwise. This approach has two main advantages. The first one is that we can use the White heteroscedasticity correction procedure. The second consists in the ability to test stability not only for the complete regression but also for each parameter. This method induces an increase in the number of regressors (only two dummies for the market model but eight dummies for the 4-State Models) that could be detrimental in case of few data. Fortunately this wasn t the case. We also note that the joint test no longer corresponds to a Fisher test but to a test. (10) 4. Empirical Results Table [1] exhibits the explanatory power of the 4 models tested in our study. As anticipated, the Market Model has the worst explanatory power with an average adjusted R 2 (thereafter ) of 27.47% whereas the use of the 4-State Model increases the explanatory power to 61.74%, and to 63.7% with the rotated 4-State Model. 7
Table 1: Explanatory Power In fact, we know that the more we introduce parameters in the regression, the better the adjustment quality obtained. But we can also verify that by using rotation we better fit the data. For instance, Figure 4 indicates results obtained with the two models for Cap Gemini: With our sample, the best adjustment for the Market Model was obtained with Axa and it is lower that the worst obtained with the two other models. The max criterion also concludes to a best explanatory power with the rotated model. The complete results are reported in annex 2. Figure 5 presents one example of the quality adjustment. Figure 4: Scatter Plot: Market Model and Rotated 4-State Model. The three 4-State Models perform better than the traditional Market Model which is common sense. Table 2: Diversification Effect 8 We now turn to the diversification effect. The same comparison is made and the results are summarised in Table [2]. The following conclusions emerge from Table [2] which summarises the results of the analysis of the non-systematic risk by the use of the standard error of estimate as a proxy of the aforementioned risk. Firstly, the 4-State Model always reduces the unsystematic risk. Secondly, the rotated version improves the former results, as expected. We can see that using the 4-State Model leads to a lower non-systematic risk than the Market Model. In fact, the reduction is by 27.7%. This percentage becomes 57.28% when the rotated version of the model is used. The variability of the standard error of estimate with the Market Model is about 0.016 while the 4-State Model gives 0.0119. Therefore, moving from the Market Model to the 4-State Model
reduces the range of the non-systematic risk. We notice an improvement of this result with the rotated version which diminishes the value to only 0.0097. The standard deviation of the standard error of estimate also decreases. To summarise, we find a decrease in the idiosyncratic risk with the three versions of the 4-state Model. The final effect we want to analyse corresponds to an improvement of the parameters stability. As reported above, one of the main problems with the Market Model is the fact that the β s are unstable. Figure 5: Fitted Values : Market Model, Unrotated and Rotated 4-State Model. Table 3: Parameters Stability Table [3] summarises parameters stability within our two sub-periods. We distinguish between two kinds of stability. The first is an individual parameter s stability. This criterion makes less sense with multi-factorial models. Therefore, we analyse the simultaneous stability of all the betas for each model. We can see that in the case of Market Model only 10 betas over 34 were stable. We conclude there is better stability using 4-State Model because it increases this number to 18. We also conclude there is better modelisation with the rotated version of the 4-State Model with 23 stable betas. While the β s were all significant for the Market Model, the 4-State Models exhibits more insignificance. Most of the time we could not conclude that β 2 s and β 4 s are different from zero with the unrotated 4-State Model and β 1 s and β 3 s with the rotated one. From the above explanation we are able to rank the four models that we tested. As we can see on the Table 4, the Rotated 4-state model succeeds in all the three performance conditions we have tested for whereas the traditional market model appears to be the worst. 9
Table 4: Ranking of the four models We claim the proposed model allows us better explain the betas stability. We can directly use the betas to predict the two possible expected returns on the assets. However in order to choose between these two outcomes we need to know the evolution of both market and asset. In our case, the use of the 4-State Model requires knowledge of the level of the market return together with the value of the asset return. That is we need to know the value that we want to predict! Therefore, in order to correct this drawback we propose incorporating a memory effect into the aforementioned model. This could be done by changing the definition of our four states. The different states will be defined as follows: in State 1, the return on the market is non-negative and the asset is bullish; in State 2, the return on the market is non-negative and the asset is bearish; in State 3, the return on the market is negative and the asset is bearish; in State 4, the return on the market is negative and the asset is bullish. Knowing the situation of the asset (bearish or bullish), the values of the different betas, and the expected return on the market, we can with our model directly predict the expected return on the asset. This procedure will be used and tested in our next paper. 5. Conclusion This paper extends to the French case the results obtained by Norsworthy et al. in the U.S. case when comparing the asset pricing model with a four-way partition of daily returns (4-State Model) and the usual Market Model. Results of our empirical study suggest there is a strong evidence to support our claim that there is an increase in the explanatory power when switching from Market Model to 4-State Model. The adjusted R 2 is multiplied by 2.5 on average. We also record an improvement of the diversification effect. With our sample, 4-State Model (rotated and unrotated) always leads to a better diversification effect. Our sample exhibits a gain in betas stability when switching from Market Model to 4-State Model. As previously suggested the rotation also improves the results. Our analysis identifies areas where improvement is possible and this improvement generates more diversification and therefore reduces non-systematic risk. 10
A. Stocks and Index statistics B. Adjustment quality 11
12 C. Diversification effects
D. Parameters Stability 13
14
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