Estimation of Risk Factors for Models with a Zero-Beta Portfolio Seung C. Ahn Department of Economics W.P. Carey School of Business Arizona State University Tempe, AZ 85287 USA [email protected]

Alex R. Horenstein Department of Economics W.P. Carey School of Business Arizona State University Tempe, AZ 85287, USA [email protected]

This Version: December 5th, 2008

Abstract The return on the equally-weighted portfolio (EWP), as well as that on the value-weighted portfolio, is often used as a proxy for the market return. Intertemporal Arbitrage Pricing Models and Black’s (1972) CAPM predict the presence of a zero-beta portfolio, whose return has unitary betas. Our major finding is that when asset returns are regressed on the EWP and other variables highly correlated with the risk-related factors, the EWP acts as a proxy for the zero-beta portfolio (ZBP), even if the EWP itself is more highly correlated with risk-related factors than with the ZBP. As a consequence, the betas of the EWP have small cross-sectional variation and fail to capture systematic risks of individual asset returns. Applying the principal component method to a large number of individual stock returns data, we estimate the number of latent factors in stock returns and the time-series of the ZBP and excess market return (EM). Our empirical finding is consistent with Black’s CAPM (1972). When the estimated ZBP is used to compute excess returns, abnormal returns of individual stocks substantially decrease. The excess return of the EWP over the estimated ZBP explains individual stock returns as well as the three factor model of Fama and French (1993). In addition, our result explains why the betas of the excess market return in their model have a small cross-sectional variation and fail to capture systematic risks successfully: The SMB and HML factors are highly correlated with the risk factor. Thus, the excess market return acts as a proxy for the ZBP.

Key words: factor models, zero-beta portfolio, asset pricing, Fama-French model. JEL classification: G11, G12, C01 Acknowledgments We are grateful to Stephan Dieckmann and Crocker Liu for their helpful comments on this paper. All remaining errors are of course our own.

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List of Acronyms EWP=Equally-Weighted market Portfolio VWP=Value-Weighted market Portfolio ZBP=Zero-Beta market Portfolio HML=High Minus Low Fama-French factor SMB=Small Minus Big Fama-French factor PCA=Principal Component Analysis

Introduction The main contribution of this paper is showing why the EWP and the VWP, commonly used as proxies for the unobserved mean-variance efficient market portfolio, fail to capture the systematic risk of stock returns in multifactor asset pricing models. This happens because the EWP1 captures the mean value of the model (the ZBP plus the mean values of the loadings in the risk-factor times the risk-factor itself) while the dispersion on the betas is captured by the other factors correlated with the underlying risk-factor like the HML and SMB factors from Fama-French (1993). This makes the EWP act as if it were the ZBP (a portfolio with unitary loadings uncorrelated with systematic risk) even though the EWP is more related to the market portfolio than to the ZBP. As a consequence, the EWP shows little dispersion on the betas, which might be interpreted as a failure of the market portfolio to account for systematic risk. Figure 1 show this result explicitly. In this figure we draw the relationship between the loadings on the EWP (horizontal axis) and its mean excess returns over the risk free rate (vertical axis) for two sets of assets using the following 2 regressions: (i) ri − rf = α i + βiEWP ( EWP − rf ) + ε i (ii) ri − rf = α i + β iEWP ( EWP − rf ) + β iSMB SMB + β iHML HML + ε i Equation (i) represents a standard CAPM using the EWP as a proxy for the market portfolio while equation (ii) is the CAPM augmented with the Fama-French SMB and HML factors. Panel a.(i) in Figure 1 shows the result using as assets the 100 Size and Book to Market portfolios constructed by Fama-French on equation (i) while panel a.(ii) 1

In this paper we will focus on the EWP as a proxy for the market portfolio, but all the results hold for the VWP.

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shows the result for the same assets using equation (ii). In panels b.(i) and b.(ii) we used the 25 Size and Momentum portfolio constructed by Fama-French on the same two equations as before. The time period spanned is January 1970- December 2006.2 [Figure 1] We can clearly observe in the figure that the dispersion on the EWP loadings is drastically reduced when the HML and SMB factors are added to the regression. We also show in this paper that the true data generating process for the stock returns is more likely to include the ZBP as in Black’s (1972) CAPM or Connor and Korajczyk’s (1989) intertemporal-APT than just the risk-free rate as is usually assumed in empirical work. The ZBP generates problems when using PCA to extract the common factors because it is included in every factor. Since using PCA implies extracting orthogonal factors (normalized eigenvectors), the ZBP will be extracted and its factor loadings will be one minus the sum of all the other factors’ loadings. The rest of the extracted factors will be biased since they will be normalized raw returns instead of excess returns over the ZBP. We are able to solve this problem by extracting the real factor from the time-demeaned data. We also show how to estimate the ZBP using the principal component method, which is another important contribution of this paper. Although the ZBP can be considered a weak factor given its low signal to noise ratio (low explanatory power of the cross-section of stock returns by itself), not taking it into account in the time-series regression leads to a worse fit of the data by the model, generating more intercepts significantly different than zero (which means having a model that produces more abnormal returns).3 Identifying the number of common factors that explain the common variations is one of the major tasks of factor analysis. In this paper we test the number of common factors in stock returns using the new tests developed in Ahn and Horenstein (2008) finding evidence for the presence of one factor with heterogeneous loadings and one with constant loadings. These tests are based on the behavior of two adjacent eigenvalues from the second-moment matrix of excess stock returns. 2

We used 96 out of the 100 Size and Book to Market portfolios for which the data has no missing values. In unreported results we found that adding the estimated zero-beta portfolio to the time-series regression of individual stocks returns results in a lower number of intercept different than zero in nine out of a total of thirteen samples used.

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Brown (1989) found that eigenvalue-based estimation methods might be falsely favorable to a CAPM model even when there are k>1 equally important factors. Another contribution of this paper is to show that this problem can be solved by simply calculating the number of factors from the time-demeaned data (equivalent to subtracting the sample EWP from each observation). We do the latter and we still find one factor. The limitation of this procedure is that we can only estimate the number of factors with heterogeneous loadings since a factor with constant loadings will be eliminated when the time-mean is subtracted from the data. We are able to relate this finding to the presence of a zero-beta portfolio (a factor with constant loadings). Given all the above results we propose the use of a single factor model in the spirit of Black’s (1972) CAPM where the excess returns of the assets over the estimated ZBP are regressed against the excess return of the EWP over the estimated ZBP. We show that this model performs as well as the Fama-French Model and tends to generate less intercepts different than zero when using individual returns. When using a set of portfolios constructed by Fama-French based on different characteristics, their model does better with portfolios constructed based on Size and Book to Market while the model proposed in this paper fits the Industry Portfolios better. In previous studies, Shanken (1985, 1986) developed a multivariate test for portfolio efficiency in the presence of an unknown ZBP. His test requires the time dimension (T) to be larger than the cross-sectional dimension (N). He is able to obtain limited information on the ZBP as his test assumes that the ZBP is constant. In this paper we extract ZBP using PCA, exploiting the information contained in large panels of data without constraints on either N or T, as long as both are large. We are able to extract the complete time-series of the ZBP. Charles Trzcinka (1986) used eigenvalues calculated from sequentially larger covariance matrices of stock returns to test for the number of common factors. This was a natural way to do the test given that Chamberlain and Rothschild (1983) showed that if the data is generated by k common factors then k eigenvalues of the covariance matrix of

N response variables grow unbounded as N increases while the rest remain bounded. Trzcinka (1986) found that the first eigenvalue dominates the covariance matrix suggesting that a single factor model is enough to fit the data. Although he could not 4

reject the presence of two to five common factors, it appeared that more than one factor was not justified by the data in his paper. The main difference between his tests and the one used in this paper is that the estimators developed in Ahn and Horenstein (2008) are consistent estimators for the number of factors while the tests used in Trzcinka (1986) estimate the number of factors indirectly, for example by testing at what point the eigenvalues become indistinguishable from each other. His tests also required the use of more restrictive assumptions which we are not going to discuss for brevity reasons. Bai and Ng (2002) and Onatski (2006) also developed consistent estimators for the number of factors. The main difference between Ahn and Horenstein (2008) estimators and Bai and Ng (2002) and Onatski (2006) is that the former do not require the use of a penalty functions. A problem with the penalty functions is that they can be arbitrarily chosen by the econometrician leading to different results in finite samples depending on the value chosen. Furthermore, in finite samples the PCp estimators of Bai and Ng (2002) and the one from Onatski (2006) seem to be more sensitive to maximum number of factors to test for. There are several other empirical papers that have studied the use of PCA to test the APT model or CAPM model, but as far as we know, none of them has addressed the problem of using PCA in the presence of the ZBP. For example, Chan et al. (1998) studied which factors are important for driving the common variation in stock returns. They compared factors extracted using PCA with other factors proposed in the literature which they categorized into fundamental factors (e.g. Fama-French (1993)), technical factors (e.g. Jegadeesh and Titman (1993)), macroeconomic factors (e.g. Chen, Roll and Ross (1986)), and market factors (value and equally-weighted CRSP index). They confirmed the fact that the first PC is related to the EWP, and they concluded that it captures the market portfolio. Chan et al. (1998) also showed that beyond the second or third PC the other components are statistically irrelevant in explaining the cross-section of stock returns. In this paper we do not use the extracted factors explicitly in the timeseries regression but only to estimate the ZBP. Once the ZBP is estimated, we use the excess return of the EWP over the estimated ZBP as our only risk-factor. Thus, based on the classification of Chan et al. (1998), the model proposed in this paper can be considered a mix between a PCA model and a market factor model. 5

Preliminaries a) Asset Pricing Models with a Factor Structure The one-period CAPM developed by William Sharpe (1964), John Lintner (1965), Jack Treynor (1962) and Jan Mossin (1966) shows that in the presence of a risk free rate, unlimited borrowing and lending, and agreement about the opportunity set faced among all the agents, all assets can be priced with respect to the non-diversifiable risk captured by the excess return on a mean-variance efficient market portfolio. The pricing equation can be stated as follows:

E (ri ) = rf + β i ( E (rm ) − rf ) where ri is the return on asset i, rf is the risk-free rate of return and rm is the return on a mean-variance efficient market portfolio. βi is the sensitivity of asset i to the nondiversifiable common risk (rm − rf ) . The econometric version of the model that has been widely tested is

ri − rf = α i + β i (rm − rf ) + ε i where εi is the idiosyncratic risk that can be diversified by aggregating assets into portfolios. If the market portfolio is mean-variance efficient then the αi’s should be equal to zero which implies the absence of abnormal returns. The main problem with this equation is that, as stated by Richard Roll (1977) in his critique to the tests of the CAPM, it is very difficult to identify the market portfolio; therefore, only proxies can be used. Fama and French (2004) suggested the use of the VWP as proxy for the market portfolio, while Chen, Roll and Ross (1986) suggested that if any market portfolios is priced it would be the EWP. Another issue is that the risk free rate is also unknown and proxies need to be used for it too. Black (1972) showed that under the assumption that there is no riskless asset or at least none on which short positions are allowed (with no short sales constraint in the risky assets), the linear pricing relation also holds true by replacing the risk free rate with a new mean-variance efficient portfolio uncorrelated with the market portfolio. This portfolio is called the zero-beta portfolio (ZBP or r0). The pricing equation in this case is as follows 6

E (ri ) = r0 + βi (rm − r0 ) By assuming a factor structure generating the returns on asset prices, Ross (1977) developed the APT model in which pricing by no arbitrage might lead to a multifactor pricing equation. Connor and Korajczyk’s (1989, CK from now on) developed a discrete time competitive equilibrium version of Ross’s APT. The econometric version of this model nests all the previously mentioned models. The key distinction between this model and the APT is that the corporate dividends, rather than the stock returns, are assumed to follow a factor structure. Using recursive competitive equilibrium to solve the model CK are able to derive endogenously the k-factor structure of the model. The main three differences between the CK model and Ross’s APT are that in the CK model (i) the betas have a time subscript, (ii) the intertemporal risk premia may vary through time, and (iii) the zero-beta portfolio has a long-run relationship with respect to the one period riskless return. In fact, a k-factors APT becomes a k+1-factor intertemporal-APT when the time dimension is added to the model and the zero-beta portfolio does not disappear with the existence of a risk-free asset. Equation (7) of CK states the econometric version of their model as follows:

ri ,t − rf ,t −1 = (r0,t − rf ,t −1 ) + β i1,t −1 (rm1,t − r0,t ) +
+ β k1,t −1 (rmk ,t − r0,t ) + ε i where rm1,…rmk are portfolios orthogonal to each other that capture common sources of risk and r0 is the return on a zero-beta portfolio uncorrelated with the other portfolios. Ferson and Harvey (1991) and Evans (1994) showed that time-varying risk premia are more important than time varying betas. Ferson and Korajczyk (1995) showed that allowing betas to vary through time does not improve the performance of asset pricing models in explaining predictability of returns. We will not address the issue of time-varying betas in this paper. Since we will extract the common factors using principal components analysis, we are assuming the data generating might be as follows4

rit − rft −1 = (r0t − rft −1 ) + βi (rm1t − r0t ) + ε it

(1)

Equation (1) considers the excess market portfolio as the only risk-factor, but it can be expanded to fit a multifactor model if needed. 4

CK (1989) gives the necessary assumption that needs to be added to the model in order to have constant betas.

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b) Extracting the Factors using PCA in the presence of a non-observable ZBP Chamberlain and Rothschild (1983) proved that the common factors from an approximate factor model can be consistently estimated using PCA. If there are k-factors, the eigenvectors from the second moment matrix corresponding to the largest k eigenvalues are consistent estimators of the true underlying factors as the number of observations N goes to infinity. If we know the zero-beta portfolio, then we can subtract it from the individual returns’ observations and equation (1) becomes:

rit − r0t = β i (rmt − r0t ) + ε i

(2)

Note that knowing the risk free rate for extracting the real factors in the intertemporalAPT is irrelevant since it does not appear in equation (2). However, not knowing the ZBP generates problems. If we do not know the ZBP and we use raw returns, the econometric model from which we extract the principal components becomes

rit = (1 − βi )r0t + β i rmt + ε i

(3)

The derivation of the intertemporal-APT implies that the portfolios r0 and rm are orthogonal to each other. This implies that the only factor that we might correctly estimate using principal components is the zero-beta portfolio although we might not be able to distinguish it from the rest as it will have heterogeneous loadings.

Comment #1: estimating the factors using PCA from the raw data generates a biased estimator of the risk-factor. Using the estimated risk factor rmt in a time-series regression will give us biased intercepts.

Note that in the true model all the stocks load the same in the zero-beta portfolio. To be more precise, in an OLS regression in the true model all the stocks should have a unitary beta as coefficient on the ZBP. Given that we cannot rule out the existence of the ZBP, the problem is how to extract the real risk factors from the data using PCA. Note that N

EWPt = N −1 ∑ rit = r0 + β (rmt − r0 ) + ε t i =1

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(4)

N

N

i =1

i =1

where β = N −1 ∑ β for j=1,…k and ε t = N −1 ∑ ε it . Equation (4) shows that the EWP is a noisy estimation of the real underlying riskfactor. Then adding other variables also correlated with the underlying risk factor will cause the loadings on the EWP to show less cross-sectional variation.

Comment #2: adding variables related to the underlying risk factor in a model that uses the EWP as a proxy for the market portfolio will reduce the cross-sectional variation of the EWP’s betas.

If we subtract (4) from (2) we obtain the following pricing equation: rit − EWPt = ( βi − β )(rmt − r0t ) + (ε it − ε t )

(5)

Comment #3: the factors extracted from the time-demeaned dataset are the true factors.

Comment #4: since the data has been time demeaned, the loadings on the real factors in equation (5) will be distributed around zero.

Comment #5: the factor extracted from the time-demeaned data is a linear combination of the first and second factor extracted from the raw data (see equation (3)).

In order to obtain loadings that are not dispersing around zero, we need to recover the ZBP, the process for which is shown in the empirical section of this paper. Our empirical analysis will be based in testing the following model: rit − rˆ0t = α i + β i ( EWPt − rˆ0t ) + ε it

(6)

where rˆ0t is the estimated ZBP. By equation (4), if ε t → 0 as expected when N → ∞ , and rˆ0t → r0t , then ( EWP − rˆ0 t ) → β (rmt − r0t ) . Thus, the risk factor we propose is an estimation of the true one rescaled by β . 9

Finally, suppose that the true data generating process comes a model without the ZBP but with a risk-free rate. In this case, it is easy to see that the extracted factors from the excess returns of the time demeaned data should the same as those extracted from the not demeaned data.

Data The data on individual stocks returns has been retrieved from the CRSP database. We used monthly returns including dividends. We divided the individual stock data into three periods: 1970-1978, 1979-1992 and 1993-20065, each period starting in January and ending in December. This division was made in order to have a balanced panel with more than 1,000 stocks for each subdivision and more than 100 observations for each stock in each sample. In order to be included in our analysis, a stock must have information for every month included in the sample. We used common stocks traded in the NYSE, Nasdaq and AMEX, excluding REITs and ADRs. We also excluded stocks that had a return of more than 300% in a given month, since we are trying to estimate common factors and such growth is not common but probably due to idiosyncrasies. However, when we calculated the EWP, we used all the stocks in the database (with or without a full data sample for the entire time period), excluding only REITs and ADRs. The following table presents the summary statistics for the different samples used, where T is the time length and N is the amount of stocks in each sample. [Table I] We also used the data that is available at Kenneth French’s webpage. From his dataset we used the 100 Size and Book to Market portfolios, 25 Size and Book to Market portfolios, 49 Industry portfolios, 30 Industry portfolios, 25 Size and Momentum Portfolios and also the three Fama-French factors they calculate: excess market return, SMB and HML. The risk free rate was also downloaded from Kenneth French’s webpage, which is the one month Treasury bill rate.

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We also tried the following divisions of the time span 1970-2006, 1970-1987 and 1988-2006 or 1970-74, 1975-79, 1980-84, 1985-89, 1990-94, 1995-99 and 2000-06. The results did not vary enough with respect to those presented in the paper in order to use this division. Some tests did slightly better and some slightly worse. The three sample presented seem to be a good average of the results.

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Empirical Results a) Number of Factors Before estimating the common factors using PCA we need to know how many factors to extract. Knowing the number of factors to extract is one of the main questions to answer when analyzing factor models using statistical methods. For this purpose we use the methodology developed in Ahn and Horenstein (2008). Brown (1989) criticized the use of eigenvalue based methods to estimate the number of factors and showed that they can be favorable to a one factor CAPM even when there are k>1 equally important factors. In the appendix we show that Brown’s critique can be solved by simply using the time-demeaned data to calculate the number of factors. However, once we subtract the time-mean from the response variables, we are also eliminating the possibility of finding a factor with constant loadings, and only the number of the factor with homogenous loadings can be estimated. Let us define µ1 ≥ µ2 ≥ ... ≥ µT (≥ 0) to denote the sample ordered eigenvalues of the second moment matrix of the time-demeaned response variables (in our case the matrix of response variables is the T × N matrix R of excess returns). Ahn and Horenstein (2008) proposed the following two consistent estimators for the number of factors:

µ Eigenvalue Ratio estimator kER = arg max k ≤ kmax k . µk +1

ln( µk* )  Growth Ratio estimator kGR = arg max k ≤kmax ln( µk*+1 ) where µk* = (ΣTj = k µ j ) / ( ΣTj = k +1µ j ) and kmax is a number proportional to min(N,T) greater than the true number of factors r. Table II shows the results of estimating the number of factors using the two estimators previously defined.6 The first results we present in the table correspond to the 100 Fama-French portfolios based on Size and Book to Market.7

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Unreported results on raw data find one factor as reported in previous studies. We used 96 portfolios for which the data was complete for the period 1970-2006 out of the 100 portfolios in the dataset.

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The second set of results is obtained from estimating the number of factors on individual stocks and randomly generated portfolios. Each portfolio contains 10 stocks, and they are randomly assigned to it. Every stock is included in only one portfolio. We generate 1,000 sets of random portfolios and present the average number of factors found and their frequencies. We test the number of factors with randomly generated portfolio due to the fact that idiosyncratic noise is reduced when grouping assets into portfolios. However, this reduction in idiosyncratic variance is not free and comes at the cost of reducing the number of cross-section observations from where the number of factors is being estimated. [Table II] The results in Table II suggest strong evidence for the presence of one common factor with heterogeneous loadings. We found two factors in some samples of the FamaFrench portfolios, but when we take into account the January effect as suggested by Daniel and Titman (1997) we end up with one factor in the entire sample and every subsample except for the period 1979-1992.

b) ZBP vs. risk-free rate If the true model is one that includes the risk-free rate and not the ZBP, when we extract the factors from excess-returns the data generating process is

ri − rf = α i + β i (rm − rf ) + ε i The time demeaned version of this model is

ri − EWP = α i + ( β i − β )(rm − rf ) + (ε i − ε ) Note that in this case, the factor (rm − rf ) extracted from both models should be the same. Thus, if we regress the first principal component from the time-demeaned data (dPC1) on the first and second principal components from the raw data (PC1 and PC2), PC1 should have all the information contained in dPC1, and PC2 should be meaningless (since by construction it is orthogonal to PC1 and thus to dPC1 too). On the other hand, if excess returns are used over the risk-free rate in a model with a ZBP, the data generating process would be

ri − rf = α i − rf + (1 − β i )r0 + β i rm + ε i 12

while when time demeaning the model would be

(ri − rf ) − ( EWP − rf ) = α i + ( β i − β )(rm − r0 ) + (ε i − ε ) where EWP = N −1 ∑ i =1 ri = r0 + β (rm − r0 ) + ε . Note that in this case dPC1 will be N

explained by both PC1 and PC2. Table III shows the results of regressing dPC1 on PC1 and PC2. As we can clearly see, dPC1 is explained almost entirely by PC1 and PC2 with both components being highly significant, ruling out a model with only the risk-free rate. [Table III]

c) The EWP as a ZBP It is a known fact that the first principal component (PC1) is captured by the EWP and has been used as a proxy for the market portfolio (see for example Connor and Korajczyk (1988)). However, when we include in the regression another factor correlated with the real market portfolio, the EWP acts as a proxy for the ZBP. We provide a graphic intuition in the introduction of this paper by adding the HML and SMB FamaFrench factors to a regression with the EWP. We previously claimed that the true factor can be extracted from the timedemeaned data. Given this result we use the following model to show that the EWP acts as a ZBP:

rit = α i + βi 0 EWPt + βi1dPC1 + εi ,t where EWP is calculated from all the common stocks on the CRSP database, as long as they are not REITs or ADRs, dPC1 captures (rmt − r0t ) , βi captures ( βi − β ) and εi ,t captures (ε i ,t − ε t ) . If the EWP acts as a ZBP, then β i 0 should be approximately equal to one for all i when dPC1 is added to the time-series regressions. The β i 0 ’s should also show more dispersion when the returns are regressed only against EWP instead of EWP and dPC1. Finally, we should expect the mean values of βi to be close to zero. We show all this in Table IV where we test two models, one with only the EWP as regressor (plus a constant) and the other adding dPC1. In both cases we run two regressions. The first N regression is just to calculate the sample mean β0 = N −1 ∑ i =1 βˆi 0 . In the second

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regression we calculate how many βˆi 0 are different than β0 using the usual t-stats at a 5% level of confidence. We also report in Table IV the value of the sample mean β0 and the N sample mean β1 = N −1 ∑ i =1 βˆi1 .

[Table IV] It can be seen in Table IV that the loadings on the EWP show more variation with respect to the mean value β0 when returns are regressed only against the EWP. When adding dPC1 to the regression, most of the stocks and portfolios have loadings in the EWP that are not significantly different than β0 . The table also shows that once dPC1 is added to the regression on individual stocks and portfolios, the loadings on the EWP become very close to one and the loadings on dPC1 close to zero. These results show that once the real factor is included, the EWP behaves as if it were a zero-beta portfolio. In an unreported result we find that the loadings on dPC1 are negative around 30% of the time, depending on the sample and the time period used. Now we are going to show that SMB and HML are correlated with the risk-factor dPC1. Table V reports the R2 from regressing dPC1 on SMB and HML separately. It shows that dPC1 is highly correlated with both SMB and HML. This is the reason why in many cases the EWP is unable to explain the cross-section of stock returns when regressed together with variables highly correlated with the risk factor. [Table V] One problem of using the EWP portfolio in the same regression together with dPC1 is that the loadings on the latter become positive and negative. We are able to deal with this problem by calculating the ZBP, which is the topic of the next section.

d) Estimation of the ZBP When we extract principal components from the raw data, PC1 is the one correlated with the market portfolio. We know that dPC1 is the real factor, which is the market portfolio minus the ZBP. However, since the principal component estimators are normalized eigenvectors, we do not know the real scale of each of these factors, so we cannot subtract PC1 from dPC1 to obtain the ZBP.

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Another alternative would be use PC2 as the ZBP, but again, PC2 is a normalized factor, so we cannot subtract it from the returns data in order to estimate excess returns over the ZBP. Another problem with PC2 is that it is a factor with a very low signal to noise ratio.8 In unreported results the signal to noise ratio of PC2 is less than 0.06 on average. This means that PC2 explains a little more than one twentieth of the total variation in stock returns. Ahn and Horenstein (2008) showed that these types of factors (called weak factors) might be a noisy estimation of the real factor and the information contained in the real factor might be spread among many principal components. Thus, given that the ZBP is a weak factor and that PC2 is a normalized eigenvector whose true scale we do not know, we propose to estimate the ZBP in two steps. In the first step we extract the second through the fifth principal component from the cross-sectional demeaned data9 (cPC2 thru cPC5). Then we regress the raw data on a constant and the extracted factors ( rit = α i + β cPC 2,t cPC 2t + ... + β cPC 5,t cPC 5t + ε it ) and obtain the fitted values ( rˆit = βˆcPC 2,t cPC 2t + ... + βˆcPC 5,t cPC 5t ). The time-mean of those fitted values is our first step estimation of the ZBP which we called r0 ( r0t = N −1 ∑ i =1 rˆit ). N

In the second step, we used r0 and dPC1 to estimate the ZBP that is contained in the EWP (see equation (4)). The final estimation of the ZBP is equal to

rˆ0 = EWP − βˆdPC1dPC1 where βˆdPC1 is the coefficient estimated from EWP = α + β r r0 + β dPC1dPC1 + ε it . By using this procedure we are able to obtain an estimation of the ZBP that has the same scale of the stock returns; thus, we can use it to test the following zero-beta CAPM equation:

rit − rˆ0t = α i + β i ( EWPt − rˆ0 t ) + ε i

(7)

e) Performance of the zero-beta CAPM

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The signal to noise ratio is defined as the part explained by the factor (the R2 of a regression of the data in a constant and the factor alone) divided by the total part left unexplained (one minus the R2 of the data regressed on a constant and all the factors). 9 We cross-sectional demeaned data to avoid problems that may arise with the existence of individual effects. This means that we subtract from each observation its own mean.

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In this section we study the performance of the model stated in equation (7) on two dimensions: (i) how many abnormal returns, or intercepts different from zero, it generates) and (ii) how well it explain the common variation in stock-returns using the adjusted R2 as a measure of goodness of fit. We use as a benchmark for comparison the risk-free rate CAPM, the three-factor model of Fama-French (1993) and a modification of the later that includes the ZBP instead of the risk-free rate. We add the ZBP to the Fama-French model because in order to compare R2 we need the response variables to be the same. Thus, we use the following three models for comparison:

rit − rft = α i + β i ( EWPt − rft ) + ε i

(8)

rit − rˆ0 t = α i + βVWP0 (VWPt − rˆ0 t ) + β SMB SMB + β HML HML + ε it

(9)

rit − rft = α i + βVWP (VWPt − rft ) + β SMB SMB + β HML HML + ε it

(10)

For the four models, (7), (8), (9) and (10) we report the number of intercepts different than zero at a 5% level and the R2. However, a comparison of R2 is only relevant between models (7) and (9) or (8) and (10). [Table VI] When using individual stocks as dependent variables, the zero-beta CAPM (model 7) dominates the other two Fama-French models in terms of generating fewer abnormal returns. The risk-free rate CAPM has fewer intercepts different than zero in the first period but many more in the other two periods. Among the two Fama-French models, the standard one (model 10) generates slightly fewer abnormal returns than the one that takes into account the ZBP (model 9). When comparing adjusted R2, the Fama-French model with the ZBP dominates the zero-beta CAPM. It is important to notice that the zero-beta CAPM R2s are much closer to the zero-beta Fama-French than the risk-free rate CAPM is to the standard Fama-French. Given these results, we suggest the use of the zero-beta CAPM in individual stocks over the risk-free rate CAPM. When analyzing the models using the portfolios constructed by Fama-French, the Fama-French models dominates both CAPM models in terms of R2 for every portfolio and also in terms of zero intercept for most portfolios based on Size and Book to Market. We should expect this to happen since the SMB and HML factors have been developed to have maximum correlation with these portfolios. As before when using individual stock 16

returns, the R2s from the zero-beta CAPM are a lot closer to those from the zero-beta Fama-French than the risk-free rate CAPM R2s are to those of the standard Fama-French. When analyzing Industry portfolios and those created based on Momentum and Size, the zero-beta CAPM generates fewer intercepts different from zero for the last two sample periods than the Fama-French models. When comparing the risk-free rate CAPM with the zero-beta CAPM in terms of intercepts different than zero no clear conclusions can be made from these portfolios. The use of portfolios sorted by characteristics known to the researcher has been criticized in the literature (see Ahn D.H. et al. (2006)), but we do not address that issue in this paper as we are only trying to show the usefulness of the zerobeta CAPM. Given the results of Table VI, we can say that none of the benchmark models dominate the zero-beta CAPM while the risk-free rate CAPM is the one lagging in performance.

Conclusion Including variables related to the underlying risk-factor to the CAPM might lead to the wrong conclusion that the market factor fails to capture the non diversifiable risk. This happens because the proxy for the market portfolio (like the equally-weighted market portfolio) captures the mean value of the risk while the dispersion of the beta is captured by the other variables. Although we do not rule out the existence of a multifactor model, we do not find evidence for the existence of more than one common factor driving stock returns. Brown (1989) suggested that using eigenvalue based method to test for the number of factors may result in results biased towards a CAPM, but we show how this bias can be avoided to still find one factor. We are able to relate our result to the existence of a zero-beta portfolio and propose a method to estimate its complete timeseries. Finally, we suggest the use of a zero-beta CAPM and compare its performance against a CAPM with a risk-free rate, the three-factor Fama-French (1993) model and a three-factor Fama-French model modified to include the estimated zero-beta portfolio. The proposed zero-beta CAPM seems to fit the data as well as the Fama-French models and better than the CAPM with a risk-free rate.

17

References Ahn, D.H, Jennifer C. and R. Dittmar (2006): “Basis Assets”, working paper. Ahn, S.C. and A. Horenstein (2008): “Eigenvalue Ratio Test for the Number of Factors”, Working Paper, Arizona State University. Black, Fischer (1972): “Capital Market Equilibrium with Restricted Borrowing”, The Journal of Business, 45, 444-455. Bai, J., and S. Ng (2002): “ Determining the number of factors in approximate factor models”, Econometrica , 191-221. Brown, Stephen (1989): “The Number of Factors in Security Returns”, The Journal of Finance, 44, 1247-1262. Chamberlain, G., and M. Rothschild (1983): “Arbitrage, factor structure, and meanvariance analysis on large asset markets”, Econometrica, 51, 1281-1304 Chan L., Karceski, J. and J. Lakonishok (1998): “The Risk and Return from Factors”, The Journal of Financial and Quantitative Analysis, 33, 159-188. Chen, N., Roll, R. and S. Ross (1986): “Economic Forces and the Stock Market”, Journal of Business, 59, 383-403.

Connor, G. and R. Korajczyk (1988): “Risk and Return in an Equilibrium APT: Application of a New Test Methodology””, Journal of Financial Economics, 21, 255-289. Connor, G. and R. Korajczyk (1989): “An Intertemporal Equilibrium Beta Pricing Model”, The Review of Financial Studies, 2, 373-392. Daniel K. and S. Titman (1997): “Evidence on the Characteristics of Cross Sectional Variation in Stock Returns”, The Journal of Finance, 52, 1-33. Evans, Martin (1994): “Expected Returns, Time Varying Risk, and Risk Premia”, The Journal of Finance, 49, 655-679. Fama, E. and K. French (1993): “Common risk factors in the returns on stocks and bonds”, Journal of Financial Economics, 33, 3-56. Fama, E. and K. French (2004): “The Capital Asset Pricing Model: Theory and Evidence”, Journal of Economics Perspectives, 18, 25-46. Ferson, W. and C. Harvey (1991): “The Variation of Economic Risk Premiums”, The Journal of Political Economy, 99, 385-415. 18

Ferson, W. and R. Korajczyk (1995): “Do Arbitrage Pricing Models Explain the Predictability of Stock Returns?”, The Journal of Business, 68, 309-349. Harding (2008): “Explaining the single factor bias of arbitrage pricing models in finite samples”, Economics Letters, 99, 85-88. Jegadeesh, N. and S. Titman (1993): “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency”, The Journal of Finance, 48, 65-91. Lintner, John (1965): “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Review of Economic and Statistics, 47, 13-37. Mossin, Jan (1966): “Equilibrium in a Capital Asset Market”, Econometrica, 35, 768-783. Onatski, Alexei (2006): “Determining the number of factors from empirical distribution of eigenvalues”, Working Paper, Columbia University. Press, S. James (1972): “Applied Multivariate Analysis” (Holt, Rinehart and Winston, New York). Roll, Richard (1977): “A Critique to the Asset Pricing Theory’s Tests’ Part I: On Past and Potential Testability of the Theory”, Journal of Financial Economics, 4, 129-176. Ross, Stephen (1976): “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory, 13, 341-360. Shanken, Jay (1985): “Multivariate Test of the Zero-Beta Portfolio”, Journal of Financial Economics, 14, 327-348. Shanken, Jay (1986): “Testing Portfolio Efficiency when the Zero-Beta Rate is Unknown”, The Journal of Finance, 41, 269-276. Sharpe, William (1964): “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk”, The Journal of Finance, 19, 425-442. Treynor, Jack: “Toward a Theory of Market Value and Risky Assets”, Unpublished manuscript. Final version in Asset Pricing and Portfolio Performance, 1999, Robert A. Korajczyk, ed., London: Risk Books, 15-22. Trzcinka, Charles (1986): “On the Number of Factors in the Arbitrage Pricing Model”, Journal of Finance, 41, 347-368.

19

Appendix Solution to Brown’s (1989) single factor bias Brown (1989) studied the implications of the following simple model: x j = F ′λ j + ε j

where j=1,…,N. F = ( f1 , f 2 ,..., fT )′ is a T×r matrix of common factors where

f t = ( f1t , f 2t ,..., f rt )′ , λ j = (λ j1 , λ j 2 ,..., λ jr )′ is the r×1 vector of factor loadings for variable j, and ε j is the idiosyncratic component of the variable. Brown also assumed that the λ ’s are iid N (1, σ λ2 ) . Furthermore, he assumed that the covariance matrix of the idiosyncratic term is σ ε2 I N and the factors are all iid with variance σ 2f . He showed that with this simple setup that is not ruled out by the APT, the first factor seems to be more important than the others even though all the factors are equally important (all have the same variance). Given the above assumptions, the covariance matrix of the response variables is:

Σ x = ΛΛ′σ 2f + σ ε2 I N As pointed by Brown, calculating the eigenvalues of Σ x implies: Σ x − µ I N = ΛΛ 'σ 2f + σ ε2 I N − µ I N = ΛΛ′ − µ * I N = 0

µi − σ ε2 where µi * = . σ 2f The nonzero eigenvalues of ΛΛ ′ are the same as those from Λ ′Λ where:

Λ′Λ = ( N − 1)Σˆ λ + N λλ ′ = ( N − 1)σ λ2 I r + N ⊗ 1r because λ is the vector of cross sectional means from the population factor loadings. Given that Λ ′Λ is an intraclass structure10, we get the following result for the eigenvalues:

10

b a  An intraclass matrix of order r × r is defined as    and its eigenvalues are given by:  b a 

µ1 = a + (r − 1)b µ 2,...,r = a − b For more details see Press (1972), page 29.

20

µ1 = σ 2f ( N − 1)σ λ2 + rN  + σ ε2 µ2,...,r = σ 2f ( N − 1)σ λ2  + σ ε2

µr +1,..., N = σ ε2 Note that: ∂µ1 = σ 2f (σ λ2 + r ) ∂N

∂µ2,...,r ∂N

= σ 2f σ λ2

µr +1,..., N

=0

∂N

The above result implies that as N increases the eigenvalue corresponding to the first factor increases faster than those of the other factors generating the one factor bias in finite samples studied in Brown (1989). 11 This fact led him to conclude that an eigenvalue based test might not be useful to test for the number of factors in factor models. As we show next, the above problem can be easily solved by time demeaning the data. Too see this, let’s define: xi t = N −1Σ Nj =1 x jt

which under the previous assumption implies xi t = Ft ′1 . Then, x jt = x jt − xi t = Ft ′λ j + ε jt where λ j = λ j − 1 . Now, the population mean of λ j is zero, which implies that the population eigenvalues from Σ x are:

µ1,...,r = σ 2f ( N − 1)σ λ2  + σ ε2

µr +1,..., N = σ ε2 Now all the population eigenvalues corresponding to the common factors increase at the same rate as N increases. Moreover, note that if we want to estimate the number of factors using ER(k) as developed in Ahn and Horenstein (2008), all the ratios converge to

11

Harding (2008) provided an approximation to this bias and the corresponding sampling distribution.

21

one except the one corresponding to the true number of factors, µr / µr +1 , that is equal to

σ 2f ( N − 1)σ λ2  + 1 ,then lim ( µ r / µ r +1 ) → ∞ . 2  N →∞ σε A graphical representation of these findings is shown in Figure 2. We generate data according to model (1) with r=3,

ftj ~ N (0, 0.065) , λij = 1 + ϕij where

ϕij ~ N (0, 0.1) and ε it ~ N (0,1) for t=1,…T, i=1,..,N and j=1,..,r. With this setup, the average R2 of the factors regressed on the generated xj is approximately .19 which is close to the minimum R2 found in the stock-returns samples used in this paper. This means that the signal to noise ratio of each of the three factors is approximately 1/12. We generated 50 different samples in which ( N , T ) = (1,000,150). From each of the samples, we drew subsamples of ( N , T ) = (20,150), (40,150), and so on, up to (1,000,150) (note that T is fixed at 150 while N increases from 20 to 2000 in steps of 20). Then, for each subsample, we calculate the ER(k) estimator and we also calculate the first ten eigenvalues from XX’. We then average all the results. Panel a.1) of Figure 2 shows the average values of ER(k) and the eigenvalues of XX’ as N increases for the case where the data is not time demeaned. As shown in Brown (1989), only the first eigenvalue seems relevant although the data was generated with three equally important factors. Panel a.2) shows the behavior of ER(k) in the same situation and only one factor is captured by the estimator. Panel b.1) shows the evolution of the eigenvalues when the data has been time-demeaned and we can clearly observe that the first three eigenvalues grow faster than the rest. Panel b.2) shows the evolution of the ER(k) criterion and shows that the only criterion that grows unbounded is ER(3), correctly estimating the number of factors used to generate the data after N equals 740. [Figure 2]

22

Table I Individual Stock Return Data 1970-1978

1979-1992

1993-2006

N

1384

1640

1855

T

108

168

168

Table II Number of Factors in Different Datasets

100 Size and B/M portfolios

108

kˆER 2 (1)

kˆGR 2 (1)

96

168

2 (2)

2 (2)

96

168

1 (1)

1 (1)

96

444

2 (1)

2 (1)

1384 108

1

1

1640 168

1

1

1855 168

1

1

N

T

96

(January 1970- December 1978)

100 Size and B/M portfolios (January 1979- December 1992)

100 Size and B/M portfolios (January 1993- December 2006)

100 Size and B/M portfolios (January 1970- December 2006)

Individual Stocks (January 1970- December 1978)

Individual Stocks (January 1979- December 1992)

Individual Stocks (January 1993- December 2006)

1000 Random Portfolios

138

108

(January 1970- December 1978)

1000 Random Portfolios

164

168

(January 1979- December 1992)

1000 Random Portfolios

185

168

1.001

1.001

[999,1,0,0]

[999,1,0,0]

1.18

1.17

[867,81,51,1]

[879,75,45,1]

1

1

(January 1993- December 2006) [1000,0,0,0] [1000,0,0,0] *Values in parenthesis correspond to the amount of factors calculated omitting the month of January. ** Values in brackets correspond to the frequency that 1 to 4 factors were captured in the random portfolios. *** we used 96 out of the 100 FF portfolios. This is because there are 4 portfolios that have missing values for the time spanned.

23

Table III dPC1 = γ o + γ 1 PC1 + γ 2 PC 2 + ε γ0 1970-1978 1979-1992 1993-2006

γ1

γ2

R2 .975

-0.02

0.84

1.12

(-0.23)

(52.73)

(35.93)

-0.43

-1.78

-0.39

(-1.18)

(-29.26)

(-19.12)

-0.13

-0.33

1.38

(-3.38)

(-93.14)

(64.13)

.883 .988

T-stats are in parenthesis.

Table IV N Different βˆi 0 than the β0 = N −1 ∑ i =1 βˆi 0 at 5% and value of β0 and β1 for the following

two models: Without dPC1: ri ,t = α i + β i 0 rEWP ,t + εi ,t With dPC1: ri ,t = α i + β i 0 rEWP ,t + βi1dPC1 + εi ,t N

Without dPC1 β % βˆ ≠ β 0

individual 1970-1978 individual 1979-1992 individual 1993-2006 100FF 1970-1978 25FF 1970-1978 25szFF 1970-1978 49iFF 1970-1978 30iFF 1970-1978 100FF 1979-1992 25FF 1979-1992 25szFF 1979-1992 49iFF 1979-1992 30iFF 1979-1992 100FF 1993-2006 25FF 1993-2006 25szFF 1993-2006 49iFF 1993-2006 30iFF 1993-2006

1384 1640 1855 96 25 25 49 30 96 25 25 49 30 96 25 25 49 30

i0

.95 .86 .67 .86 .84 .86 .78 .71 .87 .85 .87 .83 .79 .73 .71 .76 .52 .51

55% 46% 59% 72% 72% 64% 69% 67% 60% 68% 56% 45% 60% 77% 84% 80% 57% 53%

0

β

With dPC1 % βˆ ≠ β

β1

1.07 .97 .89 1.07 1.06 1.11 1.12 1.10 1.05 1.05 1.04 1.01 1.01 .91 .91 .98 .88 .88

27% 22% 33% 42% 52% 44% 28% 30% 11% 12% 12% 28% 30% 30% 28% 56% 39% 43%

0.01 0.007 0.014 0.018 0.018 0.021 0.029 0.032 0.012 0.012 0.011 0.014 0.012 0.012 0.013 0.014 0.024 0.025

0

i0

0

Calculations reported for the random portfolios are the mean result over 1000 sets of randomly created portfolios with 10 stocks in each of them. 100 FF are 96 out of the 100 Size and Book to market portfolios constructed by Fama-French. The rest of the Fama-French portfolios used are the 25FF that are the 25 Size and Book to Market portfolios, 25szFF that are 25 portfolios constructed on Size and Momentum and 49iFF and 30iFF that are the 49 industrial portfolios and 30 industrial portfolios respectively. The equally weighted portfolio is constructed with all the common shares included in CRSP excluding REITs and ADRs.

24

Table V Model A: dPC1 = γ o + γ 1SMB + ε Model B: dPC1 = γ o + γ 1 HML + ε Model A R2

Model B R2

1970-1978

.81

.4

1979-1992

.47

.23

1993-2006

.49

.60

Table VI R and percentage of α i ≠ 0 at 5% level of confidence for the following two models 2

Model 7: rit − rˆ0t = α i + β i ( EWPt − rˆ0 t ) + ε i Model 8: rit − rft = α i + β i ( EWPt − rft ) + ε i Model 9: rit − rˆ0 t = α i + β iVWP (VWPt − rˆ0t ) + β iSMB SMB + β iHML HML + ε it Model 10: rit − rft = α i + β iVWP (VWPt − rft ) + β iSMB SMB + β iHML HML + ε it N

T

Model 7 Adj. R2

αi ≠ 0 Individual Stocks* 1970-1978 Individual Stocks* 1979-1992 Individual Stocks* 1993-2006 25 Size and B/M portfolios* 70-78*** 100 Size and B/M portfolios* 70-78*** 49 Industry Portfolios* 70-78*** 30 Industry Portfolios* 70-78*** 25 Size and Momentum* 70-78*** 25 Size and B/M portfolios* 79-92*** 100 Size and B/M portfolios* 79-92*** 49 Industry Portfolios* 79-92*** 30 Industry Portfolios* 79-92*** 25 Size and Momentum* 79-92*** 25 Size and B/M portfolios* 93-06*** 100 Size and B/M portfolios* 93-06*** 49 Industry Portfolios* 93-06*** 30 Industry Portfolios* 93-06*** 25 Size and Momentum* 93-06***

1384 1640 1855 25 96 49 30 25 25 96 49 30 25 25 96 49 30 25

108 168 168 108 108 108 108 108 168 168 168 168 168 168 168 168 168 168

78 48 47 8 19 11 6 9 8 14 6 6 10 6 18 2 1 5

.399 .263 .135 .877 .827 .658 .698 .871 .824 .765 .606 .613 .804 .666 .595 .322 .331 .676

Model 8 Adj. R2

αi ≠ 0 10 141 99 6 16 7 2 10 8 23 3 4 10 8 24 2 0 9

.248 .152 .096 .810 .762 .570 .600 .807 .765 .705 .529 .533 .751 .583 .500 .250 .252 .583

The dPC1 used in all the datasets is extracted from the individual stocks data. * Results are the average of all the individuals R2. ** Results are the average over 1000 different sets of random portfolio. Each portfolio in each set contains 10 stocks. *** Results for the other time periods on these portfolios are available upon request.

25

Model 9 Adj.R2

αi ≠ 0 137 111 67 3 4 7 3 6 7 14 12 11 14 5 10 5 3 5

.430 .292 .161 .945 .881 .731 .775 .908 .944 .863 .722 .744 .895 .864 .741 .444 .473 .783

Model 10 Adj.R2

αi ≠ 0 116 90 61 0 4 5 3 8 6 14 9 9 14 8 15 6 3 9

.391 .256 .156 .937 .865 .722 .767 .906 .934 .845 .680 .704 .878 .870 .746 .457 .487 .767

Figure 1 Dispersion of the equally-weighted portfolio’s (EWP) loadings. January 1970-December 2006 a.(i) 100 B/M and Size portfolios: CAPM

b.(i) 25 Size and Momentum portfolios: CAPM 0.016

0.014

0.014

0.012

0.012 0.01

0.01

mean excess return

mean excess return

0.008

0.006

0.004

0.008

0.006

0.004

0.002

0.002

0

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

-0.002

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.002

-0.004

-0.004

β EWP

β EWP

a.(ii) 100 B/M and Size portfolios: CAPM+SMB and HML

b.(ii) 25 Size and Momentum portfolios: CAPM+SMB and HML

0.014

0.016

0.012

0.014

0.012 0.01

0.01

mean excess return

mean excess return

0.008

0.006

0.004

0.008

0.006

0.004

0.002 0.002

0 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

-0.002

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.002

-0.004

-0.004

β EWP

β EWP

* The EWP has been constructed using the CRSP database and includes all the common stocks traded in the NYSE, Nasdaq and AMEX, excluding REITs and ADRs. The Fama-French portfolios have been downloaded from Kenneth French’s webpage.

26

Figure 2 Simulation of Brown’s (1989) model with raw and time-demeaned data a.1) Eigenvalues from raw data

a.2) ER(k) from raw data

90000

20

80000

18

16 70000

14

Eigenvalue 1

60000

Eigenvalue 2

ER(1) ER(2) ER(3) ER(4) ER(5) ER(6) ER(7) ER(8)

12

Eigenvalue 3

50000

Eigenvalue 4

10

Eigenvalue 5

40000

8

Eigenvalue 6 30000

Eigenvalue 7 6

Eigenvalue 8 20000

Eigenvalue 9

ER(9) ER(10)

4

Eigenvalue 10 10000

2

0

40 100 160 220 280 340 400 460 520 580 640 700 760 820 880 940 1000 1060 1120 1180 1240 1300 1360 1420 1480 1540 1600 1660 1720 1780 1840 1900 1960

40 120 200 280 360 440 520 600 680 760 840 920 1000 1080 1160 1240 1320 1400 1480 1560 1640 1720 1800 1880 1960

0

N

N

b.1) Eigenvalues from time-demeaned data

b.2) ER(k) from time-demeaned data 1.3

5000

4500

1.25 4000

3500

Eigenvalue 1

1.2

ER(1) ER(2)

Eigenvalue 2

3000

Eigenvalue 3 Eigenvalue 4

2500

ER(3) ER(4) ER(5) ER(6) ER(7) ER(8) ER(9) ER(10)

1.15

Eigenvalue 5 2000

Eigenvalue 6 Eigenvalue 7

1.1

1500

Eigenvalue 8 Eigenvalue 9

1000

Eigenvalue 10

1.05

500

0

N

27

1960

1880

1800

1720

1640

1560

1480

1400

1320

1240

1160

1080

920

1000

840

760

680

600

520

440

360

280

200

40

N

120

40 120 200 280 360 440 520 600 680 760 840 920 1000 1080 1160 1240 1320 1400 1480 1560 1640 1720 1800 1880 1960

1