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Arbitrage Pricing Models for two Scandinavian stock markets
OMEGA Int J of Mgmt Scn. Vol 17, No 5. pp 437-447, 1989
0305-0483/89 $300+000 Copyright ~ 1989 Pergamon Press pk
Pnnted m Great Britain All rights reserved
Arbitrage Pricing Models for Two Scandinavian Stock Markets R OSTERMARK ]~bo Academy, Fmland (Recewed Not,ember 1988; m revisedform Aprd 1989) This l m l ~ examines the Arbitrage Pricing M ~ i e b (APM) for Finnish and Swedish data. The te~flng is based oa weekly returns of portfolio-nggr~ated data, partly to achieve multivariate normality, partly to dampen omller efl'eets. Data of beta-ranked portfolin~ reed asewbere in testing the Capital Amet Prklall Model (CAPM) and the Arbitrnge Pricing Theory (APT) m used here too and i ~ v i d ~ a ~ ~ for dominm~ testing of CAPM and APT. Following ~ the dominance relaflomhip is analyzed by the D a v i d ~ - M K k l m m a test tad tiu Pmt~dor Odds Ratio test. The Indicate that APT daminates CAPM in b o ~ eountrlu. The F i a n ~ ~ ~ out to be more volatile than the Swedish. Furthermore, the multiple factor model is seen to bare relatively more power In Finolab tium In S w e d ~ conditions. The results are combtant with ln~vious evidence, that the single factor model (CAPM) tends to be more powerful In explaining Swedish than Finnish stock returns. Since the t e s t i q is breed oa portfolJo48greatted weekly returns series, the results do not ~ l y agree with t h e ~ for daily returns at the Individual a~et level.
Key words--dominance between APM and CAPM, Dav,dson-Mackmnon test, Scandmavtan evtdence
1. M O T I V A T I O N
I~ THE STUDY, the dominance relationship between the Arbitrage Pricing Model (APM) and the Capital Asset Pricing Model (CAPM) with Finnish and Swedish data is compared. The models are estimated with equivalent index series. A complete description of the databases is given in [11]. The study replicates [4, 53] within a two-country setting. Whereas in [49] all hsted Finnish stocks over the period 19701983 were used in the testing, m this study, we use the subsets of continuously listed stocks in the Helsinki Stock Exchange (HeSE) over 1970-1987 and the Stockholm Stock Exchange (StSE) over 1977-1987. By admitting only continuously listed stocks, we eliminate problems with missing data. The study continues our work on Portfolio Efficiency of Capital Market Models presented in [49, 51-53]. The APT, originally formulated by Ross [38, 39] and extended by others [9, 10, 21] is based on the assumption that each asset return is linearly related to a set of common factors and its own idiosyncratic disturbance (own risk). There is some encouraging empirical evidence on the 437
model, as [11, 20, 33, 35, 37]. Several authors, e.g. [13, 30, 43], have derived and/or tested various versions of the international asset pricing model (IAPM). It is of interest to note that according to the inter-battery factor analysis of [! 3], the number of factors common to a pair of countries ranged from one to five. They concluded that, if the number of international common factors reflects the degree of economic integration, the United States can be said to be highly economically integrated with Canada, the United Kingdom and Japan, and least integrated with France. How the inter-battery factor analysis works on Scandinavian data remains to be studied. While empirical tests of the APT are usually carried out on security markets with a large amount of assets ( > 1000), there is little experience on how the APT performs on thin security markets. Admittedly, the fundamental assumptions under which the APT is expected to hold are fairly discouraging when it comes to thin markets with limited possibilities of arbitrage and diversification; the theory is designed for a well diversified, frictionless and perfectly competitive economy. However, the APT might
438
Ostermartr--Arbltrage Prtcmg Models
hold even on thin security markets if an (asymptotically) multivariate normal space could be constructed by properly transforming and aggregating the underlying series. Such a space would allow comparative testing of competing theories such as APT and CAPM, which explicitly assumes multivariate normal returns. Furthermore, even if normally distributed returns are not reqmred by the APT, the need for a multivariate normal space is emphasized in empirical testing. The maximum likelihood estimation method, using a likelihood function of a multivariate gaussian distribution, is preferable since more is known about its statistical properttes [38]. In the present study we test APT with Finnish and Swedish data. In particular, we are interested in the dominance relationship between APT and CAPM, derived by [6,40,41]. Whereas the former is a multifactor approach, the latter is a two-parameter model. The CAPM relates the return on the individual asset linearly to the market return, by a risk premium (beta) in excess of the riskless return. In empirical testing of the CAPM, a two-stage procedure (in the spirit of [17]) is followed [49]. First, the beta coefficients of each asset (or portfolio) are estimated by Ordinary Least Squares (OLS). The standard errors are then computed from the &fference between observed and estimated returns. Second, the beta coefficients and standard errors are used in a cross-sectional regression procedure to obtain the final CAPM estimates. If the CAPM is vahd, the risk-return relationship will be linear, with a positive trade-off between risk and return. Linearity is tested by augmenting the empirical form of the model by a squared beta component [14, 28]. Sufficiency is tested by including the standard errors of the OLS-estimates (step 1) in the cross-sectional regression (step 2). In the present study, we test the dominance relationship between a k-factor APM (k 611,5]) and the CAPM augmented by the squared beta component. The testing is based on weekly returns of a set of portfolio aggregated data, partly to achieve (approximate) multivariate normality, partly to dampen outlier effects. The testing of the dominance relationship is bound to produce superiority of APM over CAPM, when increasing the number of factors sufficiently. Thus, the results m the present study are only indicative of the relative power of the two theories.
We nottce that the APT has been tested on daily returns data of the New York Stock Exchange despite a high skewness of individual daily returns. [37] found that the results of their APT tests improved when every second observa. tion was used. They argued that the effect was due to skewness in daily data and the effect of nonsynchronous trading. [32] suggested that daily data are best modeled by an instable variance process, as it is well knogn that daily returns are improperly described by the normal distribution [18]. Adding the econometric problems due to splitting assets into groups and computing subcovariance matrices for these groups to accommodate computer limitations, the results of, for example, [37] may be severely biased. While there is no doubt as to the internal consistency of the Arbitrage Pricing Theory, the procedures for empirically testing it on large security markets may be criticized. If daily return series are significantly non-normal, tests based on the normality assumption, with untransformed data, should be avoided. If all individual assets cannot be simultaneously captured within the APT framework because of current computer constraints, we are forced to perform the testing either on a subset of the assets or on some portfolios (see [22]). However, then the results cannot be generalized to apply to a set of individual securities, unless further assumptions are made, e.g. that the individual assets or portfolios used m model testing represent the total population of securities in the stock exchange. The generalization efforts naturally involve other issues, such as the validity of the criteria applied when selecting assets for model testing, the correctness of portfolio formation procedures, etc. The discussion reflects some problems arising from the observed anomalies on thin security markets in particular (see [4]). Consequently, instead of attempting to demonstrate the universal validity of a theoretical construct, our investigation aims to demonstrate the empirical performance of the APT framework compared to the CAPM in a suitably transformed space only. Insofar as the level of aggregation is concerned, it has been shown that the daily returns series on the HeSE and StSE are non-normal (e.g. [4, 12]). Finnish and Swedish daily stock returns are affected by the following disturbances: seasonaliues (weekday, month of the year and announcement effects), firm size, under-estimated betas m the CAPM
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prices to those based on the opening trading price and found no statistically significant differences between the autocorrelations for the 13 most traded stocks on the exchange over 1982-1983. The Stockholm Stock Exchange (StSE, Sweden) database was acquired from Aktmdata Ab (Helsinki) by funds granted by the Foundation for Supporting Capital Market Research m Finland. The database consists of daffy price index series of the mdividual stocks over the 1977-1987 time period. The price indices are computed in the same way as the Finnish mdices. The database consists of 371 stocks covering 2737 days in the exchange. The Fmmsh and Swedish data bases used m this study are completely comparable with respect to index 2. THE DATA AND number computation. The databases are ITS TRANSFORMATION summarized in Table 1. The Helsinki Stock Exchange (HeSE, FinIn subsequent testing, both data sets are land) price index database was obtained from transformed logarithmically. The empirical disKansallis-Osake-Pankki (a Finnish Commer- tribution of average weekly logarithmic price cial Bank) for this specific research. The data- indices is fairly consistent with the normality base (KOP Stock Market Indices) consists of assumption [27, 49, 53]. The logarithm of the daily price indices from the beginning of Febru- price index is a good approximation of returns, ary 1970 up to the end of 1987 for each stock assuming continuous compounding (see [14], listed on the HeSE, except for Porin Puuviila p. 304, footnote 13): let P, = Pie'.'. denote (listed 01/1973-02/1974). By the end of 1983 the price path from time 0 to time t and let some 70 stocks were quoted on the exchange, I, = P,/Po denote the corresponding pnce index whereas the number increased to some 100 by Then r, = In(It)= i n ( P , ) - In(P0) is the return the end of 1987. The index series are based on of the asset. average trading prices for each trading day. The Following [45], we use weekly returns in the prices are corrected for cash dividends, splits, testing. Initially, we studied the relationship stock dividends and new issues. For days when between the average empirical distribution and no trade in a stock took place, the quoted bid a set of transformation contenders with both price is used. If both trading and bid quotations Finmsh and Swedish data. In parucular, we are lacking, the previous quoted price (trading demonstrated that the returns of portfolios or bid) is used (the number of such prices is 8 formed in the Fama and Macbeth procedure are in all). This principle is justified by the results of approximately normal with both data sets, as [4, p. 64]. He compared the first order serial measured by the normality test statistic of [7] correlations based on returns from average daily (see [I, 2, 3, 8]). In [49], we also observed that
and non-normal return distnbutions. Empirical evidence on the nature of the serial correlation of daily stock returns observed on Scandinavian stock exchanges is summarized by [5]. [48] demonstrated that both univariate and multivariate (econometric) models could be built for the Finnish stock market. The authors used a monthly market index over the period 1975-1984 to construct ARIMA models and multiple regression models. In general, very good short-term forecasts were produced with their models. In general, the econometric models outperformed the univariate models. The work of [48] was extended in [53] to the level of all individual assets (see [50]).
TAble I. Data descnpUon The Kansalhs-Osake-Pankkt (KOP) weekly ,ndtces of asset pnces on the HeSE. and equivalently computed mdtces on the StSE. With HeSe data" 1970-1987. The enure period (826 weeks) ,s dmded into n,ne partly overlapping porffoho formaUon panods With StSE data 19"/7-1987. The enure panod (526 weeks) ts d,vlded into five partly overlapping porffol,o formaUon panods The porffoho formation procedure ts eaphuned below (See also [89] for a detaded description of the test,rig schedule) Days when n o t r a n m c U o n s o c c u r are proxled by Average bKl quotauons Mining ~ J : Price index adJUsted for All capital changes and dtvldeods. The impact of"cash d,vtdends on the variance Data ~it: of' Fmmsh returns ,s neshlpble (see [49]). since the d,vtdend pohcy of" the hsted compar.cs ,s almost invariable over the long term No. of eoatimmetsly listed stocks: H©SE :36. StSE 93
..~,m'c e :
No. of l~rtfo#os:
HeSE 12, StSE 19
440
Ostermark~Arbitrage Pricing Models
the Fama and Macbeth procedure is very powerful m reducing non-normahty, since almost all individual asset returns were significantly nonnormal. Thus, the use of portfolio aggregated data in subsequent testing is justified. 3. THE ARBITRAGE PRICING THEORY AND ITS EMPIRICAL TESTING Whereas the CAPM is based on the assumption that security returns are linearly dependent on a single common factor--the return on the market portfolio--the APT is more general: the latter allows simultaneous recognition of the return generating effects of several factors, not just one, as in the CAPM. In fact, the CAPM may be considered as a special case of the APT. The former is reduced to the latter by defining the CAPM-beta as the weighted average of the betas in the k-factor model, with the security's sensitivities to the factors as weights [19, p. 340]. In the present section we give an intuitive presentation of the APM. Since we wish to test the model empirically, a discussion of the mechanics of the empirical form of the model will suffice. (See e.g. [14, 16, 32, 36,45] for a rigorous derivation and discussion.) The basic assumption in the APT is that the returns on any securities are linearly dependent on k independent factors (see Appendix A). The empirically important result of the APT is the fact that only the risks reflected in the covariance matrix of the (transformed) asset returns series have non-zero factor prices. Thus, if a priced factor could be determined by some other method, even after accounting for the effect of relevant factors, the APT would be rejected [i 1]. Such extraneous factors could possibly be derived from variables such as firm size and own risk. We notice Chen's [11] emphasis on the non-priced factors in an investor's decision-making. The possible existence of non-priced factors is important in order to understand the causal structure of concrete investment decisions, yet irrelevant for empirical testing of priced factors. The potential richness of the APM may be conjectured by noticing that the return generating process of any security in the economy is determined by the same k + ! factors for, f), . . . . . fk,. Given the (true) values of these factors, the expected return of the i:th security is obtained as the sum of the intercept (f0,) and
the sensitivities (l,j) to the k remaining factors. Ilk --- 1, the APM reduces to a one factor model comparable to the CAPM. 3.1 Example Assume that we have been able to determine that, at time t, the return generating process of the securities in the observed market place is governed by two factors with values f), ==0.2, f2, = 0.5. Assume also that the factor sensitivities of the i:th security have been estimated as 1,~ --- 0.22, 1,: = 0.35. Then the expected return of the i:th asset would equal 0.2 x 0.22+0.5 x 0.35 = 0.219, or 21.9%. Given perfectly competitive and fricUonless capital markets, the best forecast for the return on the t:th asset at t + 1, made at time t would be 21.9%. Furthermore, if we are able to unravel the economic forces represented by factors I and 2, we have a complete economic model of security market pricing in the observed economy. A difficulty with the APM is, however, that the factors are derived mathematically from the covariance structure of the observed security returns: There is no way of determining the exact number of relevant factors a priorz, hence we are forced to rationalize our results ex post, that is, to try to find an economic meaning-content to the k + ! factors obtained for the particular economy. Compared to the CAPM, the gain in explanative power of multidimensional analysis is obtained at the expense of difficulties in interpretation. The APM is estimated by factor analysis, using the sample covariance matrix as computed from the observed stock returns by the standard covariance formula [23, 25, 26, 29]. It is assumed that the theoretical covariance between the rates of return on any two stocks is given by [38]:
Cove(R,, R,) -- I,, 5, o2 ) + " ' +
i
Factor analysis makes the working assumption that the individual factor variances o2(fh) are equal to i.00, all h, and then finds that set of factor loadings (10) for each stock that will make theoretical covariances correspond as closely as possible to the sample covariances Covs(R,, R~). In other words, factor analysis seeks optimal values for l,j such that Covr(R ,, Rj) - Covs(R,, Rj). The procedure guarantees that the k + 1 factors absorb the maximal explanative potential in the covariance matrix, which is the key to explaining variation of security returns. After obtaining estimates of the factor betas, the next
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step is to estimate the values of the factor prices fj for all factors. This is usually done by crosssectionally relating the factors to average returns [11, 36]. 4. EMPIRICAL RESULTS
4.1 The range of sigMflcant factors
441
all subintervals. On the other hand, 90% of variance was explained by three factors for all subintervals. Bartlett's chi square statistic for testing the equality of the remaining k - r factors implied that the factors were unique. However, as the relative standard deviation for Bartlett's statistic tends to increase for the tail of remaining (~<4) components, the last factors are not necessarily unique for some subintervals. On the basis of the results of the principal component analysis we fixed the range of factors to be tested at k ¢ [1, 5] for both datasets. Since the Swedish dataset is almost three times larger than the Finnish, ~t might seem conceivable that the number of significant factors in StSE could exceed five. However, according to the evidence presented by [25], no more than five significant factors could reasonably be expected with US data. In fact, their study implied only two chances m 100 that at least six factors were present in the data.
A criucal issue in empirical testing of the APT is the number of factors. [16] report that the number of significant factors in standard chi squared tests tends to increase as more securittes are analyzed. In an extension by [9], it was shown that if a k - factor model is valid, then k eigenvalues of the covariance matrix of returns grow large as the number of securities increases. Using this result on the US market, [45] found evidence that there is one large factor and no obvious way to choose more than one The possibility of having more than one significant factor was not ruled out, however. As admitted by [45], it is quite possible that, say, five factors could provide a proper model of 4.2 Estimated factor Ioadings asset pricing in the US market, even if the data The factor ioadings were computed on the were consistent with only one exploding eigen- correlation matrices for each subinterval with value. The results of [9] and [45] suggest that the both Finnish and Swedish data. The averages of proper number of factors cannot be determined the means and standard deviations for the loadin a straightforward fashion. We applied princi- ings for the I- and 5-factor models are presented pal component analysis to determine an interval in Table 2. [Detailed results of the unrotated that may reasonably be expected to contain the loadings for all models of the 90 (HeSE) and 50 true number of factors. Initially, the 90 covari- (StSE) subintervals may be obtained from the ance matrices for the beta-derived portfolios author on request.] Table 2 shows considerable variation in the constructed over partly overlapping estimation subintervals within the estimation and test loadings of the correlation matrices of the periods were computed. The resulting covari- subintervals. The Finnish loadings turn out to ance cube was then subjected to principal com- be more volatile than the Swedish. One reason ponent analysis, one matrix at a time. The for this variation might be the instability of the results indicated that, with Finnish data, at least portfolios for consecutive subintervals. The four factors are needed to account for 95% of portfolio addresses, i.e. the beta-rankings of the total variance of the underlying variates. For individual assets, vary over time (see [49]). As some covariance matrices n o more than three the content of portfolios at given risk categories, factors were needed to account for 95% of measured by their beta ranking, changes, so variance, whereas five factors explained 95% for does the factorial structure of the corresponding
Table 2. Average of Ioadmge with Fmmsh and Swedish data
Me.an St dev
I 0774 0304
Mean St dev
I 0 893 0 199
( I ) A ~ r q r load.s wit~ Fia~A dma F~'tor H i b e r : 2 3 4 5 0 139 0058 0049 0018 0.374 0220 0 166 0 117 (b ) Atvralle I o ~ s wklt Swfdish data Facto¢ atnnber:. 2 3 4 5 0 056 0 041 - 0 034 - 0 023 0268 0 171 0 104 0079
442
Ostermark--Arbttrage Prtcmg Models
correlation matrices. Another issue is that the factors of two correlation matnces are not comparable (see [37], p. 1099). Because of thesea problems we consider a test of the equivalence of the factorial structure across the subintervals by standard statistical techniques (e.g. HoteUing's T 2 statistic, see [34]) to be unnecessary: the considerable variation in the individual factor Ioadings already indicates that the factor structures are unequal. Table 2 also reveals that the multiple factor model tends to have relatively more power with Finnish than with Swedish data. The results may indicate that the single factor model (CAPM) is more powerful in explaining Swedish than Finnish stock returns. In Table 3 we present the means and standard deviations for some important statistics. The Tucker and Lewis [46] reliability coefficient (pstatistic) for the proportion of variance explained indicates significance of at least five factors with both data sets. The coefficient is an estimate of the ratio of explained to total variation in the data (see [46] for the expression). The p-statistic is lower for the Swedish data set than for the Finnish. Thus, the potential power of the ApT-framework seems to be higher with Finnish data. The chi square statistic tests that all remaining parameters associated with additional factors are zero. The statistics indicate, that some variation is contained in the remaining factors.
Table 4 Results of Davtdson~Mackmnon test for all test IX'nods. based on the extended CAPM (Appendix A) and I- and 5-factor ApT-models HeSE
/~o of Factors: ~0 ";, s(:/0) s('/,) t(~) tfir) Ar2 .g'
StSE
1
5
I
0007 0 149 0017 0311 4214 4 546 0 076 2457
0003 0017 0043 0109 0674 I 457 0 009 I 341
00004 0098 0066 0174 0043 3 979 0 083 4445
5 -0007 0026 0046 0153 - I 067 I 182 0 032 3 237
q'he regress,ons are s~gmficant at the 5% nsk level m all cases but the 5.factor APT-model with HesE data. The values of the -,,-coeffictent suggest that APT is superior to CAPM with both data sets
The t-statistics should be interpreted with caution because of the nature of the underlying
series and the testing procedure. The statistics give a descriptive measure of the strength of the obtained estimates; they cannot be used, however, to determine their exact level of significance. This would be a somewhat meaningless activity in any case, since the asymptotic standard errors of equation (5), Appendix A, are underestimated (see [! 1], p. 1401). The results of the DM-test for all test periods are shown in Table 4. The regressions are significant at the 5% risk in all cases but the 5-factor APT-model with HeSE data. The values of the afcoefficient suggest that APT is superior to CAPM with both data sets. The a,-term is not as close to zero as observed by Chen [11] with US data. However, Chen's [11] dominance test was based on the standard empirical form of the CAPM. A weaker APT-dominance (non-zero as and a:) may indicate that not all priced factors are recognized: the squared beta disturbance may be incompletely captured by the APT-model for k ÷ [2, 5]. The accuracy of the competing models is indicated by the Mean Absolute Percentage Error, computed over all test periods in Table 5. With both data sets, even the I-factor APM ts 2.8--4 times more accurate than the CAPM. In order to support an evaluation of statistical significance, we recorded the number of significant parameter estimates in the individual
Table 3 Key staUsucs of 5-factor models with Finnish and Swedish stock data
Table 5 Mean absolute percentage error averaged over all test pertods
4.3 Dominance testing Following [11] on US data, the Davidson and Mackinnon (DM) test [15] was run for test periods that were not included in the subintervals used to estimate the factor loadings of the APT and the cross-sectional regression coefficients of the CAPM. The t-statistic for the a-coefficients is estimated from the standard formula [28]: t(ot*,) = a~,x/n/s(fz,)
(n =, n u m b e r o f test periods).
Tucker and Lewss p
HeSE
Chl square
Statistic:
HeSE
StSE
HeSE
StSE
Mean St dev
0 89 0 05
0 80 0 04
76 36 26 38
561 40 68 37
No o//actors: MAPE (APT) MAPE (CAPM)
StSE
I
5
I
5
0012 0051
0007
0017 0049
0007
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APPENDIX
Table 6 Proportton of s,gmficant parameter esumates m md,vtdual DM-regresstonso ",-level 10%
HeSE I-factor APT 5-factor APT
0 97 0 80
0 82 0 90
0 68 0 86
It can be proved that, in a well diversified economy, the expected rate of return on stock i at time t may be expressed as ([14, 38]):
E(R,,)=fo,+t,~f.+ cross-sectional regressions at the 10% level for both data sets. The results are presented in Table 6. The hypothesis of a non-pure APTdominance is corroborated in Table 6: the 5factor APM is unable to force the intercept to zero both with Finnish and Swedish data. For the latter, the APT-dominance is seen to be somewhat weaker. (The consistency of the Davidson and Mackinnon test was also checked by the Bayesian Posterior Odds Ratio (see [11]). All ratios were strongly in favor of APT m both countries). 5. CONCLUSION In the present study, the dominance relationship between the Arbitrage Pricing Model (APM) and the Capital Asset Pricing Model (CAPM) with Finnish and Swedish data was analyzed. The models were estimated with eqmvalent index series. The APT-tests were earned out on weekly returns series, because of significant anomalies in daily series. The testing schedule was constructed in the spirit of Fama and Macbeth, which provided a convenient basis for comparing the APT and CAPM frameworks. The Finnish loadings turned out to be more volatile than the Swedish. Furthermore, the multiple factor model tends to have relatively more power with Finnish than with Swedish data. The results suggest that the single factor model (CAPM) tends to be more powerful in explaining Swedish than Finnish stock returns. The results corroborate the hypothesis of non-pure APT-dominance: the 5-factor APM is unable to force the intercept to zero with both Finnish and Swedish data. For the latter, the APT-dominance is seen to be somewhat weaker. The conststency of the Davidson and Mackinnon test was checked by the Bayesian Posterior Odds Ratto. All ratios were strongly in favor of APT m both countries. The results suggest that APT tends to absorb the explanatory power of CAPM, given a sufficient number of factors.
OME 17 ~ " D
A
A PM Test Procedure
StSE 0 70 0 91
443
+t~L,.
(1)
where. l,j = The sensitivity of asset 't' to factor j (factor loading) f0, = Zero-beta factor. fj,--jth priced factor. It ts assumed, that the k +1 vector (f0,, f~, . . . . . f~,) is multivariate normal. Various theoretical papers [16, 24,42] have significantly weakened the assumptions necessary to derive equation (1). In testing the APM we use the same portfolio formation procedure as used in [49] (and the same steps are followed as m [28]), where, the individual assets were assembled m equally weighted portfohos of the same size, using OLS-betas as ranking criteria for each formation period. The empincal market line was estimated by cross-sectional regression for each esttmation interval wtthin the formatton time intervals, with model parameters updated every five weeks [17]. With Finnish data, 240 weeks are used initmlly to compute OLS-betas and 12 portfolios, each containing 3 stocks, are formed on the basis of the ranked values for these estimates. The next 140 weeks are then used to recompute the individual OLS-betas. The resulting portfolio betas enter the cross-sectional regression (2) below. The procedure is repeated every five weeks, so that the database is continuously expanded with new price information during the testing period from week 381 to 426. The beta estimates are recomputed every five weeks. The whole process is repeated nine times, corresponding to the formation periods. As each formation period spans ten estimation and testing periods, there are 90 OLS-regressions and cross-sectional regressions involved besides the nine OLS formauon models. With Swedish data, 140 weeks are used initially to compute OLS-betas, and 19 portfolios018 of which contain 5, and 1 containing 3 stocks--are formed on the basis of the ranked values for these estimates. The next 140 weeks
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Ostermark--Arbttrage Prtcmg Models
are then used to recompute the individual OLSbetas. The resulting portfolio betas enter the cross-sectional regression (2). As with Finnish data, the procedure is repeated every five weeks, so that the database is continuously expanded with new price information during the testing period from week 201 to 381. The beta estimates are recomputed every five weeks. The whole process is repeated five times, giving 50 crosssectional regressions [14, 28]" R,,, = A, + A,B,, + A , ~ + ~3,s(e,,) + ,,, (2) where Rp, = Return on portfolio p at time t. /il, = Rm,- R/, is the difference between the market rate of return (/~m,) and the riskless rate of return (Rf,). s(ep) =estimated standard error of time series model for Rpt. np,= random noise term.
(5)
We use the test procedure of Chert [11] to estimate the APT ([17,37]. See [49] for a description of the computer system): (1)
We consider the application of the above scheme for testing the APT justified on the following grounds ([49, 53]): (1)
Individual asset returns of a thin security market may be biased due to infrequent trading. By portfolio formation, outlier disturbance is expected to be duly eliminated or at least sufficiently dampened (of. [7]). This is indicated by the fact that the empirical distribution of portfolio returns complies well to normality [53]. Portfolio formation is a frequently used technique for eliminating outlier effects [1, 19, 25, 14, p. 101].
(2) The APT and CAPM performance may be directly compared by using the same testing schedule and portfolios in both frameworks (see [41] for the theoretical linkage between APT and CAPM). (3)
By repeating the estimation procedure for a sufficiently large number of times, it is possible to obtain important information on the stability of APT-estimates analogous to that of the CAPM-estimates, even if the factors for different covariance matrices are not easily compared.
(4)
Computer constraints justify an aggregation procedure that can be
utthzed and refined in the future, when the number of assets in the Scandinavian Stock Markets increases. Since the portfolios are ranked according to the level of risk as measured by OLS-betas, some factor(s) may be expected to capture the risk element. Portfolio rankings and the factor loadings for such factor(s) should be positively correlated over time. Thus, the portfolio rankings may be used to extract information on the nature of at least some factor(s).
Compute the set of covariance matrices (covariance cube) correspondmg to each estimation interval, using the ume series of the beta-ranked portfolios with equally weighted assets. The portfolios are expected to act as super-assets, mimicking the performance of the underlying assets so that unbiased estimates of the model parameters can be secured [22].
(2) Perform a principal component analysis on each matrix of the covariance cube in order to determine the expected number of significant factors in (I). (3)
Estimate the factor loadings for each portfolio over all estimation intervals. The factor loadmgs matrix ~s estimated by a two-stage procedure: First, initial estimates of the factor loadings are obtained by the noniterative image method of Kaiser [26, p. 226]. Second, initial estimates are used as input for a maximum likelihood estimation of factor loadings (see [23, 25, 29, 44, 46]). The image factor pattern is calculated on the assumption that the ratio of the number of factors to the number of observed variables is near zero. Consequently, a good estimate for the unique variance of normalized variables is given as one minus the
On.a,
Vol. 17, N o 5
squared multiple correlation of the variable under consideration with all variables in the covariance matrix. (4)
(5)
Estimate the risk premia (factor scores) for all test periods using the ioadings for the portfolios of the corresponding estimation intervals. The factor scores are estimated by a generalized least squares regression method also applied in [37]. Analyze the dominance relationship between the APT and the CAPM using the Davidson and Mackinnon [15] test and the Posterior Odds Statistic [47].
Consider the APM and CAPM equations (1) and (2) in matrix form at time "t', solved for the same set of portfolios [49]: R, = ~dL + e, = >
e, = R, - ~dL (AFT),
(3)
where R~ = p x 1 vector of stock returns. f, = vector of factor scores, estimated by the regression method for each test point t. The test points are not included in the corresponding estimation intervals. B d = p x k matrix of estimated factor Ioadings. e, = residual vector for time point t. R, ,= fldfi , + n, = >
n, = R, - f l d i ,
(CAPM),
(4)
where
f I # ¢ R p~3
is a 3 x l coefficient vector based on (2), as estimated by the Fama and Macbeth [17] crosssectional regression method for dffil . . . . . q. has entries corresponding to equation (2) for each estimation period. Thus, row 'i' of ~# is ~# ffi (1, ~1, ~ . The quadratic term has been included, since its ~-coefficient was significant in the Fama and Macbeth test made on Finnish data in [33]. By recognizing the quadratic term in the empirical market line, we try to overcome the problem of underestimated and/or heteroscedastic
445
residual variances of the basic CAPM [1 l, p. 1402]. R,, n, ¢ R p* ~ are the observed (transformed) index and residual vectors respectively. The issue of the dominance relationship between APT and CAPM involves the problem of discriminating among non-nested alternative models. One approach to the problem has been suggested by Davidson and Mackinnon [15]. Consider the expected values for the APT and CAPM equations: ~cA,,, = a,a,.
We form the following cross-sectional regression model over all test periods: ~,=%+
t~,(CAPM) +(1 --at) ~,(AFT)
(5)
Define v,ffi~,- ~A~ and V!= ~CAm~_~ ' ~ Then the restricted parameters model can be estimated as: f,= ~ + ~sf~. (6) If APT were to dominate over CAPM, we would expect ~,t to be close to zero, whereas the contrary dominance relation would produce an at--value close to unity. The Davidson and Mackinnon Test (6) is biased toward an asymptotically underestimated standard error of at. However, by performing the test (6) for a sufficiently large number of time intervals, the mean and standard error of at may be estimated directly from its own time series [11, p. 1201]. In consequence, some evidence in favor of either theory can be secured by this procedure. In order to check the consistency of the Davidson and Mackinnon test, we will also compute the Bayesian Posterior Odds Ratio, following Chen [I 1] on US data. The ratio is given by given by the following formula [48, p. 306-312]: ODDS=.(ESS~cAm~/ESS~,n)#~.N ~[c~M]-~'r~z, (7)
where ESS = error sum of squares; N = number of observations; k ffi dimension of the respective model. The ODDS-statistic indicates the posterior odds in favour of APT over CAPM. Since we possess no prior knowledge of the odds, we assume that APT and CAPM are equally probable a priori.
446
Ostermark--Arbttrage Prtcmg Models
ACKNOWLEDGEMENTS
The advsce of Professor Letf Nordberg (Department of StatlsUcs, Abo Academy) ts gratefully acknowledged. I thank MSC. (Phil) Irmeh Enksson and BA Christopher Grapes for revmng my English style. The testmg procedure was programmed m FORTRAN with access to the IMSLhbrary on ~bo Academy's VAX 8800 computer by the author The databases used m the study were graven by Kansalhs-Osake-Pankks, a Fmmsh commercial bank. and Aktiedata Ltd The author Is responsible for any errors m the text.
REFERENCES I. Barnett V and Lewts T (1978) Outhers m Stattstwal Data. Wdey, NY 2, Bern A and Jarque C. (1982) Model specification tests a simultaneous approach. J. Econometrtcs 20(2), 59-82 3, Bern A and Kannan S (1986) An adjustment procedure for predlctmg systemattc rtsk J appl Econometrtcs 1(3), 317-332 4 Berglund T (1986) Anomahes m stock returns on a thin security market (dtssertatton) Svenska Handelsh6gskolan. Helsmgfors. 5. Bcrglund T and Wahlroos B and Ornmark A (1985) The weak form efficiency of the Finnish and Scandinawan Stock Exchanges A comparatsve Note on thin trading Scand J. Economics 4, 521-530 6. Black F (1972) Capital market equihbnum with restricted borrowing. J Bus. 45(3), 444-454 7 Bowman KO (1975) Ommbus test contours for departures from normahty based on bj and b: Bwmetnca 62(2). 243-250 8. Box GEP and Cox DR (1964) An analysis on transformauon (wtth discussion). J R stattst Soc Ser. B 26, 211-252 9. Chamberlain G and Rothschdd M (1983) Arbttrage, factor structure and mean-variance analysis on large asset markets Econometrtca 51(5) (September), 1281-1301. 10 Chamberlain G (1983) Funds, Factors and diversification tn arbitrage pricing models. Econometrlca 51(5), 1305-1323. II Chen N-F (1983) Some empirical tests of the theory of arbitrage pncmg J Fm 38(5), 1393-1414. 12 Claesson K (1987) Effektt~tteten pd Stockholms Fondb6rs Ekonomtska Forskningsmstltutet vtd Handelsh6gskolan I Stockholm 13. Cho CD, Eun CS and Scnbet LW 0986) Internattonal arbitrage pncmg theory: An empmcal investigation. J Fro. XLI (2), June, 313-329. 14. Copeland TE and Weston (1983) Fmanctal Theory and Corporate Policy. 2nd edn Addison-Wesley. MA. 15 Davldson R and Mackmnon J (1981) Several tests for model spec~ficatton m the presence of alternative hypotheses. Econometrwa 49(3). 781-793. 16. Dhrymes PJ, Friend I and Gultekm NB (1984) A critical reexamination of the empirical evidence on the arbitrage pricing theory J Fm XXXIX (2). June, 323-346. 17 Fama E and Macbeth J (1973) Risk, return and equihbnum- empmcal tests J. Political econ. 81(3), (May-June) 607-636. 18 Fama E (1976) Foundauons of Finance. Basic Books. NY 19. Foster G (1986) Financial Statement Analysts, 2nd edn. Prentice-Hall. Englewood Chffs. NJ
20 Gnnblatt M and Tttman S (1987) The relation between mean-variance efficiency and arbitrage pncmg J Bus 60(1). 97-112 21 Huberman G 0982) Arbttrage pricing theory A simple approach J Econ Theory 28, 183-191 22 Huberman G, Kandel S and Stambaugh (1987) Mimicking portfohos and exact arbitrage pricing J Fin. 42(I). !-9 23 IMSL Libraries (1987) FORTRAN Subroutines for Mathemattcal and Statistical Analysts Houston, "IX. 24 Ingersoll J (1984) Some results m the theory of arbttrage pncmg. J Finance 39(5) (September). 1021-1039 25 J6resko[ KG (1977) Factor analysts by least-squares and maxtmum-hkelihood methods In Stanst:cal Methods for Dtgttal Computers (Edited by Kurt Enslem, Anthony Ralston and Herbert S Wdf), pp 125-153 Wdey, NY 26 Kaiser HF (1963) Image analysts In Problems in Measuring Change (Edtted by C Hams) Umv Wtsconsm Press, Madtson, WI 27 Kmg BF (1966) Market and mdustry factors m stock price behavtor J Bus 39(1, Pt 2), 134-190 28 Korhonen A (1977) Stock prices, mformatton and the effictency of the finmsh stock market Emptncal tests (dtssertatton). Helsmkt School of Economtcs 29 Lawley DN and Maxwell AE (1971) Factor Analysts as a Stattst~cal Method. 2nd edn. Eisevter. New York 30 Lmtner J 0965) The Valuatton of Rtsk Assets and the Selection of Rtsky Investments m Stock Portfolios and Capttal Budgets Rev Econ stat:st 47(1)(February), 13-37 31 Litzenberger R and Ramaswamy K (1979) The effect of personal taxes and dividends on capital asset pnces theory and empmcal evsdence J fin Econ 7(2) (June), 163-195 32 Perry P (1982) The time variance relationship of security returns Imphcattons for the return generating stochastic prices J Fm 37(3) (June), 847-869 33 Pettway RH and Jordan BD (1987) APT vs CAPM esttmates of the return-generatmg function parameters for regulated pubhc uulittes. J fin Res X, (3), (Fall), 227-238 34 Press JS (1972) Apphed Multtvartate Analysts Holt (Rmehart & Winston) NY 35 Remganum M (1981) The arbitrage pricing theory Some empirical results J Fm 36(2), 313-321 36 Roll R (1977) A critique of the asset pricing theory's tests J Fm Econ 4(2, Pt l)(March). 120--176 37 Roll R and Ross SA (1980) An emptncal mvestlgatton of the arbitrage pncmg theory J Fin 35(5) (December), 1073-1101 38 Ross SA (1976) The arbttrage theory of capttal asset pncmg J econ Theory 13(I) (December), 341-360 39 Ross SA (1977) Return, risk and arbitrage (Edtted by Friend I and Btckler J), In Rtsk and Return m Finance Vol I Balhnger, Cambridge, MA 40 Sharpe WF (1964) Capttal asset prices A theory of market eqmhbnum under condttsons of risk J Fm 19(5). 425--442 41 Sharpe WF (1985) Investments, 3rd edn. Prenucc-Hall, Englewood ChiTs, NJ 42. Stambaugh R (1983) Arbttrage pricing with reformation J Fro. Econ. 12(3), (November), 357-369 43. Stultz R (1981~ A model of mternattonal asset pncmg The J Fm Econ. 9(4) (December). 383--406 44. Thomson GH (1951) The Factortal Analysts of Human Abthty London Unlv Press 45 Trzcmka C (1986) On the number of factors m the arbitrage pricing model J Fin XLI, (2), June, 347-368 46 Tucker LR (1973) A rehablhty coeffictent for maxt-
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47 48 49
50
mum hkehhood factor analysis Psvchometrtka 38(1), (March), 1-10 Zellner A (1971) An lntroductwn to Bayes:an Inference m Econometrws Wiley, NY Virtanen I and Yh-Olh P (1987) Forecasting stock market prices on a thin security market. Omega 15(2), 145--155 (~sterrnark R (1987) On the relationship between the empirical market hne on the Fmmsh Stock Exchange and international evidence Fmmsh J Bus econ 6(3), 244--259 Ostermark R and H6glund° R (1988) The Cartesian ARIMA Search Algorithm Abo Academy, Ser A 261 Presented at The Nordtc Congress on Mathematzcal
447
Stattsttcs. Abo. June. and The E:ghth lnternattonal Sympostum on Forecasting, Amsterdam. June 51 Ostermark R (1988) Fuzzy Linear Constraints m The Capital Asset Pricing Model. Fu::y Sets and $;stems 29 52. Ostermark R (1988) Portfolio Efiiclencyof a"Dynamic Capital Asset Pnang Model Emptncal Evidence on Fmmsh and Swedish Stock Returns. Abo Academy, Ser A 271 53 Ostermark R (1988) Predictabdlty of Fmmsh and Swedish stock returns Omega 17(3), 223-236
Ralf Ostermark, Department of Business Admm:stratwn, ,,ibo Academy, Henrlksgatan 7, 20500 ,~bo 50, Finland
ADDIIIF_~S FOR CORRI~PONDENCE. D r
0305-0483/89 $300+000 Copyright ~ 1989 Pergamon Press pk
Pnnted m Great Britain All rights reserved
Arbitrage Pricing Models for Two Scandinavian Stock Markets R OSTERMARK ]~bo Academy, Fmland (Recewed Not,ember 1988; m revisedform Aprd 1989) This l m l ~ examines the Arbitrage Pricing M ~ i e b (APM) for Finnish and Swedish data. The te~flng is based oa weekly returns of portfolio-nggr~ated data, partly to achieve multivariate normality, partly to dampen omller efl'eets. Data of beta-ranked portfolin~ reed asewbere in testing the Capital Amet Prklall Model (CAPM) and the Arbitrnge Pricing Theory (APT) m used here too and i ~ v i d ~ a ~ ~ for dominm~ testing of CAPM and APT. Following ~ the dominance relaflomhip is analyzed by the D a v i d ~ - M K k l m m a test tad tiu Pmt~dor Odds Ratio test. The Indicate that APT daminates CAPM in b o ~ eountrlu. The F i a n ~ ~ ~ out to be more volatile than the Swedish. Furthermore, the multiple factor model is seen to bare relatively more power In Finolab tium In S w e d ~ conditions. The results are combtant with ln~vious evidence, that the single factor model (CAPM) tends to be more powerful In explaining Swedish than Finnish stock returns. Since the t e s t i q is breed oa portfolJo48greatted weekly returns series, the results do not ~ l y agree with t h e ~ for daily returns at the Individual a~et level.
Key words--dominance between APM and CAPM, Dav,dson-Mackmnon test, Scandmavtan evtdence
1. M O T I V A T I O N
I~ THE STUDY, the dominance relationship between the Arbitrage Pricing Model (APM) and the Capital Asset Pricing Model (CAPM) with Finnish and Swedish data is compared. The models are estimated with equivalent index series. A complete description of the databases is given in [11]. The study replicates [4, 53] within a two-country setting. Whereas in [49] all hsted Finnish stocks over the period 19701983 were used in the testing, m this study, we use the subsets of continuously listed stocks in the Helsinki Stock Exchange (HeSE) over 1970-1987 and the Stockholm Stock Exchange (StSE) over 1977-1987. By admitting only continuously listed stocks, we eliminate problems with missing data. The study continues our work on Portfolio Efficiency of Capital Market Models presented in [49, 51-53]. The APT, originally formulated by Ross [38, 39] and extended by others [9, 10, 21] is based on the assumption that each asset return is linearly related to a set of common factors and its own idiosyncratic disturbance (own risk). There is some encouraging empirical evidence on the 437
model, as [11, 20, 33, 35, 37]. Several authors, e.g. [13, 30, 43], have derived and/or tested various versions of the international asset pricing model (IAPM). It is of interest to note that according to the inter-battery factor analysis of [! 3], the number of factors common to a pair of countries ranged from one to five. They concluded that, if the number of international common factors reflects the degree of economic integration, the United States can be said to be highly economically integrated with Canada, the United Kingdom and Japan, and least integrated with France. How the inter-battery factor analysis works on Scandinavian data remains to be studied. While empirical tests of the APT are usually carried out on security markets with a large amount of assets ( > 1000), there is little experience on how the APT performs on thin security markets. Admittedly, the fundamental assumptions under which the APT is expected to hold are fairly discouraging when it comes to thin markets with limited possibilities of arbitrage and diversification; the theory is designed for a well diversified, frictionless and perfectly competitive economy. However, the APT might
438
Ostermartr--Arbltrage Prtcmg Models
hold even on thin security markets if an (asymptotically) multivariate normal space could be constructed by properly transforming and aggregating the underlying series. Such a space would allow comparative testing of competing theories such as APT and CAPM, which explicitly assumes multivariate normal returns. Furthermore, even if normally distributed returns are not reqmred by the APT, the need for a multivariate normal space is emphasized in empirical testing. The maximum likelihood estimation method, using a likelihood function of a multivariate gaussian distribution, is preferable since more is known about its statistical properttes [38]. In the present study we test APT with Finnish and Swedish data. In particular, we are interested in the dominance relationship between APT and CAPM, derived by [6,40,41]. Whereas the former is a multifactor approach, the latter is a two-parameter model. The CAPM relates the return on the individual asset linearly to the market return, by a risk premium (beta) in excess of the riskless return. In empirical testing of the CAPM, a two-stage procedure (in the spirit of [17]) is followed [49]. First, the beta coefficients of each asset (or portfolio) are estimated by Ordinary Least Squares (OLS). The standard errors are then computed from the &fference between observed and estimated returns. Second, the beta coefficients and standard errors are used in a cross-sectional regression procedure to obtain the final CAPM estimates. If the CAPM is vahd, the risk-return relationship will be linear, with a positive trade-off between risk and return. Linearity is tested by augmenting the empirical form of the model by a squared beta component [14, 28]. Sufficiency is tested by including the standard errors of the OLS-estimates (step 1) in the cross-sectional regression (step 2). In the present study, we test the dominance relationship between a k-factor APM (k 611,5]) and the CAPM augmented by the squared beta component. The testing is based on weekly returns of a set of portfolio aggregated data, partly to achieve (approximate) multivariate normality, partly to dampen outlier effects. The testing of the dominance relationship is bound to produce superiority of APM over CAPM, when increasing the number of factors sufficiently. Thus, the results m the present study are only indicative of the relative power of the two theories.
We nottce that the APT has been tested on daily returns data of the New York Stock Exchange despite a high skewness of individual daily returns. [37] found that the results of their APT tests improved when every second observa. tion was used. They argued that the effect was due to skewness in daily data and the effect of nonsynchronous trading. [32] suggested that daily data are best modeled by an instable variance process, as it is well knogn that daily returns are improperly described by the normal distribution [18]. Adding the econometric problems due to splitting assets into groups and computing subcovariance matrices for these groups to accommodate computer limitations, the results of, for example, [37] may be severely biased. While there is no doubt as to the internal consistency of the Arbitrage Pricing Theory, the procedures for empirically testing it on large security markets may be criticized. If daily return series are significantly non-normal, tests based on the normality assumption, with untransformed data, should be avoided. If all individual assets cannot be simultaneously captured within the APT framework because of current computer constraints, we are forced to perform the testing either on a subset of the assets or on some portfolios (see [22]). However, then the results cannot be generalized to apply to a set of individual securities, unless further assumptions are made, e.g. that the individual assets or portfolios used m model testing represent the total population of securities in the stock exchange. The generalization efforts naturally involve other issues, such as the validity of the criteria applied when selecting assets for model testing, the correctness of portfolio formation procedures, etc. The discussion reflects some problems arising from the observed anomalies on thin security markets in particular (see [4]). Consequently, instead of attempting to demonstrate the universal validity of a theoretical construct, our investigation aims to demonstrate the empirical performance of the APT framework compared to the CAPM in a suitably transformed space only. Insofar as the level of aggregation is concerned, it has been shown that the daily returns series on the HeSE and StSE are non-normal (e.g. [4, 12]). Finnish and Swedish daily stock returns are affected by the following disturbances: seasonaliues (weekday, month of the year and announcement effects), firm size, under-estimated betas m the CAPM
Omega, Vol 17, No 5
439
prices to those based on the opening trading price and found no statistically significant differences between the autocorrelations for the 13 most traded stocks on the exchange over 1982-1983. The Stockholm Stock Exchange (StSE, Sweden) database was acquired from Aktmdata Ab (Helsinki) by funds granted by the Foundation for Supporting Capital Market Research m Finland. The database consists of daffy price index series of the mdividual stocks over the 1977-1987 time period. The price indices are computed in the same way as the Finnish mdices. The database consists of 371 stocks covering 2737 days in the exchange. The Fmmsh and Swedish data bases used m this study are completely comparable with respect to index 2. THE DATA AND number computation. The databases are ITS TRANSFORMATION summarized in Table 1. The Helsinki Stock Exchange (HeSE, FinIn subsequent testing, both data sets are land) price index database was obtained from transformed logarithmically. The empirical disKansallis-Osake-Pankki (a Finnish Commer- tribution of average weekly logarithmic price cial Bank) for this specific research. The data- indices is fairly consistent with the normality base (KOP Stock Market Indices) consists of assumption [27, 49, 53]. The logarithm of the daily price indices from the beginning of Febru- price index is a good approximation of returns, ary 1970 up to the end of 1987 for each stock assuming continuous compounding (see [14], listed on the HeSE, except for Porin Puuviila p. 304, footnote 13): let P, = Pie'.'. denote (listed 01/1973-02/1974). By the end of 1983 the price path from time 0 to time t and let some 70 stocks were quoted on the exchange, I, = P,/Po denote the corresponding pnce index whereas the number increased to some 100 by Then r, = In(It)= i n ( P , ) - In(P0) is the return the end of 1987. The index series are based on of the asset. average trading prices for each trading day. The Following [45], we use weekly returns in the prices are corrected for cash dividends, splits, testing. Initially, we studied the relationship stock dividends and new issues. For days when between the average empirical distribution and no trade in a stock took place, the quoted bid a set of transformation contenders with both price is used. If both trading and bid quotations Finmsh and Swedish data. In parucular, we are lacking, the previous quoted price (trading demonstrated that the returns of portfolios or bid) is used (the number of such prices is 8 formed in the Fama and Macbeth procedure are in all). This principle is justified by the results of approximately normal with both data sets, as [4, p. 64]. He compared the first order serial measured by the normality test statistic of [7] correlations based on returns from average daily (see [I, 2, 3, 8]). In [49], we also observed that
and non-normal return distnbutions. Empirical evidence on the nature of the serial correlation of daily stock returns observed on Scandinavian stock exchanges is summarized by [5]. [48] demonstrated that both univariate and multivariate (econometric) models could be built for the Finnish stock market. The authors used a monthly market index over the period 1975-1984 to construct ARIMA models and multiple regression models. In general, very good short-term forecasts were produced with their models. In general, the econometric models outperformed the univariate models. The work of [48] was extended in [53] to the level of all individual assets (see [50]).
TAble I. Data descnpUon The Kansalhs-Osake-Pankkt (KOP) weekly ,ndtces of asset pnces on the HeSE. and equivalently computed mdtces on the StSE. With HeSe data" 1970-1987. The enure period (826 weeks) ,s dmded into n,ne partly overlapping porffoho formaUon panods With StSE data 19"/7-1987. The enure panod (526 weeks) ts d,vlded into five partly overlapping porffol,o formaUon panods The porffoho formation procedure ts eaphuned below (See also [89] for a detaded description of the test,rig schedule) Days when n o t r a n m c U o n s o c c u r are proxled by Average bKl quotauons Mining ~ J : Price index adJUsted for All capital changes and dtvldeods. The impact of"cash d,vtdends on the variance Data ~it: of' Fmmsh returns ,s neshlpble (see [49]). since the d,vtdend pohcy of" the hsted compar.cs ,s almost invariable over the long term No. of eoatimmetsly listed stocks: H©SE :36. StSE 93
..~,m'c e :
No. of l~rtfo#os:
HeSE 12, StSE 19
440
Ostermark~Arbitrage Pricing Models
the Fama and Macbeth procedure is very powerful m reducing non-normahty, since almost all individual asset returns were significantly nonnormal. Thus, the use of portfolio aggregated data in subsequent testing is justified. 3. THE ARBITRAGE PRICING THEORY AND ITS EMPIRICAL TESTING Whereas the CAPM is based on the assumption that security returns are linearly dependent on a single common factor--the return on the market portfolio--the APT is more general: the latter allows simultaneous recognition of the return generating effects of several factors, not just one, as in the CAPM. In fact, the CAPM may be considered as a special case of the APT. The former is reduced to the latter by defining the CAPM-beta as the weighted average of the betas in the k-factor model, with the security's sensitivities to the factors as weights [19, p. 340]. In the present section we give an intuitive presentation of the APM. Since we wish to test the model empirically, a discussion of the mechanics of the empirical form of the model will suffice. (See e.g. [14, 16, 32, 36,45] for a rigorous derivation and discussion.) The basic assumption in the APT is that the returns on any securities are linearly dependent on k independent factors (see Appendix A). The empirically important result of the APT is the fact that only the risks reflected in the covariance matrix of the (transformed) asset returns series have non-zero factor prices. Thus, if a priced factor could be determined by some other method, even after accounting for the effect of relevant factors, the APT would be rejected [i 1]. Such extraneous factors could possibly be derived from variables such as firm size and own risk. We notice Chen's [11] emphasis on the non-priced factors in an investor's decision-making. The possible existence of non-priced factors is important in order to understand the causal structure of concrete investment decisions, yet irrelevant for empirical testing of priced factors. The potential richness of the APM may be conjectured by noticing that the return generating process of any security in the economy is determined by the same k + ! factors for, f), . . . . . fk,. Given the (true) values of these factors, the expected return of the i:th security is obtained as the sum of the intercept (f0,) and
the sensitivities (l,j) to the k remaining factors. Ilk --- 1, the APM reduces to a one factor model comparable to the CAPM. 3.1 Example Assume that we have been able to determine that, at time t, the return generating process of the securities in the observed market place is governed by two factors with values f), ==0.2, f2, = 0.5. Assume also that the factor sensitivities of the i:th security have been estimated as 1,~ --- 0.22, 1,: = 0.35. Then the expected return of the i:th asset would equal 0.2 x 0.22+0.5 x 0.35 = 0.219, or 21.9%. Given perfectly competitive and fricUonless capital markets, the best forecast for the return on the t:th asset at t + 1, made at time t would be 21.9%. Furthermore, if we are able to unravel the economic forces represented by factors I and 2, we have a complete economic model of security market pricing in the observed economy. A difficulty with the APM is, however, that the factors are derived mathematically from the covariance structure of the observed security returns: There is no way of determining the exact number of relevant factors a priorz, hence we are forced to rationalize our results ex post, that is, to try to find an economic meaning-content to the k + ! factors obtained for the particular economy. Compared to the CAPM, the gain in explanative power of multidimensional analysis is obtained at the expense of difficulties in interpretation. The APM is estimated by factor analysis, using the sample covariance matrix as computed from the observed stock returns by the standard covariance formula [23, 25, 26, 29]. It is assumed that the theoretical covariance between the rates of return on any two stocks is given by [38]:
Cove(R,, R,) -- I,, 5, o2 ) + " ' +
i
Factor analysis makes the working assumption that the individual factor variances o2(fh) are equal to i.00, all h, and then finds that set of factor loadings (10) for each stock that will make theoretical covariances correspond as closely as possible to the sample covariances Covs(R,, R~). In other words, factor analysis seeks optimal values for l,j such that Covr(R ,, Rj) - Covs(R,, Rj). The procedure guarantees that the k + 1 factors absorb the maximal explanative potential in the covariance matrix, which is the key to explaining variation of security returns. After obtaining estimates of the factor betas, the next
Omega, Vol 17, No 5
step is to estimate the values of the factor prices fj for all factors. This is usually done by crosssectionally relating the factors to average returns [11, 36]. 4. EMPIRICAL RESULTS
4.1 The range of sigMflcant factors
441
all subintervals. On the other hand, 90% of variance was explained by three factors for all subintervals. Bartlett's chi square statistic for testing the equality of the remaining k - r factors implied that the factors were unique. However, as the relative standard deviation for Bartlett's statistic tends to increase for the tail of remaining (~<4) components, the last factors are not necessarily unique for some subintervals. On the basis of the results of the principal component analysis we fixed the range of factors to be tested at k ¢ [1, 5] for both datasets. Since the Swedish dataset is almost three times larger than the Finnish, ~t might seem conceivable that the number of significant factors in StSE could exceed five. However, according to the evidence presented by [25], no more than five significant factors could reasonably be expected with US data. In fact, their study implied only two chances m 100 that at least six factors were present in the data.
A criucal issue in empirical testing of the APT is the number of factors. [16] report that the number of significant factors in standard chi squared tests tends to increase as more securittes are analyzed. In an extension by [9], it was shown that if a k - factor model is valid, then k eigenvalues of the covariance matrix of returns grow large as the number of securities increases. Using this result on the US market, [45] found evidence that there is one large factor and no obvious way to choose more than one The possibility of having more than one significant factor was not ruled out, however. As admitted by [45], it is quite possible that, say, five factors could provide a proper model of 4.2 Estimated factor Ioadings asset pricing in the US market, even if the data The factor ioadings were computed on the were consistent with only one exploding eigen- correlation matrices for each subinterval with value. The results of [9] and [45] suggest that the both Finnish and Swedish data. The averages of proper number of factors cannot be determined the means and standard deviations for the loadin a straightforward fashion. We applied princi- ings for the I- and 5-factor models are presented pal component analysis to determine an interval in Table 2. [Detailed results of the unrotated that may reasonably be expected to contain the loadings for all models of the 90 (HeSE) and 50 true number of factors. Initially, the 90 covari- (StSE) subintervals may be obtained from the ance matrices for the beta-derived portfolios author on request.] Table 2 shows considerable variation in the constructed over partly overlapping estimation subintervals within the estimation and test loadings of the correlation matrices of the periods were computed. The resulting covari- subintervals. The Finnish loadings turn out to ance cube was then subjected to principal com- be more volatile than the Swedish. One reason ponent analysis, one matrix at a time. The for this variation might be the instability of the results indicated that, with Finnish data, at least portfolios for consecutive subintervals. The four factors are needed to account for 95% of portfolio addresses, i.e. the beta-rankings of the total variance of the underlying variates. For individual assets, vary over time (see [49]). As some covariance matrices n o more than three the content of portfolios at given risk categories, factors were needed to account for 95% of measured by their beta ranking, changes, so variance, whereas five factors explained 95% for does the factorial structure of the corresponding
Table 2. Average of Ioadmge with Fmmsh and Swedish data
Me.an St dev
I 0774 0304
Mean St dev
I 0 893 0 199
( I ) A ~ r q r load.s wit~ Fia~A dma F~'tor H i b e r : 2 3 4 5 0 139 0058 0049 0018 0.374 0220 0 166 0 117 (b ) Atvralle I o ~ s wklt Swfdish data Facto¢ atnnber:. 2 3 4 5 0 056 0 041 - 0 034 - 0 023 0268 0 171 0 104 0079
442
Ostermark--Arbttrage Prtcmg Models
correlation matrices. Another issue is that the factors of two correlation matnces are not comparable (see [37], p. 1099). Because of thesea problems we consider a test of the equivalence of the factorial structure across the subintervals by standard statistical techniques (e.g. HoteUing's T 2 statistic, see [34]) to be unnecessary: the considerable variation in the individual factor Ioadings already indicates that the factor structures are unequal. Table 2 also reveals that the multiple factor model tends to have relatively more power with Finnish than with Swedish data. The results may indicate that the single factor model (CAPM) is more powerful in explaining Swedish than Finnish stock returns. In Table 3 we present the means and standard deviations for some important statistics. The Tucker and Lewis [46] reliability coefficient (pstatistic) for the proportion of variance explained indicates significance of at least five factors with both data sets. The coefficient is an estimate of the ratio of explained to total variation in the data (see [46] for the expression). The p-statistic is lower for the Swedish data set than for the Finnish. Thus, the potential power of the ApT-framework seems to be higher with Finnish data. The chi square statistic tests that all remaining parameters associated with additional factors are zero. The statistics indicate, that some variation is contained in the remaining factors.
Table 4 Results of Davtdson~Mackmnon test for all test IX'nods. based on the extended CAPM (Appendix A) and I- and 5-factor ApT-models HeSE
/~o of Factors: ~0 ";, s(:/0) s('/,) t(~) tfir) Ar2 .g'
StSE
1
5
I
0007 0 149 0017 0311 4214 4 546 0 076 2457
0003 0017 0043 0109 0674 I 457 0 009 I 341
00004 0098 0066 0174 0043 3 979 0 083 4445
5 -0007 0026 0046 0153 - I 067 I 182 0 032 3 237
q'he regress,ons are s~gmficant at the 5% nsk level m all cases but the 5.factor APT-model with HesE data. The values of the -,,-coeffictent suggest that APT is superior to CAPM with both data sets
The t-statistics should be interpreted with caution because of the nature of the underlying
series and the testing procedure. The statistics give a descriptive measure of the strength of the obtained estimates; they cannot be used, however, to determine their exact level of significance. This would be a somewhat meaningless activity in any case, since the asymptotic standard errors of equation (5), Appendix A, are underestimated (see [! 1], p. 1401). The results of the DM-test for all test periods are shown in Table 4. The regressions are significant at the 5% risk in all cases but the 5-factor APT-model with HeSE data. The values of the afcoefficient suggest that APT is superior to CAPM with both data sets. The a,-term is not as close to zero as observed by Chen [11] with US data. However, Chen's [11] dominance test was based on the standard empirical form of the CAPM. A weaker APT-dominance (non-zero as and a:) may indicate that not all priced factors are recognized: the squared beta disturbance may be incompletely captured by the APT-model for k ÷ [2, 5]. The accuracy of the competing models is indicated by the Mean Absolute Percentage Error, computed over all test periods in Table 5. With both data sets, even the I-factor APM ts 2.8--4 times more accurate than the CAPM. In order to support an evaluation of statistical significance, we recorded the number of significant parameter estimates in the individual
Table 3 Key staUsucs of 5-factor models with Finnish and Swedish stock data
Table 5 Mean absolute percentage error averaged over all test pertods
4.3 Dominance testing Following [11] on US data, the Davidson and Mackinnon (DM) test [15] was run for test periods that were not included in the subintervals used to estimate the factor loadings of the APT and the cross-sectional regression coefficients of the CAPM. The t-statistic for the a-coefficients is estimated from the standard formula [28]: t(ot*,) = a~,x/n/s(fz,)
(n =, n u m b e r o f test periods).
Tucker and Lewss p
HeSE
Chl square
Statistic:
HeSE
StSE
HeSE
StSE
Mean St dev
0 89 0 05
0 80 0 04
76 36 26 38
561 40 68 37
No o//actors: MAPE (APT) MAPE (CAPM)
StSE
I
5
I
5
0012 0051
0007
0017 0049
0007
Omega, Vol 17, No 5
APPENDIX
Table 6 Proportton of s,gmficant parameter esumates m md,vtdual DM-regresstonso ",-level 10%
HeSE I-factor APT 5-factor APT
0 97 0 80
0 82 0 90
0 68 0 86
It can be proved that, in a well diversified economy, the expected rate of return on stock i at time t may be expressed as ([14, 38]):
E(R,,)=fo,+t,~f.+ cross-sectional regressions at the 10% level for both data sets. The results are presented in Table 6. The hypothesis of a non-pure APTdominance is corroborated in Table 6: the 5factor APM is unable to force the intercept to zero both with Finnish and Swedish data. For the latter, the APT-dominance is seen to be somewhat weaker. (The consistency of the Davidson and Mackinnon test was also checked by the Bayesian Posterior Odds Ratio (see [11]). All ratios were strongly in favor of APT m both countries). 5. CONCLUSION In the present study, the dominance relationship between the Arbitrage Pricing Model (APM) and the Capital Asset Pricing Model (CAPM) with Finnish and Swedish data was analyzed. The models were estimated with eqmvalent index series. The APT-tests were earned out on weekly returns series, because of significant anomalies in daily series. The testing schedule was constructed in the spirit of Fama and Macbeth, which provided a convenient basis for comparing the APT and CAPM frameworks. The Finnish loadings turned out to be more volatile than the Swedish. Furthermore, the multiple factor model tends to have relatively more power with Finnish than with Swedish data. The results suggest that the single factor model (CAPM) tends to be more powerful in explaining Swedish than Finnish stock returns. The results corroborate the hypothesis of non-pure APT-dominance: the 5-factor APM is unable to force the intercept to zero with both Finnish and Swedish data. For the latter, the APT-dominance is seen to be somewhat weaker. The conststency of the Davidson and Mackinnon test was checked by the Bayesian Posterior Odds Ratto. All ratios were strongly in favor of APT m both countries. The results suggest that APT tends to absorb the explanatory power of CAPM, given a sufficient number of factors.
OME 17 ~ " D
A
A PM Test Procedure
StSE 0 70 0 91
443
+t~L,.
(1)
where. l,j = The sensitivity of asset 't' to factor j (factor loading) f0, = Zero-beta factor. fj,--jth priced factor. It ts assumed, that the k +1 vector (f0,, f~, . . . . . f~,) is multivariate normal. Various theoretical papers [16, 24,42] have significantly weakened the assumptions necessary to derive equation (1). In testing the APM we use the same portfolio formation procedure as used in [49] (and the same steps are followed as m [28]), where, the individual assets were assembled m equally weighted portfohos of the same size, using OLS-betas as ranking criteria for each formation period. The empincal market line was estimated by cross-sectional regression for each esttmation interval wtthin the formatton time intervals, with model parameters updated every five weeks [17]. With Finnish data, 240 weeks are used initmlly to compute OLS-betas and 12 portfolios, each containing 3 stocks, are formed on the basis of the ranked values for these estimates. The next 140 weeks are then used to recompute the individual OLS-betas. The resulting portfolio betas enter the cross-sectional regression (2) below. The procedure is repeated every five weeks, so that the database is continuously expanded with new price information during the testing period from week 381 to 426. The beta estimates are recomputed every five weeks. The whole process is repeated nine times, corresponding to the formation periods. As each formation period spans ten estimation and testing periods, there are 90 OLS-regressions and cross-sectional regressions involved besides the nine OLS formauon models. With Swedish data, 140 weeks are used initially to compute OLS-betas, and 19 portfolios018 of which contain 5, and 1 containing 3 stocks--are formed on the basis of the ranked values for these estimates. The next 140 weeks
444
Ostermark--Arbttrage Prtcmg Models
are then used to recompute the individual OLSbetas. The resulting portfolio betas enter the cross-sectional regression (2). As with Finnish data, the procedure is repeated every five weeks, so that the database is continuously expanded with new price information during the testing period from week 201 to 381. The beta estimates are recomputed every five weeks. The whole process is repeated five times, giving 50 crosssectional regressions [14, 28]" R,,, = A, + A,B,, + A , ~ + ~3,s(e,,) + ,,, (2) where Rp, = Return on portfolio p at time t. /il, = Rm,- R/, is the difference between the market rate of return (/~m,) and the riskless rate of return (Rf,). s(ep) =estimated standard error of time series model for Rpt. np,= random noise term.
(5)
We use the test procedure of Chert [11] to estimate the APT ([17,37]. See [49] for a description of the computer system): (1)
We consider the application of the above scheme for testing the APT justified on the following grounds ([49, 53]): (1)
Individual asset returns of a thin security market may be biased due to infrequent trading. By portfolio formation, outlier disturbance is expected to be duly eliminated or at least sufficiently dampened (of. [7]). This is indicated by the fact that the empirical distribution of portfolio returns complies well to normality [53]. Portfolio formation is a frequently used technique for eliminating outlier effects [1, 19, 25, 14, p. 101].
(2) The APT and CAPM performance may be directly compared by using the same testing schedule and portfolios in both frameworks (see [41] for the theoretical linkage between APT and CAPM). (3)
By repeating the estimation procedure for a sufficiently large number of times, it is possible to obtain important information on the stability of APT-estimates analogous to that of the CAPM-estimates, even if the factors for different covariance matrices are not easily compared.
(4)
Computer constraints justify an aggregation procedure that can be
utthzed and refined in the future, when the number of assets in the Scandinavian Stock Markets increases. Since the portfolios are ranked according to the level of risk as measured by OLS-betas, some factor(s) may be expected to capture the risk element. Portfolio rankings and the factor loadings for such factor(s) should be positively correlated over time. Thus, the portfolio rankings may be used to extract information on the nature of at least some factor(s).
Compute the set of covariance matrices (covariance cube) correspondmg to each estimation interval, using the ume series of the beta-ranked portfolios with equally weighted assets. The portfolios are expected to act as super-assets, mimicking the performance of the underlying assets so that unbiased estimates of the model parameters can be secured [22].
(2) Perform a principal component analysis on each matrix of the covariance cube in order to determine the expected number of significant factors in (I). (3)
Estimate the factor loadings for each portfolio over all estimation intervals. The factor loadmgs matrix ~s estimated by a two-stage procedure: First, initial estimates of the factor loadings are obtained by the noniterative image method of Kaiser [26, p. 226]. Second, initial estimates are used as input for a maximum likelihood estimation of factor loadings (see [23, 25, 29, 44, 46]). The image factor pattern is calculated on the assumption that the ratio of the number of factors to the number of observed variables is near zero. Consequently, a good estimate for the unique variance of normalized variables is given as one minus the
On.a,
Vol. 17, N o 5
squared multiple correlation of the variable under consideration with all variables in the covariance matrix. (4)
(5)
Estimate the risk premia (factor scores) for all test periods using the ioadings for the portfolios of the corresponding estimation intervals. The factor scores are estimated by a generalized least squares regression method also applied in [37]. Analyze the dominance relationship between the APT and the CAPM using the Davidson and Mackinnon [15] test and the Posterior Odds Statistic [47].
Consider the APM and CAPM equations (1) and (2) in matrix form at time "t', solved for the same set of portfolios [49]: R, = ~dL + e, = >
e, = R, - ~dL (AFT),
(3)
where R~ = p x 1 vector of stock returns. f, = vector of factor scores, estimated by the regression method for each test point t. The test points are not included in the corresponding estimation intervals. B d = p x k matrix of estimated factor Ioadings. e, = residual vector for time point t. R, ,= fldfi , + n, = >
n, = R, - f l d i ,
(CAPM),
(4)
where
f I # ¢ R p~3
is a 3 x l coefficient vector based on (2), as estimated by the Fama and Macbeth [17] crosssectional regression method for dffil . . . . . q. has entries corresponding to equation (2) for each estimation period. Thus, row 'i' of ~# is ~# ffi (1, ~1, ~ . The quadratic term has been included, since its ~-coefficient was significant in the Fama and Macbeth test made on Finnish data in [33]. By recognizing the quadratic term in the empirical market line, we try to overcome the problem of underestimated and/or heteroscedastic
445
residual variances of the basic CAPM [1 l, p. 1402]. R,, n, ¢ R p* ~ are the observed (transformed) index and residual vectors respectively. The issue of the dominance relationship between APT and CAPM involves the problem of discriminating among non-nested alternative models. One approach to the problem has been suggested by Davidson and Mackinnon [15]. Consider the expected values for the APT and CAPM equations: ~cA,,, = a,a,.
We form the following cross-sectional regression model over all test periods: ~,=%+
t~,(CAPM) +(1 --at) ~,(AFT)
(5)
Define v,ffi~,- ~A~ and V!= ~CAm~_~ ' ~ Then the restricted parameters model can be estimated as: f,= ~ + ~sf~. (6) If APT were to dominate over CAPM, we would expect ~,t to be close to zero, whereas the contrary dominance relation would produce an at--value close to unity. The Davidson and Mackinnon Test (6) is biased toward an asymptotically underestimated standard error of at. However, by performing the test (6) for a sufficiently large number of time intervals, the mean and standard error of at may be estimated directly from its own time series [11, p. 1201]. In consequence, some evidence in favor of either theory can be secured by this procedure. In order to check the consistency of the Davidson and Mackinnon test, we will also compute the Bayesian Posterior Odds Ratio, following Chen [I 1] on US data. The ratio is given by given by the following formula [48, p. 306-312]: ODDS=.(ESS~cAm~/ESS~,n)#~.N ~[c~M]-~'r~z, (7)
where ESS = error sum of squares; N = number of observations; k ffi dimension of the respective model. The ODDS-statistic indicates the posterior odds in favour of APT over CAPM. Since we possess no prior knowledge of the odds, we assume that APT and CAPM are equally probable a priori.
446
Ostermark--Arbttrage Prtcmg Models
ACKNOWLEDGEMENTS
The advsce of Professor Letf Nordberg (Department of StatlsUcs, Abo Academy) ts gratefully acknowledged. I thank MSC. (Phil) Irmeh Enksson and BA Christopher Grapes for revmng my English style. The testmg procedure was programmed m FORTRAN with access to the IMSLhbrary on ~bo Academy's VAX 8800 computer by the author The databases used m the study were graven by Kansalhs-Osake-Pankks, a Fmmsh commercial bank. and Aktiedata Ltd The author Is responsible for any errors m the text.
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Ralf Ostermark, Department of Business Admm:stratwn, ,,ibo Academy, Henrlksgatan 7, 20500 ,~bo 50, Finland
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