B & E Review Vol. 7 No.1 1994 - 1996

THE ARBITRAGE PRICING THEORY AND COMMON STOCK RETURNS

Pablo F. Mangaran, Jr.

The Arbitrage Pricing Theory (APT) originally proposed by Ross (1976) may be conceived a an attractive alternative to the Capital Asset Pricing Model (CAPM) (Sharpe, 1964; lintner, 1965; Masin , 1966; and Black, 1972) for explaining the relationship between risk and return .

The major assumption of the theory is that the returns of a large number of assets can be broken into two components:

o the systematic risk which is nondiversifiable and which can be measured as exposure to a small number of common factors; and

o the unsystematic or the idiosyncratic risk which can be elimi­nated through efficient portfolio diversification .

From the major assumption , both the CAPM and the APT recog­nize that every asset must be compensated only according to its system­atic risk . There are two major differences, however.

o In the CAPM, the systematic risk of an asset is defined to be its covariability with the market portfolio . In the APT, the systematic risks are defined to be the covariability with sev­eral economic/financial factors.

o The CAPM requires the economy to be in equilibrium while the APT requires only that the economy has no arbitrage opportunities .

Several researchers have been made on the APT. The first major research in support of the APT is that made by Roll and Ross (RR) (1980) . While other researchers are made basically using the same set of data

66 A~!8ITRAGE PRICING THEORY

and meth cdology used by RR , results range from strong agreement to

disagreement relative to the research findings although RR 's main find­ings rem2lin unchallenged: At most, three to five identifiable common factors systematically affect common stock returns and that their asso­ciation risk premia are more or less positive.

(This is a , in abbreviated version of the author's DBA dissertation which was suc­cessfully dl!fended April 1993 before the faculty of the University of the Philippines College of Business Administration).

OBJECTIVES OF THE STUDY

The APT is a relatively new and different approach to determ ining asset pric'9s . Considered as a more general and flexible theory than the CAPM, the APT suggests that multiple factors are involved in the return generatin!l process.

The major objectives of this study are:

o to test the applicability of the APT using comm on stock re­turns traded in the coun try 's Manila Stock Exchange; and

a to present the different theoretical and empirical issues sur­rounding th e su bj ect matter

THE STRII CT FACTOR MODEL OF THE APT

The APT developed by Ross (1976) is a new and different approach in determining asset prices . It is based on the law of one price: similar good cannot sell at different prices . This implies that se curities with sim ilar risks must have the same expected return

APT assumes that th e rate of return on any security can be repre­sented by a strict linear k-factor model of the form

where R, E(R) F.

e. I

i = 1,2, .. ,n assets R

one period rate of return for asset i expected (mean) rate of return of asset i

(1 )

fa ctor k common to the returns of all assets under co n­sideration and has a mean of ze ro sensit ivi ty (facto r load ings ) of asset i's return to move­ments in th e co mm on factor F. residual noise of asset i's return which has expected value of zero and vian ce VAR e,

PABLO F. MANGARAN, JR. 67

APT assumes that investors are risk averse who prefer more wealth to less wealth . Investors have homogeneous beliefs that the random return for the set of assets being considered is governed by the linear k­factor model of the form given in (1). The model is derived under the usual assumptions of perfectly competitive and frictionless capital mar­kets. Moreover, the model requires that the number of assets n must be greater than the number of factors k.

The economic argument of the APT is simple. In equilibrium, all portfolios that can be selected from among the set of n assets and which satisfy the conditions of using no wealth and having no risk must also earn no return on average. These portfolio are called arbitrage portfo­lios.

Using the economic argument of the APT and linear algebra, the relationship given in (1) may be shown to result to

(2)

where ao = riskless rate of return a

k = risk premium or the price of risk to the kth factor; and

bik = sensitivity of the returns on the ith security to the kth factor

Relationship (2) is the central conclusion of the APT and is the subject of the present empirical testing.

EMPIRICAL TESTS OF THE APT

The usual empirical test of the strict factor model of the APT follow a two-step procedure which employs two different phases of statistical activities :

o using factor analysis, estimate the expected returns and the factor coefficients from time series data on individual asset returns; and

o using regression analysis, use these estimates to test the basic cross-sectional pricing conclusion (2), i.e., are the ex­pected returns from these assets consistent with the factors derived in the first step.

The study by Gehr (1975) is considered as the first empirical study of the APT and also lends support to the validity of the APT

68 ARBITRAGE PRICING THEORY

The RR (1980) study is considered the classic literature in APT. The RR study shows that:

o about three or possibly four risk factors systematically af­fect security returns ; and

o only two risk factors are "priced", that is, the impact of these two factors are significantly reflected in the security prices.

Overall, the empirical evidence provided support to the validity of the APT.

Other studies in support of the APT are those made by Chen (1983), Cho , Elton and Gruber (1984) , Bower, Bower and Lugue (1984), and Lehman and Modest (1988).

The studies made by Reinganum (1981) and Dhrymes, Friend, Gultekin and Gultekin (1985) are not supportive of the APT.

Other than the RR procedure, recent literature contain a number of tests to empirically estimate the APT model. Brown and Weinstein (1983) propose the bilinear paradigm, while Cho (1984) employees inter-battery factor analysis . Jobson (1982) proposes the miltivariate linear regres­sion mod,:!I, while Connor and Korajczyk (1986) develop the Jensen coef­ficient and the Treynor-Black appraisal ratio to reset the implications of the APT. Lehman and Modest (1988) show the utilization of portfolio to mimic factor returns .

Finance experts also propose alternative approach to testing the APT. Farna and Macbeth (1973) and Sharpe (1982) propose the specifi­cation a priori of a set of attributes that affect stock returns. Meantime, Oldfield and Rogalski (1981), Elton, Gruber and Rentzler (1983), Chen , Roll and Ross (1986), Fogler, John and Tipton (1981), and Burmeister and McE lroy (1988) propose the specification of a set of influences or indices which enter the return-generating process.

METHODOLOGY AND DATA USED

This paper initially utilizes the usual empirical tests of the APT ini­tiated by RR (1980) explained earlier and which are followed practically by most APT researchers . In addition , general tests of the APT have been done which involve the introduction of other explanatory economic/ financial '1ariable/s and a test of the hypothesis that the corresponding coefficients are zero .

The evidence was developed from monthly return data from 1972 to 1981 and from 1982 to 1991 for all common stocks actively traded in the Manila Stock Exchange.

The 91-day Treasury Bill rate was used as an estimate of the risk­free rate .

PABLO F. MANGARAN, JR. 69

ANALYSIS AND INTERPRETATION OF RESULTS

For the 1972 to 1981 period, the monthly returns for 31 common stocks with 49 monthly observations, and for the 1982 to 1991 period, 22 common stocks with 60 monthly observations were considered. Inac­tive, short-lived securities, and those with greater than 24 missing monthly observations were excluded in the study. The reduced number of monthly observations was due to the listwise deletion treatment applied by the factor analytic procedure on missing observations or on those months with no transactions.

To be able to estimate the elements of the matrix of factor loadings and the diagonal matrix, the earlier formed correlation matrices were analyzed separately using the maximum likelihood factor analysis proce­dure.

After applying the different diagnostic procedures, it was found that four factors were adequate to account for the intercorrelations of the com­mon stocks considered during each of the two periods of study.

In order to obtain a meaningful summary of the data, the maxi­mum factor loadings for each period were rotated using the varimax or­thogonal method.

For 1972 to 1981, the following factors were identified:

Small Board Factor. Fifteen out of 16 securities in the small board loaded heavily on this factor.

Big Board Factor. Stocks belonging to the commercial and industrial issues as well as mining issues loaded heavily on this factor.

Mineral Factor. This factor clearly describes factors 3 and 4. The 1970's was characterized by the frenzied activity in the mineral sector. Mining and oil securities were the issues of the decade.

For the 1982 to 1991 period, the appropriate labels were:

Big Board Factor. Stocks listed in the big board loaded heavily on this factor.

Mining Factor. This factor clearly describes factors two and three. Min­ing issues loaded heavily on this factor.

Lepanto Factor. Lepanto Consolidated Mining Corporation was the lone firm present in this factor, hence the label.

Once the expected returns and the factor loadings were estimated, the next step is to test for the significance of the risk factor coefficient and the intercept.

For the first part , the following excess return cross-sectional re­gression equation derived from (2) was used for each of the two periods considered :

70 ARBITRAGE PRICING THEORY

(3)

The dependent variable is defined as the stock's average monthly return and the appropriate riskless rate of return for the period. The riskless rate of return was assumed equal to the average monthly return of the 91-day Treasury rate.

When the cross-sectional regression equation (3) without the inter­cept term was applied, results showed that two factors were significant for each of the two periods. The significant risk premium coefficients were thoBe for FACTOR 1 and FACTOR 3 for the first period and FAC­TOR 1 and FACTOR 4 for the second period.

In order to be able to compare the relative importance of the re­gressors, the computed standardized estimate for each parameter was used. From Table 1A, FACTOR 1 was the biggest contributor for the first period, while FACTOR 4 was the largest contributor for the second pe­riod . These strongest contributions were also significant risk premium coefficients during their respective periods.

When the ANOVA was applied to test the hypothesis that the factor risk premia are all null, results showed significant results during the two periods. The adjusted coefficient of multiple determination were high for the two periods.

Thus, when the risk-free rate was assumed equal to the mean monthly return of the 91 -day Treasury Bill rate, two risk premia were found significant for each of the two periods of study. These were FAC­TOR 1 and FACTOR 3 for the 1972 to 1981 period and FACTOR 1 AND FACTOR 4 for the 1982 to 1991 period .

When the excess return equation (3) with intercept term was at­tempted, results in Table 1 B showed that only one factor was significant for each of the two periods. FACTOR 3 and FACTOR 4 were the signifi­cant risk premium coefficients for the first and second periods, respec­tively. Using the ANOVA test, only the first period factor risk premium coefficients were found significant.

Results also showed that the intercept term was found to be sig­nificant during the first period only. This means that during the 1972 to 1981 period, common stocks provided higher return than the average risk-free rate. Please note that during the 1982 to 1991 period, the 91-day Treasury Bill rate was at its highest particularly the more than 30% average yield from July 1984 to June 1985 and the more than 20% aver­age return during 1990 and 1991 .

Despite the cited deficiency that the factor risk premium estimates may not be statistically independent when the risk-free rate was esti­mated directly using the cross-sectional regression equation (2) , this pro­cedure was also pursued to determine if this methodology could provide additional evidence on the findings , that is, two factor risk premia for each period were significant to explain the common stock returns .

PABLO F. MANGARAN, JR. 71

Table 1. Summary of Results for the Cross-Sectional Regression Analy-sis of the Mean Security Returns on Factor Loadings When the Risk-Free Rate was Assumed Equal to the Mean Monthly Return of 91-day Treasury Bill Rate

VARIABLE PARAMETER COMPUTED STANDARDIZED ESTIMATE T-VALUE ESTIMATE

A. NO INTERCEPT 1972 to 1981

FACTOR1 0.023473 5.274*** 0.64224 FACTOR2 -0 .007501 -0.716NS -0.08683 FACTOR3 -0 .051686 -3.353*** -0.41154 FACTOR4 0.020206 1.130NS 0.13922

F = 10.478*** R2 == 0.6082 R2a = 0.5501 Standard Error == 0.0165

1982 to 1991

FACTOR1 0.018527 1.804* 0.39296 FACTOR2 0.008319 0.462NS 0.10179 FACTOR3 0.000725 0.050NS 0.00792 FACTOR4 0.034160 2.642** 0.44572

F 6.619*** R2 == 0.5953 R2a == 0.5053 Standard Error 0.0178

B. WITH INTERCEPT 1972 to 1981

INTERCEPT 0.018306 1.836* FACTOR1 -0.002159 -0.148NS

FACTOR2 -0.000086 -0.008NS

FACTOR3 -0 .056614 -3.769*** FACTOR4 0.017483 1.016NS

F == 3.730** R2 == 0.3646 R2a 0.2668 Standard Error == 0.1580

72 ARBITRAGE PRICING THEORY

1982 to 991

INTERCEPT FACTOR1 FACTOR2 FACTOR3 FACTOR4

0.012690 0.003153 0.003374 0.003829 0.027680

F = 0.879NS

F~2a = -0 .0235

• - signifh:ant at the .10 level •• - significant at the .05 level ••• - significant at the .01 level NS - not significant

0.977NS

0.171NS -0.160NS

0.253NS

1.758*

R2 = 0.1714 Standard Error = 0.0183

Results shown in Table 2 showed that only FACTOR 3 was signifi­cant during the 1972 to 1981 period. On the other hand, FACTOR 4 was the only significant risk premium during the 1982 to 1991 period.

ThE! intercept term or the estimate of the risk-free rate was found to be signific;antly greater than zero for the two periods.

ThEl t-test was again applied to determine if the intercept term or the estimate of the risk-free rate was equal to the equivalent risk-free rate using the average return of the 91-day Treasury Bill rate during the period. I~esults showed significant result only during the first period . This impl ied that common stocks provided higher yields than the aver­age risk-free rate during the 1972 to 1981 period.

Results of the ANOVA showed that the factor risk premia were sig­nificant only during the first period.

Thus, when the risk-free rate was directly estimated from the cross­sectional regression model, only one factor was found priced for each of the two periods considered. FACTOR3 and FACTOR4 were the signifi­cant contributions during the 1972 to 1981 and from the 1982 to 1991 periods, respectively.

A plausible explanation may be the earlier cited deficiency that the risk premia estimates could not be statistically independent when the risk-free rate was estimated directly from the regression model.

A lower number of priced risk premia was likewise found by RR (1980) when they estimated the risk-free rate directly from the cross­sectional regression model similar to (2) .

Analysis of residuals, influence diagnostics and multicollinearity diagnostics were also performed on the data set to determine whether the assumptions of the regression model were satisfied. Results of the various tests available using the SAS computer package showed that no possible errors or assumptions of the model were violated using the avail­able data set.

PABLO F. MANGARAN, JR. 73

Earlier, it was shown that common stock returns depended on as many as two factors. In the language of the APT, these statistically sig­nificant risk factors were considered "priced", meaning, their impact was reflected in the common stock prices to some statistically significant ex­tent. Thus, no other relevant economic or financial variables should have any effect on the estimates of expected rates of returns.

Table 2. Summary of Results for the Cross-Sectional Regression Analy­sis of the Mean Security Returns on Factor Loadings When the Risk­Free Rate was Estimated Directly from the Cross-Sectional Regression Model

VARIABLE PARAMETER COMPUTED STANDARDIZED ESTIMATE T-VALUE ESTIMATE

1972 to 1981

INTERCEPT 0.027526 2.761*** FACTOR1 -0.002159 -0.148NS -0.02488 FACTOR2 -0.000086 -0.008NS -0 .00133 FACTOR3 -0.056614 -3.769*** -0 .60826 FACTOR4 0.017483 1.1016NS 0.16071

F = 3.730** R2 = 0.3646 R2a = 0.2668 Standard Error = 0.0158

1982 to 1991

INTERCEPT 0.028360 2.184** FACTOR1 0.003153 0.171NS 0.04178 FACTOR2 -0.003374 -0.160NS 0.03556 FACTOR3 0.003829 0.253NS 0.05630 FACTOR4 0.027680 1.758* 0.42056

F = 0.879NS R2 = 0.1714 R2a = -0 .0235 Standard Error = 0.0183

* - significant at the .1 0 level ** - significant at the .05 level *** - significant at the .01 level NS - not significant

74 ARBITRAGE PRICING THEORY

To determine whether other economic or financial variable adds to the explanation of security returns , the respecified model is of the form

E(R,) = ao + ajbj1 + ... + akbjk + ak+1 P 1 + ej

where P 1 represents the extraneous variable .

(4)

On€> such specific alternative was the total variance of the indi­vidual stock return represented by the stock's standard deviation . An­other alternative considered was the square root of the diagonal elements of the residual matrix determined during the first stage of the factor ana­lytic procE!dure.

Using the individual standard deviation of stock returns as the ex­traneous variable, the ANOVA result showed highly significant results for the two periods considered. This result suggested that the APT may be false .

However, a review of available literature showed that skewness in the distribution of individual security returns was well documented as a possible source of a spurious effect of the own variance on expected returns (Miller and Scholes, 1972).

An analysis of the distribution of individual security returns were indeed highly skewed. Of the 31 stocks considered in the first period, 25 were significantly positively skewed , two negatively skewed , while the other four stocks were approximately normal. For the second period, of the 22 stocks considered, 20 were significantly positively skewed, while the other two stocks were approximately normal.

Studies made by RR (1980), Drhymes, Friend, Gultekin and Gultekin (1985), Drhymes (1984), and another by Drhymes, Friend, Gultekin and Gultekin ('1985) showed that the individual standard deviation significantly affected the expected returns in a number of cases. However, when re­finements were introduced such as using non-contemporaneous periods instead of contemporaneous periods, the percent of rejection likewise declined .

WhEln the square root of the diagonal elements of the residual vari­ance was used separately as another extraneous variable , result showed that it had no effect in the determination of the asset's expected returns . This result was in support of the APT.

SUMMARY AND CONCLUSIONS

Combinations of maximum likelihood factor analytic t~chn i que with varimax rotation of factors and regress ion analysis were used to analyze common stocks traded in the Manila Stock Exchange. Monthly returns for 31 stocks during the 1972 to 1981 period and 22 stocks during the 1982 to 1991 period were considered.

In the factor analysis part , results showed that four factors were found sufficient to account for the intercorrelations among the sto ck re­turns .

PABLO F. MANGARAN, JR. 75

Results of the excess return cross-sectional regression analysis showed that, at most, two factors were found to significantly explain com­mon stock returns. These significant factors were FACTOR1 and FAC­TOR3 for the first period, and FACTOR1 and FACTOR4 for the second period. Results of the various diagnostics showed that the assumptions of the regression model were satisfied.

When the standard deviation of the individual common stocks were used as an added predictor, analysis for both periods showed significant results which were contrary to the APT model. When further investi­gated, the positive skewness of the stock returns was traced as the pos­sible reason for the significant results .

When the square root of the residual variance was used as the extraneous variable, analysis showed insignificant results for both peri­ods which were support to the APT.

Therefore, it is safe to conclude that, at most; two risk factors exist that systematically influence common stock returns. In the language of the APT, these statistically significant factors are "priced", meaning their impact is significantly reflected in the asset prices.

RECOMMENDATIONS

1.Since the multifactor APT has been confirmed to apply to Philip­pine stocks, the APT should now be used as an additional tool in asset valuation.

2.Exogeneous economic theories such as macroeconomic theory or the theory of the firm should likewise be considered to develop risk factors .

3.The APT should also be used to predict or explain numerous re­sults that are not predictable under the single-beta CAPM .

4. Future research should explore the applicability of the rotated factor loadings on the tests of the APT.

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Brown Stephen J. and Mark I. Weinstein, 1983, A New Approach to Test­ing Asset Pricing Models: The Bilinear Paradigm , Journal of Fi ­nance 38, 711-743.

Burmeister, Edwin and Marjorie B. McElroy, 1988, Joint Estimation of Factor Sensitives and Risk Premia for the Arbitrage Pricing Theory, Journal of Finance 43, 721-733

76 AI1BITRAGE PRICING THEORY

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PABLO F. MANGARAN, JR . 77

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