1、The Quarterly Review of Economics and Finance 53 (2013) 175187Contents lists available at SciVerse ScienceDirectThe Quarterly Review of Economics and Financejo ur nal home page: /locate/qrefOrthogonalized factors and systematic risk decompositionRudolf F. Klein a, , Victor K. Chowb,1

2、a Sweet Briar College, Department of Economics, 134 Chapel Road, Sweet Briar, VA 24595, United Statesb West Virginia University, Division of Economics and Finance, P.O. Box 6025, Morgantown, WV 26506, United Statesa r t i c l ei n f oa b s t r a c tArticle history:Received 19 June 2010Received in re

3、vised form 26 January 2013 Accepted 19 February 2013Available online 7 March 2013JEL classication:G11 G12 G14Keywords: Orthogonalization Systematic risk Decomposition Fama-French Model Asset pricingIn the context of linear multi-factor models, this study proposes an egalitarian, optimal and unique p

4、rocedure to nd orthogonalized factors, which also facilitates the decomposition of the coefcient of determination. Importantly, the new risk factors may diverge signicantly from the original ones. The decomposition of risk allows one to explicitly examine the impact of individual factors on the retu

5、rn variation of risky assets, which provides discriminative power for factor selection. The procedure is experi- mentally robust even for small samples. Empirically we nd that even though, on average, approximately eighty (sixty-ve) percent of style (industry) portfolios volatility is explained by t

6、he market and size factors, other factors such as value, momentum and contrarian still play an important role for certain port- folios. The components of systematic risk, while dynamic over time, generally exhibit negative correlation between market, on one side, and size, value, momentum and contra

7、rian, on the other side. 2013 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.1. IntroductionUnder the traditional single-factor Sharpe (1964) and Lintner (1965) Capital Asset Pricing Model (CAPM), the market beta cap- tures a stocks systematic ris

8、k for all rational, risk-averse investors. Therefore, a decomposition of the market beta is sufcient to break down the systematic risk of a stock.2 For example, Campbell and Vuolteenaho (2004) break the market beta of a stock into a bad component, that reects news about the markets future cash ows,

9、and a good component, that reects news about the mar- kets discount rates. In an earlier paper, Campbell and Mei (1993) show that the market beta can be decomposed into three sub- betas that reect news about future cash ows, future real interest rates and a stocks future excess returns, respectively

10、. Acharya and Pedersen (2005) develop a CAPM with liquidity risk by divid- ing the market beta of a stock into four sub-betas that reect Corresponding author. Tel.: +1 434 321 4773.E-mail addresses: (R.F. Klein), (V.K. Chow).1 Tel.: +1 304 293 7888.the impact of

11、 illiquidity costs on the systematic risk of an asset. Researchers frequently apply decompositions of the market beta to examinethesizeand/orbook-to-marketanomalies.Althoughbeta- decompositions are useful to describe the structure and source of systematicvariation ofreturnson risky assets, theyareco

12、mplicated under multi-factor frameworks. For instance, Campbell and Mei (1993) show that one complication is due to the possible covari- ance between the risk price of one factor and the other factors, which prevents identifying a neat linear relationship between the overall beta of an asset and its

13、 beta of news about future cash ows.The purpose of this paper is to develop an optimal procedure to identify the underlying uncorrelated components of common factors, by a simultaneous and symmetric orthogonal transforma- tion of sample data, such that the linear dependence is removed and the system

14、atic variation of stock returns becomes decom- posable. We empirically compare our approach with two popular orthogonalization methods, Principal Component Analysis (PCA) and the Gram-Schmidt (GS) process, and unsurprisingly nd that our technique has the essential advantage of maintaining maxi- mum

15、resemblance with the original factors.32 According to the CAPM, the systematic risk is measured as (j RM)2 . Since the market factor is the only priced risk factor faced by all investors, j is sufcient todetermine the systematic risk.3 For instance, Baker and Wurgler (2006, 2007) employ PCAtodevelop

16、 measures of investor sentiment, shownto have signicant effectsonthe cross-section ofstock1062-9769/$ see front matter 2013 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rig hts reserved. /10.1016/j.qref.2013.02.003176R.F. Klein, V.K. Chow / The

17、Quarterly Review of Economics and Finance 53 (2013) 175187In the past two decades, one of the most extensively researched areas in nance has concentrated on alternative common risk factors, in addition to market risk, that could characterize the cross-section of expected stock returns. Fama and Fren

18、ch (1992, 1993, 1996, 1998) document that a companys market capitaliza- tion, size, and the companys value, which is assessed by ratios of book-to-market (B/M), earnings to price (E/P) or cash ows to price (C/P), together predict the return on a portfolio of stocks with much higher accuracy than the

19、 market beta alone, or the tradi- tional CAPM.4 In addition to the size and value effects, Jegadeesh and Titman (1993), Jegadeesh and Titman (2001), Rouwenhorst (1998), and Chan, Jegadeesh, and Lakonishok (1996) report that short-term past returns or past earnings predict future returns.Average retu

20、rns on the best prior performing stocks (i.e., the win- ners) exceed those of the worst prior performing stocks (i.e., the losers), attesting the existence of momentum in stock prices. Con- versely, De Bondt and Thaler (1985, 1987) detect a contrarian effect by which stocks exhibiting low long-term

21、past returns outperform stockswithhighlong-termpastreturns. De Bondtand Thaler(1985, 1987), Chopra, Lakonishok, and Ritter (1992), and Balvers, Wu, and Gilliland (2000) suggest a protable contrarian strategy of buying the losers and shorting the winners.Consequently, for the determination of the ret

22、urn generatingprocess for risky assets, one needs to consider more than just the marketriskfactor. Forthisreason, multi-factormarket modelshave been widely employed by both academics and practitioners. Under the multi-factor framework, the expected excess return on a risky asset is specied as a line

23、ar combination of beta coefcients and expected premia of individual factors. Fama and French (1993) emphasize that, if there are multiple common factors in stock returns, they must be in the market return, as well as in other well-diversied portfolios that contain these stocks. This indicates that r

24、eturns on common factors must be, to some degree, corre- lated with the market and with each other. Consequently, in a multiple linear regression setting, although the beta coefcient cor- responding to an individual factor provides a sensitivity measure of an assets return to the factors variation,

25、it is not sufcient to assess the systematic variation of the assets return with respect to that factor. The volatility of an assets return is determined jointly not only by the betas, but also by the variances and covariances of the factors premia. Therefore, determining the factors underlying uncor

26、related components helps us achieve a clearer identication of the separate roles of common factors in stock returns.This paper proposes an optimal simultaneous orthogonal trans- formation of factor returns. The data transformation allows us to identify the underlying uncorrelated components of commo

27、n fac- tors. Specically, the inherent components of factors retain their variances, but their cross-sectional covariances are equal to zero. Moreover, a multi-factor regression using the orthogonalized fac- tors has the same coefcient of determination, R-square, as that using the original, non-ortho

28、gonalized factors. Importantly, themixture of the multiple common factors, an orthogonalization of the market factor is necessary so that it can capture common varia- tion in returns left from other factors such as size or value. We argue that not only the market factor, but all factors need to be o

29、rthog- onally transformed to eliminate any dependence among them. Although Fama and Frenchs (1993) orthogonalization procedure for the market factor is straightforward, it cannot be extended to eliminate the correlations between all variables in a model, with- out generating two related biases. Firs

30、tly, similarly to GS, it leaves one factor (call it leader) unchanged. Secondly, it is a sequential (i.e., order-dependent) procedure. Therefore, a different selection of the leader or a different orthogonalization sequence generates differ- ent transformation results. Our method avoids these two bi

31、ases by construction.Using Monte Carlo simulations, we demonstrate that our ortho- gonal transformation isrobust, in that itproduces precise estimates of the population systematic risk even for small samples. By apply- ing our methodology to some of the Kenneth Frenchs style and industry portfolios,

32、 we show empirically that the systematic return variation can now be unequivocally allocated to the common factors.5 We nd that, over a time period from January 1931 to December 2008, the market and size factors are the largest sources of systematic risk, while other factors such as value, momentum

33、and contrarian play relatively small roles in stock volatility.The paper is organized as follows. In Section 2, after explaining why the systematic risk decomposition isproblematic under multi- factor models, we present our procedure of symmetric orthogonal transformation and risk decomposition. In

34、Section 3, we illustrate the procedure empirically, using monthly U.S. Research Returns Data obtained from Kenneth Frenchs Data Library, for the time interval January 1931December 2008. The nal section of the paper provides concluding remarks.2. Orthogonalization procedureSuppose a risky asset js re

35、turn generating process is linearly determined by a set of K common factors (fk), such as market (RM), size (SMB), value (HML), momentum (Mom), and long-term reversal (Rev), as shown in the following general linear factor model.Ktj ttrj = j + k f k + j ,(1)k=1where fk are assumed to be uncorrelated

36、with the residual term (j), but not with each other. For instance, the market factor is a mix- ture of the multiple common factors, while the factor-mimicking portfolios of size, value, momentum and contrarian are all formed using securities in the same market, and thus their returns are not uncorre

37、lated.sThe systematic return variation ( 2) of asset j can then be mea-sured ascthoeefovceieranltl ovfoldaetitleitrymoinf aatiroinsk(ythaessreatt)ioisoaf msyesatseumreatoicf vthaeriasytisotnemto-K Katic risk of an asset. Therefore, disentangling the R-square based on factors volatilities and their c

38、orresponding betas enables us to decompose the systematic risk. For that, we need to extract the core, standalone components of common factors. Fama and French (1993) clearly demonstrate that since the market return is a2 =k l Cov(f k, fl ),(2)sjj jl=1 k=1sj jwhile the coefcient of determination, R-

39、square, is the ratio of sys- tematic variation to total return variation ( 2 / 2).returns. Boubakri and Ghouma (2010) remove the multicollinearity between their variables using the Gram-Schmidt algorithm.It is important to note that under the multi-factor framework, systematic risk depends not only

40、on the beta coefcients but also on the factors variancecovariance. Thus, beta coefcients alone are4 Fama and French (1992, 1996, 1998) show that the investment strategy of buy- ing the Small Value stocks and shorting the Big Growth stocks produces positive returns.5 Kenneth Frenchs Data Library is l

41、ocated at / pages/faculty/ken.french/data library.html.R.F. Klein, V.K. Chow / The Quarterly Review of Economics and Finance 53 (2013) 175187177inappropriate measures of systematic variation. One of the goals of this paper is to develop a decomposable systematic risk meas

42、ure.sAs shown in (2), 2 is not decomposable into individual sys-jtematic risk components, due to the covariance between factors.6 Kennedy (2008, p. 46), among others, points out that the total R- square cannot be allocated unequivocally to each explanatory variable, unless we have zero multicollinea

43、rity between the vari- ables. To be able to achieve zero multicollinearity, thus eliminating the impact of covariances, we employ an orthogonal transforma- tion which helps us identify the underlying components of all factors. We argue that, even though numerous orthogonalization techniques are avai

44、lable, the optimal alternative for nding the proper orthogonal proxies of the original factors is the so-called symmetric procedure of Schweinler and Wigner (1970) and Lwdin (1970), as adapted in this paper (henceforth denoted as SW/L).7 For instance, thepopular Principal Component Analysis, oftenus

45、edfordimensionality reduction, though similar to our procedure in some aspects, cannot offer by itself a meaningful one-to-one and onto correspondence from the original to the orthogonalized set of fac- tors, if the number of variables is larger than two. And even in the cases where we have only two

46、 explanatory variables, if they are sig- nicantly correlated, it is difcult to maintain a strong resemblance to the original variables once the transformation is performed. Table 1 considers all the possible combinations of pairs out of the ve factors mentioned above and, using two measures of simil

47、i- tude, compares SW/L, PCA and the classical GS process. A major deciency of this last procedure is that it requires a choice of the initial starting factor, which will not be transformed, thus failing to give all factors equal footing. We are nevertheless interested to see whether the deviations i

48、n the other factor are greater or less than those generated by the rst two methods.Panel A presents the correlation coefcients between the orig-inal and their orthogonalized counterparts. It is easily noticeable that SW/L outperforms both PCA and GS. For instance, its lowest correlation of 0.945 (be

49、tween Rev and orthogonalized Rev) is more than double the minimum correlation for PCA, 0.450, and about 20 percent larger than that of GS, 0.787, all for HML & Rev.8 Also, given that GS does not modify the rst factor, which can be interpreted as perfect correlation, while the correlations correspond

50、ing to the secondfactoraregreater thanthelowercoefcients for PCA, wecan conclude that GS performs better than PCA in all ten cases. PanelB reports, on a comparative scale, the Frobenius norm of the T Kmatrix whose elements are the deviations of the orthogonalizedfrom the original sets of data where,

51、 in these cases, T = 936 months (from January 1931 to December 2008) and K = 2 factors.9 Since we want the original factors to be modied as little as possible, the results conrm the superiority of our symmetric procedure, which has the lowest (square root of the) sum of squared deviations, in all te

52、n cases (i.e., the lowest Frobenius norm). Similar to Panel A, the principal component scores are further from the values of the original variables, compared to the GS transformation. This is very severe especially when the original variables show signicant cor- relation. That is why some researcher

53、s, for ease of interpretation of the factor loading pattern, prefer to perform rotations, once the PCA is completed.6 For simplicity and without loss of generality, henceforth we will refer to com- mon factors only as factor portfolio returns, but the model can also be applied to other systematic fa

54、ctors (e.g., macro-economic or fundamental variables).7 The actual procedure is presented in Section 2.1.8 HML and Rev, in their original form, exhibit the highest correlation among allTable 1Method comparison with respect to resemblance criteria.Panel A: Correlation coefcients between the original

55、and theorthogonalized factorsSW/LPCAGS(1)(2)(3)(4)(5)(6)(7)RM & SMB0.9920.9780.9860.8741.0000.941RM & HML0.9960.9900.9870.9221.0000.970RM & Mom0.9870.9820.9090.7091.0000.946RM & Rev0.9950.9890.9880.9201.0000.968SMB & HML0.9990.9990.8690.9131.0000.994SMB & Mom0.9960.9980.9570.9901.0000.991SMB & Rev0.

56、9760.9790.5900.8711.0000.908HML & Mom0.9720.9840.7460.9521.0000.925HML & Rev0.9470.9450.9040.4501.0000.787Mom & Rev0.9940.9900.9700.8821.0000.975Panel B: Frobenius norm values of the deviation matrixSW/LPCAGSRM & SMB30.51857.87735.994RM & HML22.17151.23929.428RM & Mom38.748128.79754.975RM & Rev22.82

57、450.52928.768SMB & HML7.46570.00917.464SMB & Mom12.92036.67530.671SMB & Rev31.769123.19747.165HML & Mom37.32788.97463.458HML & Rev51.105116.38171.404Mom & Rev21.66163.15828.691This table considers all the possible combinations of pairs out of ve stock-market factor portfolios: RM, SMB, HML, Mom and Rev. For each pair of original factors, their orthogonalized counterparts are computed. In Panel A, each of the columns (2)(7) reports the correlation coefci