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Journal Of Investment Management, Vol. 13, No. 4, (2015), pp. 71–86

© JOIM 2015

EFFICIENTLY COMBINING MULTIPLESOURCES OF ALPHA

Jose Menchero∗ and Jyh-Huei Lee†

In this article, we examine the question of efficiently combining multiple sources of alpha.We begin with a comparison of the various methods used by practitioners for constructingportfolios that capture a single alpha signal. These methods are broadly categorized aseither: (a) simple factor portfolios, (b) pure factor portfolios, or (c) minimum-volatilityfactor portfolios. We then derive an equation that shows the optimal alpha weights giventhe expected returns and covariance matrix of the alpha signals. We provide a discussionon how the required inputs can be estimated in practice, and conclude with an empiricalexample to illustrate these effects.

1 Introduction

Alpha represents the opportunity for investors to“beat the market.” For active managers, the searchfor alpha is usually regarded as the most importantcomponent of the investment process. Nonethe-less, poor implementation of even the best-qualityalpha signal may result in mediocre risk-adjustedperformance.

A crucial question, therefore, concerns how toconstruct a portfolio that captures a particular

∗Jose Menchero is Founder and CEO of Menchero PortfolioAnalytics Consulting.†Jyh-Huei Lee is Director of Quantitative Equity PortfolioManagement at TIAA-CREF.

alpha signal. Various portfolio construction tech-niques are used in practice to accomplish this task.One focus of our paper is to compare and contrastthese various techniques.

A common approach for capturing an alpha sig-nal is to create a simple factor portfolio. Thisis constructed by directly overweighting stockswith positive alpha, and underweighting thosewith negative alpha. While such a portfolio cer-tainly captures the alpha factor,1 it will also haveincidental exposure to many other risk factors.These exposures typically add unnecessary riskto the portfolio, without improving the expectedperformance.

A more risk-aware approach to capturing an alphasignal is to tilt the portfolio toward the alpha

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72 Jose Menchero and Jyh-Huei Lee

factor, while hedging away exposures to all otherrisk factors. We refer to this as the pure factorportfolio. While this portfolio has the reassuringquality that it has exposure only to the alpha fac-tor, it does not represent the theoretical optimalimplementation of the signal.

The most efficient implementation of the alphasignal is given by the minimum-volatility factorportfolio.2 As discussed by Grinold and Kahn(2000), the minimum-volatility factor portfoliohas the lowest volatility of all portfolios with unitexposure to the factor. Applied to an alpha factor,therefore, the minimum-volatility factor portfoliohas the highest expected information ratio.

An important characteristic of minimum-volatilityfactor portfolios is that they have non-zero expo-sures to all risk factors. Many portfolio managersregard these exposures as non-intuitive, and maytherefore set tight constraints on these factorexposures as part of the optimization process.We argue against blindly imposing such con-straints. One of the aims of our paper is to buildthe intuition behind these factor exposures, andto demonstrate the practical benefit in volatilityreduction that can be achieved by allowing theserisk factors to hedge the risk of the alpha factor.

In practice, most portfolio managers employ morethan a single source of alpha. Another criticalquestion, therefore, is how to optimally com-bine multiple alpha signals. Qian et al. (2007)provided one solution to this problem in termsof information coefficients of the signals andthe covariance matrix of these information coef-ficients. In Qian’s approach, the informationcoefficient is defined as the cross-sectional cor-relation between the forecasts and the subsequentreturns. Our approach is quite different in thatwe solve for the optimal combination of weightsin terms of the expected returns and covariancematrix of minimum-volatility factor portfolios.

Another salient feature of Qian’s approach is thatthe optimal portfolio is constrained to have zeroexposure to all risk factors. This contrasts withour approach, where we explicitly allow for suchexposures in order to hedge the risk of the alphafactor.

Another method for combining multiple sourcesof alpha was described by Grinold (2010). In thisapproach, one constructs efficient portfolios foreach alpha signal and then scales them to unitvolatility. The optimal weights are then solved forin terms of the correlation matrix and informationratios of these unit-volatility portfolios.

Our approach to signal weighting is similar toGrinold’s. The focus of our paper, however, isvery different. The main objective of Grinold’sarticle was to provide a framework for incorpo-rating the effects of transaction costs for signalswith different levels of information turnover.By contrast, the focus of our paper is how toestimate the required inputs for the optimiza-tion process itself. More specifically, we provideexplicit detail showing how to estimate the riskand expected returns for each signal within aself-consistent framework.

The remainder of this paper is organized asfollows. We begin with a detailed descriptionof the various types of factor portfolios. Next,we provide definitions for “alpha” factors and“risk” factors. These definitions are essentialfor understanding how risk factors can be usedto improve the risk-adjusted performance of aportfolio. More specifically, we show that riskfactors can be used to hedge alpha factors, with-out impacting expected returns. We then presentempirical results to compare and contrast the riskand return characteristics of these various factorportfolios. Finally, we present our approach forcombining multiple alpha signals, together withan illustrative example. Mathematical details areprovided in the technical Appendix.

Journal Of Investment Management Fourth Quarter 2015


E FFiciently Combining Multiple Sources of Alpha 73

2 Capturing a single alpha source

A common source of confusion and ambiguity inthe realm of factor modeling comes from genericusage of the term “factor.” To avoid such confu-sion, it is important to clearly distinguish betweenfactor exposures and factor returns. In funda-mental factor models, the exposures representintuitive attributes of individual stocks. Thesemight correspond to industry or country member-ship, or to some other stock characteristics suchas beta, book-to-price ratio, or alpha.

When we speak of factor returns, by contrast, werefer to the returns of portfolios that are derivedfrom the factor exposures via some definedalgorithm. These factor portfolios are generallylong/short, although not always dollar neutral.In this article, we examine three distinct typesof factor portfolios. Each is characterized by thesame unit exposure to the factor in question. Whatdistinguish them are their exposures to the otherfactors.

2.1 Simple factor portfolios

The first type we consider is the simple factor port-folio, which may be regarded as a simple tilt onthe factor. This portfolio is constructed by directlytaking long positions in stocks with positive expo-sure to the factor, and short positions in stockswith negative exposure. Mathematically, this isequivalent to performing a univariate regressionagainst the factor in question, with the result-ing weights being given by Equation (A.5) of thetechnical Appendix.

If the factor has some degree of collinearity withother factors—as is typically the case—then theportfolio construction process guarantees that theresulting portfolio will take “inadvertent” tilts onthose factors as well. For instance, if positivemomentum stocks tend to have high beta, then themomentum simple factor portfolio will also have

positive exposure to the beta factor. Similarly, ifenergy stocks have recently performed well, thenthe simple momentum factor portfolio will also beoverweight in the energy sector. Such unintendedbets generally add risk to the portfolio, withoutenhancing the expected return.

2.2 Pure factor portfolios

By contrast, pure factor portfolios have unitexposure to the factor in question, with zeroexposure to all other factors. Mathematically,pure factor portfolios are formed by multivariatecross-sectional regression, with the resultingweights given by Equation (A.8) of the techni-cal Appendix. Pure factor portfolios are powerfulconstructs because they provide a means of disen-tangling the confounding effects of collinearity.

Before proceeding to a concrete example, we firstdescribe the set of factors employed in our study.We focus our analysis on the Barra Global EquityModel (GEM3), which contains the followingfour types of equity factors: (a) a world factor, rep-resenting the regression intercept, to which everystock has unit exposure, (b) multiple country fac-tors, spanning developed and emerging markets,with exposures given by 0 or 1, (c) 34 industryfactors, again with exposures given by 0 or 1, and(d) 11 style factors, which are standardized to bemean zero and unit standard deviation.

The estimation universe for the GEM3 Modelis the constituents of the MSCI All CountryWorld Investable Market Index (ACWI IMI), abroad market index spanning both emerging anddeveloped markets. Factor returns are estimatedfor each period by cross-sectional regression ofstock returns against the start-of-period factorexposures. Specific returns represent the portionof stock return that cannot be explained by the fac-tors. The GEM3 Model assumes that the specificreturns are mutually uncorrelated, thus implyingthat the model factors fully capture all sources of

Fourth Quarter 2015 Journal Of Investment Management



equity return covariance. Adetailed description ofthe GEM3 Model is provided by Morozov et al.(2012).

Note that the GEM3 factor structure contains twoexact collinearities. Namely, the sum of columnvectors corresponding to the industry factor expo-sures replicates the exposure to the world factor,with the same identity holding for country fac-tors. To obtain a unique regression solution, wemust impose two constraints on the factor returns:Every period, we set the cap-weighted averageindustry and country factor returns to zero. Tosolve for the pure factor portfolio weights, we usesquare root of market capitalization as the regres-sion weights. Further details are provided in thetechnical Appendix.

As discussed by Menchero (2010), the pure fac-tor portfolio corresponding to the world factoressentially represents the cap-weighted worldportfolio. Pure country factor portfolios are 100percent long the particular country and 100 per-cent short the world portfolio, with zero netweight in every industry and zero exposure toevery style. In other words, pure country factorportfolios capture the performance of the countrynet of the market, industries, and styles. Similarly,pure industry factor portfolios capture the perfor-mance of the industry net of the market, countries,

and styles. Pure style factor portfolios have unitexposure to the particular style, and zero exposureto all other countries, industries, and styles.

2.3 Minimum-volatility factor portfolios

The third kind of factor portfolio that we consideris the minimum-volatility factor portfolio. Thisrepresents the unique portfolio that has the lowestpredicted volatility among the set of all portfolioswith unit exposure to the factor. Mathemati-cally, this portfolio is formed by mean–varianceoptimization, with the weights being given byEquation (A.9) of the technical Appendix. Notethat the minimum-volatility factor portfolio hasnon-zero exposure to all other factors. Theseexposures, however, are designed to reduce theoverall risk of the portfolio by hedging the factorin question.

Turning now to a concrete example, we con-sider the minimum-volatility factor portfolio forthe GEM3 momentum factor on December 31,2012. The asset covariance matrix was obtainedfrom the GEM3 risk model. The universe usedto construct the portfolio is the constituents ofMSCI ACWI IMI Index. In Table 1, we attributeportfolio risk to the GEM3 factors using thex-sigma-rho risk attribution formula, as described

Table 1 Risk attribution analysis on 31-Dec-2012 for the GEM3 momentumminimum-volatility factor portfolio. For illustrative purposes, only a representativesubset of the GEM3 factors is exhibited.

Factor Exposure Volatility Correlation Risk contrib

World 0.04 11.25 0.00 0.00Banks 0.06 3.66 0.00 0.00Momentum 1.00 2.91 0.80 2.34Beta 0.02 4.40 0.00 0.00Size 0.04 1.11 0.00 0.00USA −0.02 3.64 0.00 0.00




by Menchero and Davis (2011). For illustrativepurposes, we report only a small subset of the fac-tors. In the x-sigma-rho scheme, the risk contri-bution is given by the product of the exposure (x)to the return source, the volatility (sigma) of thereturn source, and the correlation (rho) betweenthe return source and the overall portfolio. For theexample at hand, the return sources represent theGEM3 pure factor portfolios and the overall port-folio is the minimum-volatility momentum factorportfolio. As shown in Equation (A.10) of thetechnical Appendix, the minimum-volatility fac-tor portfolio is uncorrelated with all of the purefactor portfolios, except its own. Consequently, adefining characteristic of minimum-volatility fac-tor portfolios is that the pure factors contributezero to their risk.

Note that the exposure to the momentum factor,by construction, is exactly 1. The other exposures,however, are non-zero. We argue that these expo-sures are intuitive and serve to reduce portfoliovolatility. For instance, Table 1 shows that theminimum-volatility momentum factor portfoliohas small positive exposures to both the worldfactor (0.04) and the beta factor (0.02). On theanalysis date, both of these factors were nega-tively correlated with the pure momentum factorportfolio. As a result, positive exposure to thesefactors helps hedge the risk of the momentumfactor.

The fact that the pure factors contribute zero tothe risk of the minimum-volatility factor port-folio does not imply that they are ineffective atreducing portfolio risk. On the contrary, it is pre-cisely because of these factor exposures that thecorrelation of the pure momentum factor is only0.80 with the momentum minimum-volatility fac-tor portfolio. Note that the correlation of the puremomentum factor portfolio with itself is exactly1.0; therefore, the correlation of 0.80 leads to asignificant reduction in portfolio volatility.

3 Alpha factors and risk factors

When building equity multi-factor risk models,the primary objective is to identify factors thatcapture and explain equity return co-movement.A “good” risk factor is therefore one that is veryvolatile and explains cross-sectional differencesin stock returns. For risk model construction pur-poses, it is largely irrelevant whether the factorhas any return premium associated with it. Forinstance, an industry grouping may represent agood risk factor if stocks within the industry tendto strongly co-move. This does not imply, how-ever, that the industry factor portfolio will havea significant drift (i.e., earn “abnormal” returns)across time.

For portfolio construction purposes, by contrast,it is crucial to identify which factors representalpha factors, and which do not. In other words,only a small subset of the risk factors may exhibitsignificant drift across time. We define an “alpha”factor as one for which the pure factor portfoliohas non-zero expected return. A“good” alpha fac-tor therefore is one that has a strong drift and highinformation ratio. Conversely, the other risk fac-tors (i.e., those that do not qualify as alpha factors)are defined to have expected returns of zero.

It is worth mentioning that while not all risk fac-tors are expected to be alpha factors, there isstrong reason to believe that the converse is nottrue. That is, if markets are efficient, any sourceof alpha should have a systematic risk associatedwith it.

Determining which risk factors represent alphafactors and which do not requires a combinationof empirical analysis and subjective insight. Oneof the criteria that we will employ is that the alphafactor must exhibit persistence. Therefore, weadopt the view that to qualify as a potential alphafactor, the realized information ratio should differsignificantly from zero.




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ResidualVola lity

Figure 1 Histogram of information ratios for (a)country factors, (b) industry factors, and (c) style fac-tors. The 95 percent confidence level is indicated bythe dashed vertical lines. The sample period runs fromJanuary 1997 through December 2012.

In Figure 1, we present histograms of the real-ized information ratios for the GEM3 pure factorportfolios over a 16-year period spanning January1997 to December 2012. The factor returns wereestimated using monthly cross-sectional regres-sions with MSCI ACWI IMI as the estimationuniverse. Even if the true information ratio werezero, the realized information ratio would neverbe exactly zero due to sampling error. The con-fidence interval can be established by assumingnormality and a true information ratio of zero; inthis case the realized information ratio over the16-year period should fall between −0.5 and 0.5with the probability of 95 percent.

In Panel (a) of Figure 1, we plot the histogramof realized information ratios for the 48 GEM3country factors representing both developed andemerging markets. Only one country (Colombia)fell outside of the confidence interval. This is notmuch different from what would be expected bychance. In this paper, therefore, we will assumethat country factors do not represent alpha factors.

In Panel (b) of Figure 1, we show the realizedinformation ratios for the 34 GEM3 industry fac-tors. In this case, four factors fall outside of theconfidence interval. This is only slightly morethan the five percent that would be expected bypure chance. Pharmaceuticals and biotechnologywere statistically significant on the positive side,whereas as consumer durables and constructionwere significant on the negative side. For someof these, however, we find that the outcomewas dominated by brief spurts of performance.For instance, the best-performing industry fac-tor was biotechnology, which was characterizedby a short window of spectacular performance in2001. In other words, it was not persistent driftthat caused biotechnology to fall outside of theconfidence interval, but rather a one-time eventthat is unlikely to repeat itself in the future. Inthis paper, therefore, we adopt the position thatindustry factors are not alpha factors.

In Panel (c) of Figure 1, we present the histogramof realized information ratios for the 11 GEM3style factors. In this case, six of the factors lieoutside of the confidence interval—much greaterthan the five percent that would be expected bychance. We further observe that several of theselie well outside of the confidence interval. Forinstance, earnings yield and momentum had real-ized information ratios between 1 and 2. Dividendyield and book-to-price also performed well, withinformation ratios between 0.5 and 1.0. On thenegative side, size and residual volatility hadsignificant information ratios.




These results are also consistent with the aca-demic literature. For instance, Basu (1977)showed that high earnings-to-price stocks havehistorically outperformed on a risk-adjustedbasis. Similarly, Banz (1981) documented the sizeeffect, in which small-cap stocks tend to outper-form their large-cap counterparts. The momen-tum effect, in which recent “winners” outperform“losers,” was described by Jegadeesh and Titman(1993). More recently, Ang et al. (2009) demon-strated the low-volatility anomaly in which highvolatility stocks have underperformed.

We might reasonably conclude that all six repre-sent valid alpha factors. Nonetheless, for exposi-tional simplicity, we reduce the list to just three. Inthis study, we consider only earnings yield, div-idend yield, and book-to-price as alpha factors;we assume the other eight style factors have zeroexpected returns.

4 Empirical comparison

In this section, we compare the risk and returncharacteristics of the three types of factor port-folios. In Figure 2, we plot the cumulativeperformance of the simple, pure, and minimum-volatility factor portfolios for the earnings yield

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Figure 2 Cumulative performance of simple, pure,and minimum-volatility factor portfolios for theGEM3 earnings yield factor.

factor. All portfolios are constructed using MSCIACWI IMI as the universe and are rebalancedon a monthly basis. Our first observation is thatall three portfolios had similar cumulative per-formance over the 16-year sample period. Thevolatilities, by contrast, were dramatically dif-ferent. Over the sample period, the annualizedvolatilities of the simple, pure, and minimum-volatility factor portfolios were 4.63 percent, 2.24percent, and 1.28 percent, respectively. In otherwords, the minimum-volatility factor portfoliocaptured the return most efficiently, whereas thesimple factor portfolio exhibited much higherrisk.

To illustrate the differences between these factorportfolios, it is particularly instructive to exam-ine the Internet Bubble period. We see that thesimple factor portfolio exhibited a spectaculardrop during 1999 and early 2000, followed byan equally impressive rebound beginning in April2000. Qualitatively, the pure earnings yield factorportfolio followed a similar pattern, except thatthe returns were less extreme. Most remarkably,the minimum-volatility factor portfolio “sailedthrough” the Internet Bubble virtually unper-turbed. It is worth stressing that the return con-tribution from the earnings yield factor itself wasidentical in all three cases, since each portfo-lio had the same unit exposure to the factor.The starkly differing behaviors of these portfolioswere therefore due to differences in exposures toother factors.

To gain further insight into these results, wereport in Table 2 the portfolio exposures to a rep-resentative subset of GEM3 factors at the startof April in 2000. We see that the simple factorportfolio had large negative exposures to bothbeta and momentum. These exposures are intu-itive, given that stocks with high earnings yieldhad strongly underperformed the broad market(i.e., negative momentum) and tended to have




Table 2 Exposures of earnings yield factor portfolios to GEM3 factors, as of 31-Mar-2000. Forillustrative purposes, only a subset of GEM3 factors are reported. The simple factor portfoliohad positive exposures to “old-economy” industries and negative exposures to “new-economy”industries; for the minimum-volatility factor portfolios, the reverse holds.

Factor Simple exposure Pure exposure Min-vol exposure

Beta −0.347 0.000 0.089Momentum −0.554 0.000 0.014Earnings Yield 1.000 1.000 1.000Dividend Yield 0.565 0.000 0.327Book-to-Price 0.579 0.000 0.183

Food Beverage 0.018 0.000 −0.011Diversified Financials 0.019 0.000 −0.011Construction 0.018 0.000 −0.026

Telecommunications −0.054 0.000 0.024Internet −0.031 0.000 0.016Software −0.041 0.000 0.018

low beta. Also note that the simple earningsyield factor had large positive exposures to the“old-economy” industries (e.g., food & beverage,diversified financials, and construction), whileit had large negative exposures to the “new-economy” industries (e.g., telecommunications,Internet, and software). This example illustratesthat the risk of the earnings yield simple factorportfolio was not driven by the earnings yield fac-tor itself, but rather by the “incidental” exposuresto other factors.

The factor exposures of the pure earnings yieldfactor portfolio are very straightforward: By defi-nition, the portfolio has unit exposure to itself andzero exposures to all other factors. As the purefactor portfolio has no incidental exposures, theperformance is entirely attributed to the earningsyield factor itself.

The factor exposures of the minimum-volatilityfactor portfolio are the most interesting to inter-pret. In April 2000, the pure earnings yield factorreturns were positively correlated with the old-economy industries, and negatively correlated

with the new-economy industries. As a result, wefind that the minimum-volatility factor portfoliohad positive exposure to the new-economy indus-tries and negative exposure to the old-economyindustries in April 2000. These served as excel-lent hedges for the risk of the earnings yield factor,and explain the “smooth sailing” that the portfolioexperienced during this otherwise stormy period.

In Figure 3, we plot the cumulative performanceof the simple, pure, and minimum-volatility fac-tor portfolios for the dividend yield factor. Again,we used the MSCI ACWI IMI as the universeand rebalanced monthly. Qualitatively, the resultsare very similar to the earnings yield factor inFigure 2. Namely, the annualized returns of thethree portfolios were roughly comparable over theentire sample period, whereas the volatilities weredrastically different. Note also the striking simi-larity between the behavior of the earnings yieldand dividend yield simple factor portfolios dur-ing the Internet Bubble period. Again, this wasmainly due to similar “incidental” exposures toother factors.




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Figure 3 Cumulative performance of simple, pure,and minimum-volatility factor portfolios for theGEM3 dividend yield factor.

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Figure 4 Cumulative performance of simple, pure,and minimum-volatility factor portfolios for theGEM3 book-to-price factor.

In Figure 4, we plot the cumulative performanceof the book-to-price factor. Once more, we usedthe MSCI ACWI IMI universe and rebalancedthe portfolios monthly. Qualitatively, the resultsare again very similar to those for the earningsyield and dividend yield factors. That is, the netreturns over the sample period were quite similar,whereas the volatilities were starkly different.

In Table 3, we summarize results for the return,risk, and information ratio of the three typesof factor portfolios for earnings yield, dividendyield, and book-to-price factors. As previouslynoted, the annualized returns of the simple,pure, and minimum-volatility factor portfoliowere comparable. For earnings yield, the sim-ple factor portfolio slightly outperformed theother two, whereas the minimum-volatility fac-tor portfolio performed best for dividend yieldand the pure factor portfolio had the highestreturns for the book-to-price factor. Nonethe-less, in each case, the return difference betweenthe top-performing and bottom-performing factorportfolio was rather small (typically 50–60 bps).

While the annualized returns were quite simi-lar, the volatilities were dramatically different.In each case, the volatility of the pure fac-tor portfolio was less than half the volatility ofthe corresponding simple factor portfolio. Sim-ilarly, the volatilities of the minimum-volatilityfactor portfolios were roughly 40 percent lower

Table 3 Summary return, risk and information ratios for earnings yield, dividend yield, and book-to-pricefactors. For each style factor, we report results for simple, pure, and minimum-volatility factor portfolios. Thesample period runs from January 1997 through December 2012.

ReturnRiskIR

(Earnings yield)

Simple Pure Min-vol

3.05 2.59 2.494.63 2.24 1.280.67 1.25 2.11

(Dividend yield)

Simple Pure Min-vol

1.71 1.27 1.884.47 1.55 0.980.39 0.67 1.74

(Book-to-price)

Simple Pure Min-vol

1.30 1.74 1.664.66 1.81 1.090.31 0.96 1.50




than the volatilities of their pure factor portfoliocounterparts.

Also reported in Table 3 are the informationratios3 for the three different types of fac-tor portfolios. Not surprisingly, we find thatthe minimum-volatility factor portfolios had thehighest information ratios over the sample period.This was primarily due to their lower volatili-ties. Conversely, the lowest information ratiosoccurred for the simple factor portfolios, owingto their much higher volatilities.

5 Combining multiple sources of alpha

Next, we turn our attention to the case in whichthe alpha factor is composed of several distinctalpha signals. For this study, we take the alpha fac-tor to be a weighted combination of the earningsyield, dividend yield, and book-to-price signals.Our task is to construct the maximum informa-tion ratio portfolio combining these three alphasources.

In Equation (A.21) of the technical Appendix,we derive the intuitive result that the optimalportfolio is simply a weighted combination ofthe minimum-volatility portfolios for each alphasignal taken separately. We therefore treat eachminimum-volatility factor portfolio as a sepa-rate “asset” and use mean–variance optimizationto solve for the optimal weights. This, in turn,requires an asset covariance matrix and a set ofexpected returns.

Our first task is to construct the 3 × 3 assetcovariance matrix. As discussed in the technicalAppendix, our approach is to compute the covari-ance matrix directly using the GEM3 model. Thisis a straightforward exercise, since we know theholdings of each minimum-volatility portfolio.

Our next task is to estimate the expectedreturns for each minimum-volatility factor port-folio. Observe that the factor exposures of the

minimum-volatility factor portfolio are known.For example, these are shown in Table 2 for theearnings yield factor. Note that only the alpha fac-tors will contribute to the expected return of anyportfolio. That is, since the other risk factors, bydefinition, have zero expected return, they do notcontribute to the expected portfolio return.

To illustrate our technique, consider the follow-ing simple example. Suppose that the expectedreturns of the pure factor portfolios were known.For the sake of the argument, say that the expectedreturns were 300 bps for earnings yield, 100 bpsfor dividend yield, and 200 bps for book-to-price.From Table 2, we saw that the exposure of theearnings yield minimum-volatility factor portfo-lio was exactly 1.0 to earnings yield, 0.327 fordividend yield, and 0.183 for book-to-price. Theexpected return of the earnings yield minimum-volatility factor portfolio therefore would havebeen 370 bps. This is attributed as 300 bps fromthe earnings yield factor itself, plus 33 bps fromdividend yield (0.327 × 100), and another 37 bpsfrom book-to-price (0.183 × 200).

The final remaining task is to obtain the expectedreturn of each pure alpha factor. To accomplishthis, we follow a two-step process. First, we esti-mate the information ratio of each pure factorportfolio. We use the first five years of data tomake our first estimate in January 2002. Everymonth, we add a new observation to our expand-ing window and update the information ratioestimate. By construction, such information ratioswill evolve slowly across time. The second stepis to scale the information ratio of the pure alphafactor by its GEM3 predicted volatility to obtainthe expected return.

We do not make any claim that the proceduredescribed above represents the “only” way oreven the “best” way to compute the expectedreturns and covariance matrix of the minimum-volatility factor portfolios. Rather, we believe




that our approach provides a reasonable andtransparent method useful for illustrative pur-poses. Sophisticated asset managers will certainlyemploy prudent judgment and their investmentinsights to refine these methods according to theirown views.

6 Empirical results

With the covariance matrix and expected returnsof the three minimum-volatility factor portfo-lios determined, we can now easily compute theoptimal set of portfolio weights. These weights,plotted in Figure 5, are quite stable. The mainexception was during the “Quant Meltdown” ofAugust 2007, which saw a sudden decline in theweight of the earnings yield portfolio. Over thecourse of the month, the volatility of the earningsyield factor nearly tripled, whereas the volatili-ties of dividend yield and book-to-price increasedmore modestly. From a mean–variance perspec-tive, therefore, earnings yield temporarily becamerelatively less attractive. As the volatility of theearnings yield factor receded in the months fol-lowing the Quant Meltdown, we see that theweight of the earnings yield factor increasedaccordingly.

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Figure 5 Weights of minimum-volatility factor port-folios versus time.

The weights in Figure 5 are also quite intu-itive when one considers the ex ante informationratios of the minimum-volatility factor portfolios.These are plotted in Figure 6 for the three alphasignals, as well as for the composite alpha fac-tor. We see that earnings yield typically had thehighest information ratio among the three sig-nals, so it is not surprising that it should receivethe largest weight in the optimization. Similarly,the dividend yield factor typically had the low-est information ratio and received the smallestweight. Figure 6 also illustrates the benefits ofdiversifying across multiple alpha sources. Thatis, the ex ante information ratio for the com-posite factor was significantly greater than thestand-alone information ratios of the individualsignals.

In Table 4, we report the risk attribution for theoptimal portfolio on analysis date 31-Dec-2012.As before, we show only a subset of representativefactors. We see that only the three alpha factorscontribute to portfolio risk, whereas the otherrisk factors contribute zero. This is consistentwith the property that for an unconstrained opti-mal portfolio, the expected return contribution

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Figure 6 Ex ante information ratio of minimum-volatility portfolios for individual signals and com-posite alpha versus time.




Table 4 Risk attribution of optimal minimum-volatility factor portfolio, for analysisdate 31-Dec-2012.

Factor Exposure Volatility Correlation Risk contrib

Earnings Yield 0.559 1.01 0.60 0.34Book-to-Price 0.338 1.16 0.40 0.16Dividend Yield 0.317 0.82 0.17 0.05

World −0.004 11.25 0.00 0.00Banks 0.005 3.66 0.00 0.00Momentum 0.054 2.91 0.00 0.00USA 0.010 3.64 0.00 0.00

is proportional to the risk contribution for everycomponent of the portfolio. Since the other riskfactors have zero expected return, it followsthat they must contribute zero to the risk of theunconstrained optimal portfolio.

In Figure 7, we plot the cumulative performanceof the optimal portfolio, represented by the solidblue line. We see that the portfolio earned a con-sistently positive return, with the exception of theperiod from mid-2007 until early 2009, duringwhich time it was flat. Over the full sample period,the annualized return was 2.25 percent, with avolatility of only 87 bps, leading to an impressive

Year

2002 2004 2006 2008 2010 2012

Cum

ula

ve R

etur

n (P

erce

nt)

-10

0

10

20

30

40

50

60

70

80

Total Por olioAlpha FactorsOther Risk FactorsSpecific

Figure 7 Return attribution of optimal minimum-volatility factor portfolio. The annualized return was2.25 percent, with a volatility of 87 bps, leading to aninformation ratio of 2.78.

information ratio of 2.78. Of course, this analysisdoes not include the effect of trading costs. Nei-ther does it take into account real-world invest-ment constraints such as long-only portfolios withmonthly turnover limits. These constraints wouldserve to further reduce the information ratio thatcould be achieved in practice.

In Figure 7, we also plot the return contributionfrom three distinct sources: (a) the three alphafactors, (b) all other risk factors, and (c) the stock-specific component. We see that virtually all ofthe return came from the alpha factors, consistentwith our portfolio design objective. The returncontribution of the other risk factors was essen-tially flat over the entire sample period, reflectingthe notion that these factors do not exhibit drift.It is also interesting to note that from July 2007until February 2009, the alpha factors detractednearly 600 bps from performance. Over this sametime period, the other risk factors contributedmore than 500 bps, thus largely mitigating thedrawdown.

While the other risk factors made little impacton the cumulative returns of the portfolio, theywere quite effective at reducing overall volatilitylevels. This is illustrated in Figure 8, which repre-sents a scatterplot of monthly return contributionsfrom alpha factors and all other risk factors. Therealized correlation was −0.52, indicating that the




Alpha Factor Contribu on (Percent)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Oth

er R

isk F

acto

rs (P

erce

nt)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Figure 8 Scatterplot of monthly return contributionsfrom alpha factors and from other risk factors, fromJanuary 2002 through December 2012. The realizedcorrelation was −0.52. The regression fit through thepoints is also indicated.

other risk factors were highly effective at hedg-ing the risk of the alpha factors. Under normalassumptions, the standard error of the correla-tion estimate is approximately 0.087. Thus, therealized correlation was also highly statisticallysignificant.

7 Summary

We have discussed several approaches for captur-ing the return premium of an alpha factor. Simplefactor portfolios tilt on the factor by overweight-ing positive alpha stocks and underweightingthose with negative alpha. Such portfolios typi-cally contain “incidental” exposures that may addconsiderable risk without enhancing the returnprofile. Another approach is to construct a purefactor portfolio by neutralizing exposures to allother factors. The most efficient implementa-tion is given by the minimum-volatility factorportfolio, which exploits correlations betweenrisk factors and alpha factors to reduce overallvolatility levels.

We also presented a methodology for combiningmultiple sources of alpha. We showed that theoptimal portfolio is a weighted combination ofthe minimum-volatility factor portfolios for eachalpha signal. We provided a detailed examplethat illustrates our technique as well as builds theintuition behind these portfolios.

Appendix A

A.1 Simple factor portfolios

Simple factor portfolios are formed by univariatecross-sectional regression,

rn = fw + XnkfSk + en, (A.1)

where rn is the return of stock n, fw is the worldfactor return (the regression intercept), Xnk is thestock exposure to factor k, fS

k is the simple factorreturn, and en is the residual return. We assumethat the exposures are standardized to be meanzero,

N∑n=1

wnXnk = 0, (A.2)

where wn is the regression weight of stock n andN is the total number of stocks. We use regressionweights that are proportional to the square root ofmarket capitalization. The factor exposures arestandardized to have unit standard deviation,

N∑n=1

wnX2nk = 1. (A.3)

Under this specification, the regression coeffi-cient for the world factor is given by

fw =N∑

n=1

wnrn, (A.4)

which represents the regression-weighted meanreturn of the universe. The return of the simple




factor portfolio is given by

fSk =

N∑n=1

(wnXnk)rn. (A.5)

In other words, the weight of each stock inthe simple factor portfolio is simply the productof the regression weight and the factor expo-sure. Equation (A.3) immediately shows that thesimple factor portfolio has unit exposure to theunderlying factor.

A.2 Pure factor portfolios

Pure factor portfolios are formed by multivariatecross-sectional regression,

rn =K∑

k=1

XnkfPk + un, (A.6)

where fPk is the return of the pure factor, un is the

stock-specific return, and K is the total numberof factors. The GEM3 factor structure containstwo exact collinearities. Thus, we impose twoconstraints in order to obtain a unique regres-sion solution: (a) the cap-weighted industry factorreturns sum to zero, and (b) the cap-weightedcountry factor returns sum to zero. These con-straints are embodied in the restriction matrix R,with dimensionality K × (K − 2). Following theapproach of Menchero (2010), we define a K×N

matrix

�P = R(R′X′WXR)−1R′X′W, (A.7)

where W is the regression weighting matrixwhose diagonal elements are given by wn, andwhose off-diagonal elements are zero. The purefactor returns are given by

fPk =

N∑n=1

�Pknrn. (A.8)

That is, the kth row of matrix �P gives the stockweights in the pure factor portfolios. Each purefactor style portfolio has unit exposure to the

underlying factor, and zero exposure to all otherfactors.

A.3 Minimum-volatility factor portfolios

Minimum-volatility factor portfolios are formedby mean–variance optimization.

�MVk = V−1Xk

X′kV

−1Xk

, (A.9)

where V is the N × N asset covariance matrix,and Xk is the kth column vector of factor expo-sure matrix X. The denominator in Equation (A.9)ensures that the resulting portfolio has unit expo-sure to the underlying factor. We obtain theasset covariance matrix using the GEM3 riskmodel. Minimum-volatility factor portfolios havethe interesting property that they are uncorre-lated with all pure factor portfolios, except theirown underlying factor. This can be easily seen asfollows:

cov(�Pm, �MV

k ) = (�Pm)′V(V−1Xk)

X′kV

−1Xk

. (A.10)

If k �= m, Equation (A.10) shows that the covari-ance is zero, since pure factor portfolio m has zeroexposure to factor k. According to the x-sigma-rho formula, described by Menchero and Davis(2011), the risk contribution is proportional to thecorrelation with the portfolio. Consequently, purefactor portfolios contribute zero to the risk of theminimum-volatility factor portfolio. For the casek = m, of course, the contribution is non-zero.

A.4 Alpha factors and risk factors

We define an alpha factor to be any factorwhose pure factor portfolio has non-zero expectedreturn,

E[fP(α)k ] = αP

k , (A.11)

where αPk denotes the expected return of the pure

alpha factor portfolio. The other risk factors areassumed to have an expected return of identically




zero,

E[fP(σ)k ] = 0. (A.12)

Segmenting factors into alpha factors (α) andthe other risk factors (σ), Equation (A.6) can berewritten as

rn =Kα∑k=1

X(α)nk f

P(α)k +

Kσ∑k=1

X(σ)nk f

P(σ)k + un,

(A.13)

where Kα is the total number of alpha factors andKσ is the total number of other risk factors. Theoptimizer can exploit covariances between thealpha factors and the other risk factors to reduceportfolio volatility. Furthermore, since the otherrisk factors have zero expected return, the volatil-ity reduction comes without incurring a price inexpected portfolio performance.

A.5 Combining multiple alpha signals

We write the composite alpha signal as a linearcombination of the individual alpha signals,

αn =Kα∑k=1

vkX(α)nk . (A.14)

Our task is to solve for the coefficients vk.The optimal portfolio is given by the standardexpression,

h = V−1α

λ, (A.15)

where λ is the risk-aversion parameter and α isthe N ×1 vector whose elements are given by αn.Note that since h has the lowest volatility for agiven level of alpha, it represents the maximuminformation ratio portfolio. Substituting Equation(A.14) into Equation (A.15), we obtain

h = 1

λV−1

[Kα∑k=1

vkX(α)k

], (A.16)

which can be rewritten as

h = 1

λ

[Kα∑k=1

vkV−1X

(α)k

]. (A.17)

The minimum-volatility portfolio for alpha signalk can be expressed as an N × 1 vector

hk = V−1X(α)k

sk, (A.18)

where sk is a scalar determined by Equation (A.9),

sk = X′(α)k V−1X

(α)k . (A.19)

Now substituting Equation (A.18) into Equation(A.17), we find

h = 1

λ

[Kα∑k=1

vkskhk

]. (A.20)

This states that the optimal portfolio is a linearcombination of minimum-volatility factor port-folios for the individual alpha signals. Therefore,we can write the optimal portfolio as

h =Kα∑k=1

wkhk, (A.21)

where wk is the weight of minimum-volatilityfactor portfolio k. Comparing Equation (A.21)with Equation (A.20), we can relate the portfolioweights to the alpha weights

vk = λwk

sk. (A.22)

The portfolio weights wk can be solved via mean–variance optimization. The “assets” in this caserepresent the minimum-volatility factor portfo-lios. Consequently, we have reduced the opti-mization problem from N dimensions to the muchsmaller dimensionality Kα of the alpha factors.

To apply mean–variance optimization, we requirethe asset covariance matrix and the asset expected




returns. We compute the asset covariance matrixdirectly

Gkl = h′kV hl, (A.23)

where V is the N × N asset covariance matrixfrom the GEM3 risk model.

To estimate the asset expected returns, first notethat the exposure XMV

km of minimum-volatility fac-tor portfolio k to factor m is known. Since onlyalpha factors contribute to the expected return ofany portfolio, we can write

αMVk =

Kα∑m=1

XMVkm αP

m. (A.24)

A reasonable way to obtain the expected return ofthe pure alpha factor is to estimate the informationratio of the pure alpha factor portfolio over a trail-ing window and then to multiply this informationratio by the predicted volatility of the pure factorportfolio. The optimal set of weights is then givenby the standard solution,

w = 1

λG−1αMV . (A.25)

Reverting to summation notation, this can beexpressed as

wk = 1

λ

Kα∑j=1

G−1kj αMV

j . (A.26)

Plugging Equation (A.26) into Equation (A.22)allows us to solve for the weights vk

vk = 1

sk

Kα∑j=1

G−1kj αMV

j . (A.27)

This expresses the optimal set of alpha weights interms of quantities that can be easily computed orestimated. Also note that the final result is inde-pendent of the risk-aversion parameter, as shouldbe expected.

Notes1 In this article, we use the terms “alpha signal” and “alpha

factor” interchangeably.2 Grinold and Kahn (2000) also refer to this as the

“characteristic” factor portfolio.3 To avoid undue influence from periods of high volatil-

ity, we compute the information ratios from monthlyz-scores. This explains why the information ratios maydeviate slightly from the conventional information ratio,defined as annualized return divided by annualized risk.The conventional information ratios are easily computedfrom Table 3.

References

Ang, A., Hodrick, R., Xing, Y., and Zhang, X. (2009).“High Idiosyncratic Volatility and Low Returns: Interna-tional and Further U.S. Evidence,” Journal of FinancialEconomics 91, 1–23.

Banz, R. (1981). “The Relationship Between Return andMarket Value of Common Stock,” Journal of FinancialEconomics 9, 3–18.

Basu, S. (1977). “Investment Performance of CommonStocks in Relation to Their Price-Earnings Ratio: A Testof the Efficient Market Hypothesis,” Journal of Finance32 (June), 663–682.

Grinold, R. (2010). “Signal Weighting,” Journal of Portfo-lio Management 36(4) (Summer), 24–34.

Grinold, R. and Kahn, R. (2000). Active Portfolio Manage-ment (New York: McGraw-Hill).

Jegadeesh, N. and Titman, S. (1993). “Returns to Buy-ing Winners and Selling Losers: Implications for StockMarket Efficiency,” Journal of Finance 48, 65–92.

Menchero, J. (2010). “The Characteristics of Factor Portfo-lios,” Journal of Performance Measurement 15(1) (Fall),52–62.

Menchero, J. and Davis, B. (2011). “Risk Contribution isExposure Times Volatility Times Correlation,” Journal ofPortfolio Management 37(2) (Winter), 97–106.

Morozov, A., Wang, J., Borda, L., and Menchero, J. (2012).“The Barra Global Equity Model Empirical Notes,” MSCIModel Insight (January).

Qian, E., Hua, R., and Sorensen, E. (2007). QuantitativeEquity Portfolio Management (Boca Raton: Chapman &Hall).

Keywords: Portfolio construction; alpha estima-tion; mean-variance optimization



journal of investment anagement joim · our approach to signal weighting is similar to grinold’s....

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