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OSSIAM RESEARCH TEAM

Efficient portfolio: market beta and beyond

May, 23, 2012

WHITE PAPER

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Efficient portfolio: market beta andbeyond

Bruno Monnier and Ksenya Rulik

May, 23, 2012,This is the submitted version of the following article : Efficient Portfolio: Market Beta and Beyond, B.

Monnier and K.Rulik, Guide to European Investing, Spring 2012, Institutional Investors Inc.

Abstract

Bruno MonnierQuantitative [email protected]

Ksenya Rulik, PhD, CFAHead of Quantitative [email protected]

The rise of alternative beta investment strategies is a recenttrend that positions itself on the top of another powerfultrend: the growth of passive “beta” investing. Passive in-vesting in market indices began almost 40 years ago andreceived a major support from the financial theory, as themarket capitalization-weighted portfolio was claimed to bethe most efficient one by the Sharpe-Lintner Capital AssetPricing Model (CAPM) back in the 1960s.Both theory and empirical research have since then accu-mulated plenty of evidence that questions the validity ofthe CAPM and efficiency of the market portfolio. With the“one-for-all” market portfolio solution under attack, a fam-ily of new ideas on efficient equity investing, the alternativebeta strategies, is proposed to investors. This range of in-vestment ideas is very heterogeneous though its common de-nominator is the attempt to fix the inefficiencies discoveredin the market capitalization weighted portfolios.In this paper we review the reasoning behind the efficiencyof the market portfolio, and its flaws. We then discuss therationales behind the competing alternative beta investmentapproaches.


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Market Beta Story

”The market portfolio is the only efficient port-folio” teaches us the Capital Asset PricingModel (CAPM). The CAPM, introduced bySharpe [1964] and Lintner [1965], has largelyshaped the financial investment theory in thelast half century. Its conclusions are taught inmost, if not all, finance classrooms. Now thata whole generation has lived by it, its mainpractical conclusion “the market portfolio isthe only efficient portfolio” is engraved in ourminds. It means that any investor intendingto make the best possible investment decisionwill opt for an investment in the market portfo-lio, which is a portfolio where the risky assetsare weighted by their market capitalizations.According to the CAPM, this is the portfo-lio that delivers the maximum Sharpe ratio, inother words no other investment offers betterreward for a given risk target.

The immediate consequences of the marketportfolio’s efficiency are far-reaching. Investorswould no longer need to spend time and re-sources in looking for a tailor made allocation:the best “one-for-all” choice is readily avail-able. The CAPM proves that it is impossibleto consistently beat its peers, and it furtherimplies that trying to outperform the rest ofthe market will generally result in underper-forming it. Originally an indicator of generalsentiment about the economy, the market cap-italization weighted portfolio, has become aninvestment objective for managers, a bench-mark they should at least track closely, thebest allocation they should emulate.

Conveniently, buy-and-hold investing in a cap-weighted mix of assets is one of the easiestways of investing. Once acquired, a share ofthe market portfolio will remain efficient with-out the need of active management or periodicportfolio reviews. This led to the birth of thepassive investment era. The share of passive

investing has grown1 to 13.2% since the firstpassive mutual fund following S&P 500 indexwas launched by Vanguard in 1975.But the CAPM conclusions are not limited tothe efficiency of the market portfolio. It alsoprovides a pricing model where expected assetreturns are proportional to the portion of sys-tematic risk the stocks carry. The CAPM re-lation between the market sensitivities (betas)and the expected returns is the foundation ofthe alpha-beta separation in performance anal-ysis. The expected return of any security isgiven by the following equation:

E [Ri] = Rf + βi(E[Rm]−Rf ) (1)

where E [Ri] is a one-period expected return ofan asset E[Rm] is the market return, Rf is thereturn of the risk-free asset, and βi is the mar-ket sensitivity of the asset, i.e. the regressionslope of the asset’s excess return on the excessreturn of the market.By analogy, when analyzing realized perfor-mances, one can determine the alpha, the por-tion of the performance that cannot be ex-plained by the aggregate market:

α = (Ri −Rf )− βi(Rm −Rf ) (2)

The CAPM argues that the expected value ofα is zero if measured over a sufficiently longperiod.

Reasoning behind the market portfo-lio efficiency

We will follow the reasoning behind theCAPM. Starting from the hypothesis of marketefficiency, we will highlight how the model’s as-sumptions come into play. It will also provideus with a basis on which to discuss the model’slimitations.

1Based on estimate of index fund and industryglobal assets in 2010, including ETF. Source: C.Philips, ”The Case for Indexing”, Vanguard researchpaper, 2011.


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CAPM assumptions and conclusionsThe CAPM’s assumptions

i. investing is a costless and smooth process (no transaction costs, no taxation, possibilityto buy fractions of shares);

ii. Investors differentiate investments only by their expected returns and variances.When confronted to a choice, they prefer more profits and less risk;

iii. All investors share the same expectations and these expectations are correct;iv. Every investor has the ability to borrow or lend any amount at the risk free rate.

If the above assumptions are true, the CAPM implies that:

i. The market portfolio is an efficient portfolio: its allocation provides the only optimalmix of risky assets;

ii. For each asset, its expected return follows a simple linear relationship with theexpected return of the market portfolio. This relationship depends solelyon the regression slope of the assets returns on the return of themarket portfolio: the market beta.

First of all, the CAPM assumes that the mar-kets we operate in are efficient. This is crucialin order to rule out arbitrage trades as eligi-ble investments. Market efficiency (not yet theCAPM-market portfolio efficiency) means thatinvestors act rationally and rely on the avail-able information to form their view about theoutcomes of the various investments at theirdisposal. The information must quickly bespread in the marketplace and investors mustbe able to act on it. If these two conditionsare fulfilled, capital markets will reach an op-timal state, an “efficient” state. The efficiencyof capital markets is thus a measure of howpromptly and rationally they react to new in-formation.

Dissemination of information generally hap-pens through a trading process. Informedtraders are in a better position to assess thetrue value of a stock than uninformed traders.If they have the liberty to act on their views,they will drive the market prices toward the

fair values and information will spread in themarket place as the price moves. When mar-kets are efficient, there is no arbitrage trade leftand the market as a whole is informed. If a dis-symmetry in information does arise, it wouldonly exist for a short span of time when arbi-trageurs would work to bring the market pricesback the equilibrium. The resulting prices are“fair” in the sense that two informed partieswould willingly enter in an exchange trade atthis price. Thus by assuming markets are ef-ficient, we also assume that market prices arecorrect at any time.

Once the agents are sufficiently informed, it isnecessary to formulate their attitude towardsinvestments. Here the CAPM is based on therational investor profile, defined in the work ofMarkowitz on optimal portfolio selection thatappeared few years before the work of Sharpeand actually inspired him to construct a pric-ing model for a rational investor. In this setupinvestors think of an investment only in terms


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of its final payoff. For instance, an investorwill not consider the reluctance he can have inthe activities of a certain company, but will fo-cus only on the financial aspects of the trade.The preferences of such an investor can be de-scribed through a utility function, which as-signs scores to the payoff of an investment. Arational agent will seek to maximize the util-ity derived through an investment. Because ofthe uncertainty surrounding the actual termi-nal values of the investments, one has to thinkin terms of expected utility and integrate in theproblem the probability distribution of termi-nal values of the considered investment. To ar-rive to the Markowitz ”mean-variance” frame-work, one needs to assume that either the util-ity function is quadratic, or the return distri-bution is normal. Then only two parametersof the payoff’s distribution matter: its mean –the expected return on the investment – andits variance – a measure of the expected risk.Finally, our rational investor acts responsiblywhen allocating the capital at his disposal. Itmeans assigning more value to the more prof-itable investment when choosing between twoequally risky opportunities, and more value tothe less risky investment when choosing be-tween two equally profitable opportunities. Wecall this behavior responsible because this in-vestor will only choose to bear more risk ifcompensated for it. In the Markowitz mean-variance scheme, a set of efficient risky portfo-lios can be built, each efficient portfolio repre-senting an allocation with a maximal return fora given level of risk. Then, an investor with agiven risk tolerance simply has to pick up fromthis efficient frontier a portfolio that is suitablein terms of risk.Further, the CAPM showed that in the pres-ence of a risk-free asset the efficient frontier ofrisky portfolios becomes redundant. There is aunique portfolio on the efficient frontier that issuperior to all the other portfolios. It is calledthe tangent portfolio, as it is located on the tan-

gent line connecting the risk-free asset and theefficient frontier. This portfolio provides thehighest Sharpe ratio, and thus all rational in-vestors should hold a mix of this portfolio andthe risk-free assets if they want to maximizetheir risk-to-reward ratio.

Example 1: Risky investments We firststudy the case of an investor that must allo-cate his capital between two risky investments.Based on expectations about the return andvariance of these two opportunities, we cancompute expected means and variances for anycombination of the investments. Using expec-tations from the table [1], we can draw the ex-pectations of all portfolios based on differentallocations between these two assets.

Among the different allocations on the Figure1, one is of particular importance: the onewhich has the lowest expected variance, port-folio C. This portfolio divides possible alloca-tion in two sets: the segment AC, ”inefficientallocation” and the segment CB ”efficient al-location”. A portfolio belongs to the efficientset, if no other portfolio shows higher return forthe same amount of risk. A rational investorwould never choose a portfolio outside of thisefficient set. The portfolio C would be chosenby an investor focusing solely on risk minimiza-tion. The portfolio B would be chosen by aninvestor focusing more on the expected returnand ready to assume the corresponding risk.All other portfolios on the segment CB corre-spond to optimal portfolios for investors withintermediate levels of risk aversion.

Example 2: Risk-free and risky assetsWe introduce cash as a third asset. This bringsleverage as a new parameter in the choice ofthe optimal portfolio. In order to derive theCAPM conclusions, we will assume an investorhas full control on its leverage, being able toborrow and lend unlimited amounts of cash at


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Asset Expected Return(Risk free rate 3%)

Expected Volatility Expected Correla-tion

A 5% 12%50%

B 7% 15%

Table 1: Expectations about the assets A and B

Figure 1: Case of two risky assets without borrowing or lending. On top, the representationof the efficient portfolios in the ”mean-variance” space, on the bottom, the representation ofefficient portfolios in the ”allocation space”

a risk free rate.

The presence of a risk-free asset modifies theopportunity set available to the investor. Asillustrated on the Figure 2, given a portfolio P,all combinations of P and the risk free asset R(leveraged versions of P) will fall on the linepassing through R and P in the mean varianceplane. This line is often called a ”Capital Al-location Line” (CAL). Portfolios on the sameCAL will share the same Sharpe ratio. To get

an investment with a certain level of risk, onenow has at disposal several opportunities: in-vest in a risky portfolio with the given level ofrisk, invest in a leveraged version of a less riskyportfolio, or invest in a de-leveraged version ofa more risky portfolio.

As one sees on the Figure 3, among all theCALs there is one line that lays above all theothers: the one that is a tangent line to thefrontier of risky portfolios. It crosses the fron-


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Figure 2: Illustration of a Capital Allocation Line in the ”mean-variance” space

Figure 3: Case of two risky assets with unlimited borrowing and lending. On the left, the rep-resentation of efficient portfolios in the ”mean-variance” space, on the right, the representationof efficient portfolios in the ”allocation space”


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tier only in one point, the portfolio T, calledthe tangent portfolio. The tangent portfoliothus has the highest Sharpe ratio among allthe risky portfolios, and a combination of theportfolio T with the risk-free rate, the CapitalMarket Line, is the new efficient investmentset. All portfolios on the new efficient frontierRT are leveraged versions of the portfolio T.

Now the conclusions of the CAPM unfoldquickly if we assume uniformity in the borrow-ing power and expectations of all investors. Ifall investors share the same expectations theywill all deduct the same allocation for the tan-gent portfolio T and the associated CapitalMarket Line. If all investors can freely borrowor lend money, they will further agree that allefficient portfolios are leveraged versions of T.All investors would thus invest in a combina-tion of the portfolio T and the risk free asset.The market portfolio, M, - the aggregate of allinvestors’ portfolios – will itself be identical tothe portfolio T. The market portfolio is thusefficient.

Starting from the hypothesis of efficient mar-kets, we have had to make a series of assump-tions in order to reach the conclusion of theCAPM. It is these assumptions that we haveto keep in mind when using the powerful toolsprovided by CAPM. We cannot take shortcutsbecause once an assumption is violated, thefinal result, namely the efficiency of marketportfolio, does not hold any more.

Market Beta Critics

From its debut, the CAPM and efficient mar-ket portfolio standpoint have firmly enteredthe investment community’s mind. Spectacu-lar growth of passive investments, performanceanalysis based on alpha-beta separation, build-ing financial forecasts from the market betas,these are only some obvious signs of the tri-umph of the CAPM and its applications. At

the same time, academics and market prac-titioners undertook numerous empirical stud-ies testing the predictions of the CAPM andthe efficient market hypothesis. These studieschallenged many of the CAPM conclusions.

From a theoretical standpoint the attack onthe CAPM came first of all through the unreal-istic nature of some its assumptions. The real-world markets can be hardly approximated bya homogeneous group of investors with simi-lar views and no investor-specific constraints,leaving apart the investors rationality ques-tioned by the behavioral finance. Anotherweak CAPM point turned out to be the repli-cability of the ”true” market portfolio. In-deed, CAPM advocates efficiency of a globalmarket portfolio, aggregating all the possiblerisky holdings investors can have. In reality in-vestors can access only ”market proxies”, oftenin the form of regional equity indices. In 1975Richard Roll even stated that the true mar-ket portfolio is unobservable. Finally, an im-portant pricing anomaly was discovered in the1990s, following the work of Fama and French[1992]. It is widely admitted nowadays thatrisk factors other than the market impact stockreturns, such as value, size, momentum, andlater on, volatility.

What if CAPM assumptions fail

Among the assumptions one has to make in or-der to derive the conclusions of the CAPM, isa conjecture that investors could borrow un-limited amounts of cash at a risk free rate.This was crucial in the demonstration of mar-ket portfolio efficiency as it allowed us to findthe one and only one risky portfolio that allinvestors could agree upon. It can be shown,see for example Markowitz [2005], that if thisassumption does not hold the market portfoliois no longer neither optimal nor efficient. Weuse the same example of two risky assets. Thistime we will limit the leverage between 0 and


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2. We represent all available investment op-portunities in the ”mean variance” space (Fig-ure 4, left panel), and determine the efficientset. The efficient frontier is no longer reducedto the Capital Market Line and the tangentportfolio T. Because leverage is limited, in-vestors that seek returns above those providedby the twice leveraged version of T, the port-folio TT, will select portfolios that lie on thesegment [TT,BB], where BB is a twice lever-aged version of the portfolio B. In the alloca-tion space (Figure 4, right panel), it is evenclearer that the efficient frontier is divided intwo distinct groups: the portfolios on segment[R,TT] and the portfolios on segment [TT,BB].Respectively, we can distinguish two aggregateportfolios: M1, the average of all portfolios onthe segment [R,TT], and M2, the average of allportfolios of the segment [TT,BB]. M1 and M2also belong to their respective segments, i.e.are themselves efficient. The market portfoliois an aggregate of the portfolios M1 and M2.In the allocation space, it will lay on the seg-ment [M1,M2], out of either efficient subgroup.In this case, the market portfolio is inefficientand no rational investor would want to investin it.

Roll’s Critique

The famous Roll’s critique (1975) stated thatthe true market portfolio is unobservable, be-cause in order to assemble it one should in-clude all existing risky assets, covering all pos-sible securities, but also privately held com-panies and even human capital. Surprisingly,notwithstanding this critic the optimality ofthe true market portfolio was ”granted” to allportfolios weighted by market capitalization,even small ones consisting for example of re-gional large-cap stocks.

Empirical tests of CAPM

Starting from the 1970s, researchers maderepetitive empirical tests to probe the valid-ity of the CAPM conclusions. A special focuswas on testing the formula [1] and its predic-tions, namely that market beta is related to apositive premium over the risk-free rate, andthat the cross-sectional variations in stock re-turns are explained by their market betas. Anexcellent review of this research can be foundin Fama and French [2004].As Fama and French indicate, the tests consis-tently rejected the equivalence between stockexpected returns and their CAPM estimationsbased on market betas. A great amount of vari-ations in stock returns was found to be unre-lated to market betas. This important researchleads to the discovery of new pricing factors,summarized in the Fama and French 3-factormodel (Fama, French [1992]) and later gener-alizations.

Noisy Market Hypothesis

Another direction of critics was the efficiency ofmarket prices, the so called ”Noisy market hy-pothesis”. Rejecting the Efficient Market Hy-pothesis, i.e. accepting that the market val-uations contain deviations from the true un-derlying values, it can be shown mathemati-cally (see for example Hsu [2006]), that themarket portfolio ceases to be the most effi-cient one. In this case one could build otherportfolios with weights that are (to some ex-tent) independent of the market valuations andthat will outperform the market portfolio whenthe market valuations effectively mean-revertto the ”true” values. This result motivates re-searchers and practitioners to search for the al-ternative weighting schemes that will not con-tain the errors due to the market ”noise”.


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Figure 4: Case of two risky assets with limited borrowing and lending. On the left, the repre-sentation of efficient portfolios in the ”mean-variance” space, on the right, the representationof efficient portfolios in the ”allocation space”

Empirical Features of the MarketPortfolio

Even without the use of complex mathe-matics, the investment community agrees onsome properties of the market capitalizationweighted portfolios that are in contradictionwith an intuitive definition of an efficient port-folio. Market-capitalization weighted portfo-lios often are very concentrated. A small frac-tion of mega-cap companies can represent adominant part of the portfolio, introducing astrong asymmetry in the weight repartition.There is a tendency in the market-cap weightedportfolios to overweight the stocks that re-cently outperformed the market, resulting in atrend-following behavior. In its extreme formthis leads to the formation of market bub-bles, like the dot-com bubble of the late 90sin the US. From 1993 to 1999 the Informa-tion Technology sector in the S&P 500 indexgrew 5-fold, from 5.9% to 29.2%, and then itsweight was halved in 20022 to 14.31%. Onemore feature is the tendency of market cap-

2Source: Standard & Poor’s

italization weighted portfolios to overweightgrowth stocks, that makes the portfolio tiltedtowards growth factor and away from the valuefactor. Altogether, the arguments and evi-dence of non-efficiency of the ”market” portfo-lio are rather convincing. Still, building ”bet-ter” portfolios proves to be a very hard task,as is demonstrated by the statistics of activeasset managers’ performances with respect totheir market-cap weighted benchmarks (see forexample Philips [2011]). Indeed, the famous”zero-sum game” argument is often evoked toprotect the passive market-following, sayingthat an average investor in the active fundsgets exactly the market return before the feesare paid. Or, if we put it differently, for anywinning manager there should be a losing man-ager on the other side of the trade.

However, there are ways that are currently ex-plored by the researchers and practitioners tosystematically exploit the inefficiencies of themarket portfolio and the failures of the CAPM.It is worth emphasizing again that in the world”beyond the CAPM” the relations among riskand return characteristics are not necessarily


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linear, and no universal answer exists for anyinvestor on how to build an efficient portfolio.

In the next section we review some of the mostwidespread ”alternative beta” approaches toinvesting. The words ”alternative beta” indi-cate that the methods in question do not aimat delivering alpha by incorporating uniquemanager knowledge and skills or superior re-turn forecasts. Rather the alternative beta ap-proaches focus on finding completely system-atic rule-based solutions to correct the market-capitalization weighted portfolios, and in thissense are passive (or ”beta”) investments aswell.

Alternative Beta Story

The family of alternative beta, or alternativeweighting strategies, is very heterogeneous. Byalternative beta we mean the strategies (orportfolios) that are broad and not restrictedto some specific investment theme or risk fac-tor (e.g. a bet on an industrial sector, or avalue portfolio). The common theme in thealternative beta space is to improve the ineffi-ciencies of market capitalization weighted port-folios and offer more efficient investment solu-tions, possibly with specific investor believesor constraints in mind. In this sense, the alter-native beta portfolios compete with the pas-sive market portfolio just in one aspect: be-ing an efficient investment objective. Theseare not ”market thermometers” as the market-capitalization weighted portfolios are, and arenot meant to replace the market portfolio inother areas, as underlying of derivative con-tracts for example.We consider here the rationales for some ofthe alternative weighting approaches, namelyequal weighting, minimum variance, risk-parity, economic-scale weighting and diver-sity weighting. There are more approaches inthis family, for more complete review of risk-

based alternative beta approaches we invite thereader to see Lee [2010].

Equal Weighting

It is natural to begin with the equal weightedapproach for several reasons. First of all, theidea of allocating the same amount to everyasset in the portfolio is one of the oldest andby far the simplest one. This strategy was thefirst to emerge in the passive investing: thefirst passive investment account was launchedin 1973 by Wells Fargo and was actually animplementation of an equal-weight portfolioof NYSE-listed stocks that was later switchedto the market-capitalization weights becauseof operational reasons. The first ”alternativebeta” ETF was following the S&P 500 EqualWeight index, it was launched by Rydex in2003. Finally, the equal-weighted portfolio, or”1/N”, is a widespread performance analysisbenchmark in the academic literature. Theequal weight benchmarks are easier to con-struct, especially if the access to the historicaldata on market capitalization of stocks is lim-ited. Moreover, in some cases, for example theconstruction of arbitrage strategies, the mar-ket capitalization weights are just not relevantas benchmarks.

The rationale for the equal weighted portfo-lio is the search for maximal diversification inabsence of reliable information on stocks’ fu-ture risk and returns. The equal-weight al-location is simple and easy to read, and re-sults in spreading the investment bets evenlyacross the investment universe. The return ofthe equal weighted portfolio is also easy to in-terpret: it is an arithmetic average of all thereturns in the portfolio.

Naturally, the equal weighted portfolio cor-rects the ”mega-cap” concentration bias of thecapitalization-weighted portfolios and avoidsthe trend-following behavior, as the portfoliois rebalanced periodically to restore the equal


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weights and the past outperformance of a stockrelative to the basket does not lead to a su-perior weight for it. Consequently, the equalweighted portfolio will not follow a marketbubble. With 62 Internet companies in theS&P index in late 1999, the total weight ofthe IT sector in the S&P 500 equal weight wasof the order of 62/500 = 12.4 %, that is muchlower than the 29.2 % in the S&P 500 portfo-lio3.

The risk profile of the equal weighted portfolioresembles a lot to that of the capitalization-weighted portfolio. Equal weightings bringsneither significant increase nor decrease inportfolio long-term volatility with respectto the market-cap counterparts, and equalweighted portfolio generally remains very cor-related to the market-capitalization weightedportfolio. To get further insight on possiblerisks, one should pay attention to the definitionof the investment universe that is used to buildan equal weighted portfolio. If the universeitself has some significant asymmetries, theywill be reflected in the resulting equal weightallocation. For example, a global universe of5000 stocks will contain a dominant numberof small cap stocks that will then dominatethe equal weighted portfolio. Or, if there aresignificant differences in the number of stocksacross industries, some industries as a conse-quence will be underweighted and others over-weighted, notwithstanding their weight in theeconomy.

Possible driver of outperformance of the equalweight strategy with respect to the market capindex is the correction of mega-cap bias. It isoften referred as a small-cap bias ”driver” ofthe equal-weighted portfolio performance, butit is true only when the equal weighted portfo-lio is built on a universe that includes a signif-icant amount of small-cap stocks. Obviously,the large-cap or blue-chip equal-weighted port-

3Source: Standard & Poor’s

folios could not be said to have a small-capbias.

Our research on European equal weightedportfolios showed that reducing the weight ofthe large-cap sector and correcting the mega-cap bias inside this sector were the most sig-nificant sources of relative outperformance ofSTOXX Europe 600 Equal Weight index overits market cap counterpart in the period 2003-2010 (see Monnier and Rulik [2011]).

One additional performance driver being dis-cussed is the periodic rebalancing that leadsto reducing positions in stocks that outper-formed and increasing the position of stocksthat underperformed, i.e. a contrarian behav-ior. However, up to our knowledge there is noempirical evidence of the positive performanceachieved in this way and in our study we foundno significant contribution of this effect to thetotal excess return of the equal weighted port-folio over the market cap portfolio.

Minimum Variance

As was discussed above, the market-capitalization weighted portfolio in theCAPM framework is thought as an efficientoptimized portfolio. This means that it can beconstructed via a mean-variance optimizationalgorithm that takes as an input consensusmarket forecasts for future returns and thecovariance among the stocks. Note that themarket-capitalization weighted portfolio isactually an ”active” portfolio in the mean-variance sense. It incorporates implicit futurereturn forecasts contained in the stocks’market capitalization and the market weightsare optimal only if these forecasts are the bestthat one can have. There is only one optimalportfolio on the mean-variance frontier that istruly ”passive” in its objective and remainson the frontier for any configuration of returnforecasts, the Minimum Variance portfolio.

Minimum variance is an optimal allocation


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that is constructed by minimizing portfoliovariance. The minimum variance constructiondoes not use stocks’ expected returns as in-puts, and relies only on the covariance matrix.Usually the minimum variance portfolio is de-picted as the outmost left point on the mean-variance frontier (see the Figure 1), the lessrisky and the less performing one. But thispicture is very misleading, since it is condi-tional on some non-homogeneous forecast offuture returns. If such a forecast happens tobe wrong, the ex-post performance of the min-imum variance might be well above the ex-anteoptimal portfolios that have higher risk. If ex-post the stocks have similar returns, the min-imum variance portfolio will take the place ofthe efficient tangent portfolio, giving the high-est Sharpe ratio.This makes minimum variance portfolio a purerisk-based solution. The fact that the mini-mum variance construction does not use returnforecasts implies that there is no impact of thereturn forecasting error on the resulting alloca-tion. As was shown in the academic literature(see for example Chopra and Ziemba [1993]),the negative impact of the return forecastingerrors was one of the main causes of disappoint-ing performance of portfolios constructed withthe mean-variance optimization. Of course,there are errors that are inevitably containedin the estimation of covariance matrix, andthe minimum variance allocation is sensitive tothese. Still, the distortions these errors bringto the allocation are much smaller than in thecase of forecasted returns, and covariances areby far more stable and predictable in time thanreturns are.Contrary to the equal weighted portfolio, theminimum variance approach adds value viaportfolio construction. Because of the sub-tleties of covariance estimation and the choiceof constraints that mitigate the covariance esti-mation risk, the minimum variance methodolo-gies can differ significantly from one provider

to another. The explicit use of constrainedoptimization also offers investors an inter-esting opportunity to customize their port-folio by including specific constraints, suchas maximal weight per stock or per sector,constraints neutralizing exposure to unwantedrisk factors, etc. . . In this sense, the minimumvariance framework represents a flexible toolfor an investor who does not have specificviews on future returns and tries to achieverisk efficiency while maintaining proper risk-management constraints or objectives.There is a growing body of evidence on per-formance and characteristics of the minimumvariance investing. Over the past three yearsMSCI, DAX, STOXX and FTSE have createdminimum variance versions of their benchmarkequity indices. In addition, a growing amountof academic research is dedicated to mini-mum variance portfolios, where usually some”test” minimum variance portfolios are stud-ied. All these portfolios share some importantproperties, as reduced ex-post portfolio volatil-ity with respect to the market-capitalizationbenchmarks, and low market beta.If the CAPM were right, the low market betashould have implied smaller return for theminimum variance strategies than that of themarket-capitalization benchmark. However,the evidence is showing the opposite situation,the strategies have added significant positiveexcess return at least over the last 10 years.In part, the empirical outperformance of min-imum variance strategies can be explained bythe fact that in a multi-period setting the extravolatility reduces the total multi-period return.Think of an ”annualized” geometric average (amulti-period return) versus an arithmetic aver-age (a one-period return), the former is alwayssmaller and the magnitude of the difference isproportional to the half of the return variance.But the multi-period argument still does nottell the whole story. The major part of theoutperformance of the minimum variance port-


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folios comes from the fact that empiricallyeven in one-period setting the low volatilitystocks were found to perform better, or atleast not worse, than the high volatility stocks.This means that even arithmetic average ofone-month returns is not lower for the low-risk stocks, contrary to what one would ex-pect from the CAPM predictions. The ”low-volatility anomaly” in the cross-section of stockreturns was reported repeatedly by Haugen,Baker [1991, 2008], who studied the relationbetween the expected return and the risk mea-sures like total return volatility and idiosyn-cratic volatility (the volatility of a residual inthe factor model of expected stock returns).In a later study Ang et al [2006] has con-firmed a negative relation between stock re-turns in the cross-section and their sensitiv-ity to the market volatility factor (VIX). Thismeans that the stocks that react strongly tothe movements in market volatility (and tendto be themselves more volatile) in general un-derperformed the stocks with lower aggregaterisk sensitivity. They have also found a neg-ative relation among stock returns and theiridiosyncratic (residual) volatilities that can beonly in part explained by their sensitivities tothe aggregate market risk.

More research continues in the ”low-volatilityanomaly” direction, and we did not meanto make a complete overview of the subjecthere. Even before a consensus of this researchemerges, a low-volatility and minimum vari-ance investing represent an interesting oppor-tunity. These are examples of a failure of theCAPM prediction that more risk should bepaid off with more return.

Risk Parity or Equal Risk Contribu-tion

Risk parity proposes a risk-based portfolio con-struction method unrelated to the Markowitz-like optimization. The risk parity allocation

needs only a covariance matrix as an input, asthe minimum variance, but instead of runninga full-fledged optimization the method uses anad-hoc rule and assigns the weights in such away as to equalize the contributions of all theassets to the portfolio variance. No close-formanalytical solution is available to this problemin a general case, but the studies (see for ex-ample Maillard, Roncalli and Teiletche [2010])show that a solution always exists. In thecase when all pairwise correlations are equal,the risk-parity weights will be precisely in-versely proportional to the stocks’ volatilities(wi ∼ 1/σi).

The risk parity method leads to a very intuitiveallocation: all assets contribute equally to theportfolio risk. It appeals to the investors thatare comfortable with the use of risk budgetingrules. The scheme can be also very flexible,as one can redefine the rule for example by re-quiring parity among risk contributions of in-dustrial sectors rather than individual stocks.

The risk of equal risk contribution portfo-lios is ”in between” the market capitalizationportfolios and minimum variance portfolios.The stocks with lower volatilities tend to havehigher weights in this scheme, but the over-all volatility reduction is not as big as for theminimum variance portfolios because in therisk parity portfolio all the stocks are included,even those with very high volatility. Naturally,the risk parity portfolio tends to be well di-versified as it includes all the stocks from itsinvestment universe. By overweighting low-risk stocks, the risk parity allocation is well-positioned to profit from the ”low-volatilityanomaly” as the minimum variance portfoliosdo.

Economic Scale Weighting

Here we turn to a different breed of alternativebeta strategies that are not based on risk man-agement considerations or portfolio optimiza-


15

tion, but rather attempt to add value by moreefficient forecast of the stocks valuations thanthat of the market consensus. This is a con-venient way to wrap investor valuations into asystematic investment strategy.

The noise market hypothesis that we alreadymentioned above, assumes that market valua-tions come with noise that negatively impactsthe ex-post performance of the market capital-ization weighted portfolios (see Hsu [2006] andTreynor [2005]). Then, the weighting schemesthat are not dependent on market valuationsare also free from the market valuation noise,and under certain assumptions this leads tooutperformance of market-valuation indifferentallocations. This rationale lies behind the so-called “economic scale” weights of the Funda-mental Index introduced in 2005 by ResearchAffiliates.

Economic scale weighting uses companies’ fun-damentals – sales, cash flows, book value anddividends – to construct the stocks valuations,as an alternative to that of the market con-sensus. As argued in Arnott et al [2010] theerrors in such valuations will be independentfrom the errors present in the stock prices,so when the stocks undervalued or overvaluedby the market will have their price correctedthe fundamental index portfolio will not suffer,but the market-capitalization weighted portfo-lio will (see for example Hsu [2006]). This argu-ment is not without flaws, since the outperfor-mance of the Fundamental index depends on acrucial assumption, namely that the errors inthe “economic scale” valuations are indepen-dent of the market weights. As Kaplan [2008]showed, this is not generally the case and ifomitting the error independency assumptionthe outperformance of the Fundamental indexportfolio is no longer guaranteed.

Diversity Weighting

One more approach that can be understoodalong the lines of noisy market hypothesisis the diversity-based weighting based on theDiversity measure introduced by Fernholz in1999. The idea of this alternative weight-ing scheme is to “smooth” extreme bets inthe market capitalization weighted portfolio bytargeting a greater “diversity”, a measure ofportfolio concentration. The higher the port-folio diversity, the more evenly the weightsare spread among the stocks in the portfo-lio. Mathematically, enhancing the diversityamounts to a smoothing power transformationof the market cap weights (wi → wp

i , 0 < p <1). Such a transformation decreases the gapbetween the biggest and the smallest weightsof the market portfolio and in this way helps toreduce the magnitude of errors in the marketvaluations. Fernholz et al [2005] argue that ifone waits long enough, the diversity-weightedportfolio will outperform the market cap port-folio since the stocks with the biggest marketcapitalization weights will not be able to con-tinue their ”excess growth” and their weightswill be eventually reduced. This weightingscheme is implemented in the Intech Diversityindex, which is a diversity-weighted version ofthe S&P 500 index.

The fundamentals-weighted and diversity-weighted portfolios have some very attrac-tive features that the equal weighted andother risk-based portfolios do not have. Be-ing based on economic size or directly on mar-ket weights, these portfolios enjoy almost thesame liquidity and capacity that the market-capitalization weighted portfolio. Finally, allthe alternative beta schemes discussed aboveare purely systematic strategies, needing pe-riodic adjustments as stock prices and funda-mentals change.


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Alternative beta examples

In the last decade, and even more so re-cently, index providers and asset managershave striven to propose alternatives to marketcapitalization weighted indices. For many suchindices a backtested history is available, thatallows to compare the classical capitalization-weighted indices to some alternative marketbeta indices over at least the last ten years.

As an illustration we consider the performanceof three European Indices: the Stoxx R© Eu-rope 600 market-capitalization weighted index(MC), the Stoxx R© Europe 600 Equal Weightindex (EW) and the iStoxx R© Europe MinimumVariance index (MV). MC is a free float marketcapitalization weighted index which will serveas our market portfolio proxy. EW and MVrepresent two alternative beta strategies high-lighted in the previous section – respectivelyEqual Weighted, and Minimum Variance.

0.1 Risk and performance profile ofEqual Weight & Minimum Vari-ance indices

We present the data on a ten year period rang-ing from 31/12/2001 to 31/12/2011 for thethree indices. Figure 5 represents the evolu-tion of the level of the three indices.

Starting with the relative performance of theEW versus the MC, we witness a risk adjustedoutperformance over the period of study, i.e.a positive alpha. With a strong correlation tothe market portfolio and comparable volatility,the beta of the equal weight index is close toone. Other measures of riskiness are also po-sitioned at comparable levels, for instance themaximum drawdown is 58.69% for the MC and64.01% for the EW. The main driver of theoutperformance of the EW is its size-neutralallocation: companies with relatively smallercapitalization receive a significant weight whilein the MC portfolio they are dwarfed by the in-

vestment in widely capitalized companies. Per-haps counter intuitively, the active componentof the strategy – periodically resetting equalweighting – is not a critical element in the per-formance (as we discussed in the previous sec-tion); while this is an essential task to performfor the relevance of the strategy.

The MV index has a radically different riskprofile. The track record clearly shows a largereduction of the drawdowns. It exhibits a 40%volatility reduction and a corresponding betaof 0.54 over the period. Though its exposure tomarket risk is reduced, its performance stays inline with our market portfolio proxy throughmost of the periods, in contradiction to theCAPM conclusions that would predict it tounderperform. The MV benefits from its ex-posure to low volatility stocks which have con-sistently beaten their beta-based expectations.

If we plot these indices on the mean–varianceplane, it is even clearer that the proxy of themarket portfolio can be improved upon. Al-ternatives to market capitalization can providebetter risk and return profiles.

According to the formula (1), one can cal-culate the expected returns for the EW andMV indices, based on their market beta. Us-ing monthly returns over the considered periodwe find that sensitivity to the market capital-ization index (market beta) is of 1.14 for theEW and 0.57 for the MV. With the monthlyrisk-free rate of 0.1925% and average monthlymarket returns of 0.1818% over the period,the CAPM would give the expected return of0.1803% for the EW and 0.1867% for the MV.The realized average returns for the two alter-native indices were much higher: 0.41% for theEW and 0.57% for the MV.

Conclusion

As one sees, the alternative beta approachesdo not give a unique solution on how to fix the


17

Figure 5: Evolution of the Stoxx R© Europe 600, Stoxx R© Europe 600 Equal Weight and iStoxx R©

Europe Minimum Variance Indices.Source: Bloomberg

Market index Equal Weight MinimumVariance

Annualized Performance 0.73% 2.96% 6.25%

Volatility (annualized) 21.66% 21.34% 13.02%

Max Drawdown -58.69% -64.01% -39.71%

Sharpe Ratio* -7.21% 3.11% 30.41%

Correlation vs Benchmark - 96.77% 89.69%

Beta - 95.34% 53.90%

Annual Alpha - 2.15% 4.80%

Table 2: Statistics of alternative beta strategies and the market index benchmark over theperiod 2001-2010. EONIA used as risk free rate for computations. Source: Bloomberg (Data),Ossiam (Computation)

problem with the market portfolio inefficiency.Rather these strategies go back to the investorbeliefs and constraints, attempting to build ef-ficient portfolios depending on investors’ ob-jectives and views.

Our review of alternative beta approaches inby no means exhaustive, and we address thereader to Arnott et al [2010] and Lee [2010] for

more information.

References

Ang Andrew, Hodrick Robert J., Xing Yuhangand Zhang Xiaoyan, ”The Cross-Section ofVolatility and Expected Returns”, The Journal


18

Figure 6: Mean variance scatter plot based on 2001-2011 annualized data. Source: Bloomberg(Data), Ossiam (Computation)

of Finance, Vol. 61, No. 1, 2006, pp. 259-299.

Arnott Rob, Kaleskik Vitali, Moghtader Pauland Scholl Craig, ”Beyond Cap Weight.”,Journal of Indexes, January/February 2010.

Chopra Vijay K. and Ziemba William T., ”TheEffect of Errors in Means, Variances and Co-variances on Optimal Portfolio Choice”,Journal of Portfolio Management, Vol,19, No.2, 1993, pp.6-11.

Fama Eugene F. and French Kenneth R.,”The Cross-Section of Expected Stock Re-turns.” Journal of Finance, 47:2, pp.427-465.

Fama Eugene F. and French Kenneth R.,”The Capital Asset Pricing Model: Theoryand Evidence.”, Journal of Economic Perspec-tives, Vol. 18, No. 3, 2004, pp. 26.46.

Fernholz Robert, ”On the Diversity of EquityMarkets.”, Journal of Mathematical Eco-nomics 31, 1999, pp. 393-417.

Fernholz Robert, Karatzas Ioannis, Kar-daras Constantinos, ”Diversity and RelativeArbitrage in Equity Markets.”, Finance &Stochastics, Vol. 9 No. 1, 2005.

Haugen Robert A. and Baker Nardin L., ”CaseClosed”, Handbook of Portfolio Construction:Contemporary Applications of Markowitztechniques.”, Springer, 2010.

Haugen Robert A. and Baker Nardin L., ”TheEfficient Market Inefficiency of Capitalization-Weighted Stock Portfolios”, Journal of Port-folio Management, Vol. 17, No. 3, 1991, pp.35-40


19

Hsu Jason C., ”Cap-Weighted Portfoliosare Sub-optimal Portfolios”, Journal of In-vestment Management, Vol.4, No. 3, 2006,pp.1-10.

Kaplan Paul D., “Why Fundamental Indexa-tion Might – or Might Not – Work.” FinancialAnalysts Journal, Vol. 64, No. 1, 2008, pp.32-39.

Lee Wai, ”Risk-based Asset Allocation: ANew Answer to an Old Question?”, Journal ofPortfolio Management, Vol. 37, No. 4, 2011,pp. 11-28.

Lintner John, “The Valuation of Risk Assetsand the Selection of Risky Investments inStock Portfolios and Capital Budgets.” Re-view of Economics and Statistics, vol. 47, no.1 (February), 1965, pp. 13–37.

Maillard Sebastien, Roncalli Thierry andTeiletche Jerome, ”The Properties of EquallyWeighted Risk Contribution Portfolios”, TheJournal of Portfolio Management, Vol. 36,No. 4, 2010, pp. 60-70.

Markowitz H.M., ”Market Efficiency: A The-oretical Distinction and So What?”, FinancialAnalysts Journal, Vol. 61, No. 5, 2005, pp.17-30.

Monnier Bruno and Rulik Ksenya, ”Behind thePerformance of Equally Weighted Indices”,Ossiam research paper, May 2011.

Philips, Christopher B., ”The Case for Index-ing.” Vanguard research paper, February 2011.

Roll, Richard. 1977. “A Critique of the As-set Pricing Theory’s Tests, Part I: On Pastand Potential Testability of the Theory.” Jour-nal of Financial Economics, vol. 4, no. 2

(March):129–176.Sharpe, William F. 1964. “Capital AssetPrices: A Theory of Market Equilibrium un-der Conditions of Risk.” Journal of Finance,vol. 14, no. 3 (September):425–441.

Treynor J., ”Why Market-Valuation-Indifferent Indexing Works”, FinancialAnalysts Journal, Vol. 61, No. 5, 2005, pp.65-69.


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