A note on the returns from minimum variance investing

Bernd Scherer 1

EDHEC Business School, London, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 July 2010Received in revised form 3 May 2011Accepted 10 June 2011Available online 30 June 2011

Disappointed with the performance of market weighted benchmark portfolios yet skepticalabout the merits of active portfolio management, investors in recent years turned to alternativeindex definitions. Minimum variance investing is one of these popular concepts. I show in thispaper that the portfolio construction process behind minimum variance investing implicitlypicks up risk-based pricing anomalies. In other words the minimum variance tends to hold lowbeta and low residual risk stocks. Long/short portfolios based on these characteristics havebeen associated in the empirical literature with risk adjusted outperformance. This papershows that 83% of the variation of the minimum variance portfolio excess returns (relative to acapitalization weighted alternative) can be attributed to the FAMA/FRENCH factors as well as tothe returns on two characteristic anomaly portfolios. All regression coefficients (factorexposures) are highly significant, stable over the estimation period and correspond remarkablywell with our economic intuition. The paper also shows that a direct combination of marketweighted benchmark portfolio and risk based characteristic portfolios will provide astatistically significant improvement over the indirect pickup via the minimum varianceportfolio.

© 2011 Elsevier B.V. All rights reserved.

JEL classification:G11C14

Keywords:Minimum variance portfolioSHARPE-ratioRisk based anomaliesMarket capitalization weighted portfolioBootstrapping

1. Introduction

Disappointed with the performance of market weighted benchmark portfolios yet skeptical about the merits of active portfoliomanagement, investors in recent years turned to alternative index concepts.2 The notion of passive investments originallyreserved for capitalization weighted indices3 got redefined to also cover rule based index construction. To distinguish rule basedindices from systematic quantitative strategies the axioms of capital market theory needed to be replaced by other investmentcriteria, that – like axioms – more or less everybody could agree with.

Minimum variance investing or more broadly risk based investing is one of these “obvious” concepts.4 It has been inspired byearly work from Haugen and Baker (1991).5 For the period covering the years 1972 to 1989 the authors found that a minimumvariance portfolio would outperform the Wilshire 5000 at lower risk. A vast number of studies followed their original paper. Forthe US stock market Chan et al. (1999), Schwartz (2000), Jagannathan and Ma (2003) and Clarke et al. (2006) found both higherreturns and lower realized risks for the minimum variance portfolio (MVP) versus a capitalization weighted benchmark (MWP).

Journal of Empirical Finance 18 (2011) 652–660

E-mail address: bernd.scherer@edhec.edu.1 Board member of the London Quant group.2 See Johnson (2008) and Appel (2008) for investment press treatments.3 Grinold (1992) was among the first to outline the potential inefficiency of market capitalization weighted portfolios.4 Fundamental indexing as popularized by Arnott et al. (2005) is another concept that recently won some traction within the investment community.5 To the author's knowledge Acadian Asset Management, AXA Rosenberg, Invesco, LGT Capital Management, MSCI BARRA, Robeco, SEI, State Street Global

Advisors and Unigestion are running minimum variance index concepts.

0927-5398/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jempfin.2011.06.001

Contents lists available at ScienceDirect

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j ourna l homepage: www.e lsev ie r.com/ locate / jempf in

For global equity markets Geiger and Plagge (2007), Nielsen and Aylusubramanian (2008) and Poullaouec (2008) all findqualitatively similar results.6

While there is ample empirical evidence for the outperformance of the MVP there is also little theoretical justification. Thesuccess of the MVP cannot be rooted in portfolio theory according to which investors should hold the maximum SHARPE-ratioportfolio (MSP). Investors wanting to reduce risk should add cash to theMSP rather than investing into theMVP or low beta stocksinstead. In the authors view, the minimization of risk is – on its own – a meaningless objective. The same applies to relatedconcepts that try to maximize “diversity” as in Fernholz et al. (1998) or Choueifaty and Coignard (2006)7 or to minimizeconcentration as in King (2007).8 What all papers have in common is that they followmechanical portfolio construction rules thatdo not directly rely on expected return information but rather use risk information (volatility, factor based correlation models,etc.) instead. It however should be clear that mechanical trading strategies cannot lead to outperformance9 in a market withoutstructure (a market where no characteristics based or autocorrelation exists). My conjecture is that the portfolio constructionprocess behind minimum variance investing implicitly picks up risk based pricing anomalies. If that is true, minimum varianceinvesting will be a clumsy and indirect process to benefit from. Investor would be better advised to directly decide if, when and towhat degree they want to invest into long/short anomaly portfolios on top of a market weighted benchmark.

This paper shows – using a standard multifactor regression with HAC (heteroskedasticity and autocorrelation consistent)adjusted errors – that 84% of the variation of excess returns can be attributed to the Fama and French (1992) factors and thereturns of two characteristic portfolios leveraging on widely published risk anomalies picked up by the construction of minimumvariance portfolios. All regression coefficients (factor exposures) are stable over the estimation period as confirmed by the Nyblom(1989)–Hansen (1992) test and correspond remarkably well with economic intuition. The paper also shows – using bootstrappingmethods – that a combination of market weighted benchmark portfolio and risk based characteristic portfolios will provide astatistically significant improvement over the minimum variance portfolio. In other words investors would be better off addinganomaly portfolios to the maximum Sharpe-Ratio portfolio instead.

Section 2 will describe the MVP and its market capitalization weighted (benchmark) portfolio (MWP) used in this paper.Section 3 economically motivates the explanatory variables used to explain the return difference between MVP and MWP.Section 4 provides the mathematical proof for the conjecture that the MVP will prefer low residual volatility and low beta stocks.Section 5 documents the explanatory power and statistical significance of the chosen right hand side variables related to the MVPportfolio construction process. Finally Section 6 tests whether a direct exploitation of these risk based pricing anomalies offerslarger and statistically significant SHARPE-ratios than the MVP itself.

2. Returns on the minimum variance portfolio

The calculation of the MVP is subject to many decisions. What statistical model should be used for covariance matrixestimation? What is the eligible universe of securities? What are the set of constraints for the optimization process? While thiscontains many interesting empirical research questions10 their answers are somewhat subject to data mining, i.e. the search forthe best in sample specification. Instead of creating an own index I therefore use theMSCI BARRAminimum volatility index for theUSmarket which is a commercially available minimum variance index created on the basis of BARRA risk models. While this indexwill to some degree be data mined (as every index) it resembles an actual real world investment opportunity.

According to the technical document of MSCI BARRA the minimum volatility index is rebalanced semi-annually using thestocks in the MSCI US equity index as eligible universe. The document also states that in order to ensure investability theoptimization (ex ante risk minimization) is heavily constrained. However, while there are position, sector and BARRA risk factorconstraints, factor neutrality is not targeted. Our sample covers the period from January 1998 to December 2009. We use monthlytotal return data. Sample period and frequency equally apply to all subsequent series. Given that all index constituents for theMSCIUS minimum volatility index are drawn from the MSCI US equity index it is natural to use the MSCI US as a market cap weightedbenchmark portfolio as it is the likely investment alternative.

Fig. 1 plots the excess return of the MVP — defined as the difference in log returns between the MSCI US minimum volatilityindex and its cap weighted relative, the MSCI US equity index against time. In bull markets (shaded area) the excess returns tendon average to fall below zero, while they tend to come out above zero in bear markets. This is confirmed by our calculations inTable 1 that split up descriptive statistics between bull and bear markets. Over the full period the returns of MVP and MWP arealmost indistinguishable and virtually zero which is a consequence of the telecom bubble crash (starting in 2000) and the creditcrisis crash (starting at the end of 2007) that left the equity market risk premium close to zero over this decade.11 As expected theMVP has a lower volatility than theMWP over all periods. Interestingly the null hypothesis of normally distributed returns for bothequity benchmarks cannot be rejected in either bull or bear markets, while it is strongly rejected for the whole sample period. Thisprovides evidence for the existence of two equity market regimes. Both are normally distributed on their own, but non-normal in

6 A related but different stand of the literature compares the minimum variance portfolio with mean variance optimized alternatives. Kritzman et al. (2010)and DeMiguel et al. (2009) both find that optimized portfolios outperform the minimum variance portfolio if risk premia can be measured with some reliability.

7 Both papers view the average variance of individual stocks relative to portfolio variance as a driver of portfolio returns. This is close in spirit to the long/shortresidual risk portfolio. In fact Scherer (2007) has shown that the approach by Fernholz ceases to produce alpha if this portfolio is added to FAMA/FRENCH factors.

8 The author introduces the HERFINDHAL index as a portfolio optimization constraint.9 I define out-performance as risk adjusted return versus a benchmark portfolio.

10 There is already a large literature on “improved” covariance optimization. See Connor et al. (2010) for a good review.11 This would make it interesting to look at earlier decades, i.e. to backwards expand the BARRA time series.

653B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

combination. This should not be too surprising as we know that amixture of normal distributions can generate any shaped form ofmarginal distribution. Sharpe ratios for this period are close to zero because of the negative impact of the credit crisis on both timeseries. The next section will motivate the set of explanatory variables that is assumed to explain the variance of excess returns inFig. 1.

3. Portfolio mathematics of the minimum variance portfolio

So far, I simply conjectured that the minimum variance portfolio is likely to invest into low residual risk and low beta stocks.This section provides the analytical proof. We start with a set of definitions.

In a CAPM world with k assets, the k×k covariance matrix, Ω, can be decomposed into:

Ω = ββTσ2m + diag σ2

1 ;σ22 ;…;σ2

k

� �= ββTσ2

m + Σ: ð1Þ

Here,Σ, denotes the k×k diagonal matrix of residual variances, σ12, σ2

2,…, σk2, on themain diagonal, β represents a k×1 vector of

security betas, β1, β2,…, βk and σm2 represents the market variance. The inverse of Ω is given by12:

Ω−1 = Σ−1− σ2m

1 + κbbT

: ð2Þ

I define κ=σm2 ∑ i=1

k biβi and bi =βiσ2i. The k×1 vector of optimal portfolio weights w for the minimum variance portfolio

represents a characteristic portfolio:

w =Ω−1ιιTΩ−1ι

ð3Þ

where the k×1 vector ι of ones, denotes the characteristics. Minimum variance portfolio risk amounts to σmv2 =

1

ιTΩ−1ι, where ι

represents a vector of ones. Substitute Eq. (2) in Eq. (3) to arrive at:

w = σ2mv Σ−1ι− σ2

m

1 + κbbTι

!: ð4Þ

12 See for example Sorensen et al. (2007, p. 40).

Fig. 1. Monthly excess returns of MSCI BARRA US minimum volatility portfolio versus MSCI US equity portfolio for January 1999 to December 2009. The shadedareas show the bull markets (1999: 1–2000: 10 and 2003: 2–2007: 10 and 2009: 3–2009: 12) defined by ex post identification turning points.

654 B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

We can now calculate the optimal holding wjas an element of the right hand side k×1 vector in Eq. (4).

wj = σ2mv

1σ2j

− σ2m

1 + κbj∑

ki = 1bi

!

=σ2mv

σ2j

1− σ2m

1 + κbjσ

2j ∑

ki = 1bi

!

=σ2mv

σ2j

1− σ2m

1 + κβj∑

ki = 1bi

!

=σ2mv

σ2j

1−βjσ2m

1 + κ∑k

i = 1bi

" # !ð5Þ

The term σ2m

1 + κ ∑ki = 1bi is difficult to interpret13 but close to one for realistic parameterizations.14 We then arrive at an

expression for the optimal weight of stock jin the MVP:

wj≈σ2mv

σ2j

1−βj

� �: ð6Þ

The optimal weight on asset j is high if residual risk σj2 is small. Low residual risk assets will ceteris paribus obtain a positive

weight in the minimum variance portfolio. At the same time a low β (below one) will also create a positive portfolio weight. Thisproofs the earlier conjecture that the minimum variance portfolio is likely to pick up low beta and low residual risk stocks.

Note that Eq. (6) shows the optimal weight for stock j if the optimizer constructing the minimum variance portfolio is fed witha single factor risk model. A model of this type imposes the assumption that all residual covariance terms are zero (otherwise wewould still have to calculate a large scale k×k residual covariance matrix). This assumption is most certainly wrong (residuals areglued together by industry membership, size and other stock characteristics) and a model of this type would severelyunderestimate out of sample portfolio risk (residual risk gets diversified away too quickly for in sample risk calculations). For anyother employed risk model the optimizer would try to remove systematic risks or shift exposures to systematic risks that exhibitsmaller volatility (large value stocks for typical multifactor risk models).15 However, given that the market risk factor is thedominant factor in security returns we conjecture it should be a mayor driver. Applying the matrix inversions lemma to a generalrisk model will lead to no meaningful interpretation as risk factors are highly correlated.

4. Data and methodology

The central hypothesis of this paper is that the excess returns of the minimum variance portfolios relative to a cap weightedindex (with the same investment universe) are a function of risk related factor portfolios. These risk based characteristic portfoliosare directly related to the mechanical rule the minimum variance portfolio is constructed from. If we can identify these risk factors

13 We could also give the term an average beta meaning realizing that σ2m

1 + κ ∑ki = 1bi =

σ2m∑k

i = 1bi1 + σ2

m∑ki = 1biβi

.

14 While this is ultimately an empirical question, we can get some ideas on this by simulating the data. Suppose σm=0.2 and σj=0.3+N(0, 0.05). Let usfurther assume that individual stock market betas are normally distributed with mean 1 and volatility 0.2. This means that 95% of the betas range between 0.6and 1.4 and coincide with what we know from studies of beta as a driver of cross sectional returns. Assuming independence between beta and residual risk, wecalculate σ2

m1 + κ ∑k

i = 1bih i

= 0:96 which is close to one. Residual risk has very little influence on this adjustment factor but the distribution of stock market betas

is crucial. For large dispersion of betas the adjustment factor is significantly lower than one.15 Our empirical results confirm this conjecture. The MVP has significant loadings to large cap value stocks.

Table 1Descriptive statistics.

All periods Bull markets Bear market

Minimum variance MSCI US Minimum variance MSCI US Minimum variance MSCI US

Mean 0.027 0.013 1.019 1.342 −2.027 −2.738Volatility 3.560 4.773 2.712 3.436 4.209 5.903t-value 0.088 0.031 3.547 3.685 −3.157 −3.041Sharpe-ratio 0.008 0.003 0.376 0.391 −0.481 −0.464Skewness −0.736 −0.641 0.301 0.146 −0.621 −0.107Kurtosis 1.936 0.784 0.214 −0.287 0.717 −0.387Minimum −0.146 −0.177 −0.055 −0.058 −0.146 −0.177JB-test (p-value) 0.000 0.002 0.469 0.733 0.159 0.839

The table shows basic statistics for the MVP and the MWP excess returns for all periods (1999: 1 to 2009: 12), bull markets (1999: 1–2000: 10 and 2003: 2–2007:10 and 2009: 3–2009: 12).

655B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

we can build a better portfolio by directly leveraging on the risk based anomalies picked up by minimum variance investing.Alternatively we can use the estimated loadings on risk factors to project the long term performance differential from minimumvolatility investing. This section will motivate the right hand side (explanatory) variables in our factor model used to explain thevariation of excess returns.

We aim for a parsimonious set of economically well motivated time series. All right hand side variables are also excess returns,i.e. self-financing long/short portfolios. This allows us to treat the intercept in these regressions as an “alpha” term. Table 2provides a summary of descriptive statistics for the data used in this article.

4.1. Market returns

Theminimumvariance portfoliowill – ex ante – always exhibit amarket beta smaller than one. It will ex-ante provide a smallerstandard deviation and as correlations can never exceed one, it must exhibit a smaller beta than unity. One would therefore expecta significantly negative beta exposure of the MVP excess returns with respect to market returns. In bull markets the minimumvolatility portfolio is likely to underperform while it tends to outperform in bear markets. This is confirmed by our results in Fig. 1as well as the numbers in Table 1. This paper uses monthly data for the market risk premium from Ken French's website for theperiod January 1999 to December 2009.16

4.2. Value factor returns

Low volatility equity investing is usually associated with value strategies as detailed by Dibartolomeo (2007). We wouldtherefore expect a significantly positive value bias for the minimum variance portfolio relative to a cap-weighted portfolio. Valuestocks in turn have historically provided superior risk adjusted returns for decades. Their skewed distributions (proxying fordefault risk) will let investors demand a higher risk premium than the CAPM prescribes and the minimum volatility portfolio willbenefit. I use monthly data for the value premium (HML), from Ken French's website for the period January 1999 to December2009 and expect HML to load positively on excess returns.

4.3. Size factor returns

Large stocks tend to bemore diversified, both geographically as well as with respect to business lines than small companies. Byconstruction the MVP will prefer these diversified (lower risk) companies. I use the size premium (SMB) from Ken French'swebsite for the period January 1999 to December 2009 andwould expect a positive size bias, i.e. theMVP should load positively onSML.

So far the set of explanatory variables is standard. I complete the above factor return series with two risk based anomalyportfolios that are strongly related with the way the MVP is constructed.

Table 2Summary for explanatory data.

Type Characteristic portfolio Factor return

Name Beta Residual volatility HML SMB Market

Symbol Rβ Rσ RHML RSMB RMKT

Description Beta and cash neutral portfolio oflong low and short high beta stocks

Long/short portfolio of long lowresidual volatility and short highresidual volatility stocks

Value factor Size factor Market factor

Source Own calculation Own calculation FAMA/FRENCH FAMA/FRENCH FAMA/FRENCHMean 1.691 1.224 0.321 0.578 0.022Volatility 9.678 10.042 4.975 3.617 4.754t-value 2.007 1.401 0.742 1.836 0.053Sharpe-ratio 0.175 0.122 0.065 0.160 0.005Skewness −0.065 −0.715 −0.169 0.466 −0.607Kurtosis 1.173 4.007 4.552 1.978 0.622Minimum −0.242 −0.418 −0.208 −0.116 −0.172JB-test (p-value) 0.02 0.00 0.321 0.578 0.022

The table shows basic descriptive statistics for monthly data from January 1999 to December 2009. HML (Value), Size (SMB) and Market risk premium are takenfrom K. French's website while the long/short portfolio for beta and residual risk is calculated as described above.

16 I use http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html for the collection of Fama and French (1992) factor data.

656 B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

4.4. Small beta versus large beta returns

One expects – ceteris paribus – that the MVP is likely to rather invest into low beta stocks than into high beta stocks as provenin Eq. (6). Low systematic risk is simply attractive in creating a MVP. To pick this effect up I construct a beta and cash neutrallong/short portfolio that is long (equal weighted) the 20% stocks with the lowest betas and short the 20% stocks with the highestbetas for stocks in the S&P1500 universe (the same universe BARRA calibrates their risk model for the variance minimizationon).17 Each stock beta is calculated from a regression of asset returns versus the S&P1500 using three years of daily data. Myconjecture is that the excess returns of the MVP will have exposure to this portfolio. Since the early days of the CAPM it is a wellknown empirical regularity that high beta stocks tend to earn less thanwhat their beta implies. Low beta stocks on the other sidetend to earn more than their beta implies. In other words the cross sectional relation between beta and return is too flat asdescribed in Fama andMacBeth (1973) or Black et al. (1972). More recently Thomas and Shapiro (2009) find, that “high beta hasnot been compensated, value has been compensated and is associated with low beta and low stock-specific volatility has beencompensated”. The estimated beta for the low beta portfolio is βlow=0.599while the estimated beta on the high beta portfolio is

βhigh=1.799. This results in a hedge ratio of h =βhigh

βlow= 3:005, which in turn yields a time series for a beta and cash neutral

anomaly series: Rbeta=h(Rbeta, low−c)−(Rbeta, high−c), where c denotes the risk free rate. As low beta stocks tend to be valuestocks (low net present value of growth opportunities creates little economic sensitivities) I expect a positive value loading ofRbeta against the FAMA/FRENCH value factor. A recent study by Guo and Savickas (2010) finds that high beta stocks tend tounderperform. To confirm my prior I regress the beta anomaly portfolio returns against the FAMA/FRENCH factors. The resultsare documented on the left hand side of Table 3. In accordancewith the academic literature I find that the beta anomaly portfolio(cash and beta neutral portfolio of long low beta stocks and short high beta stocks) outperforms the linear best trackingcombination of FAMA/FRENCH factors by 1.6% per month. This result is significant with a (HAC adjusted) t-value of 2.21, i.e. it issignificant at the 5% level. The regressions in Eqs. (4) and (5) also confirm our intuition that the beta anomaly portfolio exhibitssignificant value bias with a t-value of 2.573 for the value beta. There is no significant size exposure.

4.5. Residual risk anomaly

Everything else equal, the minimum variance portfolio will also find stocks with low residual risk attractive to invest as provenin Eq. (6). Ang et al. (2006) find that stocks with high residual risk exhibit too low CAPM-adjusted returns. Blitz and Vliet (2007)demonstrates that investors tend to overpay for volatility – possibly because of leverage restrictions – which leaves low volatilitystocks unattractive. Guo and Savickas (2010) confirm their results and find further that the low residual risk premium relatesstrongly to the value premium.18 Value stocks tend to have low volatilities, while growth stocks tend to have high volatilities. Tocapture this effect I construct a portfolio that is long the 20% stocks with the lowest residual risk (the residuals come from aregression of equity return against the S&P1500 and a constant using 3 years of daily data) and short the 20% stocks with thehighest residual risks.19 Both of these portfolios are beta neutralized and rebalanced monthly.20 Results from regressing thereturns of the residual risk anomaly portfolio against the FAMA/FRENCH factors yields are shown on the right hand side of Table 3.

17 The data have been created during the author's research visit at CapitalIQ.18 Note, that older empirical evidence by Tinic and West (1986) and Malkiel and Yu (1997) find that stocks with high residual volatility earn higher returns.19 Ideally one would want to build long/short portfolios on the basis of the BARRA risk model used to construct the MVP. However, the CAPITALIQ risk model(see Scherer et al., 2010) is a close competitor to the BARRA model for the same universe of US stocks, so our results should be reasonably close.20 Our monthly rebalancing frequency relative to the semiannual rebalancing frequency of the MSCI BARRA min volatility index is likely to reduce theexplanatory power of our model. Our conjecture is that the results will be even stronger if the rebalancing frequency is matched.

Table 3Factor loadings for characteristic portfolios.

Return Beta anomaly Residual risk anomaly

Symbol Rβ Rσ

Regression # (1) (2) (3) (4) (5) (1) (2) (3) (4) (5)

α 0.017(0.008)**

0.017(0.008)**

0.015(0.008)*

0.019−(0.008)**

0.016(0.007)**

0.012(0.009)

0.012(0.009)

0.009(0.008)

0.021(0.007)***

0.019(0.006)***

βMKT 0.000(0.264)

−0.001(0.244)

0.000(0.255)

0.308(0.201)

βHML 0.690(0.247)***

0.659(0.256)***

0.913(0.322)***

0.689(0.206)***

βSMB −0.400(0.274)

−0.240(0.296)

−1.600(0.285)***

−1.559(0.258)***

R2

0.00 0.00 0.11 0.01 0.11 0.00 0.00 0.19 0.33 0.46DW 1.72 1.72 1.94 1.68 1.92 2.13 2.13 2.22 1.87 2.06

Each characteristic portfolio is regressed against a set of explanatory variables, i.e. the FAMA/FRENCHmarket risk premium (MKT), the factor for value (HML) andthe size factor (SMB). Variable significant at the 1%, 5% and 10% level are marked by ***, ** and *.

657B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

Again HAC adjusted t-values are in brackets. In accordance with the previously quoted literature the residual risk portfolio offers ahighly significant risk premium of about 2% per month relative to a FAMA/FRENCH three factor model. The exposure to value issignificantly positive while the size exposure is significantly negative. In other words low residual risk stocks tend to be large capvalue stocks.

4.6. Regression based testing

In order to explain the MVP excess returns versus the MWP I run a simple multifactor regression of (RMVP−RMWP) against thedescribed factor and risk based anomaly portfolios RMKT,RHML, RSMB, Rβ and Rσ. The regression coefficients can be interpreted as theweights of a multifactor portfolio that most closely tracks excess returns. All standard tests on parameter significance and stabilitycan now be applied to see how well the parsimonious model for excess returns fits the data.

5. Results

This section explains the variance of MVP excess returns by a set of factor portfolios. In order to better understand the MVPexposures, I first run six regressions. In each, I regress MVP excess returns against a single explanatory factor. The results aredocumented in columns (2) to (6) in Table 4. All factors are highly significant with the signs matching our economic priors. TheMVP market beta is around 0.65 and highly significant. Given that both risk anomaly portfolios are market neutralized and HMLand SMB have little covariance with the market portfolio the beta estimate remains almost the same over various specifications.Factor loadings on Market and SML are negative confirming our previous intuition that the MVP is likely to hold large diversifiedstocks and will by design exhibit a negative beta bias. Next we investigate the explanatory power of the FAMA/FRENCH model incolumn (7). What holds individually also holds collectively and the adjusted R2 rises from 51% for the market factor alone to 73% ifHML and SMB are added. However we would like to contrast this with a regression where HML and SMB are instead replaced byour two risk based anomaly portfolios in column (8). Again all variables are highly significant with the correct sign. Moreinteresting is the higher R2 of our anomaly based model versus the FAMA/FRENCH model. This means that the two anomalyportfolios proxy for more than the FAMA/FRENCH portfolios. It is an important insight as we know that the beta anomaly portfoliopicks up both positive value and negative size exposure, so a high R2 on its own would not have been too surprising. The final testcomes in column (9)where all factor portfolios as well as the anomaly portfolios enter as right hand variables. This can be seen as anesting of models (7) and (8). All independent variables again come out highly significantwith the right conjectured signs. Neithermodel (7) nor (8) can therefore be seen as alternatives. Both specifications complement each other with a (marginally) higher R2

of 84% for specification (9).21 Finally I want to test the stability of the above results. One can use split sample or predictive failuretests. However they usually split the estimation period in arbitrary ways and are therefore suspect to the chosen split point. This isnot satisfactory and I therefore employ the Nyblom (1989)–Hansen (1992) parameter stability test that does not rely on theselection of potential break points and produces two types of statistic: a joint test statistic and individual test statistics. Results forour final model (9) are given in Table 5.

The null hypothesis of parameter constancy (versus the alternative of parameters following a random walk) is rejected if the(individual/joint) test statistics is significant, i.e. p-values are low. All parameters are jointly constant and none of the individual p-values falls below the 5% significance level. In other words our model specification does not suffer from parameter instability.22

Table 4Results from various linear regressions with excess returns (returns MVP minus MWP) as dependent variable against a variety of explanatory variables.

Dependent variable MVP excess return

Regression specification (1) (2) (3) (4) (5) (6) (7) (8) (9)

α 0.001(0.002)

0.001(0.001)

0.000(0.002)

0.003(0.002)

−0.001(0.002)

−0.001(0.002)

−0.001(0.001)

−0.000(0.001)

−0.000(0.001)

βMKT −0.349(0.039)***

−0.312(0.029)***

−0.349(0.037)***

−0.329(0.029)***

βHML 0.143(0.069)**

0.141(0.030)***

0.075(0.020)***

βSMB −0.370(0.047)***

−0.209(0.038)***

−0.113(0.045)**

ββ 0.105(0.030)***

0.064(0.014)***

0.042(0.026)**

βσ 0.231(0.01)***

0.126(0.023)***

0.055(0.019)***

R2

0.00 0.51 0.08 0.32 0.18 0.29 0.73 0.79 0.83DW 1.85 2.01 1.81 1.86 1.64 1.69 1.99 1.73 1.82

The table also reports adjusted R2 and DURBIN/WATSON test statistic. Variable significant at the 1%, 5% and 10% level are marked by ***, ** and *.

21 All results remain unchanged when we use robust regressions instead. If anything the statistical significance of most regression coefficients becomes larger.22 Note, that the individual stability test for value (HML) yielded a p-value only marginally above 5%. If we plot the recursive residuals of model (9) we seestrong variation around mid 2008 where the value factor has been particularly volatile as a result of the credit crisis.

658 B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

One caveat in this analysis is that the anomaly portfolios have been leveraged long/short portfolios, while the employed MVPportfolio is constrained to long only investments. While some of this difference is mitigated as we try to explain excess returns ofMVP versusMWP (essentially a long/short portfolio) I would expect that the results documented in this paper get stronger if shortpositions are allowed in the MVP as the MVP is likely to load more heavily on the above anomalies.

6. Building a better portfolio

One of the central conjectures of this paper is that investors can achieve a higher SHARPE-ratio than the minimum varianceportfolio by directly identifying the risk based pricing anomalies the minimum variance portfolio draws upon. This would bepreferable to (clumsily) following the minimum variance portfolio (in the hope of an indirect pickup of these anomalies).

In order to show the superiority of directly exploiting risk based anomalies I compare the SHARPE-ratio of a portfolio investedby one third each into the MSCI US, the beta anomaly, and the residual risk anomaly portfolio with the SHARPE-ratio of theminimum volatility portfolio.23 Given the small sample, the non-normality and autocorrelation of the underlying time series Ifollow Ledoit andWolf (2008) to use bootstrapping rather than the Jobson and Korkie (1981) test. It is well known that the latter isnot valid if we deviate from iid normal returns. My test consists of bootstrapping the return series of (RMSCI−c)+Rβ+Rσ togetherwith the series (RMV−c) 10,000 times. In effect each run will draw 132 returns with replacement and calculate the difference inSHARPE-ratio between both portfolios. This will preserve both cross correlation as well as non-normality. However, given thesubstantial autocorrelation in the data (Ljung–Box Q — Statistics of 20.4, i.e. a p-value of 5% for autocorrelation up to lag 12 in thereturns of the minimum volatility portfolio) I also use block-wise bootstrapping with random block length. To arrive at an

expected block length of n, the algorithm takes the next value in the historic series with probability 1−1n. With probability

1nthe

algorithm randomly starts a new block.For the resulting 10000 SHARPE-ratio differences I calculate the (1−α%) confidence interval as the interval between the

α%2

and 1−α%2

bias corrected fractiles.24 The results are shown in Table 6. It shows the upper and lower values for the bootstrapped

99%, 95% and 90% confidence interval. If these intervals (bootstrapped from the empirical data) do not cover zero, the SHARPE ratiodifference (in the empirical data) is statistically significant. As an example, the 95% confidence interval for an expected block sizeof 12 covers a SHARPE-ratio difference from 0.05 to 0.24. In other words the difference in SHARPE-ratio is statistically significant atthe 95% confidence level.

The above test confirms our initial intuition: It is preferable to directly leverage on the above anomalies rather than trying toimplicitly participate via the MVP. The bootstrapping results are similar to the naive Jobson and Korkie (1981) test for thedifference between two SHARPE-ratios in the Memmel (2003) adjusted version. The z-value for this test is given by

z =SRnew−SRMVPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1T 2 1−ρð Þ + 1

2SR2

new + SR2MVP−SRnewSRMVP 1 + ρ2� �� � s = 2:03 where SRnew, SRMVP are the SHARPE-ratios of the new

portfolio (excess returns on MSCI US plus Beta plus Residual Risk anomaly) as well as the MVP, ρ=0.6 denotes the correlationbetween both excess return series and T=132 equals the number of observations. This indicates a statistical significance at the97.5% confidence.

23 I follow the habit of comparing SHARPE-ratios simply out of convention with the before quoted literature. However it should be noted that (even ignoringnon-normality) it is not permissible to compare investments on the basis of their SHARPE-ratio. The SHARPE-ratio is a stand alone measure that ignores aninvestments portfolio contribution. For example: investors are willing to hold even negative SHARPE-ratio assets if they show negative covariance with theirremaining assets. This is the gist of modern portfolio theory. Long bonds for example have small individual SHARPE-ratios (large risk and little risk premium overcash). However, long bonds are a widely held asset by almost all investors due to their negative correlation with equities. They are essential recession hedges thatpay out if the marginal utility from consumption is high.24 To correct for potential bias in the average bootstrapped SHARPE-ratio difference versus the actual SHARP ratio difference, the sample mean is replaced withthe actual SHARPE-ratio difference.

Table 5Results from Nyblom (1989)–Hansen (1992) stability test for regression (9) in Table 4.

Test-statistic p-value

Joint 1.63400 0.12047Variance 0.18170 0.29536α 0.12237 0.46802MKT 0.03429 0.95833HML 0.56677 0.05734SMB 0.22757 0.21398β 0.17131 0.31884σ 0.16662 0.33211

The null hypothesis of parameter stability is jointly rejected, even though the loadings on thevalue factor (HML) seem to exhibit instability.

659B. Scherer / Journal of Empirical Finance 18 (2011) 652–660

7. Conclusion

Investing in the minimum variance portfolio instead of investing into a cap weighted market portfolio is not a clever forecastfree strategy. It simply picks up risk based pricing anomalies, so investors in the MVP need some conviction on the persistence ofthese risk based pricing anomalies. Most importantly this paper shows that 83% of the variation of theminimum variance portfolioexcess returns can be attributed to the FAMA/FRENCH factors as well as the returns on two characteristic anomaly portfolios. Theserisk based characteristic portfolios are particularly successful in explaining the performance of the MVP and on their own almostcrowd out the FAMA/FRENCH factors. All regression coefficients (factor exposures) are highly significant, stable over theestimation period and correspond remarkably well with our economic intuition. The paper also shows that a direct combination ofmarket-cap weighted benchmark and risk-based characteristic portfolios, provides investors a statistically significantimprovement over the indirect pickup via the minimum variance portfolio simply because the risk based characteristic portfoliosare amore efficient way to exploit these anomalies. This also puts some concern on results by Jagannathan andMa (2003) that linkthe superior performance of a long only constrained minimum variance portfolio with Bayesian shrinkage. The returns on MVPand characteristic portfolios are both out of sample and are both subject to the same type of estimation error, i.e. build from riskinformation only.

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Table 6Bootstrapped difference in Sharpe-ratios.

Confidence interval

99% 95% 90%

Lower Upper Lower Upper Lower Upper

Expected block size 1 0.004 0.287 0.026 0.264 0.043 0.2432 0.005 0.288 0.027 0.264 0.043 0.2434 0.028 0.229 0.043 0.248 0.058 0.2298 0.036 0.266 0.049 0.246 0.062 0.26712 0.038 0.266 0.053 0.245 0.064 0.230

The table shows the upper and lower values for the 99%, 95% and 90% confidence intervals. As an example, the 95% confidence interval for an expected block size of12 covers a SHARPE-ratio difference from 0.05 to 0.24. In other words the difference in SHARPE-ratio is statistically significant at the 95% confidence level.

660 B. Scherer / Journal of Empirical Finance 18 (2011) 652–660