Mechanics of minimum variance investment approach · Minimum variance portfolios constructed via mathematical optimization allow to signi cantly reduce equity risk. Min-imum variance

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Mechanics of minimum variance investment approach

June, 09, 2011

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Mechanics of minimum varianceinvestment approach

Bruno Monnier and Ksenya Rulik

June, 09, 2011

Abstract

Bruno MonnierQuantitative [email protected]

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

Minimum variance portfolios constructed via mathematicaloptimization allow to significantly reduce equity risk. Min-imum variance optimization, though bearing a certain levelof complexity, cannot be regarded anymore as a ”black box”process. We will argue in this paper that properly designedminimum variance investment approach can be relevant forinvestors seeking risk-efficient passive equity allocation, thatit can be made transparent to investors by giving them moredisclosure and intuition on the resulting optimized alloca-tion and the choices made in by the optimization procedure.We discuss here how the minimum variance engine worksand how to structure the optimization procedure in order tokeep the minimum variance portfolio diversified.


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Towards risk efficiency in pas-sive investing

Since Harry Markowitz seminal work (1952)the investment community embraced the con-cept of optimal portfolio. While simple inits objective - delivering the best risk/returntrade-off - building such a portfolio is a daunt-ing task requiring implicit or explicit assump-tions about future risk and returns. As a con-sequence, the resulting allocation remains op-timal only ex-ante, and its ex-post performanceis seriously compromised by forecasting errors.

Since the 1960s, the capital asset pricingmodel (CAPM) brought an elegant solutionto the optimization problem, arguing that themost efficient portfolio is necessarily a broadmarket portfolio weighted by market capital-ization of stocks. This corresponds to perform-ing mean-variance optimization with market-implied forecasts of risk and returns. Theoret-ically backed by the powerful efficient marketframework, this idea gained an enormous in-fluence and reshaped the equity investing in-dustry over the past 35 years. Mean-varianceframework suggests another definition of pas-sive equity investing: an investment processaiming at providing an access to the equitymarket or one of its segments, that does notrequire return forecasts for the stocks insidethe chosen universe. The objective of ”newpassive” approach is to shape the portfolio insuch a way that the equity exposure gives thebest possible risk-adjusted performance.

Market-capitalization weighted portfolio isactually an ”active” portfolio in the mean-variance sense. It is reliant on implicit futurereturn forecasts contained in the stocks relativecapitalization and is optimal only if this fore-cast is the best that one can have. There is onlyone optimal portfolio on the mean-variancefrontier that is truly ”passive” in its objec-tive, that is the Minimum Variance Portfolio

(MVP). MVP is an optimal portfolio that isconstructed by minimizing portfolio variance.The minimum variance construction does notuse stocks’ expected returns as inputs, and re-lies only on the covariance matrix.

Usually the MVP is depicted as the outmostleft point on the mean-variance frontier, theless risky and the less performant one. Butthis picture is very misleading, since it is con-ditional on some non-homogeneous forecast offuture returns. If such a forecast happens to bewrong, the ex-post performance of the MVPmight be well above the ex-ante optimal port-folios that have higher risk. If ex-post thestocks will have similar returns, the MVP willtake the place of the efficient tangent portfolio,giving the highest Sharpe ratio.

Figure 1: Widespread but often misleading il-lustration of the Minimum Variance Portfolio

There is a growing body of evidence on per-formance and characteristics of the ”new pas-sive” minimum variance investing. Over thepast three years MSCI and DAX have createdminimum variance versions of their benchmarkequity indices. In addition, a growing amountof academic research is dedicated to minimumvariance portfolios, where usually some testMVP are studied. All these portfolios sharesome important properties, as reduced ex-post


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portfolio volatility with respect to the market-capitalization benchmarks. But the differencesin portfolio construction across different min-imum variance strategies can be significant1.This stems from the fact that the Minimumvariance methodology is not unique, and it isdifficult for investors to get insight on the im-plications of minimum variance portfolio choiceand the resulting stock allocation.

Technically, to construct a minimum-variance portfolio one needs a forecast of thecovariance matrix and a mean-variance opti-mization engine. The latter is a mathemati-cal procedure that decides which stocks to pickand what weights to give them to obtain thelowest possible volatlity of the overall portfo-lio. The mechanics of this choice, especiallyin realistic cases where different portfolio con-straints are involved, is by no means intuitivefor investors. The widely spread belief thatsuch a technique tends to pick low-volatility orlow-beta stocks is relevant but far from fullydescribing the resulting allocation of a mini-mum variance portfolio.

We will argue in this paper that properlydesigned minimum variance investment ap-proach can be relevant for investors seekingrisk-efficient passive equity allocation, that itcan be made transparent to investors by givingthem more disclosure and intuition on the re-sulting portfolios and the choices made in con-structing the mean-variance procedures, and,finally, that this approach can be consistentlyapplied in a systematic investment mode.

1According to the respective index guides, MSCIMinimum Volatility US index portfolio is forced to con-tain well above 60 stocks, while the DAXplus MinimumVariance US index is limited to maximum 50 stocks.

Reducing portfolio variance is afeasible goal

A natural question that investors ask is: is itpossible to maintain a full exposure to the eq-uity market while mitigating risk? An accu-rate view on the future risk structure is veryimportant here. Risk profiles of single stocksas well as the structure of stocks comovementsare resumed in the variance-covariance matrix.Minimum variance technique combines the in-formation on expected risk levels to build an al-location with the minimum expected risk pos-sible.

The dominant approach is to infer the futurestocks variances and covariances from histori-cal data. While subject to estimation errors,the historical method gives a relevant short andmedium-term picture of the future risk. Stockvolatility levels are highly persistent2, and thecovariance structure is persistently dominatedby the broad market risk factor3.

Once the estimation of covariance matrix isbuilt 4, various constraints, that we will discussbelow in greater detail, allow to adapt the tech-nique to specific investor situations (in partic-ular, limit short sales) and incorporate ex-anterisk management features in the portfolio (e.g.limiting concentration on single stocks).

Growing evidence, based on live records ofminimum variance products and historical sim-

2This stems from the well-documented stylized factsabout stocks returns (see e.g. Cont, 2001), such as per-sistence in absolute return values and volatility cluster-ing.

3The dominance of the first eigenvalue of correlationmatrix is discussed for example in the Bouchaud andPotters (2005)

4There are various techniques of error reduction incovariance estimation at one’s disposal: the examplesare covariance shrinkage (see e.g. Ledoit, Wolf, 2003),use of risk factor models (the example is the MSCIBarra risk estimation framework), or noise filtering withthe random matrix theory (For a recent review seeBouchaud, 2009).


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ulations, shows that volatility reduction usingminimum variance technique is significant5.

The other question that is on everyone’smind is whether this new portfolio will performbetter than the market-cap weighted bench-mark. Is there a monetary advantage on con-centrating solely on the least volatile equityconfigurations? This question is out of the sub-ject of this paper, since we do not want to delveinto estimating future relative or absolute re-turns, that is a huge and controversial subjecton its own. The only point that we want tomention here is that historical evidence did notsupport the CAPM conjecture of performancereduction for low-volatility stocks with respectto high-risk stocks 6. On the other hand, thereis no clear understanding of why the low-riskstocks could have outperformed the market.

When we incorporate estimated stock re-turns into Markowitz optimization, the effi-ciency of the resulting allocation will be largelydetermined by the structure of this estimatesand errors contained in it7, and to much lesserextent by the design of the optimization prob-lem. Instead, when assuming homogeneous re-turn estimates, the design of the optimizationproblem plays much greater role, and allowsto build an allocation that will maintain itspromise: exhibiting lower risk in the futurewhile giving full exposure to equity market.

So, what happens inside the minimum vari-ance optimizer?

5For example, the simulated MSCI MinimumVolatility World index had around 30% reduction involatility with respect to the market-capitalizationweighted index over the period 1995-2007, as docu-mented by MSCI BARRA Research (2008)

6See for example A.Ang et al (2006)7see for example Chorpa, Ziemba 1993

Closer look at minimum vari-ance optimization

We discuss here, step-by-step, the main ingre-dients of the minimum variance optimizationproblem. At each step of the discussion wedescribe the impact that each new element ofoptimization design has on the optimized allo-cation.

First we need to introduce some notations:we refer to σ as single-stock volatility, ρ - a cor-relation between two stocks and Σ - the covari-ance matrix that groups single-stock volatili-ties and pairwise correlations. Stocks’ weightswe denote by w. Now we can formulate min-imum variance objective as follows (in matrixnotations):

minw

~w′Σ~w, (1)

The above is a minimization problem, thatis quadratic in the weights w. Standalone,this problem gives an empty solution. Sinceno return expectations are given, the naturalchoice is to invest 100% of the portfolio in cash(w∗ = 0), shrinking the portfolio variance tothe absolute zero.

The minimum variance mechanism needs ad-ditional parts to make the approach work.These parts are formulated as portfolio con-straints, that we will add one by one to showtheir role.

Budget constraint:∑wi = X%

This constraint forces the optimizer to pro-duce a non-empty solution, with X% net longinvesting in stocks. All existing minimum vari-ance strategies employ it, usually with X =100%. Thus the budget constraint is consid-ered to be inherent to the minimum variance


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optimization problem, and this setup is oftenreferred to as unconstrained.

In the general case the result is a long-shortportfolio, where long investment exceeds theshort side by the specified amount of X%. Thelong side of the portfolio will tend to be popu-lated by less risky stocks and the short by morerisky ones.

Exact long-short configuration will dependon the structure of the covariance matrix. Forexample, if the stocks are all uncorrelated (ρ =0), there will be no short positions since allstocks will have a potential to provide addi-tional risk reduction by diversification. Thereis another simple particular case of correla-tion structure that demonstrates well the stockselection: the case of equal pairwise correla-tions. We denote different stock volatilities asσi, where i goes from 1 to N, the number ofstocks in the group. The pairwise correlation,that we assume to be equal, we denote by ρ. 8

A useful relation that follows from the exactoptimization solution in this case is:

σi < H(σ, ρ,N)⇒ wi > 0 (2)

σi > H(σ, ρ,N)⇒ wi < 0 (3)

with H(σ, ρ,N) ≡ H(σ)

(1 +

1

N ρ− 1

N

)where H(σ) = N(

∑Ni=1 1/σi)

−1, a harmonicaverage of the volatilities, and H(σ, ρ,N) ≥H(σ) is a modified harmonic average, thatis adjusted upwards by a function of corre-lation. This relation allows to infer whichstocks will be assigned to the long portfolioside. Volatility of each stock will be comparedto the threshold (represented by the modifiedharmonic average). The above formula coversalso the case of uncorrelated stocks, where thethreshold is shifted to infinity and all the stocks

8This situation is a very rough approximation of theequity market, where normally the correlations amongstocks are positive and significant.

are assigned to the long side. For another ex-treme case ρ→ 1, the group is divided exactlyby the harmonic average of volatilities. Whenthe correlation ρ is growing from one extreme(0) to the other (1), the optimizer is assigningmore and more stocks to the short portfoliostarting from the more volatile ones, until ittransfers there around 50% of the group (theexact percentage depends on where the har-monic average is situated).9

Another important piece of information onthe resulting allocation is the form of theweights assigned to the stocks. The weightswill be inversely proportional to the stocks’variance, and an exact formula is known in thiscase.10

The situation is more complex in the generalcase of non-equal pariwise correlations, and theoptimizer will add to the portfolio some stockswith consistent volatility that appear weaklycorrelated to the rest of the selected group.

The long-short allocation gives interestinginsights on the mechanics of the minimum vari-ance optimizer, but is hardly suitable for in-vestors. Note, that single stock weights arenot restricted and nothing prevents them fromassuming very high absolute values.

The next step is crucial to adopt the mini-mum variance process to the needs of a long-only investor.

Short sale constraint:

wi ≥ 0

9Harmonic average is always smaller that arithmeticaverage. The distance between the two is related tothe standard deviation of the sample as well as to theoutliers.

10The general solution in the matrix form is w =Σ−1 ~u~u′ Σ ~u , where ~u is a vector of ones. In the case of con-stant correlation the formula becomes: wi = 1

σMV[ 1σ2i−

ρ1+(N−1) ρ

1σi

∑j

1σj

] = 1σMV

1σ2i

[1 − σiH(σ,ρ,N)

], where

σMV is the portfolio variance.


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This is a constraint that all the existing ”pas-sive” minimum variance strategies have. Itsrole is critical, since this constraint forces theoptimizer to assign only non-negative positionsto stocks, and actually it dictates the optimizerto restrict the portfolio on the ”zero-weightboundary”, often by excluding a great amountof the initial universe from the portfolio.

In this case no close-form solution is avail-able for analysis in the constant correlationcase, but actually a modified version of theabove relation holds. The optimizer implicitlydefines a certain threshold (which is differentfrom the long-short case above), that decideswhether a certain stock will receive zero weightor not. The position of this threshold deter-mines how many stocks will be included in theportfolio.

One can imitate the optimization by pickingthe stocks one by one, starting from the leastvolatile ones, and comparing their volatility tothe modified harmonic average of the stocksalready admitted to the portfolio. If thenew stock has volatility below the threshold,it will be added to the portfolio, otherwise theprocedure stops and no new stocks are added.This works like a test on remaining diversifica-tion potential of adding more stocks, and thethreshold in this procedure is changing afteradding a new stock in the portfolio.

Below we give a graphical representation ofthe stock selection process:

This example takes a universe of 100 stockswith volatilities that correspond to recent 6-months volatilities of the 100 largest US stocks,and with constant pairwise correlations. Sev-eral moving threshold are constructed us-ing the modified harmonic average introducedabove, and assuming different correlation lev-els. To construct each moving threshold theformula is applied progressively to growing setsof stocks, sorted by their volatility in descend-ing order. All the stocks with volatility situ-ated below the threshold will be selected in the

optimized portfolio. The example shows thatfrom 10 to 20 stocks among the 100 largest USstocks will be selected by the optimizer, de-pending on correlation level. This means thatonly a small part of the initial group passesthe selection. This result is somewhat disap-pointing, since the portfolio is very concen-trated. It gets worse when one looks insidethis small portfolio and discovers that the threeleast volatile stocks got a combined weight of40% of the portfolio.

One of the problems leading to the ex-treme concentration is the presence of a cer-tain amount of estimation error. The mini-mum variance approach appears to be not sen-sitive to the overestimation of volatility (”tail”events), but instead suffers from the underes-timation of risk that happens for some stocks.Indeed, the harmonic average of volatilitiesthat appeared in above examples is sensitiveto the presence of very low values in the sam-ple. As we mentioned already, there are sev-eral methods to improve the situation. Oneconsists in applying various techniques that re-duce the estimation noise and improve the sta-bility of the estimated variance (e.g. shrinkingapproaches or variance estimation using factormodels). The other way out is to introducefurther constraints on portfolio weights in theoptimization problem.


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Additional Constraints

This is the point where additional con-straints are brought onto the stage to fix theconcentration problem and guarantee variousrisk management and allocation restrictionsthat an investor might have. The most im-portant examples are linear constraints:

1. limit the maximal weight of one stock,

2. ensure a certain minimum weight for eachstock in the universe,

3. limit single sector or country exposures,

4. constrain possible sector/country expo-sures around broad market exposures...

The exact combination of these additionalconstraints is what makes the difference amongvarious minimum variance indices and prod-ucts. Still, the mechanics of the optimizerremain largely untouched by the new con-straints. It was shown 11 that applicationof constraints can be translated into certaintransformations of the initial covariance ma-trix. In particular, an optimization problemwith no short sale constraint (wi ≥ 0) that usesa covariance matrix Σ is identical to the tra-ditional unconstrained optimization problemwith a shrinked covariance matrix.12 Thereare shrinkage transformations that correspondto other (linear) constraints, for example max-imal weight constraint13.

11see Jagannathan and Ma (2003).12 The new covariance matrix can be given as Σ̃ =

Σ−~λ ~u−~u~λ, where ~λ is the vector of Lagrange multipli-ers found in the constrained optimization. This shrink-age corresponds to reducing outliers in the matrix.

13see e.g. Roncalli, 2010

How to keep minimum varianceportfolio diversified

How do the weight bounds impact the port-folio efficiency, and do they allow to achievethe desired diversification? For example, toensure that a certain number of stocks will bepresent in the portfolio, one can set a maximalweight per stock wmax, such as 1/wmax ∼ K,where K is the minimal number of stocks thatwill be present in the portfolio. But from thepoint of view of covariance shrinkage, the im-pact of linear constraints on covariance matrixcan be quite substantial, since the resulting al-location could have a lot of weights that lie onthe wmax bound. This is somewhat worrying,given that the empirical covariance has predic-tive power of the future level of risk. In thisregard, portfolio optimization design that has asmooth impact on covariance matrix and pre-serves the relative riskiness of the stocks willbe preferable.

One very interesting solution comes from theidea to apply quadratic, instead of linear, con-straints on the optimal portfolio. The newconstraints were first proposed by DeMiguelet al (2009) under the name of 2-norm con-straints, and shown to have several nice prop-erties. Here we consider a quadratic equalityconstraint, that is formulated as:

~w′ ~w =∑i

w2i =

1

H(4)

The impact of this constraint on the covari-ance matrix is quite intuitive: it correspondsto a classical covariance shrinkage towards anidentity matrix. Indeed, adding the new con-straint to the objective function with a La-grange multiplier, one sees that the shrinkedcovariance matrix is simply

Σ̃ = Σ + λ Id (5)


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This shrinkage method is used in the James-Stein estimator14. The target shrinkage ma-trix - the identity matrix, is biased with re-spect to the ”true” covariance, but allows toefficiently reduce the estimation noise. Thedegree of shrinkage depends on the parameterλ, that comes from the solution of the con-strained optimization problem. The bigger thevalue of λ, the stronger is the impact of theconstraint. In the limit of λ→∞ the shrinkedcovariance goes towards the identity matrix,that implies a very simple solution for the op-timization problem: an equi-weighted portfo-lio (if compatible to all the additional con-straints in the problem). So, the higher theLagrange multiplier, the closer the solution isto the equal weights.

The degree of shrinkage will depend on thevalue of the constraint target appearing onthe right-hand side. One can note that theform of this constraint is familiar: the sum ofsquared weights corresponds to the Herfindahl-Hirschman index (HHI), that measures theconcentration of firms in the industries, to as-sess the degree of competition. The bigger thevalue of HHI, the higher is the concentration.This analogy suggests similar interpretation ofthe quadratic constraint in the optimizationproblem: it merely represents a diversificationtarget in terms of portfolio concentration. Itis very easy to induce a minimum number ofstocks that will be held in a portfolio that hasa certain diversification (HHI) level. If thediversification target is set, for example, at1H = 0.02, then the portfolio cannot containless than 1/0.02 = 50 stocks.15

This diversification target approach providesa way to ensure meaningful portfolio diversi-fication in a way that the covariance matrixstructure is modified smoothly. Integrating

14See James, Stein (1961).15It is easier to formulate the diversification target

directly in terms of number of stocks, i.e. specifyingH = 50 rather than 1

H= 0.02.

this constraint in the minimum variance op-timization problem, allows to design a system-atic strategy that has a desired diversificationlevel, without saturating a maximal weightconstraint.

Conclusion

We considered here the main ingredients ofthe minimum-variance equity investing. This”new passive” systematic approach does notrely on forecasts of stock returns and uses asinput only the covariance matrix. The portfo-lio resulting from minimum-variance optimiza-tion has the smallest ex-ante volatility, and ex-hibits a significant reduction in ex-post risk aswell, with respect to the market-capitalization-weighted benchmark.

The portfolio optimization problem behindthe minimum-variance portfolio is not uniquelydefined. Different constraints can be inte-grated into the optimization setup, that couldchange significantly the properties of the re-sulting optimal allocation. The most impor-tant constraints, such as short sale ban, canbe combined with restrictions on the maximalpossible weight for single stock, as well as dif-ferent constraints on groups of stocks (for ex-ample, on the sector level). The constraintshelp to control the composition of minimumvariance portfolio, setting up risk-managementlimits that ensure certain level of diversifica-tion of factor exposure. However the con-straints necessarily transform the informationcontained in the covariance matrix, and over-constrained configurations could have very fewinformation from the initial covariance matrixleft.

An alternative to the use of bounding linearconstraints is to ensure sufficient diversifica-tion via targeting a certain portfolio 2-norm,that is a sum of squared portfolio weights.This constraint ensures that a certain mini-


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mum number of stocks will be present in theportfolio, this minimum number of stocks be-ing just the inverse of the portfolio norm tar-get. This diversification target constraint in-duces a smooth transformation of initial covari-ance matrix, that corresponds to James-Steincovariance shrinkage with an identity matrix.

References

[1] J.P. Bouchaud and M.Potters, FinancialApplications of Random Matrix Theory: ashort review, arXiv q-fin 0910.1205, 2009.

[2] J.P. Bouchaud and M.Potters, Theory ofFinancial Risk and Derivative Pricing,Cambridge University Press, 2nd edition,2005.

[3] R. Cont, Empirical properties of asset re-turns: stylized facts and statistical issues,Quantitative Finance, vol.1, pp. 223-236,2001.

[4] V. DeMiguel, L. Garlappi, F.J. Nogales,and R. Uppal, A Generalized Approach toPortfolio Optimization: Improving Perfor-mance by Constraining Portfolio Norms,Management Science, vol.55, pp.798-812,2009.

[5] R. Jagannathan and T. Ma, Risk Reduc-tion in Large Portfolios: Why Imposing theWrong Constraints Helps, Journal of Fi-nance, vo.58, pp.1651-1684, 2003.

[6] W. James and C. Stein, Estimation withquadratic loss, Proceedings of 4th Berke-ley Symposium on Probability and Statis-tics, University of California Press, Berke-ley, pp.452-460, 1961.

[7] O. Ledoit and M. Wolf, Improved estima-tion of the covariance matrix of stock re-turns with an application to portfolio selec-

tion, Journal of Empirical Finance, vol.10,pp.603-621, 2003.

[8] H. Markowitz, Portfolio Selection, TheJournal of Finance, Vol.7, pp. 77-91, 1952.

[9] T. Roncalli, Understanding the Im-pact of Weights Constraints in Port-folio Theory, working paper at SSRN:http://ssrn.com/abstract=1761625, 2011.


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Mechanics of minimum variance investment approach · Minimum variance portfolios constructed via mathematical optimization allow to signi cantly reduce equity risk. Min-imum variance