Estimation of the Mean–Variance PortfolioModel

• In the mean–variance framework, the optimal port-folio weight vector, x⋆, is a function of the investor’spreference parameters, c, and the first two momentsof the return distribution, i.e., the mean vector µand the covariance matrix Σ, that is,

x⋆ = x(c,µ,Σ). (1)

• As µ and Σ are not directly observable, they haveto be estimated, using either subjective judgement(e.g., based on fundamental information) or histori-cal data (i.e., statistical methods).

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• If we plug-in the estimates of means and cova-riances, µ̂ and Σ̂, into our solution for the optimalweight (1), we get estimates of the optimal portfolioweights, x̂⋆,

x̂⋆ = x(c, Σ̂, µ̂).

• The estimation error of the parameter estimates istransferred to the portfolio weights.

• Therefore, estimated optimal weights are almostcertainly different from the true optimal weights.

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Sample Moments

• The most straightforward method to estimate µand Σ is simply to use the sample mean and thesample covariance matrix,

µ̂ =1

T

T∑t=1

Rt, Σ̂ =1

T − 1

T∑t=1

(Rt−µ̂)(Rt−µ̂)′,

where T is the number of (usually monthly) returnobservations for each asset.

• These estimators are rarely used in practice.

• Portfolio weights calculated on the basis of thesample moments have been documented to be quiteimprecise for realistic sample sizes.

• This is the case in particular if the number ofassets under consideration is large, relative to thenumber of historical return observations (a commonsituation in practice).

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• Concerning the volatility part, as the number ofassets increases, the number of distinct elements ofthe covariance matrix increases at a quadratic rate,as it has N(N + 1)/2 independent elements (apartfrom the requirement of positive definiteness).

• For example, with 500 assets, the covariance matrixinvolves 125250 unique elements.

• As optimal portfolio weights depend on all theseparameters, the portfolio weights are estimated witha lot of error.

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• Moreover, the portfolio weights tend to be rathersensitive to the inputs (expected returns and varian-ces/covariances).

• That is, small changes in the inputs may producerelatively large changes in the estimated optimalportfolio weights.

• Unfortunately, the most extreme (and thus “in-fluential”) estimated coefficients are likely to beextreme simply because they have large estimationerrors.

• Extreme (“unrealistic”) and unstable portfolioweights may be the result.

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• A potential remedy is to impose some structure onthe covariance matrix to reduce the large fractionof random noise in the sample moments.

• This gives rise to a more parsimonious para–metrization of the covariance matrix which, ho-pefully, helps to efficiently filter out the systematicinformation from historical correlations, and thusto reduce estimation error and improve forecastingperformance.

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• Candidate methods for improved (statistical) esti-mation of risk premiums include, for example,

– shrinkage estimation,– incorporation of economic models (as in the ap-

proach of Black and Littermann).

• To structure covariance matrices, factor models (orindex models) are frequently used.

• The factors may be observable

– observable (linear regression)– extracted directly from the returns using a sta-

tistical technique such as principal componentanalysis (PCA)

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Simulation Experiment (similar to Jobsonand Korkie (1980))

• Consider monthly returns of 24 stocks in the DAXover the period 1996–2001.

• Compute the sample mean and covariance matrixfrom these returns and take these estimates as the“true” values (to have a “realistic” example).

• Then simulate from the “true” return distribution,assuming iid normality.

• For each simulated sample, compute the samplemean and the sample covariance matrix, and com-pute the tangency portfolio using these quantities(assuming rf = 0).

• Then, using these estimated tangency portfolios, wecompute the associated Sharpe Ratio implied by thisweight vector and the true moments.

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• Realized Sharpe Ratios are on average considerablylower than the corresponding measure for the truetangency portfolio, indicating substantial economicloss due to estimation error.

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−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

35

40

45T = 36, i.e. 3 years of monthly data

Sharpe Ratio (rf = 0)

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

35

40

45T = 60, i.e. 5 years of monthly data

Sharpe Ratio (rf = 0)

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

30

35

40

45T = 120, i.e. 10 years of monthly data

Sharpe Ratio (rf = 0)

true

true

true

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Factor (Index) Models

• The first and simplest index model is the single–index model (SIM) proposed by Sharpe (1963).

• This is based on the observation that, when themarkets go up (bull markets), most stocks tend toincrease in price, and when markets go down (bearmarkets), most stocks tend to decrease in price.

• Thus, there seems to be a strong common (market)factor driving the joint movement of asset returns.

• That is, one important reason why asset returnsare correlated is because of a common response tomarket changes.

• In the SIM, it is assumed that the only reason whystocks vary together, systematically, is because oftheir common comovement with the market.

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• The return of asset i, i = 1, . . . , N , is described by

Ri = αi + βiRM + ϵi, i = 1, . . . , N ;(2)

E(ϵi) = 0, Var(ϵi) = σ2ϵi, i = 1, . . . , N ;(3)

Cov(ϵi, ϵj) = 0, i ̸= j. (4)

• Expected return and variance of the market returnwill be denoted by E(RM) = µM and Var(RM) =σ2M , and we assume

Cov(RM , ϵi) = 0, i = 1, . . . , N. (5)

• This structure implies that

E(Ri) = αi + βiµM , i =, . . . , N, (6)

Var(Ri) = β2i σ

2M + σ2

ϵi, i = 1, . . . , N, (7)

Cov(Ri, rj) = βiβjσ2M , i, j = 1, . . . , N, i ̸= j.(8)

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• For the covariance structure of the returns, given by(8), Assumption (4) is crucial, as it implies that theonly reason for asset i and asset j moving togetheris their joint dependence on the market return rMt.

• The first part of (7) is also often referred to asthe systematic risk (which is related to the generaltendency of the market), whereas the second part isthe unsystematic (idiosyncratic, specific) risk, whichis not related to market factors.

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• In contrast to the market–related, systematic risk,the specific risk can be diversified away.

• Consider an equally, weighted portfolio, i.e., a port-folio with weights xi = 1/N , i = 1, . . . , N .

• Then the portfolio variance is, assuming the SIMcorrectly describes the covariance structure,

σ2p =

1

N2

N∑i=1

(β2i σ

2M + σ2

ϵi) +

1

N2

N∑i=1

∑j ̸=i

βiβjσ2M

=

1

N2

N∑i=1

N∑j=1

βiβj

σ2M +

1

N2

N∑i=1

σ2ϵi

=

(1

N

N∑i=1

βi

)2

σ2M +

1

N2

N∑i=1

σ2ϵi.

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• Now

1

N2

N∑i=1

σ2ϵi≤

max{σ2ϵ1, . . . , σ2

ϵN}

N

N→∞−→ 0,

provided the variances of the unsystematic risks arebounded.

• Hence, for large N ,

σ2pt ≈

(1

N

N∑i=1

βi

)2

σ2M = β

2

pσ2M ,

where

βp =1

N

N∑i=1

βi

is the portfolio’s β.

• That is, the market risk cannot be diversified away.

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Multi-Index Models

• The single–index model is straightforwardly exten-ded to k factors.

• In this case, with k factors fi, i = 1, . . . , k,

Ri = αi +

k∑j=1

βijfi + ϵi, i = 1, . . . , N,

or, in obvious notation,

r = α+Bf + ϵ.

The covariance matrix of returns is

Σ = BΣfB′ +Σϵ,

where Σf is the covariance matrix of the factors,which may or may not be diagonal.

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• For example, in industry index models, we start withthe basic SIM and add industry indices to captureindustry effects.

• E.g., two steel stocks will most likely have positivecorrelation between their returns, even after theeffects of the market have been removed.

• To avoid noisy estimates, it may be reasonable thatreturns of each firm are affected only by the marketand one industry factor (in contrast to all industryfactors).

• For firm i in industry j, we then have

Ri = αi + βimIm + βijIj + ϵi, (9)

where Im and Ij are the market and industry factors,respectively.

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• Assuming for the moment that the factors are un-correlated (Σf is diagonal), the covariance betweensecurities i and ℓ is then

σiℓ = βimβℓmσ2m + βijβℓjσ

2Ij

for firms in the same industry j, and

σiℓ = βimβℓmσ2m

for firms in different industries.

• macroeconomic variables, country–specific variables

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• In contrast to the sample covariance matrix, thematrix provided by an index model will usually benonsigular.

• For example, when N = 1000, we would need morethan 1000 observations on past return vectors toget a nonsingular sample covariance matrix.

• So data requirements are also reduced when indexmodels are used.

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Common Correlation Models (CCMs)

• Such models have been proposed by Elton andGruber (1973).

• They are extremely simple but tend to outperformthe historical correlation matrix.

• The idea is that “historical data only contain infor-mation concerning the mean correlation coefficientsand that observed pairwise differences from theaverage are random or sufficiently unstable“ (EG,1973).

• Thus, the best way to forecast future correlations isto use the average of the observed historical samplecorrelation coefficients.

• In the simplest case, this reduces the number of pa-rameters to be estimated for the correlation matrixfrom N(N − 1)/2 to one.

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More Sophisticated CCMs

• The idea of a common correlation between all pairsof assets may be too restrictive, and grouping tech-niques will be more appropriate (e.g., common cor-relation among stocks from the same industry).

• For example, if there were two industries, steelsand chemicals, we may assume that the correlationbetween all steels is constant (ρSS) and the corre-lation between all chemicals is constant (ρCC), butpotentially different from ρSS.

• Also, the correlation between members of the steelgroup and those of the chemical group is constant(ρCS).

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• In international portfolio diversification, the Natio-nal Mean Model has been shown to be potentiallyuseful, where every intra–country pairwise correlati-on is constant, and every correlation between assetsof two countries, say A and B, is equal to ρAB.

• This was first observed by Eun and Resnick (1984).

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