Statistical inference of the efficient frontier for dependent asset returns

Stat Papers (2009) 50:593–604DOI 10.1007/s00362-007-0108-x

REGULAR ARTICLE

Statistical inference of the efficient frontierfor dependent asset returns

Taras Bodnar · Wolfgang Schmid ·Taras Zabolotskyy

Received: 25 February 2007 / Revised: 8 November 2007 / Published online: 12 December 2007© Springer-Verlag 2007

Abstract In the paper we consider the three characteristics of the efficient frontier.These characteristics are estimated by substituting the unknown parameters by the sam-ple counterparts. Assuming that the asset returns follow a stationary Gaussian processit is shown that the estimated characteristics are asymptotically normally distributed.This result is used to determine the joint asymptotic distribution of the estimated port-folio return and the estimated portfolio variance in the case of the expected utilityportfolio and the tangency portfolio, respectively.

Keywords Asset allocation · Mean–variance efficient frontier · Optimal portfolios ·Asymptotic normality

1 Introduction

Since the seminal paper of Markowitz (1952) the mean–variance analysis has becomean important tool for both practitioners and researchers in finance. This theory is basedon the consideration of the portfolio with the smallest risk for a given level of theexpected portfolio return. It was shown by Merton (1972) that all these portfolios arelying on a parabola in the mean–variance space which is known as the efficient frontier.Its location in the mean–variance space is uniquely determined by three characteristics.These quantities depend on the first two moments of the asset returns process.

T. Bodnar (B) · W. Schmid · T. ZabolotskyyEuropa-Universität Viadrina, GroßeScharrnstraße 59, 15230 Frankfurt (Oder), Germanye-mail: [email protected]

W. Schmide-mail: [email protected]

T. Zabolotskyye-mail: [email protected]

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594 T. Bodnar et al.

Because these parameters are unknown in practice they have to be estimated.Substituting the parameters by suitable estimators we obtain estimators of the charac-teristics of the efficient frontier and the portfolio weights. Recently, several papers dealwith the statistical properties of the estimated portfolio weights. Jobson and Korkie(1980) derived asymptotic formulas for the moments of the estimated weights of thetangency portfolio. Britten-Jones (1999) presented a finite sample test for the weightsof this portfolio. Kempf and Memmel (2006) calculated a conditional distribution ofthe estimated weights of the global minimum variance portfolio. Okhrin and Schmid(2006) analyzed the distributional behavior of the sample portfolio weights for sev-eral optimality criteria in the case of a finite and infinite sample size. Moreover, theyshowed that the sample weights of the Sharpe portfolio do not possess a first momentat all. Schmid and Zabolotskyy (2007) extended this result. They proved that theredoes not exist an unbiased estimator for the weights of the Sharpe optimal portfolio.While in most papers the parameters of the return process are estimated by the samplecounterparts, Okhrin and Schmid (2007) provided an overview and a comparison ofseveral other estimation procedures.

Another possibility to characterize a portfolio is to consider not directly the weightsbut the expected return and the variance of the portfolio. Using the sampleestimators, Bodnar and Schmid (2007) derived the exact densities of the estimatedexpected return and the estimated variance of the portfolios obtained by maximizing theexpected quadratic utility function and the tangency portfolio, respectively. Theirresults directly lead to distributional statements about the estimated three character-istics of the efficient frontier. This result is a generalization of the findings of Jobsonand Korkie (1980), who calculated the first two moments of the estimated character-istics.

In the above cited papers it is mainly assumed that the asset returns areindependent and normally distributed. These assumptions are very restrictive andthey have been heavily criticized in financial literature. Asset returns are usually notindependent (see, e.g., Engle 1982; Conrad and Kaul 1988; Lo and MacKinley 1991;Mech 1993). Up to now there are only a few papers dealing with the estimated port-folio weights for dependent asset returns. Okhrin and Schmid (2006) derived theasymptotic distribution of the estimated portfolio weights for several optimality cri-teria assuming that the asset returns follow a stationary process. In this paper wefocus on another important problem. We calculate the joint asymptotic distributionof the estimated characteristics of the efficient frontier. Moreover, we determine thejoint asymptotic distribution of the estimated expected portfolio return and the esti-mated portfolio variance for the expected utility portfolio and the tangency portfolioas well.

The paper is organized as follows. In the next section we give a brief introductioninto portfolio theory and present basic formulas for the characteristics of the efficientfrontier and the portfolio weights. Our main results about the asymptotic behaviorare presented in Sect. 3. In Sect. 4 we consider the special case of a vector autore-gressive moving average process of order (1,1). A short summary is presented inSect. 5. The paper ends with an Appendix where the proofs of the obtained results aregiven.

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Statistical inference of the efficient frontier for dependent asset returns 595

2 Optimal portfolios and the efficient frontier

Let Xt denote the k-dimensional vector of asset returns taken at time point t . Inthe following it is assumed that {Xt } follows a k-dimensional stationary process. Itsmean vector is denoted by µ. �(h) = Cov(Xt+h, Xt ) = (γi j (h)) stands for thecross-covariance matrix at lag h. Let �(0) be positive definite.

Depending on the investor’s preferences (utility function, attitude towards the risk,etc.) different optimal portfolios can be constructed. The relative fraction of the i thasset in the portfolio is denoted by wi . Let w = (w1, . . . , wk)

′ and 1′w = 1, where 1denotes the k-dimensional vector of ones. This means that the whole wealth is sharedbetween the selected assets. We consider the possibility of short-selling by allowingthe weights to take negative values.

In the following we distinguish between four types of optimal portfolios. The firstone is based on the weights that maximize the expected quadratic utility function. Theweights of this portfolio are given by

wEU = �(0)−111′�(0)−11

+ α−1 Rµ with R = �(0)−1 − �(0)−111′�(0)−1

1′�(0)−11. (1)

α is the coefficient of the risk aversion. We refer to the portfolio with the weights (1)as the EU portfolio. The expected return and the variance of the EU portfolio are equalto

REU = µ′�(0)−111′�(0)−11

+ α−1µ′ Rµ and VEU = 1

1′�(0)−11+ α−2µ′ Rµ. (2)

The second optimal portfolio considered in the paper is known as the tangencyportfolio. The weights of the portfolio are obtained by maximizing w′(µ − r f 1) −αw′�(0)w subject to 1′w = 1 assuming that the investor’s wealth is shared betweenthe risky assets. Here r f is a risk-free rate and µ − r f 1 is a vector of excess returns.The expected return and the variance are given by

RT =r f + (µ − r f 1)′�(0)−1(µ − r f 1)

1′�(0)−1(µ − r f 1)and VT = (µ − r f 1)′�(0)−1(µ − r f 1)

(1′�(0)−1(µ − r f 1))2 .

(3)

The portfolio has been discussed for example by Ingersoll (1987, p. 87) andBritten-Jones (1999). For r f = 0 the tangency portfolio transforms to the optimalportfolio with the highest Sharpe ratio (see Cochrane 1999; MacKinley and Pastor2000; Lo 2002). We refer to this portfolio as the SR portfolio.

Another approach is to consider not only the optimal portfolios but the wholeefficient frontier of optimal portfolios. Merton (1972) showed that in order to determinethe efficient frontier, which appears to be an upper part of a parabola in the mean–variance space, the analyst has to calculate the following three parameters

a = µ′�(0)−1µ, b = 1′�(0)−1µ, and c = 1′�(0)−11. (4)

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Note that the Eq. (2) is nothing else as the parametric equation of the efficient frontierin case of no risky asset. Solving the equation we obtain

(R − RGMV)2 = µ′ Rµ(V − VGMV), (5)

where

RGMV = µ′�(0)−111′�(0)−11

and VGMV = 1

1′�(0)−11(6)

are the expected return and the variance of the global minimum variance portfo-lio (GMVP). The GMVP is the portfolio with the smallest variance. Its weights areobtained from the weights of the EU portfolio assuming a fully risk averse investor(α = ∞), i.e.

wGMV = �(0)−111′�(0)−11

. (7)

There is one-to-one transformation between the two parameter sets (4) and {RGMV,

VGMV, s} with s = µ′ Rµ being the slope coefficient of the parabola. Because the latterset permits a better financial interpretation we use it as the parameters of the efficientfrontier in the mean–variance space.

3 Asymptotic theory for the estimators of the optimal portfolio characteristics

Usually in practice µ and �(0) are unknown, and have to be estimated based onprevious data (see, e.g., Stambaugh 1997; Bodnar and Schmid 2007). Because ofthis the investor cannot determine his investment policy without estimating the meanvector and the covariance matrix of the asset returns. We make use of the followingestimators

µ = 1

n

n∑

j=1

X j and �(0) = 1

n − 1

n∑

j=1

(X j − µ)(X j − µ)′. (8)

of µ and�(0). Then the characteristics of the efficient frontier and the above consideredoptimal portfolios are estimated by substituting the unknown parameters µ and �(0)

by the estimators µ and �(0). For example, the estimator for the slope coefficient ofthe parabola is

s = µ′ Rµ with R = �(0)−1 − �(0)−111′�(0)−1

1′�(0)−11. (9)

In order to distinguish between the parameters and the estimators all estimators aremarked by the symbol ’.’. This leads to REU, VEU, RGMV, VGMV, RT , VT , etc.

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Bodnar and Schmid (2007) derived the finite sample densities of the estimators forthe expected returns and the variances of the optimal portfolios considered in Sect. 2assuming the asset returns to be independently normally distributed. The formulas ofthe densities are based on special mathematical functions, like the confluent hyperge-ometric function, the Bessel function, etc. The computations of the densities can beeasily carried out using standard mathematical software packages.

Because the independency assumption of the asset returns is rather restrictive andit is critically discussed in financial literature it is desirable to have results for moregeneral models. Here we derive the asymptotic distributions of the estimators if theasset returns follow a stationary Gaussian process.

Let θ = (µ′, vech(�(0))′)′ with vech(A) = (a11,. . ., ak1,. . ., aii ,. . ., aki ,. . ., akk)′

for an arbitrary symmetric matrix A = (ai j ). In the following we make use of theoperator vec defined by vec(A) = (a11, . . . , ak1, . . . , a1i , . . . , aki , . . . , akk)

′. For theproperties of the operators vech and vec we refer to Harville (1997). Moreover, letDk be a k2 × k(k + 1)/2 duplication matrix such that Dkvech(A) = vec(A) andD+

k = (D′kDk)

−1D′k with the property D+

k vec(A) = vech(A).It has been shown by several authors under various conditions that the estimator

θ = (µ′, vech(�(0))′)′ of θ is asymptotically normally distributed with mean vector

θ and covariance matrix � (see, e.g., Hannan 1970; Brockwell and Davis 1991). Fora stationary Gaussian process it follows that

� =(∑+∞

h=−∞ �(h) 0k×k(k+1)/2

0k(k+1)/2×k D+k (Ik2 + Kk)

(∑+∞h=−∞(�(h) ⊗ �(h))

)D+ ′

k

)(10)

provided that the both series converge. Ik2 is the identity matrix of order k2 and Kk isa commutation matrix (Harville 1997). The symbol ⊗ denotes the Kronecker product.

The estimators for the characteristics of the optimal portfolios presented in Sect. 2can be considered as continuous differentiable functions depending on θ . Let g(θ) =(g1(θ), g2(θ), g3(θ))′ = (RGMV, VGMV, s)′. The application of Proposition 6.4.3 ofBrockwell and Davis (1991) leads to

√n(g(θ) − g(θ))

d→ N (0, G′�G), (11)

where G is a k(k + 3)/2 × 3-dimensional vector with G = (∂gi/∂θ j ). Note that thepartial derivatives ∂gi/∂θ j can be determined using the rules for matrix differentiation(see Magnus and Neudecker 1999).

Using this approach the joint asymptotic density of the estimators for the parametersof the efficient frontier can be calculated. Our result is summarized in Theorem 1. Theproof of Theorem 1 is given in the Appendix.

Theorem 1 Let {Xt } be a stationary Gaussian process with mean µ andcross-covariance matrix �(h) at lag h. Let �(0) be positive definite. We assume that

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the infinite sums in (10) converge. Let n > k. Then it follows that

√n

⎛

⎝

⎛

⎝RGMV

VGMVs

⎞

⎠ −⎛

⎝RGMVVGMV

s

⎞

⎠

⎞

⎠ d→ N⎛

⎝

⎛

⎝000

⎞

⎠ ,

⎛

⎝σ 2

1 σ12 σ13

σ12 σ 22 σ23

σ13 σ23 σ 23

⎞

⎠

⎞

⎠ ,

where

σ 21 = V 2

GMV

+∞∑

h=−∞

(qh(1, 1) + 2R2

GMVqh(1, 1)2 + qh(1, 1)qh(µ,µ)

+ qh(1,µ)2 − 4RGMVqh(1, 1)qh(1,µ)),

σ 22 = 2V 4

GMV

+∞∑

h=−∞qh(1, 1)2,

σ 23 =

+∞∑

h=−∞

(4µ′ R�(h)Rµ + 2R4

GMVqh(1, 1)2 + 2qh(µ,µ)2

+ 4R2GMV

(qh(1, 1)qh(µ,µ) + 2qh(1,µ)2

)− 8R3

GMVqh(1, 1)qh(1,µ)

− 8RGMVqh(1,µ)qh(µ,µ)),

σ12 = 2V 3GMV

+∞∑

h=−∞

(RGMVqh(1, 1)2 − qh(1, 1)qh(1,µ)

),

σ13 =+∞∑

h=−∞

(2VGMV1′�(0)−1�(h)Rµ − 2R3

GMVVGMVqh(1, 1)2

− 2RGMVVGMV(qh(1, 1)qh(µ,µ) + 2qh(1,µ)2)

+ 6R2GMVVGMVqh(1, 1)qh(1,µ) + 2VGMVqh(1,µ)qh(µ,µ)

),

σ23 =−V 2GMV

+∞∑

h=−∞

(2R2

GMVqh(1, 1)2−4RGMVqh(1, 1)qh(1,µ)+2qh(1,µ)2)

and qh(c1, c2) = c′1�(0)−1�(h)�(0)−1c2 for some vectors c1 and c2.

In case of stationary Gaussian processes with �(h) = dh�(0) the asymptoticcovariance matrix in Theorem 1 is a diagonal matrix. This result is immediatelyobtained from the above representations. It includes all stationary vector autoregres-sive moving average processes whose coefficient matrices are diagonal matrices withthe same diagonal elements.

Our next aim is to derive the joint asymptotic distribution of the estimated expectedreturn and the estimated variance of the EU portfolio and the tangency portfolio,respectively. Because (REU, VEU) and (RT , VT ) are functions of the characteristics of

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the efficient frontier, i.e. RGMV, VGMV, and s, such results can be obtained by applyingTheorem 1.

Now let

a11 = 1 − sVGMV

(RGMV − r f )2 , a12 = s

(RGMV − r f ), a13 = VGMV

(RGMV − r f ),

a21 = −2V 2GMVs

(RGMV − r f )3 , a22 = 1 + 2VGMVs

(RGMV − r f )2 , a23 = V 2GMV

(RGMV − r f )2 .

Corollary 1 Let {Xt } satisfy the conditions of Theorem 1 and let n > k. Then itfollows that

(a)

√n

((REU

VEU

)−

(REUVEU

))d→ N

((00

),

(σ 2

EU;1 σEU;12

σEU;12 σ 2EU;2

)),

where

σ 2EU;1 = σ 2

1 + 2α−1σ13 + α−2σ 23 , σ 2

EU;2 = σ 22 + 2α−2σ23 + α−4σ 2

3 ,

σEU;12 = σ12 + α−1σ23 + α−2σ13 + α−3σ 23 .

(b)

√n

((RT

VT

)−

(RT

VT

))d→ N

((00

),

(σ 2

T ;1 σT ;12

σT ;12 σ 2T ;2

)),

where

σ 2T ;1 = a2

11σ21 + a2

12σ22 + a2

13σ23 + 2a11a12σ12 + 2a11a13σ13 + 2a12a13σ23,

σ 2T ;2 = a2

21σ21 + a2

22σ22 + a2

23σ23 + 2a21a22σ12 + 2a21a23σ13 + 2a22a23σ23,

σT ;12 = a11a21σ21 + a12a22σ

22 + a13a23σ

23 + (a11a22 + a12a21)σ12

+ (a11a23 + a13a21)σ13 + (a12a23 + a13a22)σ23.

4 Examples

In this section we consider a special case of a stationary Gaussian process. Wedetermine the asymptotic distributions given in Theorem 1 and Corollary 1 for a vectorautoregressive moving average (VARMA) process of order (1,1) which is given by

Xt − µ = �(Xt−1 − µ) + εt − �εt−1.

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The vectors {εt } are assumed to be independent and normally distributed with mean0 and covariance matrix �. Moreover, we assume that �1 = φI and �1 = θI.

In this case it holds that (see, e.g., Reinsel 1997, p. 36)

�(h) = φh−1η�(0), h = 1, 2, . . . and η = (φ − θ)(1 − φθ)

1 + θ2 − 2φθ.

Applying Theorem 1 we get

σ 21 = VGMV(1 + s) + 2ηVGMV

1 + ηs + φ

1 − φ2 , σ 22 = 2V 2

GMV

(1 + 2η2

1 − φ2

),

σ 23 = 4s + 2s2 + 2η

(4s

1 − φ+ 2ηs2

1 − φ2

),

and σ12 = σ13 = σ23 = 0.Although the asset returns are no longer independently distributed the estimator

for the global minimum variance, the estimator for the expected return of the globalminimum variance portfolio, and the estimator for the slope parameter of the parabolaare asymptotically independent.

From Corollary 1 we obtain

σ 2EU;1 = σ 2

1 + α−2σ 23 , σ 2

EU;2 = α−3σ 23 , σEU;12 = σ 2

2 + α−4σ 23 ,

σ 2T ;1 = a2

11σ21 + a2

12σ22 + a2

13σ23 , σ 2

T ;2 = a221σ

21 + a2

22σ22 + a2

23σ23 ,

σT ;12 = a11a21σ21 + a12a22σ

22 + a13a23σ

23 .

Next the derived asymptotic densities are compared with the exact densities givenin Bodnar and Schmid (2007). We consider the estimators for the expected returnand the variance of the EU portfolio. Assuming {Xt } to be independent and normallydistributed, Bodnar and Schmid (2007) proved that the exact densities of REU and VEUare given by

f REU(x) = n(n − k + 1)

(k − 1)(n − 1)

∞∫

0

n

(RGMV + α−1 y,

1 + nn−1 y

nVGMV

)(x)

× fk−1,n−k+1,n s

(n(n − k + 1)

(n − 1)(k − 1)y

),

fVEU(x) = n(n − k + 1)

(k − 1)VGMV

α2x∫

0

gn−k((n − 1)(x − y/α2)/VGMV)

× fk−1,n−k+1,n s

(n(n − k + 1)

(n − 1)(k − 1)y

)dy.

Here fm1,m2,λ(.) denotes the density of the noncentral F-distribution with m1 and m2degrees of freedom and noncentrality parameter λ. The symbols gm(.) and n(µ, σ 2)(.)

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-2 -1 1 2x

0.2

0.4

0.6

0.8

f(x)

Fig. 1 Exact (n = 120 solid black line, n = 240 solid gray line) and asymptotic densities of√

n(REU −REU). For the calculation the annualized asset returns are used

stand for the densities of the χ2-distribution with m degrees of freedom and theunivariate normal distribution with mean µ and variance σ 2.

In order to compare these density functions with the asymptotic densities obtainedin Corollary 1 the three parameters of the efficient frontier have to be defined. We put

RGMV = 0.0145664, VGMV = 0.0010165, and s = 0.227957. (12)

These quantities are the estimates of the corresponding parameters. They are obtainedby using monthly data from Morgan Stanley Capital International for the equity mar-kets returns of five developed countries (UK, Germany, USA, Canada, and Switzer-land) for the period from July 1994 to June 1999. The coefficient of risk aversion istaken equal to 5, i.e. α = 5.

In Fig. 1 the asymptotic density of√

n(REU − REU) is compared with the exactdensities for different sample sizes (n = 120—solid black line, n = 240—solid grayline). The results for

√n(VEU − VEU) are given in Fig. 2. In both cases the annualized

expected return and the annualized variance are considered. We observe an accurateapproximation for both sample sizes.

5 Summary

Applying the theory of Markowitz the investor is faced with the problem that theparameters of the asset returns process are unknown and have to be estimated. Recentlyseveral papers studied the problem in more details. In the present paper we use thesample mean and the sample covariance matrix to estimate the parameters. Using theseestimators we derive the joint asymptotic distribution of the estimated characteristicsof the efficient frontier assuming that the asset returns follow a stationary Gaussianprocess. This result is used to derive the joint asymptotic distribution of the estimatedportfolio return and the estimated portfolio variance in the case of the EU portfolioand the tangency portfolio.

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-0.9 -0.6 -0.3 0.3 0.6 0.9x

0.5

1

1.5

2f(x)

Fig. 2 Exact (n = 120 solid black line, n = 240 solid gray line) and asymptotic densities of√

n(VEU −VEU). For the calculation the annualized asset returns are used

Appendix

In this section the proofs of the theorems are given. We make use of the followingresult from Harville (1997, p. 368)

∂(vec(�(0)−1))′

∂vech�(0)= −D′

k(�(0)−1 ⊗ �(0)−1)D+k

′D′

k . (13)

The next properties of the duplication and commutation matrices are important forthe proofs of Theorems:

(i) DkD+k = 1

2 (Ik2 + Kk),(ii) Kk = K′

k = K−1k ,

(iii) Kk(A ⊗ a) = (a ⊗ A) for an arbitrary k × n matrix A and an arbitrary k × 1vector a.

Proof of Theorem 1 (a) First, we derive the formula for σ 21 . It follows that G′

1 =((∂g1(θ)/∂µ)′, (∂g1(θ)/∂vech(�(0)))′)=((∂ RGMV/∂µ)′, (∂ RGMV/∂vech(�(0)))′),where

∂ RGMV

∂µ= ∂(µ′�(0)−11)

∂µ

1

1′�(0)−11= �(0)−11

1′�(0)−11

and

∂ RGMV

∂vech(�(0))= ∂(µ′�(0)−11)

∂vech�(0)

1

1′�(0)−11− ∂(1′�(0)−11)

∂vech�(0)

µ′�(0)−11(1′�(0)−11)2

= ∂(vec�(0)−1)′

∂vech�(0)(VGMV(1 ⊗ µ) − RGMVVGMV(1 ⊗ 1)).

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The application of the formula (10) leads to

G′1�G1 = S11 + S12,

where

S11 =(

∂ REU

∂µ

)′ ( +∞∑

h=−∞�(h)

)(∂ REU

∂µ

)

=(

�(0)−111′�(0)−11

)′ ( +∞∑

h=−∞�(h)

)(�(0)−11

1′�(0)−11

)

= V 2GMV

+∞∑

h=−∞1′�(0)−1�(h)�(0)−11.

From (13) we get

S12 =(

∂VEU

∂vech(�(0))

)′ ( +∞∑

h=−∞(�(h) ⊗ �(h))

)(∂VEU

∂vech(�(0))

)

= (VGMV(1 ⊗ µ) − RGMVVGMV(1 ⊗ 1))′

×( − D′k

(�(0)−1 ⊗ �(0)−1)D+

k′D′

k

)′D+k (Ik2 + Kk)

×( +∞∑

h=−∞(�(h) ⊗ �(h))

)D+ ′

k

( − D′k

(�(0)−1 ⊗ �(0)−1)D+

k′D′

k

)

× (VGMV(1 ⊗ µ) − RGMVVGMV(1 ⊗ 1))

= (VGMV(1′ ⊗ µ′) − RGMVVGMV(1′ ⊗ 1′)

)

×(Ik2 + Kk)

( +∞∑

h=−∞

(�(0)−1�(h)�(0)−1 ⊗ �(0)−1�(h)�(0)−1)

)

× (VGMV(1 ⊗ µ) − RGMVVGMV(1 ⊗ 1)),

where we use the properties (i) and (iii) and the fact that (Ik2 + Kk)2 = 2(Ik2 + Kk).

The application of the property (iii) yields

S12 = V 2GMV

+∞∑

h=−∞

(2R2

GMV

(1′�(0)−1�(h)�(0)−11

)2

+((

1′�(0)−1�(h)�(0)−11)(

µ′�(0)−1�(h)�(0)−1µ)

+ (1′�(0)−1�(h)�(0)−1µ)2)

− 4RGMV(1′�(0)−1�(h)�(0)−11)(1′�(0)−1�(h)�(0)−1µ).

Putting the formulas for S11 and S12 together the result of Theorem 1a follows.

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For calculating of σ 22 , σ 2

3 , σ12, σ13, and σ23 we use the following results

∂VEU

∂µ= 0,

∂VEU

∂vech(�(0))=−∂(1′�(0)−11)

∂vech�(0)

1

(1′�(0)−11)2 = ∂(vec�(0)−1)′

∂vech�(0)

−(1 ⊗ 1)

(1′�(0)−11)2 ,

∂s

∂µ= ∂(µ′ Rµ)

∂µ= 2Rµ,

∂s

∂vech(�(0))= −∂(vec�(0)−1)′

∂vech�(0)

((µ′�(0)−11)2

(1′�(0)−11)2 (1 ⊗ 1)

− 2µ′�(0)−111′�(0)−11

(1 ⊗ µ) + (µ ⊗ µ)

).

References

Bodnar T, Schmid W (2007) Mean–variance portfolio analysis under parameter uncertainty (submitted forpublication)

Britten-Jones M (1999) The sampling error in estimates of mean–variance efficient portfolio weights.J Finance 54:655–671

Brockwell PJ, Davis RA (1991) Time series: theory and methods. Springer, New YorkCochrane JH (1999) Portfolio advice for a multifactor world. NBER working paper 7170Conrad J, Kaul G (1988) Time-variation in expected returns. J Bus 61:409–425Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K.

inflation. Econometrica 50:987–1008Hannan EJ (1970) Multiple time series. Wiley, New YorkHarville DA (1997) Matrix algebra from a statistician’s perspective. Springer, New YorkIngersoll JE (1987) Theory of financial decision making. Rowman & Littlefield Publishers, SavageJobson JD, Korkie B (1980) Estimation of Markowitz efficient portfolios. J Am Stat Assoc 75:544–554Kempf A, Memmel C (2006) Estimating the global minimum variance portfolio. Schmalenbach Bus Rev

58:332–348Lo AW (2002) The statistics of Sharpe ratio. Financ Anal J 58:36–52Lo A, MacKinley AC (1991) An econometric analysis of non-synchronous trading. J Econom 45:181–212MacKinley AC, Pastor L (2000) Asset pricing models: implications for expected returns and portfolio

selection. Rev Financ Stud 13:883–916Magnus JR, Neudecker H (1999) Matrix differential calculus with applications in statistics and economet-

rics. Wiley, New YorkMarkowitz H (1952) Portfolio selection. J Finance 7:77–91Mech T (1993) Portfolio return autocorrelation. J Financ Econ 34:307–344Merton RC (1972) An analytical derivation of the efficient frontier. J Financ Quant Anal 7:1851–1872Okhrin Y, Schmid W (2006) Distributional properties of optimal portfolio weights. J Econom 134:235–256Okhrin Y, Schmid W (2007) Comparison of different estimation techniques for portfolio selection. Adv

Stat Anal 91:109–127Reinsel GC (1997) Elements of multivariate time series analysis. Springer, New YorkSchmid W, Zabolotskyy T (2007) On the existence of unbiased estimators for the portfolio weights obtained

by maximizing the Sharpe ratio (to appear in Adv Stat Anal)Stambaugh RF (1997) Analyzing investments whose histories differ in length. J Financ Econ 45:285–331

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Statistical inference of the efficient frontier for dependent asset returns