Higher order moments of the estimated tangency portfolio weights

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1 WORKING PAPER 0/07 Higher order moments of the estimated tangency portfolio weights Farrukh Javed, Stepan Mazur, Edward Ngailo Statistics ISSN Örebro University School of Business 70 8 Örebro SWEDEN

2 Higher order moments of the estimated tangency portfolio weights Farrukh Javed a, Stepan Mazur a,, Edward Ngailo b,c a Department of Statistics, Örebro University School of Business, SE-708 Örebro, Sweden b Department of Mathematics, Stockholm University, SE-069 Stockholm, Sweden c Department of Mathematics, Dar es salaam University College of Education, Tanzania Abstract In this paper we consider the estimated weights of tangency portfolio. The returns are assumed to be independently and multivariate normally distributed. We derive analytical expressions for the higher order non-central and central moments of these weights. Moreover, the expressions for mean, variance, skewness and kurtosis of the estimated weights are obtained in closed-forms. Finally, we complement our result with an empirical study where we analyze a portfolio with actual returns of eight financial indexes listed in NASDAQ stock exchange. JEL Classification: C0, C44 ASM Classification: 6H99, 9G0 Keywords: Tangency portfolio, higher order moments, Wishart distribution Corresponding author. address: stepan.mazur@oru.se.

3 Introduction The fundamental goal of the portfolio theory as devised by Markowitz 95) is to determine efficient way of portfolio allocation. The mean-variance optimization technique plays a central role in allocating the investments among different asset. According to it, the investor allocates the wealth among risky assets by maximizing the expected return based on a given level of risk or by minimizing the risk for a given level of expected returns. The trade-off between the risk and return of the portfolio is at the heart of portfolio theory, which seeks to find optimal allocations of the investor s initial wealth to the available assets. The tangency portfolio is the portfolio that consists of both risky and risk-free assets. In order to have entire understanding of the conditions and processes in a portfolio, the study on statistical properties of the tangency portfolio is crucial and unavoidable. Statistical properties of the estimated tangency portfolio weights are intensively discussed in the literature. Britten-Jones 999) developed an F -test for the tangency portfolio weights, while Bodnar 009) delivered sequential monitoring procedures for the tangency portfolio weights. The univariate density of the tangency portfolio as well as its asymptotic distribution under the assumption of independently and multivariate normally distributed returns are obtained by Okhrin and Schmid 006). Later on, Bodnar and Okhrin 0) derived the explicit density of the linear transformation of the estimated weights and suggested several exact tests of general linear hypothesis about the elements of the portfolio weights. Kotsiuba and Mazur 05) derived the approximate density function formula for the weights which is based on the Gaussian integral and the third order Taylor expansion. Bodnar et al. 07) extended the results by Bodnar and Okhrin 0) in the setting when both the population and the sample covariance matrices are singular. Moreover, they established the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. Finally, Bauder et al. 07) studied the distributional properties of the weights of the tangency portfolio from the Bayesian point of view. The aim of the present paper is to derive the higher order moments of the sample weights of the tangency portfolio in closed-forms when the returns are assumed to be independently and multivariate normally distributed. As argued by Okhrin and Schmid 006), the knowledge of portfolio weights leads to information about the expected portfolio return and the variance of the portfolio return. Since the expected portfolio returns play a crucial role in most financial theories, the knowledge of higher moments of the estimated portfolio weights can be helpful in learning about the expected portfolio return as well as the portfolio variance. Similarly, via deriving expressions for skewness and kurtosis of estimated weights, we would be able to understand the tail and asymmetric behaviour of wealth distribution. Moreover, Okhrin and Schmid 006) shows that the moments of the optimal portfolio weights are very sensitive to changes in the moments of

4 stock returns. The obtained expressions for the higher moments of the estimated portfolio weights can, therefore, be very informative to account for tail risks in making portfolio strategies. We have obtained explicit expressions for partial cases such as the mean, variance, skewness, and kurtosis. In particular, we take a look at the skewness and kurtosis for measuring the deviation from the normal distribution. This paper is organized as follows. In Section, we establish preliminary results which we use in proving main results in Section 3. In Section 3, we deliver explicit formulas for the higher order non-central and central moments of the estimated tangency portfolio weights. Moreover, we derive the mean, variance, skewness and kurtosis in closed-forms. The results of the empirical study are given in Section 4, while Section 5 summarizes the paper. Preliminary Results In this section, we present the preliminary results, which are used in proving our main results of Section 3. Let us note that our results are complementary to the ones obtained in Bodnar and Okhrin 0), Kotsiuba and Mazur 05) and Bodnar et al. 06). Let A W k n, Σ), i.e., the random matrix A has a k-dimensional Wishart distribution with n degrees of freedom and a positive definite covariance matrix Σ. We assume that n > k, implying that the matrix A is non-singular. Also, let z N k µ, λσ), it means that the random vector z has the k-dimensional normal distribution with mean vector µ and covariance matrix λσ, where λ > 0 is a constant. In Lemma, we present a stochastic representation for l T A z, where l is a k- dimensional vector of constants. Let us note that this result can be found in Bodnar and Okhrin 0). Below, the symbol Fd, d, s) stands for the non-central F-distribution with d and d degrees of freedom and the non-centrality parameter s, while the symbol d stands for the equality in distribution Lemma. Let A W k n, Σ), n > k and z N k µ, λσ) with λ > 0 and positive definite Σ. Furthermore, let A and z be independent and l be a k-dimensional vector of constants. Then the stochastic representation of l T A z is given by l T A z d u l T Σ µ λ ) ) λk ) n k u 3 l T Σ lu ) where u χ n k, u N 0, ) and u 3 Fk, n k, s) with s µ T R l µ/λ and R l Σ Σ ll T Σ /l T Σ l. The random variables u, u and u 3 are mutually independently distributed. From Lemma we get that a stochastic representation of l T A z is given in terms of 3

5 independently distributed χ, a standard normal and a non-central F random variables. Let us recall that a confluent hypergeometric function F a; b; x) is defined by F a; b; x) a b aa ) x x bb )! a) k x k b) k k!, ) k0 where a) k and b) k are Pochhammer symbols cf. Abramowitz and Stegun 984, Chapter 3)). In the next theorem, we consider the higher order moments of l T A z. Theorem. Let A W k n, Σ), n > k and z N k µ, λσ) with λ > 0 and positive definite Σ. Furthermore, let A and z be independent and l be a k-dimensional vector of constants. Then the r-th order moment of l T A z is given by E [ l T A z) r] l T Σ µ) r for n k > r with c m n k )...n k r ) r/ ) r j)! j j j! j l T Σ µ ) r j λl T Σ l ) j j m k m ))... k ) s n k m ))... n k) e F m k ; k ; s ), where s µ T R l µ/λ and R l Σ Σ ll T Σ /l T Σ l. Proof. From Lemma we obtain that ) ) j c m 3) m E [ [ l T A z) r] ) )) r ] E l T Σ λk ) µ λ u n k u 3 l T Σ lu [ ) r ] [ ) ) r ] E E l T Σ λk ) µ λ u n k u 3 l T Σ lu, where the last equality follows from the fact that u is independent of u and u 3. Since u χ n k, we get that /u Inv χ n k. From Glen 07, p. 8) it follows that r ) Γ n k r ) [ ) r ] E u Γ n k r Γ ) n k r n k )... n k r)γ n k r ) n k )...n k r ), n k > r. This result follows from the property Γx) x )Γx ). Using the well-known binomial formula see Biggs 979, p. 9)) and the fact that u 4

6 and u 3 are independent, we obtain that [ ) ) r ] E l T Σ λk ) µ λ n k u 3 l T Σ lu r ) E r l T Σ µ ) ) ) i r i λk ) λ i n k u 3 l T Σ lu i0 r ) r l T Σ µ ) r i l T Σ l ) [ ) ] i/ [ ] i/ E u i λk ) E λ i n k u 3. i0 Let us note that the odd moments of the standard normal distribution are equal to zero, i.e. E [ ] u j 0 for j {0,,,... }, while the even moments are given by E[u j ] j)! j j! for j, c.f. Walck 996, Chapter 34.)). It leads us to E [ r i0 l T Σ µ λ ) ) r ] λk ) n k u 3 l T Σ lu ) r l T Σ µ ) r i λl T Σ l ) i/ [ E u i i l T Σ µ) r l T Σ µ) r r/ j r/ j r j [ ] E k ) ] i/ n k u 3 ) l T Σ µ ) r j λl T Σ l ) [ j [ ] E u j E k ) ] j n k u 3 ) r j)! l T Σ µ ) r j λl T Σ l ) [ j E k ) ] j j j j! n k u 3. Applying the binomial formula again we get E [ k ) ] j n k u 3 [ j ) ) ] m j k E m n k u 3 m0 j ) ) m j k E [u m 3 ]. m n k m0 For m it holds that see Walck 996, Chapter 3.)) 5

7 ) m k c m : E [u m 3 ] n k Γ n k m ) Γ Γ k m ) ) n k Γ ) e s k F k m ))... k ) s n k m ))... n k) e F m k m k ; k ; s ) ; k ; s ). 4) Finally, putting all the terms together we get the statement of the theorem. Now we consider an explicit formula for the higher oder central moments of l T A z which is given in the next corollary. Corollary. Let A W k n, Σ), n > k and z N k µ, λσ) with λ > 0 and positive definite Σ. Furthermore, let A and z be independent and l be a k-dimensional vector of constants. Then the r-th order central moment of l T A z is given by E [ l T A z E[l T A z] ) r] κ ) r l T Σ µ) i i/ j r i ) r i ) i j)! l T Σ µ ) i j λl T Σ l ) j j j j! for n k > r with κ n k lt Σ µ and κ ) r i n k )...n k i ) j ) ) j c m m m c m k m ))... k ) s n k m ))... n k) e F m k ; k ; s ), where s µ T R l µ/λ and R l Σ Σ ll T Σ /l T Σ l. Proof. From Theorem it follows that that κ : E [ l T A z ] n k lt Σ µ. 5) Using the binomial formula and properties of the mathematical expectation, we obtain E [ [ ) l T A r ] r ] r z κ E ) κ ) r i l T A z) i 6) i i0 r ) r κ ) r ) r i κ r i E [ l T A z) i]. 7) i i 6

8 Finally, applying Theorem we get the statement of the corollary. In the following corollary, we deliver the expressions of the second order central moment, the third order central moment, and the fourth order central moment for l T A z in closed-forms without using the confluent hypergeometric function. Corollary. Let A W k n, Σ), n > k and z N k µ, λσ) with λ > 0 and positive definite Σ. Furthermore, let A and z be independent and l be a k-dimensional vector of constants. Also, let s µ T R l µ/λ with R l Σ Σ ll T Σ /l T Σ l Then a) the second order central moment of l T A z is given by for n k > 3 with E[l T A z E[l T A z]) ] d 0) l T Σ µ) d 0) l T Σ l, d 0) n k n k)n k ) n k 3) and d 0) λn ) µ T Σ µ n k)n k )n k 3) ; b) the third order central moment of l T A z is given by for n k > 5 with E[l T A z E[l T A z]) 3 ] d ) l T Σ µ) 3 d ) l T Σ µ l T Σ l d ) d ) 6 n k ) 3 n k 3)n k 5), λ n k ) n k 3)n k 5) s k n k ) ; c) the fourth order central moment of l T A z is given by E[l T A z E[l T A z]) 4 ] d 3) l T Σ µ) 4 d 3) l T Σ µ) l T Σ l d 3) 3 l T Σ l) for n k > 7 with d ) 3[n k )n k 5)n k 7) n k ) n k 9)], n k ) 4 n k 3)n k 5)n k 7) d ) 6λ c )[n k ) n k 3)n k 7)] n k ) 3 n k 3)n k 5)n k 7), d ) 3 3λ c c ) n k )n k 3)n k 5)n k 7), 7

9 with c s k n k and c s s k )k ). n k)n k ) Proof. From Theorem and Walck 996, Chapter 3.) we obtain that c c k n k E[u 3] s k n k, ) k E[u n k 3] s s k )k ). n k)n k ) Using Corollary, we get that the second order central moment of l T A z is given by E[l T A z E[l T A z]) ] κ ) l T Σ µ) i i/ j i ) i j)! l T Σ µ ) i j λl T Σ l ) j j j j! κ κ n k lt Σ µ n k )n k 3) [ l T Σ µ) λ s k ) ] l T Σ l n k [ ] n k )n k 3) l T Σ µ) n k ) λ s k ) l T Σ l n k )n k 3) n k d 0) l T Σ µ) d 0) l T Σ l with d 0) and d 0) as defined in the formulation of the corollary. ) κ ) i i n k )...n k i ) j ) ) j c m m m 8

10 In order to derive the third order central moment of l T A z, it holds that E [ l T A z E[l T A z] ) 3 ] κ ) 3 l T Σ µ) i i/ j 3 i ) 3 i ) i j)! l T Σ µ ) i j λl T Σ l ) j j j j! κ ) 3 i n k )...n k i ) j ) ) j c m m κ 3 3κ n k lt Σ 3κ [ µ l T Σ µ) λ c )l T Σ l ] n k )n k 3) [ l T Σ µ) 3 3λ c )l T Σ µ l T Σ l ] n k )n k 3)n k 5) [ ] n k ) 3 3 n k ) n k 3) l T Σ µ) 3 n k )n k 3)n k 5) [ ] 3λ n k )n k 3)n k 5) n k ) n k 3) s k ) l T Σ µ l T Σ l n k d ) l T Σ µ) 3 d ) l T Σ µ l T Σ l with d ) and d ) which are defined in the statement of the corollary. Finally, we derive the fourth order central moment of l T A z. From Corollary, we m 9

11 have E [ l T A z E[l T A z] ) 4 ] κ ) 4 l T Σ µ) i i/ j n k ) 4 lt Σ µ) 4 4 i ) 4 i ) i j)! l T Σ µ ) i j λl T Σ l ) j j j j! 4κ 3 n k lt Σ µ κ ) 4 i n k )...n k i ) j ) ) j c m m m 6κ n k )n k 3) lt Σ µ) 6λ c )κ n k )n k 3) lt Σ 4κ l n k )n k 3)n k 5) lt Σ µ) 3 λ c )κ n k )n k 3)n k 5) lt Σ µ l T Σ l n k )... n k 7) lt Σ µ) 4 6λ c ) n k )... n k 7) lt Σ µ) l T Σ l 3λ c c ) n k )... n k 7) lt Σ l) [ 6 n k ) 3 n k 3) 3 n k ) 4 4 n k ) n k 3)n k 5) ] l T Σ µ) 4 n k )... n k 7) [ 6λ c ) n k ) 3 n k 3) λ c ) n k ) n k 3)n k 5) ] 6λ c ) l T Σ µ) l T Σ l n k )... n k 7) 3λ c c ) n k )... n k 7) lt Σ l) d ) l T Σ µ) 4 d ) l T Σ µ) l T Σ l d ) 3 l T Σ l) with d ), d ), and d ) 3 as defined in the formulation of the corollary. It completes the proof of the corollary. 3 Main Results We consider a portfolio that consists of k assets. Let x t x t,..., x kt ) T be the k- dimensional vector of log-returns of these assets at time t,..., n. The weight of the ith asset in the portfolio is denoted by w i and let w w,, w k ) T be the vector of weights. Let the mean vector of the asset returns be denoted by µ and the covariance matrix by Σ which is assumed to be positive definite. Notice that the mean-variance portfolio problem is equivalent to maximizing the expected quadratic utility. Since the risk is usually measured by the variance of the portfolios return, the optimal portfolio 0

12 without a risk-free asset is obtained by minimizing the portfolio variance for a given level under the constraint w T k where k denotes the vector of ones. However, if short selling is allowed and a risk-free asset, with return r f, is available, then part of investor s wealth is invested into the risk-free asset, whereas the rest of the wealth is invested into the portfolio from the efficient frontier. The return of risky assets is given as µ p w T µ r f k ) r f with the variance σp w T Σw. Maximizing the utility leads us to the tangency portfolio weights which are given by w T P α Σ µ r f ). 8) where α denotes the investor s attitude towards risk. The tangency portfolio lies on the intersection of the mean-variance frontier and the tangency line drawn from the portfolio consisting of the risk-free asset Ingersoll 987)). Throughout the paper it is assumed that the asset returns x,..., x n are independent and identically normally distributed, i.e. x t N k µ, Σ), t,..., n. Since µ and Σ are unknown parameters the investor cannot determine w T P. In practice, these parameters need to be estimated. Here we use the corresponding sample estimators for µ and Σ expressed as x n n j x j and S n n x i x)x i x) T. j Replacing µ and Σ with x and S in 8), we obtain the sample estimator ŵ T P of tangency portfolio weights w T P, i.e. ŵ T P α S x r f ). In this paper we focus on the linear combination of the tangency portfolio weights. In particular, we are interested in θ l T w T P α l T Σ µ r f ), where l is a k-dimensional vector of constants. Then the sample estimator of θ is given by ˆθ l T ŵ T P α LS x r f ). We now utilize fundamental results obtained in Theorem and Corollary to derive explicit expressions for the higher order non-central and central moments of ˆθ l T ŵ T P.

13 Theorem. Let x,..., x n be identically and independently distributed random vectors with x N k µ, Σ), k < n and Σ > 0. Also, let l be a k-dimensional vector of constants, l n )/αl, µ µ r f k, and s n µ T R l µ with R l Σ Σ l lt Σ / l T Σ l. Then a) the r-th order moment of ˆθ is given by µ r : E[ˆθ r ] n k )... n k r) r/ ) r j)! l lt ) r j lt l) j T Σ µ) r Σ µ Σ j n) j j! j j ) ) )] m j k c m, n k > r m n k m where c m k m ))... k ) s n k m ) )... n k ) e F m k ; k ; s ). b) the r-th order central moment of ˆθ is given by µ r : E[ˆθ µ ) r ] µ ) r r i l T Σ µ) i j m ) r µ ) r i i n k )... n k i) i/ j ) i j)! lt ) i j lt l) j Σ µ Σ j n) j j! ) )] m c m ) j k m n k Proof. From Muirhead 98, Chapter 3) we have that x N k µ, n Σ ), V : n )S W k n, Σ), moreover, x and V are independently distributed. Since ˆθ l T ŵ T P α l T S x r f k ) n α lt V x r f k ). the rest of the proof follows from Theorem and Corollary., n k > r. The following corollary delivers the expressions of the mean and the variance for ŵ T P.

14 Corollary 3. Let x,..., x n be identically and independently distributed random vectors with x N k µ, Σ), k < n and Σ > 0. Also let µ µ r f k and δ n µ T Σ µ. Then the mean and the variance of ŵ T P are given by and E[ŵ T P ] n n k w T P Var[ŵ T P ] 0) w T P w T T P 0) Σ with 0) n k)n ) n k )n k ) n k 4) and 0) n ) n δ) nα n k )n k )n k 4). Proof. From Corollary we get the first two moments of ˆθ which are given by and E[ˆθ] n n k θ Var[ˆθ] 0) θ 0) l T Σ l with 0) n k)n ) n k )n k ) n k 4) and 0) n ) n δ) nα n k )n k )n k 4). Moreover, since l is an arbitrary vector of constants we get the statement of the corollary. In the next corollary, we derive the expressions for skewness and the kurtosis of ˆθ l T ŵ T P. Corollary 4. Let x,..., x n be identically and independently distributed random vectors with x N k µ, Σ), k < n and Σ > 0. Also, let l be a k-dimensional vector of constants, l n )/αl, µ µ r f k, and s n µ T R l µ with R l Σ Σ l lt Σ / l T Σ l. Then the skewness ˆθ is given by Skewness[ˆθ] ) θ 3 ) ) θl T Σ l 0) θ ) 3/ 0) l T Σ l 3

15 with 0) and 0) which are defined in Corollary 3, and ) ) 6n ) 3 n k ) 3 n k 4)n k 6), n ) 3 α nn k ) n k 4)n k 6) s k ), n k while the kurtosis of ˆθ is expressed as where with Kurtosis[ˆθ] 3) θ 4 3) θ l T Σ l 3) 3 l T Σ l) ) 0) θ 0) l T Σ l). 3) 3n )4 [n k)n k 6)n k 8) n k ) n k 0)], n k ) 4 n k 4)n k 6)n k 8) 3) 6 c )n ) 4 [n k ) n k )n k 8)], α nn k ) 3 n k 4)n k 6)n k 8) 3) 3 c c )n ) 4 3 α 4 n n k )n k 4)n k 6)n k 8), c s k n k and c s s k )k ) n k )n k 3). Proof. The skewness of ˆθ is given by Skewness[ˆθ] µ 3 [ ] 3/ µ 0) 3 θ Varˆθ) 0) l T Σ l) 3/, where Varˆθ) is obtained from Corollary 4. From Corollary it follows that µ 3 d ) l T Σ µ) 3 d ) l T Σ µ l T Σ l, where d ) d ) 6 n k ) 3 n k 4)n k 6) s k ) nn k ) n k 4)n k 6) n k with s n µ T R l µ and R l Σ Σ l lt Σ / l T Σ l. Furthermore, µ3 can be rewritten 4

16 in the next form µ 3 ) d n ) 3 θ 3 ) d α n ) 3 θl T Σ l ) θ 3 ) θl T Σ l, ) ) where and are the same as in the formulation of the corollary. Putting all above together we get the skewness of ˆθ. We later move on and derive the explicit formula for the kurtosis of ˆθ. It holds that Kurtosis[ˆθ] Using Corollary we get µ 4 [ ] µ 0) 4 θ Varˆθ) 0) l T Σ l). µ 4 d 3) l T Σ µ) 4 d 3) l T Σ µ) lt Σ l d3) 3 l T Σ l) where d 3) 3[n k)n k 6)n k 8) n k ) n k 0)], n k ) 4 n k 4)n k 6)n k 8) d 3) 6 c )[n k ) n k )n k 8)] nn k ) 3 n k 4)n k 6)n k 8), d 3) 3 c c ) 3 n n k )n k 4)n k 6)n k 8), with c s k n k and c s s k )k ) n k )n k 3). Moreover, µ 4 can be rewritten as µ 4 3) θ 4 3) θ l T Σ l 3) 3 l T Σ l) 3) 3), where, and the proof of the corollary. 3) 3 are the same as in the formulation of the corollary. It completes 4 Empirical Illustration In this section we present the results of empirical study where we show how theoretical results of the previous section can be applied to real data. We consider weekly data of k 8 financial indexes which are listed in NASDAQ stock exchange. Their abbreviated symbolic names are RCMP, IXTS, IXCO, TRAN, INDS, NBI, IXIC, BANK. The data are 5

17 taken for the period from August 007 to April 07. Weekly log returns on each index have been considered. The weekly log returns on the three-month US treasury bill are used as the risk-free rate. The risk aversion coefficient α is taken as 50. Table presents the mean, variance, skewness, and kurtosis for the densities of the estimated tangency portfolio weights. The sample size n is set to be {50, 00, 50, 500}. We observe that some expected values are negative, which means that we have a short sales for corresponding financial index. The variance of weights decreases significantly with the increase in sample size n. Moreover, when n increases the skewness tends to approach 0, while the kurtosis remains closer to 3 in all cases, thereby, indicating that the distribution of estimated weights satisfies the normality assumption. Through this empirical exercise, we confirm that with a larger sample size the distribution of the tangency portfolio weights can be nicely approximated by the normal distribution. n50 RCMP IXTS IXCO TRAN INDS NBI IXIC BANK Mean Variance Skewness Kurtosis n00 Mean Variance Skewness Kurtosis n50 Mean Variance Skewness Kurtosis n500 Mean Variance Skewness Kurtosis Table : Mean, variance, skewness and kurtosis of the estimator for the eight financial indexes from NASDAQ stock exchange. 5 Conclusions In this paper we study higher order moments of the estimated tangency portfolio weights obtained under the assumption of normally and independently distributed returns. In particular, we derive the higher order non-central and central moments of estimated weights 6

18 that depend on the confluent hypergeometric function. Moreover, we provide the expressions of the mean, variance, skewness and kurtosis in a closed-forms without using the confluent hypergeometric function. The results are supported by an empirical study, where such expressions are evaluated for the actual returns for eight financial indexes listed in NASDAQ stock exchange. References Abramowitz, M. and Stegun, I. A. 984). Pocketbook of mathematical functions. Verlag Harri Deutsch, Frankfurt Main). Bauder, D., Bodnar, T., Mazur, S., and Okhrin, Y. 07). Bayesian inference for the tangent portfolio. Technical report, Stockholm University. Biggs, N. L. 979). The roots of combinatorics. Historia Mathematica, 6: Bodnar, O. 009). Sequential surveillance of the tangency portfolio weights. International Journal of Theoretical and Applied Finance, 6): Bodnar, T., Mazur, S., and Parolya, N. 06). Central limit theorems for functionals of large dimensional sample covariance matrix and mean vector in matrix-variate skewed model. arxiv preprint arxiv: Bodnar, T., Mazur, S., Podgórski, K., and Tyrcha, J. 07). Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory. Technical report, Stockholm University. Bodnar, T. and Okhrin, Y. 0). On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory. Scandinavian Journal of Statistics, 38:3 33. Britten-Jones, M. 999). The sampling error in estimates of mean-variance efficient portfolio weights. The Journal of Finance, 54: Glen, A. G. 07). Computational Probability Applications, volume 47 of International Series in Operations Research & Management Science, chapter On the Inverse Gamma as a Survival Distribution, pages Springer International Publishing, Switzerland. Ingersoll, J. E. 987). Theory of Financial Decision Making. Rowman & Littlefield Publishers. Kotsiuba, I. and Mazur, S. 05). On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian random vector. Theory of Probability and Mathematical Statistics, 93:

19 Markowitz, H. 95). Portfolio selection. Journal of Finance, 7:77 9. Muirhead, R. J. 98). Aspects of Multivariate Statistical Theory. Wiley, New York. Okhrin, Y. and Schmid, W. 006). Distributional properties of portfolio weights. Journal of Econometrics, 34: Walck, C. 996). Hand-book on statstical distributions for experimentalists. Internal Report SUF PFY/96 0, Stockholm University. 8