Statistical Estimation of Optimal Portfolios depending on Higher Order ...

Statistical Estimation of Optimal Portfolios depending on Higher Order Cumulants Hiroshi Shiraishi Waseda University Masanobu Taniguchi∗ Waseda University

Abstract When financial returns are supposed to be vector-valued non-Gaussian stationary processes, optimal portfolios for general utility function depend on their higher order cumulants. In this paper we assume that the concerned optimal portfolio g depends on the mean µ, the variance Σ and the third order cumulants c(3) of a non-Gaussian return process, i.e., g = g(µ, Σ, c(3) ). Then the asymptotic distribution of a sample version estimator gˆ of g is derived. Traditional optimal portfolios are usually written in the form of g∗ = g(µ, Σ, 0). We compare the asymptotic mean squares error (MSE) of gˆ with that of a sample version estimator gˆ∗ for g∗ when the return process is contiguous to a Gaussian stationary process. Some sufficient conditions for the cases M SE(ˆ g ) ≤ M SE(ˆ g∗ ) and M SE(ˆ g ) > M SE(ˆ g∗ ) are given. We neumerically evaluate the magnitude of M SE(ˆ g∗ )−M SE(ˆ g ) for an ARMA return, which shows an interesting feature of the two estimators. Finally an application for actual financial data will be provided.

JEL classsification: C22; G11 Key words and phrases: optimal portfolio; return process; non-Gaussian linear process; spectral density; asymptotic efficiency; contiguity

1.

Introduction

In the usual theory of portfolio analysis, optimal portfolios are determined by the mean µ and the variance Σ of the portfolio return. Several authors proposed estimators of optimal ˆ for independent portfolios as functions of the sample mean µ ˆ and the sample variance Σ returns of assets (e.g.Jobson and Korkie (1980 and 1989)). However, empirical studies show that financial return processes are often dependent and non-Gaussian. Under the nonGaussianity, if we consider general utility function U [·], the expected utility should depend on higher order moments of the return. From this point of view, several authors proposed ∗ Corresponding author ; Department of Mathematical Sciences, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan; e-mail: [email protected].

1

portfolios including higher order moments of the return. However, in the literature there has been no study on the asymptotics of estimators for these optimal portfolios. Therefore, in this paper, denoting the optimal portfolios by a function g = g(µ, Σ, c(3) ) of µ , Σ and higher order cumulants c(3) , and traditional optimal portfolios by a function g∗ = g(µ, Σ, 0) of µ and Σ, we discuss the asymptotics of estimators gˆ and gˇ for g and g∗ , respectively, when the return is a vector-valued non-Gaussian stationary process {X(t)}. The asymptotic distributions of gˆ and gˇ are derived. Since, for non-Gaussian returns, gˆ is trivially better than gˇ, we evaluate the mean squares errors M SE(ˆ g ) and M SE(ˇ g ) for gˆ and gˇ, respectively, when {X(t)} is contiguous to a Gaussian process. Sufficient conditions for the cases M SE(ˆ g ) ≤ M SE(ˇ g ) and M SE(ˆ g ) > M SE(ˇ g ) are given. Then we investigate this result numerically. Also numerical examples for an actual financial data are provided, and they support {ˆ g } rather than {ˇ g }. The paper is organized as follows. Section 2 describes optimal portfolios as a function g(ca1 , ca2 a3 , ca4 a5 a6 ) of ca1 , ca2 a3 and ca4 a5 a6 . Section 3 gives the asymptotic distribution of gˆ. Section 4 presents some sufficient conditions which gˆ is better than gˇ, and vice versa, when {X(t)} is contiguous to a Gaussian process. Numerical examples are provided for the theoretical results. Finally, we examine our approach for real financial data.

2.

Optimal Portfolios depending on higher order cumulants

Suppose the existence of a finite number of assets indexed by i, (i = 1, . . . , p). Let X(t) = (X1 (t), . . . , Xp (t))0 denote the random returns on p assets at time t. Since it is empirically observed that {X(t)} is non-Gaussian, assuming the stationarity and the existence of third order cumulant of {X(t)}, we write ca1 a2 a3

c

a4 a5 a6

c

= cum(Xa1 (t)) = cum(Xa2 (t), Xa3 (t)) = cum(Xa4 (t), Xa5 (t), Xa6 (t)).

Let us now suppose that there exists a risk-free asset. We denote by X0 (t) its return. Let α0 and α = (α1 , . . . , αp )0 be portfolio weights at time t. Then the return of portfolio based on X(t) and X0 (t) is X(t)0 α + X0 (t)α0 = Y (t), (say), whose higher order cumulants are written as cY1 (t) = cum(Y (t)) = ca1 αa1 + X0 (t)α0 , cY2 (t) = cum(Y (t), Y (t)) = ca2 a3 αa2 αa3 , cY3 (t)

= cum(Y (t), Y (t), Y (t)) = ca4 a5 a6 αa4 αa5 αa6 ,

where Einstein’s summation convention, that is, ”Summation is automatically taken without the symbol Σ for those indices which appear twice in one term once as a subscript and once as a superscript”, is assumed throughout this paper.

2

The investment problem for an investor is to maximize the expected utility of portfolio returns at the end of the period. For a utility function U (·), the expected utility can be represented as; E[U (Y (t))] ≈

U [E(Y (t))] +

1 2 D U [E(Y (t))]E(Y (t) − E(Y (t)))2 2!

1 3 D U [E(Y (t))]E(Y (t) − E(Y (t)))3 3! 1 1 U [cY1 (t)] + D2 U [cY1 (t)]cY2 (t) + D3 U [cY1 (t)]cY3 (t), 2! 3! +

=

(1)

if we approximate it by Taylor expansion of order 3. The approximate optimal portfolio may be written as ( max {the right hand side of (1)} , α0 ,α (2) Pp subject to α0 + i=1 αi = 1. Shiraishi and Taniguchi (2006) gave the asymptotic distribution of portfolio estimators based on the sample mean vector and the sample covariance matrix for non-Gaussian dependent return processes and also addressed the problem of asymptotic efficiency for a class of estimators. However, as we said, the optimal portfolio for (2) depends on the third-order cumulants of the return processes. Therefore, we introduce an optimal portfolio based on third order cumulants g(ca1 , ca2 a3 , ca4 a5 a6 ), and evaluate effect of the third order cumulants. Here it should be noted that the coefficient α0 and α satisfy the restriction Pp α0 + i=1 αi = 1. Then we have only to estimate α. Hence we assume that the function g(·) is p-dimensional, i.e., g : g(ca1 , ca2 a3 , ca4 a5 a6 ) → Rp .

(3)

This paper addresses the problem of statistical estimation for g(ca1 , ca2 a3 , ca4 a5 a6 ), and discusses effects of the third order cumulant.

3.

Asymptotic Theory

Let the return process {X(t) = (X1 (t), . . . , Xp (t))0 ; t ∈ Z} be the p-vector linear process generated by ∞ X X(t) = G(j)²(t − j) + µ, t ∈ Z, (4) j=0

where ²(t)’s are p-dimentional stationary process such that E{²(t)} = 0 and E{²(s)²(t)0 } = δ(s, t)K, with K a nonsingular p by p matrix; G(j)’s are p by p matrices; {µ = (µ1 , . . . , µp )} is a p vector; and the components of X, ², G and µ are all real. Assuming that {²(t)} has all order cumulants, let Qea1 ,...,aj (t1 , . . . , tj−1 ) be the joint jth order cumulant of ²a1 (t), . . . , ²aj (t + tj−1 ) and assume

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Assumption 1

For each j = 1, 2, 3, . . . , p X

∞ X

|Qea1 ...aj (t1 , . . . , tj−1 )| < ∞.

t1 ,...,tj−1 =−∞ a1 ,...,aj =1

˜ ea ,...,a (λ1 , . . . , λj−1 ) defined by The process {²(t)} has the jth-order spectral density Q 1 j ˜e Q a1 ,...,aj (λ1 , . . . , λj−1 ) µ ¶j−1 X 1 = exp{−i(λ1 t1 + . . . + λj−1 tj−1 )}Qea1 ,...,aj (t1 , . . . , tj−1 ). 2π t ,...,t 1

j−1

We further make the following assumption. Assumption 2 ∞ X

kG(l)k < ∞,

l=0

where kAk denotes the Euclidean norm of a matrix A. Now let QX a1 ,...,aj (t1 , . . . , tj−1 ) be the joint j-th order cumulant of Xa1 (t), . . . , Xaj (t + tj−1 ). Then, the process {X(t)} is a j-th order stationary process and has the cumulant spectral density matrix {fa1 ...aj (λ1 , . . . , λj−1 )} which is representable as fa1 ...aj (λ1 , . . . , λj−1 ) p X ˜ ea ...a (λ1 , . . . , λj−1 ), ka1 b1 (λ1 + · · · + λj−1 )ka2 b2 (−λ1 ) · · · kaj bj (−λj−1 )Q = 1 j b, ...,bj =1

P∞ where kab (λ) = l=0 Gab (l)eiλl , and Gab (l) is the (a,b)th element of G(l). For a partial realization {X(1), . . . , X(n)}, we introduce n

cˆa1

=

1X Xa (s) n s=1 1

cˆa2 a3

=

1X (Xa2 (s) − cˆa2 )(Xa3 (s) − cˆa3 ) n s=1

cˆa4 a5 a6

=

1X (Xa4 (s) − cˆa4 )(Xa5 (s) − cˆa5 )(Xa6 (s) − cˆa6 ). n s=1

n

n

Write θ = (ca1 , ca2 a3 , ca4 a5 a6 )0 and θˆ = (ˆ ca1 , cˆa2 a3 , cˆa4 a5 a6 )0 . Then dim θ = dim θˆ = p + q + r with q = p(p + 1)/2 and r = p(p + 1)(p + 2)/6. 4

Theorem 1 Under Assumptions 1 and 2, ´ √ ³ D n θˆ − θ → N (0, Ω) , where the element of Ω corresponding to the covariance between c(♦) and c() is denoted by V {(♦), (¤)}, and

V {(a1 ), (a01 )}

2πfa1 a01 (0) a1 , a01 ∈ L1 Z π V {(a2 , a3 ), (a01 )} = 2π fa2 a3 a01 (λ, 0)dλ (a2 , a3 ) ∈ L2 , a01 ∈ L1 −π Z Z π 0 V {(a4 , a5 , a6 ), (a1 )} = 2π fa4 a5 a6 a01 (λ1 , λ2 , 0)dλ1 dλ2 (a4 , a5 , a6 ) ∈ L3 , a01 ∈ L1 −π Z Z π 0 0 V {(a2 , a3 ), (a2 , a3 )} = 2π fa2 a3 a02 a03 (λ1 , λ2 , −λ2 )dλ1 dλ2 −π Z π © ª + 2π fa2 a02 (λ)fa3 a03 (−λ) + fa2 a03 (λ)fa3 a02 (−λ) dλ =

−π

Z Z Z V {(a4 , a5 , a6 ), (a02 , a03 )}

=

(a2 , a3 ), (a02 , a03 ) ∈ L2

π

fa4 a5 a6 a02 a03 (λ1 , λ2 , λ3 , −λ3 )dλ1 dλ2 dλ3

2π −π

Z Z Z Z V {(a4 , a5 , a6 ), (a04 , a05 , a06 )}

=

2π Z Z Z

+ +

2π Z Z

+

−π π

2π Z Z

2π

(a4 , a5 , a6 ) ∈ L3 , (a02 , a03 ) ∈ L2

π

fa4 a5 a6 a04 a05 a06 (λ1 , λ2 , λ3 , λ4 , −λ3 − λ4 )dλ1 · · · dλ4

X

−π ν1 Z π X −π ν2 π

X

−π ν3

fai1 ai2 ai3 ai4 (λ1 , λ2 , λ3 )fai5 ai6 (−λ2 − λ3 ) fai1 ai2 ai3 (λ1 , λ2 )fai4 ai5 ai6 (λ3 , −λ2 − λ3 )dλ1 dλ2 dλ3

fai1 ai2 (λ1 )fai3 ai4 (λ2 )fai5 ai6 (−λ1 − λ2 )dλ1 dλ2 (a4 , a5 , a6 ), (a04 , a05 , a06 ) ∈ L3 ,

where (ai1 , . . . , ai6 ) shows an arbitrary permutation of (a4 , a5 , a6 , a04 , a05 , a06 ) and ν1

= { all the combinations of (ai1 , . . . , ai6 ) satisf ying ai5 ∈ {a4 , a5 , a6 }, ai6 ∈ {a04 , a05 , a06 }}

ν2 ν3

= { all the combinations of (ai1 , . . . , ai6 ) satisf ying ai1 ∈ {a4 , a5 , a6 }, ai3 ∈ {a04 , a05 , a06 }} = { all the combinations of (ai1 , . . . , ai6 ) satisf ying ai1 , ai3 , ai5 ∈ {a4 , a5 , a6 }}. For g given by (3) we impose the following.

Assumption 3

The function g(θ) is continuously differentiable.

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ˆ For this we have the following As a unified estimator for optimal portfolios we use g(θ). result. Theorem 2 Under Assumptions 1,2 and 3, √

D

ˆ − g(θ)) → N (0, (Dg)Ω(Dg)0 ) , n(g(θ)

where Dg = {∂i g j (θ); i = 1, . . . , p + q + r, j = 1, . . . , p}.

4.

Effect of third order cumulants

In Section 3 we saw the asymptotics of the portfolio estimator gˆ ≡ g(ˆ ca1 , cˆa2 a3 , cˆa4 a5 a6 ) based on higher order cumulants of the return process. In the literature, estimators based on cˆa1 and cˆa2 a3 of the form gˇ ≡ g∗ (ˆ ca1 , cˆa2 a3 ) have been used (e.g. Jobson and Korkie (1980), Basak, Jagannathan and Sun (2002) and Shiraishi and Tanigushi (2006)). Here we compare our estimator gˆ with the traditional one gˇ when the return process is contiguous to a Gaussian process in the sense of Assumption 3 below. The asymptotic mean squares errors for gˆ and gˇ are evaluated. Then the results give conditions that gˆ is better than gˇ, and vice versa. Now we consider a case that {X(t)} of (4) is non-Gaussian but contiguously close to a Gaussian process i.e, Assumption 4 ( (i) (ii)

QX a4 a5 a6 (t1 , t2 ) P∞

=

√1 ha4 a5 a6 n³ ´ O √1n

(t1 = t2 = 0) (f or any t1 6= 0 or t2 6= 0)

Pp

t1 ,...,tk =−∞

e a1 ,...,ak =1 |Qa1 ···ak (t1 , . . . , tk−1 )| = O

Then, the asymptotic covariance matrix Ω in  Ω11 0 Ω =  0 Ω22 0 0

³

√1 n

´ , for any k ≥ 3.

Theorem 1 becomes  0 0 , Ω33

where the elements of Ω11 , Ω22 and Ω33 are V {(a1 ), (a01 )}

=

V {(a2 , a3 ), (a02 , a03 )}

=

2πfa1 a01 (0) a1 , a01 ∈ L1 Z π © ª 2π fa2 a02 (λ)fa3 a03 (−λ) + fa2 a03 (λ)fa3 a02 (−λ) dλ −π

Z Z V {(a4 , a5 , a6 ), (a04 , a05 , a06 )}

=

2π

π

X

−π ν3

(a2 , a3 ), (a02 , a03 ) ∈ L2 fai1 ai2 (λ1 )fai3 ai4 (λ2 )fai5 ai6 (−λ1 − λ2 )dλ1 dλ2 (a4 , a5 , a6 ), (a04 , a05 , a06 ) ∈ L3 .

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Write θ∗ θˆ∗ g∗

= (ca1 , ca2 a3 ) = (ˆ ca1 , cˆa2 a3 )0 : g∗ (ca1 , ca2 a3 ) → Rp ,

and Dg

=

(∂gj /∂ca1 , ∂gj /∂ca2 a3 , ∂gj /∂ca4 a5 a6 ) ≡ (Dg a1 , Dg a2 a3 , Dg a4 a5 a6 )

Dg∗

=

(∂gj /∂ca1 , ∂gj /∂ca2 a3 ) ≡ (Dg∗a1 , Dg∗a2 a3 ),

where g∗ (ca1 , ca2 a3 ) ≡ g(ca1 , ca2 a3 , 0) Then, we have the following result. Theorem 3 Under Assumptions 1 - 4, µ µ √ Ω11 D a 4 a5 a6 0 a4 a5 a6 ˆ n(g∗ (θ∗ ) − g(θ)) → N − (Dg )h , (Dg∗ ) 0

0 Ω22

¶

¶ 0

(Dg∗ )

,

where ha4 a5 a6 is the r-dimentional row vector with ha4 a5 a6 , (a4 , a5 , a6 ) ∈ L3 . ˆ and g∗ (θˆ∗ ), respectively. Then from Theorem 3, we have, Let gˆ and gˇ denote g(θ) Theorem 4 Suppose that Assumptions 1-4 hold. Let M SE(ˆ g) = M SE(ˇ g)

=

lim n E {(ˆ g − g)(ˆ g − g)0 }

n→∞

lim n E {(ˇ g − g)(ˇ g − g)0 } .

n→∞

Then, we have tr {M SE(ˆ g )} ≤ tr {M SE(ˆ g )}

tr {M SE(ˇ g )}

> tr {M SE(ˇ g )}

© ª 0 if tr Dg a4 a5 a6 (Ω33 − ha4 a5 a6 ha4 a5 a6 )Dg a4 a5 a6 0 ≤ 0 ª © 0 if tr Dg a4 a5 a6 (Ω33 − ha4 a5 a6 ha4 a5 a6 )Dg a4 a5 a6 0 > 0.

Example I Now we evaluate the difference between tr {M SE(ˆ g )} and tr {M SE(ˇ g )} for the return process {X(t) = (X1 (t), X2 (t))} generated by X1 (t) − d1 X1 (t − 1) X2 (t)

= ²(t) − d2 ²(t − 1) + µ1 = µ2 ,

where E{²(t)} = 0, V ar{²(t)} = σ 2 . We consider the following two optimal portfolios g(c1 , c2 , c3 ) and g∗ (c1 , c2 ); ³ 1 g(c1 , c2 , c3 ) = max U [αc1 + (1 − α)µ2 ] + D2 U [αc1 + (1 − α)µ2 ](α2 c2 ) α 2! ´ 1 + D3 U [αc1 + (1 − α)µ2 ](α3 c3 ) 3! µ ¶ 1 2 1 2 2 1 2 1 g∗ (c , c ) = max U [αc + (1 − α)µ2 ] + D U [αc + (1 − α)µ2 ](α c ) α 2! 7

where

= =

1 x1−γ (CRRA power utility) 1−γ cum{X1 (t)} cum{X1 (t), X1 (t)}

=

cum{X1 (t), X1 (t), X1 (t)}.

U [x] = c1 c2 c3

For d1 = −0.9(0.1)0.9, d2 = 0.5, µ1 = 0.1, µ2 = 0.01, σ 2 = 1, γ = 0.5 and c3 = 10(1/10) , . . . , 10(10/10) , we plotted ∆M SE = tr {M SE(ˇ g )} − tr {M SE(ˆ g )} in Figure 1. Figure 1 is about here. From Figure 1, it is seen that as |d1 | tends to 1, ∆M SE becomes quite small, and also as c3 decreases, ∆M SE becomes slightly small. For d1 = 0.5, d2 = −0.9(0.1)0.9, µ1 = 0.1, µ2 = 0.01, σ 2 = 1, γ = 0.5 and c3 = 10(1/10) , . . . , 10(10/10) , we plotted ∆M SE in Figure 2. Figure 2 is about here. From Figure 2, we observe that if d2 & −1, ∆M SE decreases. For d1 = 0.5, d2 = 0.5, µ1 = 0.1, µ2 = 0.01, σ 2 = 1(1)10, γ = 0.5 and c3 = 10(1/10) , . . . , 10(10/10) , we plotted ∆M SE in Figure 3. Figure 3 is about here. From Figure 3, we can see that as σ 2 increases, ∆M SE decreases. Summarizing the above we get the following conclusions : (i) {ˆ g } is better than {ˇ g } as the third order cumulant c3 increases. (ii) {ˆ g } is worse than {ˇ g } as the absolute value of AR coefficient d1 tends to 1. (iii) {ˆ g } is worse than {ˇ g } as the MA coefficient d2 tends to −1. (iv) {ˆ g } is better than {ˇ g } as the variance of innovation decreases. Example II We construct the two optimal portfolio estimators gˆ and gˇ for three financial series X1 (t), X2 (t) and X3 (t) which are selected from seven financial series SEIYU, MITSUBISHI Motors, TOYOTA Motors, SANYO, SUMITOMO Metals, NIPPON Steel and TOSHIBA’s monthly returns with length 131. For each T = 1, . . . , 30, we calculated the optimal portfolio estimators gˆ and gˇ for CRRA power utility based on the observations X(t) ≡ (X1 (t), X2 (t), X3 (t))0 , t = T, . . . , T + 99. Denote gˆ and gˇ by gˆT and gˇT , respectively. Then we calculated M SE1 =

30 1 X 0 (ˆ gT X(T + 1) − gˆ0 X)2 30 T =1

M SE2 =

30 1 X 0 (ˇ gT X(T + 1) − gˆ0 X)2 30 T =1

P30

1 where gˆ0 X = 30 ˆT0 X(T + 1). Table 1 plots M SE1 and M SE2 for six combinations T =1 g of these financial series.

8

Table 1 is about here. From this table, we can see that the values of M SE1 are almost smaller than those of M SE2, which implies that gˆ is better than gˇ.

References [1] Angelo, R. and Laurent, F. (2003) How to Price Hedge Funds: From Two- to FourMoment CAPM. EDHEC Business School Discussion Paper. [2] Basak, G. & Jagannathan, R. & Sun, G. (2002) A direct test for the mean variance efficiency of a portfolio. Journal of Economic Dynamics and Control 26, 1195-1215. [3] Brockwell, P. J. and Davis, R. A. (1991) Time Series:Theory and Methods. New York: Springer. [4] Brillinger, D. R. (1981) Time Series:Data Analysis and Theory, expanded. San Francisco: Holden-Day. [5] Fuller, W. A. (1996) Introduction to Statistical Time Series, second edition. New York: Willy. [6] Hosoya, Y. and Taniguchi, M. (1982) A Central Limit Theorem for Stationary Processes and the Paremeter Estimation of Linear Processes The Annals of Statistics. 10, 132-153. [7] Jaksa, C. , Vassilis, P. and Fernando, Z. (2005) Optimal Portfolio Allocation with Higher Moments. Discussion Paper. [8] Jobson, J. D. and Korkie, B. (1980) Estimation for Markowitz Efficient Portfolios. Journal of the American Statistical Association . 75, 544-554. [9] Jobson,J.D.& Korkie,B.(1989) A Performance Iterpretation of Multivariate Tests of Asset Set Intersection, Spaning, and Mean-Variance Efficiency. Journal of Financial and Quantitative Analysis 24,185-204. [10] Magnus, J. R. and Neudecker, H. (1988) Matrix Differential Calculus with Apprications in Statistics and Econometrics . New York: Wiley [11] Shiraishi, H. and Taniguchi, M. (2006) Statistical Estimation of Optimal Portfolios for non-Gaussian Dependent Returns of Assets. Waseda University Time Series Discussion Paper. [12] Taniguchi, M & Kakizawa, Y. (2000) Asymptotic Theory of Statistical Inference for Time Series. New York: Springer.

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Figure 1: ∆M SE

Figure 2: ∆M SE

10

Figure 3: ∆M SE

Table 1: SEIYU MITSUBISHI.M TOYOTA.M SANYO SUMITOMO.M NIPPON.S

asset MITSUBISHI.M TOYOTA.M SANYO SUMITOMO.M NIPPON.S TOSHIBA

TOYOTA.M SANYO SUMITOMO.M NIPPON.S TOSHIBA SEIYU

11

MSE1 0.003747 0.007252 0.008900 0.008424 0.008591 0.009167

MSE2 0.003946 0.008264 0.007723 0.008434 0.008876 0.009406