Statistical Estimation of High-Dimensional Portfolio Hiroyuki Oka*, Hiroshi Shiraishi

Keio University Graduate School of Science and Technology

Abstract

Methods

Results

References Bai, Z., Liu, H., and Wong, W.-K. (2009). Enhancement of the applicability of markowitz’s portfolio optimization by utilizing random matrix theory. Mathematical Finance, 19(4):639–667. El Karoui, N. et al. (2010). High-dimensionality effects in the markowitz problem and other quadratic programs with linear constraints: Risk underestimation. The Annals of Statistics, 38(6):3487–3566. Bodnar, T., Parolya, N., and Schmid, W. (2014). Estimation of the global minimum variance portfolio in high dimensions. arXiv preprint arXiv:1406.0437. Fujikoshi, Y., Ulyanov, V. V., and Shimizu, R. (2010). Multivariate Statistics: High-Dimensional and Large-Sample Approximations (Wiley Series in Probabilityand Statistics). Wiley, 1 edition. Glombek, K. (2012). High-Dimensionality in Statistics and Portfolio Optimization. Josef Eul Verlag Gmbh.

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Background

Future Work

We introduce a Markowitz’s mean-variance optimal portfolio

estimator from d � n data matrix under high dimensional set-

ting where d is the number of assets and n is the sample size.

When d/n converges in (0, 1), we show inconsistency of the

traditional estimator and propose a consistent estimator.

Let X be a asset returns (r.v.), and w be portfolio weights.

Then, optimal portfolio weights are the solution of following

optimization problem.

��

�maxw�Rd

u(w) = E[w�X] � 12� Var(w�X)

subject to w�1d = 1

Here, � is a positive constant depending on individual investor.

And then, using expressions E[X] = µ and Var(X) = �,

expected value and variance of optimal portfolio are expressed

as follows.

Expected value µopt and variance �2opt of optimal portfolio

return are expressed as follows.

µopt(�; �) = �

��1 � �2

2

�3

�+

�2

�3

�2opt(�; �) = �2

��1 � �2

2

�3

�+

1

�3

A set of (�opt, µopt) is called ”e�cient frontier”. Here, � is

� =

�

��1

�2

�3

�

� =

�

�µ���1µ1�

d ��1µ1�

d ��11d

�

�

Fig.1 E�cient frontier：Following figure1 shows portfolio plot. Black points (·) are the portofolios in feasible area which can obtain by

changing value of w. And red circles (�) shows optimal portfolios which can obtain by changing value of �. This figure shows that rational

investor prefer lower risk when mean is same value, and higher return when risk is same value.

Purpose：estimate e�cient frontier in high dimension

• d：fix, n � � � S�1 is consistent

• n, d � � � S�1 is inconsistent

To estimate optimal portfolio, we should estimate optimal

portfolio parametor �. It is assumed that n data vectors

X1, . . . , Xn is following unknown distribution which has mean

vector µ, and covariance matrix �.

Then, �̃, estimator of parmetor �, is defined as follows.

�̃ =

�

���̃1

�̃2

�̃3

�

�� =

�

��X̄�S�1X̄

1d�S�1X̄

1d�S�11d

�

��

For some mathematical argument, we put some assumptions.

So, we would like to derive asymptotic property of optimal

portfolio on the following assumption.

n � �, d � �,d

n� � � (0, 1)

µ̂opt(�; �) = �

��̂1 � �̂2

2

�̂3

�+

�̂2

�̂3

, �̂2opt(�; �) = �2

��̂1 � �̂2

2

�̂3

�+

1

�̂3

Fig.2 Consistency of estimators：Fig.2 shows histgrams of generated �̃1 and �̂1. Red part is histgram

of �̂1, and blue part is histgram of �̃1. In order of n = 100, 500, 1000 from the left, and this shows that

�̂1 converge to true value �1

Fig.3 Asymptotic normality of�

n(�̂1 � �1)：Fig.3 shows histgram of�

n(�̂1 � �1) and blue curve line

of asymptotic normal distribution. In order of n = 100, 500, 1000 from the left, and this shows that�

n(�̂1 � �1) converge to objective normal distribution.

Simulation Study

• fix d/n = 0.8

• increase n = 100, 500, 1000, and assume Xti.i.d.� Nd(µ,�)

• components of µ are devided [�1, 1] into d equal parts

• � has diagonal components 1、and the others 0.5

In this condition, generate �̃1 and �̂1 in 10000 times and confirm theoretical results.

When an investor invest in d financial products, he consider

how maximize the portfolio return for a given level of risk,

defined as variance. Define optimal portfolio as follows.

Because E[X] = µ and Var(X) = � is generally unknown, it

is considered that optimal portofolio should be estimated by

d � n data matrix (X1, . . . , Xn). We estimate µ by sample

mean vector X̄, and � by sample covariance matrix S.

X̄ =1

n

n�

t=1

Xt, S =1

n

n�

t=1

(Xt � X̄)(Xt � X̄)�

In these days, because of expansion of market scale, the num-

ber of assets d grows bigger. But, it is known that the bigger

dimension size d grows, the worse estimator S�1 becomes.

We introduce (n, d)-asymptotic properties of estimators of op-

timal portfolio parametor �. It is known that when

X1, . . . , Xni.i.d.� (µ,�) and satisfy previous assumption 1�4,

�̃ converges following value as n goes to infinity.

�̃1a.s.� 1

1 � ��1 +

�

1 � �, �̃2

a.s.� 1

1 � ��2, �̃3

a.s.� 1

1 � ��3

This shows that estimator using X̄ and S is overestimated.

So, we need to correct estimators. We propose the following

estimator �̂.

Result1 　 Define estimator �̂ = (�̂1, �̂2, �̂3)� as following

expressions.

�̂1 =

�1 � d

n

��̃1�

d

n, �̂2 =

�1 � d

n

��̃2, �̂3 =

�1 � d

n

��̃3

Then, �̂ is consistent estimator of �.

This estimator �̂ has asymptotic normality.

Result2　X1, . . . , Xni.i.d.� (µ,�) satisfy previous assump-

tion 1�4. Then,�

n(�̂ � �) converges to normal distribution

as n goes to infinity.

�n(�̂ � �)

D� N3(0,�) ((n, d)-asymptotic)

In this, � is following matrix.

� =1

1 � �

�

��2�2

1 + 4�1 + 2� � �2�1�2 �2

2 + �1�3 + �3 �2�1�3 2�2�3 2�2

3

�

��

Using this estimator �̂, we make e�cient frontier estimators

µ̂opt and �̂2opt as follows.

These e�cient frontier estimators µ̂opt and �̂2opt has consis-

tency.

Result3　 µ̂opt and �̂2opt satisfy previous assumption 1�4.

Then, µ̂opt and �̂2opt converge to following values as n goes

to infinity.

µ̂opta.s.� µopt, �̂2

opta.s.� �2

opt ((n, d)-asymptotic)

• To derive asymptotic nomality of µ̂opt and �̂2opt

• To derive confidential interval and test of e�cient frontier

• To analyze e�cient frontier from actual stock price data

Assumption1 Let Z1, . . . , Zni.i.d.� (0, Id). Assume that entries of Zt are independent with 4 + � moment.

Assumption2 Then, data vectors Xt can be expressed as Xt = �12 Zt + µ. (µ � Rd,� > 0)

Assumption3 d is expressed with n, and d/n � � � (0, 1) (n � �) is satisfied.

We call this limit operation ”(n, d)-asymptotic”.

Assumption4 Assume that � converge the following constants �1, �2, �3.

�1 � �1, �2 � �2, �3 � �3 (d � �)

However, it is satisfied that �1, �3 > 0, �2 � R, �1�3 � �22 > 0.