Robust Optimization of the Investment Portfolio under Uncertainty Conditions

Journal of Environmental Treatment Techniques

2019, Special Issue on Environment, Management and Economy, Pages: 1093-1098

J. Environ. Treat. Tech.

ISSN: 2309-1185

Journal web link: http://www.jett.dormaj.com

Robust Optimization of the Investment Portfolio

under Uncertainty Conditions

Alexey G. Isavnin, Damir R. Galiev, Anton N. Karamyshev , Ilnur I. Makhmutov

Kazan Federal University, Naberezhnye Chelny Branch, Naberezhnye Chelny, Russia

Received: 13/09/2019

Accepted: 22/11/2019

Published: 20/12/2019

Abstract

Of the goal of this study is to investigate the assessment of expected return parameters µ and return covariance matrices Σ in modern

portfolio investment tasks. These parameters are used in almost all modern portfolio investment models, including the classic mean-

variance Markowitz model, Black-Litterman model, “smart ”models. In practice, they are difficult to evaluate correctly, since the

parameter values change every day. However, the quality of investment portfolio depends precisely on these parameters. The quality

of investment portfolio is understood as a combination of risk and profitability parameters. Number of methodologies are used in this

article to reduce the uncertainty of these parameters. The main idea of these methods is to reduce the sensitivity of resulting optimal

portfolios to uncertain input parameters. In other words, if the parameter values µ and Σ change slightly, the final portfolio shall not

radically change its structure. According the results gained in this article, one asset will not be able to dominate the final portfolio.

Chopra offers using the James-Stein estimate for the expected averages, while Black and Litterman use the Bayesian estimate µ and Σ

(

taking into account the expert opinions). There are also selection methods and scenarios that are described in detail, for example, in.

Of all these methods, the Black-Litterman model is most often used in practice.

Keywords: Black-Litterman model, Portfolio investment, Analysis, Uncertainty, Risk

Introduction

SOCP is a class of tasks that lies between linear

programming (LP) and semi-definite programming (SDP).

Quadratic programming problems, problems with hyperbolic

constraints, etc. are examples of SOCP class. SOCP can be

solved more efficiently than SDP. There are suitable numerical

methods for solving SOCP, which are implemented in some

software packages. In this work, we used the SEDUMI library

Robust optimization principles reduce the impact of the

problems described above. To do this, one shall first determine

the interval of possible parameter values µ and Σ. The value

interval is called an indefinite set of these parameters. The final

task will be solved for the "worst" case. As a result, the

investor will be able to see the guaranteed level of portfolio

income with the “worst ”development of events. Quite often,

VaR (Value at Risk) indicator is used as a criterion for the

an addition to the MATLAB complex for solving the

problems of SOCP and SDP class.

worst" case (1). Similar approaches are proposed in (2, 3, 4,

In this work, to give the model the robustness property, the

worst case optimization method will be used, and the risk of

capital loss will be introduced by defining VaR restrictions and

restrictions on the investment portfolio structure. The idea of

making models robust by optimizing the worst case is

described in (7, 8, 9). To optimize the worst case scenario, one

, 6).

Text of Article

To solve the problem of constructing optimal portfolios

(

without taking into account the uncertainty of parameters), the

Lagrange method or the Kuhn-Tucker theorem are used (if

there are restrictions on the portfolio structure). When solving

a robust optimization problem for the “worst ”case from an

indefinite set, the use of these methods is inefficient. Instead,

the task can be reduced to the class of the second order cone

problems (SOCP):

shall first specify the many possible portfolio returns S and

covariance matrices S . This set is called the "indefinite set" in

the literature. The scheme for generating indefinite sets for

returns and covariances is as follows:

퐿

푈

ꢁ

푈

휇 ≤휇 ≤휇 , ∀푖

ꢁ

(

퐿

ꢂ ≤ꢂꢁ푗 ≤ꢂ , ∀푖,ꢃ

ꢁ푗 ꢁ푗

푇

ꢁ

푀푖푛{푓 푥|ꢀ‖퐴푥+푏‖≤푐 푥+푑,ꢀ푖=1,...,푁}.

(1)

ꢁ

Corresponding Author: Anton N. Karamyshev, Kazan Federal University, Naberezhnye Chelny Branch, Naberezhnye Chelny,

Russia. E-mail: antonkar2005@yandex.ru.

093

Journal of Environmental Treatment Techniques

2019, Special Issue on Environment, Management and Economy, Pages: 1093-1098

ꢁ

ꢛ휇 푥 + ꢛ 훽푥 − ꢛ 훽푥

ꢁ

ꢁ ꢁ

퐿

푈

퐿

ꢁ

휇 =(휇 +휇 )/2,ꢀ훽 =(휇 −휇 )/2,

ꢁ

ꢁ::ꢏꢜ<0

ꢁ:ꢏꢜꢝ0

ꢛ(휇 푥 −훽|푥|)

ꢁ ꢁ ꢁ ꢁ

퐿

ꢁ푗

푈

ꢁ푗

푈

퐿

ꢁ

ꢂ =(ꢂ +ꢂ )/2,ꢀ훿 =(ꢂ −ꢂ )/2,휇 −

ꢁ

푗

ꢁ푗

훽 ≤휇 ≤휇 +훽, ∀푖

ꢁ

ꢁ푗

ꢂ −훿 ≤ꢂ ≤ꢂ +훿 , ∀푖,ꢃ

0 푇

푇

ꢁ푗

=ꢌ휇 ꢍ 푥−훽 |푥|

푆 =ꢄ휇:휇 −훽≤휇≤휇 +훽,훽≥ꢅꢆ

푇

푚

ꢎ푎푥ꢙ푥 훴푥ꢚ=ꢎ푎푥ꢛꢂ 푥푥

ꢁ푗 ꢁ 푗

푆 =ꢄ훴:훴 −훥≤훴≤훴 +훥,훥≥ꢅꢆ.

ꢘ

푣

ꢁ

,푗

ꢛ (ꢂ −훿 )푥푥

The formed worst-case optimization problem shall be

reduced to SOCP form, after which a robust statement of the

original problem will be obtained. Until recently, modern

portfolio theory formed by G. Markowitz as far back as 1952

remained almost the only quantitative method for solving the

portfolio analysis problem. The main idea of this theory is as

follows. Let there be n types of assets from which the investor

can form a portfolio. Capital is distributed between assets in

ꢁ푗

ꢁ푗 ꢁ 푗

ꢁ,푗:ꢏꢜꢏꢞ<0

ꢛ (ꢂ +훿 )푥푥

ꢁ

푗

ꢁ푗 ꢁ 푗

ꢁ

,푗:ꢏꢜꢏꢞꢝ0

=ꢛꢂꢁ푗푥ꢁ푥푗+ꢛ훿ꢁ푗ꢟ푥ꢁ푥푗ꢟ

ꢁ,푗

ꢛꢂ 푥푥 +ꢛ훿 |푥|ꢟ푥ꢟ

ꢁ푗 ꢁ 푗 ꢁ푗 ꢁ 푗

ꢇ

shares 푥,ꢀꢅ≤푥 ≤1,ꢀ∑ 푥 =1. Assets are characterized

ꢁ

ꢁꢈꢉ ꢁ

ꢁ

,푗

ꢁ,푗

by efficiencies R

, which are random variables with known

푇 0

푇

푥 훴 푥+|푥| 훥|푥|

i i

mathematical expectations MR =휇, and covariance matrix

훴=ꢊ푐표ꢋꢌ푅,푅ꢍꢊ. Markowitz problem is formulated as

follows:

ꢉ

푇

ꢒ

ꢁ

푗

0 푇

푇

푀푎푥ꢠꢌ휇 ꢍ 푥−훽 |푥|− 훾푥 훴푥−

ꢏ

푇

ꢉ

ꢒ

푇

훾|푥| 훥|푥||퐼 푥=퐶ꢡ.

푇

휆

퐼 푥=1

푇

ꢎ푎푥ꢐꢑ휇 푥− 푥 훴푥ꢓꢔ

ꢖ.

(3)

ꢇ

ꢏ

ꢒ

푥∈푅

ꢕ

As a result, the robust task will take the following form:

Although the Harry Markowitz model may seem attractive

and well-grounded from a theoretical point of view, a number

of problems arise in its practical application. Application of the

Markowitz model in the Russian market also showed its

inconsistency (9). Main disadvantages of the Markowitz

model:

ꢉ

0 푇

푇

푀푎푥ꢣꢌ휇 ꢍ 푥−훽 |푥|− 훾ꢤ−

ꢏ,ꢢ,휏

ꢒ

(5)

퐼^푇푥=퐶

ꢉ

ꢒ

푇 0

훾ꢥꢦꢤ≥푥 훴 푥 ꢧ.

푇

▪

The model does not take into account the fundamental

and other factors of profitability;

The model does not allow for taking into account the

uncertainty levels for individual assets;

With a slight change in the input parameters, one can

get a result that is very different from the previous one

instability);

In the absence of restrictions on the assets structure,

ꢥ≥|푥| 훥|푥|

▪

The Telser model is a logical continuation of the

Markowitz model. The main difference and advantage of this

model in contrast to the classical statement of the problem of

choosing the optimal portfolio is to control the risk of capital

loss using the VaR indicator. The model itself has the

following formulation:

▪

(

▪

there is a large number of negative weights in the final

portfolio.

Let us compose a robust model, having previously

performed a number of transformations:

푉푎푅훼 =−휇푝−푧훼ꢂ푝

ꢪ

ꢮ

푇

휇 =휇 푥

푝

ꢫ

ꢒ

푇

ꢎ푎푥 휇 ꢂ =푥 훴푥

(6)

푝 푝

ꢏ

ꢩ ꢫ

ꢭ

ꢬ

퐼^푇푥=퐶0

_푥_∈_푅ꢇ

ꢕ

푇

푀푎푥ꢐꢎ푖푛[휇 푥− 훾푥 훴푥]|퐼 푥=퐶 ꢖ

ꢨ

ꢏ

ꢗ,ꢘ

(

푇

푀푎푥ꢐꢎ푖푛ꢙ휇 푥ꢚ− 훾ꢎ푎푥ꢙ푥 훴푥ꢚ|퐼 푥=퐶 ꢖ

However, this model also inherits the main drawback of

the classical approach - a strong instability to the input

parameters. We compose a robust model according to the

above definitions and prerequisites by performing preliminary

transformations:

ꢏ

ꢗ

ꢘ

퐼^푇푥=퐶

푇

푀푎푥ꢯꢎ푖푛ꢙ휇 푥ꢚꢀꢰ

ꢳ

ꢏ

ꢗ

ꢎ_ꢗ푎_,_ꢘ푥ꢙ푃(푅푝 ≤−푉푎푅ꢱ)ꢚ≤ꢲ

(

푇

0 푇

푇

ꢎ푖푛ꢙ휇 푥ꢚ=ꢌ휇 ꢍ 푥−훽 |푥|

ꢗ

094

Journal of Environmental Treatment Techniques

2019, Special Issue on Environment, Management and Economy, Pages: 1093-1098

ꢎ푎푥ꢙ푃(푅 ≤−푉푎푅 )ꢚ≤ꢲꢀ

subjective opinion, (퐾×퐾); 훱 –equilibrium return vector,

푝

ꢱ

ꢗ,ꢘ

(푁×1); Q –vector of subjective views, (퐾×1).

푇

−

푉푎푅 −휇 푥

ꢱ

Uncertainty of subjective views is reflected in the error

vector 휀, whose elements are normally distributed with an

average of 0 and a matrix 훺. Thus, the final values of

subjective opinions have the form of 푄+휀.

⇔

ꢀꢎ푎푥

≤푧 ⇔

훼

ꢗ,ꢘ

푇

√

푥 훴푥

푇

−

푉푎푅 −ꢎ푖푛휇 푥

ꢱ

ꢗ

≤

푧 ꢀ

훼

푇

ꢎ푎푥√푥 훴푥

ꢘ

푇

^푄ꢉ

휀ꢉ

⋮

⇔

≤

ꢀ−ꢎ푖푛휇 푥−푧 ꢎ푎푥ꢴ푥 훴푥

훼

ꢗ

ꢘ

푄+휀=ꣀ ⋮ ꣁ+ꣀ ꣁ.

푄_푘

(12)

푉푎푅 ⇔

ꢱ

휀

푘

0 푇 푇

푇 0 푇

ꢌ휇 ꢍ 푥+훽 |푥|−푧 ꢴ푥 훴 푥+|푥| 훥|푥|≤푉푎푅

훼 ꢱ

−

⇔

Error vector elements 휀, usually nonzero. Variations 휔of

the error vector elements 휀form a diagonal covariance matrix

훺 and demonstrate the uncertainty measure of subjective

views. The matrix is diagonal, because subjective opinions are

independent of each other according to the model assumptions.

0,5

‖

ꢌ훴 ꢍ 푥‖

0 푇 푇

ꢷꢵ≤ꢌ휇 ꢍ 푥+훽 |푥|+푉푎푅 .

ꢱ

−푧_훼ꢵꢶ

0,5

훥 |푥|‖

As a result, the robust task will take the following form:



₁



0 

푇

푀푎푥ꢣꢌ휇 ꢍ 푥−



  0





ꢏ

(

13)



(

퐼^푇푥=퐶

 0

_k







푇

_‖_ꢌ_훴₀_ꢍ0,5푥‖

훽 |푥|ꢀꢦ

ꢧ.

ꢷꢵ≤ꢌ휇 ꢍ 푥+훽 |푥|+푉푎푅

ꢱ

푇

−

푧_훼ꢵꢶ

_훥0,5|푥|‖

‖

T here are several methods for determining matrix elements

훺(2, 10).

The Black-Litterman model was first published by Fisher

Black and Robert Litterman from Goldman Sachs

The values of returns for subjective views, located in the

column vector Q, are introduced into the model using the

matrix P. The presence of the influence of each subjective

opinion is reflected in the line vector of dimension 1×푁.

Thus, we get the matrix P of dimension 퐾×푁for K views:

(2,18,19,20). They proposed a theory of "equilibrium

approach". Moreover, equilibrium is understood as an

idealized state in which demand is equivalent to supply.

According to the authors, “natural forces”, the functioning of

which eliminates the deviation from equilibrium, function in

the economic system. Equilibrium returns are calculated by the

formula:

^ꣂꢉ,ꢉ ⋯ ꣂ_ꢉ_,_ꢇ

푃=ꣀ ⋮ ⋱ ⋮ ꣁ.

(14)

ꣂ

⋯ ꣂ

푘,ꢉ

푘,ꢇ

훱=ꢸ훴푤_푚_푘_푡

(9)

The final formula is as follows:

where 훱 - equilibrium return vector; ꢸ - risk aversion

coefficient; 훴- covariance matrix of historical returns; 푤_푚_푘_푡

푤=휇ꢌꢸ훴ꢍ^ꢹ^ꢉ

(15)

market capitalization vector of each asset relative to the

capitalization amount of assets in the portfolio. The coefficient

Let us make a robust model. In this case, due to the vector

formation features for estimating future returns, the use of the

previous schemes is unacceptable. To give robustness, we

introduce restrictions on the portfolio structure, as well as

introduce VaR restrictions to control the risk of capital loss:

ꢸ

characterizes the investor’s willingness to sacrifice the value

of expected portfolio return in order to reduce its risk:

퐸ꢌ푟ꢍꢹ푟ꢺ

ꢸ

(10)

ꢻ^ꢼ

퐼^푇푥=1

ꢪ

ꢮ

where ꢽꢌꢾꢍ- expected market return, ꢾ - risk-free interest

퐴푋≤푏

푥 훴푥≤푠

ꢿ

ꢫ

ꢒ

푇

rate, ꢂ =푤 훴푤

- market portfolio dispersion. Let us

푚푘푡

푇

ꢎ푎푥 휇 푥ꢀ

(16)

consider the Black-Litterman formula for the posterior return

vector (7). It is a key point before calculating the final portfolio

푃ꢑ푌≤푉푎푅 ꢌ푌ꢍꢓ>ꣂ

ꢏ

ꢩ

푝

ꢭ

ꢬ

ꢫ

푌∈푅

(

6). Let K is the number of subjective opinions, N - the number

ꢨ

푥∈푅^ꢇ

ꢕ

of assets.

We preliminary carry out a series of transformations

according to the method proposed in (11):

ꢹꢉ

ꢹꢉ ꢹꢉ

ꢹꢉ

휇=ꢙꢌꢥ훴ꢍ +푃′훺 푃ꢚ ꢙꢌꢥ훴ꢍ 훱+푃′훺 푄ꢚ

(11)

Where µ –new (posterior) mixed return vector (푁×1); ꢥ

scaling factor; 훴 –return covariance matrix with dimension

푇

푃ꢑ푌≤푉푎푅 ꢌ푌ꢍꢓ>ꣂ⇔푃ꢌ휉 푥≥−훽ꢍ≥ꣂ

(17)

–

푝

(푁×푁); P –dimension matrix (퐾×푁), which identifies

assets for which the investor has a subjective opinion; 훺 –

diagonal covariance matrix with confidence levels for each

095

Journal of Environmental Treatment Techniques

2019, Special Issue on Environment, Management and Economy, Pages: 1093-1098

푇

휉 푥−휇 푥 −훽−휇 푥

consider the results of experiments with a flat trend. It can be

seen that profitability increases for all models, with an increase

in risk (Fig. 1, 2). However, robust models have higher returns

at approximately the same risk levels. Consequently, the

quality of models is increased.

Similarly, let us consider the results of experiments in the

Russian market with a ascending trend. It can be seen that

profitability increases for all models, with an increase in risk

(Fig. 3, 4). However, in case of ascending trend, there is a high

return on portfolios both with standard and robust formulations

of the problem. The quality of robust models is slightly higher

than the quality of models in a standard setting. Again, we can

see that the Black-Litterman model dominates, while the

classical mean-variance model and the Telser model behave

roughly the same. It depends on several reasons.

Similarly, Sharp coefficients were calculated for

ascending trend for various risk levels (Fig. 6). Firstly,

forecasts from analytical departments with adequate

forecasting ability were used (12, 13, 16, 17). Secondly, in the

robust formulation of the problem, additional restrictions were

introduced on the portfolio structure, which made it possible

to maintain the diversification level at higher risks. We draw

attention to the behavior of the Sharpe coefficient at various

risk levels. Sharp values for lateral trend are shown below (Fig.

푇

푃ꢌ휉 푥≥−훽ꢍ=푃ꣃ

≥

꣄

푇

푥 훴푥

푇

√

푥 훴푥

푇

−

훽−휇 푥

1−퐹 ꣃ

꣄

ꢌꢏꢍ

푇

√

푥 훴푥

푇

−훽−휇 푥

꣄≥ꢅ.9ꣅ⇔퐹 ꣃ ꣄

ꢌꢏꢍ

−

훽−휇 푥

−퐹 ꣃ

ꢌꢏꢍ

푇

√푥 훴푥

√

푥 훴푥

≤

ꢅ.ꢅꣅ

꣇

ꢹ

꣆ꢹꢗ ꢏ

ꢹ

ꢉ

푇

≤퐹 ꢌꢅ.ꢅꣅꢍ=휇 푥+

ꢏ꣇ꢘꢏ

ꢌꢏꢍ

ꢴ

ꢹ

ꢉ

푇

퐹 ꢌꢅ.ꢅꣅꢍ√푥 훴푥≥−훽.

ꢌꢏꢍ

As a result, the robust task will take the following form:

_퐼푇_푥₌₁

ꢪ

ꢮ

퐴푋≤푏

ꢫ 푇

푥 훴푥≤푠

푇

ꢎ푎푥 휇 푥ꢀ

(18)

푇

ꢹꢉ

ꢌꢏꢍ

푇

ꢏ

휇 푥+퐹 ꢌꢅ.ꢅꣅꢍ√푥 훴푥≥−훽

ꢩ

ꢭ

ꢬ

ꢫ

푌∈푅

ꢇ

ꢕ

ꢨ

푥∈푅

Profitability is one of the most important indicators of

portfolio management efficiency, indicating management

efficiency. But it is impossible to judge the quality of

management strategy using only profitability. In addition to

profitability, there is a downside - risk, neglect of it in

assessing effectiveness can distort the real state of things. In

this work, Sharpe and Schwager coefficients were used to

assess the effectiveness of investment portfolio.

In total, several experiments were carried out as part of the

work. Time interval: 01.07.2010

conducting experiments on constructing optimal portfolios

using the described models, we used data on daily stock quotes

traded on the MICEX. The experiments were conducted on the

Russian market with ascending and flat trends. Let us first

5).

Methods

In the course of the study, the authors applied the following

methods:

1. Selective analysis of specialized literature with a high

citation index on the topics indicated in the article title. In

particular, we considered the Lagrange method, Kuhn-Tucker

theorem, modern portfolio theory of G. Markowitz, Telser

model, and Black-Litterman model.

–01.02.2011. When

Figure 1: Risks and returns of portfolios with a flat trend

Figure 2: Risks and returns of portfolios with a flat trend

096