=Paper=
{{Paper
|id=Vol-1623/paperme6
|storemode=property
|title=On the Equivalence of Optimality Principles in the Two-Criteria Problem of the Investment Portfolio Choice
|pdfUrl=https://ceur-ws.org/Vol-1623/paperme6.pdf
|volume=Vol-1623
|authors=Victor Gorelik,Tatiana Zolotova
|dblpUrl=https://dblp.org/rec/conf/door/GorelikZ16
}}
==On the Equivalence of Optimality Principles in the Two-Criteria Problem of the Investment Portfolio Choice==
https://ceur-ws.org/Vol-1623/paperme6.pdf
On the Equivalence of Optimality Principles
in the Two-Criteria Problem
of the Investment Portfolio Choice
Victor Gorelik1 and Tatiana Zolotova2
1
Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia
2
Financial University under the Government of the Russian Federation
Abstract. In this paper, we examine the problem of finding an optimal portfo-
lio of securities by using the probability function of portfolio risk as a constraint.
We obtain the value of the risk coefficient for which the problem of maximiz-
ing the expectation of the portfolio return with a probabilistic risk function
constraint is equivalent to the maximizing the linear convolution of the criteria
”expectation – variance”. The positive correlation of portfolios returns that are
solutions of different optimization problems is proved.
Keywords: optimality principles, expectation, variance, correlation, convolu-
tion of the criteria
1 Introduction
The problem on the choice of optimality principles of investor behavior in the stock
market has been discussed in an extensive literature (for example, [1], [3], [5-7], [9], [12]).
In this case, the development of optimality criteria for a securities portfolio involves
solving the issue on the relationship between the return and risk of the portfolio. A
static formulation of the problem on the portfolio selection initially was proposed by
Markowitz [9]. In his studies, Markowitz used for the risk assessment a risk function
defined in the metric l22 (variance). Then Markowitz [10] stated the problem on the
selection of an optimal portfolio as the problem of minimizing the difference between
the variance and the expectation of the portfolio return. In addition, in the same book
the problem of maximizing the expected return under a constraint on the variance is
considered. The problem of minimizing the variance under the constraint on the return
was also considered. Solutions of all these problems are efficient portfolios.
Sharpe et al. [12] has proposed to consider the probability risk functions (VAR)
for finding the optimal portfolio of securities. This trend has been developed in recent
papers ([1], [4], [8], [13]).
In the papers of Gorelik and Zolotova ([1], [2]) the problem on portfolio selection was
considered as the problem of maximizing a linear convolution of criteria ”expectation—
variance” with a weight factor (risk coefficient). By the convexity of the set of attainable
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
On the Equivalence of Optimality Principles 597
values for the expectation and variance of portfolios (in the ”north-west” direction) it
gives necessary and sufficient conditions for the Pareto optimality, i.e., any problem
whose solution is an effective portfolio is equivalent to a given problem at a certain
value of risk coefficient.
In this paper, we consider two statements of the problem on portfolio selection: the
problem of maximizing the expectation with probabilistic risk function in a constraint
and the problem of maximizing the linear convolution of the criteria ”expectation—
variance” with a weight factor. We show that the optimal choice in the problem with
probabilistic risk function in a constraint leads to one of the efficient portfolios corre-
sponding to a definite value of the risk coefficient at the variance in the problem of
maximizing the linear convolution of the criteria ”expectation—variance”. We explic-
itly find also the value of weight factor in terms of the known initial parameters of the
problem. An example of finding the optimal portfolio of shares of Russian companies is
given. It illustrates the obtained results and demonstrates some of the characteristics of
methods for optimal portfolio selection. In addition, a positive correlation of portfolios
returns as the decisions of the various productions of optimization problems is proved.
That allows us to speak not only about formal equivalence of models under fixed initial
data but also on their uniform response to market fluctuations.
2 The equivalence of the problem of maximizing the
expectation with probabilistic risk function in a
constraint and the problem of maximizing the convolution
”expectation—standard deviation”
We assume that the stock market is characterized by the vector of expectations of
financial instruments r̄ = (r̄1 , ..., r̄i , ..., r̄n ) and the covariance matrix K. We assume
that the behavior of an investor whose control is the vector x (a portfolio of securities)
is based on this information. Components of the portfolio are proportions of funds
invested in financial instruments of the final list (i = 1, ..., n).
Define an optimal portfolio as a solution of the problem of maximizing the expec-
tation of the portfolio return, provided that the probability of a negative random value
of the portfolio return does not exceed a given, sufficiently small value:
max r̄x, P (rx ≤ 0) ≤ ε, xe = 1, (1)
x
where ε is a given sufficiently small positive value, e = (1, ..., 1), and P is the probability.
Hereinafter, there is no distinction in notation of row vectors and column vectors; we
assume that these vectors comply with the requirements of multiplication of matrices
and vectors. The problem of selecting an optimal portfolio in (1) and problems discussed
below implies the presence of short sales that take place through securities lending,
which would then be repaid by the same securities (this is reflected in the absence of
the nonnegative conditions for the components of the vector x ).
We show that problem (1) is reduced to a problem of convex programming and its
solution coincides with the solution of the problem of maximizing the linear convolution
598 V. Gorelik, T. Zolotova
of the criteria of the expectation and the standard deviation of the random portfolio
return for some weight coefficient of the standard deviation. Consider the problem
1
max r̄x, kr̄x ≥ (xKx) /2 , xe = 1, (2)
x
for which the Lagrange function
1
L(x, λ) = r̄x + λ(kr̄x − (xKx) /2 ) (3)
is defined on the set X = {x|xe = 1}, λ is the Lagrange multiplier, k is a positive
coefficient.
Lemma 1. If the convex programming problem (2) has a solution x0 and the corre-
sponding Lagrange multiplier is positive, λ0 > 0, i.e., (x0 , λ0 ) is a saddle point of the
function (3), then x0 is a solution of the problem
λ0 1
max[r̄x − 0
(xKx) /2 ], xe = 1. (4)
x 1+λ k
Proof. By the convexity, problem (2) is equivalent to the maximizing the Lagrange
1
function L(x, λ0 ) = r̄x+λ0 (kr̄x−(xKx) /2 ) on the set X, where the Lagrange multiplier
λ0 provides a minimum of the function L(x0 , λ). Since λ0 > 0, we see that the inequality
1
constraint in problem (2) at the optimal point becomes active: kr̄x0 = (x0 Kx0 ) /2
(otherwise, this constraint would be insignificant). After a transformation we obtain
L(x,λ0 ) λ0 1/2
1+λ0 k = r̄x − 1+λ0 k (xKx) . Then the equivalent problem takes the form (4), which
was required.
Theorem 1. Let {ri } be a system of random variables each of which has a normal
distribution, r̄i be the expectations, K = (σij )n×n be the covariance matrix, and let
the conditions of the lemma hold. Then the solution of problem (1) coincides with
the solution of the problem of maximizing the linear convolution of the criteria of the
expectation and the standard deviation of the random portfolio return:
1
max[r̄x − α1 (xKx) /2 ], (5)
x∈X
0
−1
λ
where α1 = 1+λ 0 d , d = (Φ (1−2ε))−1 , Φ(·) is the Laplace function, λ0 is the value of
the Lagrange multiplier in problem (2).
Proof. We prove that problem (1) is reduced to problem (2) under these assumptions.
∫0 −(t−m)2
The random variable rx is normally distributed, i.e., P (rx ≤ 0) = σ√12π −∞ e 2σ2 dt,
1
where m = r̄x is the expectation of the random portfolio return and σ = (xKx) /2
is the standard deviation of the random variable rx. We transform the expression
∫ y −t2
forP (rx ≤ 0) by using the Laplace function Φ(y) = √22π 0 e 2 dt: P (rx ≤ 0) =
∫0 −(t−m)2 ∫ −m −z 2
√1 e 2σ 2 dt = √12π −∞σ
e 2 dz, where z = t−m or t = m + σ z. We have
σ 2π −∞ σ
∫0 −z 2 ∫m −z 2
P (rx ≤ 0) = √2π −∞ e 2 dz − √2π 0σ e 2 dz = 2 + Φ(0) − Φ( m
1 1 1
σ ) = 2 − 2 Φ( σ ). By
1 1 m
On the Equivalence of Optimality Principles 599
−1
the condition, 12 − 12 Φ( m
σ ) ≤ ε, therefore, σ ≥ Φ
m
(1 − 2ε) or m
σ
≤ (Φ−1 (1 − 2ε))−1 .
−1 −1
Introducing the notation (Φ (1−2ε)) = d, we arrive at the problem of convex pro-
gramming (2), in which k = d:
1
max r̄x, dr̄x ≥ (xKx) /2 , xe = 1. (6)
x
By the lemma 1, problem (6) is equivalent to problem (4), i.e., problem (1) is
equivalent to (4).
If we define an optimal portfolio from the solution of the problem of maximizing
the linear convolution of the criteria of the expectation and the standard deviation of
the random portfolio return,
1
max[r̄x − α1 (xKx) /2 ], (7)
x∈X
0
λ
where α1 > 0 is the risk coefficient, then for α1 = 1+λ 0 d problem (7) coincides with
0
problem(4). By the lemma, λ > 0, and the Laplace function takes positive values, and
λ0 λ0
hence 1+λ 0 d > 0. Therefore, problem (7) for α1 = 1+λ0 d is equivalent to the initial
problem (1). The theorem is proved.
The optimality conditions for the problem(7) due to the presence of the square root
in the objective function are complicated. It is therefore desirable to find a connection
of the problem (1) with a suitable convolution ”expectation - variance”.
3 The equivalence of the problem of maximizing the
expectation with probabilistic risk function in a constraint
and the problem of maximizing the linear convolution of
the criteria ”expectation — variance”
Now we find an optimal portfolio as a solution of the problem of maximizing the linear
convolution of the expectation and variance criteria for the portfolio return with the
weight coefficient α > 0:
max[r̄x − α(xKx)], xe = 1. (8)
x
As it is known, the covariance matrix K is nonnegative definite, and therefore (1)
is a convex programming problem. Let assume that the matrix K is no degenerate
(detK ̸=0). We examine the following question: in which case solutions of problems (1)
and (8) coincide.
Theorem 2. Let x0 be a solution of problem (1), the optimal value of the Lagrange
multiplier in problem (2) is positive, λ0 > 0, and the covariance matrix K = (σij )n×n
is strongly positive definite. Then there exists a value of the weight coefficient α in
problem (8) such that the solutions of problems (1) and (8) coincide.
600 V. Gorelik, T. Zolotova
Proof. We denote the expected return of a portfolio r̄x0 at the solution point of problem
(1) by rp0 , i.e., r̄x0 = rp0 , and consider two equivalent problems
min xKx, r̄x ≥ rp0 , xe = 1, (9)
x
and
1
min (xKx) /2 , r̄x ≥ rp0 , xe = 1. (10)
x
Note that problems (9) and (10) are equivalent for any rp . By the convexity of the
Pareto set in the space ”expectation–standard deviation”, the point x0 as a solution
of problem (5), i.e., the maximum of the linear convolution of these criteria, is Pareto-
1
optimal. Therefore, the minimum in problem (10) cannot be less than (x0 Kx0 ) /2 .
0
However, it satisfies the constraint in problem (10); therefore, x is a solution of problem
(10) and the equivalent problem (9). By the convexity, problem (9) is equivalent to the
problem of minimizing the Lagrange function L1 (x, λ01 ) = xKx + λ01 (rp0 − r̄x) on the set
X for some value of the Lagrange multiplier λ01 , and this value λ01 provides the maximum
of the function L1 (x0 , λ1 ). This problem is equivalent to the problem of minimizing the
function xKx − λ01 r̄x on the set X. Obviously, λ01 > 0, since in the opposite case
the problem is reduced to the minimizing of the variance, i.e., the constraint r̄x ≥ rp0
becomes insignificant. We set α = λ10 , then α > 0 and the solutions of problems (8)
1
and (9) coincide and hence x0 is a solution of problem (8), which was required.
Theorem 2 proves the existence of a value of the risk coefficient α in problem (8)
for which solutions of problems (1) and (8) coincide. However, Theorem 2 allows one to
find the risk coefficient only by solving problem (9). In the following assertion (Theorem
3), we obtain a value of the risk coefficient α.
The Lagrange function for problem (8) is L(x, λ) = r̄x − α(xKx) + λ(1 − xe).
Optimality conditions of portfolio lead to a system of linear algebraic equations: r̄ −
2α(Kx0 )−λe = 0, x0 e = 1, which solution gives the optimal portfolio for the problem
(8):
K −1 e −1 eK −1 r̄ −1 1
x0 (α) = + (K r̄ − K e) . (11)
eK −1 e eK −1 e 2α
We show that the weight coefficient α can be expressed through the parameters of the
problem (1).
Theorem 3. Let the conditions of Theorem 2 be satisfied. If in problem (8) the weight
coefficient α satisfies the equation
( −1 )2 ( −1 ) ( −1
)
r̄)2
4(1 − r̄K e
eK −1 e d2 )α2 − 4d2 r̄K e
eK −1 e r̄K −1 r̄ − (eK eK −1 e α+
( −1 2
) ( −1 2
)2 (12)
+ r̄K −1 r̄ − (eK r̄)
eK −1 e − d2 r̄K −1 r̄ − (eK r̄)
eK −1 e = 0,
where d = (Φ−1 (1−2ε))−1 , ε > 0, then solutions of problems (1) and (8) coincide.
On the Equivalence of Optimality Principles 601
0
λ
Proof. By Theorem 1, solutions of problems (1) and (5) coincide for α1 = 1+λ 0d ,
1
moreover, by the lemma, the condition λ0 > 0 leads to dr̄x0 = (x0 Kx0 ) /2 in prob-
lem (2), to which problem (1) is reduced. Thus, a solution of problem (8) defin-
ing a portfolio (11) is also a solution of problem (1) if the following relation holds
1
dr̄x0 (α) = (x0 (α)Kx0 (α)) /2 . Using (11) we have
( −1 ( ) )2 ( −1 )2
−1 (eK −1 r̄)2
(dr̄x0 (α))2 = r̄K −1
eK e
e
d + r̄K r̄ − −1
eK e
d
2α = r̄K e
eK −1 e d2 +
( −1 ) ( −1
) ( −1
) 2 2
r̄)2 d2 r̄)2
+ r̄K e
eK −1 e r̄K −1 r̄ − (eK
eK −1 e α + r̄K
−1
r̄ − (eK
eK −1 e
d
4α2 ,
x0 (α)Kx
(
0
(α) = ) ( )
K −1 e −1
K −1 e −1
= −1 + (K −1 r̄ − eK r̄
eK −1 e K
−1 1
e) 2α K eK −1 e + (K
−1
r̄ − eK r̄
−1 K
−1 1
e) 2α =
( eK e −1
) ( −1
)eK e
−1
eK −1 e + (r̄ − eK −1 e e) 2α K eK −1 e + (r̄ − eK −1 e e) 2α =
e eK r̄ 1 e eK r̄ 1
=
−1 −1 −1 −1
−1 −1
−1 e (r̄ − eK −1 e e) α + (r̄ − eK −1 e e)(K r̄ − eK
K e eK r̄ 1 eK r̄ r̄
= 1 + (eK ) eK −1 e K e) 4α1 2 =
−1 2
= 1 + r̄K −1 r̄ − (eK r̄)
eK −1 e
1
4α2 .
Equating the right-hand sides of these equalities, we obtain:
( −1 )2 ( −1 ) ( ) 2
−1 (eK −1 r̄)2
r̄K e
−1
eK e d 2
+ r̄K e
eK e −1 r̄K r̄ − −1
eK e
d
α+
( −1
) 2 2 ( )
2
(eK −1 r̄)2
+ r̄K −1 r̄ − (eK −1
eK e
r̄) d
4α 2 = 1 + r̄K −1
r̄ − −1
eK e
1
4α2 .
After a simple transformation we have
( −1 )2 ( −1 ) ( −1
) 2
r̄)2
1 − r̄K e
−1 e d2 − r̄K e
−1 e r̄K −1 r̄ − (eK −1 e
d
+
(( eK eK
) (
eK
)
α)
−1 −1 2
r̄)2 r̄)2
+ r̄K −1 r̄ − (eK
eK −1 e − d2 r̄K −1 r̄ − (eK
eK −1 e
1
4α2 = 0.
Multiplying both sides by 4α2 , we obtain the quadratic equation (12). The theorem is
proved.
Example 1. Find the connection between the problems (1) and (8) for a portfolio of
shares of ”Aeroflot”, ”MTS” and ”Megaphone”, using statistical data of stock prices
of these companies for the period from January 2013 to January 2014 [15].
To solve these problems we use a specialized program ([11], [14]), written in VB.NET
programming language. This program determines the structure of the portfolio, its re-
turn mean and standard deviation for the various models and statistics data, selected
by user. Shares returns of companies under consideration were determined using the
daily closing prices of trading sessions. For shares of ”Aeroflot”, ”MTS” and ”Mega-
phone” we have the vector of expectations of shares returns r̄ = (0, 967; 0, 189; 0, 327)
and covariance matrix
0, 65 0, 466 −0, 18
K = 0, 466 1, 678 −0, 189 .
−0, 18 −0, 189 0, 379
602 V. Gorelik, T. Zolotova
Let ε = 0, 2 in problem (1). Then 1 − 2ε = 0, 6 and by the table of values of the
Laplace function we have Φ−1 (1−2ε) = 0, 85 and hence d = (Φ−1 (1−2ε))−1 = 1, 176.
Solving problem (6), which is equivalent to (1) by Theorem 1, for d = 1, 176 we obtain
an optimal portfolio x0 = (13, 85; −7, 518; −5, 332). The risk coefficient α in Problem
(8) found as the solution of Eq. (12) for d = 1, 176, is α = 0, 036. Then by formula (11)
we obtain the following composition of the portfolio: x0 = (13, 042; −7, 064; −4, 978).
The approximate coincidence of the solutions of problems (1) and (8) can be explained
by the fact that the solution of problem (6) with nonlinear constraints obtained by
using the software Mathcad turns out very inaccurate. Note that the second root of
the quadratic equation (12) is α = 0, 792, according to the (11) it gives the portfolio
(0, 923; −0, 263; 0, 34) which does not coincide with the solution of the problem (6).
High volatility (shares of the company ”MTS” has dispersion 167.8%) leads to the
fact that for small ε constraints of the problem (6) are not satisfied. Suppose that in
the problem(1) ε = 0, 001, then 1 − 2ε = 0, 998 and using the table of the Laplace
function values we have Φ−1 (1−2ε) = 3, 09, d = (Φ−1 (1−2ε))−1 = 0, 324. The problem
(6) at d = 0, 324 has no solutions.
4 Studying the correlation dependency of return rates of
optimal portfolios
Let’s calculate the correlation moments of return rates of the optimal portfolios found
based on different models. In a view of the above theorems it is enough to select
different values of the factor α in the problem (8). First, we’ll calculate the covariance
matrix cov(x1 , x2 ) of the return rates of two arbitrary portfolios consisting of x1 and
x2 . Let M stand for the mean of a random value, rx is the return rate of the portfolio
taking random values, r̄x is the expected return rate of the portfolio. According to the
definition of covariance we get the following:
cov(x1∑
, x2 ) = M [(rx∑1 − r̄x1 )(rx2 − r̄x2 )] =
n n ∑n ∑n
= M [(∑ i=1 ri x1i − i=1 ∑ r̄i x1i )( i=1 ri x2i −
∑
2
i=1 r̄i xi )] =
n n n
= M [ i=1 (ri − r̄i )xi i=1 (ri − r̄i )xi ]M [ i,j=1 (ri − r̄i )(rj − r̄j )x1i x2j ] =
1 2
∑n ∑n
= i,j=1 M [(ri − r̄i )(rj − r̄j )]x1i x2j = i,j=1 Kij x1i x2j .
Thus, the covariance of random values of return rates of two portfolios is calculated
through the components of these portfolios with the help of the following formula:
cov(x1 , x2 ) = x1 Kx2 . (13)
Theorem 4. Covariance cov(x01 , x02 ) of the two optimal portfolios is positive.
Proof. Let’s demonstrate that if detσ̸=0, then eK −1 e > 0. ⟨Kx, x⟩ ≥ 0 implies that
⟨K −1 x, x⟩ ≥ 0 ∀x, where ⟨·, ·⟩ is the inner product of the vectors. Indeed, let’s take the
equation Kx = ζ. If we multiply the equation by x, we have ⟨Kx, x⟩ = ⟨ζ, x⟩ ≥ 0. On
the other hand, x = K −1 ζ and ⟨x, ζ⟩ = ⟨K −1 ζ, ζ⟩ ≥ 0 ∀ζ. As it is commonly known, the
minimal eigenvalue of the symmetric matrix K −1 equals the minimum of the quadratic
form ⟨K −1 x, x⟩ on a unit sphere ⟨x, x⟩ = 1. Suppose that ∃x̃ : ⟨K x̃, x̃⟩ = 0, therefore,
On the Equivalence of Optimality Principles 603
the minimal eigenvalue µmin = 0. Then, the characteristic equation det(K −1 −µE) = 0,
where E is the diagonal identity matrix, produces det K −1 = 0 if µmin = 0. Here we
reach a contradiction. It means that ∀x ⟨x, x⟩ = 1, ⟨K −1 x, x⟩ > 0. For the vector ẽ =
√e belonging to the unit sphere, it holds true that ⟨K −1 ẽ, ẽ⟩ > 0 or n−1 ⟨K −1 e, e⟩ > 0,
n
i.e. eK −1 e > 0.
A structure of an optimal portfolio (11) can be presented as follows
x0 (β) = C0 + C1 β, (14)
1
where β = 2α and C0 = (C01 , . . . , C0j , . . . , C0n ), C1 = (C11 , . . . , C1j , . . . , C1n ) are
determined according to the following formulas
K −1 e eK −1 r̄ −1
C0 = −1
, C1 = K −1 r̄ − K e. (15)
eK e eK −1 e
Consider two optimal portfolios x01 and x02 , which are determined from the solution of
the problem (8) at different values of the parameter α or, in accordance with the above
notation, the parameter β. According to (14), the structure of the optimal investment
portfolios looks as follows: x01 = C0 +C1 β1 and x01 = C0 +C1 β2 respectively. It follows
from (13) and (15) that the covariance of the two portfolios returns is cov(x01 , x02 ) =
x01 Kx02 = (C0 + C1 β1 )K(C0 + C1 β2 ) = C0 KC0 + (C1 KC1 )β1 β2 + (C0 KC1 )β2 +
(C1 KC0 )β1 .
Since the matrix K is symmetric, we have the equality C0 KC1 = C1 KC0 and the
following expression for covariance:
cov(x01 , x02 ) = C0 KC0 + (C1 KC1 )β1 β2 + (C0 KC1 )(β1 + β2 ). (16)
Using the properties of the inner product and (15), we have
−1 −1 −1 −1
−1 −1
C0 KC1 = eK K e
−1 e K(K r̄ − eK r̄
−1 e K
eK−1
K e
e) = eK −1 e (r̄ − eK −1 e e) =
eK r̄
−1 −1
r̄)(eK −1 e)
= eK1−1 e (⟨K −1 e, r̄⟩ − eK eK r̄⟨K e,e⟩
−1 e ) = eK1−1 e (eK −1 r̄ − (eK eK −1 e ) = 0.
Then, (16) will look as follows: cov(x01 , x02 ) = C0 KC0 + (C1 KC1 )β1 β2 .
Since the matrix K is non-negatively defined, then C1 KC1 ≥ 0 and C0 KC0 =
K −1 e K −1 e ⟨K −1 e,e⟩ 1
eK −1 e K eK −1 e = (eK −1 e)2 = eK −1 e > 0. It means that the inequality
cov(x , x ) > 0 holds true for two optimal portfolios x01 and x02 , Q.E.D.
01 02
Comment. In the absence of short selling a similar result holds under the additional
assumption of strict positive definiteness of the covariance matrix K.
Example 2. For the data of Example 1 determine the covariance of the two portfolios,
which are solutions of problems (1) and (8). It was shown above, that the solution
of the problem (1) at d = 1, 176 is x0 = (13, 85; −7, 518; −5, 332). Risk factor α
in the problem (8) at d = 1, 176 is α = 0, 036, and by (11) the structure of the
portfolio is x0 (0, 036) = (13, 042; −7, 064; −4, 978). Let now α = 1 (i.e., consider
the model of Markowitz [10]), then the solution of the problem (8) is the portfolio
x0 (1) = (0, 804; −0, 196; 0, 392). According to (13) we have a positive correlation of
optimal portfolios cov(x0 , x0 (1)) = x0 Kx0 (1) = 4, 991.
604 V. Gorelik, T. Zolotova
5 Conclusion
We have considered the problem of finding an optimal portfolio of securities using the
probabilistic function of portfolio risk. We have found the value of the risk coefficient
in the model ”expectation–variance” at which the problem of maximizing the expected
return with the probabilistic risk function in a constraint is equivalent to the problem
of maximizing the linear convolution of criteria ”expectation–variance”. Thus, if we
use the model with a probabilistic risk function for the search of an optimal portfolio,
the results of this study make it possible to determine the equivalent ratio of the
investor to risk (the risk coefficient). The convex programming problem (6), to which
the problem (1) is reduced at first, is inconvenient from the computing point of view;
this is related to the type of nonlinear constraints that make it difficult to find an exact
solution analytically, whereas numerical methods provide an approximate solution with
large errors. The problem (8) is computationally most convenient because it is reduced
to a system of linear equations. The results obtained in the present paper allow one to
solve problem (8) instead of (1) for certain values of the parameters of these problems.
Positive covariance of different portfolios returns means that a particular investor
control has a property of stability in a sense, that using various two-criteria decision-
making models, he obtains portfolios which random returns tend to vary in the same
direction. So we can talk about the robustness of the two-criteria model of portfolio
formation (with the use of sub-optimization or convolution of criteria of mathematical
expectation, variance, standard deviation, VAR). This result (Theorem 4), as well as
the existence of a weighting factor for which the solutions of problems (1) and (8)
coincide (Theorem 2) are valid for any law of return distribution.
Quantitative estimates of the parameters in Theorems 1 and 3 were obtained under
the assumption of a normal distribution of returns. It should be noted that in the
simulation of some processes in the economy and finance ”heavy-tailed” distributions
of random variables are used (eg. Pareto). However, the normal distribution is often
most convenient for modeling of random processes. Moreover, according to the central
limit theorem of probability theory, linear combination of a sufficiently large number of
comparable variance random variables with any laws of distribution is approximately
normal distributed. In addition, in this study, the formulations of problems exclude
right tails.
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