The problem for choice of an optimum investment portfolio is considered. The square-law form of risk is presented as two-multiple convolution of ÐºÐ¾Ð²Ð°Ñ€Ð¸Ð°Ð½Ñ‚Ð½Ð¾Ð³Ð¾ tensor of the covariance matrix and kontravariant vector of weights. By means of reduction of covariance matrix to the diagonal form, the problem by definition of optimum structure of a portfolio is solved: simple expressions for a minimum of risk and optimum distribution of the weights providing this minimum are received.tensor, convolution, invariants, risky assets, portfolio, covariance matrix, kontravariant vector, optimum structural potentials, relative optimum structural potentials

Alexander Milnikov

Mikheil Mamistvalov

English

Research Papers in Economics

One method of solution of an optimum investment 
portfolio problem for risky assets
Alexander MILNIKOV
Mikheil MAMISTVALOV
Abstract:The problem for choice of an optimum investment portfolio is 
considered. The square-law form of risk is presented as two-multiple convolution 
of covariant tensor of the covariance matrix and contravariant vector of weights. 
By means of reduction of covariance matrix to the diagonal form, the problem by 
definition of optimum structure of a portfolio is solved: simple expressions for a 
minimum of risk and optimum distribution of the weights providing this minimum 
are received.
Keywords:  tensor,  convolution,  invariants,  risky  assets,  portfolio, 
covariance matrix, contravariant vector, optimum structural potentials, relative 
optimum structural potentials
The  primary  goal  in  the  theory  of  a  choice  of  an  optimum 
investment portfolio can be reduced to a multidimensional problem of 
minimization of a quadratic form with constraints [1, 2]. Here the basic role 
covariance matrix g plays. It reflects a set of relationships among n shares.  ij  
The vector of weights w, characterizing distribution of invested means, 
should satisfy equation                 
In this case the mentioned problem of minimization looks as follows.
It is required to minimize the quadratic form representing a square 
of total risk of the given portfolio.
 (1)
with constraint
 (2)
Thus, the decision of the formulated problem will allow defining 
such distribution of weights of an investment portfolio, which minimizes 
risk. The received distribution we refer as optimum structure of a portfolio. 
We underline, that we do not consider shares with zero risk.
Alexander MILNIKOV is a professor in the Faculty of Information Tecnologies and Engineering at 
International Black Sea University, Georgia
Mikheil MAMISTVALOV is a senior student in the Finance and Banking Direction at International 
Black Sea University, Georgia
å =1
i w
j i
ij w w g =
2 s
å = - 0 1 i w
66
IBSU Scientific Journal      2 (1), 2008It is known, that 
symmetric matrix, therefore its eigen values are always non negative, and 
eigen vectors orthogonal [3,4]. We will consider it as bivalent covariant 
metric tensor, then total risk (is more exact, its square) of n shares, written 
down in the form of (1),  is possible to be considered  as double convolution 
of metric tensor g with contravariant  vector of weights w.  ij
If (1) is a convolution, it is invariant. The last means that total risk 
does not depend on system of coordinates that allows restating a problem of 
calculating the diagonal form of tensor g . We will define the diagonal form  ij
of tensor g for what we are calculating its eigen values and corresponding  ij 
eigen vectors. Thanks to simmetricity and positive definiteness of g it's not  ij  
2 difficult to do it. We will refer eigen values as µ (i=1,2, …, n), that is they  i 
are represented  in the form of squares of positive numbers that is possible 
thanks  to  mentioned  property  of  eigen  values  of  positively  defined 
symmetric  matrixes. Among  eigen  values  can  be  multiple  ones,  that 
essentially  changes  nothing.  It  is  easy  to  define  also  eigen  vectors, 
corresponding to their eigen values. If we normalize them and write down 
decomposition  of  already  normalized  eigen  vectors  according  to  the 
i· vectors of current basis, we will receive matrix  p where the top index  ·j   
indicates the number of coordinate of eigen vector (number of a line in 
i· matrix P), and bottom - vector's  number. The matrix  p can be considered  ·j  
as the operator of transition from old basis to new one. As new basis is 
orthonormal than the transition matrix (it consists of coordinates of these 
T vectors) appears to be the own orthogonal matrix (P*P  = E u |P| = 1). 
Coordinates of contravariant vectors are transformed, as it is known, by 
means of a matrix transposed and inverse to R. However, considering the 
T -1 orthogonality of Р, we have (P )  = P , that is in our case contarvariant and 
covariant vectors are transformed by means of the same matrix.  Notice that 
this matrix represents also bivalent, but mixed tensor (once covariant - the 
bottom index and once contravariant the-top index), that allows to consider 
Р as the operator of n-dimensional space.
 In the new basis the matrix of tensor g will have a diagonal form: 
(3)
i Proceeding from the aforesaid, the coordinates u of contarvariant 
 g being a covariance matrix, is positively defined  ij 
{
m l
m l lm
j
m
i
l ij
l g p p g
=
¹
×
×
×
× = ¢ =
2
0
m
67
IBSU Scientific Journal      2 (1), 2008vector of weights w in new basis
(4)
It's obvious that we also have inverse transformation
(5)
Thus we have received the transformed initial problem of optimization:
to minimize
(6),
at restriction
(7),
which after a regrouping will become
(8).
It is easy to write down Lagrange function of a problem (5) - (6)
Whence we have
1 (8 )
and
2 (8 ).
1 From (8 )
(9)
j i
j
i w p u
×
× =
i j
i
j u p w
×
× =
2 2 2 ) (
i
i u m s =
0 1
1 1
= - åå
= =
×
×
n
i
n
j
i j
i u p
0 1
1 1
= - å å
= =
×
×
n
i
n
j
j
i
i p u
) 0 1 ( ) ( ) , (
1 1
2 2 = - - = å å
= =
×
×
n
i
n
j
j
i
i i
i p u u u l m l f
0 2
) , (
1
2 = - =
¶
¶ å
=
×
×
n
j
j
i
i
i i p u
u
u
l m
l f
0 1
) , (
1 1
= - =
¶
¶ å å
= =
×
×
n
i
n
j
j
k
i p u
u
l
l f
2
1
2 i
n
j
j
i
i
p
u
m
lå
=
×
×
=
68
IBSU Scientific Journal      2 (1), 20082 And substituting in (8 ) it is received 
(10)
Let's enter notations
1 (11 )
and
2 (11 )
1 2 Substituting (10) in (9) and using (11 ) and (11 ), we have for optimum 
weights
(12)
where
Now it is possible to calculate a minimum of risk
(13)
Using transformation (5), it is possible to write down optimum structure for 
initial variables
(14)
Considering,  that  in  similar  notations  the  first  the  top  index 
changes, it is visible that constant       is the sum of elements of i-th column 
of transposed matrix Р, or the sum of elements of i-th line of matrix Р, that is 
the sum of co-ordinates of i-th eigen vector of covariance matrixes g. 
As  these  constants  completely define  optimum  structure  of  an 
investment  portfolio  it  is  possible  to  name  them  optimum  structural 
å
å
=
=
×
×
=
n
i i
n
j
j
i p
1
2
2
1
2
) (
1
m
l
å
=
×
× =
n
j
j
i i p
1
e
å
=
=
n
i i
i
1
2
2
m
e
e
e
p
m
e
e em
i
i
i
i
n
j
j
i
i
op
p
u = = =
å
=
×
×
2 2
1 1
2
i
i i
m
e
p =
e
m s
1
) (
2 2 2
min = =
i
op i u
e
p
i
j
i
i
op p w
×
× =
i e
69
IBSU Scientific Journal      2 (1), 2008i potentials of a portfolio, constants π- relative optimum structural potentials 
2 (they are expressed in terms of variances µ ), and constant     which is  i   
inverse to  the minimum risk,  - the maximum risk.
References
1. Elton E. J. and Martin J.G., (1991), “Modern Portfolio Theory and Investment 
Analysis”, New York: John Wiley
2. Gordon J. Alexander and Jack Clark Francis, (1986), “Portfolio Analysis”, 
New York: Prentice Hall
3.  Kostrikin A.I.,  Manin  JU.I.,  (1980),  “Linear  algebra  and  geometry”,  М: 
Moscow State University Publishing house
4. Postnikov M.M., (1982), “Lektsii po geometrii. Linear algebra. A semestre II”, 
М: Nauka
e
70
IBSU Scientific Journal      2 (1), 2008

Lektsii po geometrii. Linear algebra. A semestre II”,

Linear algebra and geometry”, М: Moscow State

Portfolio Analysis”,

One Method of Solution of an Optimum Investment Portfolio Problem for Risky Assets

http://journal.ibsu.edu.ge/index.php/ibsusj/article/download/47/40

One Method of Solution of an Optimum Investment Portfolio Problem for Risky Assets

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