On the Equivalence of Optimality Principles in the Two-Criteria Problem of the Investment Portfolio Choice

Abstract

In this paper, we examine the problem of nding an optimal portfolio 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 maximizing 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
• convolution of the criteria

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 de ned 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 nding 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 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 corresponding to a de nite value of the risk coefficient at the variance in the problem of maximizing the linear convolution of the criteria "expectation|variance". We explicitly nd also the value of weight factor in terms of the known initial parameters of the problem. An example of nding 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 xed initial data but also on their uniform response to market uctuations. 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 nancial instruments r = (r1; :::; ri; :::; rn) 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 nancial instruments of the nal list (i = 1; :::; n).

De ne an optimal portfolio as a solution of the problem of maximizing the expectation 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: where " 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 re ected 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 for which the Lagrange function is de ned on the set X = fxjxe = 1g, coefficient. Lemma 1. If the convex programming problem ( 2 ) has a solution x0 and the corresponding Lagrange multiplier is positive, 0 > 0, i.e., (x0; 0) is a saddle point of the function ( 3 ), then x0 is a solution of the problem 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 ( 2 ) ( 3 ) ( 4 ) ( 5 ) Proof. By the convexity, problem ( 2 ) is equivalent to the maximizing the Lagrange function L(x; 0) = rx+ 0(krx (xKx)1=2) on the set X, where the Lagrange multiplier 0 provides a minimum of the function L(x0; ). Since 0 > 0, we see that the inequality constraint in problem ( 2 ) at the optimal point becomes active: krx0 = (x0Kx0)1=2 (otherwise, this constraint would be insigni cant). After a transformation we obtain L1(+x; 00k) = rx 1+ 00k (xKx)1=2. Then the equivalent problem takes the form ( 4 ), which was required.

Theorem 1. Let frig be a system of random variables each of which has a normal distribution, ri be the expectations, K = ( ij )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 p2 P (rx where 1 = 1+ 00d , d = ( 1(1 2")) 1, ( ) is the Laplace function, 0 is the value of the Lagrange multiplier in problem ( 2 ).

Proof. We prove that problem ( 1 ) is reduced to problem ( 2 ) under these assumptions. The random variable rx is normally distributed, i.e., P (rx 0) = ∫ 0 e (t2 m2)2 dt,

1(xKx)1=2]; 1 p12 ∫0m e 2z2 dz = 12 + (0) ( m ) = 12 the condition, 12 12 ( m ) ", therefore, m 1(1 2") or m ( 1(1 2")) 1. Introducing the notation ( 1(1 2")) 1 = d, we arrive at the problem of convex programming ( 2 ), in which k = d: 1(xKx)1=2]; where 1 > 0 is the risk coefficient, then for 1 = 1+ 00d problem ( 7 ) coincides with problem( 4 ). By the lemma, 0 > 0, and the Laplace function takes positive values, and 0 0 hence 1+ 0d > 0. Therefore, problem ( 7 ) for 1 = 1+ 0d 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 nd a connection of the problem ( 1 ) with a suitable convolution "expectation - variance". ( 6 ) ( 7 ) ( 8 ) 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 nd 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 > 0:

Theorem 2. Let x0 be a solution of problem ( 1 ), the optimal value of the Lagrange multiplier in problem ( 2 ) is positive, 0 > 0, and the covariance matrix K = ( ij )n n is strongly positive de nite. Then there exists a value of the weight coefficient in problem ( 8 ) such that the solutions of problems ( 1 ) and ( 8 ) coincide. Proof. We denote the expected return of a portfolio rx0 at the solution point of problem ( 1 ) by rp0, i.e., rx0 = rp0, and consider two equivalent problems and min xKx;

x min (xKx)1=2; x rx

rp0; xe = 1; rx rp0; xe = 1:

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 Paretooptimal. Therefore, the minimum in problem ( 10 ) cannot be less than (x0Kx0)1=2 . However, it satis es the constraint in problem ( 10 ); therefore, x0 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; 01) = xKx + 01(rp0 rx) on the set X for some value of the Lagrange multiplier 01, and this value 01 provides the maximum of the function L1(x0; 1). This problem is equivalent to the problem of minimizing the function xKx 01rx on the set X. Obviously, 01 > 0, since in the opposite case the problem is reduced to the minimizing of the variance, i.e., the constraint rx rp0 becomes insigni cant. We set = 10 , then > 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 in problem ( 8 ) for which solutions of problems ( 1 ) and ( 8 ) coincide. However, Theorem 2 allows one to nd the risk coefficient only by solving problem ( 9 ). In the following assertion (Theorem 3), we obtain a value of the risk coefficient .

The Lagrange function for problem ( 8 ) is L(x; ) = rx (xKx) + (1 xe). Optimality conditions of portfolio lead to a system of linear algebraic equations: r 2 (Kx0) e = 0; x0e = 1; which solution gives the optimal portfolio for the problem ( 8 ):

K 1e eK 1e x0( ) = + (K 1r

can be expressed through the parameters of the ( 9 ) ( 10 ) ( 11 ) ( 12 ) Theorem 3. Let the conditions of Theorem 2 be satis ed. If in problem ( 8 ) the weight coefficient satis es the equation 4(1 + (rK 1r (reKK 11ee )2 d2) 2 4d2 (reKK 11ee ) (rK 1r (eK 1r)2 ) eK 1e + (eK 1r)2 ) eK 1e 1(1 2")) 1, " > 0, then solutions of problems ( 1 ) and ( 8 ) coincide. Proof. By Theorem 1, solutions of problems ( 1 ) and ( 5 ) coincide for 1 = 1+ 00d , moreover, by the lemma, the condition 0 > 0 leads to drx0 = (x0Kx0)1=2 in problem ( 2 ), to which problem ( 1 ) is reduced. Thus, a solution of problem ( 8 ) de ning a portfolio ( 11 ) is also a solution of problem ( 1 ) if the following relation holds drx0( ) = (x0( )Kx0( ))1=2. Using ( 11 ) we have (drx0( ))2 = (reKK 11ee d + (rK 1r + (reKK 11ee ) (rK 1r

(eeKK 11re)2 ) 2d )2 = (reKK 11ee )2 d2+ (eeKK 11re)2 ) d2 + (rK 1r (eeKK 11re)2 )2 4d22 ; x0( )Kx0( ) = = (eKK 11ee + (K 1r = (eKe 1e + (r = 1 + eKK 11ee (r = 1 + (rK 1r eeKK 11re K 1e) 21 )K (eKK 11ee + (K 1r eeKK 11re K 1e) 21 ) = eeKK 11re e) 21 )K 1 (eKe 1e + (r eeKK 11re e) 1 + (r eeKK 11re e)(K 1r (eK 1r)2 ) 412 : eK 1e eeKK 11re e) 21 ) = eK 1 eK 1re K 1e) 412 = Equating the right-hand sides of these equalities, we obtain: (reKK 11ee )2 d2 + (reKK 11ee ) (rK 1r + (rK 1r (eeKK 11re)2 )2 4d22 = 1 + (rK 1r (eK 1r)2 ) d2 + eK 1e (eK 1r)2 ) 412 : eK 1e After a simple transformation we have 1 +

(reKK 11ee )2 d2 (( rK 1r

(reKK 11ee ) (rK 1r (eK 1r)2 ) eK 1e Multiplying both sides by 4 2, 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 "Aero ot", "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 return 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 "Aero ot", "MTS" and "Megaphone" we have the vector of expectations of shares returns r = (0; 967; 0; 189; 0; 327) and covariance matrix

0 0; 65 0; 466 0; 18 1 K = @ 0; 466 1; 678 0; 189 A.

0; 18 0; 189 0; 379

Let " = 0; 2 in problem ( 1 ). Then 1 2" = 0; 6 and by the table of values of the Laplace function we have 1(1 2") = 0; 85 and hence d = ( 1(1 2")) 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 in Problem ( 8 ) found as the solution of Eq. ( 12 ) for d = 1; 176, is = 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 = 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 " constraints of the problem ( 6 ) are not satis ed. Suppose that in the problem( 1 ) " = 0; 001; then 1 2" = 0; 998 and using the table of the Laplace function values we have 1(1 2") = 3; 09, d = ( 1(1 2")) 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 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, rx is the expected return rate of the portfolio. According to the de nition of covariance we get the following: cov(x1; x2) = M [(rx1 rx1 )(rx2 rx2 )] = == MM [[(∑∑nin=1 rixi1ri)x∑i1 in∑=1nrixi1)(∑rini=)1xi2r]iMxi2[∑∑nin=1 rixi2)] =

i=1(ri i=1(ri i;j=1(ri ri)(rj = ∑in;j=1 M [(ri ri)(rj rj )]xi1xj2 = ∑in;j=1 Kij xi1xj2: rj )xi1xj2] =

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) = x1Kx2: ( 13 ) Theorem 4. Covariance cov(x01, x02) of the two optimal portfolios is positive. Proof. Let's demonstrate that if det ̸=0, then eK 1e > 0. ⟨Kx; x⟩ 0 implies that ⟨K 1x; x⟩ 0 8x, where ⟨ ; ⟩ is the inner product of the vectors. Indeed, let's take the equation Kx = . If we multiply the equation by x, we have ⟨Kx; x⟩ = ⟨ ; x⟩ 0. On the other hand, x = K 1 and ⟨x; ⟩ = ⟨K 1 ; ⟩ 0 8 . As it is commonly known, the minimal eigenvalue of the symmetric matrix K 1 equals the minimum of the quadratic form ⟨K 1x; x⟩ on a unit sphere ⟨x; x⟩ = 1. Suppose that 9x~ : ⟨Kx~; x~⟩ = 0, therefore, the minimal eigenvalue min = 0. Then, the characteristic equation det(K 1 E) = 0, where E is the diagonal identity matrix, produces det K 1 = 0 if min = 0. Here we reach a contradiction. It means that 8x ⟨x; x⟩ = 1; ⟨K 1x; x⟩ > 0. For the vector e~ = e pn belonging to the unit sphere, it holds true that ⟨K 1e~; e~⟩ > 0 or n 1⟨K 1e; e⟩ > 0, i.e. eK 1e > 0.

A structure of an optimal portfolio ( 11 ) can be presented as follows ( 14 ) ( 15 ) x0( ) = C0 + C1 ; where = 21 and C0 = (C01; : : : ; C0j ; : : : ; C0n), C1 = (C11; : : : ; C1j ; : : : ; C1n) are determined according to the following formulas

K 1e C0 = eK 1e ; C1 = K 1r Consider two optimal portfolios x01 and x02, which are determined from the solution of the problem ( 8 ) at different values of the parameter or, in accordance with the above notation, the parameter . According to ( 14 ), the structure of the optimal investment portfolios looks as follows: x01 = C0 +C1 1 and x01 = C0 +C1 2 respectively. It follows from ( 13 ) and ( 15 ) that the covariance of the two portfolios returns is cov(x01; x02) = x01Kx02 = (C0 + C1 1)K(C0 + C1 2) = C0KC0 + (C1KC1) 1 2 + (C0KC1) 2 + (C1KC0) 1.

Since the matrix K is symmetric, we have the equality C0KC1 = C1KC0 and the following expression for covariance: cov(x01; x02) = C0KC0 + (C1KC1) 1 2 + (C0KC1)( 1 + 2): (16) Using the properties of the inner product and ( 15 ), we have

C0KC1 = eKK 11ee K(K 1r eK 1 = eK1 1e (⟨K 1e; r⟩ eK 1erK⟨KeK1e1e1;reeK⟩) =1ee)K=1 1eeKK(e1K1ee (r1 r eeKK 11re e) = (eK 1r)(eK 1e) ) = 0: eK 1e Then, (16) will look as follows: cov(x01; x02) = C0KC0 + (C1KC1) 1 2.

Since the matrix K is non-negatively de ned, then C1KC1 0 and C0KC0 = eKK 11ee K eKK 11ee = (⟨eKK 11ee;e)2⟩ = eK1 1e > 0. It means that the inequality cov(x01; x02) > 0 holds true for two optimal portfolios x01 and x02, Q.E.D. Comment. In the absence of short selling a similar result holds under the additional assumption of strict positive de niteness 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 in the problem ( 8 ) at d = 1; 176 is = 0; 036, and by ( 11 ) the structure of the portfolio is x0(0; 036) = (13; 042; 7; 064; 4; 978). Let now = 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 )) = x0Kx0( 1 ) = 4; 991.

Conclusion

We have considered the problem of nding 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 rst, is inconvenient from the computing point of view; this is related to the type of nonlinear constraints that make it difficult to nd 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 decisionmaking 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 nance "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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