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SOLVING OF INVESTMENT PORTFOLIO MANAGEMENT PROBLEM

УДК 517.977

В работе рассматриваются задачи динамического управления инвестиционным портфелем. Учитываются критерий максимальных финансовых потерь за период управления портфелем и максимальный ожидаемый доход на конец периода. Рассматриваются задачи управления портфелем с учетом ограничений на риск и с учетом потребления. Все рассматриваемые модели сводятся к задаче целочисленного линейного программирования.

Бұл жұмыста инвестициялық қоржынды динамикалық басқару есептері қарастырылады. Қоржынды басқару мезгілінде қаржылардың максималды азайуын және басқару соңында максималды күтілетін кіріс белгілері ескеріледі. Тәуекел шектеулерін және тұтыныстарды ескере отырып қоржынды басқару есептері де қарастырылады. Барлық қарастырылған модельдер бүтін санды сызықты программалау есебіне келтіріледі.

ABSTRACT

That work addresses the issues of dynamic management of the investment portfolio. Criteria of maximum financial losses during the portfolio management period and maximum expected income at the end of the period are considered. Portfolio management problems under risk limitations and consumptions are examined. All models considered are reduced to the problems of the integer linear programming.

Keywords: investment, dynamic management, portfolio, maximum financial, risk models, integer linear programming

PROBLEM DEFINITION

Let’s consider way of risk restriction, in the first time this method was described in portfolio theory of Markowitz H. [1]. The main idea concludes in risk diversification effect, which composed of concept existence of financial instruments with negative price correlation. In terms of independent model of risk, to the same event of an environment the price Pt1 and Pt2 of tools react accordingly increase r and fall f .So, forming a portfolio of both kinds of tools in the necessary proportions, on the average, at event A occurrence, this portfolio does not depreciate. This problem is a problem of optimization of structure of a portfolio with criterion of a maximum mean of increase in value of a portfolio (profitability) for one period at the set risk level. Markowitz has entered a measure of value of risk or potential losses as a dispersion of profitability of a portfolio. This statement it is limited to (“buy-and-hold”) type of management. In practice are much more extended dynamic (multiperiod) strategy of management by a portfolio. Besides, the criterion of risk as to income dispersion is represented to inadequate representations of portfolio managing directors about risk. Here to them and their clients the criterion of the maximum financial losses which are possible during management of a portfolio is more essential. The portfolio managing director specifies algorithm of purchases and sales of financial tools on which portfolio re-structuring during each period of time is carried out.

LET’S CONSIDER FOLLOWING PROBLEMS

Problem 1. Management of a portfolio on strategy (“buy-and-hold”) by criterion of efficiency «the maximum expected income on the end of the period of management» with restriction on «risk the maximum financial losses in the management period».

Problem 2. In managerial process there is a possibility of withdrawal by the managing director of resources from a portfolio for consumption.

Problem 3. In managerial process the problem of indivisibility of financial tools is considered. Portfolio formation occurs dynamically taking into account arriving and consumed in the period of management of resources.

Problem 4. Problem of optimum dynamic placing of resources on Kelly for trading strategy by criterion of some set speed of a gain of the capital. [2, 3].

For the solving of these problems we investigate a question of management of a portfolio by criterion of financial losses. We will consider a management problem an investment portfolio of the economic subject on an interval [t0, t0+T] with initial own capital C0. The increase in the capital as a result of change of cost of an investment portfolio Xi is

(1)

Let’s introduce following definitions

Definition 1. Loss function of the portfolio over a period of time [0, T] is called a function

Thereby function of losses is defined on separate realization of process of change of market cost of a portfolio Ct and is equal maximum reached by the time of size of falling of cost of an investment portfolio after in the past this cost has reached the peak.

Process of change of a market price of the tool is the historical scenario of the price, and historical scenarios are samples of an accessible database for the period (initial -i and final-f),

with reference t0 and length T which depends on parameter.

By changing reference t0 (changing scenarios Pt in time windows t [t0, t0+T] we receive artificial set of realizations of the same process Ct on unique realization of process {Pt, t [ti, tf]}. This method is called (butt strep).

Using this method we model forming by the economic subject of a portfolio during the different moments of time and in different modes of the market. It is possible to analyze function of losses in two time cuts – along one scenario LF(t), t0 [tj, tg-T] by parameter t, t0, is fixed and on the set of scenarios t0 [ti, ts-T] for any specific moment t (I.e. under one scenario of process and on one set of scenarios for fixed t, the left end t0 is not fixed). We will be interested in distribution of function of losses under scenarios and its characteristics – an average, extreme values and quantile.

Definition 2. Losses at worst case are defined as

WLF=max{LF(t)} (2)

0 ≤t ≤T

These are losses in the worst case observed in the history.

Definition 3. Mean losses

(3)

Here there is an averaging both along one scenario - t, and under all scenarios- t0.

Definition 4. Losses with the risk 1- , [0,1] analogue of a measure Value-at-Risk are defined as corresponding quantile level of distribution LF(t0;t)

(4)

Here ω( ) – the top border of potential losses, such (1- ) of LF(t) that is less than values ω( ). For example, if we interesting in 95% quantile then losses will exceed LRLF (95%) only in =5% cases.

Problems of a investment portfolio management taking into account restrictions on risk

Let’s consider S–the market on which are available n investment tools. Strategy of management by a portfolio assumes seed capital placing in volume C0 on the S market at the moment of time t0 to proportionally shares xi, i=1,2,.....n in I th tool for term T=tT-t0. The portfolio is not re-structured till the moment tT=t0+T, i.e. strategy of management of type “buy-and-hold” is accepted. If rti=Sti - St-i– an increment of the prices of i th tool in the S market for one period [t-1,t] then efficiency of a portfolio on the end of horizon T is defined as

.

Where vectors

The second criterion of management is restriction of function of losses. It is possible to assume following restrictions

MLF(x) ≤ w1C0 - restriction of the maximum losses         (5)

MLLF(x) ≤ w2C0 - restriction on average losses              (6)

LRLF(x, ) ≤ w3C0 - restriction on losses with risk          (7)

Appetite” or risk “avoiding” is set by experts in the form of “weight” multipliers w1, w2,w3 [0,1]. In practice also can be set in the form of restrictions on the maximum and minimum size of investments into each tool of S – the market.

The problem of a choice of an optimum portfolio {xi} solves in a rectangular parallelepiped

(8)

The management purpose is maximization of efficiency of a portfolio E(Xt) → max by the time of end of investments tT at one of three types of restrictions of losses in the form of (5), (6) or (7) and restrictions on admissible strategy (8). From here we have following problems of mathematical programming

The numerical solving of problems (9), (10) and (11)

For discretization of problems (9), (10) and (11) we will break a time interval [t0, t0 + T] to M equals parts i.e. we have So r (ti) = ri- discrete vector and function of losses registers in the shapeEfficiency criterion function registers as

Then problems (9), (10) and (11) have a following view

(12)


In last formula this (g(x))*=max {0, g(x)} designation is used.

The following theorem is fair.

The theorem. Problems (12) – (14) are problems of linear programming

The proof. Criterion and restrictions are on construction linear functional. Problems (12) – (14) are reduced to a standard problem of linear programming by introduction of additional variables and can be solved numerically standard simplex-method.

Management of a portfolio taking into account consumption

Let’s consider a problem about multistage management of a portfolio taking into account possibility of withdrawal of a part of resources from a portfolio in the period of management for the purpose of their consumption by managing director. Such situation arises, when in parallel with investment activity there is a budget of expenses and additional investments of resources.

Problem statement

Let’s assume that 1. The condition of a portfolio of the investor is characterized throughout all period of planning: 1.1 number of units (open positions) financial actives on each type of actives during each period of time; 1.2 financial (monetary, resource) streams from the investor (a stream of payments) and to it (a stream of receipts) or the budget for each period; in 1.3 size of an authorised capital stock and in volume of extra loans on В - the market on an initial stage. 2. The condition of the markets is characterized by the current prices of actives during each moment of time. 3. The purpose of the investor is to maximize total market cost of the portfolio (i.e. cost of set of all actives in the prices for the final period of planning) at preservation of financial stability in all investment period.

The requirement of performance of a condition of financial stability means that during each moment of time the investor should have possibility to finance the obligations on В - the market.

Construction of mathematical model

Let’s notice that the model will be not closed in the sense that for its practical application it is necessary to know forecasts of movement of the prices in the markets of actives used by the investor that represents other group of problems of the financial analysis. It is necessary to know not absolute values of the prices, and forecasts of sizes of their change – volatility. The generalised investment portfolio considered here, really gets lines of the present portfolio or a backpack (see a mathematical problem about a backpack), in which the investor during each period «stacks so much actives, how many can carry away». Physically the model represents two caches – a cache of a portfolio and a budgetary cache – united on resources and co-operating in time.

The entrance model parameters (problem parameters): n>2 – number of types of the generalised actives (financial tools); T>2 – number of the periods of planning; P0 - Volume of loans in a portfolio of the investor when t=t0, r0 - the interest rate on loans, % for the period, I (tk), D(tk) a stream of receipts (Income) and payments (Deposit) during the moment t=tk, AC -the authorised capital (authorised capital); vi(t0)=Vi initial volume of brave investments in a type i active (initial risk investment) L(tj)=Lj insurance limits on a brave part of a portfolio (insurance limit) pi(tj)=pij, i=1,...,n, j=0,...,T market prices of an active of type at the moment of time.

Characteristics of an investment portfolio (variable problems) vi(tj)=vij, i=1,...,n, j=1,...,T number of units (open positions) active of type i at the moment of time t=tj actives are considered as the indivisible vij Z+ Short positions are inadmissible.


An interval (cache) by the time t=tj or cost of risk free portfolio part.

– cost of a brave part of a portfolio at the moment t=tj of time.

Problem statement

1. To maximise cost of the generalised portfolio of the investor by the time T and performance of restrictions of insurance limits and liquidity restrictions (financial stability). 2. To construct strategy {vij } of placing of the resources, steady concerning following parameters of a problem: 2.1. Market prices of actives {pij }; 2.2 streams of resources {I(tk) - D(tk)} k=1,...,T.

Then the mathematical model looks like model of linear integer programming

Where (15) – criterion function (criterion), (16) – restrictions of liquidity (T pieces), (17) – restrictions of insurance limits (pieces), (18) – restrictions on «short sales», (19) – restrictions integer.

Number of variables of a problem – nT, number of restrictions of a problem– 2T.

Let’s notice that in practice the number of variables of a problem is much more number of restrictions of a problem, i.e. the task in view is a problem of integer linear programming of the big dimension. Such problems belong to the class difficult, i.e. application of a method of linear programming of type a simplex-method expensive. But, we will notice that variable problems can be divided. We will notice that

.

Then the initial problem (15) – (19) looks like


Last problem we will write down in the vector-matrix form. For this purpose we will enter following designations: v = (v11, v21,...vn1,...,v1j,...,vnj,...,v1T,...vnT)’- vector a column

- vector a line

- increment of vector ;  

;

From here we receive


Or in standard form


With new designations


consist of T blocks with n+2 elements, view of elements

finally we receive

(20)

The problem (20) is a typical problem of integer linear programming. [4, 5].

 

REFERENCES

1. Markowitz H. Portfolio Selection // Journal of Finance, 7, No. 1, March 1952. P. 152–164.

2. Shakenov R.K. About hedging of a portfolio of securities on (B, S)- the market. The bulletin of al-Farabi, a series of the mathematician, the mechanic, computer science. № 2 (57), 2008. P. 60–71.

3. Orynbay M.S., Shakenov R.K., Shakenov K.K About the numerical decision of some problems of risk with distribution of type of Pareto. The international 11th interuniversity conference on mathematics and the mechanics, devoted to the 10 anniversary of the Euroasian national university of L.N.Gumilev. 25–on May, 26th, Astana. Theses of reports. Astana, 2006. P. 152.

4. Vasilev F.P Numerical methods of the decision of extreme problems. Мoscow. The Science. 1980. P. 520.

5. Kalikhman I.L. The collection of problems on mathematical programming. The 2nd edition, processed and added. Moscow. The Higher School, 1975. P 270.