# Optimization of the investment portfolio, Statement of the problem, Formalization of the model - International financial management

## Optimization of the investment portfolio

The problem of capital rationing arises always when the company's funds are limited (a very trivial situation!), and at the same time it can not (or does not want) to raise additional funds for the implementation of its investment projects. Due to such budget constraints, not all investment projects that meet the NPV and IRR, criteria can be accepted unconditionally. Selection of projects in the investment portfolio is associated here with the procedure for optimizing the capital investment budget. The question in this case is how to optimally distribute the available capital in a limited amount between alternative investment decisions. This problem can be translated into the area of ​​mathematical programming problems, which we will try to illustrate on the simplest conditional example.

Suppose that a small company is created, whose business consists in leasing to other companies of cars and trucks. The owner of the company assumed that his business will be carried out for three years, all lease contracts will last exactly this term and will be concluded immediately after the company is established. It is known that the net discounted cash flow for each car and lorry will be \$ 200 and \$ 500, respectively. All cars will be purchased at the beginning of the first year of the firm's operation at prices of \$ 400 and \$ 300 for trucks and cars, respectively. The sequence of payments for purchased cars is as follows: at the end of the first year, a fee of \$ 200 for each automobile should be paid for a car, 100 for a freight car; at the end of the second year, the final calculation for the equipment purchased (\$ 100 and \$ 300, respectively) should be carried out. According to the owner of the company, the available cash will be 40 thousand dollars in the first year and 30 thousand - in the second. What number of cars and trucks should the owner of the firm purchase in order to maximize the net present value of this investment?

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The solution of this problem can be found using the apparatus of mathematical programming.

Denoting by x 1 and x 2 the required number of cars and trucks, the economic-mathematical model the maximization of net present value can be written as follows:

The solution of the formalized model with the help of the linear programming device gives the optimal solution

In addition to the optimal plan, this method also makes it possible to calculate the so-called dual resource estimates, which are the increments of the objective function, obtained as a result of increasing the available volumes of corresponding resources by one unit. With the help of a comparative analysis, the most scarce resources are identified, which have the greatest significance for the company from the point of view of optimizing the considered objective function of the model.

Taking into account the significant specificity (and certain complexity!) of mathematical formalization of economic systems and tasks, we will analyze one more example related to the optimization of the conditional investment portfolio.

The international company considers four projects as possible objects for investment. The net present value for the projects under consideration A, B, C, D is (thousand dollars) - 230, 200, 190 and 220, respectively. Projects can be implemented within one year and require quarterly funding. The required amounts of investment and available funds for this purpose in each quarter are presented in Table. 6.10.

Table 6.10

Optimization of the investment portfolio

 Investment project Quarterly need for funds, thousand dollars I II III IV A 108.0 108.0 135.0 135.0 In 94.5 121.5 121.5 148.5 From 67.5 94.5 121.5 148.5 D 121.5 108.0 94.5 81.0 Available Tools 300.0 320.0 360.0 370.0

Determine which of the presented in Table. 6.10 projects the company should select in the investment portfolio and what amounts of funds it will need in each quarter for their implementation, if the company's goal is to maximize the total amount of the present value.

## Model formalization

By designating the projects analyzed by the Boolean variables , the model for optimizing its investment portfolio can be represented as follows:

The solution of this model allows to obtain an optimal plan , which means that the company should include in the investment portfolio projects A, C and D). The net present value of the company will be thousand dollars with quarterly costs: 297.0; 310.5; 351.0; 364.5 thousand dollars

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