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Секция «Математические методы моделирования, управления и анализа данных»
УДК 519.6
ИССЛЕДОВАНИЕ И АНАЛИЗ ЗАКУПОК ИНВЕСТИЦИОННЫХ ПРОДУКТОВ НА ОСНОВЕ МОДЕЛИ ЛИНЕЙНОГО ПРОГРАММИРОВАНИЯ
Ма Чжаньцзюнь Научный руководитель - Л. А. Казаковцев
Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
E-mail: levk@bk.ru
Покупка инвестиционных продуктов часто сопровождается определенными рисками. Чем выше доход, тем больше риск. Как получить максимальную выгоду при минимизации общего риска - это задача программирования с несколькими целями. В этой статье используется фиксированный уровень риска для построения математической модели, позволяющей преобразовать ее в задачу линейного программирования и решить ее.
Ключевые слова: математическое моделирование, линейное программирование.
RESEARCH AND ANALYSIS OF PURCHASING INVESTMENT PRODUCTS BASED
ON LINEAR PROGRAMMING MODEL
Ma Zhanjun Scientific supervisor - L.A. Kazakovtsev
Reshetnev Siberian State University of Science and Technology 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037, Russian Federation
E-mail: levk@bk.ru
The purchase of investment products is often accompanied by certain risks. The higher the return, the greater the risk. How to obtain the maximum benefit while minimizing the overall risk is a multi-objective programming problem. This paper uses a fixed risk level to construct a mathematical model to convert it into a linear programming problem and solve it.
Keywords: mathematical modeling, linear programming.
1. Problem Description
There are n types of investment products si (i=1, 2,..., n) available in the market, and the current amount of funds M (sufficient) is used as the initial capital for a period. The average return of buying si for these n products in the period is ri, the risk investment rate qi, and the overall risk is measured by the largest risk in si.
When purchasing an investment product si, a certain transaction fee will be paid. The rate is pi. When the purchase amount does not exceed the given value ui, the transaction fee is calculated
according to ui. Assuming that the bank deposit interest rate in the same period is r0, there is no transaction fee, There is no risk (r0=5%).
Design a portfolio investment plan that maximizes net income and minimizes total risks. 2. Basic Assumptions
1. The initial investment capital is quite large, for ease of calculation, suppose M=1;
2. The more diversified the investment, the smaller the total risk;
3. The total investment risk is measured by the largest risk in the investment project;
4. Various investment risks are independent of each other;
5. During the investment period, ri, pi, and qi are fixed values and are not affected by unexpected factors;
6. Net income and overall risk are only affected by ri, pi, qi, and not affected by other factors. 3. Symbol Description
1. si means the i-th investment project, i=0,1,...,n; where s0 means deposit in the bank;
2. ri, pi, and qi respectively represent the average rate of return of si, transaction fee rate, and risk loss rate, i=0,...,n; where p0=0, q0=0;
3. ui represents the fixed amount of si transaction fee, i=0,1,...,n;
4. xi represents the funds of the investment project si, i=0,1,...,n;
5. a represents the degree of investment risk;
6. Q represents the overall return. 4. Model building
1. The overall risk is measured using the largest risk in the investment si, namely:
2.
The transaction fee paid to purchase si is a piecewise function:
3. Since ui is small relative to the investment amount M, piui is smaller, so the net income can be simplified as (ri-pi)xi.
4. Make the net income as large as possible and the overall wind risk as small as possible. This is a multi-objective planning model.
Initial objective function :
The above is a multi-objective programming problem, which is transformed into a linear programming [1, 2] problem.
Fixed risk level and optimized returns
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Given a limit a of risk, the maximum risk -W , multi-objective planning becomes a goal linear programming [3-5], namely:
5. Model solving and analysis
Use the linprog() function in matlab to achieve.
Relevant data when n=4
Table 1
si ri(%) qi(%) p(%) u(%)
Si 28 2.5 1 103
S2 21 1.5 2 198
S3 23 5.6 4.5 52
S4 25 2.6 6.5 40
Since a makes any given degree of risk, starting from a = 0, a cyclic search is performed with a step of 0.001.
Fig. 2. Experimental results
Through experiments, we can see that the greater the risk, the greater the return. There is a turning point near a=0.006. On the left side of this point, when the risk increases little, the profit grows quickly. On the right side of this point, the risk increases greatly but the profit increases Slow, so for investors who have no special preference for risk and return, the turning point of the curve should be selected as the optimal portfolio, a=0.006.Q=20%.
So the final investment plan is x0=0, x1=0.24, x2=0.4, x3=0.1091, x4=0.2212, risk a=0.006, return Q=0.2019.
6. Conclusion
For this problem, this is only one of the models. It cannot be said that this is the best model. The method of solidifying returns and reducing risks can also be used to turn multi-objectives into linear programming. In any case, obtaining the greatest benefits and reducing risks are the main research goals. Finding a relatively better model among many models is a further goal.
References
1. Gill P E, Murray W, Saunders M A. Primal-dual methods for linear programming. Mathematical Programming, Vol. 70 (13), 1995.
2. Jiang Qiyuan, Xie Jinxing, Ye Jun. Mathematical Model (Fourth Edition) [M]. Beijing: Higher Education Press, 2010.
3. Chen Dongyan, Li Dongmei, Wang Shuzhong: Mathematical Modeling, Science Press, 2007.
4. Cao Xiwang: Mathematical Models in Management Science, Peking University Press, 2006.
5. Fang Daoyuan, Wei Mingjun: Mathematical Modeling: Method Guide and Case Analysis, Zhejiang University Press, 2011.
© Ma Zhanjun, 2021
