UDC 336.76
UDC 519.852
This study investigates the use of the Markowitz portfolio optimization model to construct an investment portfolio with the lowest possible level of risk while maintaining a predefined return target. The research is grounded in Modern Portfolio Theory, which examines the relationship between expected returns and the risk associated with financial assets. To ensure an adequate degree of diversification, the analysis is based on stocks representing different sectors of the economy. The study includes the calculation of logarithmic returns, volatility indicators, and the covariance matrix of asset returns. Using the obtained data, an equally weighted portfolio was evaluated and subsequently optimized through the mean-variance framework and constrained optimization techniques. Numerical computations were performed using Microsoft Excel. The findings indicate that portfolio risk can be significantly reduced without compromising the desired level of expected return. The results confirm the practical value of mathematical methods in investment management and highlight the effectiveness of the Markowitz approach for portfolio decision-making in financial markets.
Markowitz model, investment portfolio, optimization, risk, return, diversification, covariance, financial mathematics
1. Markowitz H. Portfolio Selection // The Journal of Finance. — 1952. — Vol. 7, No. 1. — P. 77–91.
2. Elton E. J., Gruber M. J. Modern Portfolio Theory and Investment Analysis. — 9th ed. — Wiley, 2014.
3. Bodie Z., Kane A., Marcus A. Investments. — 11th ed. — McGraw-Hill Education, 2018.
4. Fabozzi F. J., Gupta F., Markowitz H. The Legacy of Modern Portfolio Theory // The Journal of Investing. — 2002. — Vol. 11, No. 3. — P. 7–22.
5. Chiang A. C., Wainwright K. Fundamental Methods of Mathematical Economics. — McGraw-Hill, 2005.
6. Luenberger D. G. Investment Science. — Oxford University Press, 2013.
7. Reilly F. K., Brown K. C. Investment Analysis and Portfolio Management. — 10th ed. — Cengage Learning, 2011.
8. Benninga S. Financial Modeling. — 4th ed. — MIT Press, 2014.
9. Hull J. C. Risk Management and Financial Institutions. — 5th ed. — Wiley, 2018.
10. Sharpe W. F. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk // The Journal of Finance. — 1964. — Vol. 19, No. 3. — P. 425–442.
11. Fama E. F. Efficient Capital Markets: A Review of Theory and Empirical Work // The Journal of Finance. — 1970. — Vol. 25, No. 2. — P. 383–417.
12. Ross S. A., Westerfield R., Jaffe J. Corporate Finance. — 11th ed. — McGraw-Hill Education, 2016.
13. Jorion P. Value at Risk: The New Benchmark for Managing Financial Risk. — 3rd ed. — McGraw-Hill, 2007.
14. Campbell J. Y., Lo A. W., MacKinlay A. C. The Econometrics of Financial Markets. — Princeton University Press, 1997.
15. DeMiguel V., Garlappi L., Uppal R. Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? // The Review of Financial Studies. — 2009. — Vol. 22, No. 5. — P. 1915–1953.
16. Matematicheskoe modelirovanie anomal'nyh dohodnostey kriptovalyut pod vozdeystviem infopovodov / A. A. Sidorov, T. A. Burceva, E. S. Darda, O. R. Paraskevopulo // Moskovskiy ekonomicheskiy zhurnal. – 2026. – T. 11, № 2. – S. 204-241. – DOIhttps://doi.org/10.55186/2413046X_2026_11_2_27. – EDN ZESDHX.
17. Sidorov, A. A. Sravnitel'nyy analiz dvuh kombinatornyh metodov vychisleniya matematicheskogo ozhidaniya nailuchshego ranga v zadache o razmeschenii / A. A. Sidorov, R. U. Astaf'ev // Opticheskie tehnologii, materialy i sistemy (Optoteh - 2025) : Mezhdunarodnaya nauchno-tehnicheskaya konferenciya, Moskva, 08–12 dekabrya 2025 goda. – Moskva: MIREA - Rossiyskiy tehnologicheskiy universitet, 2026. – S. 1125-1131. – EDN ACIABJ.
18. Astaf'ev, R. U. Imitacionnoe modelirovanie slozhnyh ierarhicheskih sistem / R. U. Astaf'ev // NAUKA XXI VEKA: VYZOVY, STANOVLENIE, razvitie : sbornik statey XXIV Mezhdunarodnoy nauchno-prakticheskoy konferencii, Petrozavodsk, 24 noyabrya 2025 goda. – Petrozavodsk: Mezhdunarodnyy centr nauchnogo partnerstva «Novaya Nauka» (IP Ivanovskaya I.I.), 2025. – S. 287-291. – EDN QTROAE.
