June 2026 (published: 29.05.2026)
Number 2(65)
UDC 336.767, 519.86, 330.322
DOI 10.17586/2310-1172-2026-19-2-13-22
Optimization model of an international investment portfolio on the example
of companies in the Pharma 4.0 sector
Reference for citation: Kopylova E.V., Stepchenkova O.S. Optimization model of an international investment portfolio on the example of companies in the Pharma 4.0 sector. Scientific journal NRU ITMO. Series «Economics and Environmental Management». 2026. № 2. Р. 13-22. DOI: 10.17586/2310-1172-2026-19-2-13-22.
Abstract. The purpose of this article is to develop a step-by-step algorithm for optimizing an international investment portfolio based on the classical Harry Markowitz theory and to test it on the example of four public companies in the Pharma 4.0 sector – Microsoft, Roche, Novartis, AstraZeneca, representing key segments of the digital transformation of the pharmaceutical industry. Monthly stock quotes of these companies for the period 2024–2025, converted to a single base currency, were used as a statistical sample. To achieve this goal, average monthly returns, standard deviations (volatility) and a covariance matrix reflecting the relationships between asset returns were calculated. Based on the data obtained, a tangent portfolio with the maximum Sharpe ratio was constructed using matrix calculus methods (covariance matrix inversion), and the efficiency of the resulting portfolio was assessed by comparing its characteristics with an equally weighted portfolio. The calculations showed that the proposed approach is accessible, does not require specialized software, and demonstrates a statistically significant improvement in the risk-return ratio. The study revealed a negative covariance between the technology company Microsoft and the European pharmaceutical giants Roche and Novartis, which indicates the presence of a natural hedging effect within the Pharma 4.0 ecosystem and confirms the hypothesis of the possibility of effective diversification even within a single industry with correct consideration of internal relationships. The optimal asset weights, expected monthly return and portfolio risk were determined; it was found that the optimized portfolio significantly outperforms the equally weighted portfolio in terms of the Sharpe ratio. The obtained research results can be used by private investors for independent formation of an investment portfolio based on any historical data, and also serve as a methodological basis for further research in the field of portfolio optimization in relation to high-tech sectors of the economy.
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Keywords: investment portfolio, Markowitz theory, tangent portfolio, Sharpe ratio, Pharma 4.0, personalized medicine, portfolio diversification
References:
1. Скрипниченко М.В. Портфельные инвестиции: Учебное пособие. - СПб: Университет ИТМО, 2016 – 40 с.
2. Andreas M. Fischer, Rafael P. Greminger, Christian Grisse, Sylvia Kaufmann,Portfolio rebalancing in times of stress, Journal of International Money and Finance, Volume 113, 2021, 102360, ISSN 0261-5606.
3. Bannier C.E., Bofinger Y. & RockB. The risk-return tradeoff: are sustainable investors compensated adequately?. J Asset Manag 24, 165–172 (2023).
4. Brar J., Braun J., Hare W., & Wang, D.(2025). A simulation analysis of returns-risk portfolio optimization models // Communications in Statistics-Simulation and Computation, 1–28.
5. Sangyup Choi, Jiri Havel, Geopolitical risk and U.S. foreign portfolio investment: A tale of advanced and emerging markets // Journal of International Money and Finance, Volume 151, 2025, 103253, ISSN 0261-5606.
6. James N., Menzies M., & Chin E.(2022). On financial market correlation structures and diversification benefits across and within equity sectors.
7. Morandi Stagni R., & Santaló J. (2025). Investors' attention and the paradox of technologically related diversification: Evidence of stock market mispricing. Strategic Management Journal, 46(10), 2432–2466.
8. Luo J., Guo J., & Li H.(2021). An Analysis of Cross-sectional Investment Portfolio with the Consideration of Risk and Return. E3S Web of Conferences 235, 01036 (2021).
9. Доржиева В.В.Цифровая трансформация промышленности в условиях внешних ограничений (на примере фармацевтической промышленности): Научный доклад. – М.: Институт экономики РАН, 2023. – 54 с.
10. Baoyang Ding, Pharma Industry 4.0: Literature review and research opportunities in sustainable pharmaceutical supply chains, Process Safety and Environmental Protection, Volume 119, 2018, P. 115-130, ISSN 0957-5820.
11. Mastrantonas A., Kokkas P., ChatzopoulosA. et al. Identifying the effects of Industry 4.0 in the pharmaceutical sector: achieving the sustainable development goals. Discov Sustain 5, 460 (2024).
12. Vincent Junior Phiri, Ilham Battas, Atae Semmar, Hicham Medromi, Fouad Moutaouakkil, Towards enterprise-wide pharma 4.0 adoption, Scientific African, Volume 28, 2025, e02771, ISSN 2468-2276.
13. TechSci Research. Pharma 4.0 Market – Global Industry Size, Share, Trends, Opportunity, and Forecast, 2021-2031F [Электронныйресурс] / TechSci Research. 2026. 186 p. URL: https://www.giiresearch.com/report/tsci1961334-pharma-market-global-industry-size-share-trends.html(датаобращения: 10.03.2026).
14. The Business Research Company. Pharma 4.0 Global Market Report 2026 [Электронныйресурс] / The Business Research Company. 2026. 250 p. URL: https://www.giiresearch.com/report/tbrc1957612-pharma-global-market-report.html(датаобращения: 07.03.2026).
15. Value Market Research. Global Personalized Medicine Market Size, Share, Trends & Growth Analysis Report 2026-2034 [Электронныйресурс] / Value Market Research. 2026. 176 p. URL: https://www.giiresearch.com/report/vmr1949077-global-personalized-medicine-market-size-share.html(датаобращения: 07.03.2026).
16. Macrotrends. Microsoft – 40 Year Stock Price History [Электронныйресурс] / Macrotrends. 2026. URL: https://www.macrotrends.net/stocks/charts/MSFT/microsoft/stock-price-history(датаобращения: 06.03.2026).
17. OnVista. AktuelleKennzahlenRocheAktie[Электронныйресурс] / OnVista. 2026. URL: https://www.onvista.de/aktien/kennzahlen/Roche-Aktie-CH0012032048 (датаобращения: 06.03.2026).
18. OnVista. AktuelleKennzahlenNovartisAktie[Электронныйресурс] / OnVista. 2026. URL: https://www.onvista.de/aktien/kennzahlen/Novartis-Aktie-CH0012005267 (датаобращения: 06.03.2026).
19. Macrotrends. AstraZeneca – 33 Year Stock Price History [Электронныйресурс] / Macrotrends. 2026. URL: https://www.macrotrends.net/stocks/charts/AZN/astrazeneca/stock-price-history(датаобращения: 06.03.2026).
20. Беширов М.Д. Диверсификация как инструмент управления рисками инвестиционного портфеля / М.Д. Беширов // Московский экономический журнал. – 2020. № 4. С. 66. DOI 10.24411/2413-046X-2020-10252.
21. Мартемьянова К. Д. Формирование инвестиционного портфеля методом дж. Тобина // Хроноэкономика. 2017. №6 (8).
22. Chen S., Yang R., Hou X. & Zhao J.(2024). Global Value Chain and the Changing Roles of Country and Industry Effects in International Portfolio Diversification. Complexity, 2024, Article ID 5530604. DOI:1155/2024/5530604.
23. Huynh T.L.D., Hille E. & Nasir M.A.(2020). Diversification in the Age of the 4th Industrial Revolution: The Role of Artificial Intelligence, Green Bonds and Cryptocurrencies. Technological Forecasting and Social Change, 159, 120188.
24. Elsayed A.H., Billah M., Goodell J.W. & Hadhri S. (2024). Examining connections between the fourth industrial revolution and energy markets. Energy Economics, 133, 107476.
25. El Khoury R., Alshater M.M. & Yoon S.M. (2023). Multidimensional connectedness among the fourth industrial revolution assets. Borsa Istanbul Review, 23(4), 776–789.
26. Demiralay S., Gencer H.G. & Bayraci S. (2021). How do Artificial Intelligence and Robotics Stocks co-move with traditional and alternative assets in the age of the 4th industrial revolution? Implications and Insights for the COVID-19 period. Technological Forecasting and Social Change, 171, 120989.
