Markowitz portfolio optimization through pairs trading cointegrated strategy in long-term investment

  • Alessia Naccarato
  • Andrea Pierini
  • Giovanna FerraroEmail author
S.I.: Recent Developments in Financial Modeling and Risk Management


This work aimed to solve the problem of Markowitz portfolio optimization for a long-term horizon investment, through the pairs trading cointegrated strategy. Such a strategy allowed us to identify the prices and returns of each stock on the basis of a cointegration relationship estimated by means of the Vector Error Correction Model (VECM). Once the returns had been established, the Markowitz allocation problem among the pairs was solved by minimizing the portfolio risk. We proposed to determine the optimal allocation for each stock as a linear combination of the allocation coefficients calculated for each pair and the cointegration coefficients estimated by means of the VECM model. The proposed strategy was applied to three pairs of real cointegrated stocks belonging to the European financial sector. The results obtained were compared with those from five methods, proposed in the scientific literature, by means of a bootstrap simulation experiment.


Markowitz portfolio Pairs trading Cointegration Vector Error Correction Model 

JEL Classification

C15 C32 G11 G17 



Authors would like to thank the anonymous referees for their careful reviews of this paper.


  1. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723.Google Scholar
  2. Alexander, G. (1976). The derivation of efficient sets. Journal of Financial and Quantitative Analysis, 11(5), 817–830.Google Scholar
  3. Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (1994). Time series analysis: Forecasting and control. Englewood Cliff: Prentice Hall.Google Scholar
  4. Bradley, M. D., & Jansen, D. W. (2004). Forecasting with a nonlinear dynamic model of stock returns and industrial production. International Journal of Forecasting, 20(2), 321–342.Google Scholar
  5. Brandt, M. W., & Kang, Q. (2004). On the relationship between the conditional mean and volatility of stock returns: A latent VAR approach. Journal of Financial Economics, 72(2), 217–257.Google Scholar
  6. Broussard, J. P., & Vaihekoski, M. (2012). Profitability of pairs trading strategy in an illiquid market with multiple share classes. Journal of International Financial Markets, Institutions and Money, 22(5), 1188–1201.Google Scholar
  7. Burmeister, E., & McElroy, M. B. (1988). Joint estimation of factor sensitivities and risk premia for the arbitrage pricing theory. The Journal of Finance, 43(3), 721–733.Google Scholar
  8. Byrne, B., & Lee, S. (2004). Different risk measures: Different portfolio compositions? Journal of Property Investment & Finance, 22(6), 501–511.Google Scholar
  9. Caldeira, J. F., & Moura, G. V. (2013). Selection of a portfolio of pairs based on cointegration: A statistical arbitrage strategy. Review Brazilian Finance, 11(1), 49–80. as (Online), Rio de Janeiro,Google Scholar
  10. Calvo, C., Ivorra, C., & Liern, V. (2012). On the computation of the efficient frontier of the portfolio selection problem. Journal of Applied Mathematics, 2012(105616), 1–25.Google Scholar
  11. Campbell, J. Y. (1987). Stock returns and the term structure. Journal of Financial Economics, 18(2), 373–399.Google Scholar
  12. Campbell, J. Y., Grossman, S. J., & Wang, J. (1993). Trading volume and serial correlation in stock returns. The Quarterly Journal of Economics, 108(4), 905–939.Google Scholar
  13. Campbell, R., Huisman, R., & Koedijk, K. (2001). Optimal portfolio selection in a value-at-risk framework. Journal of Banking and Finance, 25(9), 1789–1804.Google Scholar
  14. Campbell, J. Y., & Viceira, L. M. (1999). Consumption and portfolio decisions when expected returns are time varying. The Quarterly Journal of Economics, 114(2), 433–492.Google Scholar
  15. Campbell, J. Y., & Viceira, L. M. (2001). Strategic assets allocation : Portfolio choice for long-term investors, Clarendon lectures in economics (p. 272). Oxford: Oxford University Press.Google Scholar
  16. Casella, R. L. (2002). Statistical inference (p. 660). Pacific Grove: Duxbury Advanced Series.Google Scholar
  17. Chen, N.-F., Roll, R., & Ross, S. A. (1986). Economic forces and stock market. The Journal of Business, 59(3), 383–403.Google Scholar
  18. Chiu, M. C., & Wong, H. Y. (2011). Mean-variance portfolio selection of cointegrated assets. Journal of Economic Dynamics and Control, 35(8), 1369–1385.Google Scholar
  19. Chiu, M. C., & Wong, H. Y. (2018). Robust dynamic pairs trading with cointegration. Operation Research Letters, 46(2), 225–232.Google Scholar
  20. Chung, P. J., & Liu, D. J. (1994). Common stochastic trends in Pacific Rim stock markets. The Quarterly Review of Economics and Finance, 34(3), 241–259.Google Scholar
  21. Connor, G. (1995). The three types of factor models: a comparison of their explanatory power. Financial Analysts Journal, 51(3), 42–46.Google Scholar
  22. Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7(1), 1–26.Google Scholar
  23. Elliot, R. J., Van Der Hoek, J., & Malcolm, W. P. (2007). Pairs trading. Quantitative Finance, 5(3), 271–276.Google Scholar
  24. Elton, E., Gruber, M., & Rentzler, J. (1983). The arbitrage pricing model and returns on assets under uncertain inflation. The Journal of Finance, 38(2), 525–537.Google Scholar
  25. Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: representation, estimation and testing. Econometrica, 55(2), 258–276.Google Scholar
  26. Fama, E. F. (1998). Determining the number of priced state variables in the ICAPM. Journal of Financial and Quantitative Analysis, 33(2), 217–231.Google Scholar
  27. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(3), 3–56.Google Scholar
  28. Gârleanu, N., & Pedersen, L. H. (2013). Dynamic trading with predictable returns and transaction costs. The Journal of Finance, 68(6), 2309–2340.Google Scholar
  29. Gasser, S. M., Rammerstorfer, M., & Weinmayer, K. (2017). Markowitz revised: social portfolio engineering. European Journal of Operation Research, 258(3), 1181–1190.Google Scholar
  30. Gatev, E., Goetzmann, W. N., & Rouwenhorst, G. (2006). Pairs trading: performance of a relative-value arbitrage rule. The Review of Financial Studies, 19(3), 797–827.Google Scholar
  31. Green, R. C. (1986). Positively weighted portfolio on the minimum-variance frontier. The Journal of Finance, 41(5), 1051–1068.Google Scholar
  32. Griffin, J. M. (2002). Are the Fama and French factors global or country specific? Review of Financial Studies, 15(3), 783–803.Google Scholar
  33. Harvey, A. C. (1989). Forecasting, structural time series models and the Kalman Filter (p. 572). Cambridge: Cambridge University Press.Google Scholar
  34. Hasuike, T., Katagiri, H., & Ishii, H. (2009). Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets and Systems, 160(18), 2579–2596.Google Scholar
  35. Huang, X. (2007). Portfolio selection with fuzzy returns. Journal of Intelligent and Fuzzy Systems, 18(4), 383–390.Google Scholar
  36. Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica, 59(6), 1551–1580.Google Scholar
  37. Johansen, S. (1995). Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford University Press.Google Scholar
  38. Leung, P.-L., Ng, H.-Y., & Wong, W.-K. (2012). An improved estimation to make Markowitz’s portfolio optimization theory users friendly and estimation accurate with application on the US stock market investment. European Journal of Operational Research, 222(1), 85–95.Google Scholar
  39. Levy, H. (1973). The demand for assets under conditions of risk. The Journal of Finance, 28(1), 79–96.Google Scholar
  40. Luenberger, D. G. (1998). Investment science (p. 494). New York: Oxford University Press.Google Scholar
  41. Lütkepohl, H. (2005). New introduction to multiple time series analysis. Berlin: Springer.Google Scholar
  42. Mangram, M. E. (2013). A simplified perspective of the Markowitz portfolio theory. Global Journal of Business Research, 7(1), 59–70.Google Scholar
  43. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.Google Scholar
  44. McCausland, W. J., Miller, S., & Pelletier, D. (2011). Simulation smoothing for state-space models: A computational efficiency analysis. Computational Statistics and Data Analysis, 55(1), 199–212.Google Scholar
  45. Mukherjee, T. K., & Naka, A. (1995). Dynamic relations between macroeconomic variables and the Japanese stock market: An application of a vector error correction model. The Journal of Financial Research, 18(2), 223–237.Google Scholar
  46. Naccarato, A., & Pierini, A. (2014). BEKK element-by-element estimation of a volatility matrix. A portfolio simulation. In P. Cira & S. Marilena (Eds.), Mathematical and statistical methods for actuarial sciences and finance (pp. 145–148). Cham: Springer.Google Scholar
  47. Perlin, M. S. (2009). Evaluation of pairs-trading strategy at the Brazilian financial market. Journal of Derivatives and Hedge Funds, 15(2), 122–136.Google Scholar
  48. Pla-Santamaria, D., & Bravo, M. (2013). Portfolio optimization based on downside risk: A mean-semivariance efficient frontier from Dow Jones blue chips. Annals of Operations Research, 205(1), 189–201.Google Scholar
  49. Pole, A. (2007). Statistical arbitrage. Hoboken: Wiley.Google Scholar
  50. Priestley, R. (1996). The arbitrage pricing theory, macroeconomic and financial factors, and expectations generating processes. Journal of Banking and Finance, 20(5), 869–890.Google Scholar
  51. Salah, H. B., Chaouch, M., Gannoun, A., & De Peretti, C. (2018). Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier. Annals of Operations Research, 262(2), 653–681.Google Scholar
  52. Sanei, M., Banihashemi, S., & Kaveh, M. (2016). Estimation of portfolio efficient frontier by different measures of risk via DEA. International Journal of Industrial Mathematics, 8(3), 10. Article ID IJIM-00460.Google Scholar
  53. Sharpe, W. F. (1992). Asset allocation: Management style and performance measurement. Journal of Portfolio Management, 30(10), 7–16.Google Scholar
  54. Sharpe, W. F. (1994). The Sharpe Ratio. Journal of Portfolio Management, 21(1), 49–58.Google Scholar
  55. Soleimani, H., Golmakani, H. R., & Salimi, M. H. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36(3), 5058–5063.Google Scholar
  56. Tsay, R. S. (2010). Analysis of financial time series (p. 720). Hoboken: Wiley.Google Scholar
  57. Vidyamurthy, G. (2004). Pairs trading: quantitative methods and analysis. Hoboken: Wiley.Google Scholar
  58. Zhao, Z., & Palomar, D. P. (2018). Mean-reverting portfolio with budget constraint. IEEE Transactions on Signal Processing, 66(9), 2342–2357.Google Scholar
  59. Zopounidis, C., Doumpos, M., & Fabozzi, F. J. (2014). Preface to the special issue: 60 years following Harry Markowitz’s contributions in portfolio theory and operations research. European Journal of Operational Research, 234(2), 343–345.Google Scholar

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Authors and Affiliations

  1. 1.Department of EconomicsRoma Tre UniversityRomeItaly
  2. 2.Department of Enterprise EngineeringUniversity of Rome Tor VergataRomeItaly

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