Markowitz Portfolio Theory

  • Arlie O. Petters
  • Xiaoying Dong
Part of the Springer Undergraduate Texts in Mathematics and Technology book series (SUMAT)


We introduce Harry Markowitz’s mathematical model for how to distribute an initial capital across a collection of risky securities to create an efficient portfolio, namely, one with the least risk given an expected return and largest expected return given a level of portfolio risk. This chapter covers: the set up of the Markowitz portfolio model, which includes modeling security returns, the issue of multivariate normality, weights, short selling, portfolio return, portfolio risk, and portfolio log returns; two-security portfolio theory; the efficient frontier for N securities with and without short selling; the global minimum-variance portfolio, diversified portfolio, and Mutual Fund Theorem; utility functions and utility maximization; and diversification.


  1. [1]
    Bodie, Z., Kane, A., Marcus, A.: Investments, 9th edn. McGraw-Hill, New York (2011)Google Scholar
  2. [2]
    Capiński, M., Zastawniak, T.: Mathematics for Finance. Springer, New York (2003)zbMATHGoogle Scholar
  3. [3]
    Durrett, R.: Probability: Theory and Examples, 4th edn. University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  4. [4]
    Fine, T.: Probability and Probabilistic Reasoning for Electrical Engineering. Pearson Prentice Hall, Upper Saddle River (2006)Google Scholar
  5. [5]
    Frigyik, B., Kapila, A., Gupta, M.: Introduction to the Dirichlet distribution and related processes. University of Washington Electrical Engineering Technical Report, Number UWEETR-2010-0006 (2010)Google Scholar
  6. [6]
    Goldberg, L.: A review of “Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume I)” by Harry Markowitz and Kenneth Blay. (2014)
  7. [7]
    Goodman, J.: Statistical Optics. Wiley Classics Library Edition. Wiley, New York (2000)Google Scholar
  8. [8]
    Graham, J., Smart, S., Megginson, W.: Corporate Finance. South-Western Cengage Learning, Mason (2010)Google Scholar
  9. [9]
    Grinold, R., Kahn, R.: Active Portfolio Management. McGraw-Hill, New York (2000)Google Scholar
  10. [10]
    Ingersoll, J.: Theory of Financial Decision Making. Rowman and Littlefield, Savage (1987)Google Scholar
  11. [11]
    Jorion, P.: Value at Risk, 3rd edn. McGraw-Hill, New York (2007)Google Scholar
  12. [12]
    Korn, R.: Optimal Portfolios. World Scientific, River Edge (1997)CrossRefzbMATHGoogle Scholar
  13. [13]
    Korn, R., Korn, E.: Option Pricing and Portfolio Optimization. American Mathematical Society, Providence (2001)CrossRefzbMATHGoogle Scholar
  14. [14]
    Levy, H.: A review of “Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume I)” by Harry Markowitz and Kenneth Blay (2014). Quant. Finance 14 (7), 1141 (2014)CrossRefGoogle Scholar
  15. [15]
    Levy, H., Duchin, R.: Asset return distributions and the investment horizon. J. Portf. Manag. 30 (3), 47 (2004)CrossRefGoogle Scholar
  16. [16]
    Luenberger, D.: Investment Science. Oxford University Press, New York (1998)zbMATHGoogle Scholar
  17. [17]
    Markowitz, H.: Portfolio selection. J. Financ. Res. 7 (1), 77 (1952)Google Scholar
  18. [18]
    Markowitz, H.: Portfolio Selection. Blackwell, Cambridge (1959)Google Scholar
  19. [19]
    Markowitz, H., Blay, K.: Risk-Return Analysis: The Theory and Practice of Rational Investing, vol. I. McGraw-Hill, New York (2014)Google Scholar
  20. [20]
    Merton, R.: An analytic derivation of the efficient portfolio frontier. J. Financ. Quant. Anal. 7 (3), 1851 (1972)CrossRefGoogle Scholar
  21. [21]
    Pennacchi, G.: Theory of Asset Pricing. Pearson Addison Wesley, Boston (2008)Google Scholar
  22. [22]
    Reiley, F., Brown, K.: Investment Analysis and Portfolio Management. Dryden Press, Fort Worth (1997)Google Scholar
  23. [23]
    Roman, S.: Introduction to the Mathematics of Finance. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  24. [24]
    Ross, S.: An Elementary Introduction to Mathematical Finance. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  25. [25]
    Vanderbei, R.: Linear Programming: Foundations and Extensions, 4th edn. Springer, New York (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Arlie O. Petters and Xiaoying Dong 2016

Authors and Affiliations

  • Arlie O. Petters
    • 1
  • Xiaoying Dong
    • 1
  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations