Background & further reading

Portfolio Optimization

The following resources provide additional background on mean-variance portfolio theory, and beyond.

  • Best, M.J. (2010). Portfolio Optimization (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b17178

    In this textbook, the interplay of portfolio theory and optimization is approached at a level appropriate for a one-semester undergraduate course. It features accompanying MATLAB programs, including a self-contained quadratic programming solver. It details the practical importance of additional constraints to the classical mean-variance model, and their effects on the geometry of the efficient frontier.

  • Cornuéjols, G., Peña, J., & Tütüncü, R. (2018). Optimization Methods in Finance (2nd ed.). Cambridge: Cambridge University Press. https://doi.org/10.1017/9781107297340

    This textbook approaches mean-variance portfolio theory and practice from an optimization angle. It includes a self-contained treatment of quadratic optimization theory for portfolio optimization, and introduces mixed-integer optimization as a tool for modeling discrete decisions. The scope of this book is well beyond just classical mean-variance portfolios and includes asset pricing, multi-period models, CVaR measures (in relation to linear optimization), stochastic programming and robust optimization techniques.

  • Markowitz, H. (1952), PORTFOLIO SELECTION. The Journal of Finance, 7: 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x

    This work marks the beginning of rigorous mathematical and statistical theory of mean-variance portfolio selection (including the “expected returns–variance of returns” rule). It features intuitive, geometric arguments for the three-securities case. It is the first to describe portfolio selection as a two-stage process: (1) finding reasonable estimates for return and variance, and (2) selecting positions based on mathematical optimality conditions.

gurobipy

The following resources are helpful for getting started with and using gurobipy:

  • A one-page overview of gurobipy’s modeling functionality

    Runs through key concepts, including models, variables, constraints, solution process, and more. This is a good starting point to get an overview of gurobipy’s capabilities and common usage patterns.

  • gurobipy reference documentation

    Detailed API reference documentation for all classes and methods.

  • YouTube video on modeling with ndarray and sparse matrices

    For portfolio optimization, it is natural to express matrix and vector expressions directly as such into a gurbipy model. This video introduces gurobipy’s related modeling capabilities.

Discrete aspects of portfolio optimization

The following are a few selected research papers that touch on the interplay between portfolio optimization and discrete decision making through mixed-integer optimization:

  • Bonami P., Lejeune M. A., (2009) An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and Integer Constraints. Operations Research 57(3):650-670. https://doi.org/10.1287/opre.1080.0599

  • Chang T.-J., Meade N., Beasley J.E., Sharaiha Y.M. 2000. Heuristics for Cardinality Constrained Portfolio Optimisation. Computers & Operations Research 27, 1271–1302. https://doi.org/10.1016/S0305-0548(99)00074-X.

  • N.J. Jobst , M.D. Horniman , C.A. Lucas & G. Mitra (2001) Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints, Quantitative Finance, 1:5, 489-501. https://doi.org/10.1088/1469-7688/1/5/301