KEY POINTS

• Commonly used constraints in practice include long-only (no short-selling) constraints, turnover constraints, holding constraints, risk factor constraints, and tracking error constraints. These constraints can be handled in a straightforward way by the same type of optimization algorithms used for solving the classical mean-variance portfolio allocation problem.
• Minimum holding constraints, transaction-size constraints, cardinality constraints and round-lot constraints are also widely used in practice, but their nature is such that they require binary and integer modeling, which necessitates the use of mixed-integer and other specialized optimization solvers.
• Transaction costs can easily be incorporated in standard portfolio allocation models. Typical functions for representing transaction costs include linear, piecewise linear, and quadratic.
• Taxes can have a dramatic effect on portfolio returns; however, it is difficult to incorporate them into the classical portfolio optimization framework. Their importance to the individual investor is a strong argument for taking a multiperiod view of investments, but the computational burden of multiperiod portfolio optimization formulations with taxes is extremely high.
• For investment managers who handle multiple accounts, increased transaction costs because of the market impact of simultaneous trades can be an important practical issue and should be taken into consideration when individual clients' portfolio allocation decisions are made to ensure fairness across accounts.
• As the use of quantitative techniques has become widespread in the investment industry, the consideration of estimation risk and model risk has grown in importance. Methods for robust statistical estimation of parameters include shrinkage and Bayesian techniques.
• Portfolio resampling is a technique that uses simulation generate multiple scenarios for possible values of the input parameters in the portfolio optimization problem, and aims to determine portfolio weights that remain stable with respect to small changes in model parameters.
• Robust portfolio optimization incorporates uncertainty directly into the optimization process. The uncertain parameters in the optimization problem are assumed to vary in prespecified uncertainty sets that are selected subjectively or based on data.