When to Use Portfolio Objects Over Optimization Toolbox

While you can use Optimization Toolbox™ to solve portfolio optimization problems, Financial Toolbox™ has the Portfolio, PortfolioCVaR, and PortfolioMAD objects that you can use as well. Which tool you use depends on the problem case:

Always use a Portfolio, PortfolioCVaR, or PortfolioMAD object when the problem can most easily be written and implemented using one of these objects. This case includes problems that can be solved only with the internal solvers of the Portfolio, PortfolioCVaR, or PortfolioMAD object. The Optimization Toolbox solvers cannot directly handle these problems. For details, see Always Use Portfolio, PortfolioCVaR, or PortfolioMAD Object.
Prefer to use a Portfolio, PortfolioCVaR, or PortfolioMAD object when the problem can be modeled and implemented using both the Portfolio, PortfolioCVaR, or PortfolioMAD object and the Optimization Toolbox problem-based framework. The advantage of using a Portfolio, PortfolioCVaR, or PortfolioMAD object instead of the Optimization Toolbox is that the internal tools of these objects simplify the analysis. For details, see Preferred Use of Portfolio, PortfolioCVaR, or PortfolioMAD Object.
Opt to use Optimization Toolbox for problems that cannot be solved with the internal solvers of the Portfolio, PortfolioCVaR, or PortfolioMAD object. Some of the problems that the Portfolio, PortfolioCVaR, or PortfolioMAD object cannot solve can be addressed using the Optimization Toolbox problem-based framework. Problems that cannot be directly solved with either framework require either some restructuring of the model or an implementation of a specialized solver. For details, see Use Optimization Toolbox.

The following table summarizes the objective functions, constraints, and variables that apply in each case for solving a portfolio problem.

Case for Solving Portfolio Problem	Objective Function	Constraints	Integer (Binary) Variables
"Always" case with Financial Toolbox	Return — Gross portfolio returns or net portfolio returns Risk — Variance, CVaR, or MAD Sharpe ratio (only for mean-variance problems using `Portfolio` object) Continuous convex functions (only using the `Portfolio` object)	Return — Gross portfolio returns or net portfolio returns Risk — Variance, CVaR, or MAD (risk constraints are only supported when the objective is the return) Linear equalities Linear inequalities Tracking error (only for mean-variance problems using `Portfolio` object). Tracking error is supported with custom objectives using `estimateCustomObjectivePortfolio`. Tracking error is only supported when the objective is the return, variance, or Sharpe ratio. Turnover	Bounds on the number of assets Conditional budget constraint (an upper bound on the sum of all assets that go above a specified threshold) Conditional (semicontinuous) bounds (for example, if asset i is selected, then x_i ≥ lb_i, otherwise x_i = 0)
"Preferred" case with Financial Toolbox	Return — Gross portfolio returns or net portfolio returns Risk — Variance, CVaR, or MAD Sharpe ratio (only for mean-variance problems using `Portfolio` object) Continuous functions (only using the `Portfolio` object)	Return — Gross portfolio returns or net portfolio returns Risk — Variance, CVaR, or MAD (risk constraints are only supported when the objective is the return) Linear equalities Linear inequalities Tracking error (only for mean-variance problems using `Portfolio` object). Tracking error is supported with custom objectives using `estimateCustomObjectivePortfolio`. Tracking error is only supported when the objective is the return, variance, or Sharpe ratio.	None
Optimization Toolbox	Any other nonlinear function not mentioned in Always Use Portfolio, PortfolioCVaR, or PortfolioMAD Object and Preferred Use of Portfolio, PortfolioCVaR, or PortfolioMAD Object	Any other nonlinear function not mentioned in Always Use Portfolio, PortfolioCVaR, or PortfolioMAD Object and Preferred Use of Portfolio, PortfolioCVaR, or PortfolioMAD Object	None

Always Use `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` Object

The two general cases for always using the Portfolio, PortfolioCVaR, or PortfolioMAD object are:

Problems with both supported nonlinear constraints and conditional bounds or bounds in the number of assets.
These problems include:
- Minimum risk problems subject to constraints for return, linear equality, linear inequality, turnover, and tracking error where the supported risk measures are variance, conditional value-at-risk (CVaR), and mean-absolute-deviation (MAD)
- Maximum return problems subject to constraints for linear equality, liner inequality, turnover, risk, and tracking error where the supported risk measures are variance, CVaR, and MAD
  Tracking error is supported only for mean-variance problems using the Portfolio object. For more information on the supported constraints for a Portfolio, PortfolioCVaR, or PortfolioMAD object, see Supported Constraints for Portfolio Optimization Using Portfolio Objects.
- Custom objective minimization or maximization. The custom objective function (using estimateCustomObjectivePortfolio) must be continuous and convex subject to constraints for return, risk, linear equality, linear inequality, tracking error, and turnover.
Tracking error is supported using the Portfolio object only when the objective is the return, variance or Sharpe ratio. Sharpe ratio is supported only using the Portfolio object.
For more information on the supported constraints and nonlinear risk functions for Portfolio, PortfolioCVaR, and PortfolioMAD objects, see Supported Constraints for Portfolio Optimization Using Portfolio Objects and Portfolio Optimization Theory. The integer (binary) variables can come from either of the following sources: bounds on the number of assets that can be selected in the portfolio, setting conditional (semicontinuous) bounds on the assets, or adding a conditional budget constraint. For example, if asset i is selected, then x_i ≥ lb_i, otherwise x_i = 0. These problems cannot be solved using the Optimization Toolbox solvers. However, you can implement your own mixed-integer solver. For more information, see Mixed-Integer Quadratic Programming Portfolio Optimization: Problem-Based.
Problems with turnover constraints and sell or buy costs
Although the continuous version of these problems can be solved by the Optimization Toolbox solvers, the variable space must be manipulated to rewrite the nonsmooth constraints into equivalent smooth constraints. Given that rewriting the problem requires optimization knowledge, it is recommended to use the Portfolio, PortfolioCVaR, and PortfolioMAD objects instead.

Preferred Use of `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` Object

The general case for preferred use of the Portfolio, PortfolioCVaR, or PortfolioMAD object is:

Continuous problems with minimum risk, maximum return, and maximum Sharpe ratio that are subject to linear equality, liner inequality, turnover, and tracking error constraints.
Sharpe ratio is supported only for mean-variance problems using the Portfolio object. For more information on the supported constraints for a Portfolio, PortfolioCVaR, or PortfolioMAD object, see Supported Constraints for Portfolio Optimization Using Portfolio Objects.
Custom objective minimization or maximization. The custom objective function (using estimateCustomObjectivePortfolio) must be continuous and convex subject to constraints for return, risk, linear equality, linear inequality, tracking error, and turnover.

Tracking error is supported using the Portfolio object only when the objective is the return, variance or Sharpe ratio. Sharpe ratio is supported only using the Portfolio object.

The supported risk measures are variance, CVaR, and MAD. For more information on the supported constraints for these risk measures, see Supported Constraints for Portfolio Optimization Using Portfolio Objects, Supported Constraints for Portfolio Optimization Using PortfolioCVaR Object, and Supported Constraints for Portfolio Optimization Using PortfolioMAD Object. For all other risk measures and constraints and if tracking error is in the objective, use the Optimization Toolbox.

The advantage of the Portfolio, PortfolioCVaR, and PortfolioMAD object framework over the problem-based framework for the type of problems in the "preferred" case is that common portfolio optimization workflows are leveraged. For example, the Portfolio, PortfolioCVaR, and PortfolioMAD object framework supports the following workflows:

Estimating and plotting the efficient frontier
Exchanging the return and risk proxies from the objective function to a constraint
Solving the maximum Sharpe ratio problem
Adding bounds on the number of assets selected
Adding semicontinuous bounds
Simplifying the use of turnover constraints and sell or buy costs

Use Optimization Toolbox

The two general cases to use Optimization Toolbox is:

Problems that have nonlinear constraints other than the constraints for risk or tracking error