# MTO
## Description
Maximize expected returns relative to a benchmark and minimize expected tracking-error. Solve for the mean-tracking-error optimal portfolio (MTO). This is a quadratic programming optimizer in active space. The function allow you to specify both linear and non-linear constraints and is able to account for friction penalties such as transaction costs across assets.
## Syntax
The following describes the function signature for use in Microsoft Excel's formula bar.
```excel-formula
=MTO(mu, sigma, rho, aversion, wBenchmark, wInitial, tc, lb, ub, constraints, nonlincons)
```
### Input(s)
| Argument | Description |
| --------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| **mu** | Required. Vector of expected returns. |
| **sigma** | Required. Vector of expected risk. |
| **rho** | Required. Correlation matrix. |
| **aversion** | Optional. Scalar value for tracking aversion. If the argument is not specified, it defaults to 1. |
| **wBenchmark** | Optional. Vector of benchmark weights. This is used to evaluate the utility function in excess return space. If not specified, the function assumes a vector of zeros. |
| **wInitial** | Optional. Vector of initial weights (or your current weights). This is used to measure the friction penalties when solving for an optimal transition. If not specified, the function assumes a vector of zeros. |
| **tc** | Optional. Vector of transaction costs. If the argument is not specified, it defaults to a vector of zeros. |
| **lb** | Optional. Vector of lower bound limits. If the argument is not specified, it defaults to a vector of zeros. |
| **ub** | Optional. Vector of upper bound limits. If the argument is not specified, it defaults to a vector of ones. |
| **constraints** | Optional. Matrix of constraints, operator enumeration, and values: \begin{bmatrix}A & op \&b\end{bmatrix}
The operator enumeration is represented by op \in \begin{cases} 0: & \leq \1: & = \ 2: & \geq \end{cases}
If the argument is not specified, it will default to a fully-funded constraint.
i.e. \begin{bmatrix}1,1,…,1\_N,1,1\end{bmatrix}
|
| **nonlincons** | Optional. Matrix to specify nonlinear constraint enumeration, operator enumeration, and values: \begin{bmatrix}nonlinType & op & value\end{bmatrix}
The nonlinType enumeration is nonlinType \in \begin{cases} 0: & \text{off} \1: & \text{same risk} \ 2: & \text{same tracking-error} \end{cases}
|
### Output(s)
The function returns a vector of optimal weights $$w$$ across $$N$$assets and appends the corresponding optimization's exit flag.
$$\text{output}=\begin{bmatrix}w\_1 & w\_2 & \ldots & w\_N & \text{exitFlag}\end{bmatrix}$$
The output matrix follows the vector orientation of mu (column / row). If you have specified your inputs as column-vectors, the corresponding output matrix will be transpose of the above.
| exitFlag | Description |
| :------: | --------------------------------------------------------------------------------------------------- |
| **-2** | No feasible solution found. Check your constraints and problem definition. |
| **-1** | Unexpected interruption. |
| **0** | Number of iterations exceeded. |
| **1** | First-order optimality measure is less than tolerance threshold and the constraints were satisfied. |
| **2** | Delta in optimal weights is less than the configured numerical step size. |
| **3** | Change in the expected utility value is less than the tolerance threshold. |
| **4** | Magnitude of search direction was less than the configured threshold. |
| **5** | Magnitude of directional derivative in the search direction was less than the configured threshold. |
## Example
{% file src="" %}
Example Workbook: MTO
{% endfile %}