# MVO ## Description Maximize expected returns and minimize expected risk. Solve for a mean-variance optimal portfolio. The function allows you to specify both linear and non-linear constraints and is able to account for friction penalties (transaction costs). ## Syntax The following describes the function signature for use in Microsoft Excel's formula bar. ```excel-formula =MVO(mu, sigma, rho, aversion, 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 risk aversion. If the argument is not specified, it defaults to 1. | | **wInitial** | Optional. Vector of initial weights (or your current weights). This is used to measure the friction penalties or as a starting point should a numerical approach be necessary. If not specified, it defaults to a vector of zeros. | | **tc** | Optional. Vector of transaction costs. If the argument is not specified, it defaults to a vector zeros. | | **lb** | Optional. Vector of lower bound limits. If the argument is not specified, it defaults to a vector zeros. | | **ub** | Optional. Vector of upper bound limits. If the argument is not specified, it defaults to a vector 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. \[1, 1, \ldots , 1\_{N}, 1, 1]

| | **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 ![](https://258561627-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MK66-MGuoULhqCDXLwy%2F-MLsoAw1A3BeoKDkwJKU%2F-MLt-kXczwcWZmVXkobt%2Fimage.png?alt=media\&token=bb7a760f-ced2-4f54-9051-a58a806e4212) {% file src="" %} Example Workbook: MVO {% endfile %}