# 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 ![](/files/-MLt-kXczwcWZmVXkobt) {% file src="/files/-MLt-qotLTK9oF8WC\_X9" %} Example Workbook: MVO {% endfile %} --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://excelhelp.windhamlabs.com/functions/optimization/mvo.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.