MTO

Mean-tracking-error optimization, maximize your return in excess of a benchmark while minimizing tracking-error.

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.

=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

87KB

MTO.xlsx

Example Workbook: MTO