MVT

Optimize portfolio allocations for absolute and relative performance with this multi-goal optimization function. Maximize return while minimizing risk and tracking-error (Mean-variance-tracking-error)

Description

For investors concerned with both absolute and relative performance, the mean-variance tracking-error (MVT) objective function maximizes a portfolio’s expected return while minimizing both risk and tracking error. This function also allows for the robust specifications of linear and nonlinear constraints.

Syntax

The following describes the function signature for use in Microsoft Excel's formula bar.

=MVT(mu, sigma, rho, λ1, λ2, 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.
λ1	Optional. Scalar value for risk aversion. If not specified, it defaults to 1.
λ2	Optional. Scalar value for tracking aversion. If not specified, it defaults to 1.
wBenchmark	Optional. Vector of benchmark weights. If not specified, the function assumes a vector of zeros.
wInitial	Optional. Vector of initial weights (or your current weights). If not specified, the function assumes a vector of zeros.
tc	Optional. Vector of transaction costs. If not specified, the function assumes a vector of zeros.
lb	Optional. Vector of lower bound limits. If 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

88KB

MVT.xlsx

Example Workbook: MVT

Further Reading

Multi-goal OptimizationWindham Insights

Insights into Windham's Multi-goal Optimization approach