MTFRONTIER

Solve for the Mean-Tracking-Error efficient frontier. Optimize for multiple portfolios to evaluate return and risk trade-offs in relative return space (active management).

Description

Solve for multiple mean-tracking-error optimal portfolios on the efficient frontier. Evaluate the return and relative risk tradeoff. The function allows you to specify both linear and non-linear constraints and is able to account for friction penalties (transaction costs).

The convexity of the efficient frontier may not necessarily hold when transaction costs are present.

Syntax

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

=MTFRONTIER(P, mu, sigma, rho, wBenchmark, wInitial, tc, lb, ub, constraints)

Input(s)

Argument	Description
P	Required. Number of portfolios to solve for $P \geq 5$.
mu	Required. Vector of expected returns.
sigma	Required. Vector of expected risk.
rho	Required. Correlation matrix.
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 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 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. $\begin{bmatrix}1,1,…,1_N​,1,1\end{bmatrix}$

Output(s)

The function returns optimal weights $w$ across $N$assets for $P$portfolios. The portfolios' expected return, tracking-error, and corresponding optimization's exit flag is appended at the end of the matrix.

$\text{output} = \begin{bmatrix} w_{1,1} & w_{1,2} & \ldots & w_{1,K} & \mu_1 & \sigma_{te_1} & \text{exitFlag}_1 \\ w_{2,1} & w_{2,2} & \ldots & w_{2,K} & \mu_2 & \sigma_{te_2} & \text{exitFlag}_2 \\ \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ \\ w_{N,1} & w_{N,2} & \ldots & w_{N,K} & \mu_N & \sigma_{te_N} & \text{exitFlag}_N \end{bmatrix}$

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

83KB

MTFRONTIER.xlsx

Example Workbook: MTFRONTIER