ISORETURN

Solve for an iso-return efficient frontier. Construct an efficient frontier to evaluate the risk and tracking-error tradeoffs.

Description

Solve for multiple optimal portfolios on the efficient surface for a specified target return. Evaluate the tradeoff between absolute risk and relative risk (tracking-error).

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.

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

Input(s)

Argument	Description
P	Required. Number of portfolios to solve for, $P \geq 5$.
target	Required. Target expected returns, the target level must be between the return of minimum risk portfolio and the maximum return portfolio.
mu	Required. Vector of expected returns.
sigma	Required. Vector of expected risk.
rho	Required. Correlation matrix.
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. $\begin{bmatrix}1,1,…,1_N​,1,1\end{bmatrix}$

Output(s)

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

$\text{output} = \begin{bmatrix} w_{1,1} & w_{1,2} & \ldots & w_{1,K} & \sigma_1 & \sigma_{te_1} & \text{exitFlag}_1 \\ w_{2,1} & w_{2,2} & \ldots & w_{2,K} & \sigma_2 & \sigma_{te_2} & \text{exitFlag}_2 \\ \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ \\ w_{N,1} & w_{N,2} & \ldots & w_{N,K} & \sigma_N & \sigma_{te_N} & \text{exitFlag}_N \end{bmatrix}$

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

82KB

ISORETURN.xlsx

Example Workbook: ISORETURN