Tracking Error Constraints

PortfolioOptimisers.set_tracking_error_constraints! Function

set_tracking_error_constraints!(args...; kwargs...)

Add tracking error constraints to the JuMP optimisation model.

The fall-through method does nothing. Concrete methods dispatch on the tracking algorithm type:

• L1Norm: Enforces ‖net_X - wb * k‖₁ ≤ err * T via NormOneCone.

• L2Norm / SquaredL2Norm: Enforces a scaled L2 norm via SecondOrderCone.

• LpNorm: Enforces a scaled Lp norm via power cone.

• LInfNorm: Enforces ‖net_X - wb * k‖_∞ ≤ err * scale via NormInfinityCone.

• IndependentVariableTracking: Substitutes w - wb for w and applies the chosen risk constraint.

• DependentVariableTracking: Constrains the absolute difference between portfolio risk and benchmark risk.

The collection method iterates over all tracking errors in tres.

Mathematical definition

$$\begin{array}{rl}{t}_{te}& \ge \Vert \mathbf{X}\mathit{w}-\mathit{b}k{\Vert }_p\cdot c_p^{-1},\\ t_{te}& \le \mathrm{err}\cdot k.\end{array}$$

Where:

• $t_{te}$: Auxiliary tracking error scalar variable.

• $\mathbf{X}$: Asset returns matrix ($T\times N$).

• $\mathit{w}$: Portfolio weights vector $N\times 1$.

• $\mathit{b}$: Benchmark return vector.

• $k$: Budget scaling / homogenisation variable.

• $p$: Norm order.

• $c_p$: Norm-specific scaling factor ($T$, $\sqrt{T-d}$, etc.).

• $\mathrm{err}$: Tracking error tolerance.

Arguments

• model::JuMP.Model: The JuMP optimisation model.

• i::Integer: Constraint index for generating unique variable and constraint names.

• pr::AbstractPriorResult: Prior result providing the return matrix X.

• tr: Tracking error specification.

• opt: Optimisation estimator (required for risk-based tracking variants).

• pl: Optional phylogeny constraints.

• fees: Optional fees structure.

Returns

• nothing.

Related

• TrackingError

• RiskTrackingError

• L1Norm

• L2Norm

• LpNorm

• LInfNorm

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