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Tracking Error Constraints: private API

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: Enforces a scaled L2 norm via SecondOrderCone.
  • SquaredL2Norm: The same cone, with the bound square-rooted, because err bounds the squared error that norm_error reports.
  • 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

\[\begin{align} t_{te} &\geq \lVert \mathbf{X}\boldsymbol{w} - \boldsymbol{b} k \rVert_p \cdot c_p^{-1}\,, \\ t_{te} &\leq \mathrm{err} \cdot k\,. \end{align}\]

Where:

  • $t_{te}$: Auxiliary tracking error scalar variable.
  • $\mathbf{X}$: Asset returns matrix ($T \times N$).
  • $\boldsymbol{w}$: Portfolio weights vector $N \times 1$.
  • $\boldsymbol{b}$: 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

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PortfolioOptimisers.tracking_error_soc_factor — Function
tracking_error_soc_factor(
    f::L2Norm,
    err::Number,
    T::Integer
) -> Any

Convert a TrackingError tolerance into the bound on the second-order cone variable.

Both norms share one cone, which bounds $\lVert \mathbf{X}\boldsymbol{w} - \boldsymbol{b}k \rVert_2$. They do not share the quantity that err bounds. norm_error divides that norm by $\sqrt{T - d}$ for an L2Norm and squares it before dividing by $T - d$ for a SquaredL2Norm, so the second bound is square-rooted here to keep the model, the functor and set_risk_constraints! in agreement.

Arguments

  • f: The NormError the tracking error carries.
  • err::Number: Tracking error tolerance.
  • T::Integer: Number of observations.

Returns

  • f::Number: Upper bound on the cone variable, before the budget scaling by $k$.

Related

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