Relaxed risk budgeting

PortfolioOptimisers.RelaxedRiskBudgetingAlgorithm Type

abstract type RelaxedRiskBudgetingAlgorithm <: OptimisationAlgorithm

Abstract supertype for relaxed risk budgeting algorithm variants.

Related Types

• BasicRelaxedRiskBudgeting

• RegularisedRelaxedRiskBudgeting

• RegularisedPenalisedRelaxedRiskBudgeting

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PortfolioOptimisers.BasicRelaxedRiskBudgeting Type

struct BasicRelaxedRiskBudgeting <: RelaxedRiskBudgetingAlgorithm

Basic Relaxed Risk Budgeting formulation.

Uses the basic Second Order Cone (SOC) relaxation of the risk budgeting problem without additional regularisation.

Related Types

• RelaxedRiskBudgetingAlgorithm

• RegularisedRelaxedRiskBudgeting

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PortfolioOptimisers.RegularisedRelaxedRiskBudgeting Type

struct RegularisedRelaxedRiskBudgeting <: RelaxedRiskBudgetingAlgorithm

Regularised Relaxed Risk Budgeting formulation.

Extends the basic SOC formulation with a regularisation term to improve numerical stability.

Related Types

• RelaxedRiskBudgetingAlgorithm

• BasicRelaxedRiskBudgeting

• RegularisedPenalisedRelaxedRiskBudgeting

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PortfolioOptimisers.RegularisedPenalisedRelaxedRiskBudgeting Type

struct RegularisedPenalisedRelaxedRiskBudgeting{__T_p} <: RelaxedRiskBudgetingAlgorithm

Regularised and penalised Relaxed Risk Budgeting formulation.

Extends the regularised formulation with a penalty on deviations from target risk budgets, controlled by parameter p.

Fields

• p: Power or order parameter.

Constructors

RegularisedPenalisedRelaxedRiskBudgeting(;
    p::Number = 1.0
) -> RegularisedPenalisedRelaxedRiskBudgeting

Keywords correspond to the struct's fields.

Validation

• isfinite(p) and p > 0.

Related Types

• RelaxedRiskBudgetingAlgorithm

• RegularisedRelaxedRiskBudgeting

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PortfolioOptimisers.RelaxedRiskBudgeting Type

struct RelaxedRiskBudgeting{__T_opt, __T_rba, __T_wi, __T_alg, __T_fb} <: JuMPOptimisationEstimator

Relaxed Risk Budgeting (RRB) portfolio optimiser.

RelaxedRiskBudgeting implements a relaxed formulation of the risk budgeting problem using a Second Order Cone constraint on the portfolio variance. Unlike RiskBudgeting, it does not require a logarithmic or mixed-integer formulation, making it computationally more tractable.

Fields

• opt: JuMP optimiser configuration.

• rba: Risk budget algorithm.

• wi: Initial portfolio weights for warm-starting the solver.

• alg: Relaxed risk budgeting algorithm variant.

• fb: Fallback result or estimator.

Constructors

RelaxedRiskBudgeting(;
    opt::JuMPOptimiser = JuMPOptimiser(),
    rba::RiskBudgetingAlgorithm = AssetRiskBudgeting(),
    wi::Option{<:VecNum} = nothing,
    alg::RelaxedRiskBudgetingAlgorithm = BasicRelaxedRiskBudgeting(),
    fb::Option{<:OptE_Opt} = nothing
) -> RelaxedRiskBudgeting

Keywords correspond to the struct's fields.

Validation

• If wi is provided: !isempty(wi).

Mathematical definition

The Relaxed Risk Budgeting (RRB) formulation replaces the non-convex risk-parity constraint with a second-order cone (SOC) relaxation. Let $\mathbf{G}$ be the Cholesky factor of $\mathbf{\Sigma }$ (so ${\mathbf{G}}^{⊺}\mathbf{G}=\mathbf{\Sigma }$). Introduce auxiliary variables $\mathit{\zeta }=\mathbf{\Sigma }\mathit{w}$, $\psi \ge 0$, $\gamma \ge 0$:

$$\begin{array}{rl}\underset{\mathit{w},\psi ,\gamma ,\mathit{\zeta }}{min}& \psi -\gamma ,\\ \text{s.t.}& \mathit{\zeta }=\mathbf{\Sigma }\mathit{w},\\ & \left(\begin{array}{c}{w}_i+{\zeta }_i\\ 2\gamma \sqrt{b_i}\\ w_i-{\zeta }_i\end{array}\right)\in {\mathcal{K}}_{\mathrm{SOC}},\mathrm{\forall }i.\end{array}$$

The risk cone constraint (basic variant): $(\psi ,\mathbf{G}\mathit{w})\in {\mathcal{K}}_{\mathrm{SOC}}$, i.e. $\psi \ge \parallel \mathbf{G}\mathit{w}{\parallel }_2=\sqrt{{\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}}$.

Where:

• $\mathit{w}$: Portfolio weight vector.

• $\psi$, $\gamma$: Scalar auxiliary variables.

• $\mathit{\zeta }$: Auxiliary vector equal to $\mathbf{\Sigma }\mathit{w}$.

• $b_i$: Risk budget for asset $i$.

• $\mathbf{G}$: Cholesky factor of $\mathbf{\Sigma }$ (so ${\mathbf{G}}^{⊺}\mathbf{G}=\mathbf{\Sigma }$).

• $\mathbf{\Sigma }$: Covariance matrix.

• ${\mathcal{K}}_{\mathrm{SOC}}$: Second-order cone.

Related

• JuMPOptimisationEstimator

• RiskBudgeting

• RelaxedRiskBudgetingAlgorithm

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PortfolioOptimisers.needs_previous_weights Method

needs_previous_weights(opt::RelaxedRiskBudgeting) -> Any

Return true if the JuMP optimiser or fallback requires previous portfolio weights.

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PortfolioOptimisers.factory Method

factory(
    rrb::RelaxedRiskBudgeting,
    w::AbstractVector
) -> RelaxedRiskBudgeting

Build an updated RelaxedRiskBudgeting with all estimators that track previous weights updated via factory using w.

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PortfolioOptimisers.port_opt_view Method

port_opt_view(
    rrb::RelaxedRiskBudgeting,
    i,
    X::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    args...
) -> RelaxedRiskBudgeting

Return a cluster-sliced copy of RelaxedRiskBudgeting for asset index set i and returns matrix X.

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PortfolioOptimisers.set_relaxed_risk_budgeting_alg_constraints! Function

set_relaxed_risk_budgeting_alg_constraints!(alg, model, w, sigma, chol)

Add algorithm-specific second-order cone constraints for Relaxed Risk Budgeting.

Dispatches based on the RRB algorithm variant. Adds second-order cone constraints implementing the basic, regularised, or regularised-penalised RRB formulation.

Arguments

• alg: RRB algorithm (BasicRelaxedRiskBudgeting, RegularisedRelaxedRiskBudgeting, or RegularisedPenalisedRelaxedRiskBudgeting).

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

• w::VecJuMPScalar: Portfolio weight variables.

• sigma::MatNum: Covariance matrix.

• chol::Option{<:MatNum}: Optional pre-computed Cholesky factor.

Returns

• nothing.

Related

• RelaxedRiskBudgeting

• set_relaxed_risk_budgeting_constraints!

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PortfolioOptimisers._set_relaxed_risk_budgeting_constraints! Method

_set_relaxed_risk_budgeting_constraints!(model, ...)

Internal function to set relaxed risk budgeting constraints in the JuMP model.

Configures inequality constraints for the relaxed risk budgeting formulation, allowing small deviations from exact budget targets.

Arguments

• model: JuMP model.

• Additional relaxed risk budgeting parameters.

Returns

• nothing.

Related

• RelaxedRiskBudgeting

• _set_risk_budgeting_constraints!

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PortfolioOptimisers.set_relaxed_risk_budgeting_constraints! Function

set_relaxed_risk_budgeting_constraints!(model, rrb, pr, wb, args...)

Add Relaxed Risk Budgeting (RRB) constraints and weight variables to the JuMP model.

Dispatches based on the risk budgeting algorithm type. Configures weight variables, budget constraints, second-order cone constraints, and weight bounds.

Arguments

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

• rrb::RelaxedRiskBudgeting: RRB estimator configuration.

• pr::AbstractPriorResult: Prior result with asset moments.

• wb::WeightBounds: Weight bounds configuration.

• args...: Additional arguments (e.g. returns data for factor risk budgeting).

Returns

• Processed risk budgeting attributes.

Related

• RelaxedRiskBudgeting

• set_relaxed_risk_budgeting_alg_constraints!

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PortfolioOptimisers.optimise Function

optimise(rrb::RelaxedRiskBudgeting{<:Any, <:Any, <:Any, <:Any, Nothing},
         rd::ReturnsResult = ReturnsResult(); dims::Int = 1,
         str_names::Bool = false, save::Bool = true, kwargs...) -> RiskBudgetingResult

Run the Relaxed Risk Budgeting portfolio optimisation.

Arguments

• rrb: The relaxed risk budgeting optimiser to use.

• rd: The returns result to use. If isa(rrb.opt.pe, AbstractPriorResult), rd is not necessary if doing a standalone optimisation, but may be required/desired by fallbacks and/or clusterisation.

• dims: The dimension along which observations advance in time.

• str_names: Whether to use string names for the assets in the optimisation.

• save: Whether to save the JuMP model in the optimisation result.

• kwargs: Additional keyword arguments passed to the optimisation function.

Related

• RelaxedRiskBudgeting

• RiskBudgetingResult

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