Schur Complement Hierarchical Risk Parity

PortfolioOptimisers.SchurComplementAlgorithm Type

abstract type SchurComplementAlgorithm <: AbstractAlgorithm

Abstract supertype for Schur Complement algorithm variants.

Related Types

• NonMonotonicSchurComplement

• MonotonicSchurComplement

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PortfolioOptimisers.NonMonotonicSchurComplement Type

struct NonMonotonicSchurComplement <: SchurComplementAlgorithm

Non-monotonic Schur Complement algorithm variant for SCHRP.

Uses the raw Schur complement formula without monotonicity correction.

Related Types

• SchurComplementAlgorithm

• MonotonicSchurComplement

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PortfolioOptimisers.MonotonicSchurComplement Type

struct MonotonicSchurComplement{__T_N, __T_tol, __T_iter, __T_strict} <: SchurComplementAlgorithm

Monotonic Schur Complement algorithm variant for SCHRP.

Applies a bisection-based correction to ensure the Schur complement allocation factor $\gamma$ is monotonically increasing with cluster risk, controlled by convergence tolerance tol and maximum iterations iter.

Fields

• N: Number of bisection steps.

• tol: Convergence tolerance.

• iter: Maximum iterations (or nothing for no limit).

• strict: If true, raises an error if convergence is not achieved.

Constructors

MonotonicSchurComplement(;
    N::Integer = 10,
    tol::Number = 1e-4,
    iter::Option{<:Integer} = nothing,
    strict::Bool = false
) -> MonotonicSchurComplement

Keywords correspond to the struct's fields.

Related Types

• SchurComplementAlgorithm

• NonMonotonicSchurComplement

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PortfolioOptimisers.SchurComplementParams Type

struct SchurComplementParams{__T_r, __T_gamma, __T_pdm, __T_alg, __T_flag} <: AbstractAlgorithm

Parameters for the Schur Complement step of SCHRP.

SchurComplementParams collects the risk measure, initial allocation factor $\gamma$, positive-definite matrix correction, and monotonicity algorithm used in the Schur Complement Hierarchical Risk Parity optimisation.

Fields

• r: Risk measure (Variance or StandardDeviation) used for intra-cluster risk.

• gamma: Initial allocation factor ($0\le \gamma \le 1$).

• pdm: Positive definite matrix correction method.

• alg: Schur Complement algorithm variant.

• flag: If true, uses the Schur Complement inversion; if false, uses the direct Schur inverse.

Constructors

SchurComplementParams(;
    r::Sd_Var = Variance(),
    gamma::Number = 0.5,
    pdm::Option{<:Posdef} = Posdef(),
    alg::SchurComplementAlgorithm = MonotonicSchurComplement(),
    flag::Bool = true
) -> SchurComplementParams

Keywords correspond to the struct's fields.

Validation

• 0 <= gamma <= 1.

Related

• SchurComplementHierarchicalRiskParity

• SchurComplementAlgorithm

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PortfolioOptimisers.SchurComplementHierarchicalRiskParityResult Type

struct SchurComplementHierarchicalRiskParityResult{__T_oe, __T_pr, __T_wb, __T_clr, __T_gamma, __T_retcode, __T_w, __T_fb} <: NonFiniteAllocationOptimisationResult

Result type returned by SchurComplementHierarchicalRiskParity optimisation.

Stores the optimisation estimator, prior result, weight bounds, clustering result, Schur complement scaling parameter, return code, optimised weights, and optional fallback estimator.

Fields

• oe: Optimisation estimator used.

• pr: Prior result containing asset moments.

• wb: Weight bounds estimator.

• clr: Clustering result.

• gamma: Schur complement scaling parameter.

• retcode: Optimisation return code.

• w: Optimised asset weights.

• fb: Optional fallback optimisation estimator.

Related

• SchurComplementHierarchicalRiskParity

• NonFiniteAllocationOptimisationResult

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PortfolioOptimisers.SchurComplementHierarchicalRiskParity Type

struct SchurComplementHierarchicalRiskParity{__T_opt, __T_params, __T_fb} <: ClusteringOptimisationEstimator

Schur Complement Hierarchical Risk Parity (SCHRP) portfolio optimiser.

SchurComplementHierarchicalRiskParity extends HRP by using the Schur complement of the covariance matrix to more accurately decompose inter-cluster risk when allocating portfolio weights across the dendrogram.

Fields

• opt: Base hierarchical optimiser configuration.

• params: Schur Complement parameters (single or vector).

• fb: Fallback optimiser.

Constructors

SchurComplementHierarchicalRiskParity(;
    opt::HierarchicalOptimiser = HierarchicalOptimiser(),
    params::ScP_VecScP = SchurComplementParams(),
    fb::Option{<:OptE_Opt} = nothing
) -> SchurComplementHierarchicalRiskParity

Keywords correspond to the struct's fields.

Validation

• If params is a vector: !isempty(params).

Examples

julia> SchurComplementHierarchicalRiskParity()
SchurComplementHierarchicalRiskParity
     opt ┼ HierarchicalOptimiser
         │       pe ┼ EmpiricalPrior
         │          │        ce ┼ PortfolioOptimisersCovariance
         │          │           │   ce ┼ Covariance
         │          │           │      │    me ┼ SimpleExpectedReturns
         │          │           │      │       │   w ┴ nothing
         │          │           │      │    ce ┼ GeneralCovariance
         │          │           │      │       │   ce ┼ StatsBase.SimpleCovariance: StatsBase.SimpleCovariance(true)
         │          │           │      │       │    w ┴ nothing
         │          │           │      │   alg ┴ Full()
         │          │           │   mp ┼ DenoiseDetoneAlgMatrixProcessing
         │          │           │      │     pdm ┼ Posdef
         │          │           │      │         │      alg ┼ UnionAll: NearestCorrelationMatrix.Newton
         │          │           │      │         │   kwargs ┴ @NamedTuple{}: NamedTuple()
         │          │           │      │      dn ┼ nothing
         │          │           │      │      dt ┼ nothing
         │          │           │      │     alg ┼ nothing
         │          │           │      │   order ┴ DenoiseDetoneAlg()
         │          │        me ┼ SimpleExpectedReturns
         │          │           │   w ┴ nothing
         │          │   horizon ┴ nothing
         │      cle ┼ ClustersEstimator
         │          │    ce ┼ PortfolioOptimisersCovariance
         │          │       │   ce ┼ Covariance
         │          │       │      │    me ┼ SimpleExpectedReturns
         │          │       │      │       │   w ┴ nothing
         │          │       │      │    ce ┼ GeneralCovariance
         │          │       │      │       │   ce ┼ StatsBase.SimpleCovariance: StatsBase.SimpleCovariance(true)
         │          │       │      │       │    w ┴ nothing
         │          │       │      │   alg ┴ Full()
         │          │       │   mp ┼ DenoiseDetoneAlgMatrixProcessing
         │          │       │      │     pdm ┼ Posdef
         │          │       │      │         │      alg ┼ UnionAll: NearestCorrelationMatrix.Newton
         │          │       │      │         │   kwargs ┴ @NamedTuple{}: NamedTuple()
         │          │       │      │      dn ┼ nothing
         │          │       │      │      dt ┼ nothing
         │          │       │      │     alg ┼ nothing
         │          │       │      │   order ┴ DenoiseDetoneAlg()
         │          │    de ┼ Distance
         │          │       │   power ┼ nothing
         │          │       │     alg ┴ CanonicalDistance()
         │          │   alg ┼ HClustAlgorithm
         │          │       │   linkage ┴ Symbol: :ward
         │          │   onc ┼ OptimalNumberClusters
         │          │       │   max_k ┼ nothing
         │          │       │     alg ┼ SecondOrderDifference
         │          │       │         │   alg ┼ StandardisedValue
         │          │       │         │       │   mv ┼ MeanValue
         │          │       │         │       │      │   w ┴ nothing
         │          │       │         │       │   sv ┼ StdValue
         │          │       │         │       │      │           w ┼ nothing
         │          │       │         │       │      │   corrected ┴ Bool: true
         │      slv ┼ nothing
         │       wb ┼ WeightBounds
         │          │   lb ┼ Float64: 0.0
         │          │   ub ┴ Float64: 1.0
         │     fees ┼ nothing
         │     sets ┼ nothing
         │       wf ┼ IterativeWeightFinaliser
         │          │   iter ┴ Int64: 100
         │      brt ┼ Bool: false
         │   cle_pr ┼ Bool: true
         │   strict ┴ Bool: false
  params ┼ SchurComplementParams
         │       r ┼ Variance
         │         │   settings ┼ RiskMeasureSettings
         │         │            │   scale ┼ Float64: 1.0
         │         │            │      ub ┼ nothing
         │         │            │     rke ┴ Bool: true
         │         │      sigma ┼ nothing
         │         │       chol ┼ nothing
         │         │         rc ┼ nothing
         │         │        alg ┴ SquaredSOCRiskExpr()
         │   gamma ┼ Float64: 0.5
         │     pdm ┼ Posdef
         │         │      alg ┼ UnionAll: NearestCorrelationMatrix.Newton
         │         │   kwargs ┴ @NamedTuple{}: NamedTuple()
         │     alg ┼ MonotonicSchurComplement
         │         │        N ┼ Int64: 10
         │         │      tol ┼ Float64: 0.0001
         │         │     iter ┼ nothing
         │         │   strict ┴ Bool: false
         │    flag ┴ Bool: true
      fb ┴ nothing

Related

• ClusteringOptimisationEstimator

• HierarchicalRiskParity

• HierarchicalEqualRiskContribution

• SchurComplementParams

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PortfolioOptimisers.optimise Function

optimise(sh::SchurComplementHierarchicalRiskParity{<:Any, <:Any, Nothing},
         rd::ReturnsResult = ReturnsResult(); dims::Int = 1, kwargs...) -> HierarchicalResult

Arguments

• sh: The Schur complement hierarchical risk parity optimiser to use.

• rd: The returns result to use. If isa(hrp.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.

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

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