PortfolioOptimisers.jl

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Bootstrap Uncertainty Sets ​

PortfolioOptimisers.StationaryBootstrap Type
julia
struct StationaryBootstrap <: ARCHBootstrapSet end

Bootstrap algorithm for constructing uncertainty sets using a stationary bootstrap in time series data.

Related Types

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PortfolioOptimisers.CircularBootstrap Type
julia
struct CircularBootstrap <: ARCHBootstrapSet end

Bootstrap algorithm for constructing uncertainty sets using a circular bootstrap in time series data.

Related Types

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PortfolioOptimisers.MovingBootstrap Type
julia
struct MovingBootstrap <: ARCHBootstrapSet end

Bootstrap algorithm for constructing uncertainty sets using a moving bootstrap in time series data.

Related Types

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PortfolioOptimisers.ARCHUncertaintySet Type
julia
struct ARCHUncertaintySet{T1, T2, T3, T4, T5, T6, T7} <: BootstrapUncertaintySetEstimator
    pe::T1
    alg::T2
    n_sim::T3
    block_size::T4
    q::T5
    seed::T6
    bootstrap::T7
end

Estimator for box or ellipse uncertainty sets using bootstrap methods for time series data in portfolio optimisation.

Fields

  • pe: Prior estimator used to compute expected returns and covariance statistics.

  • alg: Uncertainty set algorithm.

  • n_sim: Number of bootstrap simulations for uncertainty set estimation.

  • block_size: Size of blocks for bootstrap resampling.

  • q: Quantile or confidence level for uncertainty set bounds.

  • seed: Optional random seed for reproducibility.

  • bootstrap: Bootstrap algorithm type.

Constructors

julia
ARCHUncertaintySet(; pe::AbstractPriorEstimator = EmpiricalPrior(),
                   alg::AbstractUncertaintySetAlgorithm = BoxUncertaintySetAlgorithm(),
                   n_sim::Integer = 3_000, block_size::Integer = 3, q::Real = 0.05,
                   seed::Union{Nothing, <:Integer} = nothing,
                   bootstrap::ARCHBootstrapSet = StationaryBootstrap())

Keyword arguments correspond to the fields above.

Validation

  • n_sim > 0.

  • block_size > 0.

  • 0 < q < 1.

Examples

julia
julia> ARCHUncertaintySet()
ARCHUncertaintySet
          pe ┼ EmpiricalPrior
             │        ce ┼ PortfolioOptimisersCovariance
             │           │   ce ┼ Covariance
             │           │      │    me ┼ SimpleExpectedReturns
             │           │      │       │   w ┴ nothing
             │           │      │    ce ┼ GeneralCovariance
             │           │      │       │   ce ┼ StatsBase.SimpleCovariance: StatsBase.SimpleCovariance(true)
             │           │      │       │    w ┴ nothing
             │           │      │   alg ┴ Full()
             │           │   mp ┼ DefaultMatrixProcessing
             │           │      │       pdm ┼ Posdef
             │           │      │           │   alg ┴ UnionAll: NearestCorrelationMatrix.Newton
             │           │      │   denoise ┼ nothing
             │           │      │    detone ┼ nothing
             │           │      │       alg ┴ nothing
             │        me ┼ SimpleExpectedReturns
             │           │   w ┴ nothing
             │   horizon ┴ nothing
         alg ┼ BoxUncertaintySetAlgorithm()
       n_sim ┼ Int64: 3000
  block_size ┼ Int64: 3
           q ┼ Float64: 0.05
        seed ┼ nothing
   bootstrap ┴ StationaryBootstrap()

Related

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PortfolioOptimisers.ucs Function
julia
ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                           <:Any, <:Any, <:Any}, X::AbstractMatrix,
    F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs box uncertainty sets for expected returns and covariance statistics using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • (mu_ucs::BoxUncertaintySet, sigma_ucs::BoxUncertaintySet): Expected returns and covariance uncertainty sets.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of expected returns and covariance using the specified bootstrap algorithm.

  • Calculates lower and upper bounds for expected returns and covariance using quantiles of bootstrapped samples.

  • Returns both sets as a tuple.

Related

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PortfolioOptimisers.ucs Function
julia
ucs(ue::ARCHUncertaintySet{<:Any, <:EllipseUncertaintySetAlgorithm, <:Any, <:Any,
                           <:Any, <:Any, <:Any}, X::AbstractMatrix,
    F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs ellipse uncertainty sets for expected returns and covariance statistics using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator. Contains prior estimator, ellipse algorithm, simulation parameters, block size, quantile, seed, and bootstrap type.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • (mu_ucs::EllipseUncertaintySet, sigma_ucs::EllipseUncertaintySet): Tuple containing ellipse uncertainty sets for expected returns and covariance.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of expected returns and covariance using the specified bootstrap algorithm.

  • Calculates deviations from prior statistics for each bootstrap sample.

  • Computes covariance matrices of deviations and constructs ellipse uncertainty sets using the specified method and quantile.

  • Returns both sets as a tuple.

Related

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PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                              <:Any, <:Any, <:Any}, X::AbstractMatrix,
       F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs a box uncertainty set for expected returns using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • mu_ucs::BoxUncertaintySet: Expected returns uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of expected returns using the specified bootstrap algorithm.

  • Calculates lower and upper bounds for expected returns using quantiles of bootstrapped samples.

  • Returns the expected returns box uncertainty set.

Related

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PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::ARCHUncertaintySet{<:Any, <:EllipseUncertaintySetAlgorithm, <:Any, <:Any,
                              <:Any, <:Any, <:Any}, X::AbstractMatrix,
       F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipse uncertainty set for expected returns using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator. Contains prior estimator, ellipse algorithm, simulation parameters, block size, quantile, seed, and bootstrap type.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • mu_ucs::EllipseUncertaintySet: Ellipse uncertainty set for expected returns.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of expected returns using the specified bootstrap algorithm.

  • Calculates deviations from prior expected returns for each bootstrap sample.

  • Computes the covariance matrix of deviations and constructs an ellipse uncertainty set using the specified method and quantile.

  • Returns the expected returns ellipse uncertainty set.

Related

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PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                                 <:Any, <:Any, <:Any}, X::AbstractMatrix,
          F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs a box uncertainty set for covariance using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator. Contains prior estimator, box algorithm, simulation parameters, block size, quantile, seed, and bootstrap type.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • sigma_ucs::BoxUncertaintySet: Covariance uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of covariance using the specified bootstrap algorithm.

  • Calculates lower and upper bounds for covariance using quantiles of bootstrapped samples.

  • Returns the covariance box uncertainty set.

Related

source
PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::ARCHUncertaintySet{<:Any, <:EllipseUncertaintySetAlgorithm, <:Any, <:Any,
                                 <:Any, <:Any, <:Any}, X::AbstractMatrix,
          F::Union{Nothing, <:AbstractMatrix} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipse uncertainty set for covariance using bootstrap resampling for time series data.

Arguments

  • ue: ARCH uncertainty set estimator. Contains prior estimator, ellipse algorithm, simulation parameters, block size, quantile, seed, and bootstrap type.

  • X: Data matrix to be resampled.

  • F: Optional factor matrix. Used by the prior estimator.

  • dims: Dimension along which to compute statistics.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • sigma_ucs::EllipseUncertaintySet: Ellipse uncertainty set for covariance.

Details

  • Computes prior statistics using the provided prior estimator.

  • Generates bootstrap samples of covariance using the specified bootstrap algorithm.

  • Calculates deviations from prior covariance for each bootstrap sample.

  • Computes the covariance matrix of deviations and constructs an ellipse uncertainty set using the specified method and quantile.

  • Returns the covariance ellipse uncertainty set.

Related

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PortfolioOptimisers.BootstrapUncertaintySetEstimator Type
julia
abstract type BootstrapUncertaintySetEstimator <: AbstractUncertaintySetEstimator end

Abstract type for estimators that construct uncertainty sets using bootstrap methods in portfolio optimisation.

Subtypes implement specific bootstrap algorithms (e.g., stationary, circular, moving block) to estimate uncertainty sets for risk or prior statistics.

Related Types

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PortfolioOptimisers.ARCHBootstrapSet Type
julia
abstract type ARCHBootstrapSet <: AbstractAlgorithm end

Abstract type for bootstrap algorithms used in constructing uncertainty sets for time series data in portfolio optimisation.

Subtypes implement specific bootstrap methods using archpy.

Related Types

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PortfolioOptimisers.bootstrap_func Function
julia
bootstrap_func(alg::ARCHBootstrapSet, block_size, X, seed)

Creates a bootstrap generator for time series data using the specified bootstrap algorithm.

Arguments

  • alg: Bootstrap algorithm type.

  • block_size: Size of blocks for resampling.

  • X: Data matrix to be resampled.

  • seed: Random seed for reproducibility.

Returns

  • Returns a Python bootstrap generator object from the arch.bootstrap package.

Details

  • Dispatches to the appropriate bootstrap algorithm based on alg.

  • Uses Python's arch.bootstrap via pyimport for stationary, circular, or moving block bootstraps.

  • The generator can be used to produce resampled datasets for uncertainty set estimation.

Related

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PortfolioOptimisers.bootstrap_generator Function
julia
bootstrap_generator(ue::ARCHUncertaintySet, X::AbstractMatrix; kwargs...)

Generates bootstrapped samples of expected returns and covariance statistics for time series data using the specified bootstrap algorithm.

Arguments

  • ue: ARCH uncertainty set estimator.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • (mus::Matrix{<:Real}, sigmas::Array{<:Real, 3}): Tuple containing the matrix of bootstrapped expected return vectors (size(X, 2) × ue.n_sim), array of bootstrapped covariance matrices (size(X, 2) × size(X, 2) × ue.n_sim).

Details

  • Uses the bootstrap algorithm specified in ue.bootstrap to generate resampled datasets.

  • For each bootstrap sample, computes the expected return and covariance using the prior estimator in ue.pe.

  • Stores the bootstrapped expected returns and covariances for uncertainty set estimation.

Related

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PortfolioOptimisers.mu_bootstrap_generator Function
julia
mu_bootstrap_generator(ue::ARCHUncertaintySet, X::AbstractMatrix; kwargs...)

Generates bootstrap samples of expected return vectors for returns data using the specified bootstrap algorithm.

Arguments

  • ue: ARCH uncertainty set estimator.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

  • mus::Matrix{<:Real}: Matrix of bootstrapped expected return vectors (size(X, 2) × ue.n_sim).

Details

  • Uses the bootstrap algorithm specified in ue.bootstrap to generate resampled datasets.

  • For each bootstrap sample, computes the expected return using the prior estimator in ue.pe.

  • Stores the bootstrapped expected returns for uncertainty set estimation.

Related

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PortfolioOptimisers.sigma_bootstrap_generator Function
julia
sigma_bootstrap_generator(ue::ARCHUncertaintySet, X::AbstractMatrix; kwargs...)

Generates bootstrap samples of covariance matrices for time series data using the specified bootstrap algorithm.

Arguments

  • ue: ARCH uncertainty set estimator.

  • kwargs...: Additional keyword arguments passed to the prior estimator.

Returns

julia
- `sigmas::Array{<:Real, 3}`: Array of bootstrapped covariance matrices (`size(X, 2) × size(X, 2) × ue.n_sim`).

Details

  • Uses the bootstrap algorithm specified in ue.bootstrap to generate resampled datasets.

  • For each bootstrap sample, computes the covariance using the prior estimator in ue.pe.

  • Stores the bootstrapped covariances for uncertainty set estimation.

Related

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