Bootstrap Uncertainty Sets

PortfolioOptimisers.StationaryBootstrap Type

struct StationaryBootstrap <: ARCHBootstrapSet

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

Related

• ARCHBootstrapSet

• CircularBootstrap

• MovingBootstrap

source

PortfolioOptimisers.CircularBootstrap Type

struct CircularBootstrap <: ARCHBootstrapSet

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

Related

• ARCHBootstrapSet

• StationaryBootstrap

• MovingBootstrap

source

PortfolioOptimisers.MovingBootstrap Type

struct MovingBootstrap <: ARCHBootstrapSet

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

Related

• ARCHBootstrapSet

• StationaryBootstrap

• CircularBootstrap

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PortfolioOptimisers.ARCHUncertaintySet Type

struct ARCHUncertaintySet{__T_pe, __T_alg, __T_n_sim, __T_block_size, __T_q, __T_seed, __T_bootstrap, __T_kwargs} <: BootstrapUncertaintySetEstimator

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

Fields

• pe: Prior estimator.

• alg: Uncertainty set algorithm.

• n_sim: Number of simulation samples.

• block_size: Block size for bootstrap sampling.

• q: Quantile parameter for uncertainty set computation.

• seed: Seed for the random number generator.

• bootstrap: Bootstrap algorithm.

• kwargs: Additional keyword arguments.

Constructors

ARCHUncertaintySet(;
    pe::AbstractLowOrderPriorEstimator = EmpiricalPrior(),
    alg::AbstractUncertaintySetAlgorithm = BoxUncertaintySetAlgorithm(),
    n_sim::Integer = 3_000,
    block_size::Integer = 3,
    q::Number = 0.05,
    seed::Option{<:Integer} = nothing,
    bootstrap::ARCHBootstrapSet = StationaryBootstrap(),
    kwargs::NamedTuple = (;),
) -> ARCHUncertaintySet

Keywords correspond to the struct's fields.

Validation

• n_sim > 0.

• block_size > 0.

• 0 < q < 1.

Examples

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 ┼ MatrixProcessing
             │           │      │     pdm ┼ Posdef
             │           │      │         │      alg ┼ UnionAll: NearestCorrelationMatrix.Newton
             │           │      │         │   kwargs ┴ @NamedTuple{}: NamedTuple()
             │           │      │      dn ┼ nothing
             │           │      │      dt ┼ nothing
             │           │      │     alg ┼ nothing
             │           │      │   order ┴ NTuple{4, Symbol}: (:pdm, :dn, :dt, :alg)
             │        me ┼ SimpleExpectedReturns
             │           │   w ┴ nothing
             │   horizon ┴ nothing
         alg ┼ BoxUncertaintySetAlgorithm()
       n_sim ┼ Int64: 3000
  block_size ┼ Int64: 3
           q ┼ Float64: 0.05
        seed ┼ nothing
   bootstrap ┼ StationaryBootstrap()
      kwargs ┴ @NamedTuple{}: NamedTuple()

Related

• BootstrapUncertaintySetEstimator

• ARCHBootstrapSet

• StationaryBootstrap

• CircularBootstrap

• MovingBootstrap

• BoxUncertaintySet

• EllipsoidalUncertaintySet

source

PortfolioOptimisers.ucs Function

ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                           <:Any, <:Any, <:Any}, X::MatNum,
    F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

Generate $M$ bootstrap samples, compute ${\hat{\mathit{\mu }}}^{(m)}$ and ${\hat{\mathbf{\Sigma }}}^{(m)}$, then take element-wise quantile bounds:

$$\begin{array}{rl}{\mu }_{lb,i}& =Q_{q/2}\left({\hat{\mu }}_i^{(m)}\right),\\ {\mu }_{ub,i}& =Q_{1-q/2}\left({\hat{\mu }}_i^{(m)}\right).\end{array}$$$$\begin{array}{rl}({\mathbf{\Sigma }}_{lb}{)}_{ij}& =Q_{q/2}\left({\hat{\mathrm{\Sigma }}}_{ij}^{(m)}\right),\\ ({\mathbf{\Sigma }}_{ub}{)}_{ij}& =Q_{1-q/2}\left({\hat{\mathrm{\Sigma }}}_{ij}^{(m)}\right).\end{array}$$

Where:

• ${\mu }_{lb,i}$, ${\mu }_{ub,i}$: Lower/upper bounds for expected return of asset $i$.

• $({\mathbf{\Sigma }}_{lb}{)}_{ij}$, $({\mathbf{\Sigma }}_{ub}{)}_{ij}$: Lower/upper covariance bounds.

• $Q_{q/2}$, $Q_{1-q/2}$: Quantile functions at level $q/2$.

• ${\hat{\mu }}_i^{(m)}$: Bootstrap mean for asset $i$ in sample $m$.

• ${\hat{\mathrm{\Sigma }}}_{ij}^{(m)}$: Bootstrap covariance element $(i,j)$ in sample $m$.

• $q$: Significance level.

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 perform the computation.

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

Returns

• mu_ucs::BoxUncertaintySet: Expected returns uncertainty set.

• sigma_ucs::BoxUncertaintySet: Covariance uncertainty set.

Details

• Computes prior statistics using the provided prior estimator.

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

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

• Returns both sets as a tuple.

Related

• ARCHUncertaintySet

• BoxUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.ucs Function

ucs(ue::ARCHUncertaintySet{<:Any, <:EllipsoidalUncertaintySetAlgorithm, <:Any, <:Any,
                           <:Any, <:Any, <:Any}, X::MatNum,
    F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

Compute bootstrap deviations ${\mathit{\delta }}_{\mu }^{(m)}={\hat{\mathit{\mu }}}^{(m)}-\hat{\mathit{\mu }}$ and ${\mathit{\delta }}_{\mathrm{\Sigma }}^{(m)}=\mathrm{vec}({\hat{\mathbf{\Sigma }}}^{(m)}-\hat{\mathbf{\Sigma }})$. Fit empirical covariances:

$$\begin{array}{rl}{\mathbf{\Sigma }}_{\mu }& =\mathrm{Cov}\left({\mathit{\delta }}_{\mu }^{(m)}\right),\\ {\mathbf{\Sigma }}_{\mathrm{\Sigma }}& =\mathrm{Cov}\left({\mathit{\delta }}_{\mathrm{\Sigma }}^{(m)}\right).\end{array}$$

Then form ellipsoidal sets:

$$\begin{array}{rl}{\mathcal{E}}_{\mu }& =\{\mathit{\mu }:(\mathit{\mu }-\hat{\mathit{\mu }}{)}^{⊺}{\mathbf{\Sigma }}_{\mu }^{-1}(\mathit{\mu }-\hat{\mathit{\mu }})\le k_{\mu }^2\}.\end{array}$$$$\begin{array}{rl}{\mathcal{E}}_{\mathrm{\Sigma }}& =\{\mathbf{\Sigma }:{\Vert {\mathbf{\Sigma }}_{\mathrm{\Sigma }}^{-1/2}\mathrm{vec}(\mathbf{\Sigma }-\hat{\mathbf{\Sigma }})\Vert }_2\le k_{\mathrm{\Sigma }}\}.\end{array}$$

Where:

• ${\mathbf{\Sigma }}_{\mu }$: Empirical covariance of bootstrap mean deviations.

• ${\mathbf{\Sigma }}_{\mathrm{\Sigma }}$: Empirical covariance of bootstrap covariance deviations (vectorised).

• $\hat{\mathit{\mu }}$, $\hat{\mathbf{\Sigma }}$: Estimated mean and covariance.

• ${\mathit{\delta }}_{\mu }^{(m)}$, ${\mathit{\delta }}_{\mathrm{\Sigma }}^{(m)}$: Bootstrap deviations for mean and covariance.

• ${\mathcal{E}}_{\mu }$: Ellipsoidal uncertainty set for expected returns.

• ${\mathcal{E}}_{\mathrm{\Sigma }}$: Ellipsoidal uncertainty set for covariance.

• $k_{\mu }$, $k_{\mathrm{\Sigma }}$: Empirically fitted scaling parameters.

Arguments

• ue: ARCH uncertainty set estimator. Contains prior estimator, ellipsoidal 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 perform the computation.

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

Returns

• mu_ucs::EllipsoidalUncertaintySet: Ellipsoidal uncertainty set for expected returns.

• sigma_ucs::EllipsoidalUncertaintySet: Ellipsoidal uncertainty set for covariance.

Details

• Computes prior statistics using the provided prior estimator.

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

• Computes deviations from prior statistics for each bootstrap sample.

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

• Returns both sets as a tuple.

Related

• ARCHUncertaintySet

• EllipsoidalUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.mu_ucs Function

mu_ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                              <:Any, <:Any, <:Any}, X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

$$\begin{array}{rl}{\mu }_{lb,i}& =Q_{q/2}\left({\hat{\mu }}_i^{(m)}\right),\\ {\mu }_{ub,i}& =Q_{1-q/2}\left({\hat{\mu }}_i^{(m)}\right).\end{array}$$

Where:

• ${\mu }_{lb,i}$, ${\mu }_{ub,i}$: Lower/upper bounds for expected return of asset $i$.

• $Q_{q/2}$, $Q_{1-q/2}$: Quantile functions at level $q/2$.

• ${\hat{\mu }}_i^{(m)}$: Bootstrap mean for asset $i$ in sample $m$.

• $q$: Significance level.

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 perform the computation.

• 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.

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

• Returns the expected returns box uncertainty set.

Related

• ARCHUncertaintySet

• BoxUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.mu_ucs Function

mu_ucs(ue::ARCHUncertaintySet{<:Any, <:EllipsoidalUncertaintySetAlgorithm, <:Any, <:Any,
                              <:Any, <:Any, <:Any}, X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

$$\begin{array}{rl}{\mathbf{\Sigma }}_{\mu }& =\mathrm{Cov}({\hat{\mathit{\mu }}}^{(m)}-\hat{\mathit{\mu }}),\\ {\mathcal{E}}_{\mu }& =\{\mathit{\mu }:(\mathit{\mu }-\hat{\mathit{\mu }}{)}^{⊺}{\mathbf{\Sigma }}_{\mu }^{-1}(\mathit{\mu }-\hat{\mathit{\mu }})\le k_{\mu }^2\}.\end{array}$$

Where:

• ${\mathbf{\Sigma }}_{\mu }$: Empirical covariance of bootstrap mean deviations.

• $\hat{\mathit{\mu }}$: Estimated mean vector.

• ${\mathcal{E}}_{\mu }$: Ellipsoidal uncertainty set for expected returns.

• $k_{\mu }$: Empirically fitted scaling parameter.

Arguments

• ue: ARCH uncertainty set estimator. Contains prior estimator, ellipsoidal 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 perform the computation.

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

Returns

• mu_ucs::EllipsoidalUncertaintySet: Ellipsoidal 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.

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

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

• Returns the expected returns ellipsoidal uncertainty set.

Related

• ARCHUncertaintySet

• EllipsoidalUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.sigma_ucs Function

sigma_ucs(ue::ARCHUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any,
                                 <:Any, <:Any, <:Any}, X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

$$\begin{array}{rl}({\mathbf{\Sigma }}_{lb}{)}_{ij}& =Q_{q/2}\left({\hat{\mathrm{\Sigma }}}_{ij}^{(m)}\right),\\ ({\mathbf{\Sigma }}_{ub}{)}_{ij}& =Q_{1-q/2}\left({\hat{\mathrm{\Sigma }}}_{ij}^{(m)}\right).\end{array}$$

Where:

• $({\mathbf{\Sigma }}_{lb}{)}_{ij}$, $({\mathbf{\Sigma }}_{ub}{)}_{ij}$: Lower/upper covariance bounds.

• $Q_{q/2}$, $Q_{1-q/2}$: Quantile functions at level $q/2$.

• ${\hat{\mathrm{\Sigma }}}_{ij}^{(m)}$: Bootstrap covariance element $(i,j)$ in sample $m$.

• $q$: Significance level.

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 perform the computation.

• 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.

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

• Returns the covariance box uncertainty set.

Related

• ARCHUncertaintySet

• BoxUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.sigma_ucs Function

sigma_ucs(ue::ARCHUncertaintySet{<:Any, <:EllipsoidalUncertaintySetAlgorithm, <:Any, <:Any,
                                 <:Any, <:Any, <:Any}, X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

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

Mathematical definition

$$\begin{array}{rl}{\mathbf{\Sigma }}_{\mathrm{\Sigma }}& =\mathrm{Cov}(\mathrm{vec}({\hat{\mathbf{\Sigma }}}^{(m)}-\hat{\mathbf{\Sigma }})),\\ {\mathcal{E}}_{\mathrm{\Sigma }}& =\{\mathbf{\Sigma }:{\Vert {\mathbf{\Sigma }}_{\mathrm{\Sigma }}^{-1/2}\mathrm{vec}(\mathbf{\Sigma }-\hat{\mathbf{\Sigma }})\Vert }_2\le k_{\mathrm{\Sigma }}\}.\end{array}$$

Where:

• ${\mathbf{\Sigma }}_{\mathrm{\Sigma }}$: Empirical covariance of bootstrap covariance deviations (vectorised).

• $\hat{\mathbf{\Sigma }}$: Estimated covariance matrix.

• ${\mathcal{E}}_{\mathrm{\Sigma }}$: Ellipsoidal uncertainty set for covariance.

• $k_{\mathrm{\Sigma }}$: Empirically fitted scaling parameter.

Arguments

• ue: ARCH uncertainty set estimator. Contains prior estimator, ellipsoidal 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 perform the computation.

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

Returns

• sigma_ucs::EllipsoidalUncertaintySet: Ellipsoidal uncertainty set for covariance.

Details

• Computes prior statistics using the provided prior estimator.

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

• Computes deviations from prior covariance for each bootstrap sample.

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

• Returns the covariance ellipsoidal uncertainty set.

Related

• ARCHUncertaintySet

• EllipsoidalUncertaintySet

• bootstrap_generator

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.BootstrapUncertaintySetEstimator Type

abstract type BootstrapUncertaintySetEstimator <: AbstractUncertaintySetEstimator

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

• ARCHUncertaintySet

source

PortfolioOptimisers.ARCHBootstrapSet Type

abstract type ARCHBootstrapSet <: AbstractAlgorithm

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

• StationaryBootstrap

• CircularBootstrap

• MovingBootstrap

source

PortfolioOptimisers.bootstrap_func Function

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

• StationaryBootstrap

• CircularBootstrap

• MovingBootstrap

• ARCHBootstrapSet

source

PortfolioOptimisers.bootstrap_generator Function

bootstrap_generator(ue::ARCHUncertaintySet, X::MatNum; 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{<:Number}: Matrix of bootstrapped expected return vectors (size(X, 2) × ue.n_sim).

• sigmas::Array{<:Number, 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 expected return and covariance using the prior estimator in ue.pe.

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

Related

• ARCHUncertaintySet

• bootstrap_func

• mu_bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.mu_bootstrap_generator Function

mu_bootstrap_generator(ue::ARCHUncertaintySet, X::MatNum; 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{<:Number}: 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

• ARCHUncertaintySet

• bootstrap_func

• bootstrap_generator

• sigma_bootstrap_generator

source

PortfolioOptimisers.sigma_bootstrap_generator Function

sigma_bootstrap_generator(ue::ARCHUncertaintySet, X::MatNum; 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

• sigmas::Array{<:Number, 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

• ARCHUncertaintySet

• bootstrap_func

• bootstrap_generator

• mu_bootstrap_generator

source