PortfolioOptimisers.jl

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Normal Uncertainty Sets ​

PortfolioOptimisers.NormalUncertaintySet Type
julia
struct NormalUncertaintySet{T1, T2, T3, T4, T5, T6, T7} <: AbstractUncertaintySetEstimator
    pe::T1
    alg::T2
    n_sim::T3
    q::T4
    rng::T5
    seed::T6
    ens::T7
end

Estimator for box or ellipsoidal uncertainty sets under the assumption of normally distributed returns in portfolio optimisation.

Fields

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

  • alg: Uncertainty set algorithm (box or ellipsoidal).

  • n_sim: Number of simulations for uncertainty set estimation.

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

  • rng: Random number generator for simulation.

  • seed: Optional random seed for reproducibility.

  • ens: Optional effective number of scenarios used for scaling the uncertainty sets.

Constructors

julia
NormalUncertaintySet(; pe::AbstractLowOrderPriorEstimator = EmpiricalPrior(),
                     alg::AbstractUncertaintySetAlgorithm = BoxUncertaintySetAlgorithm(),
                     n_sim::Integer = 3_000, q::Number = 0.05,
                     rng::Random.AbstractRNG = Random.default_rng(),
                     seed::Option{<:Integer} = nothing, ens::Option{<:Number} = nothing)

Keyword arguments correspond to the fields above.

Validation

  • n_sim > 0.

  • 0 < q < 1.

Examples

julia
julia> NormalUncertaintySet()
NormalUncertaintySet
     pe ┼ EmpiricalPrior
        │        ce ┼ PortfolioOptimisersCovariance
        │           │   ce ┼ Covariance
        │           │      │    me ┼ SimpleExpectedReturns
        │           │      │       │     w ┼ nothing
        │           │      │       │   idx ┴ nothing
        │           │      │    ce ┼ GeneralCovariance
        │           │      │       │    ce ┼ StatsBase.SimpleCovariance: StatsBase.SimpleCovariance(true)
        │           │      │       │     w ┼ nothing
        │           │      │       │   idx ┴ 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
        │           │   idx ┴ nothing
        │   horizon ┴ nothing
    alg ┼ BoxUncertaintySetAlgorithm()
  n_sim ┼ Int64: 3000
      q ┼ Float64: 0.05
    rng ┼ Random.TaskLocalRNG: Random.TaskLocalRNG()
   seed ┼ nothing
    ens ┴ nothing

Related

source
PortfolioOptimisers.ucs Function
julia
ucs(ue::NormalUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any, <:Any},
    X::MatNum,
    F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs box uncertainty sets for mean and covariance statistics under the assumption of normally distributed returns.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions 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 sets.

Details

  • Computes prior statistics using the provided prior estimator.

  • Simulates covariance matrices using the Wishart distribution.

  • Computes lower and upper bounds for covariance using quantiles of simulated values.

  • Computes upper bound for mean using the normal quantile and simulated covariance.

  • Returns both sets as a tuple.

Related

source
PortfolioOptimisers.ucs Function
julia
ucs(ue::NormalUncertaintySet{<:Any,
                             <:EllipsoidalUncertaintySetAlgorithm{<:NormalKUncertaintyAlgorithm, <:Any},
                             <:Any, <:Any, <:Any},
    X::MatNum,
    F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs ellipsoidal uncertainty sets for expected returns and covariance statistics under the assumption of normally distributed returns.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty sets.

Details

  • Computes prior statistics using the provided prior estimator.

  • Simulates mean and covariance using multivariate normal and Wishart distributions.

  • Constructs commutation matrix for covariance vectorization.

  • Computes scaling parameters for mean and covariance using the specified ellipsoidal algorithm.

  • Returns both sets as a tuple.

Related

source
PortfolioOptimisers.ucs Function
julia
ucs(ue::NormalUncertaintySet{<:Any,
                             <:EllipsoidalUncertaintySetAlgorithm{<:ChiSqKUncertaintyAlgorithm, <: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 the chi-squared scaling algorithm under the assumption of normally distributed returns.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty sets.

Details

  • Computes prior statistics using the provided prior estimator.

  • Constructs commutation matrix for covariance vectorization.

  • Computes scaling parameters for mean and covariance using the chi-squared scaling algorithm.

  • Applies diagonal processing if specified in the algorithm.

  • Returns both sets as a tuple.

Related

source
PortfolioOptimisers.ucs Function
julia
ucs(ue::NormalUncertaintySet{<: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 under the assumption of normally distributed returns, using a generic ellipsoidal algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty sets.

Details

  • Computes prior statistics using the provided prior estimator.

  • Constructs commutation matrix for covariance vectorization.

  • Computes scaling parameters for mean and covariance using the provided ellipsoidal algorithm.

  • Applies diagonal processing if specified in the algorithm.

  • Returns both sets as a tuple.

Related

source
PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::NormalUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any, <:Any},
       X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs a box uncertainty set for expected returns under the assumption of normally distributed returns.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions 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.

  • Computes the upper bound for mean using the normal quantile and simulated covariance.

  • Sets the lower bound for mean to zero.

  • Ignores additional arguments and keyword arguments except those passed to the prior estimator.

Related

source
PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::NormalUncertaintySet{<:Any,
                                <:EllipsoidalUncertaintySetAlgorithm{<:NormalKUncertaintyAlgorithm, <:Any},
                                <:Any, <:Any, <:Any},
       X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for expected returns under the assumption of normally distributed returns, using a normal scaling algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Simulates mean vectors using the multivariate normal distribution.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Computes the scaling parameter for the ellipsoidal using the normal scaling algorithm.

  • Returns an ellipsoidal uncertainty set for the mean.

Related

source
PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::NormalUncertaintySet{<:Any,
                                <:EllipsoidalUncertaintySetAlgorithm{<:ChiSqKUncertaintyAlgorithm, <:Any},
                                <:Any, <:Any, <:Any},
       X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for expected returns under the assumption of normally distributed returns, using a chi-squared scaling algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Processes the covariance matrix for positive definiteness.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Computes the scaling parameter for the ellipsoidal using the chi-squared scaling algorithm.

  • Returns an ellipsoidal uncertainty set for the mean.

Related

source
PortfolioOptimisers.mu_ucs Function
julia
mu_ucs(ue::NormalUncertaintySet{<:Any,
                                <:EllipsoidalUncertaintySetAlgorithm{<:ChiSqKUncertaintyAlgorithm, <:Any},
                                <:Any, <:Any, <:Any},
       X::MatNum,
       F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for expected returns under the assumption of normally distributed returns, using a chi-squared scaling algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • mu_ucs::EllipsoidalUncertaintySet: Expected returns uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Processes the covariance matrix for positive definiteness.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Computes the scaling parameter for the ellipsoidal using the chi-squared scaling algorithm.

  • Returns an ellipsoidal uncertainty set for the mean.

Related

source
PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::NormalUncertaintySet{<:Any, <:BoxUncertaintySetAlgorithm, <:Any, <:Any, <:Any},
          X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs a box uncertainty set for covariance under the assumption of normally distributed returns.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions 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.

  • Simulates covariance matrices using the Wishart distribution.

  • Computes lower and upper bounds for covariance using quantiles of simulated values.

  • Ensures positive definiteness of the bounds.

  • Ignores additional arguments and keyword arguments except those passed to the prior estimator.

Related

source
PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::NormalUncertaintySet{<:Any,
                                   <:EllipsoidalUncertaintySetAlgorithm{<:NormalKUncertaintyAlgorithm, <:Any},
                                   <:Any, <:Any, <:Any},
          X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for covariance under the assumption of normally distributed returns, using a normal scaling algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Simulates covariance matrices using the Wishart distribution.

  • Constructs commutation matrix for covariance vectorization.

  • Computes the scaling parameter for the ellipsoidal using the normal scaling algorithm.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Returns an ellipsoidal uncertainty set for covariance.

Related

source
PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::NormalUncertaintySet{<:Any,
                                   <:EllipsoidalUncertaintySetAlgorithm{<:ChiSqKUncertaintyAlgorithm, <:Any},
                                   <:Any, <:Any, <:Any},
          X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for covariance under the assumption of normally distributed returns, using a chi-squared scaling algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Processes the covariance matrix for positive definiteness.

  • Constructs commutation matrix for covariance vectorization.

  • Computes the scaling parameter for the ellipsoidal using the chi-squared scaling algorithm.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Returns an ellipsoidal uncertainty set for covariance.

Related

source
PortfolioOptimisers.sigma_ucs Function
julia
sigma_ucs(ue::NormalUncertaintySet{<:Any,
                                   <:EllipsoidalUncertaintySetAlgorithm{<:Any, <:Any},
                                   <:Any, <:Any, <:Any}, X::MatNum,
          F::Option{<:MatNum} = nothing; dims::Int = 1, kwargs...)

Constructs an ellipsoidal uncertainty set for covariance under the assumption of normally distributed returns, using a generic ellipsoidal algorithm.

Arguments

  • ue: Normal uncertainty set estimator.

  • X: Data matrix.

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

  • dims: Dimensions along which to perform the computation.

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

Returns

  • sigma_ucs::EllipsoidalUncertaintySet: Covariance uncertainty set.

Details

  • Computes prior statistics using the provided prior estimator.

  • Processes the covariance matrix for positive definiteness.

  • Constructs commutation matrix for covariance vectorization.

  • Computes the scaling parameter for the ellipsoidal using the provided ellipsoidal algorithm.

  • Applies diagonal processing to the covariance if specified in the algorithm.

  • Returns an ellipsoidal uncertainty set for covariance.

Related

source
PortfolioOptimisers.commutation_matrix Function
julia
commutation_matrix(X::MatNum)

Constructs the commutation matrix for a given matrix.

Arguments

  • X: Input matrix. Used to determine the dimensions of the commutation matrix.

Returns

  • cX::SparseArrays.SparseMatrixCSC{<:Number, <:Integer}: Returns a sparse commutation matrix of size (m*n, m*n) where m, n = size(X).

Details

  • The commutation matrix is used to vectorize and permute matrix elements.

  • It is constructed by reshaping and transposing the indices of the input matrix.

  • The output is a sparse matrix for efficiency.

Examples

julia
julia> PortfolioOptimisers.commutation_matrix(rand(2, 2))
4×4 SparseArrays.SparseMatrixCSC{Float64, Int64} with 4 stored entries:
 1.0   ⋅    ⋅    ⋅
  ⋅    ⋅   1.0   ⋅
  ⋅   1.0   ⋅    ⋅
  ⋅    ⋅    ⋅   1.0

julia> PortfolioOptimisers.commutation_matrix(rand(2, 3))
6×6 SparseArrays.SparseMatrixCSC{Float64, Int64} with 6 stored entries:
 1.0   ⋅    ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅   1.0   ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅   1.0   ⋅
  ⋅   1.0   ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅   1.0   ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅    ⋅   1.0

julia> PortfolioOptimisers.commutation_matrix(rand(3, 2))
6×6 SparseArrays.SparseMatrixCSC{Float64, Int64} with 6 stored entries:
 1.0   ⋅    ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅   1.0   ⋅    ⋅
  ⋅   1.0   ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅   1.0   ⋅
  ⋅    ⋅   1.0   ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅    ⋅   1.0
source