PortfolioOptimisers.jl

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Kurtosis ​

PortfolioOptimisers.Kurtosis Type
julia
struct Kurtosis{T1, T2, T3, T4, T5, T6, T7} <: RiskMeasure
    settings::T1
    w::T2
    mu::T3
    kt::T4
    N::T5
    alg1::T6
    alg2::T7
end

Represents the square root kurtosis risk measure in PortfolioOptimisers.jl.

Computes portfolio risk as the square root of the fourth central moment (kurtosis) of the return distribution, optionally using custom weights, expected returns, and a kurtosis (fourth moment) matrix. This risk measure can be evaluated using either the full or semi (downside) deviations, depending on the algorithm provided.

Fields

  • settings: Risk measure configuration.

  • w: Optional vector of observation weights.

  • mu: Optional expected returns value, vector, or VecScalar for the moment target, via calc_moment_target. If nothing it is computed from the returns series using the optional weights in w.

  • kt: Optional cokurtosis (fourth moment) matrix that overrides the prior kt when provided.

  • N: Optional integer specifying the number of eigenvalues per asset to use from the cokurtosis matrix in an approximate formulation. If nothing, the exact formulation is used.

  • alg1: Moment algorithm specifying whether to use all or only downside deviations.

  • alg2: Specifies the JuMP formulation used to encode the risk measure.

Constructors

julia
Kurtosis(; settings::RiskMeasureSettings = RiskMeasureSettings(),
                   w::Union{Nothing, <:AbstractWeights} = nothing,
                   mu::Union{Nothing, <:Real, <:AbstractVector{<:Real}, <:VecScalar} = nothing,
                   kt::Union{Nothing, <:AbstractMatrix} = nothing,
                   N::Union{Nothing, <:Integer} = nothing,
                   alg1::AbstractMomentAlgorithm = Full(),
                   alg2::VarianceFormulation = SOCRiskExpr())

Keyword arguments correspond to the fields above.

Validation

  • If mu is not nothing:

    • ::Real: isfinite(mu).

    • ::AbstractVector: !isempty(mu) and all(isfinite, mu).

  • If w is not nothing, !isempty(w).

  • If kt is not nothing:

    • !isempty(kt) and size(kt, 1) == size(kt, 2).

    • If mu is not nothing:

      • ::AbstractVector: length(mu)^2 == size(kt, 1).

      • ::VecScalar: length(mu.v)^2 == size(kt, 1).

  • If N is provided: must be positive.

JuMP Formulations

Exact

This formulation is used when N is nothing.

Approximate

This formulation is used when N is an integer, the larger the value of N, the more accurate and expensive it becomes.

Examples

julia
julia> Kurtosis()
Kurtosis
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
         w ┼ nothing
        mu ┼ nothing
        kt ┼ nothing
         N ┼ nothing
      alg1 ┼ Full()
      alg2 ┴ SOCRiskExpr()

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