Kurtosis

PortfolioOptimisers.Kurtosis — Type

struct Kurtosis{__T_settings, __T_w, __T_mu, __T_kt, __T_N, __T_alg1, __T_alg2, __T_pe} <: RiskMeasure

Represents the square root kurtosis risk measure.

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.

Mathematical definition

Let $\boldsymbol{x} = \mathbf{X} \boldsymbol{w}$ be the $T \times 1$ vector of portfolio returns, and let $\mu$ be the chosen centre (mean, weighted mean, or user-supplied value). Define the centred deviations:

\[\begin{align} \delta_t &= x_t - \mu\,. \end{align}\]

The fourth central moment (full moment) is:

\[\begin{align} m_4(\boldsymbol{w}) &= \frac{1}{T} \sum_{t=1}^{T} \delta_t^4\,. \end{align}\]

The same quantity in the $N^2 \times N^2$ cokurtosis matrix $\hat{\mathbf{K}}$, with the Kronecker product $\otimes$, is:

\[\begin{align} m_4(\boldsymbol{w}) &= (\boldsymbol{w}^\intercal \otimes \boldsymbol{w}^\intercal)\, \hat{\mathbf{K}}\, (\boldsymbol{w} \otimes \boldsymbol{w})\,. \end{align}\]

For the semi (downside) variant, only non-positive deviations contribute:

\[\begin{align} m_4^{-}(\boldsymbol{w}) &= \frac{1}{T} \sum_{t=1}^{T} \min(\delta_t, 0)^4\,. \end{align}\]

The risk is the fourth moment itself under a quadratic formulation, and its square root under a second-order cone formulation. alg2 selects between them, and alg1 selects between the full and the semi moment:

\[\begin{align} \mathrm{Kurt}(\boldsymbol{w}) &= \begin{cases} \sqrt{m_4(\boldsymbol{w})} & \text{(\texttt{SOCRiskExpr})} \\ m_4(\boldsymbol{w}) & \text{(quadratic formulation)} \end{cases}\,. \end{align}\]

Where:

$\boldsymbol{w}$: Portfolio weights vector $N \times 1$.
$\mathbf{X}$: $T \times N$ asset returns matrix.
$T$: Number of observations.
$\delta_t$: Centred portfolio return at period $t$.
$\hat{\mathbf{K}}$: $N^2 \times N^2$ cokurtosis matrix.
$\otimes$: Kronecker product.

Fields

settings: Risk measure settings.

w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

mu: Optional centre the moment is taken about, a scalar or a vector assets × 1. Also admits a Deferred Quantity — an expected returns estimator or a prior estimator that computes the centre against the optimisation's own prior, at factory time (see MuSlot and resolve_deferred_quantities). If nothing, the prior supplies it.

kt: Optional cokurtosis matrix assets^2 × assets^2. Also admits a Deferred Quantity — a cokurtosis estimator or a prior estimator that computes the matrix against the optimisation's own prior, at factory time (see KtSlot and resolve_deferred_quantities). A cokurtosis estimator supplies mu as well, from its own me, so that the tensor and the centre it was taken about come out of one object. If nothing, the prior supplies it.

N: Optional number of eigenvalues per asset for the approximate cokurtosis formulation.

alg1: First algorithm variant.

alg2: Second algorithm variant.

pe: Optional prior estimator that fills every prior-derived slot the measure leaves unstated, from a single fit. A stated slot wins. See resolve_deferred_quantities.

Constructors

Kurtosis(;    settings::RiskMeasureSettings = RiskMeasureSettings(),    w::Option{<:ObsWeights} = nothing,    mu::Option{<:MuSlot} = nothing,    kt::Option{<:KtSlot} = nothing,    N::Option{<:Integer} = nothing,    alg1::AbstractMomentAlgorithm = FullMoment(),    alg2::SecondMomentFormulation = SOCRiskExpr(),    pe::Option{<:AbstractPriorEstimator} = nothing,) -> Kurtosis

Keywords correspond to the struct's fields.

Validation

If mu is not nothing:
- ::Number: isfinite(mu).
- ::VecNum: !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:
  - ::VecNum: length(mu)^2 == size(kt, 1).
  - ::VecScalar: length(mu.v)^2 == size(kt, 1).
If N is not nothing: must be positive.

Warning

mu and kt are stated independently, so nothing makes them agree with each other. A caller who wants one consistent set names pe alone and lets it fill both from a single fit. A caller who states them by hand must make sure that they agree.

Info

kt also admits a CokurtosisEstimator or an AbstractPriorEstimator, and mu an AbstractExpectedReturnsEstimator or an AbstractPriorEstimator. Either is resolved against the optimisation's own prior — see resolve_deferred_quantities. A cokurtosis estimator in kt also supplies mu from its own me, so that the tensor and the centre it was taken about come out of one object. A deferred slot wins over pe.

JuMP Formulations

Exact

This formulation is used when N is nothing. It is the parametric model of [30].

Approximate

This formulation is used when N is an integer, the larger the value of N, the more accurate and expensive it becomes. It is the sum-of-squared-quadratic-forms model of [101].

Examples

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

Related

References

[30] D. Cajas. Convex Optimization of Portfolio Kurtosis. Available at SSRN 4202967 (2022).
[101] D. Cajas. Approximation of Portfolio Kurtosis through Sum of Squared Quadratic Forms. Available at SSRN 4472793 (2023).

source

PortfolioOptimisers.factory — Method

factory(
    r::Kurtosis,
    pr::HighOrderPrior,
    args...;
    kwargs...
) -> Kurtosis{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, _C, _D, <:AbstractMomentAlgorithm, <:SecondMomentFormulation, Nothing} where {__T_scale, __T_ub, __T_rke, _A, _B, _C, _D}

Create an instance of Kurtosis by selecting the cokurtosis matrix, expected returns, and weights from the risk-measure instance or falling back to a HighOrderPrior result.

Related

source

PortfolioOptimisers.factory — Method

factory(
    r::Kurtosis,
    pr::LowOrderPrior,
    args...;
    kwargs...
) -> Kurtosis{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, _C, _D, <:AbstractMomentAlgorithm, <:SecondMomentFormulation, Nothing} where {__T_scale, __T_ub, __T_rke, _A, _B, _C, _D}

Create an instance of Kurtosis from a LowOrderPrior result (cokurtosis matrix is not used).

Related

source

PortfolioOptimisers.port_opt_view — Method

port_opt_view(
    r::Kurtosis,
    i,
    args...
) -> Kurtosis{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, _C, _D, <:AbstractMomentAlgorithm, <:SecondMomentFormulation} where {__T_scale, __T_ub, __T_rke, _A, _B, _C, _D}

Return a view of Kurtosis r sliced to asset indices i.

Slices both the cokurtosis matrix kt and the expected returns mu for cluster-based optimisation.

Related

source

References

[30]: D. Cajas. Convex Optimization of Portfolio Kurtosis. Available at SSRN 4202967 (2022).
[101]: D. Cajas. Approximation of Portfolio Kurtosis through Sum of Squared Quadratic Forms. Available at SSRN 4472793 (2023).