Negative Skewness

PortfolioOptimisers.NegativeSkewness — Type

struct NegativeSkewness{__T_settings, __T_mp, __T_sk, __T_V, __T_alg, __T_window} <: RiskMeasure

Represents the Negative Skewness risk measure.

NegativeSkewness quantifies the portfolio's exposure to negative asymmetry in returns by computing a quadratic or SOC (second-order cone) form of the coskewness matrix. It penalises portfolio constructions that exhibit heavy left-tail behaviour.

Mathematical definition

Let $\boldsymbol{w}$ be the portfolio weight vector and $\mathbf{V}$ the negative semi-definite coskewness matrix (spectral decomposition of the negative part of the sample coskewness tensor). The Negative Skewness risk measure is:

\[\begin{align} \mathrm{NSke}(\boldsymbol{w}) &= \begin{cases} \sqrt{\boldsymbol{w}^\intercal \mathbf{V} \boldsymbol{w}} & \text{(SOC formulation)} \\ \boldsymbol{w}^\intercal \mathbf{V} \boldsymbol{w} & \text{(Quadratic formulation)} \end{cases}\,. \end{align}\]

Where:

$\mathrm{NSke}(\boldsymbol{w})$: Negative Skewness risk measure.
$\boldsymbol{w}$: Portfolio weights vector $N \times 1$.
$\mathbf{V}$: Negative semi-definite coskewness matrix (spectral decomposition of the negative part of the sample coskewness tensor).

Fields

settings: Risk measure settings.

mp: Matrix processing estimator.

sk: Optional coskewness matrix assets × assets^2. Also admits a Deferred Quantity — a coskewness estimator or a prior estimator that computes the matrix against the optimisation's own prior, at factory time (see SkSlot and resolve_deferred_quantities). A coskewness 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.

V: Optional sum of the negative spectral slices of the coskewness matrix assets × assets. Derived from sk, so it never defers: it arrives as one pair with whatever sk resolves to, and the matrix processing estimator that built it travels with it and replaces mp. Give it with a matrix sk and with neither otherwise. Stating it while sk holds a Deferred Quantity is refused at construction.

alg: Risk measure optimisation formulation algorithm.

window: Observation window. An integer selects the last window observations, and a vector of indices selects those observations.

Constructors

NegativeSkewness(;    settings::RiskMeasureSettings = RiskMeasureSettings(),    mp::AbstractMatrixProcessingEstimator = MatrixProcessing(),    sk::Option{<:SkSlot} = nothing,    V::Option{<:MatNum} = nothing,    alg::NSkeFormulations = SOCRiskExpr(),    window::Option{<:Int_VecInt} = nothing) -> NegativeSkewness

Keywords correspond to the struct's fields.

Validation

If sk is a matrix, V must be given as well, and the reverse. Both must be non-empty, with size(sk, 1)^2 == size(sk, 2) and V square.
If sk holds a Deferred Quantity, V must be nothing. The fit supplies the pair.
window is validated with assert_nonempty_nonneg_finite_val.

Warning

sk and V are a pair, and a stated V factors the sk beside it. A caller who wants one consistent pair names sk alone — a Deferred Quantity there supplies both from one fit, together with the mp that built them. A caller who states both by hand must make sure that they agree. A stated matrix is also pinned: it crosses a Cross-Validation fold or a subset view as the whole universe's answer, while a Deferred Quantity crosses unresolved and refits on the subset.

Info

sk also admits a CoskewnessEstimator or an AbstractPriorEstimator, resolved against the optimisation's own prior — see resolve_deferred_quantities. V never defers: it is derived from sk, so it travels out of that same fit. The processor that built it travels with it and replaces mp, so a later rebuild uses the same one. The measure carries one deferrable slot, so it takes no pe.

Functor

(r::NegativeSkewness)(w::VecNum)

Computes the Negative Skewness risk of a portfolio weight vector w.

Arguments

w::VecNum: Portfolio weights vector.

Examples

julia> NegativeSkewness()NegativeSkewness  settings ┼ RiskMeasureSettings           │   scale ┼ Float64: 1.0           │      ub ┼ nothing           │     rke ┴ Bool: true        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)        sk ┼ nothing         V ┼ nothing       alg ┼ SOCRiskExpr()    window ┴ nothing

Related

References

[31] D. Cajas. On the Spectral Decomposition of Portfolio Skewness and its Application to Portfolio Optimization. Available at SSRN 4540021 (2023).

source

PortfolioOptimisers.factory — Method

factory(
    r::NegativeSkewness,
    pr::HighOrderPrior,
    args...;
    kwargs...
) -> NegativeSkewness{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, <:AbstractMatrixProcessingEstimator} where {__T_scale, __T_ub, __T_rke}

Create an instance of NegativeSkewness by resolving a Deferred Quantity in sk, then falling back to a HighOrderPrior result for the coskewness matrix and its spectral decomposition.

The two are selected field by field rather than as a pair, because the constructor already refuses every mixed state: a stated sk always carries its own V, and a deferred sk always resolves to both at once. So the fallback is reached only when the measure names neither.

Related

source

PortfolioOptimisers.factory — Method

factory(
    r::NegativeSkewness,
    pr::LowOrderPrior,
    args...;
    kwargs...
) -> NegativeSkewness

Resolve a Deferred Quantity in NegativeSkewness's sk slot against a LowOrderPrior result, and otherwise return r unchanged.

Coskewness is not available on a LowOrderPrior, so there is no fallback to make. A coskewness estimator in sk needs only the returns matrix the result carries, so it resolves here all the same.

Related

source

References

[31]: D. Cajas. On the Spectral Decomposition of Portfolio Skewness and its Application to Portfolio Optimization. Available at SSRN 4540021 (2023).