Moment Risk Measures

struct FirstLowerMoment <: UnstandardisedLowOrderMomentMeasureAlgorithm

Represents the first lower moment risk measure algorithm.

Computes portfolio risk using the first lower moment, which is the negative mean of the deviations of the returns series below a target value.

Related

• UnstandardisedLowOrderMomentMeasureAlgorithm
• LowOrderMomentMeasureAlgorithm
• LowOrderMoment

References

• [98] P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977).

struct MeanAbsoluteDeviation <: UnstandardisedLowOrderMomentMeasureAlgorithm

Represents the mean absolute deviation risk measure algorithm.

Computes portfolio risk as the mean of the absolute deviations of the returns series from a target value.

Related

• UnstandardisedLowOrderMomentMeasureAlgorithm
• LowOrderMomentMeasureAlgorithm
• LowOrderMoment

References

• [99] H. Konno and H. Yamazaki. Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science 37, 519–531 (1991).

struct SecondMoment{__T_ve, __T_alg1, __T_alg2} <: LowOrderMomentMeasureAlgorithm

Represents a second moment (variance or standard deviation) risk measure algorithm.

Computes portfolio risk using the second central (full) or lower (semi) moment of the return distribution, supporting both full and semi (downside) formulations. The specific formulation is determined by the alg1 and alg2 fields, enabling flexible representation of variance, semi-variance, standard deviation, or semi-standard deviation.

Fields

• ve: Variance estimator.

• alg1: First algorithm variant.

• alg2: Second algorithm variant.

Constructors

SecondMoment(;    ve::AbstractVarianceEstimator = SimpleVariance(; me = nothing),    alg1::AbstractMomentAlgorithm = FullMoment(),    alg2::SecondMomentFormulation = SquaredSOCRiskExpr(),) -> SecondMoment

Keywords correspond to the struct's fields.

Propagated parameters

When factory is called on this type, the following @fprop-tagged fields are automatically propagated:

• ve: Recursively updated via factory.

Examples

julia> SecondMoment()SecondMoment    ve ┼ SimpleVariance       │          me ┼ nothing       │           w ┼ nothing       │   corrected ┴ Bool: true  alg1 ┼ FullMoment()  alg2 ┴ SquaredSOCRiskExpr()

Related

• LowOrderMomentMeasureAlgorithm
• AbstractVarianceEstimator
• AbstractMomentAlgorithm
• SecondMomentFormulation
• factory

References

• [11] H. Markowitz. Modern portfolio theory. Journal of Finance 7, 77–91 (1952).

struct EvenMoment{__T_p, __T_ddof, __T_alg} <: UnstandardisedLowOrderMomentMeasureAlgorithm

Represents an even-order moment risk measure algorithm.

Computes portfolio risk using the $p$-th root of the $2p$-th central (full) or lower (semi) even moment of the return distribution. Despite the potentially high moment order, even moments admit an exact power cone reformulation, keeping the optimisation formulation affine.

Fields

• p: Power or order parameter.

• ddof: Degrees-of-freedom correction.

• alg: Moment algorithm.

Constructors

EvenMoment(;    p::Integer = 2,    ddof::Integer = 0,    alg::AbstractMomentAlgorithm = FullMoment(),) -> EvenMoment

Keywords correspond to the struct's fields.

Validation

• p >= 2.
• ddof >= 0 and isfinite(ddof).

Examples

julia> EvenMoment()EvenMoment     p ┼ Int64: 2  ddof ┼ Int64: 0   alg ┴ FullMoment()

Related

• UnstandardisedLowOrderMomentMeasureAlgorithm
• LowOrderMoment
• FullMoment
• SemiMoment

References

• [100] D. Cajas. Portfolio Optimization of Even Moments using Power Cone Programming. Available at SSRN 6518258 (2026).

struct LowOrderMoment{__T_settings, __T_w, __T_mu, __T_alg} <: RiskMeasure

Represents a low-order moment risk measure.

Computes portfolio risk using a low-order moment algorithm (such as first lower moment, mean absolute deviation, or second moment), optionally with custom weights and target values. This type is used for risk measures based on mean, variance, or related statistics.

Mathematical definition

Depending on the alg field, the risk measure computes one of the following quantities.

FirstLowerMoment

The first lower moment is computed as:

\[\begin{align} \mathrm{FirstLowerMoment}(\boldsymbol{X}) &= \mathbb{E}\left[\max \circ \left(\mathbb{E}\left[\boldsymbol{X}\right] - \boldsymbol{X},\, 0\right)\right]\,. \end{align}\]

Where:

• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $\circ$: Element-wise function application.

MeanAbsoluteDeviation

The mean absolute deviation is computed as:

\[\begin{align} \mathrm{MeanAbsoluteDeviation}(\boldsymbol{X}) &= \mathbb{E}\left[\left\lvert \boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right] \right\rvert\right]\,. \end{align}\]

Where:

• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.

SecondMoment

Depending on the alg1 field the risk measure can either compute the second central moment or second lower moment.

Both central and lower moments can be formulated as quadratic moments (variance or semi-variance) or their square roots (standard deviation or semi-standard deviation). Regardless of whether they are central or lower moments, they can be formulated in a variety of ways.

Depending on the alg2 field, it can represent the variance (using different formulations in JuMP) or standard deviation.

The (semi-)variance formulations are:

• SquaredSOCRiskExpr,
• RSOCRiskExpr,
• QuadRiskExpr.

It is computed as:

\[\begin{align} \mathrm{Variance}(\boldsymbol{X}) &= \mathbb{E}\left[\left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right]\right)^2\right] \,, \\ \mathrm{SemiMoment\text{-}Variance}(\boldsymbol{X}) &= \mathbb{E}\left[\min \circ \left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right],\,0\right)^2\right] \,. \end{align}\]

Where:

• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.

The (semi-)standard deviation formulation is:

• SOCRiskExpr.

It is computed as:

\[\begin{align} \mathrm{StandardDeviation}(\boldsymbol{X}) &= \sqrt{\mathbb{E}\left[\left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right]\right)^2\right]} \,, \\ \mathrm{SemiMoment\text{-}StandardDeviation}(\boldsymbol{X}) &= \sqrt{\mathbb{E}\left[\min \circ \left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right],\,0\right)^2\right]} \,. \end{align}\]

Where:

• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.

EvenMoment

EvenMoment computes the $2p$-th central (full) or lower (semi) moment of the return distribution. Although the moment order $2p \geq 4$ can be arbitrarily high, EvenMoment is not a low-order moment in the classical sense. However, because the exponent is always even, the moment can be expressed as an iterated power of squared deviations, admitting an exact reformulation using power cone constraints. This makes the optimisation problem affine and solvable by standard conic solvers.

The full (central) even moment is computed as:

\[\begin{align} \mathrm{EvenMoment}_{p}(\boldsymbol{X}) &= \left(\frac{1}{T_d}\sum_{t=1}^{T}\left(\boldsymbol{X}_t - \mathbb{E}\left[\boldsymbol{X}\right]\right)^{2p}\right)^{1/p}\,. \end{align}\]

The semi (lower) even moment is computed as:

\[\begin{align} \mathrm{SemiMoment\text{-}EvenMoment}_{p}(\boldsymbol{X}) &= \left(\frac{1}{T_d}\sum_{t=1}^{T}\min \circ \left(\boldsymbol{X}_t - \mathbb{E}\left[\boldsymbol{X}\right],\, 0\right)^{2p}\right)^{1/p}\,. \end{align}\]

Where:

• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $T_d = T - \mathrm{ddof}$: Effective sample size after degrees-of-freedom correction.
• $p \geq 2$: Order parameter; the moment order is $2p$.
• $\circ$: Element-wise function application.

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.

• alg: Moment algorithm.

Constructors

LowOrderMoment(;    settings::RiskMeasureSettings = RiskMeasureSettings(),    w::Option{<:ObsWeights} = nothing,    mu::Option{<:MuSlot} = nothing,    alg::LowOrderMomentMeasureAlgorithm = FirstLowerMoment(),) -> LowOrderMoment

Keywords correspond to the struct's fields.

Validation

• If mu is not nothing:

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

• If w is not nothing, !isempty(w).

JuMP Formulations

Depending on the alg field, the risk measure is formulated in JuMP as follows. Every formulation measures the deviations against the target returned by calc_risk_constraint_target, which is mu when the slot holds a value and the prior's expected returns otherwise.

FirstLowerMoment

As an optimisation problem, it can be formulated as:

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & \mathbb{E}\left[\boldsymbol{d}\right] \\ \mathrm{s.t.} \quad & \boldsymbol{d} \geq \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] - \mathrm{X} \boldsymbol{w}\\ \quad & \boldsymbol{d} \geq 0 \,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations below the target.
• $\mathrm{X}$: T × N return matrix.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.

MeanAbsoluteDeviation

As an optimisation problem, it can be formulated as:

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & 2 \mathbb{E}\left[\boldsymbol{d}\right]\\ \mathrm{s.t.} \quad & \boldsymbol{d} \geq \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] - \mathrm{X} \boldsymbol{w}\\ \quad & \boldsymbol{d} \geq 0 \,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations below the target.
• $\mathrm{X}$: T × N return matrix.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.

SecondMoment

SquaredSOCRiskExpr

Represents the (semi-)variance using the square of a second order cone constrained variable.

The variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot \sigma^2\\ \mathrm{s.t.} \quad & \boldsymbol{d} = \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(\sigma,\, \boldsymbol{d}_s\right) \in K_{soc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\sigma$: Standard deviation of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $K_{soc}$: Second order cone.
• $\odot$: Element-wise (Hadamard) product.

The semi-variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot \sigma^2\\ \mathrm{s.t.} \quad & \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \geq -\boldsymbol{d} \\ \quad & \boldsymbol{d} \geq 0 \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(\sigma,\, \boldsymbol{d}_s\right) \in K_{soc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\sigma$: Standard deviation of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $K_{soc}$: Second order cone.
• $\odot$: Element-wise (Hadamard) product.

RSOCRiskExpr

Represents the (semi-)variance using a sum of squares formulation via a rotated second order cone.

The variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot t\\ \mathrm{s.t.} \quad & \boldsymbol{d} = \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(t,\, 0.5,\,\boldsymbol{d}_s\right) \in K_{rsoc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $t$: Variance of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $K_{rsoc}$: Rotated second order cone.
• $\odot$: Element-wise (Hadamard) product.

The semi-variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot t\\ \mathrm{s.t.} \quad & \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \geq -\boldsymbol{d} \\ \quad & \boldsymbol{d} \geq 0 \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(t,\, 0.5,\,\boldsymbol{d}_s\right) \in K_{rsoc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $t$: Variance of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $K_{rsoc}$: Rotated second order cone.
• $\odot$: Element-wise (Hadamard) product.

QuadRiskExpr

Represents the (semi-)variance using the deviations vector dotted with itself.

The variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot \boldsymbol{d}_s \cdot \boldsymbol{d}_s\\ \mathrm{s.t.} \quad & \boldsymbol{d} = \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $\odot$: Element-wise (Hadamard) product.

The semi-variance is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & f \cdot \boldsymbol{d}_s \cdot \boldsymbol{d}_s\\ \mathrm{s.t.} \quad & \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \geq -\boldsymbol{d} \\ \quad & \boldsymbol{d} \geq 0 \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\mu$: Minimum acceptable return.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $\odot$: Element-wise (Hadamard) product.

SOCRiskExpr

Represents the (semi-)standard deviation using a second order cone constrained variable.

The standard deviation is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & \sqrt{f} \cdot \sigma\\ \mathrm{s.t.} \quad & \boldsymbol{d} = \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(\sigma,\, \boldsymbol{d}_s\right) \in K_{soc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\sigma$: Standard deviation of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $K_{soc}$: Second order cone.
• $\odot$: Element-wise (Hadamard) product.

The semi-standard deviation is formulated as.

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{d}}{\mathrm{opt}} \quad & \sqrt{f} \cdot \sigma\\ \mathrm{s.t.} \quad & \mathrm{X} \boldsymbol{w} - \mathbb{E}\left[\mathrm{X} \boldsymbol{w}\right] \geq -\boldsymbol{d} \\ \quad & \boldsymbol{d} \geq 0 \\ \quad & \boldsymbol{d}_s = \sqrt{\boldsymbol{\lambda}} \odot \boldsymbol{d} \\ \quad & \left(\sigma,\, \boldsymbol{d}_s\right) \in K_{soc}\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $\boldsymbol{d}$: T × 1 vector of auxiliary decision variables representing deviations from the target.
• $\sigma$: Standard deviation of the portfolio returns.
• $\boldsymbol{d}_s$: T × 1 vector of scaled deviations according to observation weights.
• $\mathrm{X}$: T × N return matrix.
• $\mu$: Minimum acceptable return.
• $\boldsymbol{\lambda}$: T × 1 vector of observation weights.
• $f$: Variance correction factor from StatsBase.varcorrection. It corrects for the sample size without observation weights, and for $\boldsymbol{\lambda}$ with them. ve.corrected selects the divisor.
• $\odot$: Element-wise (Hadamard) product.
• $K_{soc}$: Second order cone.

EvenMoment

As an optimisation problem, the full even moment is formulated using a chain of power cone constraints:

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{u},\,\boldsymbol{s},\,r}{\mathrm{opt}} \quad & r \\ \mathrm{s.t.} \quad & \sum_{t=1}^{T} u_t \leq r \\ \quad & \left(u_t \cdot T_d,\, r,\, s_t\right) \in \mathcal{K}_{\mathrm{pow}}\!\left(\tfrac{1}{p}\right),\quad t = 1,\ldots,T \\ \quad & \left(s_t,\, k,\, \hat{r}_t - \mu\right) \in \mathcal{K}_{\mathrm{pow}}\!\left(\tfrac{1}{2}\right),\quad t = 1,\ldots,T\,. \end{align}\]

The semi even moment is formulated as:

\[\begin{align} \underset{\boldsymbol{w},\,\boldsymbol{u},\,\boldsymbol{s},\,\boldsymbol{d},\,r}{\mathrm{opt}} \quad & r \\ \mathrm{s.t.} \quad & \sum_{t=1}^{T} u_t \leq r \\ \quad & \left(u_t \cdot T_d,\, r,\, s_t\right) \in \mathcal{K}_{\mathrm{pow}}\!\left(\tfrac{1}{p}\right),\quad t = 1,\ldots,T \\ \quad & \left(s_t,\, k,\, d_t\right) \in \mathcal{K}_{\mathrm{pow}}\!\left(\tfrac{1}{2}\right),\quad t = 1,\ldots,T \\ \quad & \hat{r}_t - \mu + d_t \geq 0,\quad t = 1,\ldots,T \\ \quad & d_t \geq 0,\quad t = 1,\ldots,T\,. \end{align}\]

Where:

• $\boldsymbol{w}$: N × 1 asset weights vector.
• $r$: Even-moment risk variable.
• $\boldsymbol{u}$: T × 1 auxiliary variable vector.
• $\boldsymbol{s}$: T × 1 auxiliary variable vector.
• $\boldsymbol{d}$: T × 1 lower-deviation auxiliary variables, capturing returns below the target.
• $T_d$: Effective sample size.
• $k$: Budget-scaling / homogenisation variable.
• $p$: Order parameter; the moment order is $2p$.
• $\mathrm{X}$: T × N return matrix.
• $\hat{r}_t = \boldsymbol{x}_t^\intercal\boldsymbol{w}$: Portfolio return at time $t$.
• $\mu$: Target return.
• $\mathcal{K}_{\mathrm{pow}}(\alpha)$: Power cone $\{(a, b, c) : a^{\alpha}\,b^{1-\alpha} \geq |c|,\; a, b \geq 0\}$.

Functor

(r::LowOrderMoment)(w::VecNum, X::MatNum;                    fees::Option{<:Fees} = nothing)

Computes the low-order moment risk measure as defined in r using portfolio weights w, return matrix X, and optional fees fees.

Details

• r.alg defines what low-order moment to compute.
• The values of r.mu and r.w are optionally used to compute the moment target via calc_moment_target, which is used in calc_deviations_vec to compute the deviation vector.

Examples

julia> LowOrderMoment()LowOrderMoment  settings ┼ RiskMeasureSettings           │   scale ┼ Float64: 1.0           │      ub ┼ nothing           │     rke ┴ Bool: true         w ┼ nothing        mu ┼ nothing       alg ┴ FirstLowerMoment()

Related

• RiskMeasureSettings
• LowOrderMomentMeasureAlgorithm
• FirstLowerMoment
• MeanAbsoluteDeviation
• SecondMoment
• SquaredSOCRiskExpr
• RSOCRiskExpr
• QuadRiskExpr
• SOCRiskExpr
• EvenMoment

References

• [100] D. Cajas. Portfolio Optimization of Even Moments using Power Cone Programming. Available at SSRN 6518258 (2026).
• [98] P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977).
• [99] H. Konno and H. Yamazaki. Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science 37, 519–531 (1991).
• [11] H. Markowitz. Modern portfolio theory. Journal of Finance 7, 77–91 (1952).

struct ThirdLowerMoment <: UnstandardisedHighOrderMomentMeasureAlgorithm

Represents the unstandardised semi-skewness risk measure algorithm.

Computes portfolio risk using the third lower moment (unstandardised semi-skewness), which quantifies downside asymmetry by considering only the cubed deviations below a target value. The measure negates the mean cubed lower deviation, so a larger value is more downside asymmetry. This algorithm is unstandardised and operates directly on the return distribution.

Related

• UnstandardisedHighOrderMomentMeasureAlgorithm
• HighOrderMomentMeasureAlgorithm
• HighOrderMoment

References

• [98] P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977).

struct FourthMoment{__T_alg} <: UnstandardisedHighOrderMomentMeasureAlgorithm

Represents the unstandardised fourth moment (kurtosis or semi-kurtosis) risk measure algorithm.

Computes portfolio risk using the fourth central (full) or lower (semi) moment of the return distribution, depending on the provided moment algorithm. This algorithm quantifies the "tailedness" of the return distribution without normalising by the variance.

Fields

• alg: Moment algorithm.

Constructors

FourthMoment(;    alg::AbstractMomentAlgorithm = FullMoment(),) -> FourthMoment

Keywords correspond to the struct's fields.

Examples

julia> FourthMoment()FourthMoment  alg ┴ FullMoment()

Related

• UnstandardisedHighOrderMomentMeasureAlgorithm
• AbstractMomentAlgorithm

References

• [98] P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977).

struct StandardisedHighOrderMoment{__T_ve, __T_alg} <: HighOrderMomentMeasureAlgorithm

Represents a standardised high-order moment risk measure algorithm.

Computes portfolio risk using a high-order moment algorithm (such as semi-skewness or semi-kurtosis), divided by ve's variance raised to half the moment order. ve runs on the same deviations as alg, so a lower-moment alg is standardised by a semi-variance.

Fields

• ve: Variance estimator.

• alg: Moment algorithm.

Constructors

StandardisedHighOrderMoment(;    ve::AbstractVarianceEstimator = SimpleVariance(; me = nothing),    alg::UnstandardisedHighOrderMomentMeasureAlgorithm = ThirdLowerMoment(),) -> StandardisedHighOrderMoment

Keywords correspond to the struct's fields.

Examples

julia> StandardisedHighOrderMoment()StandardisedHighOrderMoment   ve ┼ SimpleVariance      │          me ┼ nothing      │           w ┼ nothing      │   corrected ┴ Bool: true  alg ┴ ThirdLowerMoment()

Related

• HighOrderMomentMeasureAlgorithm
• AbstractVarianceEstimator
• UnstandardisedHighOrderMomentMeasureAlgorithm
• factory

struct HighOrderMoment{__T_settings, __T_w, __T_mu, __T_alg} <: HierarchicalRiskMeasure

Represents a high-order moment risk measure.

Computes portfolio risk using a high-order moment algorithm (such as semi-skewness, semi-kurtosis, or kurtosis), optionally with custom weights and target values. This type is used for risk measures based on third or fourth moments of the return distribution.

Mathematical definition

Depending on the alg field, the risk measure computes the third lower moment, fourth lower (semi) moment, or fourth central (full) moment. Each can be standardised or unstandardised.

The unstandardised formulations are:

• ThirdLowerMoment,
• FourthMoment,

The standardised formulations are:

• StandardisedHighOrderMoment, which uses a variance estimator and an unstandardised high-order moment algorithm.

An odd moment order makes a lower moment negative, so the risk measure returns $(-1)^n \mu_n$. A larger value is therefore always more risk. The third lower moment is negated; the fourth moment is already non-negative and is returned unchanged.

Unstandardised Moments

All unstandardised central moments have the following formula.

\[\begin{align} \mu_n &= \mathbb{E}\left[\left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right]\right)^n\right]\,. \end{align}\]

Where:

• $\mu_n$: $n$-th central moment.
• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $n$: Moment order.

All unstandardised lower moments have the following formula.

\[\begin{align} \mu_n &= \mathbb{E}\left[\min \circ \left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right],\, 0\right)^n\right]\,. \end{align}\]

Where:

• $\mu_n$: $n$-th lower moment.
• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $\circ$: Element-wise function application.
• $n$: Moment order.

Standardised Moments

All standardised central moments have the following formula.

\[\begin{align} \mu_n &= \dfrac{\mathbb{E}\left[\left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right]\right)^n\right]}{\mathbb{E}\left[\left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right]\right)^2\right]^{n/2}}\,. \end{align}\]

Where:

• $\mu_n$: $n$-th standardised central moment.
• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $n$: Moment order.

All standardised lower moments have the following formula.

\[\begin{align} \mu_n &= \dfrac{\mathbb{E}\left[\min \circ \left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right],\, 0\right)^n\right]}{\mathbb{E}\left[\min \circ \left(\boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right],\, 0\right)^2\right]^{n/2}}\,. \end{align}\]

Where:

• $\mu_n$: $n$-th standardised lower moment.
• $\boldsymbol{X}$: T × 1 vector of portfolio returns.
• $\mathbb{E}[\cdot]$: Expected value operator, supports weighted averages.
• $\circ$: Element-wise function application.
• $n$: Moment order.

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.

• alg: Moment algorithm.

Constructors

HighOrderMoment(;    settings::RiskMeasureSettings = RiskMeasureSettings(),    w::Option{<:ObsWeights} = nothing,    mu::Option{<:MuSlot} = nothing,    alg::HighOrderMomentMeasureAlgorithm = ThirdLowerMoment(),) -> HighOrderMoment

Keywords correspond to the struct's fields.

Validation

• If mu is not nothing:

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

• If w is not nothing, !isempty(w).

Functor

(r::HighOrderMoment)(w::VecNum, X::MatNum;                    fees::Option{<:Fees} = nothing)

Computes the high-order moment risk measure as defined in r using portfolio weights w, return matrix X, and optional fees fees.

Details

• r.alg defines what high-order moment to compute.
• The values of r.mu and r.w are optionally used to compute the moment target via calc_moment_target, which is used in calc_deviations_vec to compute the deviation vector.

Examples

julia> HighOrderMoment()HighOrderMoment  settings ┼ RiskMeasureSettings           │   scale ┼ Float64: 1.0           │      ub ┼ nothing           │     rke ┴ Bool: true         w ┼ nothing        mu ┼ nothing       alg ┴ ThirdLowerMoment()

Related

• RiskMeasureSettings
• HighOrderMomentMeasureAlgorithm
• ThirdLowerMoment
• FourthMoment
• StandardisedHighOrderMoment

References

• [98] P. C. Fishburn. Mean-risk analysis with risk associated with below-target returns. The American Economic Review 67, 116–126 (1977).

factory(
    alg::StandardisedHighOrderMoment,
    w::Union{DynamicAbstractWeights, AbstractWeights}
) -> StandardisedHighOrderMoment{<:AbstractVarianceEstimator, <:UnstandardisedHighOrderMomentMeasureAlgorithm}

Return a new StandardisedHighOrderMoment with observation weights w applied to the underlying variance estimator.

Related

• StandardisedHighOrderMoment
• factory

factory(
    alg::MomentMeasureAlgorithm,
    args...;
    kwargs...
) -> SecondMoment{<:AbstractVarianceEstimator, <:AbstractMomentAlgorithm, <:SecondMomentFormulation}

Return the moment measure algorithm alg unchanged.

Identity pass-through used when a moment measure algorithm is provided in a context that calls factory.

Related

• MomentMeasureAlgorithm
• factory

factory(
    r::LowOrderMoment,
    pr::AbstractPriorResult,
    args...;
    kwargs...
) -> LowOrderMoment{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, <:LowOrderMomentMeasureAlgorithm} where {__T_scale, __T_ub, __T_rke, _A, _B}

Create an instance of LowOrderMoment by selecting observation weights, expected returns, and algorithm from the risk-measure instance or falling back to the prior result.

Related

• LowOrderMoment
• AbstractPriorResult
• factory
• nothing_scalar_array_selector

port_opt_view(
    r::LowOrderMoment,
    i,
    args...
) -> LowOrderMoment{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, <:LowOrderMomentMeasureAlgorithm} where {__T_scale, __T_ub, __T_rke, _A, _B}

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

Slices the expected returns mu for cluster-based optimisation.

Related

• LowOrderMoment
• port_opt_view
• nothing_scalar_array_view

factory(
    r::HighOrderMoment,
    pr::AbstractPriorResult,
    args...;
    kwargs...
) -> HighOrderMoment{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, <:HighOrderMomentMeasureAlgorithm} where {__T_scale, __T_ub, __T_rke, _A, _B}

Create an instance of HighOrderMoment by selecting observation weights, expected returns, and algorithm from the risk-measure instance or falling back to the prior result.

Related

• HighOrderMoment
• AbstractPriorResult
• factory
• nothing_scalar_array_selector

port_opt_view(
    r::HighOrderMoment,
    i,
    args...
) -> HighOrderMoment{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, _B, <:HighOrderMomentMeasureAlgorithm} where {__T_scale, __T_ub, __T_rke, _A, _B}

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

Slices the expected returns mu for cluster-based optimisation.

Related

• HighOrderMoment
• port_opt_view
• nothing_scalar_array_view