Base Risk Measures

All risk measures are defined as their whole names, however this can be unwieldy, so we also provide convenience aliases defined in API.

All concrete risk measures can be used as functors (callable structs) to compute their associated risk quantity, according to its risk_input_kind, or via expected_risk.

Quick-pick guide

Three legal-usage classes:

• Optimisation: JuMP formulation; usable as objective/constraint.

• Hierarchical: clustering proxy, no JuMP formulation.

• Non-Optimisation: analysis only.

Table Key:

• QP: Quadratic programming (generates a quadratic expression).

• NOC: Norm one none.

• SOC: Second order cone

• RSOC: Rotated second order cone.

• NIC: Norm infinity cone.

• EC: Exponential cone.

• PC: 3D Power cone.

• Sym: Symmetric matrix space.

• PSD: Positive semi-definite cone.

• MIP: Mixed-integer variables.

• *: Carries the requirements of its inner risk measures.

• -: Not applicable.

Type	Name (Alias)	Compatibility	Requirements	Rec. Solver
Dispersion	Variance	JuMP + Hierarchical	QP, SOC	Clarabel
Dispersion	SDP graph / Risk contribution Variance	JuMP + Hierarchical	SDP	Clarabel, SCS
Dispersion	StandardDeviation (SD)	JuMP + Hierarchical	SOC	Clarabel
Dispersion	Box UncertaintySetVariance (UcVariance)	JuMP + Hierarchical	Sym	Clarabel
Dispersion	Ellipse UncertaintySetVariance (UcVariance)	JuMP + Hierarchical	Sym, PSD, SOC	Clarabel, SCS
Dispersion	LowOrderMoment (FLM)	JuMP + Hierarchical	-	Clarabel
Dispersion	LowOrderMoment (MAD)	JuMP + Hierarchical	-	Clarabel
Dispersion	Direct / Squared SOC LowOrderMoment (SCM, SLM)	JuMP + Hierarchical	QP, SOC	Clarabel
Dispersion	Sum of Squares LowOrderMoment (SCM, SLM)	JuMP + Hierarchical	SOC, RSOC	Clarabel
Dispersion	SOC LowOrderMoment (SCM, SLM)	JuMP + Hierarchical	SOC	Clarabel
Dispersion	LowOrderMoment (ECM, ELM)	JuMP + Hierarchical	PC	Clarabel
Dispersion	HighOrderMoment (TLM, SSK, FTCM, FTLM, KT, SKT)	Hierarchical	-	-
Dispersion	Direct / Squared SOC Exact Kurtosis	JuMP + Hierarchical	QP, SOC, PSD	Clarabel, SCS
Dispersion	Sum of Squares Exact Kurtosis	JuMP + Hierarchical	SOC, RSOC, PSD	Clarabel, SCS
Dispersion	SOC Exact Kurtosis	JuMP + Hierarchical	SOC, PSD	Clarabel, SCS
Dispersion	Direct / Squared SOC Approx Kurtosis	JuMP + Hierarchical	QP, SOC	Clarabel
Dispersion	Sum of Squares Approx Kurtosis	JuMP + Hierarchical	SOC, RSOC	Clarabel
Dispersion	SOC Exact Approx Kurtosis	JuMP + Hierarchical	SOC	Clarabel
Dispersion	NegativeSkewness	JuMP + Hierarchical	QP, SOC	Clarabel
Dispersion	Square Root NegativeSkewness	JuMP + Hierarchical	SOC	Clarabel
Tail loss	Exact ValueatRisk (VaR)	JuMP + Hierarchical	MIP	Pajarito (Clarabel + HiGHS)
Tail loss	Approx ValueatRisk (VaR)	JuMP + Hierarchical	SOC	Clarabel
Tail drawdown	DrawdownatRisk (DaR)	JuMP + Hierarchical	MIP	Pajarito (Clarabel + HiGHS)
Tail drawdown	RelativeDrawdownatRisk (R_DaR)	Hierarchical	-	Clarabel
Dispersion	Exact ValueatRiskRange (VaR_RG)	JuMP + Hierarchical	MIP	Pajarito (Clarabel + HiGHS)
Dispersion	Approx ValueatRiskRange (VaR_RG)	JuMP + Hierarchical	SOC	Clarabel
Tail loss	ConditionalValueatRisk (CVaR)	JuMP + Hierarchical	-	Clarabel
Tail loss	DistributionallyRobustConditionalValueatRisk (DRCVaR)	JuMP + Hierarchical	NIC	Clarabel
Tail drawdown	ConditionalDrawdownatRisk (CDaR)	JuMP + Hierarchical	-	Clarabel
Tail drawdown	DistributionallyRobustConditionalDrawdownatRisk (DRCDaR)	JuMP + Hierarchical	NIC	Clarabel
Tail drawdown	RelativeConditionalDrawdownatRisk (R_CDaR)	Hierarchical	-	Clarabel
Dispersion	ConditionalValueatRiskRange (CVaR_RG)	JuMP + Hierarchical	-	Clarabel
Dispersion	DistributionallyRobustConditionalValueatRiskRange (DRCVaR_RG)	JuMP + Hierarchical	NIC	Clarabel
Tail loss	EntropicValueatRisk (EVaR)	JuMP + Hierarchical	EC	Clarabel
Tail drawdown	EntropicDrawdownatRisk (EDaR)	JuMP + Hierarchical	EC	Clarabel
Tail drawdown	RelativeEntropicDrawdownatRisk (R_EDaR)	Hierarchical	EC	Clarabel
Dispersion	EntropicValueatRiskRange (EVaR_RG)	JuMP + Hierarchical	EC	Clarabel
Tail loss	RelativisticValueatRisk (RVaR)	JuMP + Hierarchical	PC	Clarabel
Tail drawdown	RelativisticDrawdownatRisk (RDaR)	JuMP + Hierarchical	PC	Clarabel
Tail drawdown	RelativeRelativisticDrawdownatRisk (R_RDaR)	Hierarchical	PC	Clarabel
Dispersion	RelativisticValueatRiskRange (RVaR_RG)	JuMP + Hierarchical	PC	Clarabel
Tail loss	PowerNormValueatRisk (PNVaR)	JuMP + Hierarchical	PC	Clarabel
Tail drawdown	PowerNormDrawdownatRisk (PNDaR)	JuMP + Hierarchical	PC	Clarabel
Tail drawdown	RelativePowerNormDrawdownatRisk (R_PNDaR)	Hierarchical	PC	Clarabel
Dispersion	PowerNormValueatRiskRange (PNVaR_RG)	JuMP + Hierarchical	PC	Clarabel
Dispersion	Exact OrderedWeightsArray (OWA_GMD, OWA_RG, OWA_CVaR_RG, OWA_TG_RG, OWA_LMoment)	JuMP + Hierarchical	-	Clarabel
Dispersion	Approx OrderedWeightsArray (OWA_GMD, OWA_RG, OWA_CVaR_RG, OWA_TG_RG, OWA_LMoment)	JuMP + Hierarchical	PC	Clarabel
Tail loss	Exact OrderedWeightsArray (OWA_CVaR, OWA_TG, OWA_WR)	JuMP + Hierarchical	-	Clarabel
Tail loss	Approx OrderedWeightsArray (OWA_CVaR, OWA_TG, OWA_WR)	JuMP + Hierarchical	PC	Clarabel
Dispersion	Exact OrderedWeightsArrayRange	JuMP + Hierarchical	-	Clarabel
Dispersion	Approx OrderedWeightsArrayRange	JuMP + Hierarchical	PC	Clarabel
Drawdown	AverageDrawdown (ADD)	JuMP + Hierarchical	-	Clarabel
Drawdown	RelativeAverageDrawdown (R_ADD)	Hierarchical	-	-
Drawdown	UlcerIndex (UCI)	JuMP + Hierarchical	SOC	Clarabel
Drawdown	RelativeUlcerIndex (R_UCI)	Hierarchical	-	-
Tail drawdown	MaximumDrawdown (MDD)	JuMP + Hierarchical	-	Clarabel
Tail drawdown	RelativeMaximumDrawdown (R_MDD)	Hierarchical	-	-
Dispersion	Direct BrownianDistanceVariance (BDVariance)	JuMP + Hierarchical	QP	Clarabel
Dispersion	Sum of Squares BrownianDistanceVariance (BDVariance)	JuMP + Hierarchical	QP, RSOC	Clarabel
Tail loss	WorstRealisation (WR)	JuMP + Hierarchical	-	Clarabel
Tail loss	Range (RG)	JuMP + Hierarchical	-	Clarabel
Turnover	TurnoverRiskMeasure (TnRM)	JuMP + Hierarchical	NOC	Clarabel
Tracking	L1 Norm TrackingRiskMeasure (TrRM)	JuMP + Hierarchical	NOC	Clarabel
Tracking	L2 Norm TrackingRiskMeasure (TrRM)	JuMP + Hierarchical	SOC	Clarabel
Tracking	Squared L2 Norm TrackingRiskMeasure (TrRM)	JuMP + Hierarchical	QP, SOC	Clarabel
Tracking	Lp Norm TrackingRiskMeasure (TrRM)	JuMP + Hierarchical	PC	Clarabel
Tracking	Infinity Norm TrackingRiskMeasure (TrRM)	JuMP + Hierarchical	NIC	Clarabel
Risk tracking	Independent variable RiskTrackingRiskMeasure (RkTrRM)	JuMP + Hierarchical	*	Clarabel
Risk tracking	Dependent variable RiskTrackingRiskMeasure (RkTrRM)	JuMP + Hierarchical	NOC, *	Clarabel
Dispersion	VarianceSkewKurtosis (VSK)	JuMP + Hierarchical	Sym, PSD	SCS
Dispersion	GenericValueatRiskRange (GVaR_RG)	JuMP + Hierarchical	*	Clarabel
Ratio	RiskRatio	Hierarchical	*	*
Ratio	NonOptimisationRiskRatio (NonOptRkRatio)	-	*	*
Flat	EqualRisk	-	-	-
Dispersion	MedianAbsoluteDeviation	Hierarchical	-	-
Performance	MeanReturn	-	-	-
Performance	MeanReturnRiskRatio	-	-	-
Performance	ExpectedReturn	-	-	-
Performance	ExpectedReturnRiskRatio	-	-	-
Dispersion	ThirdCentralMoment	-	-	-
Dispersion	Skewness	-	-	-

PortfolioOptimisers.RiskMeasure Type

abstract type RiskMeasure <: OptimisationRiskMeasure

Abstract supertype for standard risk measures used in portfolio optimisation.

Subtype RiskMeasure to implement concrete risk measures that quantify portfolio risk and can be used as objectives or constraints in optimisation problems. This type ensures compatibility with the optimisation framework and enables composability with other estimators and algorithms.

Related

• OptimisationRiskMeasure

• HierarchicalRiskMeasure

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PortfolioOptimisers.HierarchicalRiskMeasure Type

abstract type HierarchicalRiskMeasure <: OptimisationRiskMeasure

Abstract supertype for hierarchical risk measures used in portfolio optimisation.

Subtype HierarchicalRiskMeasure to implement risk measures that operate on hierarchical or clustered portfolio structures. These measures are designed for use as objectives or constraints in optimisation problems that leverage asset clustering, hierarchical risk parity, or similar techniques.

Related

• OptimisationRiskMeasure

• RiskMeasure

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PortfolioOptimisers.Frontier Type

struct Frontier{__T_N, __T_factor, __T_bound} <: AbstractAlgorithm

Defines the number of points on the efficient frontier (Pareto Front).

Fields

• N: Number of points on the efficient frontier.

• factor: Scaling factor for the efficient frontier range.

• bound: What operation needs to be performed on the risk lower bound.

Constructors

Frontier(;
    N::Integer = 20,
    bound::FrontierBoundEstimator = LinearBound()
) -> Frontier

Creates a Frontier with N points, a scaling factor of 1, and the specified bound strategy. Used to set appropriate frontier bounds in variance_risk_bounds_val and second_moment_bound_val.

PortfolioOptimisers._Frontier(;
    N::Integer = 20,
    factor::Number,
    bound::FrontierBoundEstimator
) -> Frontier

Internal constructor. Keywords correspond to the struct's fields.

Validation

• N > 0.

• isfinite(factor) and factor > 0.

Examples

julia> Frontier(; N = 15)
Frontier
       N ┼ Int64: 15
  factor ┼ Int64: 1
   bound ┴ LinearBound()

Related

• FrontierBoundEstimator

• LinearBound

• SquareRootBound

• SquaredBound

• RiskMeasureSettings

source

PortfolioOptimisers.RiskMeasureSettings Type

struct RiskMeasureSettings{__T_scale, __T_ub, __T_rke} <: JuMPRiskMeasureSettings

Settings type for configuring risk measure estimators in PortfolioOptimisers.jl. Encapsulates scaling, upper bounds, and risk evaluation flags for risk measures used in optimisation routines.

Fields

• scale: Scaling factor applied to the risk measure.

• ub: Upper bound(s) for the risk measure. Can be a scalar, vector, or Frontier.

• rke: Whether to include the risk measure value in the JuMP risk expression.

Constructors

RiskMeasureSettings(;
    scale::Number = 1.0,
    ub::Option{<:RkRtBounds} = nothing,
    rke::Bool = true,
) -> RiskMeasureSettings

Creates a RiskMeasureSettings instance with the specified scale, upper bound, and risk evaluation flag.

Validation

• isfinite(scale).

• ub is validated with assert_nonempty_nonneg_finite_val.

Examples

julia> RiskMeasureSettings()
RiskMeasureSettings
  scale ┼ Float64: 1.0
     ub ┼ nothing
    rke ┴ Bool: true

Related

• JuMPRiskMeasureSettings

• RiskMeasure

• Frontier

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.HierarchicalRiskMeasureSettings Type

struct HierarchicalRiskMeasureSettings{__T_scale} <: AbstractRiskMeasureSettings

Settings type for configuring hierarchical risk measure estimators in PortfolioOptimisers.jl.

Used for HierarchicalRiskMeasure, where it is impossible to set a risk upper bound.

Fields

• scale: Scaling factor applied to the risk measure.

Constructors

HierarchicalRiskMeasureSettings(;
    scale::Number,
) -> HierarchicalRiskMeasureSettings

Creates a HierarchicalRiskMeasureSettings instance with the specified scaling factor.

Validation

• scale must be finite.

Examples

julia> HierarchicalRiskMeasureSettings()
HierarchicalRiskMeasureSettings
  scale ┴ Float64: 1.0

Related

• AbstractRiskMeasureSettings

• HierarchicalRiskMeasure

• RiskMeasureSettings

source

PortfolioOptimisers.SumScalariser Type

struct SumScalariser <: NonHierarchicalScalariser

Scalariser that combines multiple risk measures using a weighted sum.

SumScalariser aggregates a vector of risk measures by computing the weighted sum of their scaled values. The weights are specified in the scale field of RiskMeasureSettings or HierarchicalRiskMeasureSettings. This scalarisation strategy is used in portfolio optimisation routines that require a single risk value from multiple risk measures.

$$\begin{array}{rl}\varphi & =\sum _{i=1}^N{w}_i\cdot r_i.\end{array}$$

Where:

• $N$: Number of risk measures.

• $i$: Subscript denoting the i-th risk measure.

• $r_i$: i-th risk measure value.

• $w_i$: Weight of the i-th risk measure.

Related

• Scalariser

• MaxScalariser

• MinScalariser

• LogSumExpScalariser

• RiskMeasureSettings

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.MaxScalariser Type

struct MaxScalariser <: NonHierarchicalScalariser

Scalariser that selects the risk expression whose scaled value is the largest.

MaxScalariser aggregates a vector of risk measures by selecting the maximum of their scaled values. The weights are specified in the scale field of RiskMeasureSettings or HierarchicalRiskMeasureSettings. In clustering optimisations, the risk of each cluster is computed separately, so there is no coherence in which risk measure is chosen between clusters.

$$\begin{array}{rl}\varphi & =\underset{i\in (1,N)}{max}(w_i\cdot r_i).\end{array}$$

Where:

• $N$: Number of risk measures.

• $i$: Subscript denoting the i-th risk measure.

• $r_i$: i-th risk measure value.

• $w_i$: Weight of the i-th risk measure.

Related

• Scalariser

• SumScalariser

• MinScalariser

• LogSumExpScalariser

• RiskMeasureSettings

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.MinScalariser Type

struct MinScalariser <: HierarchicalScalariser

Scalariser that selects the risk expression whose scaled value is the largest.

MinScalariser aggregates a vector of risk measures by selecting the minimum of their scaled values. The weights are specified in the scale field of RiskMeasureSettings or HierarchicalRiskMeasureSettings. In clustering optimisations, the risk of each cluster is computed separately, so there is no coherence in which risk measure is chosen between clusters.

$$\begin{array}{rl}\varphi & =\underset{i\in (1,N)}{min}(w_i\cdot r_i).\end{array}$$

Where:

• $N$: Number of risk measures.

• $i$: Subscript denoting the i-th risk measure.

• $r_i$: i-th risk measure value.

• $w_i$: Weight of the i-th risk measure.

Related

• Scalariser

• SumScalariser

• MaxScalariser

• LogSumExpScalariser

• RiskMeasureSettings

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.LogSumExpScalariser Type

struct LogSumExpScalariser{__T_gamma} <: NonHierarchicalScalariser

Scalariser that aggregates multiple risk measures using the log-sum-exp function.

LogSumExpScalariser combines a vector of risk measures by applying the log-sum-exp transformation to their scaled values. The weights are specified in the scale field of RiskMeasureSettings or HierarchicalRiskMeasureSettings.

The parameter gamma controls the approximation accuracy to the maximum function: as gamma → 0, the function approaches the weighted sum; as gamma → ∞, it approaches the maximum. This behaviour is only true in JuMP-based optimisations. In clustering optimisations, each cluster's risk is computed separately, so there is no coherence between clusters.

$$\begin{array}{rl}\varphi & =\frac{1}{\gamma }\log (\sum _{i=1}^N\exp [\gamma \cdot w_i\cdot r_i]).\end{array}$$

Where:

• $N$: Number of risk measures.

• $i$: Subscript denoting the i-th risk measure.

• $r_i$: i-th risk measure value.

• $w_i$: Weight of the i-th risk measure.

• $\gamma$: Positive parameter controlling the interpolation between the weighted sum and the maximum functions.

Fields

• gamma: Log-sum-exp scalariser smoothing parameter.

Constructors

LogSumExpScalariser(;
    gamma::Number = 1.0,
) -> LogSumExpScalariser

Keywords correspond to the struct's fields.

Validation

• gamma > 0.

Examples

julia> LogSumExpScalariser()
LogSumExpScalariser
  gamma ┴ Float64: 1.0

Related

• Scalariser

• SumScalariser

• MaxScalariser

• MinScalariser

• RiskMeasureSettings

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.scalarise Function

scalarise(f, sca::Scalariser, itr; by = nothing)

Reduce per-risk-measure values into a single scalarised value.

Applies f to every element of itr (typically a vector of risk measures, or pairs thereof) and combines the results according to the scalariser. f must return a Number, an AbstractArray, or a Tuple of these, and must return freshly allocated values (no views into buffers reused across iterations), since results may be retained across iterations.

The combining rules are:

• SumScalariser: slot-wise sum of all values. f is expected to include the measure's settings.scale weight in its result.

• MaxScalariser/MinScalariser:

  • by === nothing: slot-wise (elementwise) maximum/minimum across values.

  • by given: winner-take-all — returns the single f result whose by(result) is largest/smallest (ties keep the earliest).

• LogSumExpScalariser: slot-wise logsumexp of gamma-scaled values, divided by gamma.

Arguments

• f: Per-element evaluation closure, applied to each element of itr.

• sca: Scalarisation strategy.

• itr: Iterable of risk measures (or pairs of them, when f needs the index).

• by: Optional selection key for MaxScalariser/MinScalariser; ignored by the other scalarisers.

Returns

• The scalarised value, with the same shape as the values returned by f.

Related

• Scalariser

• scalarise_combine

• scalarise_map

• scalarise_logsumexp

source

PortfolioOptimisers.scalarise_combine Function

scalarise_combine(op, a, b)

Combine two scalarised risk values slot-wise.

Numbers combine directly via op, arrays elementwise, and tuples recursively per slot. Used by scalarise to accumulate values that may be scalars, vectors, or tuples mixing both.

Related

• scalarise

• scalarise_map

source

PortfolioOptimisers.scalarise_map Function

scalarise_map(op, x)

Apply op slot-wise to a scalarised risk value.

Numbers are transformed directly, arrays elementwise, and tuples recursively per slot.

Related

• scalarise

• scalarise_combine

source

PortfolioOptimisers.scalarise_logsumexp Function

scalarise_logsumexp(vs)

Slot-wise log-sum-exp across a vector of scalarised risk values.

For a vector of numbers this is LogExpFunctions.logsumexp; for a vector of same-shaped arrays it is applied elementwise across the vector; for a vector of same-shaped tuples it recurses per slot.

Related

• scalarise

• LogSumExpScalariser

source

PortfolioOptimisers.AbstractBaseRiskMeasure Type

abstract type AbstractBaseRiskMeasure <: AbstractEstimator

Abstract supertype for all risk measure estimators in PortfolioOptimisers.jl.

Defines the interface for risk measure types, which quantify portfolio risk using various statistical or econometric methods. All concrete risk measure types should subtype AbstractBaseRiskMeasure to ensure consistency and composability within the optimisation framework.

All concrete risk measures can be used as functors (callable structs) to compute their associated risk quantity.

Related

• NonOptimisationRiskMeasure

• OptimisationRiskMeasure

• RiskMeasure

• HierarchicalRiskMeasure

source

PortfolioOptimisers.NonOptimisationRiskMeasure Type

abstract type NonOptimisationRiskMeasure <: AbstractBaseRiskMeasure

Abstract supertype for risk measures that are not intended for use in portfolio optimisation routines.

These risk measures are typically used for analysis, reporting, or diagnostics, and are not designed to be included as objectives or constraints in optimisation problems. Subtype this when implementing a risk measure that should not be selectable by optimisation algorithms.

Related

• AbstractBaseRiskMeasure

• OptimisationRiskMeasure

• RiskMeasure

• HierarchicalRiskMeasure

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PortfolioOptimisers.OptimisationRiskMeasure Type

abstract type OptimisationRiskMeasure <: AbstractBaseRiskMeasure

Abstract supertype for risk measures that are intended for use in portfolio optimisation routines.

All concrete risk measures that can be used as objectives or constraints in optimisation problems should subtype OptimisationRiskMeasure. This ensures compatibility with the optimisation framework and enables composability with other estimators and algorithms.

Related

• RiskMeasure

• HierarchicalRiskMeasure

• NonOptimisationRiskMeasure

• AbstractBaseRiskMeasure

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PortfolioOptimisers.AbstractRiskMeasureSettings Type

abstract type AbstractRiskMeasureSettings <: AbstractEstimator

Abstract supertype for all risk measure settings in PortfolioOptimisers.jl.

Defines the interface for settings types that configure the behavior of risk measure estimators. All concrete risk measure settings types should subtype AbstractRiskMeasureSettings to ensure consistency and composability within the optimisation framework.

Related

• JuMPRiskMeasureSettings

• RiskMeasureSettings

• HierarchicalRiskMeasureSettings

source

PortfolioOptimisers.JuMPRiskMeasureSettings Type

abstract type JuMPRiskMeasureSettings <: AbstractRiskMeasureSettings

Abstract supertype for risk measure settings used in JuMP-based optimisation routines.

All concrete settings types compatible with JuMP optimisation (e.g. RiskMeasureSettings, MaxRiskMeasureSettings) should subtype JuMPRiskMeasureSettings.

Related

• AbstractRiskMeasureSettings

• RiskMeasureSettings

• MaxRiskMeasureSettings

source

PortfolioOptimisers.FrontierBoundEstimator Type

abstract type FrontierBoundEstimator <: AbstractEstimator

Abstract supertype for bound-transformation strategies applied to efficient frontier bounds.

Concrete subtypes control how numeric bound values are transformed before being applied to JuMP risk expressions. All subtypes should subtype FrontierBoundEstimator.

Related

• LinearBound

• SquareRootBound

• SquaredBound

• variance_risk_bounds_val

• Frontier

source

PortfolioOptimisers.LinearBound Type

struct LinearBound <: FrontierBoundEstimator

Passes bound values through unchanged (identity transformation).

Used when the risk expression and the user-supplied bound are already in the same units (e.g. SDP variance formulation where both sides are in variance units).

Related

• FrontierBoundEstimator

• SquareRootBound

• SquaredBound

• variance_risk_bounds_val

source

PortfolioOptimisers.SquareRootBound Type

struct SquareRootBound <: FrontierBoundEstimator

Applies a square-root transformation to bound values before enforcing them.

Used when the risk expression is in standard-deviation units but the user-supplied bound is in variance units (e.g. kurtosis and negative-skewness SOC formulations).

Related

• FrontierBoundEstimator

• LinearBound

• SquaredBound

• variance_risk_bounds_val

source

PortfolioOptimisers.SquaredBound Type

struct SquaredBound <: FrontierBoundEstimator

Applies a squaring transformation to bound values before enforcing them.

Used when the risk expression is in squared units but the user-supplied bound is in linear units (e.g. kurtosis SDP formulation).

Related

• FrontierBoundEstimator

• LinearBound

• SquareRootBound

• variance_risk_bounds_val

source

PortfolioOptimisers.Scalariser Type

abstract type Scalariser <: AbstractEstimator

Abstract supertype for scalarisation strategies used to combine multiple risk measures into a single scalar value for optimisation.

Subtype Scalariser to implement different methods for aggregating risk measures. These strategies are used in portfolio optimisation routines that require a single risk value from multiple risk measures.

Related

• NonHierarchicalScalariser

• HierarchicalScalariser

source

PortfolioOptimisers.NonHierarchicalScalariser Type

abstract type NonHierarchicalScalariser <: Scalariser

Abstract supertype for scalarisation strategies that combine multiple risk measures into a single scalar value compatible with all portfolio optimisation estimators.

Subtype NonHierarchicalScalariser to implement aggregation methods that work with all optimisation estimators.

Related

• SumScalariser

• MaxScalariser

• LogSumExpScalariser

• Scalariser

• HierarchicalScalariser

source

PortfolioOptimisers.HierarchicalScalariser Type

abstract type HierarchicalScalariser <: Scalariser

Abstract supertype for scalarisation strategies that combine multiple risk measures into a single scalar value compatible only with hierarchical optimisations.

Subtype HierarchicalScalariser to implement aggregation methods that only work with hierarchical optimisation estimators.

Related

• MinScalariser

• Scalariser

• NonHierarchicalScalariser

source

PortfolioOptimisers.nothing_scalar_array_selector Function

nothing_scalar_array_selector(risk_variable::Nothing, prior_variable::Nothing)
nothing_scalar_array_selector(risk_variable::Num_ArrNum_VecScalar_DynWeights, ::Any)
nothing_scalar_array_selector(risk_variable::Nothing, prior_variable::Num_ArrNum_VecScalar_DynWeights)

Function for selecting a non-nothing value when provided by a risk measure, or fall back to a value contained in a prior result

Arguments

• risk_variable : The risk-side input.

• prior_variable : The prior-side input.

Returns

• If both inputs are nothing returns nothing.

• If risk_variable is not nothing, returns risk_variable.

• If risk_variable is nothing and prior_variable is not nothing, returns prior_variable.

source

PortfolioOptimisers.risk_measure_nothing_scalar_array_view Function

risk_measure_nothing_scalar_array_view(risk_variable, prior_variable, i)

Get a view of a risk measure's risk or prior variable for index i.

Internal helper for slicing scalar, array, or nothing risk/prior variables by index. Dispatches on the types of risk_variable and prior_variable.

Arguments

• risk_variable: Risk variable (scalar, array, or nothing).

• prior_variable: Prior variable (array or nothing).

• i: Index or range to slice.

Returns

• Sliced or unchanged value.

Related

• port_opt_view

source

PortfolioOptimisers.solver_selector Function

solver_selector(risk_solvers, slv)

Select the appropriate solver for a risk measure computation.

Returns the risk-measure-specific solver if provided, otherwise falls back to the optimiser-level solver. Returns nothing if neither is available.

Arguments

• risk_solvers: Risk-measure-specific solver(s) or nothing.

• slv: Optimiser-level solver(s) or nothing.

Returns

• Selected solver(s) or nothing.

Related

• Slv_VecSlv

• ERM

• RRM

source

PortfolioOptimisers.VecBaseRM Type

const VecBaseRM = AbstractVector{<:AbstractBaseRiskMeasure}

Alias for an abstract vector of AbstractBaseRiskMeasure elements.

Related

• AbstractBaseRiskMeasure

• VecOptRM

• VecRM

source

PortfolioOptimisers.VecOptRM Type

const VecOptRM = AbstractVector{<:OptimisationRiskMeasure}

Alias for an abstract vector of OptimisationRiskMeasure elements.

Related

• OptimisationRiskMeasure

• OptRM_VecOptRM

• VecRM

source

PortfolioOptimisers.OptRM_VecOptRM Type

const OptRM_VecOptRM = Union{<:OptimisationRiskMeasure, <:VecOptRM}

Union type accepting a single OptimisationRiskMeasure or a vector of them.

Related

• OptimisationRiskMeasure

• VecOptRM

source

PortfolioOptimisers.VecRM Type

const VecRM = AbstractVector{<:RiskMeasure}

Alias for an abstract vector of RiskMeasure elements.

Related

• RiskMeasure

• RM_VecRM

• VecOptRM

source

PortfolioOptimisers.RM_VecRM Type

const RM_VecRM = Union{<:RiskMeasure, <:VecRM}

Union type accepting a single RiskMeasure or a vector of them.

Related

• RiskMeasure

• VecRM

source

PortfolioOptimisers.RkRtBounds Type

const RkRtBounds = Union{<:Num_VecNum, <:Frontier}

Union type for risk-measure upper bound specifications.

Accepts either a scalar/vector numeric bound or a Frontier sweep configuration. Used in RiskMeasureSettings to set the upper bound field.

Related

• Frontier

• Num_VecNum

• RiskMeasureSettings

source

PortfolioOptimisers.Front_NumVec Type

const Front_NumVec = Union{<:VecNum, <:Frontier}

Union type for frontier or numeric-vector specifications used internally for risk bounds.

Related

• Frontier

• VecNum

• RkRtBounds

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PortfolioOptimisers.bigger_is_better Function

bigger_is_better(r::AbstractBaseRiskMeasure) -> Bool

Return whether a larger value of risk measure r is preferred over a smaller one.

The default implementation returns false (lower risk is better) for all AbstractBaseRiskMeasure subtypes. Ratio-based or return-like measures that should be maximised may override this method to return true.

Returns

• Bool: true if a higher value of r is preferred; false otherwise.

Related

• AbstractBaseRiskMeasure

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bigger_is_better(_::Skewness) -> Bool

Return true because higher skewness is preferred over lower skewness.

Related

• Skewness

• bigger_is_better

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PortfolioOptimisers.needs_previous_weights Method

needs_previous_weights(r::AbstractBaseRiskMeasure) -> Bool
needs_previous_weights(r::VecBaseRM) -> Bool

Return whether risk measure r requires previous portfolio weights as input.

The default returns false. Override to true for risk measures such as turnover or tracking constraints that depend on prior weights. The VecBaseRM overload returns true if any element returns true.

Returns

• Bool: true if prior weights are required; false otherwise.

Related

• AbstractBaseRiskMeasure

• VecBaseRM

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PortfolioOptimisers.RiskInputKind Type

abstract type RiskInputKind

Abstract supertype for the input-shape classification of a risk measure, used by expected_risk to decide what to feed a measure's functor.

Each concrete AbstractBaseRiskMeasure declares its kind via risk_input_kind. The three kinds correspond to the three functor call shapes:

• NetReturnsInput: r(calc_net_returns(w, X, fees)).

• WeightsReturnsFeesInput: r(w, X, fees).

• WeightsInput: r(w).

Related

• risk_input_kind

• expected_risk

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PortfolioOptimisers.NetReturnsInput Type

struct NetReturnsInput <: RiskInputKind

Input kind for risk measures whose expected risk is computed on net returns (returns after fees). The measure's functor is called as r(calc_net_returns(w, X, fees)).

Related

• RiskInputKind

• risk_input_kind

• calc_net_returns

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PortfolioOptimisers.WeightsReturnsFeesInput Type

struct WeightsReturnsFeesInput <: RiskInputKind

Input kind for risk measures whose expected risk depends on weights, the returns matrix, and fees. The measure's functor is called as r(w, X, fees).

Related

• RiskInputKind

• risk_input_kind

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PortfolioOptimisers.WeightsInput Type

struct WeightsInput <: RiskInputKind

Input kind for risk measures whose expected risk depends only on portfolio weights. The measure's functor is called as r(w).

Related

• RiskInputKind

• risk_input_kind

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PortfolioOptimisers.risk_input_kind Function

risk_input_kind(r::AbstractBaseRiskMeasure) -> RiskInputKind

Return the RiskInputKind of risk measure r, declaring what its functor consumes when expected_risk evaluates it.

There is no default: every concrete AbstractBaseRiskMeasure (other than composite measures handled by explicit expected_risk methods) must declare its kind beside its type definition. Returning one of NetReturnsInput, WeightsReturnsFeesInput, or WeightsInput. An undeclared measure throws, rather than silently routing to the wrong input shape.

Returns

• RiskInputKind: the declared input kind.

Related

• RiskInputKind

• expected_risk

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PortfolioOptimisers.supports_precomputed_returns Method

supports_precomputed_returns(r::AbstractBaseRiskMeasure) -> Bool
supports_precomputed_returns(rk::RiskInputKind, r::AbstractBaseRiskMeasure) -> Bool

Whether risk measure r has a well-defined precomputed-returns form — i.e. whether its expected risk can be evaluated on an already-reduced net-return series x alone, via the functor r(x::VecNum) (ADR 0007).

The contract is well-defined exactly when the measure's result is a function of the series alone:

• NetReturnsInput measures (quantile / drawdown families): always true — their functor is the net-returns functor.

• The moment family (LowOrderMoment, HighOrderMoment, Skewness, Kurtosis, MedianAbsoluteDeviation, ThirdCentralMoment): true iff its target is weight-independent (mu is nothing, a scalar, or a centering function); a per-asset mu (VecNum/VecScalar) reduces as dot(w, mu) and needs the weights the series no longer carries, so false.

• WeightsInput measures, tracking measures, and variance-carrying composites (VarianceSkewKurtosis): false — "risk of a bare return series" is undefined for them.

This predicate is what makes the precomputed-returns contract safe. Because a WeightsInput measure's functor r(w) shares the r(::VecNum) signature with the contract, dispatch alone cannot distinguish weights from returns; expected_risk_from_returns consults this predicate and throws an explanatory error for ineligible measures rather than silently consuming the series as weights.

Related

• expected_risk_from_returns

• risk_input_kind

• RiskInputKind

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PortfolioOptimisers.supports_precomputed_returns Method

supports_precomputed_returns(_::NetReturnsInput, _) -> Bool

Return true: NetReturnsInput measures always support precomputed returns — their functor is the net-returns functor.

Related

• supports_precomputed_returns

• NetReturnsInput

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PortfolioOptimisers.supports_precomputed_returns Method

supports_precomputed_returns(_::WeightsInput, _) -> Bool

Return false: WeightsInput measures never support precomputed returns — their functor consumes portfolio weights, not a return series.

Related

• supports_precomputed_returns

• WeightsInput

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PortfolioOptimisers.supports_precomputed_returns Method

supports_precomputed_returns(
    _::WeightsReturnsFeesInput,
    r::AbstractBaseRiskMeasure
)

Erroring tripwire for WeightsReturnsFeesInput measures that have not declared supports_precomputed_returns at their own definition site. Every such measure must declare it there: moment measures as supports_precomputed_returns(r::T) = weight_independent_target(r.mu); weights-dependent measures (tracking, variance-carrying composites) as supports_precomputed_returns(::T) = false. Reaching this leaf means a measure forgot to declare it — throws an ArgumentError with instructions rather than silently mis-routing (and the completeness test in test_09c_risk_input_kind.jl turns that into a CI failure).

Related

• supports_precomputed_returns

• WeightsReturnsFeesInput

• weight_independent_target

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PortfolioOptimisers.weight_independent_target Method

weight_independent_target(_::Nothing) -> Bool

Return true: a Nothing target is trivially weight-independent and can be evaluated on a bare return series.

Related

• supports_precomputed_returns

• weight_independent_target

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PortfolioOptimisers.weight_independent_target Method

weight_independent_target(_::Number) -> Bool

Return true: a scalar target does not require portfolio weights and can be evaluated on a bare return series.

Related

• supports_precomputed_returns

• weight_independent_target

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PortfolioOptimisers.weight_independent_target Method

weight_independent_target(_) -> Bool

Return false: the target type requires portfolio weights (e.g. a per-asset mu involves dot(w, mu)) and cannot be evaluated on a bare return series.

Related

• supports_precomputed_returns

• weight_independent_target

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PortfolioOptimisers.factory Method

factory(
    rs::AbstractBaseRiskMeasure,
    args...;
    kwargs...
) -> VarianceSkewKurtosis{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, _A, Skewness{__T_settings, __T_ve, __T_sk, __T_w, __T_mu}} where {__T_scale, __T_ub, __T_rke, _A, __T_settings, __T_ve, __T_sk, __T_w, __T_mu}

Return the risk measure rs unchanged.

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

Related

• AbstractBaseRiskMeasure

• factory

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PortfolioOptimisers.factory Method

factory(
    rs::AbstractVector{<:AbstractBaseRiskMeasure},
    args...;
    kwargs...
) -> Any

Return a new vector of risk measures with factory applied element-wise.

Related

• VecBaseRM

• factory

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PortfolioOptimisers.port_opt_view Method

port_opt_view(rs, i, X)

Get a view or subset of a risk measure for asset cluster index i.

Returns the risk measure sliced for the given cluster or asset index. Used internally in hierarchical optimisation to apply risk measures to each cluster.

Arguments

• rs: Risk measure (or vector thereof).

• i: Cluster or asset index.

• X: Data matrix (used for dimension-aware slicing).

Returns

• Sliced risk measure or the original if no slicing is needed.

Related

• AbstractBaseRiskMeasure

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PortfolioOptimisers._Frontier Function

_Frontier(; N = 20, factor, bound)

Construct a range of N evenly-spaced frontier parameter values.

Internal helper that generates a parameter grid (e.g., for risk bounds) used when sweeping the efficient frontier.

Arguments

• N: Number of frontier points (default 20).

• factor: Scaling factor for the range.

• bound: Controls whether to sweep from min-to-max or max-to-min.

Returns

• Vector of frontier parameter values.

Related

• MeanRisk

• NearOptimalCentering

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