Variance

PortfolioOptimisers.QuadRiskExpr Type

struct QuadRiskExpr <: VarianceFormulation end

Direct quadratic risk expression optimisation formulation for variance-like risk measures. The risk measure is implemented using an explicitly quadratic form. This can be in two ways.

Summary statistics

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& {\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N co-moment matrix.

Scenario-based

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& \mathit{d}\cdot \mathit{d}.\\ \text{s.t.}& \mathit{d}\in {\mathcal{S}}_w.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathit{d}$: T×1 deviations vector.

• ${\mathcal{S}}_w$: Scenario set for portfolio x.

Related Types

• VarianceFormulation

• Variance

• SOCRiskExpr

• SquaredSOCRiskExpr

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PortfolioOptimisers.SquaredSOCRiskExpr Type

struct SquaredSOCRiskExpr <: VarianceFormulation end

Squared second-order cone risk expression optimisation formulation for applicable risk measures. The risk measure is implemented using the square of a variable constrained by a second order cone.

Related

• VarianceFormulation

• QuadRiskExpr

• SOCRiskExpr

• Variance

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PortfolioOptimisers.RSOCRiskExpr Type

struct RSOCRiskExpr <: SecondMomentFormulation end

Rotated second-order cone risk expression optimisation formulation for applicable risk measures. The risk measure using a variable constrained to be in a rotated second order cone representing the sum of squares.

Related Types

• SecondMomentFormulation

• VarianceFormulation

• SOCRiskExpr

• SquaredSOCRiskExpr

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PortfolioOptimisers.SOCRiskExpr Type

struct SOCRiskExpr <: SecondMomentFormulation end

Second-order cone risk expression optimisation formulation for applicable risk measures. The risk measure is implemented using a variable constrained by a second order cone.

Related

• SecondMomentFormulation

• VarianceFormulation

• QuadRiskExpr

• SquaredSOCRiskExpr

• RSOCRiskExpr

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PortfolioOptimisers.Variance Type

struct Variance{T1, T2, T3, T4} <: RiskMeasure
    settings::T1
    sigma::T2
    rc::T3
    alg::T4
end

Represents the portfolio variance using a covariance matrix.

Fields

• settings: Risk measure configuration.

• sigma: Optional covariance matrix that overrides the prior covariance when provided. Also used to compute the risk represented by a vector.

• rc: Optional specification of risk contribution constraints.

• alg: The optimisation formulation used to represent the variance risk expression.

Constructors

Variance(; settings::RiskMeasureSettings = RiskMeasureSettings(),
         sigma::Union{Nothing, <:AbstractMatrix} = nothing,
         rc::Union{Nothing, <:LinearConstraintEstimator, <:LinearConstraint} = nothing,
         alg::VarianceFormulation = SquaredSOCRiskExpr())

Keyword arguments correspond to the fields above.

Validation

• If sigma is provided, !isempty(sigma) and size(sigma, 1) == size(sigma, 2).

JuMP Formulations

Info

Regardless of the formulation used, an auxiliary variable representing the standard deviation is needed in order to constrain the risk or maximise the risk-adjusted return ratio. This is because quadratic constraints are not strictly convex, and the transformation needed to maximise the risk-adjusted return ratio requires affine variables in the numerator and denominator.

Depending on the alg field, the variance risk measure is formulated using JuMP as follows:

QuadRiskExpr

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& {\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

SquaredSOCRiskExpr

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& {\sigma }^2\\ \text{s.t.}& {\Vert \mathbf{G}\mathit{w}\Vert }_2\le \sigma .\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\sigma$: Variable representing the optimised portfolio's standard deviation.

• $\mathbf{G}$: Suitable factorisation of the N×N covariance matrix, such as the square root matrix, or the Cholesky factorisation.

• $\Vert \cdot {\Vert }_2$: L2 norm, which is modelled as a SecondOrderCone.

Functor

(r::Variance)(w::AbstractVector)

Computes the variance risk of a portfolio with weights w using the covariance matrix r.sigma.

$$\begin{array}{rl}\mathrm{Variance}(\mathit{w},\mathbf{\Sigma })& ={\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

Arguments

• w::AbstractVector: Asset weights.

Examples

julia> w = [0.3803452066954233, 0.5900852659955864, 0.029569527308990307];

julia> r = Variance(;
                    sigma = [0.97780 -0.06400 0.84818;
                             -0.06400 3.28564 1.84588;
                             0.84818 1.84588 2.16317])
Variance
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     sigma ┼ 3×3 Matrix{Float64}
        rc ┼ nothing
       alg ┴ SquaredSOCRiskExpr()

julia> r(w)
1.3421705804186579

Related

• RiskMeasureSettings

• VarianceFormulation

• QuadRiskExpr

• SquaredSOCRiskExpr

• SOCRiskExpr

• RSOCRiskExpr

• factory(r::Variance, prior::AbstractPriorResult, args...; kwargs...)

• expected_risk

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PortfolioOptimisers.StandardDeviation Type

struct StandardDeviation{T1, T2} <: RiskMeasure
    settings::T1
    sigma::T2
end

Represents the portfolio standard deviation using a covariance matrix. It is the square root of the variance.

Fields

• settings: Risk measure configuration.

• sigma: Optional covariance matrix that overrides the prior covariance when provided. Also used to compute the risk represented by a vector.

Constructors

StandardDeviation(; settings::RiskMeasureSettings = RiskMeasureSettings(),
                   sigma::Union{Nothing, <:AbstractMatrix} = nothing)

Keyword arguments correspond to the fields above.

Validation

• If sigma is provided, !isempty(sigma) and size(sigma, 1) == size(sigma, 2).

JuMP Formulation

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& \sigma \\ \text{s.t.}& {\Vert \mathbf{G}\mathit{w}\Vert }_2\le \sigma .\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\sigma$: Variable representing the optimised portfolio's standard deviation.

• $\mathbf{G}$: Suitable factorisation of the N×N covariance matrix, such as the square root matrix, or the Cholesky factorisation.

• $\Vert \cdot {\Vert }_2$: L2 norm, which is modelled as a SecondOrderCone.

Functor

(r::StandardDeviation)(w::AbstractVector)

Computes the standard deviation risk of a portfolio with weights w using the covariance matrix r.sigma.

$$\begin{array}{rl}\mathrm{StandardDeviation}(\mathit{w},\mathbf{\Sigma })& =\sqrt{{\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

Arguments

• w::AbstractVector: Asset weights.

Examples

julia> w = [0.3803452066954233, 0.5900852659955864, 0.029569527308990307];

julia> r = StandardDeviation(;
                             sigma = [0.97780 -0.06400 0.84818;
                                      -0.06400 3.28564 1.84588;
                                      0.84818 1.84588 2.16317])
StandardDeviation
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     sigma ┴ 3×3 Matrix{Float64}

julia> r(w)
1.1585208588621345

Related

• RiskMeasureSettings

• factory(r::StandardDeviation, prior::AbstractPriorResult, args...; kwargs...)

• expected_risk

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PortfolioOptimisers.UncertaintySetVariance Type

struct UncertaintySetVariance{T1, T2, T3} <: RiskMeasure

Represents the variance risk measure under uncertainty sets. Works the same way as the Variance risk measure but allows specifying uncertainty set estimators or results. These are only used in JuMP-based optimisations because they dictate how the variance is formulated as an optimisation problem. By encapsulating the uncertainty set estimator or result, enables the use of multiple uncertainty set variances in the same optimisation model.

Fields

• settings: Risk measure configuration.

• ucs: Uncertainty set estimator or result that defines the uncertainty model for the variance calculation.

• sigma: Optional covariance matrix that overrides the prior covariance when provided. Also used to compute the risk represented by a vector.

Constructors

UncertaintySetVariance(; settings::RiskMeasureSettings = RiskMeasureSettings(),
                       ucs::Union{Nothing, <:AbstractUncertaintySetResult,
                                  <:AbstractUncertaintySetEstimator} = NormalUncertaintySet(),
                       sigma::Union{Nothing, <:AbstractMatrix{<:Real}} = nothing)

Keyword arguments correspond to the fields above.

Validation

• If sigma is provided, !isempty(sigma).

JuMP Formulations

When using an uncertainty set on the variance, the optimisation problem becomes:

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& \underset{\mathbf{\Sigma }\in U_{\mathbf{\Sigma }}}{max}{\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

• $U_{\mathbf{\Sigma }}$: Uncertainty set for the covariance matrix.

This problem can be reformulated depending on the type of uncertainty set used.

Box uncertainty set

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& \mathrm{Tr}\left({\mathbf{A}}_u{\mathbf{\Sigma }}_u\right)-\mathrm{Tr}\left({\mathbf{A}}_l{\mathbf{\Sigma }}_l\right)\\ \text{s.t.}& {\mathbf{A}}_u\ge 0\\ & {\mathbf{A}}_l\ge 0\\ & \left[\begin{array}{cc}\mathbf{W}& \mathit{w}\\ {\mathit{w}}^{⊺}& k\end{array}\right]⪰0\\ & {\mathbf{A}}_u-{\mathbf{A}}_l=\mathbf{W}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• ${\mathbf{A}}_u$, ${\mathbf{A}}_l$, $\mathbf{W}$: N×N auxiliary symmetric matrices.

• ${\mathbf{\Sigma }}_l$: N×N lower bound of the covariance matrix.

• ${\mathbf{\Sigma }}_u$: N×N upper bound of the covariance matrix.

• $k$: Scalar variable/constant.

  • If the objective risk-adjusted return, it is a non-negative variable.

  • Else it is equal to 1.

• $\mathrm{Tr}(\cdot )$: Trace operator.

Ellipse uncertainty set

$$\begin{array}{rl}\underset{\mathit{w}}{\mathrm{opt}}& \mathrm{Tr}\left(\mathbf{\Sigma }(\mathbf{W}+\mathbf{E})\right)+k_{\mathbf{\Sigma }}\sigma \\ \text{s.t.}& \left[\begin{array}{cc}\mathbf{W}& \mathit{w}\\ {\mathit{w}}^{⊺}& k\end{array}\right]⪰0\\ & \mathbf{E}⪰0\\ & \Vert \mathbf{G}\mathrm{vec}(\mathbf{W}+\mathbf{E}){\Vert }_2\le \sigma \end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

• $\mathbf{W}$, $\mathbf{E}$: N×N auxiliary symmetric matrices.

• $k_{\mathbf{\Sigma }}$: Scalar constant defining the size of the uncertainty set.

• $\sigma$: Variable representing the portfolio's variance of the variance.

• $\mathbf{G}$: Suitable factorisation of the N^2×N^2 covariance of the covariance matrix of the uncertainty set, such as the square root matrix, or the Cholesky factorisation.

• $k$: Scalar variable/constant.

  • If the objective risk-adjusted return, it is a non-negative variable.

  • Else it is equal to 1.

• $\mathrm{Tr}(\cdot )$: Trace operator.

• $\mathrm{vec}(\cdot )$: Vectorisation operator, which unrolls a matrix as a column vector in column-major order.

• $\Vert \cdot {\Vert }_2$: L2 norm, which is modelled as a SecondOrderCone.

Functor

(r::UncertaintySetVariance)(w::AbstractVector)

Computes the variance risk of a portfolio with weights w using the covariance matrix r.sigma.

$$\begin{array}{rl}\mathrm{UncertaintySetVariance}(\mathit{w},\mathbf{\Sigma })& ={\mathit{w}}^{⊺}\mathbf{\Sigma }\mathit{w}.\end{array}$$

Where:

• $\mathit{w}$: N×1 asset weights vector.

• $\mathbf{\Sigma }$: N×N covariance matrix.

Arguments

• w::AbstractVector: Asset weights.

Examples

julia> w = [0.3803452066954233, 0.5900852659955864, 0.029569527308990307];

julia> r = UncertaintySetVariance(;
                                  sigma = [0.97780 -0.06400 0.84818;
                                           -0.06400 3.28564 1.84588;
                                           0.84818 1.84588 2.16317])
UncertaintySetVariance
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       ucs ┼ NormalUncertaintySet
           │      pe ┼ EmpiricalPrior
           │         │        ce ┼ PortfolioOptimisersCovariance
           │         │           │   ce ┼ Covariance
           │         │           │      │    me ┼ SimpleExpectedReturns
           │         │           │      │       │   w ┴ nothing
           │         │           │      │    ce ┼ GeneralCovariance
           │         │           │      │       │   ce ┼ StatsBase.SimpleCovariance: StatsBase.SimpleCovariance(true)
           │         │           │      │       │    w ┴ nothing
           │         │           │      │   alg ┴ Full()
           │         │           │   mp ┼ DefaultMatrixProcessing
           │         │           │      │       pdm ┼ Posdef
           │         │           │      │           │   alg ┴ UnionAll: NearestCorrelationMatrix.Newton
           │         │           │      │   denoise ┼ nothing
           │         │           │      │    detone ┼ nothing
           │         │           │      │       alg ┴ nothing
           │         │        me ┼ SimpleExpectedReturns
           │         │           │   w ┴ nothing
           │         │   horizon ┴ nothing
           │     alg ┼ BoxUncertaintySetAlgorithm()
           │   n_sim ┼ Int64: 3000
           │       q ┼ Float64: 0.05
           │     rng ┼ Random.TaskLocalRNG: Random.TaskLocalRNG()
           │    seed ┴ nothing
     sigma ┴ 3×3 Matrix{Float64}

julia> r(w)
1.3421705804186579

Related

• RiskMeasureSettings

• Variance

• AbstractUncertaintySetResult

• AbstractUncertaintySetEstimator

• factory(r::UncertaintySetVariance, prior::AbstractPriorResult, args...; kwargs...)

• expected_risk

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PortfolioOptimisers.factory Method

factory(r::Variance, prior::AbstractPriorResult, args...; kwargs...)

Create an instance of Variance by selecting the covariance matrix from the risk-measure instance or falling back to the prior result (see nothing_scalar_array_factory).

Arguments

• r: Prototype risk measure whose settings, rc and alg fields are reused for the new instance.

• prior: Prior result providing prior.sigma to use when r.sigma === nothing.

• args...: Extra positional arguments are accepted for API compatibility but are ignored by this constructor.

• kwargs... : Keyword arguments are accepted for API compatibility but are ignored by this constructor.

Returns

• r_new::Variance: A new Variance instance.

Details

• Selects sigma using nothing_scalar_array_factory.

• Other fields are taken from r.

Related

• Variance

• nothing_scalar_array_factory

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PortfolioOptimisers.factory Method

factory(r::StandardDeviation, prior::AbstractPriorResult, args...; kwargs...)

Create an instance of StandardDeviation by selecting the covariance matrix from the risk-measure instance or falling back to the prior result (see nothing_scalar_array_factory).

Arguments

• r: Prototype risk measure whose settings, rc and alg fields are reused for the new instance.

• prior: Prior result providing prior.sigma to use when r.sigma === nothing.

• args...: Extra positional arguments are accepted for API compatibility but are ignored by this constructor.

• kwargs... : Keyword arguments are accepted for API compatibility but are ignored by this constructor.

Returns

• r_new::StandardDeviation: A new StandardDeviation instance.

Details

• Selects sigma using nothing_scalar_array_factory.

• Other fields are taken from r.

Related

• StandardDeviation

• nothing_scalar_array_factory

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PortfolioOptimisers.factory Function

factory(r::UncertaintySetVariance, prior::AbstractPriorResult, ::Any,
        ucs::Union{Nothing, <:AbstractUncertaintySetResult,
                   <:AbstractUncertaintySetEstimator} = nothing, args...;
        kwargs...)

Create an instance of UncertaintySetVariance by selecting the uncertainty set and covariance matrix from the risk-measure instance or falling back to the prior result.

Arguments

• r: Prototype risk measure whose settings and sigma fields are reused for the new instance.

• prior: Prior result providing prior.sigma to use when r.sigma === nothing.

• ::Any: Placeholder positional argument for API compatibility.

• ucs: Optional uncertainty set estimator or result to override r.ucs.

• args...: Extra positional arguments are accepted for API compatibility but are ignored by this constructor.

• kwargs...: Keyword arguments are accepted for API compatibility but are ignored by this constructor.

Returns

• r_new::UncertaintySetVariance: A new UncertaintySetVariance instance.

Details

• Selects ucs using ucs_factory.

• Selects sigma using nothing_scalar_array_factory.

• Other fields are taken from r.

Related

• UncertaintySetVariance

• AbstractUncertaintySetResult

• AbstractUncertaintySetEstimator

• ucs_factory

• nothing_scalar_array_factory

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PortfolioOptimisers.SecondMomentFormulation Type

abstract type SecondMomentFormulation <: AbstractAlgorithm end

Abstract supertype for optimisation formulations of second moment risk measures in PortfolioOptimisers.jl.

Related Types

• VarianceFormulation

• QuadRiskExpr

• SquaredSOCRiskExpr

• RSOCRiskExpr

• SOCRiskExpr

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PortfolioOptimisers.VarianceFormulation Type

abstract type VarianceFormulation <: SecondMomentFormulation end

Abstract supertype for optimisation formulations of variance-based risk measures in PortfolioOptimisers.jl.

Related Types

• QuadRiskExpr

• SquaredSOCRiskExpr

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