Power Norm X at Risk

PortfolioOptimisers.PowerNormValueatRisk Type

struct PowerNormValueatRisk{__T_settings, __T_slv, __T_alpha, __T_p, __T_w} <: RiskMeasure

Represents the Power Norm Value-at-Risk (PNVaR) risk measure.

PowerNormValueatRisk is a coherent risk measure that generalises EVaR by replacing the exponential moment-generating function with a power-norm. It is parametrised by a power $p\ge 1$ and a significance level $\alpha$, and is solved via a conic programme.

Mathematical Definition

The PNVaR at level $\alpha$ with power $p$ is:

$${\mathrm{PNVaR}}_{\alpha ,p}(\mathit{x})=\underset{\eta ,t,\mathit{w}\ge 0,\mathit{v}}{min}\{\eta +\frac{t}{\alpha T^{1/p}}:\sum _{i=1}^T{v}_i\le t,(x_i+w_i)+\eta \ge 0,(v_i,t,w_i)\in {\mathcal{K}}_{\mathrm{pow}}(1/p)\mathrm{\forall }i\},$$

where ${\mathcal{K}}_{\mathrm{pow}}(p^{\prime })=\{(a,b,c):a^{p^{\prime }}{b}^{1-p^{\prime }}\ge |c|,a\ge 0,b\ge 0\}$ is the power cone.

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• p: Power parameter ($p\ge 1$).

• w: Optional observation weights.

Constructors

PowerNormValueatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    p::Number = 2.0,
    w::Option{<:ObsWeights} = nothing
) -> PowerNormValueatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• p >= 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::PowerNormValueatRisk)(x::VecNum)

Computes the PNVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> PowerNormValueatRisk()
PowerNormValueatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
         p ┼ Float64: 2.0
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• EntropicValueatRisk

• RelativisticValueatRisk

• PowerNormValueatRiskRange

• PowerNormDrawdownatRisk

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PortfolioOptimisers.PowerNormValueatRiskRange Type

struct PowerNormValueatRiskRange{__T_settings, __T_slv, __T_alpha, __T_beta, __T_pa, __T_pb, __T_w} <: RiskMeasure

Represents the Power Norm Value-at-Risk Range (PNVaRRange) risk measure.

PowerNormValueatRiskRange computes the sum of the lower-tail PNVaR (at level alpha with power pa) and the upper-tail PNVaR (at level beta with power pb).

Mathematical Definition

$${\mathrm{PNVaRRange}}_{\alpha ,p_a,\beta ,p_b}(\mathit{x})={\mathrm{PNVaR}}_{\alpha ,p_a}(\mathit{x})+{\mathrm{PNVaR}}_{\beta ,p_b}(-\mathit{x}).$$

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• beta: Significance level for the upper tail.

• pa: Power norm parameter for the lower tail ($p_a>1$).

• pb: Power norm parameter for the upper tail ($p_b>1$).

• w: Optional observation weights.

Constructors

PowerNormValueatRiskRange(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    beta::Number = 0.05,
    pa::Number = 2.0,
    pb::Number = 2.0,
    w::Option{<:ObsWeights} = nothing
) -> PowerNormValueatRiskRange

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1, 0 < beta < 1.

• pa > 1, pb > 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::PowerNormValueatRiskRange)(x::VecNum)

Computes the PNVaR Range of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> PowerNormValueatRiskRange()
PowerNormValueatRiskRange
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
      beta ┼ Float64: 0.05
        pa ┼ Float64: 2.0
        pb ┼ Float64: 2.0
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• PowerNormValueatRisk

• EntropicValueatRiskRange

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PortfolioOptimisers.PowerNormDrawdownatRisk Type

struct PowerNormDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_p, __T_w} <: RiskMeasure

Represents the Power Norm Drawdown-at-Risk (PNDDaR) risk measure.

PowerNormDrawdownatRisk applies the Power Norm Value-at-Risk framework to the absolute drawdown series of portfolio returns.

Mathematical Definition

Define the absolute drawdown series:

$$c_t=\sum _{s=1}^t{x}_s,d_t=c_t-\underset{0\le s\le t}{max}{c}_s\le 0.$$

The Power Norm Drawdown-at-Risk is the PNVaR of the drawdown series:

$${\mathrm{PNDDaR}}_{\alpha ,p}(\mathit{x})={\mathrm{PNVaR}}_{\alpha ,p}(\mathit{d}(\mathit{x})).$$

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• p: Power parameter ($p\ge 1$).

• w: Optional observation weights.

Constructors

PowerNormDrawdownatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    p::Number = 2.0,
    w::Option{<:ObsWeights} = nothing
) -> PowerNormDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• p >= 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::PowerNormDrawdownatRisk)(x::VecNum)

Computes the PNDDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> PowerNormDrawdownatRisk()
PowerNormDrawdownatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
         p ┼ Float64: 2.0
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• PowerNormValueatRisk

• RelativisticDrawdownatRisk

• EntropicDrawdownatRisk

• RelativePowerNormDrawdownatRisk

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PortfolioOptimisers.RelativePowerNormDrawdownatRisk Type

struct RelativePowerNormDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_p, __T_w} <: HierarchicalRiskMeasure

Represents the Relative Power Norm Drawdown-at-Risk (Relative PNDDaR) risk measure for hierarchical optimisation.

RelativePowerNormDrawdownatRisk applies the Power Norm Value-at-Risk framework to the relative (compounded) drawdown series of portfolio returns.

Mathematical Definition

Define the relative drawdown series:

$$C_t=\prod _{s=1}^t(1+x_s),r{d}_t=\frac{C_t}{\underset{0\le s\le t}{max}{C}_s}-1\le 0.$$

The Relative Power Norm Drawdown-at-Risk is the PNVaR of the relative drawdown series:

$${\mathrm{RPNDDaR}}_{\alpha ,p}(\mathit{x})={\mathrm{PNVaR}}_{\alpha ,p}(\mathit{r}\mathit{d}(\mathit{x})).$$

Fields

• settings: Hierarchical risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• p: Power parameter ($p\ge 1$).

• w: Optional observation weights.

Constructors

RelativePowerNormDrawdownatRisk(;
    settings::HierarchicalRiskMeasureSettings = HierarchicalRiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    p::Number = 2.0,
    w::Option{<:ObsWeights} = nothing
) -> RelativePowerNormDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• p >= 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativePowerNormDrawdownatRisk)(x::VecNum)

Computes the Relative PNDDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativePowerNormDrawdownatRisk()
RelativePowerNormDrawdownatRisk
  settings ┼ HierarchicalRiskMeasureSettings
           │   scale ┴ Float64: 1.0
       slv ┼ nothing
     alpha ┼ Float64: 0.05
         p ┼ Float64: 2.0
         w ┴ nothing

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• PowerNormDrawdownatRisk

• RelativeRelativisticDrawdownatRisk

• RelativeEntropicDrawdownatRisk

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PortfolioOptimisers.PRM Function

PRM(
    x::AbstractVector{<:Union{var"#s19", var"#s18"} where {var"#s19"<:Number, var"#s18"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}}
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s19", var"#s18"} where {var"#s19"<:Number, var"#s18"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}},
    alpha::Number
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s19", var"#s18"} where {var"#s19"<:Number, var"#s18"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}},
    alpha::Number,
    p::Number
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s19", var"#s18"} where {var"#s19"<:Number, var"#s18"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}},
    alpha::Number,
    p::Number,
    w::Union{Nothing, DynamicAbstractWeights, AbstractWeights}
) -> Union{Float64, Vector{Float64}}

Compute the Power-Norm Risk Measure (PRM) for a vector of portfolio returns.

Solves a convex optimisation problem to compute the PRM at confidence level alpha with Lp-norm parameter p, using the specified solver(s).

Arguments

• x: Vector of portfolio returns.

• slv: Solver or vector of solvers.

• alpha: Confidence level (default 0.05).

• p: Lp-norm parameter (default 2.0).

• Additional parameters depending on the specific PRM formulation.

• kwargs...: Additional keyword arguments passed to the solver.

Returns

• PRM value (scalar).

Related

• PowerNormValueatRisk

• Slv_VecSlv

source