Power Norm X at Risk

PortfolioOptimisers.PowerNormValueatRisk Type

julia

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 $p \geq 1$ and a significance level $α$ , and is solved via a conic programme.

Mathematical definition

The PNVaR at level $α$ with power $p$ is:

\begin{aligned} {PNVaR}_{α, p} (x) & = min_{η, t, w, v} {η + \frac{t}{α T^{1 / p}} : w \geq 0, \sum_{i = 1}^{T} v_{i} \leq t, (x_{i} + w_{i}) + η \geq 0, (v_{i}, t, w_{i}) \in K_{pow} (1 / p) \forall i} . \end{aligned}

Where:

${PNVaR}_{α, p} (x)$ : Power Norm Value-at-Risk.
$x$ : Portfolio returns vector $T \times 1$ .
$α$ : Significance level (left tail probability), $α \in (0, 1)$ .
$T$ : Number of observations.
$p \geq 1$ : Power parameter.
$η$ , $t$ , $w$ , $v$ : Conic optimisation variables.
$K_{pow} (p^{'}) = {(a, b, c) : a^{p^{'}} b^{1 - p^{'}} \geq | c |, a \geq 0, b \geq 0}$ : Power cone.

Fields

settings: Risk measure settings.
slv: Solver or vector of solvers.
alpha: Quantile level for the lower tail.
p: Power or order parameter.
w: Optional portfolio weights.

Constructors

julia

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

julia

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

Computes the PNVaR of a portfolio returns vector x.

Arguments

x::VecNum: Portfolio returns vector.

Examples

julia

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

source

PortfolioOptimisers.factory Function

julia

factory(
    r::PowerNormValueatRisk,
    pr::AbstractPriorResult;
    ...
) -> PowerNormValueatRisk
factory(
    r::PowerNormValueatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> PowerNormValueatRisk

Create an instance of PowerNormValueatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

source

PortfolioOptimisers.factory Function

julia

factory(
    r::PowerNormValueatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> PowerNormValueatRisk
factory(
    r::PowerNormValueatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult};
    kwargs...
) -> PowerNormValueatRisk

Create an instance of PowerNormValueatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

source

PortfolioOptimisers.PowerNormValueatRiskRange Type

julia

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

\begin{aligned} {PNVaRRange}_{α, p_{a}, β, p_{b}} (x) & = {PNVaR}_{α, p_{a}} (x) + {PNVaR}_{β, p_{b}} (- x) . \end{aligned}

Where:

${PNVaRRange}_{α, p_{a}, β, p_{b}} (x)$ : Power Norm VaR range.
$x$ : Portfolio returns vector $T \times 1$ .
${PNVaR}_{α, p_{a}} (x)$ : Lower-tail PNVaR with parameters $(α, p_{a})$ .
${PNVaR}_{β, p_{b}} (- x)$ : Upper-tail PNVaR with parameters $(β, p_{b})$ .

Fields

settings: Risk measure settings.
slv: Solver or vector of solvers.
alpha: Quantile level for the lower tail.
beta: Quantile level for the upper tail.
pa: Power norm parameter for the lower tail.
pb: Power norm parameter for the upper tail.
w: Optional portfolio weights.

Constructors

julia

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

julia

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

Computes the PNVaR Range of a portfolio returns vector x.

Arguments

x::VecNum: Portfolio returns vector.

Examples

julia

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

source

PortfolioOptimisers.factory Method

julia

factory(
    r::PowerNormValueatRiskRange,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> PowerNormValueatRiskRange

Create an instance of PowerNormValueatRiskRange by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

source

PortfolioOptimisers.PowerNormDrawdownatRisk Type

julia

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:

\begin{aligned} c_{t} & = \sum_{s = 1}^{t} x_{s}, \\ d_{t} & = c_{t} - max_{0 \leq s \leq t} c_{s} \leq 0 . \end{aligned}

Where:

$x$ : Portfolio returns vector $T \times 1$ .
$c_{t}$ : Cumulative simple portfolio return at period $t$ .
$d_{t} \leq 0$ : Absolute drawdown at period $t$ .

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

\begin{aligned} {PNDDaR}_{α, p} (x) & = {PNVaR}_{α, p} (d (x)) . \end{aligned}

Where:

${PNDDaR}_{α, p} (x)$ : Power Norm Drawdown-at-Risk.
$x$ : Portfolio returns vector $T \times 1$ .
$α$ : Significance level (left tail probability), $α \in (0, 1)$ .
$p \geq 1$ : Power parameter.
$d (x)$ : Absolute drawdown series.

Fields

settings: Risk measure settings.
slv: Solver or vector of solvers.
alpha: Quantile level for the lower tail.
p: Power or order parameter.
w: Optional portfolio weights.

Constructors

julia

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

julia

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

Computes the PNDDaR of a portfolio returns vector x.

Arguments

x::VecNum: Portfolio returns vector.

Examples

julia

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

source

PortfolioOptimisers.factory Function

julia

factory(
    r::PowerNormDrawdownatRisk,
    pr::AbstractPriorResult;
    ...
) -> PowerNormDrawdownatRisk
factory(
    r::PowerNormDrawdownatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> PowerNormDrawdownatRisk

Create an instance of PowerNormDrawdownatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

source

PortfolioOptimisers.factory Function

julia

factory(
    r::PowerNormDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> Union{PowerNormDrawdownatRisk{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, <:AbstractVector{var"#s869"}, <:Number, <:Number} where {__T_scale, __T_ub, __T_rke, var"#s869"<:Solver}, PowerNormDrawdownatRisk{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, Solver{__T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}, <:Number, <:Number} where {__T_scale, __T_ub, __T_rke, __T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}}
factory(
    r::PowerNormDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult};
    kwargs...
) -> PowerNormDrawdownatRisk

Create an instance of PowerNormDrawdownatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

source

PortfolioOptimisers.RelativePowerNormDrawdownatRisk Type

julia

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:

\begin{aligned} C_{t} & = \prod_{s = 1}^{t} (1 + x_{s}), \\ r d_{t} & = \frac{C_{t}}{max_{0 \leq s \leq t} C_{s}} - 1 \leq 0 . \end{aligned}

Where:

$x$ : Portfolio returns vector $T \times 1$ .
$C_{t}$ : Compound wealth process at period $t$ .
$r d_{t} \leq 0$ : Relative drawdown at period $t$ .

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

\begin{aligned} {RPNDDaR}_{α, p} (x) & = {PNVaR}_{α, p} (r d (x)) . \end{aligned}

Where:

${RPNDDaR}_{α, p} (x)$ : Relative Power Norm Drawdown-at-Risk.
$x$ : Portfolio returns vector $T \times 1$ .
$α$ : Significance level (left tail probability), $α \in (0, 1)$ .
$p \geq 1$ : Power parameter.
$r d (x)$ : Relative drawdown series.

Fields

settings: Risk measure settings.
slv: Solver or vector of solvers.
alpha: Quantile level for the lower tail.
p: Power or order parameter.
w: Optional portfolio weights.

Constructors

julia

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

julia

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

Computes the Relative PNDDaR of a portfolio returns vector x.

Arguments

x::VecNum: Portfolio returns vector.

Examples

julia

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

Related

source

PortfolioOptimisers.factory Function

julia

factory(
    r::RelativePowerNormDrawdownatRisk,
    pr::AbstractPriorResult;
    ...
) -> RelativePowerNormDrawdownatRisk
factory(
    r::RelativePowerNormDrawdownatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> RelativePowerNormDrawdownatRisk

Create an instance of RelativePowerNormDrawdownatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

source

PortfolioOptimisers.factory Function

julia

factory(
    r::RelativePowerNormDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> Union{RelativePowerNormDrawdownatRisk{HierarchicalRiskMeasureSettings{__T_scale}, <:AbstractVector{var"#s869"}, <:Number, <:Number} where {__T_scale, var"#s869"<:Solver}, RelativePowerNormDrawdownatRisk{HierarchicalRiskMeasureSettings{__T_scale}, Solver{__T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}, <:Number, <:Number} where {__T_scale, __T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}}
factory(
    r::RelativePowerNormDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult};
    kwargs...
) -> RelativePowerNormDrawdownatRisk

Create an instance of RelativePowerNormDrawdownatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

source

PortfolioOptimisers.PRM Function

julia

PRM(
    x::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}}
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}},
    alpha::Number
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    slv::Union{Solver, AbstractVector{<:Solver}},
    alpha::Number,
    p::Number
) -> Union{Float64, Vector{Float64}}
PRM(
    x::AbstractVector{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<: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

source

Power Norm X at Risk ​

Power Norm X at Risk