Conditional X at Risk

PortfolioOptimisers.ConditionalValueatRisk Type

struct ConditionalValueatRisk{__T_settings, __T_alpha, __T_w} <: RiskMeasure

Represents the Conditional Value-at-Risk (CVaR) risk measure, also known as Expected Shortfall (ES).

ConditionalValueatRisk computes the expected loss given that the loss exceeds the Value-at-Risk at level alpha. It provides a coherent risk measure for tail risk quantification.

Mathematical Definition

Let $\mathit{x}=(x_1,\dots ,x_T{)}^{⊺}$ be the portfolio returns vector. The CVaR (also known as Expected Shortfall) at level $\alpha$ is the expected loss in the worst $\alpha$ fraction of scenarios:

$${\mathrm{CVaR}}_{\alpha }(\mathit{x})=\underset{\nu \in \mathbb{R}}{min}\{-\nu +\frac{1}{\alpha T}\sum _{t=1}^{T}max(-x_t-\nu ,0)\}.$$

Equivalently, it is the expected loss conditional on exceeding the VaR:

$${\mathrm{CVaR}}_{\alpha }(\mathit{x})=-\mathbb{E}[x\mid x\le -{\mathrm{VaR}}_{\alpha }(\mathit{x})].$$

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

ConditionalValueatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    w::Option{<:ObsWeights} = nothing
) -> ConditionalValueatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If w is not nothing: !isempty(w).

Functor

(r::ConditionalValueatRisk)(x::VecNum)

Computes the CVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

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

Related

• RiskMeasure

• RiskMeasureSettings

• ValueatRisk

• DistributionallyRobustConditionalValueatRisk

• ConditionalValueatRiskRange

• ConditionalDrawdownatRisk

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PortfolioOptimisers.DistributionallyRobustConditionalValueatRisk Type

struct DistributionallyRobustConditionalValueatRisk{__T_settings, __T_alpha, __T_l, __T_r, __T_w} <: RiskMeasure

Represents the Distributionally Robust Conditional Value-at-Risk (DR-CVaR) risk measure.

DistributionallyRobustConditionalValueatRisk is a robust variant of CVaR that accounts for distributional uncertainty using Wasserstein ambiguity sets. It provides robustness against model misspecification in the tails of the return distribution.

Mathematical Definition

The DR-CVaR with Wasserstein ambiguity parameter $l$ and radius $r$ is a robust upper bound on CVaR under distributional uncertainty within a Wasserstein ball of radius $r$:

$${\mathrm{DR}\text{-}\mathrm{CVaR}}_{\alpha ,l,r}(\mathit{x})={\mathrm{CVaR}}_{\alpha }(\mathit{x})+l\cdot r.$$

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• l: Wasserstein ambiguity parameter (radius scale factor).

• r: Wasserstein radius parameter.

• w: Optional observation weights.

Constructors

DistributionallyRobustConditionalValueatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    l::Number = 1.0,
    r::Number = 0.02,
    w::Option{<:ObsWeights} = nothing
) -> DistributionallyRobustConditionalValueatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• l > 0.

• r > 0.

• If w is not nothing: !isempty(w).

Functor

(r::DistributionallyRobustConditionalValueatRisk)(x::VecNum)

Computes the DR-CVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> DistributionallyRobustConditionalValueatRisk()
DistributionallyRobustConditionalValueatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     alpha ┼ Float64: 0.05
         l ┼ Float64: 1.0
         r ┼ Float64: 0.02
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• ConditionalValueatRisk

• DistributionallyRobustConditionalValueatRiskRange

• DistributionallyRobustConditionalDrawdownatRisk

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PortfolioOptimisers.RMCVaR Type

const RMCVaR{T} = Union{...}

Parameterised union of ConditionalValueatRisk and DistributionallyRobustConditionalValueatRisk sharing the same observation-weight type parameter T.

Used for unified dispatch on CVaR computation methods.

Related

• ConditionalValueatRisk

• DistributionallyRobustConditionalValueatRisk

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PortfolioOptimisers.ConditionalValueatRiskRange Type

struct ConditionalValueatRiskRange{__T_settings, __T_alpha, __T_beta, __T_w} <: RiskMeasure

Represents the Conditional Value-at-Risk Range (CVaR Range) risk measure.

ConditionalValueatRiskRange computes the difference between the lower-tail CVaR (at level alpha) and the upper-tail CVaR (at level beta), measuring the spread between downside and upside expected tail risks.

Mathematical Definition

$${\mathrm{CVaRRange}}_{\alpha ,\beta }(\mathit{x})={\mathrm{CVaR}}_{\alpha }(\mathit{x})-{\mathrm{CVaR}}_{\beta }(-\mathit{x}),$$

where ${\mathrm{CVaR}}_{\alpha }(\mathit{x})$ captures the lower-tail expected shortfall and ${\mathrm{CVaR}}_{\beta }(-\mathit{x})$ captures the upper-tail expected surplus (gain).

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• beta: Significance level for the upper tail.

• w: Optional observation weights.

Constructors

ConditionalValueatRiskRange(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    beta::Number = 0.05,
    w::Option{<:ObsWeights} = nothing
) -> ConditionalValueatRiskRange

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < beta < 1.

• If w is not nothing: !isempty(w).

Functor

(r::ConditionalValueatRiskRange)(x::VecNum)

Computes the CVaR Range of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> ConditionalValueatRiskRange()
ConditionalValueatRiskRange
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     alpha ┼ Float64: 0.05
      beta ┼ Float64: 0.05
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• ConditionalValueatRisk

• ValueatRiskRange

• DistributionallyRobustConditionalValueatRiskRange

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PortfolioOptimisers.DistributionallyRobustConditionalValueatRiskRange Type

struct DistributionallyRobustConditionalValueatRiskRange{__T_settings, __T_alpha, __T_l_a, __T_r_a, __T_beta, __T_l_b, __T_r_b, __T_w} <: RiskMeasure

Represents the Distributionally Robust Conditional Value-at-Risk Range (DR-CVaR Range) risk measure.

DistributionallyRobustConditionalValueatRiskRange computes the difference between the lower-tail DR-CVaR (at level alpha) and the upper-tail DR-CVaR (at level beta), with separate Wasserstein ambiguity parameters for each tail.

Mathematical Definition

$$\mathrm{DR}\text{-}\mathrm{CVaRRange}(\mathit{x})={\mathrm{DR}\text{-}\mathrm{CVaR}}_{\alpha ,l_a,r_a}(\mathit{x})-{\mathrm{DR}\text{-}\mathrm{CVaR}}_{\beta ,l_b,r_b}(-\mathit{x}),$$

where each DR-CVaR uses its own Wasserstein ambiguity parameters $(l_a,r_a)$ and $(l_b,r_b)$ respectively.

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• l_a: Wasserstein ambiguity scale factor for the lower tail.

• r_a: Wasserstein radius for the lower tail.

• beta: Significance level for the upper tail.

• l_b: Wasserstein ambiguity scale factor for the upper tail.

• r_b: Wasserstein radius for the upper tail.

• w: Optional observation weights.

Constructors

DistributionallyRobustConditionalValueatRiskRange(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    l_a::Number = 1.0,
    r_a::Number = 0.02,
    beta::Number = 0.05,
    l_b::Number = 1.0,
    r_b::Number = 0.02,
    w::Option{<:ObsWeights} = nothing
) -> DistributionallyRobustConditionalValueatRiskRange

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < beta < 1.

• l_a > 0, r_a > 0, l_b > 0, r_b > 0.

• If w is not nothing: !isempty(w).

Functor

(r::DistributionallyRobustConditionalValueatRiskRange)(x::VecNum)

Computes the DR-CVaR Range of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> DistributionallyRobustConditionalValueatRiskRange()
DistributionallyRobustConditionalValueatRiskRange
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     alpha ┼ Float64: 0.05
       l_a ┼ Float64: 1.0
       r_a ┼ Float64: 0.02
      beta ┼ Float64: 0.05
       l_b ┼ Float64: 1.0
       r_b ┼ Float64: 0.02
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• ConditionalValueatRiskRange

• DistributionallyRobustConditionalValueatRisk

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PortfolioOptimisers.RMCVaRRg Type

const RMCVaRRg{T} = Union{...}

Parameterised union of ConditionalValueatRiskRange and DistributionallyRobustConditionalValueatRiskRange sharing the same observation-weight type parameter T.

Used for unified dispatch on CVaR-range computation methods.

Related

• ConditionalValueatRiskRange

• DistributionallyRobustConditionalValueatRiskRange

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PortfolioOptimisers.ConditionalDrawdownatRisk Type

struct ConditionalDrawdownatRisk{__T_settings, __T_alpha, __T_w} <: RiskMeasure

Represents the Conditional Drawdown-at-Risk (CDaR) risk measure, also known as Expected Maximum Drawdown.

ConditionalDrawdownatRisk computes the expected drawdown given that the drawdown exceeds the Drawdown-at-Risk at level alpha. It provides a coherent risk measure for drawdown tail risk.

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 CDaR is the CVaR of the drawdown series $\mathit{d}=(d_1,\dots ,d_T{)}^{⊺}$:

$${\mathrm{CDaR}}_{\alpha }(\mathit{x})=\underset{\nu \in \mathbb{R}}{min}\{-\nu +\frac{1}{\alpha T}\sum _{t=1}^{T}max(-d_t-\nu ,0)\}.$$

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

ConditionalDrawdownatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    w::Option{<:ObsWeights} = nothing
) -> ConditionalDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If w is not nothing: !isempty(w).

Functor

(r::ConditionalDrawdownatRisk)(x::VecNum)

Computes the CDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

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

Related

• RiskMeasure

• RiskMeasureSettings

• DrawdownatRisk

• DistributionallyRobustConditionalDrawdownatRisk

• RelativeConditionalDrawdownatRisk

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PortfolioOptimisers.DistributionallyRobustConditionalDrawdownatRisk Type

struct DistributionallyRobustConditionalDrawdownatRisk{__T_settings, __T_alpha, __T_l, __T_r, __T_w} <: RiskMeasure

Represents the Distributionally Robust Conditional Drawdown-at-Risk (DR-CDaR) risk measure.

DistributionallyRobustConditionalDrawdownatRisk is a robust variant of CDaR that accounts for distributional uncertainty using Wasserstein ambiguity sets, applied to drawdown sequences.

Mathematical Definition

$${\mathrm{DR}\text{-}\mathrm{CDaR}}_{\alpha ,l,r}(\mathit{x})={\mathrm{CDaR}}_{\alpha }(\mathit{x})+l\cdot r.$$

Fields

• settings: Risk measure configuration.

• alpha: Significance level for the lower tail.

• l: Wasserstein ambiguity scale factor.

• r: Wasserstein radius parameter.

• w: Optional observation weights.

Constructors

DistributionallyRobustConditionalDrawdownatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    alpha::Number = 0.05,
    l::Number = 1.0,
    r::Number = 0.02,
    w::Option{<:ObsWeights} = nothing
) -> DistributionallyRobustConditionalDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• l > 0.

• r > 0.

• If w is not nothing: !isempty(w).

Functor

(r::DistributionallyRobustConditionalDrawdownatRisk)(x::VecNum)

Computes the DR-CDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> DistributionallyRobustConditionalDrawdownatRisk()
DistributionallyRobustConditionalDrawdownatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
     alpha ┼ Float64: 0.05
         l ┼ Float64: 1.0
         r ┼ Float64: 0.02
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• ConditionalDrawdownatRisk

• DistributionallyRobustConditionalValueatRisk

source

PortfolioOptimisers.RMCDaR Type

const RMCDaR{T} = Union{...}

Parameterised union of ConditionalDrawdownatRisk and DistributionallyRobustConditionalDrawdownatRisk sharing the same observation-weight type parameter T.

Used for unified dispatch on CDaR computation methods.

Related

• ConditionalDrawdownatRisk

• DistributionallyRobustConditionalDrawdownatRisk

source

PortfolioOptimisers.RelativeConditionalDrawdownatRisk Type

struct RelativeConditionalDrawdownatRisk{__T_settings, __T_alpha, __T_w} <: HierarchicalRiskMeasure

Represents the Relative Conditional Drawdown-at-Risk risk measure for hierarchical optimisation.

RelativeConditionalDrawdownatRisk computes the expected relative (compounded) drawdown given that the drawdown exceeds the Relative Drawdown-at-Risk at level alpha.

Mathematical Definition

Define the compounded wealth process and 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 CDaR is the CVaR of the relative drawdown series $\mathit{r}\mathit{d}$:

$${\mathrm{RCDaR}}_{\alpha }(\mathit{x})=\underset{\nu \in \mathbb{R}}{min}\{-\nu +\frac{1}{\alpha T}\sum _{t=1}^{T}max(-r{d}_t-\nu ,0)\}.$$

Fields

• settings: Hierarchical risk measure configuration.

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

RelativeConditionalDrawdownatRisk(;
    settings::HierarchicalRiskMeasureSettings = HierarchicalRiskMeasureSettings(),
    alpha::Number = 0.05,
    w::Option{<:ObsWeights} = nothing
) -> RelativeConditionalDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If w is not nothing: !isempty(w).

Functor

(r::RelativeConditionalDrawdownatRisk)(x::VecNum)

Computes the Relative CDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativeConditionalDrawdownatRisk()
RelativeConditionalDrawdownatRisk
  settings ┼ HierarchicalRiskMeasureSettings
           │   scale ┴ Float64: 1.0
     alpha ┼ Float64: 0.05
         w ┴ nothing

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• ConditionalDrawdownatRisk

• RelativeDrawdownatRisk

source