Entropic X at Risk

PortfolioOptimisers.EntropicValueatRisk Type

struct EntropicValueatRisk{__T_settings, __T_slv, __T_alpha, __T_w} <: RiskMeasure

Represents the Entropic Value-at-Risk (EVaR) risk measure.

EntropicValueatRisk is a coherent risk measure based on the Chernoff bound. It is an upper bound for both CVaR and VaR and is computed by solving a conic optimisation problem via an external solver.

Mathematical Definition

The EVaR is defined via the Chernoff bound as the tightest exponential upper bound on VaR and CVaR:

$${\mathrm{EVaR}}_{\alpha }(\mathit{x})=\underset{z>0}{inf}\left\{z\ln \left(\frac{M_L(1/z)}{\alpha }\right)\right\},$$

where $L_t=-x_t$ is the loss and $M_L(u)=\mathbb{E}[e^{uL}]$ is the moment-generating function. Computationally, it is solved via the conic programme:

$${\mathrm{EVaR}}_{\alpha }(\mathit{x})=\underset{t,z\ge 0,\mathit{u}}{min}\{t-z\ln (\alpha T):\sum _{i=1}^T{u}_i\le z,(-x_i-t,z,u_i)\in K_{\exp }\mathrm{\forall }i\},$$

where $K_{\exp }=\{(a,b,c):b{e}^{a/b}\le c,b>0\}$ is the exponential cone.

Fields

• settings: Risk measure configuration.

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

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

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

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::EntropicValueatRisk)(x::VecNum)

Computes the EVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

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

Related

• RiskMeasure

• RiskMeasureSettings

• RelativisticValueatRisk

• EntropicValueatRiskRange

• EntropicDrawdownatRisk

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PortfolioOptimisers.EntropicValueatRiskRange Type

struct EntropicValueatRiskRange{__T_settings, __T_slv, __T_alpha, __T_beta, __T_w} <: RiskMeasure

Represents the Entropic Value-at-Risk Range (EVaR Range) risk measure.

EntropicValueatRiskRange computes the difference between the lower-tail EVaR (at level alpha) and the upper-tail EVaR (at level beta).

Mathematical Definition

$${\mathrm{EVaRRange}}_{\alpha ,\beta }(\mathit{x})={\mathrm{EVaR}}_{\alpha }(\mathit{x})+{\mathrm{EVaR}}_{\beta }(-\mathit{x}),$$

where ${\mathrm{EVaR}}_{\alpha }(\mathit{x})$ captures the lower-tail entropic risk and ${\mathrm{EVaR}}_{\beta }(-\mathit{x})$ captures the upper-tail entropic risk (gain).

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.

• w: Optional observation weights.

Constructors

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

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1, 0 < beta < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Related

• RiskMeasure

• RiskMeasureSettings

• EntropicValueatRisk

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PortfolioOptimisers.EntropicDrawdownatRisk Type

struct EntropicDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_w} <: RiskMeasure

Represents the Entropic Drawdown-at-Risk (EDaR) risk measure.

EntropicDrawdownatRisk applies the Entropic Value-at-Risk framework to the absolute drawdown series of portfolio returns. It is a coherent risk measure providing an upper bound on both the Drawdown-at-Risk and Conditional Drawdown-at-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 EDaR is the EVaR of the drawdown series:

$${\mathrm{EDaR}}_{\alpha }(\mathit{x})={\mathrm{EVaR}}_{\alpha }(\mathit{d}(\mathit{x})).$$

Fields

• settings: Risk measure configuration.

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

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

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

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::EntropicDrawdownatRisk)(x::VecNum)

Computes the EDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

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

Related

• RiskMeasure

• RiskMeasureSettings

• EntropicValueatRisk

• RelativisticDrawdownatRisk

• RelativeEntropicDrawdownatRisk

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PortfolioOptimisers.RelativeEntropicDrawdownatRisk Type

struct RelativeEntropicDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_w} <: HierarchicalRiskMeasure

Represents the Relative Entropic Drawdown-at-Risk (Relative EDaR) risk measure for hierarchical optimisation.

RelativeEntropicDrawdownatRisk applies the Entropic Value-at-Risk framework to the relative (compounded) drawdown series of portfolio returns.

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 EDaR is the EVaR of the relative drawdown series:

$${\mathrm{REDaR}}_{\alpha }(\mathit{x})={\mathrm{EVaR}}_{\alpha }(\mathit{r}\mathit{d}(\mathit{x})).$$

Fields

• settings: Hierarchical risk measure configuration.

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

• alpha: Significance level for the lower tail.

• w: Optional observation weights.

Constructors

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

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativeEntropicDrawdownatRisk)(x::VecNum)

Computes the Relative EDaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

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

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• EntropicDrawdownatRisk

• RelativeRelativisticDrawdownatRisk

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