Entropic XatRisk Constraints

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::EntropicValueatRisk,
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> Any

Add Entropic Value-at-Risk, EVaR range, or Entropic Drawdown-at-Risk constraints to model.

Each overload uses exponential cone constraints (ExponentialCone) to encode the cumulant generating function bound. Scalar variables t, z, and per-observation variables u are introduced. EVaR and EDaR encode the single-tail bound; the range variant encodes both a lower and upper exponential cone.

Mathematical definition

Entropic Value-at-Risk via exponential cone (Ahmadi-Javid 2012):

\begin{align} \mathrm{EVaR}_\alpha(\boldsymbol{w}) &= t - z \ln(\alpha T)\,, \\ \sum_{t&=1}^T u_t \leq z\,. \end{align}

Where:

${EVaR}_{α} (w)$ : Entropic Value-at-Risk.
$t$ , $z$ : Dual variables for the exponential cone programme.
$u_{t}$ : Auxiliary exponential cone variables.
$α$ : Significance level (left tail probability), $α \in (0, 1)$ .
$T$ : Number of observations.

\begin{aligned} (- {\hat{r}}_{t} - t, z, u_{t}) & \in K_{\exp} \forall t . \end{aligned}

Where:

${\hat{r}}_{t}$ : Portfolio return at time $t$ .
$K_{\exp} = {(a, b, c) : b e^{a / b} \leq c, b > 0}$ : Exponential cone.
$u_{t}$ : Auxiliary exponential cone variables.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r: Risk measure instance.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::EntropicValueatRiskRange,
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> Any

Add JuMP risk constraints for EntropicValueatRiskRange (EVaR range) to model.

Introduces two sets of exponential cone variables for the lower-tail and upper-tail EVaR expressions and computes their difference as the range risk.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::EntropicValueatRiskRange: The EVaR range risk measure.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::EntropicDrawdownatRisk,
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> Any

Add JuMP risk constraints for EntropicDrawdownatRisk (EDaR) to model.

Uses exponential cone constraints applied to the drawdown series to encode the entropic drawdown-at-risk at confidence level r.alpha.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::EntropicDrawdownatRisk: The EDaR risk measure.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

nothing.

Related

source

Entropic XatRisk Constraints ​

Entropic XatRisk Constraints