XatRisk Constraints

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::ValueatRisk{<:Any, <:Any, <:Any, <:MIPValueatRisk},
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> VariableRef

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

The MIP overloads introduce binary variables z_var and add big-M constraints to encode the empirical quantile. The distribution overloads use closed-form z-scores computed by compute_value_at_risk_z / compute_value_at_risk_cz and add an SOC constraint. The DrawdownatRisk overload applies the MIP approach to the drawdown series.

Mathematical definition

Empirical (MIP) VaR:

\begin{aligned} z_{t} & \in {0, 1}, \sum_{t} z_{t} \leq α T, VaR \geq - {\hat{r}}_{t} - b z_{t} \forall t . \end{aligned}

Where:

$z_{t} \in {0, 1}$ : Binary indicator for tail losses.
$α$ : Significance level (left tail probability), $α \in (0, 1)$ .
$T$ : Number of observations.
$VaR$ : Value-at-Risk variable.
${\hat{r}}_{t}$ : Portfolio return at time $t$ .
$b$ : Big-M constant.

Parametric VaR (Normal/t/Laplace):

\begin{aligned} {VaR}_{α} (w) & = - μ^{⊺} w + z_{α} ∥ G w ∥_{2} . \end{aligned}

Where:

${VaR}_{α} (w)$ : Parametric Value-at-Risk.
$μ$ : Expected returns vector.
$w$ : Portfolio weights vector $N \times 1$ .
$z_{α}$ : Distribution quantile at level $α$ .
$G$ : Cholesky factor of the covariance matrix.

where $z_{α}$ is the distribution quantile at level $α$ and $G$ is the Cholesky factor of the covariance.

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: Prior result.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::ValueatRiskRange{<:Any, <:Any, <:Any, <:Any, <:MIPValueatRisk},
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> AffExpr

Add JuMP risk constraints for ValueatRiskRange using a MIP (big-M) formulation to model.

Introduces binary variables and big-M constraints to encode both the lower-tail and upper-tail empirical quantiles. The range risk expression is the difference between the two VaR expressions.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::ValueatRiskRange{<:Any, <:Any, <:Any, <:Any, <:MIPValueatRisk}: The VaR range risk measure with MIP formulation.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr: Prior result.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::ValueatRisk{<:Any, <:Any, <:Any, <:DistributionValueatRisk},
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> Any

Add JuMP risk constraints for ValueatRisk using a parametric distribution formulation to model.

Uses the closed-form z-score from compute_value_at_risk_z and adds a second-order cone constraint to bound the portfolio standard deviation. The VaR expression is -mu'w + z * g_var.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::ValueatRisk{<:Any, <:Any, <:Any, <:DistributionValueatRisk}: The VaR risk measure with distribution-based formulation.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr: Prior result.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

set_risk_constraints!(
    model::Model,
    i,
    r::ValueatRiskRange{<:Any, <:Any, <:Any, <:Any, <:DistributionValueatRisk},
    opt::RiskJuMPOptimisationEstimator,
    pr::AbstractPriorResult,
    args...;
    prefix,
    kwargs...
) -> Any

Add JuMP risk constraints for ValueatRiskRange using a parametric distribution formulation to model.

Uses closed-form z-scores from compute_value_at_risk_z and compute_value_at_risk_cz and adds a second-order cone constraint. The range risk expression is the difference between the lower-tail and upper-tail VaR expressions.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::ValueatRiskRange{<:Any, <:Any, <:Any, <:Any, <:DistributionValueatRisk}: The VaR range risk measure with distribution-based formulation.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr: Prior result.

Returns

nothing.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

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

Add JuMP risk constraints for DrawdownatRisk to model.

Introduces binary variables and big-M constraints applied to the drawdown series to encode the empirical drawdown quantile at confidence level r.alpha.

Arguments

model::JuMP.Model: The JuMP optimisation model.
i: Constraint index for unique variable and constraint naming.
r::DrawdownatRisk: The drawdown-at-risk risk measure.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr: Prior result.

Returns

nothing.

Related

source

PortfolioOptimisers.compute_value_at_risk_z Function

julia

compute_value_at_risk_z(
    dist::Distributions.Normal,
    alpha::Number
) -> Any

Compute the lower-tail z-score for a parametric VaR at significance level alpha.

Returns the complementary quantile for Normal and scaled Student-t distributions, and the closed-form expression for the Laplace distribution.

Arguments

dist: Distribution instance (Normal, TDist, or Laplace).
alpha::Number: Significance level.

Returns

z::Number: Lower-tail z-score for the parametric VaR.

Related

source

PortfolioOptimisers.compute_value_at_risk_cz Function

julia

compute_value_at_risk_cz(
    dist::Distributions.Normal,
    alpha::Number
) -> Any

Compute the upper-tail z-score for a parametric VaR at significance level alpha.

Used for the high (upper) bound in VaR range constraints. Returns the lower quantile for Normal and scaled Student-t distributions, and the closed-form expression for Laplace.

Arguments

dist: Distribution instance (Normal, TDist, or Laplace).
alpha::Number: Significance level.

Returns

z::Number: Upper-tail z-score for the parametric VaR.

Related

source

XatRisk Constraints ​

XatRisk Constraints