Relativistic XatRisk Constraints

PortfolioOptimisers.set_risk_constraints! Method

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

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

Each overload uses power cone constraints (PowerCone) to encode the Tsallis entropy-based risk measure parameterised by kappa. Auxiliary variables t, z, omega, psi, theta, and epsilon are introduced. The range variant encodes both a lower-tail and upper-tail relativistic expression.

Mathematical definition

Relativistic Value-at-Risk (Damian et al. 2023):

$$\begin{array}{rl}{\mathrm{RLVaR}}_{\alpha ,\kappa }(\mathit{w})& =t+c_{\kappa }(\alpha )z+\sum _{t=1}^T({\psi }_t+{\theta }_t),\\ c_{\kappa }(\alpha )& =\frac{(\alpha T{)}^{\kappa }-(\alpha T{)}^{-\kappa }}{2\kappa }.\end{array}$$

Where:

• ${\mathrm{RLVaR}}_{\alpha ,\kappa }(\mathit{w})$: Relativistic Value-at-Risk.

• $t$, $z$, ${\psi }_t$, ${\theta }_t$: Dual variables for the power cone programme.

• $c_{\kappa }(\alpha )$: Relativistic scaling coefficient.

• $\alpha$: Significance level (left tail probability), $\alpha \in (0,1)$.

• $\kappa$: Relativistic parameter.

encoded via power cones ${\mathcal{K}}_{1/(1+\kappa )}$ and ${\mathcal{K}}_{1/(1-\kappa )}$.

Arguments

• model::JuMP.Model: The JuMP optimisation model.

• i: Constraint index for unique variable and constraint naming.

• r: Risk measure instance with fields alpha and kappa.

• opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.

• pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

• nothing.

Related

• set_drawdown_constraints!

• set_risk_bounds_and_expression!

source

PortfolioOptimisers.set_risk_constraints! Method

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

Add JuMP risk constraints for RelativisticValueatRiskRange (RLVaR range) to model.

Uses power cone constraints for both lower-tail and upper-tail Tsallis entropy-based risk measures parameterised by kappa, then 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::RelativisticValueatRiskRange: The RLVaR range risk measure.

• opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.

• pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

• nothing.

Related

• RelativisticValueatRiskRange

• set_risk_constraints!

source

PortfolioOptimisers.set_risk_constraints! Method

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

Add JuMP risk constraints for RelativisticDrawdownatRisk (RLDaR) to model.

Uses power cone constraints applied to the drawdown series to encode the relativistic drawdown-at-risk parameterised by kappa at confidence level r.alpha.

Arguments

• model::JuMP.Model: The JuMP optimisation model.

• i: Constraint index for unique variable and constraint naming.

• r::RelativisticDrawdownatRisk: The RLDaR risk measure.

• opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.

• pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

• nothing.

Related

• RelativisticDrawdownatRisk

• set_drawdown_constraints!

• set_risk_constraints!

source