Brownian Distance Variance Constraints

PortfolioOptimisers.set_brownian_distance_variance_constraints! Function

julia

set_brownian_distance_variance_constraints!(
    model::Model,
    ::NormOneConeBrownianDistanceVariance,
    Dt::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    Dx::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}};
    prefix
)

Add auxiliary BDV constraints linking the symmetric distance matrix Dt to portfolio-return differences Dx.

The NormOneConeBrownianDistanceVariance overload adds per-element L1-norm cone constraints [sc * Dt[i,j]; sc * Dx[i,j]] in NormOneCone(2) for each upper-triangular entry. The IneqBrownianDistanceVariance overload adds global non-negativity constraints on Dt - Dx and Dt + Dx.

Mathematical definition

\begin{aligned} D_{t} (i, j) & \geq | {\hat{r}}_{i} - {\hat{r}}_{j} |, \\ BDV (w) & = \frac{1}{T^{2}} \sum_{i, j} D_{t} (i, j)^{2} - {(\frac{1}{T^{2}} \sum_{i, j} D_{t} (i, j))}^{2} . \end{aligned}

Where:

$D_{t} (i, j)$ : Distance auxiliary variable between returns at times $i$ and $j$ .
${\hat{r}}_{i}$ , ${\hat{r}}_{j}$ : Portfolio returns at times $i$ and $j$ .
$BDV (w)$ : Brownian distance variance.
$T$ : Number of observations.

Arguments

model::JuMP.Model: The JuMP optimisation model.
Dt::MatNum: Symmetric JuMP matrix variable for absolute distances.
Dx::MatNum: JuMP expression matrix for portfolio-return pairwise differences.

Returns

nothing.

Related

source

PortfolioOptimisers.set_brownian_distance_risk_constraint! Function

julia

set_brownian_distance_risk_constraint!(
    model::Model,
    ::QuadRiskExpr,
    Dt::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}},
    iT2::Number;
    prefix
) -> Any

Build the Brownian distance variance risk expression.

The QuadRiskExpr overload encodes BDV as a quadratic dot product of Dt plus a squared sum term. The RSOCRiskExpr overload introduces a scalar variable tDt bounded by a rotated second-order cone on vec(Dt), then builds the same expression with tDt in place of the quadratic dot product.

Arguments

model::JuMP.Model: The JuMP optimisation model.
Dt::MatNum: Symmetric distance matrix variable.
iT2::Number: Inverse square of the number of observations (1 / T^2).

Returns

bdvariance_risk: JuMP expression for the Brownian distance variance risk.

Related

source

PortfolioOptimisers.set_risk_constraints! Method

julia

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

Add Brownian distance variance risk constraints to model.

Constructs the pairwise return difference matrix Dx, introduces a symmetric variable matrix Dt, builds the BDV risk expression via set_brownian_distance_risk_constraint!, and links Dt to Dx via set_brownian_distance_variance_constraints!. Returns the existing expression if already present.

Arguments

model::JuMP.Model: The JuMP optimisation model.
r::BrownianDistanceVariance: BDV risk measure instance.
opt::RiskJuMPOptimisationEstimator: Risk-based optimisation estimator.
pr::AbstractPriorResult: Prior result containing the returns matrix X.

Returns

nothing.

Related

source

Brownian Distance Variance Constraints ​

Brownian Distance Variance Constraints