Stochastic Dominance API

optimize_max_return_SD(ξ, ξ_0,SDorder;p_ξ, p_ξ_0,plot=false,verbose=false, ε=1e-5,  max_ipot=50, max_iter=50, n_particles=50)

Solve a portfolio optimization problem under stochastic dominance constraints to maximize the expected return.

Arguments

• ξ::AbstractMatrix{<:Number}: Portfolio return scenarios.
• ξ_0::AbstractVector{<:Number}: Benchmark return scenarios.
• SDorder::Number}: Order of stochastic dominance.
• p_ξ::AbstractVector{<:Number}: Probability distribution over portfolio scenarios.
• p_ξ_0::AbstractVector{<:Number}: Probability distribution over benchmark scenarios.
• plot::Bool (optional, default = false): Plot the optimal asset allocation.
• verbose::Bool (optional, default = false): Print convergence details.
• ε::Float64 (optional, default = 1e-7): Convergence tolerance.
• max_ipot::Integer (optional, default = 50): Maximum number of attempts to push the objective towards maximization. (Interior point optimization)
• max_iter::Integer (optional, default = 50): Maximum iterations for optimization.
• n_particles::Integer (optional, default = 200): Number of particles for PSO.

Returns

• A tuple (x_opt, t_opt), where:
  • x_opt: Optimized portfolio weights.
  • t_opt: Auxiliary optimization variable.

Examples

# Define portfolio return scenarios (rows: assets, columns: scenarios)
ξ = [0.02  0.05 -0.01;  # Asset 1 returns under different scenarios
     0.03  0.06  0.02]   # Asset 2 returns under different scenarios

# Define benchmark return scenarios (column vector)
ξ_0 = [0.01, 0.04, 0.00]  # Benchmark returns in the same scenarios

# Define probability distributions for portfolio and benchmark scenarios
p_ξ = [0.3, 0.4, 0.3]     # Probabilities for each scenario
p_ξ_0 = [0.4, 0.4, 0.2]   # Probabilities for benchmark scenarios

# Set the order of stochastic dominance
SDorder = 2.0  # Second-order stochastic dominance


x_opt, t_opt = optimize_max_return_SD(ξ, ξ_0,SDorder;p_ξ, p_ξ_0,ε=1e-6,verbose=true)

Solves the portfolio optimization problem under stochastic dominance constraints.

optimize_min_riskreturn_SD(ξ, ξ_0,SDorder; β=0.5,r=2.0,p_ξ, p_ξ_0, ε=1e-5, plot=false, verbose=false, max_ipot=50, max_iter=50, n_particles=50)

Solve a portfolio optimization problem under stochastic dominance risk constraints

Arguments

• ξ::AbstractMatrix{<:Number}: Portfolio return scenarios.
• ξ_0::AbstractVector{<:Number}: Benchmark return scenarios.
• SDorder::S: Order of stochastic dominance.
• r::Number (optional, default = 2.0): norm of risk measure.
• β::Number (optional, default = 0.5): Confidence level for risk measure.
• p_ξ::AbstractVector{<:Number}: Probability distribution over portfolio scenarios.
• p_ξ_0::AbstractVector{<:Number}: Probability distribution over benchmark scenarios.
• ε::Float64 (optional, default = 1e-7): Convergence tolerance.
• plot::Bool (optional, default = false): Plot the optimal asset allocation.
• verbose::Bool (optional, default = false): Print convergence details.
• max_ipot::Integer (optional, default = 50): Maximum number of attempts to push the objective towards maximization. (Interior point optimization)
• max_iter::Integer (optional, default = 50): Maximum iterations for optimization.
• n_particles::Integer (optional, default = 200): Number of particles for PSO.

Returns

• A tuple (x_opt, q_opt, t_opt), where:
  • x_opt: Optimized portfolio weights.
  • q_opt: Optimized risk threshold parameter.
  • t_opt: Auxiliary optimization variable.

Examples

# Define portfolio return scenarios (rows: assets, columns: scenarios)
ξ = [0.02  0.05 -0.01;  # Asset 1 returns under different scenarios
     0.03  0.06  0.02]   # Asset 2 returns under different scenarios

# Define benchmark return scenarios (column vector)
ξ_0 = [0.01, 0.04, 0.00]  # Benchmark returns in the same scenarios

# Define probability distributions for portfolio and benchmark scenarios
p_ξ = [0.3, 0.4, 0.3]     # Probabilities for each scenario
p_ξ_0 = [0.4, 0.4, 0.2]   # Probabilities for benchmark scenarios

# Set the order of stochastic dominance
SDorder = 2  # Second-order stochastic dominance

# Set the risk parameter
β=0.5

# Set the norm of the risk measure
r=2.0

x_opt, q_opt,t_opt= optimize_min_riskreturn_SD(ξ, ξ_0,SDorder;p_ξ, p_ξ_0,β,r,ε=1e-6,verbose=true,plot=true)

Solves the portfolio optimization problem under stochastic dominance constraints.

verify_dominance(Y, X,SDorder; p_X, p_Y,ε=1e-5,verbose=false)

Verify if Y stochastically dominates X at a given stochastic oder

Arguments

• Y::AbstractVector{<:Number}: Portfolio outcomes (returns).
• X::AbstractVector{<:Number}: Benchmark outcomes.
• p_X::AbstractVector{<:Number}: Probabilities associated with X.
• p_Y::AbstractVector{<:Number}: Probabilities associated with Y.
• SDorder::Integer: Order of stochastic dominance (SD).

Keyword Arguments

• ε::Float64=1e-5: Convergence tolerance for the dominance check.
• verbose::Bool=false: If true, print detailed accuracy or residuals.

Returns

• A message indicating whether Y stochastically dominates X at order SDorder.
• true if Y stochastically dominates X at the given order.
• false otherwise.

Examples

Y = [3, 5, 7]
X = [2, 4, 6]
p_Y = [0.3, 0.4, 0.3]
p_X = [0.2, 0.5, 0.3]
SDorder = 2

verify_dominance(Y, X, SDorder; p_X, p_Y, ε=1e-8,verbose=true)

optimize_min_risk(ξ_0::AbstractVector{T}, p_ξ_0::AbstractVector{T}; 
    β::T=0.5, r::T=2, ε::Float64=1e-7, max_iter::Integer=50, 
    n_particles::Integer=100, max_ipot::Integer=50) where {T<:Number}

Performs risk minimization using Particle Swarm Optimization (PSO) to find the optimal risk threshold parameter q.

Arguments

• ξ_0::AbstractVector{<:Number}: Benchmark return scenarios.
• p_ξ_0::AbstractVector{<:Number}: Probability distribution over benchmark scenarios.

Keyword Arguments

• β::Number = 0.5: Confidence level for risk measure.
• r::Number = 2.0: Norm of the risk measure.
• ε::Float64 = 1e-7: Convergence tolerance.
• max_iter::Integer = 50: Maximum number of iterations for optimization.
• n_particles::Integer = 100: Number of particles for PSO.
• max_ipot::Integer = 50: Maximum number of attempts for interior point optimization.

Returns

• q_opt_best::Float64: The optimized risk threshold parameter.

Example

# Define benchmark return scenarios
ξ_0 = [0.01, 0.04, 0.00]  # Benchmark returns in different scenarios

# Define probability distribution for benchmark scenarios
p_ξ_0 = [0.4, 0.4, 0.2]   # Probabilities for benchmark scenarios

# Set risk parameters
β = 0.5
r = 2.0

# Solve the risk minimization problem
q_opt_best = optimize_min_risk(ξ_0, p_ξ_0; β=β, r=r, ε=1e-6, max_iter=100)

g_bar(x, ξ, ξ_0, p, p_ξ, p_ξ_0)

Compute the verification of the higher-order stochastic dominance constraint by evaluating g_p over sorted benchmark values.

Arguments

• x::AbstractVector{<:Number}: Portfolio weights (simplex).
• ξ::AbstractVector{<:Number}: Portfolio returns.
• ξ_0::AbstractVector{<:Number}: Benchmark returns.
• p::Number: The order-1 of stochastic dominance.
• p_ξ::AbstractVector{<:Number}: Probability of portfolio ξ.
• p_ξ_0::AbstractVector{<:Number}: Probability of benchmark ξ_0.

Returns

• A numeric value computed as max(max_value, 0.0), where:
  • max_value is the maximum of g_p(t, x, ξ, ξ_0, p, p_ξ, p_ξ_0) over sorted ξ_0.
  • If max_value is negative, the function returns 0.0.
  • If max_value is positive, the selected portfolio or the portfolio with assigned weights does not satisfy the dominance condition."

Examples

g_bar([0.7, 0.3], [2  3; 5 6], [2,3,4], 3, [0.5,0.5], [0.2,0.3,0.5])  

boundarymoments(x, ξ, pξ, pξ0, ξ_0, p)

Compute the boundary moments for higher-order stochastic dominance constraints by comparing the moments of the portfolio and benchmark distributions.

Arguments

• x::AbstractVector{<:Number}: Portfolio weights (simplex constraint).
• ξ::AbstractMatrix{<:Number}: Portfolio returns.
• p_ξ::AbstractVector{<:Number}: Probability distribution of the portfolio returns ξ.
• p_ξ_0::AbstractVector{<:Number}: Probability distribution of the benchmark returns ξ_0.
• ξ_0::AbstractVector{<:Number}: Benchmark returns.
• p::Real: The highest moment order. If p is an integer, boundary_moments_integer is used. If p is non-integer, boundary_moments_noninteger is used.

Returns

• A vector containing computed moment constraints:
• For k = 1: Compares the mean of the portfolio and benchmark.
• For k ≥ 2: Computes and compares the central moments.
• If the portfolio moment does not exceed the benchmark moment, the value is set to zero.
• Otherwise, the result stores the norm of the difference.

Methodology

• Integer Case (p is an integer):
• Computes moments from order 1 order p.
• Uses the function boundary_moments_integer.
• Non-Integer Case (p is a non-integer):
• Computes moments from order (0,1) order p
• Uses the function boundary_moments_noninteger.

Examples

# Example 1: Integer order moments
boundary_moments([0.7, 0.3], [2 3; 5 6], [0.5, 0.5], [0.2, 0.3, 0.5], [2, 3, 4], 3)

# Example 2: Non-integer order moments
boundary_moments([0.7, 0.3], [2 3; 5 6], [0.5, 0.5], [0.2, 0.3, 0.5], [2, 3, 4], 2.7)

expected_portfolio_return(x, ξ, p_ξ)

Compute the expected return of a portfolio given asset returns and probabilities.

Arguments

• x::AbstractVector{<:Number}: Portfolio weights (simplex).
• ξ::AbstractMatrix{<:Number}: Matrix of asset returns (each column represents a scenario).
• p_ξ::AbstractVector{<:Number}: Probability distribution over the scenarios.

Returns

• A numeric value representing the expected portfolio return

Examples

expected_portfolio_return([0.7, 0.3], [2  3; 5 6], [0.5,0.5])  
# Computes the expected portfolio return.

riskfunction_asset_allocation(x, q, ξ, r, p_ξ, β)

Calculates the risk-adjusted portfolio return based on quantile-based risk measures.

Arguments

• x: Portfolio weights (decision variables to optimize).
• q: Quantile threshold for the risk measure.
• ξ: Portfolio scenarios (matrix of asset returns).
• r: Norm parameter.
• p_ξ: Probability weights for the scenarios in ξ.
• β: Risk aversion parameter (closer to 1 indicates higher aversion).

Returns

• The risk-adjusted return, which accounts for portfolio performance and risk under extreme loss scenarios.