Mixed-Integer CVaR Portfolio Optimization Problem

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This example shows how to solve a CVaR portfolio optimization problem with constraints in the number of selected assets or conditional (semicontinuous) bounds. To solve this problem, you can use a PortfolioCVaR object along with different mixed integer nonlinear programming (MINLP) solvers.

CVaR Portfolio

Load the returns data in CAPMuniverse.mat. Then, create a PortfolioCVaR object with default constraints and a long-only portfolio whose weights sum to 1. For this example, you can define the feasible region of weights $X$ as

$X = {x | \sum_{i = 1}^{n} x_{i} = 1, x_{i} \geq 0} .$

% Load data
load CAPMuniverse.mat

% Create portfolio with default constraints
p = PortfolioCVaR(ProbabilityLevel=0.95);
p = simulateNormalScenariosByData(p,Data(:,1:12),1000,missingdata=true);
p = setDefaultConstraints(p);

Include binary variables for this scenario by setting conditional (semicontinuous) bounds. Conditional bounds are those such that $x_{i} = 0$ or $x_{i} \geq ℓ_{i}$ . In this example, $ℓ_{i} = 0.1$ for all assets.

% Set conditional bounds
condLB = 0.1;
condUB = 0.5;
p = setBounds(p,condLB,condUB,BoundType="conditional");

Use estimateFrontier to estimate a set of portfolios on the efficient frontier. The efficient frontier is a curve that shows the trade-off between the return and risk achieved by Pareto-optimal portfolios. For a given return level, the portfolio on the efficient frontier is the one that minimizes the risk while maintaining the desired return. Conversely, for a given risk level, the portfolio on the efficient frontier is the one that maximizes return while maintaining the desired risk level.

% Compute efficient frontier
p = setSolverMINLP(p,'TrustRegionCP',DeltaLowerBound=condLB);
pwgt = estimateFrontier(p)

pwgt = 12×10

         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0
    0.1000    0.1094    0.1000    0.1000         0         0         0         0         0         0
         0         0         0    0.1000    0.1662    0.2634    0.3369    0.4051    0.4647    0.5000
    0.1328    0.2387    0.3331    0.3263    0.3622    0.3543    0.3701    0.3854    0.4287    0.5000
    0.1000         0         0         0         0         0         0         0         0         0
    0.4132    0.3888    0.2755    0.2552    0.2288    0.1812    0.1110         0         0         0
         0         0         0         0         0         0         0         0         0         0
    0.2540    0.2631    0.2914    0.2185    0.2427    0.2011    0.1821    0.2095    0.1067         0
      ⋮

% Compute risk and returns of the portfolios on the efficient frontier
rsk = estimatePortRisk(p,pwgt)

ret = estimatePortReturn(p,pwgt)

Plot the weights from the frontier estimation using plotFrontier. The resulting curve is piece-wise concave with possible vertical jumps (discontinuities) between the concave intervals.

% Plot efficient frontier
plotFrontier(p,pwgt)

Changing MINLP Solvers

In the previous section, you use the default solver for estimateFrontier. However, you can solve mixed-integer portfolio problems using any of the three algorithms supported by setSolverMINLP: OuterApproximation, ExtendedCP, and TrustRegionCP. Furthermore, the OuterApproximation algorithm accepts an additional name-value argument (ExtendedFormulation) for Portfolio problems, which reformulates problems with quadratic functions to work in an extended space that usually decreases the computation time. All algorithms, including the extended formulation variation of the OuterApproximation algorithm, return the same values within numerical accuracy. The available solvers are:

OuterApproximation — The default algorithm, which is robust and usually faster than ExtenedCP
OuterApproximation with ExtendedFormulation set to true — A robust algorithm that is usually faster than other algorithms, but only supported for Portfolio object problems
ExtendedCP — The most robust solver, but usually the slowest
TrustRegionCP — The fastest algorithm, but one that is less robust and may provide suboptimal solutions

For more information on solvers for mixed-integer portfolio problems, see Choose MINLP Solvers for Portfolio Problems.

To change the MINLP solvers, use setSolverMINLP.

% Select 'TrustRegionCP' as solver
p_OA = setSolverMINLP(p,'OuterApproximation');
pwgt_OA = estimateFrontier(p_OA);
rskOA = estimatePortRisk(p,pwgt_OA);
retOA = estimatePortReturn(p,pwgt_OA);

% Select 'ExtendedCP' as solver using 'midway' cuts as 'CutGeneration'
p_ECP = setSolverMINLP(p,'ExtendedCP',CutGeneration="midway");
pwgt_ECP = estimateFrontier(p_ECP);
rskECP = estimatePortRisk(p,pwgt_ECP);
retECP = estimatePortReturn(p,pwgt_ECP);

Compare the returns and risks obtained by the portfolios on the efficient frontier from the different solvers. They are all the same within a numerical accuracy that is an absolute difference $\leq 10^{- 4}$ .

retTable = table(retOA,ret,retECP,'VariableNames',{'OA','TR','ECP'})

retTable=10×3 table
        OA            TR           ECP    
    __________    __________    __________

    0.00092881    0.00092876    0.00092876
     0.0011614     0.0011613     0.0011613
      0.001394     0.0013939     0.0013939
     0.0016265     0.0016265     0.0016265
     0.0018591     0.0018591     0.0018591
     0.0020917     0.0020917     0.0020917
     0.0023243     0.0023243     0.0023243
     0.0025569     0.0025569     0.0025569
     0.0027894     0.0027894     0.0027894
      0.003022      0.003022      0.003022

rskTable = table(rskOA,rsk,rskECP,'VariableNames',{'OA','TR','ECP'})

rskTable=10×3 table
       OA          TR         ECP   
    ________    ________    ________

    0.036808    0.036808    0.036808
    0.037658    0.037658    0.037658
    0.039808    0.039808    0.039808
    0.042256    0.042256    0.042256
     0.04527    0.045269    0.045269
    0.048489    0.048488    0.048488
    0.051963    0.051963    0.051963
    0.055783    0.055783    0.055783
    0.059828    0.059828    0.059828
    0.064705    0.064705    0.064705

% Compare risks from the different OuterApproximation formulations
norm(rskTable.OA-rskTable.TR,Inf) <= 1e-4

ans = logical
   1

Mixed-Integer CVaR Portfolio Optimization Problem

CVaR Portfolio

Changing MINLP Solvers

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