Estimate Efficient Frontiers for PortfolioCVaR Object

Whereas Estimate Efficient Portfolios for Entire Frontier for PortfolioCVaR Object focused on estimation of efficient portfolios, this topic focuses on the estimation of efficient frontiers. For information on the workflow when using PortfolioCVaR objects, see PortfolioCVaR Object Workflow.

Obtaining CVaR Portfolio Risks and Returns

This example shows how to use the functions estimatePortReturn and estimatePortRisk with a PortfolioCVaR object to provide estimates for the return (or return proxy), risk (or the risk proxy). These functions have the same input syntax but have different combinations of outputs.

Suppose that you have this following portfolio optimization problem that gave you a collection of portfolios along the efficient frontier in pwgt:

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0;
0.00408 0.0289 0.0204 0.0119;
0.00192 0.0204 0.0576 0.0336;
0 0.0119 0.0336 0.1225 ];
m = m/12;
C = C/12;

AssetScenarios = mvnrnd(m, C, 20000);

p = PortfolioCVaR;
p = setScenarios(p, AssetScenarios);
p = setDefaultConstraints(p);
p = setProbabilityLevel(p, 0.95);

pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ];

p = setInitPort(p, pwgt0);
pwgt = estimateFrontier(p);

Given pwgt0 and pwgt, use the portfolio risk and return estimation functions to obtain risks and returns for your initial portfolio and the portfolios on the efficient frontier:

prsk0 = estimatePortRisk(p, pwgt0);
pret0 = estimatePortReturn(p, pwgt0);
prsk = estimatePortRisk(p, pwgt);
pret = estimatePortReturn(p, pwgt);

You obtain these risks and returns:

display(prsk0)

prsk0 = 
0.0577

display(pret0)

pret0 = 
0.0071

display(prsk)

prsk = 10×1

    0.0404
    0.0439
    0.0526
    0.0646
    0.0786
    0.0937
    0.1121
    0.1363
    0.1638
    0.1932

display(pret)

pret = 10×1

    0.0050
    0.0061
    0.0072
    0.0083
    0.0093
    0.0104
    0.0115
    0.0126
    0.0136
    0.0147

Obtaining Portfolio Standard Deviation and VaR

This example shows how to compute standard deviations of portfolio returns and the value-at-risk of portfolios for a PortfolioCVaR object with the functions estimatePortStd and estimatePortVaR. These functions work with any PortfolioCVaR portfolios, not necessarily efficient portfolios.

For example, the following code obtains five portfolios (pwgt) on the efficient frontier and also has an initial portfolio in pwgt0. Various portfolio statistics are computed that include the return, risk, standard deviation, and value-at-risk. The listed estimates are for the initial portfolio in the first row followed by estimates for each of the five efficient portfolios in subsequent rows.

m = [ 0.0042; 0.0083; 0.01; 0.15 ];
C = [ 0.005333 0.00034 0.00016 0;
0.00034 0.002408 0.0017 0.000992;
0.00016 0.0017 0.0048 0.0028;
0 0.000992 0.0028 0.010208 ];

pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ];

p = PortfolioCVaR('initport', pwgt0);
p = simulateNormalScenariosByMoments(p, m, C, 20000);
p = setDefaultConstraints(p);
p = setProbabilityLevel(p, 0.9);

pwgt = estimateFrontier(p, 5);

pret = estimatePortReturn(p, [pwgt0, pwgt]);
prsk = estimatePortRisk(p, [pwgt0, pwgt]);
pstd = estimatePortStd(p, [pwgt0, pwgt]);
pvar = estimatePortVaR(p, [pwgt0, pwgt]);

[pret, prsk, pstd, pvar]

ans = 6×4

    0.0206    0.0454    0.0377    0.0276
    0.0949    0.0215    0.0658   -0.0096
    0.1086    0.0219    0.0738   -0.0130
    0.1223    0.0229    0.0821   -0.0161
    0.1360    0.0246    0.0907   -0.0189
    0.1497    0.0272    0.0999   -0.0205