Estimate Efficient Frontiers for PortfolioMAD Object

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

Obtaining MAD Portfolio Risks and Returns

This example shows how to use the functions estimatePortReturn and estimatePortRisk with a PortfolioMAD 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 = PortfolioMAD;
p = setScenarios(p, AssetScenarios);
p = setDefaultConstraints(p);

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

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

pwgt = 4×10

    0.8823    0.7156    0.5497    0.3854    0.2199    0.0530         0         0         0         0
    0.0427    0.1427    0.2404    0.3378    0.4365    0.5342    0.4710    0.3338    0.1681         0
    0.0392    0.0434    0.0490    0.0518    0.0553    0.0628    0.0404    0.0029         0         0
    0.0358    0.0982    0.1608    0.2250    0.2883    0.3500    0.4886    0.6633    0.8319    1.0000

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 = 
0.0252

display(pret0)

pret0 = 
0.0071

display(prsk)

prsk = 10×1

    0.0175
    0.0190
    0.0228
    0.0279
    0.0338
    0.0401
    0.0473
    0.0568
    0.0678
    0.0796

display(pret)

pret = 10×1

    0.0049
    0.0060
    0.0071
    0.0082
    0.0093
    0.0104
    0.0115
    0.0125
    0.0136
    0.0147

Obtaining the PortfolioMAD Standard Deviation

This example shows how to compute standard deviations of portfolio returns for a PortfolioMAD object with the function estimatePortStd. These functions work with any PortfolioMAD 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, and standard deviation. 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 = PortfolioMAD('initport', pwgt0);
p = simulateNormalScenariosByMoments(p, m, C, 20000);
p = setDefaultConstraints(p);

pwgt = estimateFrontier(p, 5);

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

[pret, prsk, pstd]

ans = 6×3

    0.0206    0.0300    0.0377
    0.0196    0.0321    0.0403
    0.0521    0.0365    0.0459
    0.0847    0.0480    0.0603
    0.1172    0.0629    0.0790
    0.1497    0.0796    0.0999