Obtaining Endpoints of the Efficient Frontier

Often when using a PortfolioCVaR object, you might be interested in the endpoint portfolios for the efficient frontier. Suppose that you want to determine the range of returns from minimum to maximum to refine a search for a portfolio with a specific target return. Use the estimateFrontierLimits function to obtain the endpoint portfolios.

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);
pwgt = estimateFrontierLimits(p);

disp(pwgt)

    0.8651         0
    0.0600         0
    0.0318         0
    0.0430    1.0000

The estimatePortMoments function shows the range of risks and returns for efficient portfolios: Note that the endpoints of the efficient frontier depend upon the Scenarios in the PortfolioCVaR object. If you change the Scenarios, you are likely to obtain different endpoints.

Starting from an initial portfolio, estimateFrontierLimits also returns purchases and sales to get from the initial portfolio to the endpoint portfolios on the efficient frontier. For example, given an initial portfolio in pwgt0, you can obtain purchases and sales:

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, pbuy, psell] = estimateFrontierLimits(p);
	
display(pwgt)

pwgt = 4×2

    0.8673         0
    0.0523         0
    0.0414         0
    0.0390    1.0000

display(pbuy)

pbuy = 4×2

    0.5673         0
         0         0
         0         0
         0    0.9000

display(psell)

psell = 4×2

         0    0.3000
    0.2477    0.3000
    0.1586    0.2000
    0.0610         0

If you do not specify an initial portfolio, the purchase and sale weights assume that your initial portfolio is 0.