Obtaining Efficient Portfolios for Target Risks

To obtain efficient portfolios that have targeted portfolio risks, the estimateFrontierByRisk function accepts one or more target portfolio risks and obtains efficient portfolios with the specified risks. Suppose that you have a universe of four assets where you want to obtain efficient portfolios with target portfolio risks of 12%, 14%, and 16%.

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 ];
 
 p = Portfolio;
 p = setAssetMoments(p, m, C);
 p = setDefaultConstraints(p);
 pwgt = estimateFrontierByRisk(p, [0.12, 0.14, 0.16]);

 display(pwgt)

pwgt =

    0.3984    0.2659    0.1416
    0.3064    0.3791    0.4474
    0.0882    0.1010    0.1131
    0.2071    0.2540    0.2979

Sometimes, you can request a risk for which no efficient portfolio exists. Based on the previous example, suppose that you want a portfolio with 7% risk (individual assets in this universe have risks ranging from 8% to 35%). It turns out that a portfolio with 7% risk cannot be formed with these four assets. estimateFrontierByRisk warns if your target risks are outside the range of efficient portfolio risks and replaces it with the endpoint of the efficient frontier closest to your target risk:

pwgt = estimateFrontierByRisk(p, 0.07)

Warning: One or more target risk values are outside the feasible range [ 0.0769288, 0.35 ].
	Will return portfolios associated with endpoints of the range for these values. 
> In Portfolio.estimateFrontierByRisk at 82 

pwgt =

    0.8891
    0.0369
    0.0404
    0.0336

The best way to avoid this situation is to bracket your target portfolio risks with estimateFrontierLimits and estimatePortRisk (see Obtaining Endpoints of the Efficient Frontier and Obtaining Portfolio Risks and Returns).

prsk = estimatePortRisk(p, p.estimateFrontierLimits);

display(prsk)

prsk =

    0.0769
    0.3500

This result indicates that efficient portfolios have risks that range from 7.7% to 35%.

Starting with an initial portfolio, estimateFrontierByRisk also returns purchases and sales to get from your initial portfolio to the target portfolios on the efficient frontier. For example, given an initial portfolio in pwgt0, you can obtain purchases and sales from the example with target risks of 12%, 14%, and 16%:

pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ];
p = setInitPort(p, pwgt0);
[pwgt, pbuy, psell] = estimateFrontierByRisk(p, [0.12, 0.14, 0.16]);

display(pwgt)
display(pbuy)
display(psell)

pwgt =

    0.3984    0.2659    0.1416
    0.3064    0.3791    0.4474
    0.0882    0.1010    0.1131
    0.2071    0.2540    0.2979


pbuy =

    0.0984         0         0
    0.0064    0.0791    0.1474
         0         0         0
    0.1071    0.1540    0.1979


psell =

         0    0.0341    0.1584
         0         0         0
    0.1118    0.0990    0.0869
         0         0         0

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