Obtaining Efficient Portfolios for Target Returns

To obtain efficient portfolios that have targeted portfolio returns, the estimateFrontierByReturn function accepts one or more target portfolios returns and obtains efficient portfolios with the specified returns. For example, assume that you have a universe of four assets where you want to obtain efficient portfolios with target portfolio returns of 6%, 9%, and 12%:

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 = estimateFrontierByReturn(p, [0.06, 0.09, 0.12]);

display(pwgt)

pwgt =

    0.8772    0.5032    0.1293
    0.0434    0.2488    0.4541
    0.0416    0.0780    0.1143
    0.0378    0.1700    0.3022

Sometimes, you can request a return for which no efficient portfolio exists. Based on the previous example, suppose that you want a portfolio with a 5% return (which is the return of the first asset). A portfolio that is fully invested in the first asset, however, is inefficient. estimateFrontierByReturn warns if your target returns are outside the range of efficient portfolio returns and replaces it with the endpoint portfolio of the efficient frontier closest to your target return:

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 = estimateFrontierByReturn(p, [0.05, 0.09, 0.12]);

display(pwgt)

Warning: One or more target return values are outside the feasible range [ 0.0590468, 0.18 ].
	Will return portfolios associated with endpoints of the range for these values. 
> In Portfolio.estimateFrontierByReturn at 70 

pwgt =

    0.8891    0.5032    0.1293
    0.0369    0.2488    0.4541
    0.0404    0.0780    0.1143
    0.0336    0.1700    0.3022

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

pret = estimatePortReturn(p, p.estimateFrontierLimits);

display(pret)

pret =

    0.0590
    0.1800

This result indicates that efficient portfolios have returns that range between 5.9% and 18%.

If you have an initial portfolio, estimateFrontierByReturn 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, to obtain purchases and sales with target returns of 6%, 9%, and 12%:

pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ];
p = setInitPort(p, pwgt0);
[pwgt, pbuy, psell] = estimateFrontierByReturn(p, [0.06, 0.09, 0.12]);

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

pwgt =

    0.8772    0.5032    0.1293
    0.0434    0.2488    0.4541
    0.0416    0.0780    0.1143
    0.0378    0.1700    0.3022

pbuy =

    0.5772    0.2032         0
         0         0    0.1541
         0         0         0
         0    0.0700    0.2022

psell =

         0         0    0.1707
    0.2566    0.0512         0
    0.1584    0.1220    0.0857
    0.0622         0         0

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