# Obtaining Endpoints of the Efficient Frontier

Often when using a `PortfolioMAD` 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 = PortfolioMAD; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt = estimateFrontierLimits(p); disp(pwgt)```
``` 0.8825 0 0.0419 0 0.0396 0 0.0359 1.0000 ```

Note that the endpoints of the efficient frontier depend upon the `Scenarios` in the `PortfolioMAD` 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 = PortfolioMAD; p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = setInitPort(p, pwgt0); [pwgt, pbuy, psell] = estimateFrontierLimits(p); display(pwgt)```
```pwgt = 4×2 0.8862 0 0.0413 0 0.0361 0 0.0364 1.0000 ```
`display(pbuy)`
```pbuy = 4×2 0.5862 0 0 0 0 0 0 0.9000 ```
`display(psell)`
```psell = 4×2 0 0.3000 0.2587 0.3000 0.1639 0.2000 0.0636 0 ```

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