# Obtaining Endpoints of the Efficient Frontier

Often when using a `Portfolio` 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 ]; p = Portfolio; p = setAssetMoments(p, m, C); p = setDefaultConstraints(p); pwgt = estimateFrontierLimits(p); disp(pwgt)```
``` 0.8891 0 0.0369 0 0.0404 0 0.0336 1.0000 ```

The `estimatePortMoments` function shows the range of risks and returns for efficient portfolios:

```[prsk, pret] = estimatePortMoments(p, pwgt); disp([prsk, pret])```
``` 0.0769 0.0590 0.3500 0.1800 ```

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 ]; p = Portfolio; p = setAssetMoments(p, m, C); 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 0.8891 0 0.0369 0 0.0404 0 0.0336 1.0000 ```
`display(pbuy)`
```pbuy = 4×2 0.5891 0 0 0 0 0 0 0.9000 ```
`display(psell)`
```psell = 4×2 0 0.3000 0.2631 0.3000 0.1596 0.2000 0.0664 0 ```

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