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## Plotting the Efficient Frontier for a PortfolioMAD Object

The `plotFrontier` function creates a plot of the efficient frontier for a given portfolio optimization problem. This function accepts several types of inputs and generates a plot with an optional possibility to output the estimates for portfolio risks and returns along the efficient frontier. `plotFrontier` has four different ways that it can be used. In addition to a plot of the efficient frontier, if you have an initial portfolio in the `InitPort` property, `plotFrontier` also displays the return versus risk of the initial portfolio on the same plot. If you have a well-posed portfolio optimization problem set up in a PortfolioMAD object and you use `plotFrontier`, you get a plot of the efficient frontier with the default number of portfolios on the frontier (the default number is `10` and is maintained in the hidden property `defaultNumPorts`). This example illustrates a typical use of `plotFrontier` to create a new plot:

```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); plotFrontier(p);```

The `Name` property appears as the title of the efficient frontier plot if you set it in the PortfolioMAD object. Without an explicit name, the title on the plot would be “Efficient Frontier.” If you want to obtain a specific number of portfolios along the efficient frontier, use `plotFrontier` with the number of portfolios that you want. Suppose that you have the PortfolioMAD object from the previous example and you want to plot 20 portfolios along the efficient frontier and to obtain 20 risk and return values for each portfolio:

```[prsk, pret] = plotFrontier(p, 20); display([pret, prsk]); ```
```ans = 0.0049 0.0176 0.0054 0.0179 0.0058 0.0189 0.0063 0.0205 0.0068 0.0225 0.0073 0.0248 0.0078 0.0274 0.0083 0.0302 0.0088 0.0331 0.0093 0.0361 0.0098 0.0392 0.0103 0.0423 0.0108 0.0457 0.0112 0.0496 0.0117 0.0539 0.0122 0.0586 0.0127 0.0635 0.0132 0.0687 0.0137 0.0744 0.0142 0.0806```

### Plotting Existing Efficient Portfolios

If you already have efficient portfolios from any of the "estimateFrontier" functions (see Estimate Efficient Frontiers for PortfolioMAD Object), pass them into `plotFrontier` directly to plot the efficient frontier:

```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); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = PortfolioMAD('Name', 'Asset Allocation Portfolio', 'InitPort', pwgt0); p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt = estimateFrontier(p, 20); plotFrontier(p, pwgt) ```

### Plotting Existing Efficient Portfolio Risks and Returns

If you already have efficient portfolio risks and returns, you can use the interface to `plotFrontier` to pass them into `plotFrontier` to obtain a plot of the efficient frontier:

```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 ]; AssetScenarios = mvnrnd(m, C, 20000); pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = PortfolioMAD('Name', 'Asset Allocation Portfolio', 'InitPort', pwgt0); p = setScenarios(p, AssetScenarios); p = setDefaultConstraints(p); pwgt = estimateFrontier(p); pret= estimatePortReturn(p, pwgt) prsk = estimatePortRisk(p, pwgt) plotFrontier(p, prsk, pret)```
```pret = 0.0590 0.0723 0.0857 0.0991 0.1124 0.1258 0.1391 0.1525 0.1658 0.1792 prsk = 0.0615 0.0664 0.0795 0.0976 0.1184 0.1408 0.1663 0.1992 0.2368 0.2787```