# Documentation

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# estimateFrontierByReturn

Estimate optimal portfolios with targeted portfolio returns

Use the `estimateFrontierByReturn` function with a `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` object to estimate optimal portfolios with targeted portfolio returns.

For details on the respective workflows when using these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.

## Syntax

``````[pwgt,pbuy,psell] = estimateFrontierByReturn(obj,TargetReturn)``````

## Description

example

``````[pwgt,pbuy,psell] = estimateFrontierByReturn(obj,TargetReturn)``` estimates optimal portfolios with targeted portfolio returns.```

## Examples

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To obtain efficient portfolios that have targeted portfolio returns, the `estimateFrontierByReturn` function accepts one or more target portfolio returns and obtains efficient portfolios with the specified returns. Assume 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 ```

To obtain efficient portfolios that have targeted portfolio returns, the `estimateFrontierByReturn` function accepts one or more target portfolio returns and obtains efficient portfolios with the specified returns. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio returns of 7%, 10%, and 13%.

```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 ]; rng(11); p = PortfolioCVaR; p = simulateNormalScenariosByMoments(p, m, C, 2000); p = setDefaultConstraints(p); p = setProbabilityLevel(p, 0.95); pwgt = estimateFrontierByReturn(p, [0.07 0.10, 0.13]); display(pwgt);```
```pwgt = 0.7371 0.3071 0 0.1504 0.3919 0.4396 0.0286 0.1011 0.1360 0.0839 0.1999 0.4244 ```

The function `rng`() is used to reset the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.

To obtain efficient portfolios that have targeted portfolio returns, the `estimateFrontierByReturn` function accepts one or more target portfolio returns and obtains efficient portfolios with the specified returns. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio returns of 7%, 10%, and 13%.

```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 ]; rng(11); p = PortfolioMAD; p = simulateNormalScenariosByMoments(p, m, C, 2000); p = setDefaultConstraints(p); pwgt = estimateFrontierByReturn(p, [0.07 0.10, 0.13]); display(pwgt);```
```pwgt = 0.7436 0.3147 0.0000 0.1357 0.3835 0.4422 0.0328 0.0939 0.1324 0.0879 0.2079 0.4254 ```

The function `rng`() is used to reset the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.

## Input Arguments

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Object for portfolio, specified using `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` object. For more information on creating a portfolio object, see

Target values for portfolio return, specified as a `NumPorts` vector.

### Note

`TargetReturn` specifies target returns for portfolios on the efficient frontier. If any `TargetReturn` values are outside the range of returns for efficient portfolios, the `TargetReturn` is replaced with the minimum or maximum efficient portfolio return, depending upon whether the target return is below or above the range of efficient portfolio returns.

Data Types: `double`

## Output Arguments

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Optimal portfolios on the efficient frontier with specified target returns from `TargetReturn`, returned as a `NumAssets`-by-`NumPorts` matrix. `pwgt` is returned for a `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` input object (`obj`).

Purchases relative to an initial portfolio for optimal portfolios on the efficient frontier, returned as `NumAssets`-by-`NumPorts` matrix.

### Note

If no initial portfolio is specified in `obj.InitPort`, that value is assumed to be `0` such that ```pbuy = max(0, pwgt)``` and `psell = max(0, -pwgt)`.

`pbuy` is returned for a `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` input object (`obj`).

Sales relative to an initial portfolio for optimal portfolios on the efficient frontier, returned as a `NumAssets`-by-`NumPorts` matrix.

### Note

If no initial portfolio is specified in `obj.InitPort`, that value is assumed to be `0` such that ```pbuy = max(0, pwgt)``` and `psell = max(0, -pwgt)`.

`psell` is returned for `Portfolio`, `PortfolioCVaR`, or `PortfolioMAD` input object (`obj`).

## Tips

You can also use dot notation to estimate optimal portfolios with targeted portfolio returns.

```[pwgt, pbuy, psell] = obj.estimateFrontierByReturn(TargetReturn); ```