# Documentation

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## Efficient Portfolio That Maximizes Sharpe Ratio

The Sharpe ratio is defined as the ratio

`$\frac{\mu \left(x\right)-{r}_{0}}{\sqrt{\sum \left(x\right)}}$`

where $x\in {R}^{n}$ and r0 is the risk-free rate (μ and Σ proxies for portfolio return and risk). For more information, see Portfolio Optimization Theory.

Portfolios that maximize the Sharpe ratio are portfolios on the efficient frontier that satisfy a number of theoretical conditions in finance. For example, such portfolios are called tangency portfolios since the tangent line from the risk-free rate to the efficient frontier touches the efficient frontier at portfolios that maximize the Sharpe ratio.

To obtain efficient portfolios that maximizes the Sharpe ratio, the `estimateMaxSharpeRatio` function accepts a Portfolio object and obtains efficient portfolios that maximize the Sharpe Ratio.

Suppose that you have a universe with four risky assets and a riskless asset and you want to obtain a portfolio that maximizes the Sharpe ratio, where, in this example, r0 is the return for the riskless asset.

```r0 = 0.03; 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('RiskFreeRate', r0); p = setAssetMoments(p, m, C); p = setDefaultConstraints(p); pwgt = estimateMaxSharpeRatio(p); display(pwgt);```
```pwgt = 0.4251 0.2917 0.0856 0.1977```

If you start with an initial portfolio, `estimateMaxSharpeRatio` also returns purchases and sales to get from your initial portfolio to the portfolio that maximizes the Sharpe ratio. For example, given an initial portfolio in `pwgt0`, you can obtain purchases and sales from the previous example:

```pwgt0 = [ 0.3; 0.3; 0.2; 0.1 ]; p = setInitPort(p, pwgt0); [pwgt, pbuy, psell] = estimateMaxSharpeRatio(p); display(pwgt); display(pbuy); display(psell);```
```pwgt = 0.4251 0.2917 0.0856 0.1977 pbuy = 0.1251 0 0 0.0977 psell = 0 0.0083 0.1144 0```
If you do not specify an initial portfolio, the purchase and sale weights assume that you initial portfolio is `0`.