# Example â€“ Efficient Portfolio (non-linear optimization)

One of the most important application of optimization is *portfolio management*. Several ways are used in order to determine the most efficient portfolios in various conditions. The basic type of this kind of problems are:

- how to compile the proportions of the portfolio to maximize total return while the risk does not exceed a given level.
- what would be the best composition to minimize the risk while the return exceed a given level.

But we could turn to a more sophicticated, famous method: we can calculate so-called *Sharpe-ratio*.

In our example three assets are available to invest (ATT, GMC, USX) and the risk-free asset exists as well. The returns, variances and covariances are known and return of the risk-free asset is given.

Definition of the Sharpe-ratio: It is computed as a portfolioâ€™s risk premium divided by the standard deviation for the portfolioâ€™s return.

The Sharpe Ratio is designed to measure the expected return per unit of risk for a *zero investment strategy*. The difference between the returns on two investment assets represents the results of such a strategy. The Sharpe Ratio does not cover cases in which only one investment return is involved.
A ratio was developed by Nobel Laureate Bill Sharpe to measure risk-adjusted return of an investment. It is calculated by subtracting the risk-free rate from the rate of return for a portfolio and dividing the result by the standard deviation of the portfolio returns.

The Sharpe ratio tells us whether the returns of a portfolio are due to smart investment decisions or a result of excess investor risk. This measurement is very useful because although one portfolio or fund can reap higher returns than its peers, it is only a good investment if those higher returns do not come with too much additional risk. The greater a portfolio's Sharpe ratio, the better its risk-adjusted performance has been. A lower number is worse.

Our goal therefore is to maximize the Sharpe-ratio; that's what is calculated in the objective cell.

Let see the input dataset:

The input dialog box looks like this:

### Result:

All constraints are satisfied so this is the optimal composition and the maximum equity premium.