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Example – 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 portfolio’s risk premium divided by the standard deviation for the portfolio’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.

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