# Documentation

## Overview of Performance Metrics

### Performance Metrics Types

Sharpe first proposed a ratio of excess return to total risk as an investment performance metric. Subsequent work by Sharpe, Lintner, and Mossin extended these ideas to entire asset markets in what is called the Capital Asset Pricing Model (CAPM). Since the development of the CAPM, a variety of investment performance metrics has evolved.

This section presents four types of investment performance metrics:

• The first type of metrics is absolute investment performance metrics that are called "classic" metrics since they are based on the CAPM. They include the Sharpe ratio, the information ratio, and tracking error. To compute the Sharpe ratio from data, use `sharpe` to calculate the ratio for one or more asset return series. To compute the information ratio and associated tracking error, use `inforatio` to calculate these quantities for one or more asset return series.

• The second type of metrics is relative investment performance metrics to compute risk-adjusted returns. These metrics are also based on the CAPM and include Beta, Jensen's Alpha, the Security Market Line (SML), Modigliani and Modigliani Risk-Adjusted Return, and the Graham-Harvey measures. To calculate risk-adjusted alpha and return, use `portalpha`.

• The third type of metrics is alternative investment performance metrics based on lower partial moments. To calculate lower partial moments, use `lpm` for sample lower partial moments and `elpm` for expected lower partial moments.

• The fourth type of metrics are performance metrics based on maximum drawdown and expected maximum drawdown. To calculate maximum or expected maximum drawdowns, use `maxdrawdown` and `emaxdrawdown`.

### Performance Metrics

To illustrate the functions for investment performance metrics, you work with three financial time series objects using performance data for:

• An actively managed, large-cap value mutual fund

• A large-cap market index

• 90-day Treasury bills

The data is monthly total return prices that cover a span of five years.

The following plot illustrates the performance of each series in terms of total returns to an initial \$1 invested at the start of this 5-year period:

```load FundMarketCash plot(TestData) hold on title('\bfFive-Year Total Return Performance'); legend('Fund','Market','Cash','Location','SouthEast'); hold off```

The mean (`Mean`) and standard deviation (`Sigma`) of returns for each series are

```Returns = tick2ret(TestData); Assets Mean = mean(Returns) Sigma = std(Returns, 1)```

which gives the following result:

```Assets = 'Fund' 'Market' 'Cash' Mean = 0.0038 0.0030 0.0017 Sigma = 0.0229 0.0389 0.0009 ```

 Note:   Functions for investment performance metrics use total return price and total returns. To convert between total return price and total returns, use `ret2tick` and `tick2ret`.