TreynorRatio {PerformanceAnalytics} | R Documentation

## calculate Treynor Ratio or modified Treynor Ratio of excess return over CAPM beta

### Description

The Treynor ratio is similar to the Sharpe Ratio, except it uses beta as the volatility measure (to divide the investment's excess return over the beta).

### Usage

    TreynorRatio(Ra, Rb, Rf = 0, scale = NA, modified = FALSE)

### Arguments

`Ra` | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns
`Rb` | return vector of the benchmark asset
`Rf` | risk free rate, in same period as your returns
`scale` | number of periods in a year (daily scale = 252, monthly scale = 12, quarterly scale = 4)
`modified` | a boolean to decide whether to return the Treynor ratio or Modified Treynor ratio

### Details

To calculate modified Treynor ratio, we divide the numerator by the systematic risk instead of the beta.

Equation:

(mean(Ra-Rf))/(Beta(Ra,Rb))

ModifiedTreynorRatio = (Rp - Rf)/sytematic risk

### Author(s)

Peter Carl, Matthieu Lestel

### References

http://en.wikipedia.org/wiki/Treynor_ratio, Carl Bacon, Practical portfolio performance measurement and attribution, second edition 2008 p.77

### See Also

`SharpeRatio` `SortinoRatio` `CAPM.beta`

### Examples

    data(portfolio_bacon)
    data(managers)
    round(TreynorRatio(managers[,1], managers[,8], Rf=.035/12),4)
    round(TreynorRatio(managers[,1], managers[,8], Rf = managers[,10]),4)
    round(TreynorRatio(managers[,1:6], managers[,8], Rf=.035/12),4)
    round(TreynorRatio(managers[,1:6], managers[,8], Rf = managers[,10]),4)
    round(TreynorRatio(managers[,1:6], managers[,8:7], Rf=.035/12),4)
    round(TreynorRatio(managers[,1:6], managers[,8:7], Rf = managers[,10]),4)

    print(TreynorRatio(portfolio_bacon[,1], portfolio_bacon[,2], modified = TRUE)) #expected 0.7975

    print(TreynorRatio(managers['1996',1], managers['1996',8], modified = TRUE))
    print(TreynorRatio(managers['1996',1:5], managers['1996',8], modified = TRUE))

[Package PerformanceAnalytics version 1.5.3 Index]