PerformanceAnalytics (version 1.1.0)

TreynorRatio: 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: $$TreynorRatio = \frac{\overline{(R_{a}-R_{f})}}{\beta_{a,b}}$$ $$ModifiedTreynorRatio = \frac{r_p - r_f}{\sigma_s}$$

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