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

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).

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

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

Author

Peter Carl, Matthieu Lestel

Examples


data(portfolio_bacon) 
data(managers)
round(TreynorRatio(managers[,1], managers[,8], Rf=.035/12),4) 
#> [1] 0.2528
round(TreynorRatio(managers[,1], managers[,8], Rf = managers[,10]),4) 
#> [1] 0.2428
round(TreynorRatio(managers[,1:6], managers[,8], Rf=.035/12),4) 
#>                           HAM1   HAM2  HAM3   HAM4   HAM5   HAM6
#> Treynor Ratio: SP500 TR 0.2528 0.3925 0.201 0.1209 0.0052 0.3042
round(TreynorRatio(managers[,1:6], managers[,8], Rf = managers[,10]),4)
#>                           HAM1   HAM2   HAM3   HAM4   HAM5   HAM6
#> Treynor Ratio: SP500 TR 0.2428 0.3883 0.1956 0.1144 0.0219 0.3401
round(TreynorRatio(managers[,1:6], managers[,8:7], Rf=.035/12),4) 
#>                              HAM1   HAM2   HAM3   HAM4   HAM5   HAM6
#> Treynor Ratio: SP500 TR    0.2528 0.3925 0.2010 0.1209 0.0052 0.3042
#> Treynor Ratio: EDHEC LS EQ 0.1297 0.1088 0.0776 0.0504 0.0014 0.0966
round(TreynorRatio(managers[,1:6], managers[,8:7], Rf = managers[,10]),4)
#>                              HAM1   HAM2   HAM3   HAM4   HAM5   HAM6
#> Treynor Ratio: SP500 TR    0.2428 0.3883 0.1956 0.1144 0.0219 0.3401
#> Treynor Ratio: EDHEC LS EQ 0.1242 0.1068 0.0753 0.0471 0.0060 0.1086

print(TreynorRatio(portfolio_bacon[,1], portfolio_bacon[,2], modified = TRUE)) #expected 0.7975 
#> [1] 0.7806747

print(TreynorRatio(managers['2002',1], managers['2002',8], modified = TRUE))
#> [1] -0.727545
print(TreynorRatio(managers['2002',1:5], managers['2002',8], modified = TRUE)) 
#>                              HAM1      HAM2      HAM3       HAM4     HAM5
#> Treynor Ratio: SP500 TR -0.727545 -6.045167 -2.123832 -0.4990387 -5.37482