Kurtosis

compute kurtosis of a univariate distribution

kurtosis(x, na.rm = FALSE, method = c("excess", "moment", "fisher",
  "sample", "sample_excess"), ...)

Arguments

x	a numeric vector or object.
na.rm	a logical. Should missing values be removed?
method	a character string which specifies the method of computation. These are either "moment", "fisher", or "excess". If "excess" is selected, then the value of the kurtosis is computed by the "moment" method and a value of 3 will be subtracted. The "moment" method is based on the definitions of kurtosis for distributions; these forms should be used when resampling (bootstrap or jackknife). The "fisher" method correspond to the usual "unbiased" definition of sample variance, although in the case of kurtosis exact unbiasedness is not possible. The "sample" method gives the sample kurtosis of the distribution.
…	arguments to be passed.

Details

This function was ported from the RMetrics package fUtilities to eliminate a dependency on fUtilties being loaded every time. This function is identical except for the addition of checkData and additional labeling. $$Kurtosis(moment) = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4$$ $$Kurtosis(excess) = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4 - 3$$ $$Kurtosis(sample) = \frac{n*(n+1)}{(n-1)*(n-2)*(n-3)}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_{S_P}})^4 $$ $$Kurtosis(fisher) = \frac{(n+1)*(n-1)}{(n-2)*(n-3)}*(\frac{\sum^{n}_{i=1}\frac{(r_i)^4}{n}}{(\sum^{n}_{i=1}(\frac{(r_i)^2}{n})^2} - \frac{3*(n-1)}{n+1})$$ $$Kurtosis(sample excess) = \frac{n*(n+1)}{(n-1)*(n-2)*(n-3)}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_{S_P}})^4 - \frac{3*(n-1)^2}{(n-2)*(n-3)}$$

where \(n\) is the number of return, \(\overline{r}\) is the mean of the return distribution, \(\sigma_P\) is its standard deviation and \(\sigma_{S_P}\) is its sample standard deviation

References

Carl Bacon, Practical portfolio performance measurement and attribution, second edition 2008 p.84-85

See also

skewness.

Examples

## mean -
## var -
   # Mean, Variance:
   r = rnorm(100)
   mean(r)
#> [1] 0.0707892
   var(r)
#> [1] 1.104402

## kurtosis -
   kurtosis(r)
#> [1] 0.3081869

data(managers)
kurtosis(managers[,1:8])
#>                     HAM1     HAM2     HAM3      HAM4     HAM5      HAM6
#> Excess Kurtosis 2.361589 2.379398 2.682936 0.8632077 2.314343 -0.348865
#>                 EDHEC LS EQ  SP500 TR
#> Excess Kurtosis   0.9104791 0.5598191

data(portfolio_bacon)
print(kurtosis(portfolio_bacon[,1], method="sample")) #expected 3.03
#> [1] 3.027405
print(kurtosis(portfolio_bacon[,1], method="sample_excess")) #expected -0.41
#> [1] -0.4076603
print(kurtosis(managers['1996'], method="sample"))
#>                     HAM1     HAM2     HAM3    HAM4 HAM5 HAM6 EDHEC LS EQ
#> Excess Kurtosis 4.571471 6.932463 5.039467 4.13553    0    0           0
#>                 SP500 TR US 10Y TR US 3m TR
#> Excess Kurtosis 4.893647  3.250776 4.817395
print(kurtosis(managers['1996',1], method="sample"))
#> [1] 4.571471