VaR.CornishFisher(R, p = 0.99, modified = TRUE)
modifiedVaR(R, p = 0.99)
VaR.traditional(R, p = 0.99)
VaR.mean(R, p = 0.99)
VaR(R, p = 0.99)quantile for an arbitrary historical distribution or qnorm for an idealized normal distribution. The negative return at the correct quantile (usually 95% or 99%), is the non-parametric historical VaR estimate.In a set of returns for which sufficently long history exists, the per-period Value at risk is simply the correct quantile of the period losses $\mu$ with variance $\sigma$ is denoted by:
$$VaR=\mu_{q_{.99}}$$ where $q_{.99}$ is the 99% quantile of the distribution.
fPortfolio has implemented historical mean-VaR as the VaR function.
This method is also sometimes called
When you don't have a sufficiently long set of returns to use non-parametric or historical VaR, or wish to more closely model an ideal distribution, it is common to us a parmetric estimate based on the distribution. J.P. Morgan's RiskMetrics parametric mean-VaR was published in 1994 and this methodology for estimating parametric mean-VaR has become what most literature generally refers to as VaR.traditional. See Return to RiskMetrics: Evolution of a Standard
Parametric mean-VaR does a better job of accounting for the tails of the distribution by more precisely estimating shape of the distribution tails of the risk quantile. The most common estimate is a normally distributed $R\sim N(\mu,\sigma^{2})$ return series. In this case, estimation of VaR requires the mean return $\bar{R}$, the distribution of losses $\mu$, and the variance of the returns $\sigma$. In the most common case, parametric VaR is thus calculated by
$$\sigma=variance(R)$$
$$VaR=\bar{R} - q_{.99} \cdot \sqrt{\sigma}$$
where $q_{.99}$ is the 99% quantile of the distribution. Represented in Rby qnorm(1-p)
Other forms of parametric mean-VaR estimation utilize a different distribution for the distribution of losses $\mu$ to better account for the possible fat-tailed nature of downside risk. The package VaR contains methods for simulating and estimating lognormal VaR.norm and generalized Pareto VaR.gpd distributions to overcome some of the problems with nonparametric or parametric mean-VaR calculations on a limited sample size or on potentially fat-tailed distributions. There is also a VaR.backtest function to apply simulation methods to create a more robust estimate of the potential distribution of losses. Less commonly a covariance matrix of multiple risk factors may be applied.
The limitations of mean Value-at-Risk are well covered in the literature. The limitations of traditional mean-VaR are all related to the use of a symetrical distribution function. Use of simulations, resampling, or Pareto distributions all help in making a more accurate prediction, but they are still flawed for assets with significantly non-normal (skewed or kurtotic) distributions. Favre and Galeano(2002) provide a modified VaR calculation that takes the higher moments of non-normal distributions (skewness, kurtosis) into account through the use of a Cornish Fisher expansion, and collapses to standard (traditional) mean-VaR if the return stream follows a standard distribution. This measure is now widely cited and used in the literature, and is usually referred to as
$$z_{c} = q_{p}=qnorm(p)$$
$$S= skewness(R)$$
$$K = kurtosis(R)$$
$$z_{cf}=z_{c}+\frac{(z_{c}^{2}-1)S}{6}+\frac{(z_{c}^{3}-3z_{c})K}{24}+\frac{(2z_{c}^{3}-5z_{c})S^{2}}{36}$$
$$modVaR =\bar{R} - z_{cf}\sqrt{\sigma}$$
Cornish-Fisher VaR collapses to traditional mean-VaR when returns are normally distributed. As such, the VaR.mean and VaR.traditional functions are wrappers for the VaR.CornishFisher function. The Cornish-Fisher expansion also naturally encompasses much of the variability in returns that could be uncovered by more computationally intensive techniques such as resampling or Monte-Carlo simulation.
Favre and Galeano also utilize modified VaR in a modified Sharpe Ratio as the return/risk measure for their portfolio optimization analysis, see SharpeRatio.modified for more information.
Uryasev S, Rockafellar R. Optimization of Conditional VaR. University of Florida, Working paper, 1999.
VaR.Beyond
VaR.Marginal
SharpeRatio.modified
skewness
kurtosis
VaR
VaR.gpd
VaR.norm
VaR.backtest
CVaRdata(edhec)
# first do normal VaR calc
VaR.traditional(edhec, p=.95)
# now use modified Cornish Fisher calc to take non-normal distribution into account
VaR.CornishFisher(edhec, p=.95)
# now use default p=.99
VaR.CornishFisher(edhec)Run the code above in your browser using DataLab