Risk measures come in many varieties.Â Some are mean-based, some are median-based. Some are coherent, some are not. However, most risk measures of interestÂ satisfyÂ law-invariance, translation invariance, positive homogeneity, and monotonicity.

Three examples of such areÂ Value-at-Risk (VAR), TailÂ Value-at-Risk (TVAR) (or Expected Shortfall), and RestrictedÂ Value-at-Risk (RVAR).

VAR is defined as

It represents the maximum value a distribution Â can attain within confidence level .

TVAR is defined as

which can be interpreted as the expectation of the distribution above at a given confidence level . When , the above expression reduces to the mean of the distribution.

RVAR is defined as

which is similar to TVAR, except the distribution is bounded above by a second confidence level. In R, we may calculate the above risk measures via historical simulation using the following functions:

VAR <- function(x,p){ x=sort(x) return(x[floor(length(x)*p)]) } TVAR <- function(x,p){ x=sort(x) return(mean(x[floor(length(x)*p):length(x)])) } RVAR <- function(x,p,q){ x=sort(x) return(mean(x[floor(length(x)*p):floor(length(x)*q)])) }

Note that the floor is used in the above functions. We may use the ceiling, or alternatively estimate a midpoint. The floor was chosen for simplicity.

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