VARs in R: VAR, TVAR, and RVAR

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

Three examples of such are Value-at-Risk (VAR), Tail Value-at-Risk (TVAR) (or Expected Shortfall), and Restricted Value-at-Risk (RVAR).

VAR is defined as

 \rho (x) := VAR_p(X)= \inf \{ m \in R | P(X \leq m) \geq p \}

It represents the maximum value m a distribution X can attain within confidence level p.

TVAR is defined as

 \rho (x) := \frac{1}{1-p} \int_p^1 VAR_{\alpha}(X) d \alpha= E[ X|X >\inf \{ m \in R | P(X \leq m) \geq p \}

which can be interpreted as the expectation of the distribution X above m at a given confidence level p. When p=0, the above expression reduces to the mean of the distribution.

RVAR is defined as

 \rho (x) := \frac{1}{p_2-p_1} \int_{p_1}^{p_2} VAR_{\alpha}(X) d \alpha = E[ X|\inf \{ m \in R | P(X \leq m) \geq p_2\}>X >\inf \{ m \in R | P(X \leq m) \geq p_1\}]

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){

TVAR <- function(x,p){

RVAR <- function(x,p,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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