Description Usage Arguments Details Value References Examples
The Value at Risk (VaR) of level α (α-quantile) of an event is a number attempting to summarize the risk of that event and define the worst expected loss of the event over a period of time. The Average VaR is another important measure of the risk at a given confidence level, which calculated by using the function of "rskFac".
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dat |
A numeric vector of object data. |
alpha |
Confidence level α (0<α<0.5). |
dist |
A named of distribution function which should be fitted to data values. The distibution function is selected by the name of "laplace", "logis", "gum", "t" and "norm". |
df |
degrees of freedom from a specified distribution function. |
Suppose X is random variable (rv) has distribution function (df) F. Given a confidence level α\in (0, 1), Value at Risk (VaR) of the underlying X at the confidence level α is the smallest number x such that the probability that the underlying X exceeds x is at least 1-α. In other word, if X is a rv with symmetric distribution function F (e.g., the return value of a portfolio), then VaR_{α} is the negative of the α quantile, i.e.,
VaR_{α}(X)=Q(α)=inf{x \in Real : Pr( X ≤ x )≤ α}.
where, Q(.)=F^{-1}(.).
Since, the VaR_α(X) is the nagative of α quantile in the left tail, -VaR_{1-α}(-X) is positive value of VaR in right tail.
The average VaR_α, (AVaR_α) for 0<α≤ 1 of X is defined as
AVaR_α(X)= \frac{1}{α}\int_{0}^{α}VaR(x) dx,
The AVaR is known under the names of conditional VaR (CVaR), tail VaR (TVaR) and expected shortfall.
Pflug and Romisch (2007, ISBN: 9812707409) shows the AVaR may be represented as the optimal value of the following optimization problem
AVaR_α (X) = VaR_α(X) - \frac{1}{α} E((X - VaR_α(X))^{-}).
where, (y)^{-} = min (y,0). To approximate the integral, it is given by
AVaR_α(X)=VaR_α(X)+\frac{1}{t α}∑_{i=1}^{t}max{(VaR_α(X) - X), 0},
where, t is number of observations. By considering the rv -X, the -AVaR_{1-α} in right tail is obtainable.
The values of output are "VaR", "AVaR_n" and "AVaR_p" correspond to the VaR, Average VaR in left tail, Average VaR in right tail.
Pflug and Romisch (2007, ISBN: 9812707409)
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