COMPUTING VaR AND AVaR IN INFINITELY DIVISIBLE
DISTRIBUTIONS
Young Shin Kim
Svetlozar T. Rachev
Michele Leonardo Bianchi
Frank J. Fabozzi
Abstract: In this paper we derive closed-form solutions for the cumulative distribution function
and the average value-at-risk for five subclasses of the infinitely divisible distributions:
classical tempered stable distribution, Kim–Rachev distribution, modified tempered stable
distribution, normal tempered stable distribution, and rapidly decreasing tempered stable
distribution. We present empirical evidence using the daily performance of the S&P 500 for
the period January 2, 1997 through December 29, 2006.
2000 AMS Mathematics Subject Classification: 60E07, 91B28.
Keywords and phrases: Tempered stable distribution, infinitely divisible distribution,
VaR, CVaR, AVaR.