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Computing VaR and AVaR in infinitely divisible distributions

Kim Y. S., Rachev S. T., Bianchi M. L., Fabozzi F. J.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

223--245

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.

Orderings and risk probability functionals in portfolio theory

Ortobelli S., Rachev S., Shalit H., Fabozzi F. J.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

203--234

This paper studies and describes stochastic orderings of risk/reward positions in order to define in a natural way risk/reward measures consistent/isotonic to investors’ preferences. We begin by discussing the connection between the theory of probability metrics, risk measures, distributional moments, and stochastic orderings. Then we examine several classes of orderings which are generated by risk probability functionals. Finally, we demonstrate how further orderings could better specify the investor’s attitude toward risk.