Estimation of the contribution of a component to the overall risk for a portfolio defined by the multivariate fractional Levy motion

The representation of securities portfolio in the form of multivariate fractional Levy motion is considered. This model has properties such as self-similarity, long term dependence and heavy-tails of one-dimensional distributions of portfolio components. Such properties have been discovered in empirical studies of the financial assets dynamics. One of the important problems in financial analysis is to estimate the contribution of individual components in the total risk of the portfolio. As a measure of this contribution uses the conditional average value of the individual risk components under given total risk of the portfolio. This measure of risk has very important property of of coherence.
The first results on this subject were obtained in the paper by Panjer for the case of multivariate normal distribution for possible risks of the portfolio. In our paper we give a detailed proof of this result, and generalize it in case of multivariate elliptical stable distribution. It is impossible to get explicit expressions in this situation. We propose some explicit expression for large values of the total risk. The task is solved sequentially for one-dimensional stable distribution, multivariate elliptical stable distributions and multivariate fractional Levy motion.