Taming Large Events: Optimal Portfolio Theory for Strongly Fluctuating Assets
International Journal of Theoretical and Applied Finance, 1 (1):
25--41 (1998)Abstract
We propose a method of optimization of asset allocation in the case where the stock price variations are supposed to have “fat” tails represented by power laws. General- izing over previous works using stable L ́evy distributions, we distinguish three distinct components of risk described by three different parts of the distributions of price varia- tions: unexpected gains (to be kept), harmless noise inherent to financial activity, and unpleasant losses, which is the only component one would like to minimize. The inde- pendent treatment of the tails of distributions for positive and negative variations and the generalization to large events of the notion of covariance of two random variables provide explicit formulae for the optimal portfolio. The use of the probability of loss (or equivalently the Value-at-Risk), as the key quantity to study and minimize, provides a simple solution to the problem of optimization of asset allocations in the general case where the characteristic exponents are different for each asset.