Taming large events: portfolio selection for strongly fluctuating assets
We propose a method of optimisation of asset allocation in the case where the stock price variations are supposed to have 'fat' tails represented by power laws. Generalising over previous works using stable Lévy distributions, we distinguish three distinct components of risk described by three different parts of the distributions of price variations: unexpected gains (to be kept), harmless noise inherent to financial activity, and unpleasant losses, which is the only component one would like to minimise. The independent treatment of the tails of distributions for positive and negative variations and the generalisation 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 minimise, provides a simple solution to the problem of optimisation of asset allocations in the general case where the characteristic exponents are different for each asset.