  We extend and test empirically the multifractal model of asset returns based
on a multiplicative cascade of volatilities from large to small time scales.
The multifractal description of asset fluctuations is generalized into a
multivariate framework to account simultaneously for correlations across times
scales and between a basket of assets. The reported empirical results show that
this extension is pertinent for financial modelling. The second part of the
paper applies this theory to portfolio optimisation. Our multi-scale
description allows us to characterize the portfolio return distribution at all
time scales simultaneously. The portfolio composition is predicted to change
with the investment time horizon (i.e., the time scale) in a way that can be
fully determined once an adequate measure of risk is chosen. We discuss the use
of the fourth-order cumulant and of utility functions. While the portfolio
volatility can be optimized in some cases for all time horizons, the kurtosis
and higher normalized cumulants cannot be simultaneously optimized. For a fixed
investment horizon, we study in details the influence of the number of periods,
i.e., of the number of rebalancing of the portfolio. For the large risks
quantified by the cumulants of order larger than two, the number of periods has
a non-trivial influence, in contrast with Tobin's result valid in the
mean-variance framework. This theory provides a fundamental framework for the
conflicting optimization involved in the different time horizons and quantifies
systematically the trade-offs for an optimal inter-temporal portfolio
optimization.
