  Based on a faithful representation of the heavy tail multivariate
distribution of asset returns introduced previously (Sornette et al., 1998,
1999) that we extend to the case of asymmetric return distributions, we
generalize the return-risk efficient frontier concept to incorporate the
dimensions of large risks embedded in the tail of the asset distributions. We
demonstrate that it is often possible to increase the portfolio return while
decreasing the large risks as quantified by the fourth and higher order
cumulants. Exact theoretical formulas are validated by empirical tests.
