  Any optimization algorithm based on the risk parity approach requires the
formulation of portfolio total risk in terms of marginal contributions. In this
paper we use the independence of the underlying factors in the market to derive
the centered moments required in the risk decomposition process when the
modified versions of Value at Risk and Expected Shortfall are considered.
  The choice of the Mixed Tempered Stable distribution seems adequate for
fitting skewed and heavy tailed distributions. The ensuing detailed description
of the optimization procedure is due to the existence of analytical higher
order moments. Better results are achieved in terms of out of sample
performance and greater diversification.
