Numerical approximation in the application of Risk Parity with Conditional Value at Risk in case of mixed portfolios

The 2008 financial crisis has required new methods for portfolios diversification. In the same year, Maillard, Roncalli and Teiletche (2008) suggested a method that maximizes the diversification which is called Risk Parity or Equally weighted Risk Contribution strategy. The most common method to use the Risk Parity approach is to use the standard deviation as risk measure. In this paper, we describe a method to apply Risk Parity to the Expected shortfall or also known as Conditional Value at Risk using a numerical approximation from discrete historical observation. The expected shortfall can use the advantage of being a coherent measure, not to forget that it is also a convex measure, which is very useful in the optimization. Another advantage is that the Risk Parity approach doesn’t need the estimation of the expected return as an input. Usually, the models that require the expected returns, such as the Markowitz, model have higher concentration in a smaller number of assets. This will bring a very high turnover and drawdown of the performance. The performance analysis in this paper is applied in mixed portfolios composed by stock, bonds and commodities. They show also how better this model performs in case of the crisis. We also identify not only the strong points but also the week points of these models.