# J.Franke, W. Hardle, G. Stahl - Measuring Risk in Complex Stochastic Systems

#### J.Franke, W. Hardle, G. Stahl – Measuring Risk in Complex Stochastic Systems

Since the seminal of Markowitz (1952) and Sharpe (1964) capital allocation within portfolios is based on the variance/covariance analysis. Even the introduction of Value-at-Risk in order to measure risk more accurately than in terms of standard deviation, did not chance the calculation of a risk contribution of single asset in the portfolio or its contributory capital as a multiple of the asset’s β with the portfolio. This approach is based on the assumption that asset returns are normally distributed. Under this assumption, the capital of a portfolio, usually defined as a quantile of the distribution of changes of the portfolio. Since the βs yield a nice decomposition of the portfolio standard deviation and exhibit the interpretation as an infinitesimal marginal risk contribution (or more mathematically as a partial derivative of the portfolio standard deviation with respect to an increase of the weight of an asset in the portfolio), these useful properties also hold for the quantie, i.e. for the capital.