Portfolio Risk Contribution Charts

Portfolio standard deviation can be decomposed into component risk contributions for each asset in the portfolio. These risk contributions can then be displayed in a stacked bar chart for an entire efficient frontier (map chart), or a bar chart comparing two selected portfolios (bar chart). Both charts appear as options on the Charts Worksheet.

Mathematics

The standard deviation of an N-asset portfolio P with weights w = {wi } i = 1,…,N, and assets Ai and covariance matrix Σ, with elements {σji } is defined as √(w' Σw)=(w' Σw)/√(w' Σw)=w'{Σw/√(w' Σw)}.

The vector MCR=Σw/√(w'Σw) is called the vector of marginal risk contributions.  It can be shown that the ith component of this vector is equal to the following:

 {Σw/√(w' Σw)}i=(∂σ(P))/(∂Ai )=corr(Ai,P)σi.

Thus, the risk contribution of the ith asset to the portfolio P can be expressed as wi MCRi = wi∙corr(Ai,P)∙σi, although it is far more convenient in practice to compute it using the formula (∑j [σji wj])/√(w'Σw).

More Information

Menchero, J. and Davis, B. (2011) “Risk Contribution Is Exposure Times Volatility Times Correlation: Decomposing Risk Using the X-Sigma-Rho Formula.” Journal of Portfolio Management. 37(2): 97-106.

Menchero, J., and Hu, J. (2006). “Portfolio Risk Attribution.” Journal of Performance Measurement. 10(3): 22-33.