On the Financial Interpretation of Risk Contribution: Risk Budgets Do Add Up

Abstract

There are lingering questions in the financial industry regarding the concepts of risk contribution and risk budgeting. The questions stem from both the simple belief that risks are non-additive and a lack of financial intuition behind mathematical definitions of these concepts. This paper demonstrates that these questions are misguided, by both providing and analyzing risk contribution's financial interpretation.

The interpretation is based on expected contribution to potential losses of a portfolio. We show risk contribution, defined through either standard deviation or value at risk (VaR), is closely linked to the expected contribution to the losses. In a sense, risk contribution or risk budgeting can be regarded as loss contribution or loss budgeting. We also provide empirical evidences of this interpretation using asset allocation portfolios of stocks and bonds.

Our results should dispel any doubts toward the validity of the risk contribution concept. In the case of VaR contribution, our use of Cornish-Fisher expansion method provides practitioners an efficient way to calculate risk contributions or risk budgets of portfolios with non-normal underlying returns.

Keywords: Risk management, risk budgeting, VaR contribution

JEL Classification: G10, G11