# Static and dynamic risk capital allocations with the Euler rule

## Tim J. Boonen

#### Need to know

• In static risk capital allocation problems with Value-at-Risk, the Euler rule is very volatile.
• In dynamic applications, the Euler rule is also very sensitive to measurement error.
• These findings also hold for Expected Shortfall, but are less pronounced.
• A solution is to first fit a given distribution, and then allocate risk capital theoretically.

#### Abstract

Risk capital allocations are of central importance in performance measurement. A popular solution concept in the academic literature is the Euler rule. This paper studies the volatility of the Euler rule for capital allocation in static and dynamic empirical applications with a simulated history. The Euler rule is not continuous with respect to small changes in the underlying risk capital allocation problem. We show that, when combined with value-at-risk, the Euler rule is very sensitive to empirical measurement error. The use of a known distribution with estimated parameters helps to reduce this error. The Euler rule with an expected shortfall risk measure is less volatile, but it is still more volatile than the proportional rule.