Some properties of Euler capital allocation

TL;DR

This paper explores properties of Euler capital allocation for risk measures like VaR and Expected Shortfall, highlighting non-monotonicity issues and proposing simulation methods for estimation.

Contribution

It provides new insights into the behavior of Euler capital allocation with VaR and ES, including non-monotonicity and estimation techniques using simulation and MCMC.

Findings

Capital allocation with VaR can be non-monotonous.

Using the same risk measure for allocation and evaluation is recommended.

Simulation and MCMC methods can effectively estimate risk-adjusted capital.

Abstract

The paper discusses capital allocation using the Euler formula and focuses on the risk measures Value-at-Risk (VaR) and Expected shortfall (ES). Some new results connected to this capital allocation is known. Two examples illustrate that capital allocation with VaR is not monotonous which may be surprising since VaR is monotonous. A third example illustrates why the same risk measure should be used in capital allocation as in the evaluation of the total portfolio. We show how simulation may be used in order to estimate the expected Return on risk adjusted capital in the commitment period of an asset. Finally, we show how Markov chain Monte Carlo may be used in the estimation of the capital allocation.