Some results on the CTE-based capital allocation rule

Abstract

Tasche [Tasche, D., 1999. Risk contributions and performance measurement. Working paper, Technische Universität München] introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk. Panjer [Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo, Research Report 01-15] derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman and Valdez [Landsman, Z., Valdez, E., 2002. Tail conditional expectations for elliptical distributions. North American Actuarial J. 7 (4)] generalize Panjer's result to the class of multivariate elliptical distributions.In this paper we provide an alternative and simpler proof for the CTE-based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.

Optimal Capital Allocation Principles

“…Landsman and Valdez (2003) extend these explicit capital allocation formulas to the case where risks belong to the class of multivariate elliptical distributions, for which the class of multivariate normal is a special case. Dhaene et al (2008) rederive the results of Landsman and Valdez (2003) in a more straightforward manner and apply these to sums that involve normal as well as lognormal risks. In Valdez and Chernih (2003), expressions for covariance‐based allocations are derived for multivariate elliptical risks.…”

Allocation of risk capital based on iso-entropic coherent risk measure

“…A special case is that when the risk measure is Expected Shortfall (Dhaene et al, 2008). In that case, the marginal risk contribution, if it exists, is given by…”

“…He points that the suitable allocation method satisfying all the axioms must be based on the risk measure with the property of subadditivity and positive homogeneity. Dhaene, Henrard, Landsman, Vandendorpe and Vanduffel (2008) proposes a kind of allocation rule based on CTE rule, where the distribution of risk is elliptical, and presents the approximate solutions for normal distribution and log-normal distribution. Kim and Hardy (2009) suggest an allocation rule based on solvent capacity, which satisfies the requirement of the limited liability of shareholders.…”

A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations

“…Martin (2007) suggest the Euler allocation to investigate the influence of systematic factors on the portfolio risk. Both Kurth and Tasche (2003) and Dhaene et al (2008) consider the calculation of risk contributions for VaR and ES. The focus in Kurth and Tasche (2003) lies on the calculation for classical credit portfolio models and Dhaene et al (2008) derive analytical formulas for elliptical distribution and approximations for special distributions.…”

“…Both Kurth and Tasche (2003) and Dhaene et al (2008) consider the calculation of risk contributions for VaR and ES. The focus in Kurth and Tasche (2003) lies on the calculation for classical credit portfolio models and Dhaene et al (2008) derive analytical formulas for elliptical distribution and approximations for special distributions. Ordinary Monte Carlo simulation methods are impractical for determining the Euler risk contribution of VaR and ES because these depend only on rare events of the simulation.…”