Estimation and allocation of insurance risk capital
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Date
2007-05-15T13:06:52Z
Authors
Kim, Hyun Tae
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
Estimating tail risk measures such as Value at Risk (VaR) and Conditional Tail Expectation
(CTE) is a vital component in financial and actuarial risk management.
The CTE is a preferred risk measure, due to coherence and a widespread acceptance
in actuarial community. In particular we focus on the estimation of the CTE using
both parametric and nonparametric approaches.
In parametric case the conditional tail expectation and variance are analytically
derived for the exponential distribution family and its transformed distributions.
For small i.i.d. samples the exact bootstrap (EB) and the influence function are
used as nonparametric methods in estimating the bias and the the variance of the empirical
CTE. In particular, it is shown that the bias is corrected using the bootstrap
for the CTE case. In variance estimation the influence function of the bootstrapped
quantile is derived, and can be used to estimate the variance of any bootstrapped
L-estimator without simulations, including the VaR and the CTE, via the nonparametric
delta method. An industry model are provided by applying theoretical findings
on the bias and the variance of the estimated CTE.
Finally a new capital allocation method is proposed. Inspired by the allocation
of the solvency exchange option by Sherris (2006), this method resembles the CTE
allocation in its form and properties, but has its own unique features, such as managerbased
decomposition. Through a numerical example the proposed allocation is shown
to fail the no undercut axiom, but we argue that this axiom may not be aligned with
the economic reality.
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Keywords
risk capital, capital allocation