An analysis of pricing and basis risk for industry loss warranties

Abstract

In recent years, industry loss warranties (ILWs) have become increasingly popular in the reinsurance market. The defining feature of ILW contracts is their dependence on an industry loss index. The use of an index reduces moral hazard and generally results in lower prices compared to traditional, purely indemnity-based reinsurance contracts. However, use of the index also introduces basis risk since the industry loss and the reinsured company’s loss are usually not fully correlated. The aim of this paper is to simultaneously examine basis risk and pricing of an indemnity-based industry loss warranty contract, which is done by comparing actuarial and financial pricing approaches for different measures of basis risk. Our numerical results show that modification of the contract parameters to reduce basis risk can either raise or lower prices, depending on the specific parameter choice. For instance, basis risk can be reduced by decreasing the industry loss trigger, which implies higher prices, or by increasing the reinsured company attachment, thus inducing lower prices.

Appendix

For the stochastic process as given in (2), one obtains

$$S_{1} = \exp( N( \ln( S_{0} ) + \mu_{S} - 0.5\sigma_{S}^{2},\sigma_{S}) ) = \exp( N( a,b ) ),$$

where N(a,b) denotes a normally distributed random variable with expected value a=ln (S ₀)+μ−0.5σ ² and standard deviation b=σ _S , leading to a lognormal distribution for S ₁. Given the expected value

$$E( S_{1} ) = \exp( a + 0.5b^{2} )$$

and the standard deviation

$$\sigma( S_{1} ) = \sqrt{( \exp ( b^{2} ) - 1 )E( S_{1} )^{2}},$$

σ _S and S ₀ can be obtained by transforming these equations. Using

$$a = \ln( E( S_{1} ) ) - 0.5b^{2},$$

and

$$b = \sqrt{\ln \biggl( 1 + \frac{\sigma ^{2}( S_{1} )}{ E( S_{1} )^{2} }\biggr)},$$

the standard deviation of the stochastic process σ _S and the initial nominal value S ₀ are given by

A derivation of I ₀ and σ _I for the industry loss distribution I ₁ can be done analogously.