Reply to Jiandong Ren on Their Discussion on the Paper Titled “Size-Biased Risk Measures of Compound Sums”

I am grateful to Jiandong Ren for providing readers with a unified treatment of compound sums with frequency component in the ða, b, 0Þ class of counting distributions, which is central to insurance studies. This offers a deeper understanding of the underlying structure of this family, compared to the separate treatment of the Poisson and negative binomial cases in the paper (the latter being treated as a Poisson mixture). Therefore, I sincerely thank Jiandong Ren for having supplemented the initial work with these brilliant ideas. As stressed at the end of the discussion, the Panjer algorithm is particularly useful to compute tail risk measures. In addition to exact calculations, the approximations derived by Denuit and Robert (2021) in terms polynomial expansions (with respect to the Gamma distribution and its associated Laguerre orthonormal polynomials or with respect to the Normal distribution and its associated Hermite polynomials when the size of the pool gets larger) may also be useful in the present context. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. Compound sums with ða, b, 0Þ frequency component are also considered as an application in that paper and the proposed method is compared with the well-established Panjer recursive algorithm.