Size-Biased Transform and Conditional Mean Risk Sharing, With Application to P2p Insurance and Tontines

Abstract

Using risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

Large-Loss Behavior of Conditional Mean Risk Sharing

“…In this section, we consider the situation investigated in the numerical illustration proposed by Denuit (2019) that motivated the present study. The aim is to explain the empirical findings in that paper and to provide preliminary results in the discrete case (that appear to be of independent interest).…”

“…See, for example, Abdikerimova and Feng (2019) and the references therein. The conditional mean risk sharing rule appears to be a very convenient way to distribute retained losses among participants, as shown by Denuit (2019).…”

“…The size-biased transform appears to be useful to study the conditional mean risk sharing rule, as pointed out in Denuit (2019). Given a nonnegative random variable X with distribution function F X and strictly positive expected value…”

“…According to this rule, each participant to an insurance pool contributes the conditional expectation of the loss brought to the pool, given the total loss experienced by the entire pool. The properties of the conditional mean risk allocation have been studied in Denuit (2019), and the present study originates from the empirical findings in the numerical illustration contained in that paper.…”

Investing in your own and peers’ risks: the simple analytics of P2P insurance

“…Intuitively speaking, participants pooling smaller risks should contribute less to the realized total loss. Denuit (2019a) proposed a simple and mathematically correct solution based on the conditional mean risk sharing rule introduced by Denuit and Dhaene (2012). Being based on the concept of mean value, this sharing rule turns out to be quite intuitive as most participants have at least a vague idea about averaging and its risk-reducing effect.…”