An experimental test of the descriptive adequacy of the restated diversification principle is presented. The principle postulates that risk-averse utility maximizers will pool risks for their mutual benefit, even if information is missing about the probabilities of losses. It is enough for people to assume that they face equal risks when they pool risks. The results of the experiment support the principle.

We consider the conditional mean risk allocation for an insurance pool, as defined by Denuit and Dhaene (2012). Precisely, we study the asymptotic behavior of the respective relative contributions of the participants as the total loss of the pool tends to infinity. The numerical illustration in Denuit (2019) suggests that the application of the conditional mean risk sharing rule may produce a linear sharing in the tail of the total loss distribution. This paper studies the validity of this empirical finding in the class of compound Panjer–Katz sums consisting of compound Binomial, compound Poisson, and compound Negative Binomial sums with either Gamma or Pareto severities. It is demonstrated that such a behavior does not hold in general since one term may dominate the other ones conditional of large total loss.

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.

Economic issues will be key determinants of the outcome of the Scottish referendum on independence. Pensions are a key element of the economic case for or against independence. The costs of funding pensions in an independent Scotland would be influenced by mortality risks, the costs of borrowing and the segmentation of costs and risks (i.e. pricing to Scotland's experience rather than pooled across UK experience). We compare the overall costs of providing pensions in an independent Scotland against the resources that are available to cover these costs. Scotland has worse mortality experience than the UK as a whole, and Scottish government debt is likely to attract a liquidity premium relative to UK government debt. An independent Scottish government would have to create a bond market for public debt. The liquidity premium would make pensions cheaper to buy, but taxpayers or the consumers of public services would have to pay the cost.