We design a coherent cohort-based multi-population mortality model, calibrate it to national mortality rates in the Eurozone using Human Mortality Database data, and use it to project developments in national mortality across the Eurozone. Combining this model with a stylized model of social security pensions in each country allows us to calculate the pension mortality risk in these systems and estimate the benefits of pooling it across the Eurozone. We examine three risk pools, which are all actuarially fair, but differ in how undiversifiable risk is allocated across countries. The first naïve approach allocates undiversifiable risk in proportion to GDP, a second according to a CAPM-based measure of the undiversifiable risk each country contributes to the pool and a third ensures that the aggregate benefits of diversification are shared equitably across countries using a measure we adopt. In all cases, the benefits of risk pooling increase over time as mortality uncertainty accumulates, but fall over time as cross-country correlation increases due to the long-term dominance of the mortality trend, which by assumption is shared between countries. The peak benefit occurs around 2050, with an aggregate reduction in the standard deviation of pension expenditures of around 0.11% of GDP, or 3% of pension expenditure at the 99^th percentile. We find that allocating undiversifiable risk proportional to GDP does not ensure an efficient allocation of undiversifiable risk across countries, given that different countries have markedly different pension mortality risk due to different pension system generosities as well as different mortality correlation with the Eurozone. Based on our results we propose a contract design that surmounts most of the moral hazard risks created by the pool, and suggest directions for future research.

The authors would like to thank discussants and participants at Longevity 15 conference in Washington, DC, the World Risk and Insurance Economics Congress (WRIEC 2020), an anonymous referee and the journal editors for useful comments and suggestions. All errors remain our own.

我们设计了一个基于队列的一致的多族裔死亡率模型，使用“人类死亡率数据库”数据将其校准为欧元区的国家死亡率，并将其用于预测整个欧元区的国家死亡率的发展。将该模型与每个国家的社会保障养老金的程式化模型结合起来，可以让我们计算这些系统中的养老金死亡率风险，并估算将其汇总到整个欧元区的好处。我们研究了三个风险池，它们在精算上都是公平的，但是在不同国家之间分配不可分散风险的方式有所不同。第一种幼稚的方法是按GDP比例分配不可分散的风险，第二项是根据基于CAPM的不可分散风险对每个国家的贡献进行的衡量，第三项是确保采用我们采用的衡量方法在各个国家之间公平地分享多元化的总收益。在所有情况下，随着死亡率不确定性的累积，风险汇总的好处随着时间的推移而增加，但是随着跨国相关性的增加，由于死亡率趋势的长期支配地位，风险汇总的好处随着时间的推移而下降，这在国家之间是共享的。最高收益出现在2050年左右，养老金支出的标准差总计减少约占GDP的0.11％，或99养老金支出的3％。但随着时间的流逝，由于死亡率趋势的长期支配地位，跨国相关性增加，而这种下降趋势在国家间是共享的。最高收益出现在2050年左右，养老金支出的标准差总计减少约占GDP的0.11％，或99养老金支出的3％。但随着时间的流逝，由于死亡率趋势的长期支配地位，跨国相关性增加，而这种下降趋势在国家间是共享的。最高收益出现在2050年左右，养老金支出的标准差总计减少约占GDP的0.11％，或99养老金支出的3％。^个百分点。我们发现，按比例分配不可分散的风险并不能确保在国家之间有效分配不可分散的风险，因为不同的国家由于养老金制度的慷慨程度以及与欧元区的死亡率相关性而具有显着不同的养老金死亡率风险。根据我们的研究结果，我们提出了一项合同设计，该合同设计应克服池中产生的大部分道德风险，并为未来的研究提供指导。

作者要感谢华盛顿特区举行的长寿15大会，世界风险和保险经济学大会（WRIEC 2020），匿名裁判和期刊编辑的有益的评论和建议。所有错误均由我们自行承担。