Quantitative methods for modelling and pricing dependenent insurance risks

想象一下，一堆20年期的、每年支付的共同终身年金——发给已婚夫妇。养老金领取者的未来寿命按《加拿大2000 - 2002年完整生命表》2a和2b计算，实际利率为平均值。7%，根据世界银行2008 - 2012年加拿大数据。简单的计算表明，在生命周期独立的标准假设下（与更现实的强正依赖性假设相反），合约的价格明显偏低。利差随着年龄的增长而增长，例如一对典型的65岁老人和60岁女性夫妇的利差为12%。由于池中年金的随机支付本身（相互）依赖，减值变得相当令人生畏。更普遍的是，忽视风险组合内部和之间的依赖关系，可能而且经常导致偿付能力问题，甚至导致机构破产。

Imagine a pool of 20-year term annually-paid joint life level annuities-due issued to married couples. The future lifetimes of the annuitants are per Complete Life Tables Canada 2000 – 2002, 2a and 2b, and the real interest rate is an average of .7%, as per World Bank’s 2008 – 2012 Canadian data. Simple calculations show that under the standard assumption of independence of lifetimes (as opposed to a more realistic one of strong positive dependence), the contracts are significantly underpriced. The spreads grow with age and are, for instance, 12% for a typical couple of a 65 old man and a woman aged 60. As the random payments due to the annuities in the pool, are themselves (inter)dependent, the impairment becomes rather daunting. More generally, neglecting dependencies within and among portfolios of risks can and often does lead to solvency issues and even bankruptcy of institutions. Notwithstanding, traditional actuarial models rest on the assumption of independence. Unrealistic as it is, the assumption often allows for convenient simplifications and thus guarantees a desirable level of analytic tractability. In the proposed research, I will depart from this assumption. I will formulate meaningful ways to describe dependencies amongst insurance risks and study the implications of such dependencies. I will argue that the rules assigning prices to insurance risks must take into consideration their dependencies on companion risks, as well as on other exogenous factors. I will demonstrate that the just-mentioned pricing rules do not necessarily lead to unbearable intractabilities and are worthy a try, in particular given the utterly adverse consequences of their less sophisticated alternatives.