在不确定和市场剧烈波动的时期，更难以发现适当的投资和储蓄机会，特别是在一个利率极低的体制下，例如过去十年世界大多数经济体所经历的那种体制。在这种背景下，人寿保险政策可以代表一个有趣的工具，结合退休计划，保护生物识别风险，如死亡，和财务（递延税款）回报。尽管如此，这些政策在估值、风险管理和现金流透明度方面仍面临重大挑战，因为影响收益结构及其价值的风险来源之间存在若干相互依赖关系。这种复杂性要求先进的建模方法和估价技术劳拉·巴洛塔恩斯特·埃伯林、托尔斯滕·施密特和拉吉德·泽尼丁正在创新。

In this work, we focus on a particular class of such insurance policies: variable annuities (VAs). VAs are long-term insurance policies providing a variety of benefits, the value of which can vary based on the performance of an underlying reference fund tied to the financial market. These products are quite common in North-American markets, Japan, the UK, and are increasingly present in other European markets as well. The popularity of these contracts is in general due to the fact that they offer the policyholder a participation in the growth of the economy, and at the same time the return is guaranteed by the insurer not to fall below a fixed minimum rate.

Due to their design, VAs issuers are exposed to financial and demographic risks, as well as surrender risk originated by the policyholder behaviour, which might not be completely rational from the strict financial point of view. These features pose a significant challenge for both insurance companies and policyholders in terms of understanding of the interdependencies between these risks, their appropriate quantification and their impact on the valuation and management of these policies.

Our aim is to propose a realistic integrated model of the financial and insurance risk (i.e. mortality and surrender) for the development of a market consistent valuation framework for popular VAs designs, such as point-to-point and ratchet schemes. We cast this model in a hybrid setting in order to accommodate for both the dependence between the equity and the fixed income market, and between the financial risks and surrender risk. Surrender risk is captured via an intensity-based approach, which takes into account both the policyholder personal contingencies, and the spread between the return offered by the policy and the one offered on the market for equivalent products. Finally, the model is sufficiently general to accommodate a large variety of distributions.

In this general setting, we derive explicit expressions – up to a multidimensional integral – for the valuation formulas by means of Fourier-based techniques. The dimensionality depends on the frequency with which early termination is allowed based on the terms of the contract. From the computational point of view, this dimensionality problem is resolved by means of Monte Carlo integration with importance sampling. For accurate valuation, care has to be paid to the choice of the importance sampling distribution, which is intrinsically linked to the shape of the surrender intensity function.

By means of the developed computational procedures, we explore the role of the contract parameters in controlling for the risk/return trade-off in the policy. This can prove useful to both the insurer and the policyholder in order to gain insight into the terms of the scheme. Further, we explore different constructions for the surrender intensity. This is of particular interest due to the large percentage of early terminations experienced by the issuers, and the low amount of reliable data on actual termination causes, which makes modelling particularly challenging and largely dependent on expert judgement. Therefore, our results can help with the assessment of the related model risk.

关于作者

劳拉·巴洛塔是英国伦敦大学城市卡斯商学院金融系金融数学专业的读者，德国弗赖堡高等研究院（FRIAS）的外部高级研究员。

Ernst Eberlein是弗赖堡大学数学随机学名誉教授，德国弗赖堡高等研究所（FRIAS）高级研究员。

托尔斯滕施密特是弗赖堡大学数学随机学教授，德国弗赖堡高等研究所（FRIAS）高级研究员。

拉吉德·泽伊内丁在法国巴黎多芬大学的Ceremake工作。