基于Markov模型的反向抵押贷款及其连结长期护理保险产品的定价研究

In this project,our research will focus on the pricing principles, models and algorithms for RMs (Reverse Mortgages, abbreviate RMs) and products integrating RM with LTCI (Long-term Care Insurance, abbreviate LTCI) based on Markov models from different perspectives. Firstly, we design RMs and products integrating RM with LTCI adapted to China and expound their operation mechanisms and risk managements. The Markov models are built to describe the insurance actuarial structures of RMs and products integrating RM with LTCI. Secondly, from perspective of the loan institution's risk preference, we build the utility pricing theories for RMs and products integrating RM with LTCI and the optimal investment strategies for the loan institution with the utility principle. Thirdly, from perspective of the loan institution's multiperiod operation risk, controlling the level of multiperiod operation risk, we build the multiperiod risk control pricing theories for RMs and products integrating RM with LTCI and the optimal asset allocation strategies for the loan institution. Then, from perspective of the fully competitive market, we generalize the classic pricing theories for RMs to those based on Markov models, and develop the insurance actuary pricing theory, the option pricing theory and the reinsurance pricing theory for RMs and products integrating RM with LTCI. Finally, we investigate Tianjin as an example; improve the designs of RMs and products integrating RM with LTCI; give the empirical studies on their feasibility and efficiency; develop the pricing theories for RMs and products integrating RM with LTCI adapted to China. Our research provides the pricing for RMs and products integrating RM with LTCI with a theoretical basis and technical support and give advices and strategies to provide for the aged.

本项目从多个视角出发在Markov框架下研究RM、RM连结LTCI产品的定价原理、模型和算法。首先，设计适合我国国情的RM、RM连结LTCI产品，论述其运营模式和风险管理。构建Markov模型刻画产品的保险精算结构。其次，从贷款机构的风险偏好视角出发，依据等效用原理创建基于Markov模型的效用定价理论和贷款机构的最优投资策略。然后，从贷款机构的多期运营风险视角出发，在将该风险控制在某一水平下，建立多期风险控制定价理论和贷款机构的最优资产配置策略。之后，从完全竞争市场的视角出发，将RM的经典定价理论推广到Markov框架下，建立基于Markov模型的保险精算、期权和嵌入再保险的定价理论。最后，以天津市为例开展调研，完善产品设计方案，实证研究产品实施的可行性和有效性，创建适应我国国情的RM、RM连结LTCI产品的定价理论。为定价奠定理论基础，提供技术支持。为解决我国的养老问题提供建议和对策。

反向抵押贷款（Reverse mortgage，简写为 RM）是一种专门为拥有住房的老年人提供贷款的创新型金融产品。本项目围绕RM、RM连结LTCI（Long-term care insurance，长期护理保险）产品的设计模式、定价方法和风险管理开展研究工作。首先，探寻合适的模型用于刻画RM、RM连结LTCI产品涉及的主要风险因素：房价、利率、病残和寿命风险。基于天津市二手房销售价格指数月度数据，构建房价的多种时间序列模型，采用AIC准则、BIC准则和MAPE指标评价模型的优劣，无论是在哪种评价准则之下可加AR模型均最优。利用Markov模型刻画健康、病残和死亡风险，利用正向随机微分方程刻画利率风险。其次，创建了单生命的无赎回权的RM产品的设计模式、定价原理、模型和算法。利用期望收支平衡定价原理，推导出变额年金满足的定价方程，获得了趸领、递增（递减）年金和常数年金的解析形式的定价公式，讨论了年金、趸领额和年金支付因子关于房价、利率和寿命风险所涉及的主要参数的单调性，计算出房价、利率和寿命的不同风险组合水平下的趸领、年金和年金支付因子的估值，定量评价了各风险因素对于趸领、年金和年金支付因子的影响。然后，构建了剩余寿命风险相依的双生命的RM产品的运营模式雏形、定价原理、模型和算法。获得了趸领额、联合年金、递增（或递减）年金和常数年金的显式定价公式，探讨了房价、利率、寿命、双生命的相依性、售房延时和联合年金的比例等对于联合年金的影响。夫妇寿命相依性假设下的年金高于夫妇寿命独立性假设下的年金，平均增幅大约为4%，最高增幅超过了8%。最后，建立了单生命的RM连结LTCI产品的运营模式、定价原理、模型和算法。建立了嵌入三个状态的Markov模型的期望收支平衡定价原理，构建了该组合产品的终身变额年金的定价模型，作为特例，我们获得了递增（或递减）终身年金、状态年金和常数年金的解析定价公式，给出了趸领、年金和年金支付因子的数值结果，对上述参数进行了灵敏性分析。以上三类定价模型的数值结果均表明：房价的平均收益率对于年金具有决定性的影响，利率的平均回复水平次之。