| Longevity risk refers to the risk that people’s average actual life expectancy is higher than life expectancy in the future.It can be divided into two categories:individual longevity risk and aggregate longevity risk.Individual longevity risk refers to the fact that individuals spend more than their accumulated wealth in their lifetime.Such risk can be managed by participating in the government’s social endowment insurance or purchasing commercial endowment insurance.Aggregated longevity risk refers to a group whose average life expectancy exceeds the expected life expectancy.This risk is a systematic risk that cannot be dispersed according to the law of large numbers.International research on longevity risk usually refers to aggregated longevity risk.The longevity risk studied in this paper also refers to aggregated longevity risk.With the increasing life expectancy of the population,longevity risk poses a serious challenge to the safe operation of China’s insurance system.It has become an urgent task for our government to manage longevity risk.Over the past two decades,longevity risk securitization has become an important innovation in the field of longevity risk management.It is an active area in life insurance securitization,mainly focusing on the design and pricing of longevity bonds and longevity swaps.Aiming at the problem of longevity risk in China,this paper studies the pricing of longevity risk securitization on the basis of uncertainty theory and Actuarial theory,synthesizing the theoretical knowledge of different disciplines such as finance,financial engineering and insurance.The specific research results are as follows:(i)The uncertain measure and differential equation are used to characterize the survival index,and the uncertain survival index is proposed.The uncertain Wang transform pricing method is extended to the uncertain theory framework.The pricing model of uncertain longevity bonds is established based on the uncertain one-factor Wang transform pricing method,and an example is analyzed.Firstly,from both theoretical and practical aspects,this paper comprehensively analyses that the concepts of random probability and fuzzy are not suitable for describing uncertain phenomena,and attempts to use uncertain measure and uncertain differential equation to characterize the changing characteristics of survival index.The definition of uncertain survival index is proposed,and the prediction model of uncertain survival index is constructed.Considering the interest rate risk in longevity bonds,we use the uncertain Vasicek interest rate model to derive the expression of discount factor.In this paper,Wang transform under stochastic framework is extended to the framework of uncertainty theory,and uncertain Wang transform pricing method is proposed.Then the pricing formula of uncertain longevity bonds under uncertain one-factor Wang transform pricing method is deduced,and the pricing model of uncertain longevity bonds is constructed.An uncertain longevity bond product is designed,its detailed steps and the algorithm process of price formula are given,and the applicability and rationality of the model are tested by using the data of life table of China’s life insurance industry for example analysis and sensitivity analysis.(ⅱ)Based on the uncertain survival index and uncertain one-factor Wang transform pricing method,the pricing model of uncertain longevity swap is constructed and an example is analyzed.Firstly,this paper analogizes interest rate swap,elaborates and analyses the operation mechanism of longevity swap.Based on the uncertain survival index and the uncertain one-factor Wang transform pricing method,the pricing formula of uncertain longevity swap is deduced,and the pricing model of uncertain longevity swap is constructed.This paper designs an uncertain longevity swap product,gives its detailed steps and the algorithm process of swap price,uses the data of life table of China’s life insurance industry to analyze the pricing examples,and compares the similarities and differences between uncertain longevity swap and uncertain longevity bond.(ⅲ)The uncertain renewal process with jump and the uncertain differential equation with jump are used to characterize the mortality index,and the uncertain mortality index with jump is constructed.The pricing model of uncertain longevity bonds with jump is established based on the uncertain one-factor Wang transform pricing method.Firstly,a new jumping uncertain mortality index is proposed.The jumping uncertain renewal process and the uncertain differential equation are used to describe the change of mortality rate and its thick tail characteristics of the target population,and a jumping uncertain mortality index prediction model is constructed.Five commonly used incomplete pricing methods for longevity bonds with jumps are compared comprehensively,and Wang transform pricing method is more ideal.By using the uncertain one-factor Wang transform pricing method,the pricing formulas of two kinds of longevity bonds with jump uncertainties,i.e.non-accumulative and accumulative principal repayment,are derived,and the pricing model of longevity bonds with jump uncertainties is established.Two kinds of uncertain longevity bond products with jump are designed,which are non-accumulative bond and accumulative bond with principal repayment.The uncertain longevity bond with jump is compared with uncertain longevity bond.(iv)Based on the uncertain mortality index with jump and the uncertain one-factor Wang transform pricing method,the uncertain longevity swap pricing model with jump is constructed.Firstly,this paper divides the longevity swaps with jumps into three types according to their nature:ordinary,natural hedging and triggering,and elaborates their operation mechanism.Based on the uncertain mortality index with jumps,the pricing formula of uncertain longevity swaps with jumps under the uncertain one-factor Wang transform pricing method is derived,and the pricing model of uncertain longevity swaps with jumps is constructed.The uncertain longevity swap is compared with the uncertain longevity swap and the uncertain longevity bond with jump. |
