國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
1. Introduction and Motivation
According to the publication-Sigma of Swiss Re, the life expectancy of human around the world will increase 0.2 year per year. Because the mortality rate has improved rapidly for the past decades, longevity risk has become an important topic.
In the past two decades, a wide range of mortality models have been proposed and discussed (Lee-Carter, 1992; Brouhns et al., 2002; Renshaw and Haberman, 2003;
Koissi et al., 2006; Melnikov and Romaniuk, 2006; Cairn, Blake and Dowd, 2006, 2007 ). Among them, the Lee-Carter ( LC) (1992) model is probably the most popular choice, because it is easy to implement and provides acceptable prediction errors.
Constructing delicate mortality model for the use of pricing is one solution to hedge longevity risk for both life insurance and annuity products. However, this solution is often difficult to apply into practice because of market competition. Even though insurance companies have ability to build a delicate mortality model to catch the actual future mortality improvement, they may not be able to price and sell annuity products using the mortality rate derived from this mortality model since it might be too expensive to sell these annuity products with market competition. In Lee-Carter model, we call the mortality rate without the improvement effect as period mortality rate and call the mortality rate with improvement effect as cohort mortality rate. The true mortality rate of a person is followed by cohort mortality rate, but insurance companies usually use period mortality rate to price their products. The inaccurate mortality assumption leads to major risk of insurance companies because the annuity products have longer payout period and larger liability cost than our expectation.
Another possible solution to hedging longevity risk is to use the mortality
derivatives, such as Survival Bonds and Survival Swaps. Blake and Burrows (2001) propose the concept of Survival Bond and insurance companies can hedge longevity
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
risk based on it. Cairn, Blake and Dowd (2006) propose Survival Swaps, which is a contract to exchange cash flows in the future based on the survivor indices. Although the concept of mortality derivatives are easy and convenient to use but there are still many obstacles in mortality derivative. The special purpose vehicles must pay close attention on their customers and counterparty, and that means insurance companies have to pay big transaction costs on mortality derivatives. Furthermore, we can hardly find the mortality derivatives in the present market.
Another solution is natural hedging. Insurance companies can optimize the
collocation of its products, annuities and life insurances, to hedge longevity risk. This approach can be done internally in an insurance company. Therefore it is more
convenient and practical for insurance company to hedge longevity risk by using this method. Natural hedging is a relatively new topic in actuarial field, so few papers have studied this issue. Wang, Yang and Pan (2003) investigate the influence of the changes of mortality factors and propose an immunization model to hedge mortality risks. Cox and Lin (2007) indicate that natural hedging utilizes the interaction of life insurance and annuities to a change in mortality to stabilize aggregate cash outflows.
And they drew a conclusion that natural hedging is feasible and mortality swaps make it available widely. Wang, Huang, Yang and Tsai (2010) analyze the immunization model mentioned above and use effective duration and convexity to find the optimal product mix for hedging longevity risk. However, their paper uses the same mortality rate (population mortality rate) for the pricing of both life insurance and annuity products due to lack of experience data. This is definitely not true in practice because it will lead to mispricing of both annuity and life insurance products.
Different from those previous literatures, we integrate both uncertainties of mortality rate and interest rate in our model. In addition, we use the experienced
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
mortality rates from life insurance companies rather than population mortality rates.
This experienced mortality data set includes more than 50,000,000 policies which are collected from the incidence data of the whole Taiwan life insurance companies.
Because we don’t have real annuity mortality data, we regard the experience mortality rate of life insurance policies with heavy principal repayment as annuity mortality rate.
Both experience mortality rates between with and without principal repayment are different but imperfectly correlated. Therefore, it is impossible to perfectly hedge longevity risk as Wang, Huang, Yang and Tsai (2010) under our mortality assumption.
Besides, we consider the pricing differences in those insurance products between period-mortality basis and cohort-mortality basis. Our objective is to minimize the variation of the change of total portfolio’s value and the differences between period-cohort pricing bases. On the basis of these experienced mortality rates, the proposed model in this paper provide an optimal collocation of insurance products and effectively apply the natural hedging strategy to a more general portfolio for life insurance companies.
1.1. Agenda
The paper proceeds as follows: In Section 2, we review the mortality and interest model settings used in this article and propose our portfolio model. In Section 3, we take a look at our data, parameters of mortality and interest rate model. Section 4 contains the numerical analysis of our model. Section 5 is the conclusion and suggestion.