Extending the Logistic Model for Mortality Forecasting and the Application of Mortality-Linked Securities Hong-Chih, Huang Yawen, Hwang Department of Risk Management and Insurance National Chengchi University, Tiapei, Taiwan, R.O.C.

1.Introduction Mortality risk is an important issue 1. The increasing demand for individual annuities 2. The improvement of mortality rate How to construct adequate mortality model Considering the effects of ages and years The improvement trends of different ages should be different. How to management and hedge the mortality risk Transferring the mortality risk to the financial market by issuing mortality-linked bond

1.Introduction Mortality model Extending logistic mortality model and appling this model to fit and forecast the mortality data of Taiwan, USA, and Japan Mortality risk Mortality-linked bond combining with collateral debt obligation

2.Literature review-mortality model Gompertz(1825) Makeham(1860) Heligman& Pollard(1980) Mortality model with 8 parameters Static model, these models only consider the effect of ages

2.1.Mortality model Lee-Carter(1992) Reduction Factor Model Dynamic model, these model consider the effect of age and year

2.1.Mortality model Wilmoth (1993) Lee and Chou (2002) Luciano and Vigna (2005) Tzeng and Yue (2005) Dahl and Moller (2006) Cairns, Blake & Dowd (2006)

2.2 Securitization of mortality risk Blake & Burrows (2001) derived the concept of longevity bond Swiss Re. (2003) issued the first mortality bond European investment bank (2004) issued the first longevity bond Cowley & Cummins(2005) show that securitization may increase a firm’s value Lin & Cox (2005) study and price the mortality bonds and swaps Cairns, Blake & Dowd(2006) show how to price mortality-linked financial instruments such as the EIB bond Blake et al. (2006) Introduce five types of longevity bonds

2.2 Securitization of mortality risk In this paper, we apply the extended logistic mortality model to price the longevity model. Furthermore, we introduce the structure of collateral debt obligation to the longevity model. We hope to increase the purchasing appetence of longevity bond by complicating the concept of longevity bond. Lin & Cox (2005)

3.1 Logistic mortality model senescent death ratebackground death rate Thus, this model is a dynamic model. It consider the effects of age and year. Bongaarts(2004) proposes a logistic mortality model as follows: We assume the mortality rate follows Eq(1)

3.1 Logistic mortality model We choose the MAPE (Mean Absolute Percentage Error) to measure the efficiency of fitting and forecasting. According to Lewis (1982), the standard of MAPE is described as following table:

3.2 Data and restriction Data source: 1. Taiwan: the department of statistic, ministry of interior of Taiwan ( ) 2. USA and Japan: Human mortality database ( ) Restriction: Logistic model is an increasing function which is not suitable on 0-year to 1-year old. Therefore, we fit and forecast the mortality rate of single age from 30-year to 89-year from 1982 to 2000.

3.3 Numerical analysis The fitting MAPE: Furthermore, we forecast the mortality data in 2001 for Taiwan’s female, the forecasting MAPE is %.

3.3 Numerical analysis

3.4 Improvement of logistic model Method 1 We think that people with different ages should face different background death rates on the same year. Thus, we segment into and. We also apply the same idea on and. We describe the improvement model of as follows.

3.4.1 Method 1

3.4.2 Method 2 Subsequently, we fit the mortality rate data by segment three parameters. We show that the fitting MAPEs in method 2 are decreasing. In other words, the improvement trends of different ages are different.

3.4.2 Method 2 Now, we use the extended logistic mortality model of Method 2 to forecast the mortality data on Taiwan, Japan and USA. Data: (1)Taiwan ( 2001, 30-year ~ 89-year oldhttp:// (2)Japan ( 2001~2004, 30-year ~ 89-year oldwww.mortality.org (3)USA ( 2001~2003, 30-year ~ 89-year oldwww.mortality.org The forecasting effects are good.

4. Securitization of longevity risk 4.1 Longevity bond Insurer & SPV is the survivor index. is the real survivor rate. is the payment from SPV to insurer at time t.

Not equivalent 4.1 Longevity bond SPV & Investor If SPV payment claim to insurer, then the principal of Tranche B is decreasing at time t. The principal of Tranche A will be deduct when is zero. Therefore, Tranche B is more risky than B. That is. Coupon c A Coupon c B

4.2 Numerical analysis 1.Issuing N policies of Taiwan’s female. 2.Insured age is x. 3.Insurer have to payment Ann. to the policyholder who is still alive at time. 4.Suppose the survivor index as the Taiwan’s annuity table multiplied a constant d= Using to forecast the real survivor rate. 6.The interest rate is as the following Vasicek model (1977)

4.2 Numerical analysis parameters

4.2 Numerical analysis The premium P is 3,171,936 NTD. If the initial price of longevity bond is at premium 20 percentage, which is V=9,000,000 NTD, then the fair coupon rates of Tranche A and Tranche B are 6.957% and 8.443%.

5. Conclusion 1. In this study, we extend the logistic model to fit and forecast the mortality data of Taiwan, Japan and USA. 2. We use the concept of segment to improvement the logistic model. Because the improvement trends of different ages are not equivalent. 3. We show that the fitting and forecasting effects are good in the improvement logistic mortality model. 4. We design a new LBs by introducing the structure of collateral debt obligation to increase the purchasing appetence.

Q & A