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
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Department of Risk Management and Insurance
National Chengchi University, Tiapei, Taiwan, R.O.C.
Extending logistic mortality model and appling this model
to fit and forecast the mortality data of Taiwan, USA, and
Mortality-linked bond combining with collateral debt
Mortality model with 8 parameters
Static model, these models only
consider the effect of ages
Dynamic model, these model consider the effect of age and year
Blake & Burrows (2001)
derived the concept of longevity bond
Swiss Re. (2003)
issued the first mortality bond
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
show how to price mortality-linked financial instruments such as the EIB bond
Cairns, Blake & Dowd(2006)
Blake et al. (2006)
Introduce five types of longevity bonds
Lin & Cox (2005)
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.
We assume the mortality rate follows Eq(1)
Bongaarts(2004) proposes a logistic mortality model as follows:
senescent death rate
background death rate
Thus, this model is a dynamic model.
It consider the effects of age and year.
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:
1. Taiwan: the department of statistic, ministry of interior of Taiwan
2. USA and Japan: Human mortality database
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.
The fitting MAPE:
Furthermore, we forecast the mortality data in 2001 for Taiwan’s female,
the forecasting MAPE is 5.9883%.
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.
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.
Now, we use the extended logistic mortality model of Method 2 to
forecast the mortality data on Taiwan, Japan and USA.
Data: (1)Taiwan (http://www.moi.gov.tw/stat/index.asp): 2001, 30-year ~ 89-year old
(2)Japan (http://www.mortality.org): 2001~2004, 30-year ~ 89-year old
(3)USA (http://www.mortality.org): 2001~2003, 30-year ~ 89-year old
The forecasting effects are good.
4.1.1 Insurer & SPV
is the survivor index. is the real survivor rate.
is the payment from SPV to insurer at time t.
4.1.2 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 .
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%
Q & A the Application of Mortality-Linked Securities