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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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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
1.Introduction the Application of Mortality-Linked Securities

  • Mortality risk is an important issue

    • The increasing demand for individual annuities

    • 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 introduction1
1.Introduction the Application of Mortality-Linked Securities

  • 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
2.Literature review-mortality model the Application of Mortality-Linked Securities

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
2.1.Mortality model the Application of Mortality-Linked Securities

Lee-Carter(1992)

Reduction

Factor Model

Dynamic model, these model consider the effect of age and year


2 1 mortality model1
2.1.Mortality model the Application of Mortality-Linked Securities

  • 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
2.2 Securitization of mortality risk the Application of Mortality-Linked Securities

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

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


2 2 securitization of mortality risk1
2.2 Securitization of mortality risk the Application of Mortality-Linked Securities

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.


3 1 logistic mortality model
3.1 Logistic mortality model the Application of Mortality-Linked Securities

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.


3 1 logistic mortality model1
3.1 Logistic mortality model the Application of Mortality-Linked Securities

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
3.2 Data and restriction the Application of Mortality-Linked Securities

Data source:

1. Taiwan: the department of statistic, ministry of interior of Taiwan

(http://www.moi.gov.tw/stat/index.asp)

2. USA and Japan: Human mortality database

(http://www.mortality.org)

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
3.3 Numerical analysis the Application of Mortality-Linked Securities

The fitting MAPE:

Furthermore, we forecast the mortality data in 2001 for Taiwan’s female,

the forecasting MAPE is 5.9883%.


3 3 numerical analysis1
3.3 Numerical analysis the Application of Mortality-Linked Securities


3 3 numerical analysis2
3.3 Numerical analysis the Application of Mortality-Linked Securities


3 3 numerical analysis3
3.3 Numerical analysis the Application of Mortality-Linked Securities


3 3 numerical analysis4
3.3 Numerical analysis the Application of Mortality-Linked Securities


3 4 improvement of logistic model 3 4 1 method 1
3.4 Improvement of logistic model the Application of Mortality-Linked Securities3.4.1 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
3.4.1 Method 1 the Application of Mortality-Linked Securities


3 4 1 method 11
3.4.1 Method 1 the Application of Mortality-Linked Securities


3 4 1 method 12
3.4.1 Method 1 the Application of Mortality-Linked Securities


3 4 2 method 2
3.4.2 Method 2 the Application of Mortality-Linked Securities

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 21
3.4.2 Method 2 the Application of Mortality-Linked Securities

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 securitization of longevity risk 4 1 longevity bond
4. Securitization of longevity risk the Application of Mortality-Linked Securities4.1 Longevity bond

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 longevity bond
4.1 Longevity bond the Application of Mortality-Linked Securities

4.1.2 SPV & Investor

Coupon cA

Not equivalent

Coupon cB

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 .


4 2 numerical analysis
4.2 Numerical analysis the Application of Mortality-Linked Securities

  • IssuingN policies of Taiwan’s female.

  • Insured age is x.

  • Insurer have to payment Ann. to the policyholder who is still alive at time .

  • Suppose the survivor index as the Taiwan’s annuity table multiplied a constant d=0.7.

  • Using 3.4.2 to forecast the real survivor rate.

  • The interest rate is as the following Vasicek model (1977)


4 2 numerical analysis1
4.2 Numerical analysis the Application of Mortality-Linked Securities

parameters


4 2 numerical analysis2
4.2 Numerical analysis the Application of Mortality-Linked Securities

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
5. Conclusion the Application of Mortality-Linked Securities

  • In this study, we extend the logistic model to fit and forecast the mortality data of Taiwan, Japan and USA.

  • We use the concept of segment to improvement the logistic model. Because the improvement trends of different ages are not equivalent.

  • We show that the fitting and forecasting effects are good in the improvement logistic mortality model.

  • We design a new LBs by introducing the structure of collateral debt obligation to increase the purchasing appetence.


Q & A the Application of Mortality-Linked Securities


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