Extended Logistic Model for Mortality Forecasting and the Application of Mortality-Linked Securities
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Extended Logistic Model for Mortality Forecasting and the Application of Mortality-Linked Securities. Yawen, Hwang, Assistant Professor, Dept. of Risk Management and Insurance, Feng Chia University

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Yawen, Hwang, Assistant Professor, Dept. of Risk Management and Insurance, Feng Chia University

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Yawen hwang assistant professor dept of risk management and insurance feng chia university

Extended Logistic Model for Mortality Forecasting and the Application of Mortality-Linked Securities

Yawen, Hwang, Assistant Professor, Dept. of Risk Management and Insurance, Feng Chia University

Hong-Chih, Huang, Associate Professor, Dept. of Risk Management and Insurance, National Chengchi University


1 introduction

1. Introduction

If you have 10 thousand dollars,

you will invest these money into?

Bond

Stock

v. s.

The risk attitude is different with different people.


1 introduction1

1.Introduction

Longevity

Bond

How to enhance the attractiveness of longevity bonds?

Separating it. (From the idea of collateral debt obligation )


1 introduction2

1.Introduction

How to price the longevity bonds?

Need accurate mortality model!

  • The purpose of this study:

  • Modifying the existing mortality models and providing a better mortality model

  • Improving the attractiveness of longevity bonds


2 literature review mortality model

2. Literature review-mortality model

  • Static mortality model

    • Gompertz (1825)

    • Makeham (1860)

    • Heligman & Pollard (1980)

  • Dynamic mortality model

    • Lee-Carter (1992)

    • Reduction Factor Model (1860)

    • Logistic model (Bongaarts, 2005)

    • CBD model (2006)

    • M7 model (2009)

Using two methods to modify the logistic model

Considering the cohort effect, the number of parameters are unavoidable concerns.


2 literature review securitization of mortality risk

2. Literature review- securitization of mortality risk

  • Blake & Burrows (2001)

  • Dowd & Blake (2003)

  • Cowley & Cummins (2005)

  • Blake et al. (2006)

  • Lin & Cox (2005): Wang Transformation

  • Cairns et al. (2006): CBD model

  • Cox et al. (2006): multivariate exponential tilting

  • Denuit et al. (2007): Lee-Carter model


2 literature review securitization of mortality risk1

2 Literature review- securitization of mortality risk

Lin & Cox (2005)

Special Purpose Vehicles

In this paper, we apply the extended logistic mortality models to price longevity bonds.

Furthermore, we introduce the structure of collateral debt obligation to longevity bonds.

We hope to increase the purchasing appetence of longevity bonds by designing it to encompass more than one tranche.


3 1 logistic mortality model

3.1 Logistic mortality model

We assume the mortality rate follows Eq(1)

Bongaarts(2005) proposes a logistic mortality model as follows:

Eq(1)

senescent death rate

background death rate

Thus, this model is a dynamic model.

It considers the effects of age and time.


3 2 modifying methods

3.2 Modifying methods

Method I: Segment approach (from RF model)


3 2 modifying methods1

3.2 Modifying methods

Method II: Background death rate might be related more

reasonably to age.


3 3 mortality models

3.3 Mortality models


3 4 measurement

3.4 Measurement

Measurement

1. MAPE (Mean Absolute Percentage Error)

2. According to Lewis (1982), the standard of MAPE is described as the

following table:


4 1 numerical analysis fitting

4.1 Numerical analysis - Fitting

Data:

1. USA, Japan and England & Wales: Human mortality database

2. Fitting the mortality rates of a single age range from 50-year to 89-year

from 1982 to 2000.


4 1 numerical analysis fitting1

4.1 Numerical analysis - Fitting


4 2 numerical analysis forecasting

4.2 Numerical analysis - Forecasting

  • Data:

  • Forecasting single age range from 50-year to 89-year

  • Japan, England & Wales: 2001~2006

  • USA: 2001~2005


5 1 longevity bond

5.1 Longevity bond

5.1.1 Insurer & SPV

is the survivor index. is the real survivor rate.

is the payment from SPV to insurer at time t.


5 1 longevity bond1

5.1 Longevity bond

5.1.2 SPV & Investor

Coupon cA

Not equivalent

Coupon cB

If SPV pay claim to insurer, then the principal of Tranche B is decreasing

at time t. The principal of Tranche A will deduct when is zero.

Therefore, Tranche B is more risky than A. That is .


5 1 longevity bond2

5.1 Longevity bond

USA Male

  • Survival rate index: (Insurer)

  • Lin & Cox (2005):

  • Survival rate: (SPV)

  • Modified extended logistic (beta) mortality model


5 2 numerical analysis static interest rate

5.2 Numerical analysis – Static interest rate

parameters


5 2 numerical analysis static interest rate1

5.2 Numerical analysis – Static interest rate

We issue the longevity bonds (Tranche A and B) with the price at premium 20%, which is $6,500,000.


5 2 numerical analysis static interest rate2

5.2 Numerical analysis – Static interest rate

Sensitivity analysis for SPV’s NPV

Factor: premium


5 2 numerical analysis static interest rate3

5.2 Numerical analysis – Static interest rate

Sensitivity analysis for SPV’s NPV

Factor: interest rate


5 3 numerical analysis dynamic interest rate

5.3 Numerical analysis– Dynamic interest rate


5 3 numerical analysis dynamic interest rate1

5.3 Numerical analysis– Dynamic interest rate

Sensitivity analysis for SPV’s NPV


6 conclusion

6. Conclusion

  • The proposed extended logistic models performed better forecasting efficiency than the Lee-Carter and M7 model, especially the modified extended logistic (beta) model.

  • We design LBs to encompass more than one tranche. This design offers investors more choices pertaining to their different risk preferences.

  • The SPV’s NPV are influenced by interest rate and mortality rate. SPV should carefully evaluate premium and coupon rates to control their risks.


Yawen hwang assistant professor dept of risk management and insurance feng chia university

Q & A


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