Extended Logistic Model for Mortality Forecasting and the Application of Mortality- Linked Securities Yawen, Hwang, Assistant Professor, Dept. of Ris k Management and Insurance, Feng Chia Universit y Hong-Chih, Huang, Associate Professor, Dept. of Risk Management and Insurance, National Chengch i University
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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.
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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
Hong-Chih, Huang, Associate Professor, Dept. of Risk Management and Insurance, National Chengchi University
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.Introduction
Longevity
Bond
How to enhance the attractiveness of longevity bonds?
Separating it. (From the idea of collateral debt obligation )
1.Introduction
How to price the longevity bonds?
Need accurate mortality model!
The purpose of this study:
1. Modifying the existing mortality models and providing a better mortality model
2. Improving the attractiveness of longevity bonds
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
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 risk
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.
Lin & Cox (2005)Special Purpose Vehicles
3.1 Logistic mortality model
( )
( ),
( )( )
1 ( )
t x
t xx t
t eq t
t e
senescent death rate background death rate
Thus, this model is a dynamic model.
It considers the effects of age and time.
( )( , ) ( )
1 ( )
x
x
t ex t t
t e
Bongaarts(2005) proposes a logistic mortality model as follows:
We assume the mortality rate follows Eq(1)
Eq(1)
3.2 Modifying methods
Extended Logistic (alpha) Model Extended Logistic (beta) Model ( )
, ( )( )
1
t x
x t t x
eq t
e
1
2
if 1
if 1
x seg
x seg
2
2
if
if
)(
)()(
2
1
segx
segx
t
tt
3
3
if
if
)(
)()(
2
1
segx
segx
t
tt
,
( )( )
1 ( )
x
x t x
t eq t
t e
1
2
( ) if 1( )
( ) if 1
t x segt
t x seg
1
2
if 2
if 2
x seg
x seg
3
3
if
if
)(
)()(
2
1
segx
segx
t
tt
Method I: Segment approach (from RF model)
3.2 Modifying methods
Modified-Extended Logistic (alpha) Model Modified-Extended Logistic (beta) Model ( )
, ( )( )
1
t x
x t t x
eq x
e
1
2
if 1
if 1
x seg
x seg
2
2
if
if
)(
)()(
2
1
segx
segx
t
tt
,
( )( )
1 ( )
x
x t x
t eq x
t e
1
2
( ) if 1( )
( ) if 1
t x segt
t x seg
1
2
if 2
if 2
x seg
x seg
Method II: Background death rate might be related more
reasonably to age.
3.3 Mortality models
Model Formula
M1 (Lee-Carter model) txtxxtx kq ,)2()1(
, )ln(
M2 (Reduction Factor
model) 20
0,
, )](1)][(1[)(),(t
x
tx xfxxtxRFq
q
M3 (Logistic model) ( )
( ),
( )( )
1 ( )
t x
t xx t
t eq t
t e
M4 (Extended Logistic
(alpha) model
( )
, ( )( )
1
t x
x t t x
eq t
e
M5 (Extended Logistic
(beta) model ,
( )( )
1 ( )
x
x t x
t eq t
t e
M6 (CBD model) )2()2()1()1(, log txtxtx kkqit
M7 (M7 model (Cairns et
al. 2009)) (1) (2) (3) 2 2 4
, ˆlog ( ) (( ) )x t t t t x t xit q k k x x k x x
1. The proposed extended logistic models performed better forecasting efficiency than the Lee-Carter and M7 model, especially the modified extended logistic (beta) model.
2. We design LBs to encompass more than one tranche. This design offers investors more choices pertaining to their different risk preferences.
3. The SPV’s NPV are influenced by interest rate and mortality rate. SPV should carefully evaluate premium and coupon rates to control their risks.