Embed Size (px)
Citation preview
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
M8 (Modified-Extended
Logistic (alpha) Model)
( )
, ( )( )
1
t x
x t t x
eq x
e
M9 (Modified-Extended
Logistic (beta) Model) ,
( )( )
1 ( )
x
x t x
t eq x
t e
3.4 Measurement
Measurement 1. MAPE (Mean Absolute Percentage Error)
1
1 100n
t
t t
MAPEn X
2. According to Lewis (1982), the standard of MAPE is described as the
following table:
MAPE <10% 10%~20% 20%~50% >50%
Efficiency Excellent Good Reasonable Bad
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.
Mode Fitting Parameters
M1 99
M2 7
M3 57
M4 81
M5 81
M6 118
M7 115
M8 82
M9 82
4.1 Numerical analysis - Fitting
Japan USA England & Wales
Male Female Male Female Male Female
M1 2.4373 1.9978 1.2700 1.4237 2.1147 2.3414
M2 3.3997 3.2442 2.3918 2.5961 3.3162 3.3442
M3 5.5357 10.4298 2.3538 3.9263 3.7633 4.0222
M4 4.3178 4.7741 1.9585 2.2381 3.5398 2.9301
M5 3.4756 6.5491 2.0439 2.3122 3.7800 3.2012
M6 6.6405 17.3331 3.2890 4.3897 3.9710 5.2522
M7 1.8402 2.2063 2.0100 1.6604 1.6848 1.4561
M8 2.5508 2.3856 1.9176 1.6271 2.5742 2.4618
M9 2.7307 2.4946 1.7514 1.4317 2.7457 2.5910
4.2 Numerical analysis - Forecasting
Japan USA England & Wales
Male Female Male Female Male Female
M1 8.7975 7.4732 4.3289 5.5322 9.3420 9.5546
M2 8.2058 6.7143 4.9397 3.3400 8.5763 7.1725
M3 5.3860 14.0729 5.6051 5.7011 5.9221 6.7057
M4 7.4193 7.0189 6.1529 4.6073 10.7402 7.5631
M5 5.2753 8.2599 5.5074 2.7816 9.9020 7.9593
M6 7.0305 17.6602 7.7577 7.8114 11.4362 12.5450
M7 8.1552 8.3197 7.9123 6.0636 6.5502 8.6278
M8 7.9442 5.9946 7.6088 4.9961 9.8443 7.1888
M9 4.9405 5.9710 3.1691 2.5825 5.6405 4.6914
Data:1. Forecasting single age range from 50-year to 89-year2. Japan, England & Wales: 2001~20063. USA: 2001~2005
5.1 Longevity bond
5.1.1 Insurer & SPV
t
tt
tt
t
ttt
SkS
SkSSk
SkS
ASkk
ASkSB
2
21
1
12
1
ˆ
ˆ
ˆ
)(
)ˆ(
0
is the survivor index. is the real survivor rate.
is the payment from SPV to insurer at time t.
tS tS
tB
Not equivalent
5.1 Longevity bond
5.1.2 SPV & Investor
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 .
Coupon cA
Coupon cB
BtPr
BtPr
AB cc
5.1 Longevity bond
• Survival rate index: (Insurer)
Lin & Cox (2005):
• Survival rate: (SPV)
Modified extended logistic (beta) mortality model
USA Male
5.2 Numerical analysis – Static interest rate
parameters
a 0.2 r 0.02 Interest Rate
b 0.05 0r 0.03
1k 1.02 2k 1.22
N 10,000 .Ann 1,000
x 60 T 30
Tranche A 0Pr A 6,500,000 LBs
Tranche B 0PrB 6,500,000
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.
premium VaR(95) coupon A coupon B
M-E beta Model 1,212,100 2,467,000 5.7167% 6.9531%
1k 2k mean Var(95) Max
1.02 1.22 1,246,300 2,467,000 7,064,600
- 1.25 1,257,300 2,514,200 7,064,600
- 1.28 1,262,800 2,544,400 7,064,600
1.08 1.28 337,510 830,750 7,064,600
Table 10. NPV- using mean to be premium
Min Mean VaR(90) VaR(95) Max Std. NPV<0
Insurer -4,165,800 -75,955 900,340 1,026,200 1,212,100 761,410 51.05%
Investor -6,724,700 -297,970 1,607,600 2,008,700 4,462,200 1,584,300 52.92%
All -6,133,500 -373,930 1,417,500 1,688,700 2,894,200 1,521,100 53.14%
Table 11. NPV- using VaR(95) to be premium
Min Mean VaR(90) VaR(95) Max Std. NPV<0
Insurer -2,910,900 1,178,900 2,155,200 2,281,100 2,467,000 761,410 7.34%
Investor -6,724,700 -297,970 1,607,600 2,008,700 4,462,200 1,584,300 52.92%
All -4,878,600 880,970 2,672,400 2,943,600 4,149,100 1,521,100 26.47%
5.2 Numerical analysis – Static interest rate
Sensitivity analysis for SPV’s NPV
Factor: premium
r Min Mean VaR(90) VaR(95) Max Std.
-1% -10,072,000 -2,992,300 -798,240 -452,870 990,900 1,856,300
-0.5% -7,992,100 -1,612,800 365,950 670,710 1,982,400 1,678,100
-0.2% -6,852,600 -853,700 1,007,300 1,295,500 2,540,600 1,581,600
+0% -6,133,500 -373,930 1,417,500 1,688,700 2,894,200 1,521,100
+0.2% -5,445,900 86,021 1,849,600 2,069,100 3,233,800 1,463,600
+0.5% -4,469,200 740,840 2,368,800 2,610,600 3,718,400 1,382,600
+1% -2,976,000 1,745,900 3,224,300 3,440,700 4,464,900 1,260,000
5.2 Numerical analysis – Static interest rate
Sensitivity analysis for SPV’s NPV
Factor: interest rate
5.3 Numerical analysis – Dynamic interest rate
Table 13. Premiums
Medium VaR(70) VaR(75) VaR(80) VaR(85)
1,233,800 1,658,200 1,779,900 1,923,700 2,093,500
VaR(90) VaR(95) Max. Mean Std.
2,314,500 2,645,900 8,639,600 1,288,100 761,410
Table 14. Coupon rates (%)
Min. Medium VaR(45) VaR(40) VaR(35) VaR(30)
Coupon A 3.7316 5.6858 5.5972 5.5076 5.4165 5.3152
Coupon B 3.9114 6.9052 6.7758 6.6481 6.5173 6.3836
VaR(25) VaR(20) VaR(15) Max. Mean Std.
Coupon A 5.2066 5.0735 4.9122 7.7441 5.6323 0.0062
Coupon B 6.2346 6.0734 5.8904 10.5678 6.9329 0.0099
5.3 Numerical analysis – Dynamic interest rate
Sensitivity analysis for SPV’s NPVTable 15. NPV with mean as the premium
Min Mean VaR(90) VaR(95) Max Std. NPV<0
Insurer -4,089,800 44.8145 976,340 1,102,200 1,288,100 761,410 47.26%
Investor -6,615,800 -214,750 1,682,100 2,082,600 4,526,200 1,577,600 50.98%
All -5,947,900 -214,700 1,569,000 1,839,000 3,035,900 1,514,500 49.33%
Table 16. NPV with VaR as the premium
Premium: VaR(90)
Coupon Min Mean VaR(90) VaR(95) Max Std. NPV<0
Medium -4,921,500 811,700 2,595,400 2,865,400 4,062,300 1,514,500 27.47%
VaR(40) -4,315,200 1,271,900 3,012,200 3,278,500 4,434,100 1,477,900 20.28%
Premium: VaR(95)
Coupon Min Mean VaR(90) VaR(95) Max Std. NPV<0
Medium -4,590,100 1,143,100 2,926,800 3,196,800 4,393,700 1,514,500 22.44%
VaR(40) -3,983,800 1,603,300 3,343,600 3,609,900 4,765,500 1,477,900 16.17%
VaR(25) -2,988,200 2,359,300 4,030,400 4,288,400 5,378,800 1,418,000 7.71%
6. Conclusion
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.
Q & A
