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