h. chen, s.h. cox, and j. wen longevity 5 conference september 26, 2009multivariate threshold life...

September 26, 2009Multivariate Threshold Life Table 1 H. Chen, S.H. Cox, and J. Wen

Longevity 5 Conference

Pricing Mortality-linked Securities with

Dependent Lives under Threshold Life Table

Hua Chen, Temple University

Samuel H. Cox, University of Manitoba

Jian Wen, Central University of Finance and Economics



Introduction

Extreme-age mortality modeling

Mortality improvement is a slow but persistent process

40,000 centenarians currently in the U.S.

3 million centenarians by the first decade of next century

Challenge to actuaries

since life table is usually closed earlier, say 100.

How to extrapolate extreme-age mortality and construct a reliable life table?

EVT approach Threshold life Table (Li, Hardy, Tan 2008)



Introduction

Joint survivorship of multiple lives

Independence of a pair is normally assumed The joint survival function is simply the product of the marginal survival functions

of each life.

Common risk factors for pairs of lives Genetic factors, e.g. twins Environmental factors, e.g., couples

Empirical evidence: broken heart syndrome Parkes, Benjamin, and Fitzgerald (1969), Ward (1976)Jagger and Sutton (1991)

How to capture the life dependence?

Copula function



Introduction

EVT Copula

Multivariate Threshold Life Table



EVT and Threshold Life Table

Parametric estimator

e.g. Gompertz distribution function (Frees, Carriere, Valdez 1996)

Traditional parametric methods are ill-suited to extreme probabilities

The inaccuracy and unavailability of mortality data at old ages.

Solution: EVT

“estimate extreme probabilities by fitting a model to the empirical survival function of a set of data using only the extreme event data rather than all the data, thereby fitting the tail, and only the tail”(Sanders, 2005).

)1(exp1)(

xm

eexF




For x > N

where

The Pickands-Balkema-De Hann Theorem

For sufficiently high threshold N, the excess distribution function may

be approximated by the GPD .

.

( )NF x N

, , ( )NG x

( ) { } (1 ( )) ( ) ( )NF x P X x F N F x N F N

( ) { | }NF x P X N x X N

, ,( ) (1 ( )) ( ) ( )NF x F N G x F N

1

1 (1 ( )) 1x N

F N




Li, Hardy and Tan (2008): Threshold Life Table

1

1 exp (1 ) , if

( )

1 1 ( ) 1 , if

m x

e e x N

F xx N

F N x N



Life Dependency and Copula

Dependency Measures

Parametric: e.g., Pearson correlation

Non-parametric: e.g., Spearman’s rho, Kendall’s tao

Copula

Copulas capture the dependence structure separately from the marginal distributions

Schweizer and Wolff (1981)

For any strictly increasing functions and ,

and have the same copula as and

1 2( , ) Pr( , ) ( ( ), ( ))H x y X x Y y C F x F y

1g 2g1( )g X

2 ( )g Y X Y




Archimedean copula family

where is a convex and strictly decreasing function with domain and range

such that .

))()((),( 211

21 uuuuC

]1,0(),0[ 0)1(




Frank’s copula

Spearson’s rho

Kendall’s tau

where and

1 2

1 2

1 ( 1)( 1)( , ) ln 1

1

u ue eC u u

e

)()(12

1),( 1221

DDXXS

1)(4

1),( 121

DXX

x

t

k

kk dte

t

x

kxD

0 1)(

1)()(

k

kxxDxD kk



Last-Survivor Annuity Data

Frees, Carriere and Valdez (1996)

approximately 15,000 last-survivor annuity policies 1989 - 1993.

date of birth, death (if applicable), contract initiation, and sex of each annuitant.



Modeling Algorithm

Set up the initial values of parameters

Use mortality data of males

For , find the parameters that maximize the log-likelihood f

unction;

Repeat this step for ;

The value of that gives the maximum profile log-likelihood is the optimal thr

eshold age for male. The parameter estimates corresponding to thi

s value are the optimal MLE estimates.

Replicate the same procedure for mortality data for females, and find the o

ptimal estimates and

Use Gompertzian marginals and the Frank copula to find the estimate of th

e dependence parameter ;

1 100N 1111 ,,, m

80,...,98,991 N

1N

1111 ,,, m

2222 ,,, m2N



Modeling Algorithm

Using the values of and obtained from step 1 as initial values, find the MLE estimates of these parameters, for any combination of and

The values of and that give the maximum value of log-likelihood function are the optimal threshold ages for males and females. The MLE estimates corresponding to this combination are our optimal MLE estimates.

1 1 1 1 2 2 2 2, , , , , , ,m m

1N 2N

1 2100,99,...,80 and 100,99,...,80N N

1N 2N



Estimation Results

Spearman’s rho = 0.49 and Kendall’s tau = 0.56

A positive mortality dependence between male and female



Pricing Example: Last-Survivor Annuity

Last-survivor annuity

where

Scenario analysis

Dependent lives with the threshold life table (benchmark model)

Independent lives with the threshold life table

Dependent lives without the threshold life table

Independent lives without the threshold life table

0

)1(k

yxkk

yx pra

Pr( or | and )t xyp X x t Y y t X x Y y

1 Pr( and | and )X x t Y y t X x Y y

Pr( , )1

Pr( , )

x X x t y Y y t

X x Y y

( , ) ( , ) ( , ) ( , )1

1 ( , ) ( , ) ( , )

H x t y t H x y t H x t y H x y

H x H y H x y



Pricing Example: Effect of Threshold Life Table

Ratio = annuity value with TLT/ that without TLT

Dependent lives Independent lives

Without threshold life table, the value of the last survivor annuity is underestimated.



Pricing Example: Effect of Dependence

Ratio = annuity value assuming dependence/ that assuming independence

With threshold life table Without threshold life table

Assuming independent lives, the value of the last-survivor annuity is overestimated.



Pricing Example: Overall Effect

Ratio = Dependent lives with TLT/ Independent lives without TLT

Assuming independent lives and without threshold life table, the last survivor annuity is overestimated by 5%.

50

60

70

80

50

60

70

800.94

0.95

0.96

0.97

0.98

0.99

1

Male AgeFemale Age

Ann

uity

Rat

io



Conclusion

Develop multivariate threshold life table

EVT approach

Copula approach

Apply our model to price the last-survivor annuity

Mortality-linked securities are under-priced without the threshold life table

Mortality-linked securities are over-priced assuming independent lives

Future research

How to identify an appropriate copula function?

Incorporate a stochastic process into the multivariate threshold life table.



Thanks!

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