Modelling the short-term dependence between two remaining lifetimes of a couple

Jaap Spreeuw and Xu Wang

Cass Business School

IME Conference, July 2007

Acknowledgement This project is supported financially by the

Actuarial Profession, United Kingdom.

Outline of contents

Types of dependence Instantaneous dependence Long-term dependence. Short-term dependence

Definition of types of dependence Models for dependence on two lives:

Common shock models Copula models Multiple state models

Extended multiple state model

Outline of contents Application to data set Identify type of dependence Estimate dependence parameters Further research References

Types of dependence Instantaneous dependence:

Dependence caused by common events affecting both lives at the same time.

E.g. plane crash Long-term dependence:

Dependence which is caused by a common risk environment, affecting the surviving partner for their remaining lifetime.

“Birds of a feather flock together” Short-term dependence:

The event of death of one life changes the mortality of the other life immediately, but this effect diminishes over time.

“Broken heart syndrome”.

Definition of types of dependence According to Hougaard (2000):

Long term dependence if mortality of surviving partner is constant or decreasing as a function of time elapsed since death of spouse.

Short term dependence if mortality of surviving partner is increasing as a function of time of death elapsed since death of spouse.

Models for dependence on two lives Common shock models

Suitable for instantaneous dependence. Copula models

All copulas with frailty specification (such as Clayton, Gumbel, Frank) have long-term dependence.

Almost all Archimedean copulas studied in Spreeuw (2006) (strict generator, covering entire range of positive dependence) exhibit long-term dependence.

Exception, in some cases: copula with generator

1t

t

Models for dependence on two lives Multiple state models

Diagram as in Norberg (1989) and Wolthuis (2003) :

0Both x and y alive

3Both x and y dead

1x dead, y alive

2x alive, y dead

Models for dependence on two lives Multiple state models

Model as in Denuit et al. (2001).

Special case of long-term dependence: mortality of survivor independent of time-of death of spouse.

01 01

02 02

13 13

23 23

1

1

1

1

x t

y t

y t

x t

t

t

t

t

Extended multiple state model Diagram:

1x dead, y alive

0 ≤ time since x died < t1

0Both x and y alive

3x alive, y dead

0 ≤ time since y died < t2

2x dead, y alive

time since x died ≥ t1

4x alive, y dead

time since y died ≥ t2

5Both x and y dead

Extended multiple state model Extended model would be

For lives whose partner is still alive:

01 01

03 03

1 ;

1

x t

y t

t

t

Expect positive dependence between future lifetimes, implying:

01 030; 0

Extended multiple state model Extended model would be

For lives whose partner died:

35 35 45 451 ; 1x t x tt t

Expect For widows: implies short-term

dependence, otherwise long-term dependence. Similar argument for widowers.

15 15 25 251 ; 1y t y tt t Widows:

Widowers:

15 25 35 45, , , 0

15 25

Application to data set Same data set as used by Frees et al. (1996),

Carriere (2000), and others. Eliminate same-sex couples and duplicate

contracts. Maximum period of observation: 5.005 years.

Identify type of dependence Estimates of widow(er)’s mortality as function of time

elapsed since death of partner (rounded off to nearest integer).

Compare with mortality of (wo)man whose partner is still alive.

Age x (integer), elapse e ( ): lives aged , whose partner died between and e years ago.

Estimate for each combination (x, e) mortality rate (derive exposed to risk number of death).

4, 5x x

0,..,41e

Identify type of dependence Some results for widows:

age elapse0 elapse1 elapse2 elapse3 elapse4 partner_alive65 0.0139 0.0204 0.0068 0.0123 0.0000 0.002766 0.0174 0.0147 0.0060 0.0111 0.0000 0.004667 0.0188 0.0198 0.0054 0.0105 0.0000 0.004168 0.0245 0.0220 0.0094 0.0092 0.0280 0.004369 0.0236 0.0270 0.0183 0.0081 0.0243 0.005870 0.0295 0.0321 0.0223 0.0163 0.0226 0.005771 0.0355 0.0372 0.0218 0.0158 0.0231 0.009372 0.0462 0.0347 0.0219 0.0159 0.0224 0.011873 0.0568 0.0278 0.0267 0.0160 0.0248 0.011274 0.0647 0.0335 0.0283 0.0167 0.0245 0.007375 0.0698 0.0366 0.0363 0.0180 0.0257 0.0132

Identify type of dependence Some results for widowers:

age elapse0 elapse1 elapse2 elapse3 elapse4 partner_alive70 0.1835 0.0695 0.0254 0.0556 0.0000 0.019471 0.1433 0.0501 0.0231 0.0448 0.0000 0.020172 0.1793 0.0577 0.0000 0.0410 0.0000 0.019373 0.1918 0.0727 0.0000 0.0358 0.0000 0.025474 0.1984 0.0690 0.0000 0.0325 0.0000 0.026375 0.1749 0.0693 0.0000 0.0285 0.0000 0.031076 0.2056 0.0805 0.0000 0.0000 0.1938 0.036577 0.2628 0.0962 0.0177 0.0000 0.1944 0.031678 0.3485 0.0969 0.0183 0.0000 0.2084 0.040079 0.3982 0.0805 0.0208 0.0000 0.2254 0.034980 0.4699 0.0923 0.0461 0.0000 0.2322 0.0553

Estimate dependence parameters Use Gompertz for estimation of marginal forces of

mortality. This gives for males (similar for females):

( ) /1 x mx e

Estimation by ML gives:

Parameter Male Female 9.756983 8.064262 m 86.36812 92.07142

Estimate dependence parameters

Parameters estimated by ML, given the estimates. This gives as estimate and s.e.:

_ _ˆ 1;

ˆ 1. .

_ _

i

ExitAge

x iALL EntryAge

Number of DEATHs

dx

s eNumber of DEATHs

Estimate dependence parameters Results for :

Alpha01 Alpha15 Alpha25 Alpha03 Alpha35 Alpha45mean 0.0624 5.0584 1.2363 0.1373 13.2673 0.3785no.of death 840 27 39 266 51 20SE 0.0367 1.1659 0.3581 0.0697 1.9978 0.3082

1 20.5; 0.5t t

Results for :1 21; 1t t

Alpha01 Alpha15 Alpha25 Alpha03 Alpha35 Alpha45mean 0.0624 3.3978 1.1509 0.1373 7.1853 0.4080no.of death 840 37 29 266 55 16SE 0.0367 0.7230 0.3994 0.0697 1.1037 0.3520

Other cut-off points studied as well.

Estimate dependence parameters Results in classical 4 state model :

1 2;t t

Observations: In all cases, and . This

strongly suggests short-term dependence. However, standard errors high, due to small

number of deaths (and widows/widowers).

Alpha01 Alpha15 Alpha25 Alpha03 Alpha35 Alpha45mean 0.0624 2.0142 N/A 0.1373 2.9263 N/Ano.of death 840 66 N/A 266 71 N/ASE 0.0367 0.3710 N/A 0.0697 0.4660 N/A

15 25ˆ ˆ 35 45

ˆ ˆ

Further research Analyse impact of short-term dependence on

the pricing (premium) and valuation (provisions) of standard policies on two lives, such as reversionary annuities and contingent insurance contracts.

Look into dependence on age.

References Carriere, J.F. (2000). Bivariate survival models for coupled lives.

Scandinavian Actuarial Journal, 17-31. Denuit, M. and Cornet, A. (1999). Multilife premium calculation with

dependent future lifetimes. Journal of Actuarial Practice, 7, 147-171. Frees, E.W., Carriere, J.F. and Valdez, E.A. (1996). Annuity valuation

with dependent mortality. Journal of Risk and Insurance, 63 (2), 229-261. Norberg, R. (1989). Actuarial analysis of dependent lives. Bulletin de

l'Association Suisse des Actuaires, 243-254. Spreeuw, J. (2006). Types of dependence and time-dependent association

between two lifetimes in single parameter copula models. Scandinavian Actuarial Journal (5), 286-309.

Wolthuis, H. (2003). Life Insurance Mathematics (The Markovian Model). IAE, Universiteit van Amsterdam, Amsterdam, 2nd edition.