MULTI-POPULATION MORTALITY MODELLING

Danish data and Basis Risk

Andrew Cairns

Heriot-Watt University, Scotland

and

The Maxwell Institute, Edinburgh

Joint work with

David Blake, Kevin Dowd, Malene Kallestrup-Lamb and Carsten Rosenskjold

1

Plan

• Motivation and challenges

• Danish males data

10 sub-populations grouped by affluence

• An extended CBD multi-population model

• Bayesian implementation and results

2

Motivation for multi-population modelling

A: Risk assessment

• Multi-country (e.g. consistent demographic projections)

• Males/Females (e.g. consistent demographic projections)

• Socio-economic subgroups (e.g. blue or white collar)

• Smokers/Non-smokers

• Annuities/Life insurance

• Limited data ⇒ learn from other populations

3

Motivation for multi-population modellingB: Risk management for pension plans and insurers

• Retain systematic mortality risk; versus:

• ‘Over-the-counter’ deals (e.g. longevity swap)

• Standardised mortality-linked securities

– linked to national mortality index

– < 100% risk reduction: basis risk

4

Challenges

• Data availability

• Data quality and depth

• Model complexity

– single population models can be complex

– 2-population versions are more complex

– multi-pop ......

• Multi-population modelling requires

– (fairly) simple single-population models

– simple dependencies between populations

5

A New Case Study and a New Model

• Sub-populations differ from national population

– socio-economic factors

– geographical variation

– other factors

• Denmark

– High quality data on ALL residents

– 1981-2005 available

– Can subdivide population using covariates on the

database

6

Danish Data

• What can we learn from Danish data that will inform us

about other populations?

• Key covariates

– Wealth

– Income

• Affluence = Wealth+15×Income

7

Problem

• High income ⇒ “affluent” and healthy BUT

• Low income ⇒/ not affluent, poor health

• High wealth ⇒ “affluent” and healthy BUT

• Low wealth ⇒/ not affluent, poor health

Solution: use a combination

• Affluence, A = wealth +K× income

•K = 15 seems to work well statistically as a predictor

• Low affluence, A, predicts poor mortality

8

Subdividing Data

• Males resident in Denmark for the previous 12 months

• Divide population in year t

– into 10 equal sized Groups (approx)

– using affluence, A

• Individuals can change groups up to age 67

• Group allocations are locked down at age 67

(better than not locking down at age 67)

9

Subdivided Data

• Exposures E(i)(t, x) for groups i = 1, . . . , 10

range from over 4000 down to 20

• Deaths D(i)(t, x)

range from 150 down to 6

• Crude death rates m̂(i)(t, x) = D(i)(t, x)/E(i)(t, x)

• Small groups ⇒ Poisson risk is important

10

Crude death rates 2005

60 70 80 90

0.00

20.

010

0.05

00.

200

Group 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10

Males Crude m(t,x); 2005

Age

m(t

,x)

(log

scal

e)

11

Modelling the death rates, mk(t, x)

m(k)(t, x) = pop. k death rate in year t at age x

Population k, year t, age x

logm(k)(t, x) = β(k)(x) + κ(k)1 (t) + κ

(k)2 (t)(x− x̄)

(Extended CBD with a non-parametric base table, β(k)(x))

• 10 groups, k = 1, . . . , 10 (low to high affluence)

• 21 years, t = 1985, . . . , 2005

• 40 ages, x = 55, . . . , 94

12

Model-Inferred Underlying Death Rates 2005

60 70 80 90

0.00

20.

010

0.05

00.

200

Group 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10

Males Crude m(t,x); 2005

Age

m(t

,x)

(log

scal

e)

60 70 80 90

0.00

20.

010

0.05

00.

200

Males CBD−X Fitted m(t,x); 2005

Age

m(t

,x)

(log

scal

e)

13

Modelling the death rates, mk(t, x)

logm(k)(t, x) = β(k)(x) + κ(k)1 (t) + κ

(k)2 (t)(x− x̄)

• Model fits the 10 groups well without a cohort effect

• Non-parametric β(k)(x) is essential to preserve group

rankings

– Rankings are evident in crude data

– “Bio-demographical reasonableness”:

more affluent ⇒ healthier

14

Time series modelling

• t→ t + 1: Allow for correlation

– between κ(k)1 (t + 1) and κ

(k)2 (t + 1)

– between groups k = 1, . . . , 10

• Bio-demographical reasonableness

⇒ key hypothesis: groups should not diverge

⇒ group specific period effects gravitate towards the

national trend

15

Life Expectancy for Groups 1 to 10

1985 1990 1995 2000 2005

1416

1820

2224

26Males Period LE: Age 55

1985 1990 1995 2000 2005

46

810

1214

16

Males Period LE: Age 67

Group 10Group 9Group 8Group 7Group 6Group 5Group 4Group 3Group 2Group 1

16

Mortality Fan Charts Including Parameter Uncertainty

1990 2000 2010 2020 2030 2040 2050

0.01

0.02

0.05

0.10

Group 10

Group 1

Mortality Rates: Age 75

Year

q(t,x

)

17

Simulated Group versus Population Mortality

0.02 0.04 0.06 0.10

0.02

0.04

0.06

0.08

Group 2T=2006Corr = 0.58

Tota

l q(t

,x)

Group q(t,x)

0.02 0.04 0.06 0.10

0.02

0.04

0.06

0.08

Group 2T=2010Corr = 0.72

Tota

l q(t

,x)

Group q(t,x)

0.02 0.04 0.06 0.10

0.02

0.04

0.06

0.08

Group 2T=2030Corr = 0.86

Tota

l q(t

,x)

Group q(t,x)

As T increases

• Scatterplots become more dispersed

• Shift down and to the left

• Correlation increasess

18

Forecast Correlations

• Deciles are quite narrow subgroups

• Blue collar pension plan

⇒ equal proportions of groups 2, 3, 4

• White collar pension plan

⇒ equal proportions of groups 8, 9, 10

19

Forecast Correlations

2010 2015 2020 2025 2030

0.0

0.2

0.4

0.6

0.8

1.0

Year

Cor

rela

tion Blue Collar Plan

White Collar PlanGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8Group 9Group 10

Age 75

Correlation Between Group q(t,x) and Total q(t,x)

20

Conclusions

• Development of a new multi-population dataset for Denmarkstrong bio-demographically reasonable group rankingsbased on a new measure of affluence

• Unlike multi-country dataa priori ranking of affluence-related groups

• Proposal for a simple new multi-population model

• Strong correlations over medium to long term

• Correlations depend strongly on diversity of sub-population

21

Bonus Slides

22

A specific model

κ(i)1 (t) = κ

(i)1 (t− 1) + µ1 + Z1i(t) (random walk)

−ψ(κ(i)1 (t− 1)− κ̄1(t− 1)

)(gravity between groups)

κ(i)2 (t) = κ

(i)2 (t− 1) + µ2 + Z2i(t)

−ψ(κ(i)2 (t− 1)− κ̄2(t− 1)

)where

κ̄1(t) =1

n

n∑i=1

κ(i)1 (t) and κ̄2(t) =

1

n

n∑i=1

κ(i)2 (t)

23

A specific model

κ(i)1 (t) = κ

(i)1 (t− 1) + µ1 + Z1i(t)− ψ

(κ(i)1 (t− 1)− κ̄1(t− 1)

)κ(i)2 (t) = κ

(i)2 (t− 1) + µ2 + Z2i(t)− ψ

(κ(i)2 (t− 1)− κ̄2(t− 1)

)Model structure ⇒

• (κ̄1(t), κ̄2(t)) ∼ bivariate random walk

• Each κ(i)1 (t)− κ̄1(t) ∼ AR(1) reverting to 0

• Each κ(i)2 (t)− κ̄2(t) ∼ AR(1) reverting to 0

• β(i)(x) vs β(j)(x)⇒ intrinsic group differences

24

Non-trivial correlation structure:between different ages and groups

κ(i)1 (t) = κ

(i)1 (t− 1) + µ1+Z1i(t)− ψ

(κ(i)1 (t− 1)− κ̄1(t− 1)

)κ(i)2 (t) = κ

(i)2 (t− 1) + µ2+Z2i(t)− ψ

(κ(i)2 (t− 1)− κ̄2(t− 1)

)The Zki are multivariate normal, mean 0 and

Cov(Zki, Zlj) =

vkl for i = j

ρvkl for i ̸= j

ρ = cond. correlation between κ(i)1 (t) and κ

(j)1 (t) etc.

25

Comments

• Model is very simple

– One gravity parameter, 0 < ψ < 1

– One between-group correlation parameter,

0 < ρ < 1

• Many generalisations are possible

• But more parameters + more complex computing

• This simple model seems to fit quite well.

• Nevertheless ⇒ work in progress

26

Prior distributions

• As uninformative as possible

• µ1, µ2 ∼ improper uniform prior

• {vij} ∼ Inverse Wishart

• ρ ∼ Beta(2, 2)

• ψ ∼ Beta(2, 2)

State variables and process parameters estimated using

MCMC (Gibbs + Metropolis-Hastings)

27

Posterior Distributions and 95% Credibility Intervals

−0.030 −0.020 −0.010 0.000

0.0

0.2

0.4

0.6

0.8

1.0

Kappa_1 Drift, mu_1

mu_1

Cum

ulat

ive

Pos

terio

r P

roba

bilit

y

0.0000 0.0004 0.0008 0.00120.

00.

20.

40.

60.

81.

0

Kappa_2 Drift, mu_2

mu_2

Cum

ulat

ive

Pos

terio

r P

roba

bilit

y

Note: −µ1 = global improvement rate28

Posterior Distributions and 95% Credibility Intervals

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Between Group Correlation, rho

rho

Cum

ulat

ive

Pos

terio

r P

roba

bilit

y

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.0

0.2

0.4

0.6

0.8

1.0

Gravity Parameter, psi

psi

Cum

ulat

ive

Pos

terio

r P

roba

bilit

y

29