MULTI-POPULATION MORTALITY MODELLING

Andrew Cairns

Heriot-Watt University,

and The Maxwell Institute, Edinburgh

Amsterdam, March 2015

1

Plan

• Introduction and motivation

• Modelling Danish sub-population mortality

• Economic capital

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

More diversified e.g.

• 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

Hedging and Economic Capital

Basis Risk

Two sources of basis risk

• Population basis risk

• Sub-optimal choice of hedging instrument

21

Economic capital relief using longevity options

• Population 1: national population; reference for hedge

• Population 2: hedger’s own population

• Option payoff at T based on

– Pop 1 cashflows up to T

– Estimated Pop 1 cashflows after T (commutation)

• Option ↔ Special purpose vehicle ↔ Longevity CAT bond

• BE = best estimate liability at time 0

• EC = additional Economic Capital to cover e.g. 99.5% runoff

– EC0 = EC without hedge

– EC1 = EC with index-based option hedge

22

Longevity Option

• S1(t, x) = Population 1 Survivor Index

• A1 =∑∞

t=1 vtS1(t, x) (PV Population 1 Annuity)

• Underlying index for option:

A1(T ) = EC [A1|FT ]

EC : expected cashflows after T calculated on a basis prescribed at time 0

• S2(t, x) = Population 1 Survivor Index

• A2 =∑∞

t=1 vtS2(t, x) (PV Population 2 Annuity)

23

Longevity Swaps and Option

Hedging instrument payoffs (discounted to time 0)

• Longevity swap A: A1 − EQ[A1] (cashflow hedge)

• Longevity swap B: A1(T )− EQ[A1] (payable at T )

• Bull spread, attachment points K1 < K2

B(T ) = (A1(T )−K1)+ − (A1(T )−K2)+ (payable at T )

• PV Bull spread hedge

Y (T,K1, K2, h) = A2 − h (B(T )− EQ[B(T )])

EQ ⇒ pricing measure at time 0

24

Practical issues

• Structure of the hedging instrument

• Price / risk premium payable by hedger

• Tradeoff:Hedger Counterparty

Customised Index

Full term Medium term

Uncapped payoff Limited loss

Swap Cat Bond format

• Tradeoff ⇒ basis risk: sub-optimal hedge instrument + pop.

25

Hedging Example

• Hedgers population: Blue collar plan

Equal proportions of groups 2, 3, 4

Age 65 cohort, males

• Index hedge linked to national mortality

Age 65 cohort, males

Attachment points: 13.5, 14

26

Index Based Hedge: the Underlying

12.0 12.5 13.0 13.5 14.0 14.5

0.0

0.2

0.4

0.6

0.8

1.0

Underlying PV National Annuity

Cum

ulat

ive

Pro

babi

lity

0.913

0.993

Attachment Points

27

Index Based Hedge:Payoffs

12.0 12.5 13.0 13.5 14.0 14.5

0.0

0.2

0.4

0.6

0.8

1.0

Payoff to Hedger at T

Underlying PV National Annuity

Pay

off t

o H

edge

r

12.0 12.5 13.0 13.5 14.0 14.5

0.0

0.2

0.4

0.6

0.8

1.0

Cat Bond Payoff at T

Underlying PV National AnnuityC

at B

ond

Pay

off

Cat Bond maximum loss =$0.5 Billion, Notional $1Bn⇒ $12 Billion liabilities

28

Impact of Hedging with Longevity Swap A

10.5 11.5 12.5 13.5

10.5

11.0

11.5

12.0

12.5

13.0

13.5

PV Unhedged Liability

PV

Hed

ged

Liab

ility

UnhedgedLongevity SwapOption ooOption 20 yr

11.0 11.5 12.0 12.5 13.00.

00.

20.

40.

60.

81.

0

PV Hedged Liability

Cum

ulat

ive

Pro

babi

lity

UnhedgedLongevity SwapOption ooOption 20 yr

29

Impact of Hedging with T = ∞ Bull Spread payable on last death

10.5 11.5 12.5 13.5

10.5

11.0

11.5

12.0

12.5

13.0

13.5

PV Unhedged Liability

PV

Hed

ged

Liab

ility

UnhedgedLongevity SwapOption ooOption 20 yr

11.0 11.5 12.0 12.5 13.00.

00.

20.

40.

60.

81.

0

PV Hedged Liability

Cum

ulat

ive

Pro

babi

lity

UnhedgedLongevity SwapOption ooOption 20 yr

30

Impact of Hedging with T = 20 Bull Spread

10.5 11.5 12.5 13.5

10.5

11.0

11.5

12.0

12.5

13.0

13.5

PV Unhedged Liability

PV

Hed

ged

Liab

ility

UnhedgedLongevity SwapOption ooOption 20 yr

11.0 11.5 12.0 12.5 13.00.

00.

20.

40.

60.

81.

0

PV Hedged Liability

Cum

ulat

ive

Pro

babi

lity

UnhedgedLongevity SwapOption ooOption 20 yr

31

Impact of Hedging on Economic Capital

10.5 11.0 11.5 12.0 12.5 13.0 13.5

10.5

11.0

11.5

12.0

12.5

13.0

13.5

PV Unhedged Liability

PV

Hed

ged

Liab

ility

99.5% quantiles

UnhedgedLongevity SwapOption ooOption 20 yrOption 10 yr

+Economic Capital

−Reduction in Economic Capital

11.96 → 13.08 → 12.81 → 12.59/12.9132

Challenges

• Simulation example assumes options priced at

actuarially fair value

• But options premiums might be more expensive

• Compare premium versus

value of reduction in Economic Capital

over multiple time periods

33

Summary

• Danish data allows insight into relative mortality

dynamics between socio-economic sub-populations

• Conclusions for other countries likely to be similar

• Many risk management applications

E: A.J.G.Cairns@hw.ac.uk W: www.macs.hw.ac.uk/∼andrewc

34

Bonus Slides

35

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)

36

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

37

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.

38

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

39

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)

40

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 rate41

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

42