1

MODELLING AND MANAGEMENT OF MORTALITY

RISK

Stochastic models for modelling mortality risk

ANDREW CAIRNS

Heriot-Watt University, Edinburgh

and

Director of the Actuarial Research Centre

Institute and Faculty of Actuaries

2

Actuarial Research Centre (ARC)

The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuaries’

network of actuarial researchers around the world.

The ARC seeks to deliver research programmes that bridge academic rigour with

practitioner needs by working collaboratively with academics, industry and other

actuarial bodies.

The ARC supports actuarial researchers around the world in the delivery of

cutting-edge research programmes that aim to address some of the significant

challenges in actuarial science.

www.actuaries.org.uk/ARC

3

Actuarial Research Centre (ARC)

Current research programmes (2016-2021)

• Modelling, Measurement and Management of Longevity and Morbidity Risk

• Use of Big Health and Actuarial Data for understanding Longevity and

Morbidity

• Minimising Longevity and Investment Risk While Optimising Future Pension

Plans

4

Stochastic models for modelling mortality risk: Plan

• Introduction, motivation, problems

• Modelling

– Criteria for a good model

– Comparison of 8 models

– Robustness

– Graphical diagnostics

• Applications

5

The Problem

2017: What we know as the facts:

• Life expectancy is increasing.

• Future development of life expectancy is uncertain.

“Longevity risk”

⇒ Systematic risk for pension plans and annuity providers

6

The Problem

Example: UK Defined-Benefit Pension Plans

• Before 2000:

– High equity returns masked impact of longevity improvements

• After 2000:

– Poor equity returns, low interest rates

– Decades of longevity improvements now a problem

7England and Wales males mortality (log scale)

1900 1940 1980

0.00

10.

002

0.00

4

Age = 25

Year

Mor

talit

y ra

te

1900 1940 1980

0.00

50.

010

0.02

0

Age = 45

Year

Mor

talit

y ra

te

1900 1940 1980

0.05

0.10

Age = 65

Year

Mor

talit

y ra

te

1900 1940 1980

0.2

0.4

0.8

Age = 85

Year

Mor

talit

y ra

te

8

Graphical diagnostics

• Mortality is falling

• Different improvement rates at different ages

• Different improvement rates over different periods

• Improvements are random

– Short term fluctuations

– Long term trends

• All stylised facts

9

STOCHASTIC MORTALITY

n lives, probability p of survival, N survivors

• Unsystematic mortality risk:

⇒N |p ∼ Binomial(n, p)

⇒ risk is diversifiable, N/n → p as n → ∞

• Systematic mortality risk:

⇒ p is uncertain

⇒ risk associated with p is not diversifiable

• Longevity Risk: the risk that in aggregate people live longer than

anticipated.

10

Why do we need stochastic mortality models?

Data ⇒ future mortality is uncertain

• Good risk management

• Setting risk reserves

• Regulatory capital requirements (e.g. Solvency II)

• Life insurance contracts with embedded options

• Pricing and hedging mortality-linked securities

11

Modelling

Aims:

• to develop the best models for forecasting future uncertain

mortality;

– general desirable criteria

– complexity of model ↔ complexity of problem;

– longevity versus brevity risk;

• measurement of risk;

• valuation of future risky cashflows.

12

Management

Aims:

• active management of mortality and longevity risk;

– internal (e.g. product design; natural hedging)

– over-the-counter deals (OTC)

– securitisation

• part of overall package of good risk management.

13

Modelling Stochastic Mortality

• Many models to choose from

• Limited data ⇒ model and parameter risk

• Important to take the time to analyse models thoroughly

• No single model is best for all datasets and applications

14

Model Selection Criteria• Positive mortality rates

• Consistent with historical data

• Biologically reasonable and plausible forecasts

• Robust parameter estimates and forecasts

• Straightforward to implement

• Parsimonious

• Generates sample paths

• Can include parameter uncertainty

• Cohort effect if appropriate

• Non-trivial correlation structure

• Not used as a black box

15

Consistent with historical data

• Model fit consistent with i.i.d. Poisson assumption

– goodness of fit tests

– graphical diagnostics

• Compare models using likelihoods and the

Bayes Information Criterion (BIC)

• Future versus past patterns of randomness

• Backtesting

16

Biologically reasonable and plausible forecasts

• Biologically reasonable

e.g. inverted mortality curve??

strong mean reversion??

time horizon matters

• Plausible forecasts

trend and degree of uncertainty

17

Robustness

• What happens if I change the age range?

• What happens if I add one extra calendar year?

• Revised parameter estimates and forecasts should be similar to

old

18

Not a black box

• Understand the advantages and disadvantages of each model

• Understand the limitations and assumptions of each model

• Better understanding of the model ⇒

– Better understanding of the risks

– Good risk management practice

19

Measures of mortality

• q(t, x) = underlying mortality rate in year t at age x

• m(t, x) = underlying death rate

• Assume q(t, x) = 1− exp[−m(t, x)]

Poisson model:

Exposures: E(t, x)

Actual deaths: D(t, x) ∼ Poisson (m(t, x)E(t, x))

in year t, age x last birthday

20

The Lee-Carter (1992) model

logm(t, x) = β(1)x + β(2)

x κt

Component 1: β(1)x

• Age effect

• Baseline log-mortality curve (κt = 0)

21

logm(t, x) = β(1)x + β(2)

x κt

Component 2: β(2)x κt

• Age-period component

• κt: period effect

– changes with time, t⇒ mortality improvements

• β(2)x : age effect

– dictates relative rates of improvement at different ages

22

The Lee-Carter (1992) model

logm(t, x) = β(1)x + β(2)

x κt

• Time series model for κt (e.g. random walk)

• Single κt for all ages

• Future T : St.Dev.[logm(T, x)] = β(2)x × St.Dev.[κT ]

23

Comparison of Eight Models

Cairns, et al (2009) North American Actuarial Journal

• 8 models

• Historical data

• Backtesting

• Plausibility of forecasts

24

General class of models

logm(t, x) = β(1)x κ

(1)t γ

(1)t−x + . . .+ β(N)

x κ(N)t γ

(N)t−x

OR

logit q(t, x) = β(1)x κ

(1)t γ

(1)t−x + . . .+ β(N)

x κ(N)t γ

(N)t−x

• β(k)x = age effect for component k

• κ(k)t = period effect for component k

• γ(k)t−x = cohort effect for component k

25

Lee-Carter family

logm(t, x) = β(1)x κ

(1)t γ

(1)t−x + . . .+ β(N)

x κ(N)t γ

(N)t−x

• β(k)x = non-parametric age effects

not smooth (can be smoothed)

• κ(k)t and γ

(k)t−x = random period and cohort effects

26

M1: Lee-Carter (1992) model (LC)

logm(t, x) = β(1)x + β(2)

x κ(2)t

• N = 2 components

• β(1)x , β(2)

x age effects

• κ(2)t single random period effect

• κ(1)t ≡ 1

• # parameters = 2× nages + nyears

27

Cohort Effects (e.g. Willetts, 2004)

Annual mortality improvement rates (Engl. & Wales, males)

1970 1980 1990 2000

2040

6080

Year

Age

−2%

−1%

0%

1%

2%

3%

4%

Ann

ual i

mpr

ovem

ent r

ate

(%)

28

M2: Renshaw-Haberman (2006) model (RH)

logm(t, x) = β(1)x + β(2)

x κ(2)t + β(3)

x γ(3)t−x

• N = 3 components

• β(1)x , β(2)

x , β(3)x age effects

• κ(2)t single random period effect

• γ(3)t−x single cohort effect

29

M3: Age-Period-Cohort model (APC)

logm(t, x) = β(1)x + κ

(2)t + γ

(3)t−x

• N = 3 components

• Special case of R-H model

• β(1)x age effect; β(2)

x = β(3)x = 1

• κ(2)t single random period effect

• γ(3)t−x single random cohort effect

30

Background

• M1: Lee-Carter

– First (??) stochastic mortality model

– Simple and robust

– Reasonable fit over a wide range of ages

• M2: Renshaw-Haberman

– Incorporation of a cohort effect

• M3: APC

– Roots in medical statistics, pre Lee-Carter

– Simpler and more robust than R-H

31

M4: P-splines family

Age-Period models

logm(t, x) =∑k,l

β(k)x κ

(l)t γ

(k,l)t−x

where

• β(k)x and κ

(l)t are B-spline basis functions

• γ(k,l)t−x are constant in t− x for each (k, l)

32

Background

• M4: Age-Cohort P-splines model

– Data are noisy

– Underlying m(t, x) is smooth

– Model ⇒ parsimonious, non-parametric fit

– Output: confidence intervals for underlying smooth surface

(Non-parametric generalisation of linear regression)

33

CBD family

logit q(t, x) = logq(t, x)

1− q(t, x)

= β(1)x κ

(1)t γ

(1)t−x + . . .+ β(N)

x κ(N)t γ

(N)t−x

• β(k)x = parametric age effects

pre-specified, e.g. constant, linear, quadratic in x

• κ(k)t and γ

(k)t−x = random period and cohort effects

34

M5: Cairns-Blake-Dowd (2006) model (CBD-1)

logit q(t, x) = κ(1)t + κ

(2)t (x− x) =

2∑i=1

β(i)x κ

(i)t γ

(i)t−x

• N = 2 components

• β(1)x = 1, β(2)

x = (x− x) age effects

• κ(1)t , κ

(2)t correlated random period effects

• γ(1)t−x = γ

(2)t−x ≡ 1 (model has no cohort effect)

35

Background

• M5: CBD-1

– Designed to take advantage of simple structure at higher ages

⇒ focus on pension plan longevity risk

– Two random period effects ⇒ allows different improvements at

different ages at different times

– Simple and robust, good at bigger picture

36

Case study: England and Wales males

60 65 70 75 80 85 90 95

−5

−4

−3

−2

−1

0

Age of cohort at the start of 2002

log

q_y/

(1−

q_y)

qy = mortality rate at age y in 2002

Data suggests logit qy = log qy/(1− qy) is linear

37

M6-M8: Cohort-effect extensions to CBD-1

• M6:logit q(t, x) = κ

(1)t + κ

(2)t (x− x) + γ

(3)t−x

• M7:logit q(t, x) = κ

(1)t + κ

(2)t (x− x)

+κ(3)t

{(x− x)2 − σ2

x

}+ γ

(4)t−x

• M8:logit q(t, x) = κ

(1)t + κ

(2)t (x− x) + γ

(3)t−x(xc − x)

38

Background

• M6-M8: CBD-2/3/4

– Developed during the course of the bigger study

– Build on the advantages of M1-M5

– Avoid the disadvantages of M1-M5

– Models focus on the higher ages

39

Past and Present: Modelling Genealogy

--Time

APC model (M3) - APC model (M3)

Booth et al.

Hyndman et al.

Lee-Carter (M1) -���

���

���

���3

@@@ -

Currie/Richards (M4)2-D P-splines

Eilers/MarxP-splines

��1-

DDE

���

���

-

CBD-1 (M5) -Q

QQsJJJJJ

JJJJJ - CBD-R

Mavros et al.

Renshaw-Haberman (M2)�����

CBD-2 (M6)

CBD-3 (M7)

CBD-4 (M8)

���*

��

���

-CBD-5 (M9)

Plat -���@@RAAAU

@@ ����- Multi-

population

- Multi-population

40

Quantitative Criteria

Bayes Information Criterion (BIC)

• Model k: lk = model maximum likelihood

• BIC penalises over-parametrised models

• Model k: BICk = lk − 12nk logN

– nk = number of parameters (effective)

– N = number of observations

41

Maximum Likelihood Estimation

Usual approach:• Stage 1: estimate the β(k)

x , κ(k)t , γ

(k)t−x without reference to the

stochastic models governing the period and cohort effects.

• Stage 2: fit a stochastic model to the κ(k)t and γ

(k)t−x

• Okay for large populations

• Smaller populations: exercise caution –β(k)x , κ

(k)t , γ

(k)t−x subject to estimation error

42

Alternatives to 2-stage MLE

• 1-stage MLE

– Models for κ(k)t , γ

(k)t−x specified in advance

• Full Bayesian model (e.g. Czado et al.)

– Models for κ(k)t , γ

(k)t−x specified in advance

– Output includes posterior distributions for model parameters

plus latent β(k)x , κ

(k)t , γ

(k)t−x

43

2-Stage MLE: Application to 8 Models

• England and Wales males

• 1961-2004

• Ages 60-89

• Exclusions

– 1961-1970: ages 85-89 (not available)

– 1886 cohort (unreliable exposures)

– Cohorts with 4 or fewer data points (overfitting)

44

Typical parameter estimation results: M3-APC

60 65 70 75 80 85 90

−4.

5−

4.0

−3.

5−

3.0

−2.

5−

2.0

−1.

5−

1.0

Age Effect, beta1

Age1960 1970 1980 1990 2000

−15

−10

−5

05

1015

Period Effect, Kappa2

Year1880 1900 1920 1940

−10

−8

−6

−4

−2

02

4

Cohort Effect, Gamma3

Year of Birth

45

Model Max log-lik. # parameters BIC (rank)

M1: LC -8912.7 102 -9275.8

M2: RH -7735.6 203 -8458.1

M3: APC -8608.1 144 -9120.6

M4: P-Splines -9245.9 74.2 -9372.9

M5: CBD-1 -10035.5 88 -10348.8

M6: CBD-2 -7922.3 159 -8488.3

M7: CBD-3 -7702.1 202 -8421.1

M8: CBD-4 -7823.7 161 -8396.8

46

The BIC doesn’t tell us the whole story ...

Qualitative Criteria – Graphical diagnostics

• Poisson model ⇒ (t, x) cells are all independent.

• Standardised residuals:

Z(t, x) =D(t, x)− m(t, x)E(t, x)√

m(t, x)E(t, x)

• If the data are not i.i.d.:

What do the patterns tell us?

47

Are standardised residuals i.i.d.? LC and RH models

1970 1980 1990 2000

6065

7075

8085

90

Model M1

1970 1980 1990 2000

6065

7075

8085

90

Model M2

Black ⇒ Z(t, x) < 0

48

APC and P-splines models

1970 1980 1990 2000

6065

7075

8085

90

Model M3

1970 1980 1990 2000

6065

7075

8085

90

Model M4

49

CBD-1 and CBD-2 models

1970 1980 1990 2000

6065

7075

8085

90

Model M5

1970 1980 1990 2000

6065

7075

8085

90

Model M6

50

CBD-3 and CBD-4 models

1970 1980 1990 2000

6065

7075

8085

90

Model M7

1970 1980 1990 2000

6065

7075

8085

90

Model M8

51

Are the standardised residuals i.i.d.?

More graphical diagnostics:

Scatterplots of residuals versus

• Age

• Year of observation

• Year of birth

52

M1: LC model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

53

M2: RH model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

54

M3: APC model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

55

M4: P-splines model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

56

M5: CBD-1 model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

57

M6: CBD-2 model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

58

M7: CBD-3 model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

59

M8: CBD-4 model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

60

Robustness

Want to see stability in parameter estimates

• Extra years of data

• Extra ages

• Within model hierarchy

61

M7 (CBD-3): (a) 1961 to 2004 (dots) or (b) 1981 to 2004 (solid lines).

1970 1980 1990 2000

−3.

2−

2.8

−2.

4 Kappa_1(t)

Year

1970 1980 1990 2000

0.08

00.

090

0.10

00.

110 Kappa_2(t)

Year

1970 1980 1990 2000−1

e−

03−

4 e

−04

2 e

−04

Kappa_3(t)

Year

1880 1900 1920 1940

−0.

060.

000.

040.

08

Gamma_4(t−x)

Year of birth

62

RECAP: M5: CBD-1 model

1960 1980 2000

−10

−5

05

10

Year of Observation

Sta

ndar

dise

d re

sidu

als

60 70 80 90

−10

−5

05

10

Age

Sta

ndar

dise

d re

sidu

als

1880 1910 1940

−10

−5

05

10

Year of Birth

Sta

ndar

dise

d re

sidu

als

63

M2 (RH): (a) 1961 to 2004 (dots) or (b) 1981 to 2004 (solid lines).

60 65 70 75 80 85 90

−4.

0−

3.0

−2.

0−

1.0 Beta_1(x)

Age

60 65 70 75 80 85 90

0.01

0.02

0.03

0.04

0.05

0.06

Beta_2(x)

Age

1970 1980 1990 2000

−10

−5

05

10

Kappa_2(t)

Year

60 65 70 75 80 85 90

−0.

10.

00.

10.

20.

30.

4

Beta_3(x)

Age

1880 1900 1920 1940

−10

−5

05

Gamma_3(t−x)

Year of birth

64

Robustness

• Parameter estimates should not be too sensitive to the choice of

range of ages and years.

• M2 has a possible problem

• β(3)x age effect seems to be qualitatively different for the

1961-2004 versus 1981-2004

65

Qualitative criteria or issues

• Forecast reasonableness

• More on robustness

66

Simulation models

• Up to now: historical fit only

• Forecasting requires a stochastic model

• ARIMA time series models to simulate future period and cohort

effects

⇒ Process Risk or Stochastic Risk

• Later: parameter risk and model risk

67

Simulation models

Examples:

• M1: Lee-Carter model

– period effect, κ(2)t = random walk with drift

• M7: CBD-3 model

– (κ(1)t , κ

(2)t , κ

(3)t ) = multivariate random walk with drift

– γ(4)c = AR(1) model ≡ARIMA(1,0,0)

68

Mortality Fan Charts + A plausible set of forecasts

1960 1980 2000 2020 2040

0.00

50.

010

0.02

00.

050

0.10

00.

200

AGE 65

AGE 75

AGE 85

Year, t

Mor

talit

y ra

te, q

(t,x

)

Model CBD−1 Fan Chart

69

Model risk

1960 1980 2000 2020 2040

0.00

50.

010

0.02

00.

050

0.10

00.

200

AGE 65

AGE 75

AGE 85

Year, t

Mor

talit

y ra

te, q

(t,x

)

Model CBD−1 Fan Chart

70

Model risk

1960 1980 2000 2020 2040

0.00

50.

010

0.02

00.

050

0.10

00.

200

AGE 65

AGE 75

AGE 85

Year, t

Mor

talit

y ra

te, q

(t,x

)

Combined CBD−1, CBD−3 Fan Chart

71

Model risk

1960 1980 2000 2020 2040

0.00

50.

010

0.02

00.

050

0.10

00.

200

AGE 65

AGE 75

AGE 85

Year, t

Mor

talit

y ra

te, q

(t,x

)

Combined CBD−1, CBD−3, CBD−4 Fan Chart

72

Plausibility of forecasts

• Defining “Plausible” is impossible!

• Visually: given the forecast

– are you reasonably comfortable?

– slightly uncomfortable?

– fan chart is clearly unreasonable?

73

US males 1968-2003: M8 – unreasonable forecasts

1960 1980 2000 2020 2040

0.00

20.

005

0.02

00.

050

0.20

0

x = 84

x = 75

x=65

Mor

talit

y R

ate

M8A

74

Robustness of Forecasts

• Forecasts Set 1:

– Data from 1961-2004 ⇒ β(k)x , κ

(k)t , γ

(k)t−x

– Use full set of β(k)x , κ

(k)t , γ

(k)t−x to make forecasts

• Forecasts Set 3:

• Data from 1981-2004 ⇒ β(k)x , κ

(k)t , γ

(k)t−x

• Use full set of β(k)x (30), κ

(k)t (24), γ

(k)t−x (45) to make forecasts

75

Robustness of Forecasts

• Forecasts Set 2:

– Data from 1961-2004 ⇒ β(k)x , κ

(k)t , γ

(k)t−x

– To make forecasts:

∗ Use all 30 β(k)x

∗ Use the last 24 κ(k)t only (out of 44)

∗ Use the last 45 γ(k)t−x only (out of 65)

– i.e. as if 1981-2004

76

Perfect model + large population

Forecast sets 2 and 3:

• Same β(i)x , κ

(i)t , γ

(i)t−x

• Same forecasts

Good robust model

Forecast sets 2 and 3:

• Similar β(i)x , κ

(i)t , γ

(i)t−x

• Similar forecasts

77

Robustness: e.g. M3 - Age-Period-Cohort model

1960 1980 2000 2020 2040

0.01

0.02

0.05

0.10

1961−2004 data: APC full1961−2004 data: APC limited1981−2004 data: APC

APC Model − Age 75 Mortality Rates

Year, t

Mor

talit

y ra

te

78

Robustness: e.g. M7 - CBD-3 model

1900 1940 1980

−0.

10−

0.05

0.00

0.05

0.10

Model M7

gam

ma4

1960 1980 2000 2020 2040

0.01

0.02

0.04

Age 65 Mortality Rates

Mor

talit

y ra

te

1960 1980 2000 2020 2040

0.01

0.02

0.05

0.10

Age 75 Mortality Rates

Mor

talit

y ra

te

1960 1980 2000 2020 2040

0.05

0.10

0.20

Age 85 Mortality Rates

Mor

talit

y ra

te

79

Not all models are robust: Renshaw-Haberman model

1960 1980 2000 2020 2040

0.00

50.

020

0.05

00.

200

1961−2004 data: R−H full1961−2004 data: R−H limited1981−2004 data: R−H

Model R−H (ARIMA(1,1,0)) projections

Year, t

Coh

ort e

ffect

x=65

x=75

x=85

80

Robustness Problem

• Likely reason: Likelihood function has multiple maxima

• Consequences:

– Lack of robustness within sample

– Lack of robustness in forecasts

∗ central trajectory

∗ prediction intervals

– Some sample periods ⇒ implausible forecasts

81

Parameter Uncertainty: CBD model M5 example

1960 1970 1980 1990 2000

−3.

4−

3.0

−2.

6−

2.2

2−factor model: Kappa_1(t)=1

Year, t

1960 1970 1980 1990 2000

0.08

50.

095

0.10

5

2−factor model: Kappa_2(t)

Year, t

82

κt = (κ(1)t , κ

(2)t )′

Model: Random walk with drift

κt+1 − κt = µ+ CZ(t+ 1)

• µ = (µ1, µ2)′ = drift

• V = CC ′ = variance-covariance matrix

• Estimate µ and V

• Quantify parameter uncertainty in µ and V

• Quantify the impact of parameter uncertainty

83

Application: cohort survivorship

• Cohort: Age x at time t = 0

• S(t, x) = survivor index at t

proportion surviving from time 0 to time t

S(t, x) = (1− q(0, x))× (1− q(1, x+ 1))× . . .

. . .× (1− q(t− 1, x+ t− 1))

84

90% Confidence Interval (CI) for Cohort Survivorship

65 70 75 80 85 90

0.0

0.2

0.4

0.6

0.8

1.0 Data from 1982−2002

Age

Pro

port

ion

surv

ivin

g, S

(x)

E[S(x)] with param. uncertaintyCI without param. uncertaintyCI with param. uncertainty

85

Cohort Survivorship: General Conclusions

• Less than 10 years:

– Systematic risk not significant

• Over 10 years

– Systematic risk becomes more and more significant over time

• Over 20 years

– Parameter risk begins to dominate (+ model risk)

86

Part 1: Concluding remarks

• Range of models to choose from

• Quantitative criteria are only the starting point

• Additional criteria ⇒

– Some models pass

– Some models fail

• Focus here on mortality data at higher ages

– Wider age range ⇒ CBD models less good

87

Applications – A Taster

88

Applications – Scenario Generation

Example: the Lee Carter Model

• m(t, x) = β(1)(x) + β(2)(x)κ(t)

• Choose a time series model for κ(t)

• Calibrate the time series parameters using data up to the current

time (time 0)

• Generate j = 1, . . . , N stochastic scenarios of κ(t)

κ1(t), . . . , κN(t)

89

• Generate N scenarios for the future m(t, x)

mj(t, x) for j = 1, . . . , N , t = 0, 1, 2, . . ., x = x0, . . . , x1

• Generate N scenarios for the survivor index, Sj(t, x)

• Calculate financial functions

+ variations for some financial applications.

90

0 10 20 30 40 50 60

−1.

0−

0.5

0.0

0.5

Historical Simulated

Period Effect: One Scenario

Time

Per

iod

Effe

ct, k

appa

(t)

κ(t): Generate scenario 1

91

0 10 20 30 40 50 60

−1.

0−

0.5

0.0

0.5

Historical Simulated

Period Effect: Multiple Scenarios

Time

Per

iod

Effe

ct, k

appa

(t)

Multiple scenarios

92

0 10 20 30 40 50 60

−1.

0−

0.5

0.0

0.5

Historical Simulated

Period Effect: Fan Chart

Time

Per

iod

Effe

ct, k

appa

(t)

Fan chart

93

0 5 10 15 20 25 30

0.00

60.

008

0.01

20.

016

Death Rates, Age 65: One Scenario

Time

Dea

th R

ate

(log

scal

e)

94

0 5 10 15 20 25 30

0.00

60.

008

0.01

20.

016

Death Rates, Age 65: Multiple Scenarios

Time

Dea

th R

ate

(log

scal

e)

95

0 5 10 15 20 25 30

0.00

60.

008

0.01

20.

016

Death Rates, Age 65: Fan Chart

Time

Dea

th R

ate

(log

scal

e)

96

30 35 40 45 50 55 60

6570

7580

8590

Extract Cohort Death Rates, m(t,x+t−1)

Time

Age

Annuity valuation ⇒ follow cohorts m(0, x) → m(1, x+ 1) → m(2, x+ 2) . . .

97

65 70 75 80 85 90 95 100

0.01

0.02

0.05

0.10

0.20

Cohort Death Rates From Age 65:One Scenario

Cohort Age

Dea

th R

ate

(log

scal

e)

Annuity valuation ⇒ follow cohorts m(0, x) → m(1, x+ 1) → m(2, x+ 2) . . .

98

65 70 75 80 85 90 95 100

0.01

0.02

0.05

0.10

0.20

Cohort Death Rates From Age 65:Multpiple Scenarios

Cohort Age

Dea

th R

ate

(log

scal

e)

99

65 70 75 80 85 90 95 100

0.01

0.02

0.05

0.10

0.20

Cohort Death Rates From Age 65:Fan Chart

Cohort Age

Dea

th R

ate

(log

scal

e)

100

65 70 75 80 85 90 95 100

0.0

0.2

0.4

0.6

0.8

1.0

Survivorship From Age 65:One Scenario

Cohort Age

Sur

vivo

r In

dex

(log

scal

e)

Cohort death rates −→ cohort survivorship

101

65 70 75 80 85 90 95 100

0.0

0.2

0.4

0.6

0.8

1.0

Survivorship From Age 65:Multiple Scenarios

Cohort Age

Sur

vivo

r In

dex

(log

scal

e)

102

65 70 75 80 85 90 95 100

0.0

0.2

0.4

0.6

0.8

1.0

Survivorship From Age 65:Fan Chart

Cohort Age

Sur

vivo

r In

dex

(log

scal

e)

103

17 18 19 20 21 22

010

020

030

040

050

0Cohort Life Expectancy

from Age 65

Life Expectancy From Age 65

Fre

quen

cy

17 18 19 20 21 22

0.0

0.2

0.4

0.6

0.8

1.0

Cohort Life Expectancy from Age 65

Life Expectancy From Age 65

Cum

ulat

ive

Pro

babi

lity

Cohort survivorship −→ ex post cohort life expectancy

Equivalent to a continuous annuity with 0% interest

104

14 15 16 17

020

040

060

080

0Present Value of Annuity

from Age 65

Present Value ofAnnuity From Age 65

Fre

quen

cy

13.5 14.5 15.5 16.5

0.0

0.2

0.4

0.6

0.8

1.0

Present Value of Annuity from Age 65

Present Value ofAnnuity From Age 65

Cum

ulat

ive

Pro

babi

lity

• Annuity of 1 per annum payable annual in arrears

• Interest rate: 2%

105

14 15 16 17

020

040

060

080

0

Mean 99.5% VaR+8%

Present Value of Annuity from Age 65

Present Value ofAnnuity From Age 65

Fre

quen

cy

13.5 14.5 15.5 16.5

0.0

0.2

0.4

0.6

0.8

1.0

Present Value of Annuity from Age 65

Present Value ofAnnuity From Age 65

Cum

ulat

ive

Pro

babi

lity

• Mean = $15.17 per $1 annuity; BUT

• Need $16.38 to be 99.5% sure of covering all liabilities

106

30 35 40 45 50 55 60

6570

7580

8590

Extract Period Death Rates, m(t,x+t−1)

Time

Age

Period life expectancy and related quantities

107

0 10 20 30 40 50 60

1416

1820

2224

Historical Forecast

Period Life Expectancy From Age 65By Calendar Year

Time

Per

iod

Life

Exp

ecta

ncy

108

0 10 20 30 40 50 60

0.00

50.

010

0.01

50.

020

Historical SimulatedRecalibration+

Central Forecast

Simulated

Recalibration

Death Rates, Age 65: One Scenario

Time

Dea

th R

ate

(log

scal

e)

• Valuation at time 10 ⇒

• Recalibrate model and parameters → central forecast

• Updated liability value at time 10

109

0 10 20 30 40 50 60

0.00

50.

010

0.01

50.

020

Historical SimulatedRecalibration+

Central ForecastR

ecalibration

Death Rates, Age 65: Multiple Scenarios

Time

Dea

th R

ate

(log

scal

e)

110

13.5 14.0 14.5 15.0 15.5 16.0 16.5

0.0

0.2

0.4

0.6

0.8

1.0

PV: Full RunoffPV: Valuation at Time 10PV: Valuation at Time 0

Present Value of Annuity from Age 65

Present Value ofAnnuity From Age 65

Cum

ulat

ive

Pro

babi

lity

Applications: Hedging longevity risk

111

Part 2: Concluding Remarks

• Here: Lee-Carter →m(t, x)→ application

• Modular code ⇒Model X →m(t, x); m(t, x)→ application

• Applications

– Development of simple stress tests

– Reserving

– Longevity risk transfer

• Multi-population models

112

References

• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I. (2009)

A quantitative comparison of stochastic mortality models using data from England and Wales and the United States.

North American Actuarial Journal 13(1): 1-35.

• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G., Epstein, D., and Khalaf-Allah M. (2011)

The Plausibility of Mortality Density Forecasts: An Analysis of Six Stochastic Mortality Models.

Insurance: Mathematics and Economics, 48: 355-367.

• Cairns, A.J.G., Kallestrup-Lamb, M., Rosenskjold, C.P.T., Blake, D., and Dowd, K., (2017)

Modelling Socio-Economic Differences in the Mortality of Danish Males Using a New Affluence Index.

Preprint. http://www.macs.hw.ac.uk/∼andrewc/papers/ajgc73.pdf