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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 . 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 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 England and Wales males mortality (log scale) 7

Age = 25 Age = 45 Mortality rate Mortality rate 0.005 0.010 0.020 0.001 0.002 0.004

1900 1940 1980 1900 1940 1980

Year Year

Age = 65 Age = 85 0.05 0.10 Mortality rate Mortality rate 0.2 0.4 0.8

1900 1940 1980 1900 1940 1980

Year Year 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

• Backtesting 16

Biologically reasonable and plausible forecasts

• Biologically reasonable e.g. inverted mortality curve?? strong reversion?? time horizon matters

• Plausible forecasts trend and degree of uncertainty 17

Robustness

• What happens if I change the age ?

• 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

(1) (2) log m(t, x) = βx + βx κt

(1) Component 1: βx

• Age effect

• Baseline log-mortality curve (κt = 0) 21

(1) (2) log m(t, x) = βx + βx κt

(2) Component 2: βx κt • Age-period component

• κt: period effect – changes with time, t ⇒ mortality improvements • (2) βx : age effect – dictates relative rates of improvement at different ages 22

The Lee-Carter (1992) model

(1) (2) log m(t, x) = βx + βx κt

• model for κt (e.g. random walk)

• Single κt for all ages • (2) × Future T : St.Dev.[log m(T, x)] = βx 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

(1) (1) (1) (N) (N) (N) log m(t, x) = βx κt γt−x + ... + βx κt γt−x

OR

(1) (1) (1) (N) (N) (N) logit q(t, x) = βx κt γt−x + ... + βx κt γt−x • (k) βx = age effect for component k (k) • κt = period effect for component k (k) • γt−x = cohort effect for component k 25

Lee-Carter family

(1) (1) (1) (N) (N) (N) log m(t, x) = βx κt γt−x + ... + βx κt γt−x • (k) βx = non-parametric age effects not smooth (can be smoothed)

(k) (k) • κt and γt−x = random period and cohort effects 26

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

(1) (2) (2) log m(t, x) = βx + βx κt

• N = 2 components • (1) (2) βx , βx age effects (2) • κt single random period effect (1) • κt ≡ 1

• # parameters = 2 × nages + nyears 27

Cohort Effects (e.g. Willetts, 2004)

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

4%

3%

2%

1% Age 0%

−1% 20 40 60 80

Annual improvement rate (%) −2%

1970 1980 1990 2000

Year 28

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

(1) (2) (2) (3) (3) log m(t, x) = βx + βx κt + βx γt−x

• N = 3 components • (1) (2) (3) βx , βx , βx age effects (2) • κt single random period effect (3) • γt−x single cohort effect 29

M3: Age-Period-Cohort model (APC)

(1) (2) (3) log m(t, x) = βx + κt + γt−x

• N = 3 components

• Special case of R-H model • (1) (2) (3) βx age effect; βx = βx = 1 (2) • κt single random period effect (3) • γt−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 , pre Lee-Carter – Simpler and more robust than R-H 31

M4: P-splines family

Age-Period models ∑ (k) (l) (k,l) log m(t, x) = βx κt γt−x k,l where • (k) (l) βx and κt are B-spline basis functions (k,l) • γt−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 ) 33

CBD family q(t, x) logit q(t, x) = log 1 − q(t, x) (1) (1) (1) (N) (N) (N) = βx κt γt−x + ... + βx κt γt−x • (k) βx = parametric age effects pre-specified, e.g. constant, linear, quadratic in x

(k) (k) • κt and γt−x = random period and cohort effects 34

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

∑2 (1) (2) − (i) (i) (i) logit q(t, x) = κt + κt (x x¯) = βx κt γt−x i=1 • N = 2 components • (1) (2) − βx = 1, βx = (x x¯) age effects (1) (2) • κt , κt correlated random period effects (1) (2) • γt−x = γt−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 log q_y/(1−q_y) −5 −4 −3 −2 −1 0

60 65 70 75 80 85 90 95 Age of cohort at the start of 2002 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: (1) (2) (3) logit q(t, x) = κt + κt (x − x¯) + γt−x

• M7: (1) (2) logit q(t, x) = κt + κt (x − x¯) { } (3) − 2 − 2 (4) +κt (x x¯) σx + γt−x

• M8: (1) (2) (3) logit q(t, x) = κt + κt (x − x¯) + γt−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 Currie/Richards (M4) - Multi- 1 2-D P-splines population Eilers/Marx - - DDE P-splines Hyndman et al.  3 Booth et al.  Multi- APC model (M3)  - APC model (M3) -  population  -   Lee-Carter (M1) Renshaw-Haberman (M2) @@ @ @@ - -  Plat ¨* @@R - ¨¨  A CBD-1 (M5) Q CBD-2 (M6) A J Q AU J J Qs - J J CBD-3 (M7) CBD-5 (M9) J JJ^ J CBD-4 (M8) J - CBD-R Mavros- et al. Time 40

Quantitative Criteria

Bayes Information Criterion (BIC) ˆ • Model k: lk = model maximum likelihood

• BIC penalises over-parametrised models • ˆ − 1 Model k: BICk = lk 2nk log N

– nk = number of parameters (effective) – N = number of observations 41

Maximum Likelihood Estimation

Usual approach: • (k) (k) (k) Stage 1: estimate the βx , κt , γt−x without reference to the stochastic models governing the period and cohort effects. (k) (k) • Stage 2: fit a stochastic model to the κˆt and γˆt−x • Okay for large populations • Smaller populations: exercise caution – (k) (k) (k) βx , κt , γt−x subject to estimation error 42

Alternatives to 2-stage MLE

• 1-stage MLE (k) (k) – Models for κt , γt−x specified in advance

• Full Bayesian model (e.g. Czado et al.) (k) (k) – Models for κt , γt−x specified in advance – Output includes posterior distributions for model parameters (k) (k) (k) plus latent βx , κt , γt−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

Age Effect, beta1 Period Effect, Kappa2 Cohort Effect, Gamma3 −15 −10 −5 0 5 10 15 −10 −8 −6 −4 −2 0 2 4 −4.5 −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 60 65 70 75 80 85 90 1960 1970 1980 1990 2000 1880 1900 1920 1940 Age Year 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: D(t, x) − mˆ (t, x)E(t, x) Z(t, x) = √ mˆ (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

Model M1 Model M2 60 65 70 75 80 85 90 60 65 70 75 80 85 90 1970 1980 1990 2000 1970 1980 1990 2000 Black ⇒ Z(t, x) < 0 48

APC and P-splines models

Model M3 Model M4 60 65 70 75 80 85 90 60 65 70 75 80 85 90 1970 1980 1990 2000 1970 1980 1990 2000 49

CBD-1 and CBD-2 models

Model M5 Model M6 60 65 70 75 80 85 90 60 65 70 75 80 85 90 1970 1980 1990 2000 1970 1980 1990 2000 50

CBD-3 and CBD-4 models

Model M7 Model M8 60 65 70 75 80 85 90 60 65 70 75 80 85 90 1970 1980 1990 2000 1970 1980 1990 2000 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 Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 53

M2: RH model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 54

M3: APC model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 55

M4: P-splines model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 56

M5: CBD-1 model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 57

M6: CBD-2 model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 58

M7: CBD-3 model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 59

M8: CBD-4 model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 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). Kappa_1(t) Kappa_2(t) −3.2 −2.8 −2.4 0.080 0.090 0.100 0.110 1970 1980 1990 2000 1970 1980 1990 2000

Year Year Kappa_3(t) Gamma_4(t−x) −0.06 0.00 0.04 0.08

−1 e−03 −4 e−041970 2 e−04 1980 1990 2000 1880 1900 1920 1940

Year Year of birth 62

RECAP: M5: CBD-1 model Standardised residuals Standardised residuals Standardised residuals −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 1960 1980 2000 60 70 80 90 1880 1910 1940

Year of Observation Age Year of Birth 63

M2 (RH): (a) 1961 to 2004 (dots) or (b) 1981 to 2004 (solid lines). Beta_1(x) Beta_2(x) Kappa_2(t) −10 −5 0 5 10 −4.0 −3.0 −2.0 −1.0 0.01 0.02 0.03 0.04 0.05 0.06 60 65 70 75 80 85 90 60 65 70 75 80 85 90 1970 1980 1990 2000

Age Age Year Beta_3(x) Gamma_3(t−x) −10 −5 0 5 −0.1 0.0 0.1 0.2 0.3 0.4 60 65 70 75 80 85 90 1880 1900 1920 1940

Age 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 • (3) βx 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 (2) – period effect, κt = random walk with drift

• M7: CBD-3 model (1) (2) (3) – (κt , κt , κt ) = multivariate random walk with drift (4) ≡ – γc = AR(1) model ARIMA(1,0,0) 68

Mortality Fan Charts + A plausible set of forecasts

Model CBD−1 Fan Chart

AGE 85

AGE 75

AGE 65 Mortality rate, q(t,x) 0.005 0.010 0.020 0.050 0.100 0.200 1960 1980 2000 2020 2040

Year, t 69

Model risk

Model CBD−1 Fan Chart

AGE 85

AGE 75

AGE 65 Mortality rate, q(t,x) 0.005 0.010 0.020 0.050 0.100 0.200 1960 1980 2000 2020 2040

Year, t 70

Model risk

Combined CBD−1, CBD−3 Fan Chart

AGE 85

AGE 75

AGE 65 Mortality rate, q(t,x) 0.005 0.010 0.020 0.050 0.100 0.200 1960 1980 2000 2020 2040

Year, t 71

Model risk

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

AGE 85

AGE 75

AGE 65 Mortality rate, q(t,x) 0.005 0.010 0.020 0.050 0.100 0.200 1960 1980 2000 2020 2040

Year, t 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

x = 84

x = 75

x=65 Mortality Rate

M8A 0.002 0.005 0.020 0.050 0.200

1960 1980 2000 2020 2040 74

Robustness of Forecasts

• Forecasts Set 1: ⇒ (k) (k) (k) – Data from 1961-2004 βx , κt , γt−x (k) (k) (k) – Use full set of βx , κt , γt−x to make forecasts

• Forecasts Set 3: • ⇒ (k) (k) (k) Data from 1981-2004 βx , κt , γt−x • (k) (k) (k) Use full set of βx (30), κt (24), γt−x (45) to make forecasts 75

Robustness of Forecasts

• Forecasts Set 2: ⇒ (k) (k) (k) – Data from 1961-2004 βx , κt , γt−x – To make forecasts: ∗ (k) Use all 30 βx (k) ∗ Use the last 24 κt only (out of 44) (k) ∗ Use the last 45 γt−x only (out of 65) – i.e. as if 1981-2004 76

Perfect model + large population

Forecast sets 2 and 3: • (i) (i) (i) Same βx , κt , γt−x

• Same forecasts

Good robust model

Forecast sets 2 and 3: • (i) (i) (i) Similar βx , κt , γt−x

• Similar forecasts 77

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

APC Model − Age 75 Mortality Rates Mortality rate 1961−2004 data: APC full 1961−2004 data: APC limited

0.01 0.02 0.051981−2004 0.10 data: APC

1960 1980 2000 2020 2040

Year, t 78

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

Model M7 Age 65 Mortality Rates 0.01 0.02 0.04 gamma4 Mortality rate −0.10 −0.05 0.00 0.05 0.10 1900 1940 1980 1960 1980 2000 2020 2040

Age 75 Mortality Rates Age 85 Mortality Rates 0.05 0.10 0.20 Mortality rate Mortality rate 0.01 0.02 0.05 0.10

1960 1980 2000 2020 2040 1960 1980 2000 2020 2040 79

Not all models are robust: Renshaw-Haberman model

Model R−H (ARIMA(1,1,0)) projections

x=85

x=75

x=65 Cohort effect

1961−2004 data: R−H full 1961−2004 data: R−H limited 0.005 0.020 0.0501981−2004 0.200 data: R−H

1960 1980 2000 2020 2040

Year, t 80

Robustness Problem

• Likely reason: 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

2−factor model: Kappa_1(t)=1 2−factor model: Kappa_2(t) −3.4 −3.0 −2.6 −2.2 0.085 0.095 0.105 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000

Year, t Year, t 82

(1) (2) ′ κt = (κt , κt ) Model: Random walk with drift

κt+1 − κt = µ + CZ(t + 1) ′ • µ = (µ1, µ2) = drift ′ • V = CC = - 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 Data from 1982−2002

E[S(x)] with param. uncertainty Proportion surviving, S(x) CI without param. uncertainty CI with param. uncertainty 0.0 0.2 0.4 0.6 0.8 1.0 65 70 75Age 80 85 90 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

Period Effect: One Scenario Period Effect, kappa(t) Effect, Period

Historical Simulated −1.0 −0.5 0.0 0.5 0 10 20 30 40 50 60

Time κ(t): Generate scenario 1 91

Period Effect: Multiple Scenarios Period Effect, kappa(t) Effect, Period

Historical Simulated −1.0 −0.5 0.0 0.5 0 10 20 30 40 50 60

Time Multiple scenarios 92

Period Effect: Fan Chart Period Effect, kappa(t) Effect, Period

Historical Simulated −1.0 −0.5 0.0 0.5 0 10 20 30 40 50 60

Time Fan chart 93

Death Rates, Age 65: One Scenario Death Rate (log scale) 0.006 0.008 0.012 0.016

0 5 10 15 20 25 30

Time 94

Death Rates, Age 65: Multiple Scenarios Death Rate (log scale) 0.006 0.008 0.012 0.016

0 5 10 15 20 25 30

Time 95

Death Rates, Age 65: Fan Chart Death Rate (log scale) 0.006 0.008 0.012 0.016

0 5 10 15 20 25 30

Time 96

Extract Cohort Death Rates, m(t,x+t−1) Age 65 70 75 80 85 90 30 35 40 45 50 55 60

Time Annuity valuation ⇒ follow cohorts m(0, x) → m(1, x + 1) → m(2, x + 2) ... 97

Cohort Death Rates From Age 65: One Scenario Death Rate (log scale) 0.01 0.02 0.05 0.10 0.20 65 70 75 80 85 90 95 100

Cohort Age Annuity valuation ⇒ follow cohorts m(0, x) → m(1, x + 1) → m(2, x + 2) ... 98

Cohort Death Rates From Age 65: Multpiple Scenarios Death Rate (log scale) 0.01 0.02 0.05 0.10 0.20 65 70 75 80 85 90 95 100

Cohort Age 99

Cohort Death Rates From Age 65: Fan Chart Death Rate (log scale) 0.01 0.02 0.05 0.10 0.20 65 70 75 80 85 90 95 100

Cohort Age 100

Survivorship From Age 65: One Scenario Survivor Index (log scale) Survivor Index 0.0 0.2 0.4 0.6 0.8 1.0 65 70 75 80 85 90 95 100

Cohort Age Cohort death rates −→ cohort survivorship 101

Survivorship From Age 65: Multiple Scenarios Survivor Index (log scale) Survivor Index 0.0 0.2 0.4 0.6 0.8 1.0 65 70 75 80 85 90 95 100

Cohort Age 102

Survivorship From Age 65: Fan Chart Survivor Index (log scale) Survivor Index 0.0 0.2 0.4 0.6 0.8 1.0 65 70 75 80 85 90 95 100

Cohort Age 103

Cohort Life Expectancy Cohort Life Expectancy from Age 65 from Age 65 Cumulative Probability Cumulative 0 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 17 18 19 20 21 22 17 18 19 20 21 22 Life Expectancy Life Expectancy From Age 65 From Age 65

Cohort survivorship −→ ex post cohort life expectancy

Equivalent to a continuous annuity with 0% interest 104

Present Value of Annuity Present Value of Annuity from Age 65 from Age 65 Frequency Cumulative Probability Cumulative 0 200 400 600 800 0.0 0.2 0.4 0.6 0.8 1.0 14 15 16 17 13.5 14.5 15.5 16.5 Present Value ofAnnuity Present Value ofAnnuity From Age 65 From Age 65

• Annuity of 1 per annum payable annual in arrears

• Interest rate: 2% 105

Present Value of Annuity Present Value of Annuity from Age 65 from Age 65

Mean 99.5% VaR +8% Frequency Cumulative Probability Cumulative 0 200 400 600 800 0.0 0.2 0.4 0.6 0.8 1.0 14 15 16 17 13.5 14.5 15.5 16.5 Present Value ofAnnuity Present Value ofAnnuity From Age 65 From Age 65

• Mean = $15.17 per $1 annuity; BUT

• Need $16.38 to be 99.5% sure of covering all liabilities 106

Extract Period Death Rates, m(t,x+t−1) Age 65 70 75 80 85 90 30 35 40 45 50 55 60

Time Period life expectancy and related quantities 107

Period Life Expectancy From Age 65 By Calendar Year Period Life Expectancy Life Period

Historical Forecast 14 16 18 20 22 24 0 10 20 30 40 50 60

Time 108

Death Rates, Age 65: One Scenario Recalibration

Death Rate (log scale) Simulated

Recalibration+ Historical Simulated Central Forecast 0.005 0.010 0.015 0.020 0 10 20 30 40 50 60

Time • Valuation at time 10 ⇒

• Recalibrate model and parameters → central forecast

• Updated liability value at time 10 109

Death Rates, Age 65: Multiple Scenarios Recalibration Death Rate (log scale)

Recalibration+ Historical Simulated Central Forecast 0.005 0.010 0.015 0.020 0 10 20 30 40 50 60

Time 110

Present Value of Annuity from Age 65 PV: Full Runoff PV: Valuation at Time 10 PV: Valuation at Time 0 Cumulative Probability Cumulative 0.0 0.2 0.4 0.6 0.8 1.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 Present Value ofAnnuity From Age 65

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