CICIRM 2013
Developments in Mortality and Longevity Risk
Modeling
Michael Sherris
University of New South Wales
2013 China International Conference on Insurance And
Risk Management (CICIRM 2013)
July 17th-20th, 2013
Expo Garden Hotel, Kunming, Yunnan ,China
Longevity/Mortality Models
Mortality/Longevity Models
Demographic
and Actuarial
models
Life tables
Improvement
factors
Stochastic
Financial Risk
Survival Curve Framework
models
models
Aged based nonParametric or
Parametric survival
curve
Stochastic
parameters
Risk factor
dynamics
Price of risk
Consistent
Market calibration
Longevity/Mortality Models Data
Individuals
• Risk factors: Age, sex,
smoking status,
education,
occupation, ethnicity,
income, geographical
location, marital
status
• Cause of death
• Survey data
Population by
country
• Aggregate mortality
by age, gender,
period, cohort
• Aggregate health data
• Prevalence of health
conditions
• Mortality rates by
Causes of death
Life insurance,
pension fund and
annuity pools
• Aggregation of
deaths and exposures
by underwriting risk
factors
• Effect of selection
Longevity/Mortality Model Risk
Systematic Risk
• Uncertain future mortality – age-period-cohort
trends, survival curve, risk factors, price of risk
Idiosyncratic Risk
• Pooling of individual survival risks – pool size,
assumption of i.i.d. risks
Heterogeneity
• Mortality rates and trends varying by individuals
of the same age (gender)
Agenda – Model Risks
- Cohort and forward survival curves (financial risk) versus
age-period models (demographic/actuarial/survival curve
adapted for cohort effects)
- Consistent versus inconsistent mortality curves
(dynamics and future survival curves) and parameter
stability
- Risk factors and price of risk (explicit versus ad-hoc risk
adjustment)
- Heterogeneity – multiple state models with systematic
risk versus heterogeneity only (frailty, Markov ageing
models)
- Drawing on longevity research at CEPAR, UNSW
Some of the issues – which would you prefer?
Model A
Males
Model B
Source: Shao, W., Sherris, M., and Hanewald, K., (2103), Reverse Mortgage Pricing and Capital Requirements Allowing
for Idiosyncratic House Price Risk and Longevity Risk.
Some of the issues – which would you prefer?
Model A
Females
Model B
Source: Shao, W., Sherris, M., and Hanewald, K., (2103), Reverse Mortgage Pricing and Capital Requirements Allowing
for Idiosyncratic House Price Risk and Longevity Risk.
Systematic Mortality Model Risk
Model A
Model B
• Discrete age survival function
• Cohort trends – period and
age-to-age variability and
trends
• Cohort curve generated by the
dynamics
• Multiple risk factors based on
age
• Dependence in volatility –
principal components
• Parametric survival function smoothing of age-to-age
variability
• Period trends
• Cohort curve read off projected
period curves
• Two factors - stochastic
parameters of mortality curve
• Dependence – two factors,
from smoothed curve
dynamics
Importance of cohort and forward survival curves
Source: Alai, D.H. and Sherris, M. (2012), Rethinking Age-Period-Cohort Mortality Trend Models,
Article published on line 16 Apr 2012, Scandinavian Actuarial Journal
Model Mortality Surface (age-period and cohort)
Source: C. Blackburn and M. Sherris, (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–73
Quantification of Systematic Longevity Risk
Forward
survival
curves
(cohort)
Expected
survival
curves and
pricing
Source: C. Blackburn and M. Sherris, (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–73
Consistent survival curves – 3 factor model
Three Factor Consistent HJM mortality model
Dynamics generates
consistent survival
curves
Source: C. Blackburn and M. Sherris, (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–73
Consistent model risk factors – 3 factor
estimation stability
Refitting model at
different time points
demonstrates model
consistency
How many models used in
practice have this
property?
Source: C. Blackburn and M. Sherris, (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–73
Consistent Survivor Curves – 2 versus 3 factor
Increase in
number of
factors
explains
older age
mortality
better
Source: C. Blackburn and M. Sherris, (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–73
Best Estimate Forward Survivor Curve – 2
factors
Removes need for simulations in simulations for ALM,
valuation, risk quantification
Price of Risk and Forward Survivor Curve – 2
factors
Price of risk – financial approaches versus
actuarial (Wang transform)
Wang transform gives wrong signs and magnitude
for prices of risk (offset by other parameters)
Sharpe ratio scales the survivor curve and does
not impact risk factor loadings
Price of risk - longevity risk swap pricing
Pricing differences less pronounced than for risk quantification
Price of risk versus volatility parameter risk –
survival curve
Source: Fung, M. C., Ignatieva, K. and Sherris, M., (2013), Systematic Mortality Risk: An Analysis of
Guaranteed Lifetime Withdrawal Benefits in Variable Annuities
Price of risk versus volatility parameter risk GLWB
Equity
exposure
Mortality risk
premium
Source: Fung, M. C., Ignatieva, K. and Sherris, M., (2013), Systematic Mortality Risk: An Analysis of
Guaranteed Lifetime Withdrawal Benefits in Variable Annuities
Price of risk and impact on risk based capital
Solvency capital costs versus
longevity swap for a life annuity
Varying price of risk in Model A
Incentives to hedge shorter
terms and retain tail risk
with higher prices of risk
Source: Meyricke, R. and Sherris, M. (2013), Optimal Longevity Risk Management Under Solvency II
Mortality heterogeneity - which model?
Calibration to population data versus individual
data (GLMM)
Frailty versus Markov ageing
Does not include systematic mortality risk
required
to assess
solvency/tail
risk
Source: Ramona Meyricke
and Michael Sherris
(2013), The determinants of mortality
heterogeneity and
implications for pricing underwritten annuities.
Heterogeneity model risk – longevity tail risk for
annuity fund
Fixed investment return of 3% p.a.
Mortality model
Markov
Le Bras
Vaupel
Heterogeneity
best health only
mixed
mixed w self
selection
best health only
mixed
mixed w self
selection
best health only
mixed
mixed w self
selection
Annuity
premium
16.32
14.29
14.29
15.84
14.16
14.16
16.29
14.72
14.72
Risk measures at age 110
Mean
Stdev
95% VaR
-0.07 386.09
631.73
-15.86 710.31
1176.89
5872.49 428.07
6566.69
4.24 607.33
986.31
11.56 635.70
1022.46
3105.13 613.12
4109.81
-0.88 658.73
1072.07
-1.61 673.32
1109.78
2610.51 666.36
3694.48
Premium for a life annuity of 1 p.a. and tail risk
measures for a pool of 1000 individuals aged 65.
Source: Sherris, M. and Zhou, Q. (2013), Model Risk, Mortality Heterogeneity and Implications for Solvency and
Tail Risk.
Effect of
adverse
selection
Heterogeneity model risk – investment and
longevity tail risk for annuity fund
Random investment return
Mortality model
Markov
Le Bras
Heterogeneity
best health only
state 2
state 3
state 4
state 5
mixed
mixed w self
selection
best health only
mixed
mixed w self
selection
best health only
mixed
mixed w self
selection
Annuity
premium
Risk measures at age 110
13.48
12.54
10.04
6.74
5.00
11.99
Mean
Stdev
95% VaR
-199.80
4912.11
7843.93
-198.90
4387.30
7117.17
-111.25
3192.87
5144.76
-54.63
1917.96
3131.44
-35.88
1478.46
2441.54
-132.34
4420.42
7051.55
11.99
12.95
11.84
-14675.61
-109.05
-59.61
4112.85
4901.30
4283.44
21204.18
7811.46
6883.19
11.84
13.14
12.13
-7006.90
-141.61
-112.90
4244.59
5040.23
4476.47
13922.83
8067.82
7234.56
Investmen
t risk
magnifies
longevity
risk and
impact of
selection
Vaupel
Premium for
a life annuity of 1 p.a. and tail risk measures
for a pool of 1000 individuals12.13
aged-5777.86
65 4397.70 12874.70
Results are shown for the different deterministic models of
heterogeneity.
Source: Sherris, M. and Zhou, Q. (2013), Model Risk, Mortality Heterogeneity and Implications for Solvency and
Tail Risk.
Heterogeneity model risk – impact of
systematic longevity risk
Pool
size
100
1000
10000
100000
Deterministic
Subordinated
Markov
Markov
122.66
286.21
388.23
2588.74
1216.31
25649.07
3914.59
254307.38
Without systematic risk
$500
$400
$300
$200
$100
$-
Standard deviation of the
fund at age 110 for life
annuity of 1 p.a. for best
health individuals aged 65
Fixed investment return of
3% p.a.
Stochastic model variance
of Gamma time change ν=0.095.
252627282930313233343536373839404142434445
years since start of contract
Pool 500
Pool 1000
With systematic risk
$3,000
$2,500
$2,000
$1,500
$1,000
$500
$252627282930313233343536373839404142434445
years since start of contract
Pool 500
Pool 1000
Source: Sherris, M. and Zhou, Q. (2013), Model Risk, Mortality Heterogeneity and Implications for Solvency and
Tail Risk.
Summary – key points
Mortality/longevity risk model developments – key
ideas
•
•
•
•
Model consistency and parameter stability
Tractability and ease of application
Risk factors and price of risk
Heterogeneity and data
26
Thank you for your attention
Michael Sherris
m.sherris@unsw.edu.au
School of Risk and Actuarial Studies
ARC Centre of Excellence in Population Ageing Research
University of New South Wales
Acknowledgement: ARC Linkage Grant Project LP0883398
Managing Risk with Insurance and Superannuation as Individuals
Age with industry partners PwC, APRA and the World Bank as well
as the support of the Australian Research Council Centre of
Excellence in Population Ageing Research project CE110001029.
References
Alai, D.H. and Sherris, M. (2012), Rethinking Age-Period-Cohort Mortality Trend Models, Article
published on line 16 Apr 2012, Scandinavian Actuarial Journal, DOI:
10.1080/03461238.2012.676563
Su, S. and Sherris, M. (2012), Heterogeneity of Australian Population Mortality and Implications for a
Viable Life Annuity Market, Insurance: Mathematics and Economics, 51, 2, 322–332.
Ziveyi, J, Blackburn, C., and Sherris, M. (2013), Pricing European Options on Deferred Annuities,
Insurance: Mathematics and Economics, Volume 52, Issue 2, March 2013, 300–311.
Blackburn, C. and Sherris, M., (2013), Consistent Dynamic Affine Mortality Models for Longevity Risk
Applications, Insurance: Mathematics and Economics, Volume 53, Issue 1, July 2013, Pages 64–
73 http://dx.doi.org/10.1016/j.insmatheco.2013.04.007
Meyricke, R. and Sherris, M. (2013), The determinants of mortality heterogeneity and implications for
pricing underwritten annuities, accepted Insurance: Mathematics and Economics, on-line 29 June
2013; http://www.sciencedirect.com/science/article/pii/S0167668713000887
Meyricke, R. and Sherris, M. (2013), Optimal Longevity Risk Management Under Solvency II.
Sherris, M. and Zhou, Q. (2013), Model Risk, Mortality Heterogeneity and Implications for Solvency
and Tail Risk.
Fung, M. C., Ignatieva, K. and Sherris, M., (2013), Systematic Mortality Risk: An Analysis of
Guaranteed Lifetime Withdrawal Benefits in Variable Annuities.
Shao, W., Sherris, M., and Hanewald, K., (2103), Reverse Mortgage Pricing and Capital Requirements
Allowing for Idiosyncratic House Price Risk and Longevity Risk.