2. Contents
Mortality Modelling
Contents
Deterministic Mortality Models
Deterministic Models 3
Stochastic Mortality Models
Why Stochastic Models? 6
Stochastic Mortality Models 9
Application Procedure and Model Comparison 15
Appendix 22
2
“Where there is a considerable range of possible outcomes, the FSA expects firms to use
stochastic techniques to evaluate these risks. In time, for example, longevity risk, where this
constitutes a significant risk for the firm, may fall into this category.”
-----FSA’s Regulatory Guidance for Actuaries
4. Mortality Modelling
Deterministic Projections
Scenario Tests with Different Deterministic Mortality Tables
Different mortality assumptions imply different pension benefit cash flow structures for a
specific pension scheme.
This research compares the impact on a specific pension scheme’s cash flows, presents
value and duration of ten deterministic mortality projections produced by the CMI,
including the original “92” Series projection, cohort projections and cohort projection with
1% and 2% underpins.
4
5. The table below shows that a change in mortality assumptions can have a big impact on
the PV and Duration of a pension scheme’s benefits.
*E.g. The PV for this specific scheme (calculated with a long cohort / 2% underpin mortality
table) is 13% larger than the PV calculated with “92” table, and the duration increases by
13.5%.
Scenario Tests with Different Deterministic Mortality Tables
Active Deferred Pensioners Total
PV
(£ million)
Duration
(year)
PV
(£ million)
Duration
(year)
PV
(£ million)
Duration
(year)
PV
(£ million)
Duration
(year)
"92" 195 26.8 195 25.5 598 11.1 988 17.0
SC 196 26.9 197 25.6 605 11.2 998 17.1
SC 1% 201 27.5 201 26.1 613 11.4 1,015 17.5
SC 2% 216 29.3 218 27.8 641 12.3 1,075 18.9
MC 200 27.2 201 25.9 619 11.4 1,020 17.4
MC 1% 204 27.8 205 26.4 626 11.6 1,035 17.7
MC 2% 218 29.4 220 27.9 648 12.4 1,086 18.9
LC 209 28.0 210 26.6 648 12.0 1,067 18.0
LC 1% 212 28.5 214 27.1 652 12.2 1,078 18.3
LC 2% 223 29.8 225 28.3 666 12.7 1,115 19.3
5
Accounting
basis
TPR new
proposal
Mortality Modelling
Deterministic Projections
6. Mortality Modelling
Why Stochastic Models?
Projection vs. Experience : How projections can go wrong (CMI Lee-Carter Projection)
Source: CMI Male Assured Lives
6
By using Male Assured Lives data from 1947 to 1980 to project the future mortality rate from 1980
to 2004 at age 65.
The graph shows that, by the end of 2004, the projected probability of death for males aged 65 is
about 0.015, but the realised mortality rate is about 0.009 - which is much lower than predicted.
What is worse is that projections continually overestimate the probability of death over the
estimation period; in other words, they underestimate the life expectancy substantially.
7. Graduation: Smoothing Raw Data to Remove Random Fluctuation
Source: CMI Male Assured Lives 1947-2005
Raw Death Rate Graduated Death Rate
In order to calculate the PV of liabilities, an actuarial valuation requires best estimate mortality rates
as a starting point, even if a prudent margin is being added, to evaluate liability present values; these
will be an instruction to set contribution rate as well. But we will never know when mortality
experience will go wrong and how far it may be from the “so-called” best estimates. These best
estimates are based on “smoothed” results - which means the randomness in mortality is largely
ignored.
7
Mortality Modelling
Why Stochastic Models
8. Stochastic Projections:
Allow for randomness in mortality rates
Can estimate Longevity Value at Risk (LVaR)
•Monte-Carlo Simulation:
Can show the whole range of possible future mortality rates
•Probability of each scenario
•Full distribution of future pension liabilities
Deterministic Projections:
Assume fixed mortality rates
Scenario Tests:
Cannot determine the probability of a specific scenario
•Single Value of Liability: Does not include all scenarios
• Does not show full distribution of possible future mortality
rates
Compare Deterministic and Stochastic Mortality Models
8
Mortality Modelling
Why Stochastic Models
10. Mortality Modelling
Stochastic Mortality Models
Eight Stochastic Mortality Models
10
Details of model specifications are described in Redington Longevity Technical paper, which is available upon request.
All models capture the age and period effects but they vary in the modelling approach.
Lee-Carter extension, APC, Three factor CBD, Three-factor CBD extension and Four-Factor CBD capture
the cohort effect in the model. (2, 6, 7 and 8)
1
2
3
4
5
6
7
8
11. Two Major Model Families
CBD Models: The Cairns, Blake and Dowd (CBD) model was developed by three professors in
the UK: Professor David Blake from Cass Business School, Professor Andrew Cairns from Heriot-
Watt University and Professor Kevin Dowd from Nottingham University Business School. The
CBD model was developed for and tested using mortality data from males living in England and
Wales, and has yet to be tested with data from any other countries. However, the model has
already been taken up widely by actuaries in Germany and is currently being investigated by
the CMI (Pension Institute, 2007).
Lee Carter Models: The Lee-Carter model was developed by Professors Ronald Lee and
Lawrence Carter. This model has become the “leading statistical model of mortality
forecasting in the demographic literature” in the United States (Deaton and Paxson, 2004). Lee
and Carter originally calibrated their model to use United States mortality data from 1933-
1987. Girosi and King (2007) note that the model is “now being applied to all-cause and cause-
specific mortality data from many countries and time periods, and all well beyond the
application for which it was designed” (Girosi and King, 2007).
11
Mortality Modelling
Stochastic Mortality Models
12. Two Major Model Families
Four models from the two major model families, the CBD and Lee-Carter, were tested.
Lee Carter Model: A simple one-factor model that assumes mortality improvement for
different ages has a perfect correlation. As a result, the model gives less flexibility in age
specific volatilities and usually projects less volatile future mortality rates.
CBD Model: Accounts for more factors to allow different improvements across different
ages at different periods of time. As a result, the model gives more flexibility in age specific
volatilities and often projects more volatile future mortality rates than Lee Carter model.
12
Mortality Modelling
Stochastic Mortality Models
13. Data Sample:
The data used in testing the four models was taken from the Human Mortality Database (HMD).
In order to analyse the models’ sensitivities to the choice of data sample, models were calibrated
and tested against two samples:
The first sample included population mortality data for ages 20 to 100 from 1920 to 1960;
The second test included population mortality data for ages 20 to 100 from 1963 to 1983;
The figures below show mortality rate data in the first sample is more volatile than the mortality
rate data in the second sample.
13
Mortality Modelling
Stochastic Mortality Models
14. Summary of Statistical Tests
Statistical tests: MSE, Sign and Outlier Test
• MSE and Sign Test show that the forecasting accuracy of CBD family models is more sensitive
to the choice of sample than Lee Carter family.
• The Outlier test shows that both of the models fail to capture the “hump” effect for younger
ages, and the Lee-Carter models also systematically over-estimated the mortality rates for
old ages and failed to project a proper confidence interval for old ages statistically.
Conclusion
The Two-Factor CBD model is the most appropriate model for longevity risk analysis and
management purpose, because it:
Produces stabler projections of future mortality rates;
Produces confidence intervals that cover most realised mortality rates, especially for older
ages;
Is easy to implement.
14
Mortality Modelling
Stochastic Mortality Models
16. Application Procedures
Evaluate possible solutions
Manage longevity risk in the overall LDI strategy
Quantify longevity risk
Estimate benefit cash flows with simulated mortality rates
Generate simulations on future mortality rates with calibrated model
Calibrate mortality model with historical data
16
Mortality Modelling
Applications
17. Stochastic Simulations – Mortality Rate
• By calibrating stochastic mortality models with historical data, a large number of simulated
mortality tables can be generated (e.g. 1000 tables).
• These simulated mortality tables provide a range of possible future mortality rates by incorporating
age, period and cohort effects.
• The two graphs below show the range of mortality rate simulations and a comparison with the
realised mortality rate during that period.
17
Mortality Modelling
Applications
18. Stochastic Cash Flows
For a better understanding of pension liability, cashflow analysis is largely adopted:
By using the simulated future mortality rates from previous simulations, the cashflow
structure of a generic pension scheme is projected for 1000 times.
By doing this, we can observe a range of possible cash flows implied by the two models.
This forms the basis of quantifying longevity risk for a specific pension scheme.
CBD Model Lee Carter Model
18
Mortality Modelling
Applications
19. The cashflow structures are then discounted with a flat discount curve. The two graphs
below show the distribution of pension liabilities under the different projections of the
two models.
CBD Model: Predicts a smoother and more spread out distribution of liabilities. This means
more longevity risk on the tails.
Lee Carter Model: Predicts a tighter distribution of pension liabilities. This means less
longevity risk on the tails.
Model Comparison - Distributions
19
Mortality Modelling
Applications
20. Model Comparison – Main Statistics
Two- Factor CBD Model Lee Carter Model
Two-Factor CBD Model PV Lee-Carter Model PV Short cohort Medium cohort Long cohort
(£’000,000) (£’000,000) (£’000,000) (£’000,000) (£’000,000)
Mean 1,002 998 998 1,020 1,067
S.d. 32 16 n/a n/a n/a
95% 1,054 1,026 n/a n/a n/a
5% 954 973 n/a n/a n/a
VaLR 52 27 n/a n/a n/a
As % of Total Liability 5.17% 2.73% n/a n/a n/a
20
Mortality Modelling
Applications
21. Contacts
Dawid Konotey-Ahulu | Partner Direct: +44 (0) 207 250 3415
dawid@redingtonpartners.com
Robert Gardner | Partner Direct: +44 (0) 207 250 3416
robert.gardner@redingtonpartners.com
Redington Partners LLP
13 -15 Mallow Street London EC1Y 8RD
Telephone: +44 (0) 207 250 3331
www.redingtonpartners.com
THE DESTINATION FOR ASSET & LIABILITY MANAGEMENT
Contacts
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21
22. Appendix
Model Testing
Evaluation Criteria
Mean Squared Error (MSE)
MSE is used to test the forecasting accuracy of the models defined as:
Where T is the total number of projected years, and are respectively the projected and actual
observation for age x at time i. The smaller the test statistic is the more accurate the projection is.
Sign Test
Sign test is used to test the hypothesis that the model residuals are unbiased. Mathematically, if m is the
number of positive residuals, then m should follow a binomial distribution with parameters being the
number of residuals (n) and 0.5. For test here, normal approximation is used.
Outlier Test
The outliers test is used to study the model’s capability to project confidence intervals which can cover
most of the realized mortality rates as expected. For example, a 90% confidence interval is expected to
cover 90% of the observation points.
22
23. MSE and Sign Test
MSE test shows that, in the 40 year data sample, the Lee-Carter model provides the best
projection accuracy; however, in the 20 year data sample, the CBD models performed better than
the Lee Carter models, while the three-factor CBD model provided the most accurate results.
Comparing the sign test results reveals that nearly all models based on two datasets predict
biased mortality rates; this is because the sign test statistics are all significantly larger than the
critical value of 1.96. The exception is the three-factor CBD model in the 40 year sample data
test.
In general, the two tests show that the forecasting accuracy for CBD family models is more
sensitive to the choice of sample than Lee Carter family. MSE test statistics show that the CBD
models can produce a better projection if the end-user properly selects the sample space.
23
Appendix
Model Testing
24. Outliers Test
The tables above show that, for both projection periods, the number of outliers is
greater than expected. The exception is the two-factor CBD model, which
produced significantly less outliers than expected.
24
Appendix
Model Testing
25. Outliers Test
The orange area represents outlier in right hand tail; the red area represents outlier in left
hand tail, and the green area represents observations that are within the confidence
interval.
The test results show that both of the models failed to capture the “hump” effect for
younger ages. The Lee-Carter models also systematically over-estimated the mortality rates
for old ages and failed to project a proper confidence interval for old ages.
CBD ModelLee Carter Model
25
Appendix
Model Testing
