8th July 2008

Living Longer – At What Price?

Mortality Modelling

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

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“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

Mortality Modelling

Deterministic Mortality Models

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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.

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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

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Accounting

basis

TPR new

proposal

Mortality Modelling

Deterministic Projections

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.

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.

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Mortality Modelling

Why Stochastic Models

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

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Mortality Modelling

Why Stochastic Models

Mortality Modelling

Stochastic Mortality Models

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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)

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2

3

4

5

6

7

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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).

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Mortality Modelling

Stochastic Mortality Models

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.

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Mortality Modelling

Stochastic Mortality Models

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.

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Mortality Modelling

Stochastic Mortality Models

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.

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Mortality Modelling

Stochastic Mortality Models

Mortality Modelling

Application of Stochastic Mortality Models

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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

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Mortality Modelling

Applications

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.

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Mortality Modelling

Applications

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

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

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Mortality Modelling

Applications

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

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Mortality Modelling

Applications

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

Disclaimer

Disclaimer For professional investors only. Not suitable for private customers.

The information herein was obtained from various sources. We do not guarantee every aspect of its accuracy. The information is for your private information and is for discussion purposes only. A variety of market factors and assumptions may affect this analysis, and this analysis does not reflect all possible loss scenarios. There is no certainty that the parameters and assumptions used in this analysis can be duplicated with actual trades. Any historical exchange rates, interest rates or other reference rates or prices which appear above are not necessarily indicative of future exchange rates, interest rates, or other reference rates or prices. Neither the information, recommendations or opinions expressed herein constitutes an offer to buy or sell any securities, futures, options, or investment products on your behalf. Unless otherwise stated, any pricing information in this message is indicative only, is subject to change and is not an offer to transact. Where relevant, the price quoted is exclusive of tax and delivery costs. Any reference to the terms of executed transactions should be treated as preliminary and subject to further due diligence .

Please note, the accurate calculation of the liability profile used as the basis for implementing any capital markets transactions is the sole responsibility of the Trustees' actuarial advisors. Redington Partners will estimate the liabilities if required but will not be held responsible for any loss or damage howsoever sustained as a result of inaccuracies in that estimation. Additionally, the client recognizes that Redington Partners does not owe any party a duty of care in this respect.

Redington Partners are investment consultants regulated by the Financial Services Authority. We do not advise on all implications of the transactions described herein. This information is for discussion purposes and prior to undertaking any trade, you should also discuss with your professional tax, accounting and / or other relevant advisers how such particular trade(s) affect you. All analysis (whether in respect of tax, accounting, law or of any other nature), should be treated as illustrative only and not relied upon as accurate.

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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.

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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.

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Appendix

Model Testing

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.

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Appendix

Model Testing

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 Model Lee Carter Model

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Appendix

Model Testing