On Stochastic Mortality Modeling

1
On Stochastic Mortality Modeling

• Richard Plat Belfast, 13th August

Corresponding working paper available for
download at http//ssrn.com/abstract1362487
2
Agenda

• Introduction
• Existing models
• New mortality model
• Fitting the model
• Simulation results
• Including parameter uncertainty
• Conclusions
• Appendix risk neutral specification of the model

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Introduction (1)

• Currently there is more focus of the insurance
  industry on quantifying their risks, amongst
  others driven by the introduction of Solvency 2
  in 2012.
• Important risks to be quantified are mortality
  and longevity risks.
• There is vast literature about stochastic
  mortality models, for example
• Lee and Carter (1992), Renshaw and Haberman
  (2006), Cairns et al (2006, 2007, 2008), Currie
  et al (2004) and Currie (2006)
• and lots more

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Introduction (2)

• All well known models have nice features but also
  disadvantages.
• In this presentation a stochastic mortality model
  is proposed that aims at
• combining the nice features from existing models,
  
• while eliminating the disadvantages

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Existing models (1)

• Existing models either model central mortality
  rate mx,t or initial mortality rate qx,t.
  Disadvantages are
• Lee Carter (1992)
• Disadvantages - 1-factor model, all ages
  perfectly correlated
• - No cohort effect
• - Too low uncertainty at high ages
• Renshaw Haberman (2006)
• Disadvantages - Lack of robustness (CMI(2007),
  Cairns et al (2006))
• - Trivial correlation structure

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Existing models (2)

• Cairns et al (2006)
• Cairns et al (2007) several additions on above
  formula
• Disadvantages - Only applicable to higher
  ages
• Currie (2006)
• Disadvantages - Trivial correlation structure
• - Cohort parameter mis-used for age-period
  effects
• Above the focus is on the disadvantages, but all
  the models above also have very nice features.

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Existing models (3)

• Judging the model based on criteria set by Cairns
  et al (2008)

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New mortality model (1)

• The models mentioned all have some nice features,
  for example
• ax term of the Lee-Carter model makes it suitable
  for full age ranges
• Renshaw-Haberman model addresses cohort effect
  and fits well to historical data
• Currie model has a simpler structure then
  Renshaw-Haberman, making it more robust
• The models of Cairns et al have multiple factors,
  resulting in a non-trivial correlation structure,
  while structure of the model is relatively simple
• Combining these features may eliminate the
  disadvantages.

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New mortality model (2)

• Proposed model
• Next to ax, similar to Lee-Carter, there are 4
  factors
• ?1 changes in the level of mortality for all
  ages
• ?2 allows changes in mortality to vary between
  ages
• ?3 captures specific dynamics of younger ages
• ?t-x captures the cohort effect

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New mortality model (3)

• As the other models, the model needs some
  identifiability constraints (c year of birth)
• If one is only interested in higher ages, the
  model can be reduced to

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Fitting the model (1)

• Method of Brouhns et al (2002) is used
• Model is fitted by maximising the log-likelihood
  function

R-code is availabe on the website of the
pension institute, and can be used within the
kLifemetrics toolkit of J.P. Morgan, if desired.
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Fitting the model (2)

• Comparison fit quality with existing models for
• U.S. Males, 1961-2005, ages 20-84
• England Wales, 1961-2005, ages 20-89
• The Netherlands, 1951-2005, ages 20-90
• Model are compared using BIC, a measure that
  provides a trade-off between fit quality and
  parsimony

Data comes from www.mortality.org
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Fitting the model (3)

• Comparison BIC for proposed model and existing
  models
• The proposed model gives the best fitting results
  overall for these data sets.

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Fitting the model (4)

• Including the cohort effect until the last
  observation (in this case 1980) can give weird
  projections (for all models).
• Also it is questionable whether the fitted cohort
  parameters for birth year 1945-1980 are
  persistent in the future for these birth years.
• Therefore, cohorts gt 1945 are excluded. Impact

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Fitting the model (5)

• Fitting procedure describe above leads to time
  series of the 4 factors (example U.S. Males)

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Fitting the model (6)

• Now for each of these time series the most
  favourable (measured in BIC) ARIMA process is
  fitted (U.S. Male)
• ?1 ARIMA(0,1,0)
• ?2,, ?3 and ?t-x ARIMA(1,0,0), no constant
• Commonly assumed that ?t-x is independent of the
  other processes, so this process can be fit using
  Ordinary Least Squares (OLS).
• The other process can be fit simultaneously using
  Seemingly Unrelated Regression (SUR).

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Simulation results (1)

• Cairns et al (2008) address out-of-sample
  performance of several stochastic models.
  Conclusions
• Lee-Carter, Renshaw-Haberman and Cairns et al
  (2008, model M8) did not perform in a
  satisfactory way
• Currie (2006) and Cairns et al (2006, 2007 model
  M7) did produce plausible results
• Remember that the proposed model has advantages
  over these models
• Cairns et al (2006, 2007 model M7) only
  applicable for higher ages
• Currie (2006) has a trivial correlation structure
  and fits less good to historical data

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Simulation results (2) U.S. Males
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Simulation results (3) EW Males
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Simulation results (4) N.L. Males
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Simulation results (5)

• Also robustness is tested and backtests have been
  done, with satisfactory results.
• Conclusion the proposed model gives plausible
  results and is robust.

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Including parameter uncertainty

• There exists several approaches for incorporating
  parameter uncertainty
• Using a formal Bayesian framework, see Cairns et
  al (2006)
• Simulate parameter values using the estimates and
  standard errors obtained in the fitting process
• Applying a bootstrap procedure, see Renshaw and
  Haberman (2008)
• Approach 1) is elegant, but could get very
  complex.
• Approach 2) is the most practical method.
• Approach 3) is more robust then approach 2), but
  also much more computer-intensive.

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Conclusions (1)

• All well known stochastic mortality models have
  nice features but also disadvantages.
• A stochastic mortality model is proposed that
  combines the nice features while eliminating the
  disadvantages. More specifically, the model
• fits historical data very well,
• captures the cohort effect,
• has a non-trivial (but not too complex)
  correlation structure,
• is applicable to a full age range,
• has no robustness problems,
• while the structure of the model remains
  relatively simple.

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Conclusions (2)

• Again confronting with the criteria

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Conclusions (3)

• Of the existing models, the Currie (2006) model
  seems most favourable.
• However, the advantages of the proposed model
  compared to the Currie model are
• Better fit to historical data
• Non-trivial correlation structure, which is
  important in solvency calculations
• Better applicable to a full age range, amongst
  others due to the inclusion of a separate factor
  for young ages
• Also risk neutral specification of model
  available for pricing longevity or mortality
  derivatives (see appendix).

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On Stochastic Mortality Modeling

• Richard Plat Belfast, 13th August

Corresponding working paper available for
download at http//ssrn.com/abstract1362487
27
Appendix risk neutral specification (1)

• The proposed model is set up in the so-called
  real world measure, suitable for assessing risks.
• For pricing purposes, a common approach is to
  specify a risk neutral measure Q.
• Note that the market for longevity or mortality
  instruments is currently far from complete.
  Consequence of this is that the risk neutral
  measure Q is not unique.
• Given the absence of any market data, it seems
  wise to keep the specification of the risk
  neutral process relatively simple.

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Appendix risk neutral specification (2)

• The stochastic process for ?1, ?2, ?3, ?t-x in
  the real world measure P can be specified as
• Now the proposed dynamics under Q are

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Appendix risk neutral specification (3)

• The model could be calibrated to longevity or
  mortality derivatives, for example to the
  q-forward of J.P. Morgan
• The vector ? can be solved relatively easy
• where h follows from the market prices and W is
  a matrix of weights that translates the values
  for ?1, ?2, ?3, ?t-x into values for ln(mx,t)