Stochastic Portfolio Specific Mortality and the Quantification of Mortality Basis Risk

1
Stochastic Portfolio Specific Mortality and the
Quantification of Mortality Basis Risk

• Richard Plat Belfast, 14th August

To appear in Insurance, Mathematics Economics
45 (2009) 123-132 Corresponding working paper
available for download at http//ssrn.com/abstra
ct1277803
2
Agenda

• Introduction
• Mortality rates measured in insured amounts
• Basic model
• Fitting the model
• Application to insurance portfolios
• Numerical example 1 Value at Risk
• Numerical example 2 basis risk
• Conclusions

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Introduction (1)

• 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), Currie (2006)
• Plat (2009)
• These models are tested on a long history for
  large country populations, such as U.S. or U.K.

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Introduction (2)

• However, these models are generally not directly
  applicable for insurance portfolios, because
• In practice often not enough insurance portfolio
  specific data ( years history and policies)
• Insurers more interested in mortality rates
  measured in insured amounts instead of measured
  in number of people
• Practice ? applying a deterministic portfolio
  experience factor to stochastic country mortality
  rates.
• However, it is reasonable to assume that this
  factor is also stochastic.

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Introduction (3)

• In this presentation a stochastic model is
  suggested for portfolio specific mortality
  experience.
• This process can be combined with any stochastic
  country population mortality process, leading to
  stochastic portfolio specific mortality rates.
• Applications
• Value at Risk (VaR) / Solvency Capital
  Requirement (SCR) for mortality or longevity risk
• Quantifying mortality basis risk when longevity
  or mortality is hedged

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Mortality rates measured in insured amounts

• Measurement of mortality rates in insured amounts
  is already used for a long time, starting with
  CMI (1962).
• Definition used in this paper for portfolio
  (initial) mortality rate, measured in amounts
  (age x, year t)
• Policyholders with higher insured amounts tend to
  have lower mortality rates ?

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Basic model (1)

• Aim is a stochastic model for
• where is the country population
  mortality rate
• It is desirable that the proposed model
• is as parsimonious as possible, because often
  limited data is available
• leads to an expectation of Px,t that approaches 1
  for the highest ages

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Basic model (2)

• Proposed model
• Or in vector notation

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Basic model (3)

• To ensure Px,t approaches 1, we require
• The structure of X (and the corresponding ?s)
  can be set in different ways, for example
• Principal components analysis to derive preferred
  shape of Xi
• Similar structure as Nelson Siegel model for
  yield curves
• More simple structure, using 1 factor where
  vector X is linear in age

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Fitting the model (1)

• Fitting the model requires 3 steps
• 1) Fitting the basic model
• Px,t are based on different exposures to death
  and observed deaths ? Heteroskedasticity
• Therefore Generalized Least Squares (GLS) is
  used, with (for example) square root of
  observations in group as weights
• This leads to estimates for time t (with weight
  matrix Wt)
• This leads to a time series of vector

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Fitting the model (2)

• 2) Adding stochastic behavior
• Now a (multivariate) stochastic process can be
  fitted to the time series of
• Given the often limited historical period
  available and requirement of parsimoniousness,
  process has to be simple, for example
• Correlated AR(1) or ARIMA(0,0,0) processes
• A Vector Autoregressive (1) model VAR(1)

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Fitting the model (3)

• 3) Combine with stochastic country population
  model
• If the historical data period is equal for the
  portfolio and the country population, Seemingly
  Unrelated Regression (SUR) can be used.
• This will generally not be the case. Alternative
  procedure is
• Fit equation by equation using Ordinary Least
  Squares (OLS)
• Use the residuals to estimate the elements of
  covariance matrix S

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Application to insurance portfolios (1)

• The model is applied to 2 portfolios
• Large portfolio 100.000 males, aged 65 or older,
  collective pension
• Medium portfolio 45.000 males, aged 65 or older,
  annuity portfolio
• For both portfolios, 14 years of historical data
  is available
• For these portfolios, a simple 1-factor structure
  appeared to be most favourable in terms of BIC.
  So the model used is a simple linear model, where
  the vector X is linear in age

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Application to insurance portfolios (2)

• Example of fit to actual observations

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Application to insurance portfolios (3)

• Resulting time series of
• ARIMA(0,0,0) process led to a more favourable
  BICs
• Large portfolio
• Medium portfolio

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Application to insurance portfolios (4)

• This leads to the following best estimates and
  99,5 / 0,5 percentiles in year 2016 for the
  portfolio experience mortality factor Px,t

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Numerical example 1 Value at Risk (1)

• For stochastic county population mortality the
  model of Cairns et al (2006) is used
• BE and percentiles, with stoch. and deterministic
  Px,t

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Numerical example 1 Value at Risk (2)

• Impact is determined for the following
  definitions of VaR
• 1-yr horizon, 99,5 percentile, including effect
  on BE after 1 year
• 10-yr horizon, 95 percentile, including effect
  on BE after 10 years
• Run-off of the liabilities, 90 percentile
• Impact on large portfolio

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Numerical example 1 Value at Risk (3)

• Impact on medium portfolio
• Conclusion the impact of stochastic Px,ts can
  be significant, especially for medium or smaller
  portfolios.

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Numerical example 2 basis risk (1)

• Interest in hedging mortality or longevity is
  increasing.
• Hedge derivatives are often based on country
  population mortality
• More transparent
• Chance of developing a liquid market for
  longevity or mortality
• Example q-forward (J.P. Morgan)

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Numerical example 2 basis risk (2)

• What remains is the basis risk, the risk arising
  from difference in country population mortality
  and portfolio mortality.
• This can be quantified with the proposed model.
• A minimum variance hedge is set up for the
  portfolios and the impact of the hedge on the VaR
  is measured.

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Numerical example 2 basis risk (3)

• Large portfolio
• Medium portfolio
• Conclusion hedge effectiveness is less for
  medium portfolio ? can be improved by
  periodically adjusting.

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Conclusions

• Existing mortality models are generally not
  directly applicable for insurance portfolios.
• In this presentation a stochastic model is
  suggested for portfolio specific mortality
  experience.
• Combining with a stochastic country population
  mortality process, leads to stochastic portfolio
  specific mortality rates.
• Impact on VaR / SCR and on hedge effectiveness
  can be significant.

24
Stochastic Portfolio Specific Mortality and the
Quantification of Mortality Basis Risk

• Richard Plat Belfast, 14th August

To appear in Insurance, Mathematics Economics
45 (2009) 123-132 Corresponding working paper
available for download at http//ssrn.com/abstra
ct1277803