Pricing Mortality-linked Securities with Dependent Lives under Threshold Life Table

1
Pricing Mortality-linked Securities with
Dependent Lives under Threshold Life Table

• Hua Chen, Temple University
• Samuel H. Cox, University of Manitoba
• Jian Wen, Central University of Finance and
  Economics

2
Introduction

• Extreme-age mortality modeling
• Mortality improvement is a slow but persistent
  process
• 40,000 centenarians currently in the U.S.
• 3 million centenarians by the first decade of
  next century
• Challenge to actuaries
• since life table is usually closed earlier, say
  100.
• How to extrapolate extreme-age mortality and
  construct a reliable life table?
• EVT approach Threshold life Table (Li,
  Hardy, Tan 2008)

3
Introduction

• Joint survivorship of multiple lives
• Independence of a pair is normally assumed
• The joint survival function is simply the product
  of the marginal survival functions of each life.
• Common risk factors for pairs of lives
• Genetic factors, e.g. twins
• Environmental factors, e.g., couples
• Empirical evidence broken heart syndrome
• Parkes, Benjamin, and Fitzgerald (1969), Ward
  (1976)Jagger and Sutton (1991)
• How to capture the life dependence?
• Copula function

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Introduction
EVT
Copula
Multivariate Threshold Life Table
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EVT and Threshold Life Table

• Parametric estimator
• e.g. Gompertz distribution function (Frees,
  Carriere, Valdez 1996)
• Traditional parametric methods are ill-suited to
  extreme probabilities
• The inaccuracy and unavailability of mortality
  data at old ages.
• Solution EVT
• estimate extreme probabilities by fitting
  a model to the empirical survival function of a
  set of data using only the extreme event data
  rather than all the data, thereby fitting the
  tail, and only the tail(Sanders, 2005).

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EVT and Threshold Life Table

• For x gt N
• where
• The Pickands-Balkema-De Hann Theorem
• For sufficiently high threshold N, the
  excess distribution function may
• be approximated by the GPD
  .
• 
  
  .

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EVT and Threshold Life Table

• Li, Hardy and Tan (2008) Threshold Life Table

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Life Dependency and Copula

• Dependency Measures
• Parametric e.g., Pearson correlation
• Non-parametric e.g., Spearmans rho, Kendalls
  tao
• Copula
• Copulas capture the dependence structure
  separately from the marginal distributions
• Schweizer and Wolff (1981)
• For any strictly increasing functions
  and ,
• and have
  the same copula as and

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Life Dependency and Copula

• Archimedean copula family
• where is a convex and strictly
  decreasing function with domain and range
• such that .

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Life Dependency and Copula

• Franks copula
• Spearsons rho
• Kendalls tau
• where
  and

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Last-Survivor Annuity Data

• Frees, Carriere and Valdez (1996)
• approximately 15,000 last-survivor annuity
  policies 1989 - 1993.
• date of birth, death (if applicable), contract
  initiation, and sex of each annuitant.

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Modeling Algorithm

• Set up the initial values of parameters
• Use mortality data of males
• For , find the parameters
  that maximize the log-likelihood
  function
• Repeat this step for
  
• The value of that gives the maximum profile
  log-likelihood is the optimal threshold age for
  male. The parameter estimates
  corresponding to this value are the optimal MLE
  estimates.
• Replicate the same procedure for mortality data
  for females, and find the optimal estimates
  and
• Use Gompertzian marginals and the Frank copula to
  find the estimate of the dependence parameter
  

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Modeling Algorithm

• Using the values of
  and obtained from step 1 as initial
  values, find the MLE estimates of these
  parameters, for any combination of and
• The values of and that give the
  maximum value of log-likelihood function are the
  optimal threshold ages for males and females. The
  MLE estimates corresponding to this combination
  are our optimal MLE estimates.

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Estimation Results

• Spearmans rho 0.49 and Kendalls tau 0.56
• A positive mortality dependence between male and
  female

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Pricing Example Last-Survivor Annuity

• Last-survivor annuity
• where
• Scenario analysis
• Dependent lives with the threshold life table
  (benchmark model)
• Independent lives with the threshold life table

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Pricing Example Effect of Threshold Life Table

• Ratio annuity value with TLT/ that without TLT
• Dependent lives
  Independent
  lives
• Without threshold life table, the value of the
  last survivor annuity is underestimated.

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Pricing Example Effect of Dependence

• Ratio annuity value assuming dependence/ that
  assuming independence
• With threshold life table
  Without threshold
  life table

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Pricing Example Overall Effect

• Ratio Dependent lives with TLT/ Independent
  lives without TLT
• Assuming independent lives and without threshold
  life table, the last survivor annuity is
  overestimated by 5.

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Conclusion

• Develop multivariate threshold life table
• EVT approach
• Copula approach
• Apply our model to price the last-survivor
  annuity
• Mortality-linked securities are under-priced
  without the threshold life table
• Mortality-linked securities are over-priced
  assuming independent lives
• Future research
• How to identify an appropriate copula function?
• Incorporate a stochastic process into the
  multivariate threshold life table.

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• Thanks!