Annuities and uncertain life expectancy

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Annuities and uncertain life expectancy

• Justin van de Ven and Martin Weale

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Rising and Uncertain Life Expectancy

• The interesting problems for the annuity market
  arise not because life expectancy is rising but
  because it is uncertain.
• Current estimates of cohort life expectancy are
  based on forecasts of future mortality rates
• The uncertainty surrounding these is high for
  young people and lower for old people

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Cohort Life Expectancy at Age 65 1981-2004
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Estimation of Uncertainty

• The Lee-Carter Model for mortality rate at age i
  in year t
• kt is a mortality index
• This structure preserves the shape of the
  mortality curve
• kt needs to be forecast and the uncertainty
  surrounding it gives an indication of uncertainty
  of mortality rates

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The Mortality Factor
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Forecasting the Mortality Index

• kt follows a second-order process, but even an
  equation in D2 kt is unstable and needs to be
  restricted. We restrict the sum to -0.9 and
  constant to 0. F(2,14)1.84

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Implications Male Life Expectancy at 65
Standard deviations in brackets
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Modelling the Implications of Aggregate Mortality
Risk for the Annuity Market

• Young people are affected differently from old
  people.
• Old people cannot vary their retirement savings
  while young people can.
• Consider a framework in which young people sell
  annuities to old people.

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• This structural model differs from the
  conventional capital asset pricing model.
• There all consumers are assumed to be in the same
  position the CAPM does not explain one type of
  consumer trading with another type.
• We assume that young people have no risk of death
  and old people have a rate of death which does
  not change with age.
• This allows us to construct a tractable model.

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Tontines

• In our core analysis we assume that people can
  choose between annuities and tontines.
• Investors in the latter carry the aggregate, but
  not the individual, mortality risk for
  themselves.
• The pay-out they receive depends on the mortality
  of their cohort.
• We then consider the situation where tontines are
  not available as a special case.

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A Demand Curve

• Old people can choose whether they want to carry
  the mortality risk for themselves. This depends
  on the cost of shedding it and gives us a demand
  curve.
• The degree of risk-shedding depends on the cost.
• If people cannot carry the risk for themselves
  (tontines not available) the portfolio decision
  of old people is fixed.

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A Supply Curve

• Young people can decide how much of their wealth
  to invest in annuities sold to old people,
  depending on the expected profit. This gives us a
  supply curve.
• The supply curve of the young depends on the
  risks they face when old and is derived by means
  of dynamic programming.

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• Both curves depend on aggregate mortality risk
  and the degree of risk aversion as well as on the
  real interest rate
• The supply curve also depends on the persistence
  of mortality rates
• The curves are calculated for different pay-out
  ratios on the assumption that consumers are
  rational

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Assumptions

• We assume that life expectancy at age 65 is 20
  years.
• The standard deviation is assumed to be 2 years-
  on the high side.
• The real interest rate is assumed to be 1 p.a.
• Results explore serially independent mortality
  risk and a second-order process, D2 ktnt
• The coefficient of relative risk aversion is 4.

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The Demand Curve
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The Supply Curve (no persistence)
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Supply Curve (persistence)
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Four Market Equilibria (R1 p.a.)
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Government Intervention

• The government can spread the risk across all
  future cohorts while the market just spreads it
  among the living.
• Preliminary results suggest that this is
  desirable only if the mortality rate is
  stationary.
• But we need to remember that the government can
  spread risk more than can the market.

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Conclusions

• There is no good model of mortality.
• Annuity pricing needs to be assessed in terms of
  supply and demand.
• If old people can buy tontines then annuity
  prices will be higher and demand lower than if
  they cannot.
• The most basic economics suggests that promoting
  a tontine market would raise welfare.