Survival Models and Life Contingencies

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Chapter 9

• Survival Models and Life Contingencies
• (sample extract)

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Learning Objectives

• Distribution of age of death using cumulative
  distribution function, survival function and
  density function
• Future lifetime distribution
• Actuarial notations
• Parametric survival models
• Life table
• Fractional age at death

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• Actuarial mathematics
• For analyzing and managing financial risks
• Risks depend on probabilities of survival or
  termination.
• Survival models are often related to the life
  span of a person, e.g., whole-life insurance,
  pension valuation

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9.1 Survival Distribution Function

• Let X be the random variable (RV) representing
  the age of an individual when death occurs (i.e.,
  refer to X as the age at death).
• The cumulative distribution function (CDF) of X
  is defined as
• (9.1)
• The complementary function of the CDF, called the
  survival distribution function (SDF) is defined
  as
• (9.2)

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• FX(x) is non-decreasing with FX(0) 0 and FX(?)
  1.
• SX(x) is non-increasing with SX(0) 1 and SX(?)
  0.
• If X is continuous, the probability density
  function (PDF) is given by
• (9.3)
• The force of mortality (FM), denoted by ?x, is
  defined as
• (9.4)
• FM represents the instantaneous risk of imminent
  death for an individual given that she has
  already reached age x.

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• CDF, SDF, PDF and FM are equivalent forms of
  specifying
• the distribution of X.
• If one of them is properly defined, the other
  functions can be
• uniquely determined.
• Example 9.1
• Let (a) Show that SX(x) is a proper
  survival function
• (b) Determine the corresponding CDF, PDF and
  FM.

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• Since SX(x) is a nonincreasing function with
  SX(0) 1 and
• SX(?) 0, it is a proper SDF.
• By (9.2) the CDF of X is
• By (9.3) the PDF of X is

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• By (9.4) the FM is

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• Future lifetime
• Let T(x) represent the future lifetime of an
  individual aged x, i.e., T(x) X x with X gt
  x.
• When no confusion about the present age of the
  individual is likely, we simply denote the future
  lifetime variable by T.
• The SDF of T is
• (9.5)

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• and its PDF is
• (9.6)
• Example 9.2
• If SX(x) 1 0.01x for 0 x 100, find (a)
  the median age at death m, (b) the SDF and PDF of
  T T(m), and (c) the expected future life time
  at m.

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• We need to solve SX(m) 1 0.01m 0.5.
• Hence, m 50.
• The SDF of T T(50) is
• The PDF of X is

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• The PDF of T is
• Thus the expected future lifetime at 50 is
• The PDF of T can also be obtained by
  differentiating ST(t)
• directly to obtain fX(t) dST(t)/dt 0.02

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9.2 Actuarial Notations

• Denote tqx as the conditional probability of
  death in the age interval (x, x t), given alive
  at age x (i.e., the probability that a person
  aged x will die within t years. Then
• (9.7)
• The conditional probability of surviving to age x
  t, given the person is alive at age x, is
  denoted by tpx 1 tqx. Hence,
• (9.8)

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• In the particular case of t 1, we simplify the
  above notations to qx and px.
• For the more general event that an individual
  aged x survives t years and dies within the next
  s years we denote its probability by tsqx,
  where
• (9.9)
• The suffix s may be suppressed if it is equal to
  1. Thus, tqx is the probability of an
  individual aged x surviving t years and dying
  within the next year. Hence, tqx t1qx t qx.

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• Alternatively, using the result Pr(AnBC)
  Pr(AC) ? Pr(BAnC), we have
• (9.10)
• Over a period of t s, the following chain rule
  holds
• (9.11)
• If n is an integer, it is easy to generalize
  (9.11) to
• (9.12)

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• For the conditional probability of death, we have
• (9.13)
• Using the SDF of the future lifetime random
  variable T, we have
• (9.14)

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• Example 9.5
• If iqx 0.05 for i 0, 1, , 7, calculate
  3px4.