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Lecture 17 – Associated Single Decrement Model

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					Lecture 17 – Associated Single Decrement Model

       Absolute rate of decrement

       Discuss its general relationship with probabilities in a multiple
       decrement model

       Specific relationship under constant force assumption

       Example 3

       Uniform distr. assumption under multiple decrement model
       setting




AS4422a — Lecture 17                                                       1
Absolute rate of decrement
     (j)                               (j)
    µx (t): if we pretend that µx (t) is the force of mortality in a single
    decrement model
                                  ′ (j)          t    (j)
                                                     µx (s)ds
                                t px       =e−   0


                                       ′
                                      (j)               ′
                                  t qx    = 1 − t px(j)
      ′ (j)
    t qx      is called absolute rate of decrement.
      ′ (j)
    t qxis interpreted as the net probability of decrement due to
    cause j , without competing with other causes
    The corresponding model is called the associated single
    decrement model.
                          (j)
    In comparison, t qx is the probability of decrement due to cause
    j while competing with all other causes.

AS4422a — Lecture 17                                                    2
Basic Relationship

                (τ                                       (m)
               µx ) (s) = µ(1) (s) + µ(2) (s) + · · · + µx (s)
                           x          x
                                      t (1)      (2)        (m)
                 (τ )                 0 [µx (s)+µx (s)+···+µx (s)]ds
              t px      =e       −

                                      t       (1)                    t    (2)                       t      (m)
                                          µx (s)ds                       µx (s)ds                       µx       (s)ds
                        =e       −    0                 ·e   −       0                  ···e−       0

                                 m
                                          ′
                                          (j)
                        =             t px
                                j=1

                                     (τ )            ′ (j)
                  =⇒              t px         ≤ t px
                               (τ ) (j)                      ′ (j)
                                                            (j)
             =⇒             t px µx (t)             ≤ t px µx (t)
                            1                                                                   1
                                   (τ ) (j)                    (j)              ′ (j)                   ′ (j)    (j)
          =⇒                    t px µx (t)dt           =     qx         ≤ qx           =           t px        µx (t)dt
                        0                                                                   0


AS4422a — Lecture 17                                                                                                       3
Constant force assumption

On age interval [x, x + 1), assume:

                  (j)      (j)    (τ         (τ
                 µx (t) = µx , & µx ) (t) = µx ) , for 0 ≤ t < 1




AS4422a — Lecture 17                                               4
Example 3

                                     (1)     (2)    ′ (1)    ′ (2)
                               x    qx      qx     qx       qx
                               66   0.03    0.06             *
                               67   0.04    0.07    *

                       ′ (2)        ′ (1)
       Calculate q66 and q67




AS4422a — Lecture 17                                                 5
Uniform distr. assumption for multiple decrements
       MUDD:
                          (j)         (j)
                       t qx      = t qx
                          (τ )
                       t qx
                                      (τ
                                 = t qx )




AS4422a — Lecture 17                            6

				
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