ACTEX
MLC Study ManualChapter 10 Multiple State Models
1. To state the assumptions underlying a multiple state model
2. To model disability income model and permanent disability model as
multiple state models
3. To
...
ACTEX
MLC Study ManualChapter 10 Multiple State Models
1. To state the assumptions underlying a multiple state model
2. To model disability income model and permanent disability model as
multiple state models
3. To calculate transition and occupancy probabilities for multiple state
models
4. To calculate the premiums and reserves for multiple state models
In this chapter, we discuss multiple state models. We shall see how various non-traditional
insurances can be analyzed under the framework of multiple state models.
In what follows we use the notation o(h) quite often. A function g is said to be o(h) if
Jim g(h) = 0. A function is o(h) if it shrinks to 0 quicker than h. For example, g 1(h) = Ii and
h->0 h
g2(h) = h 11 are o(h). However, g3(h) =hand g4(h) =sin hare not o(h) because the limits are I.
It is obvious from the theorem of limits that if/and g are o(h), then/+ g,f- g,fg, and cf(where
c is a constant) are also o(h). As a result, we write
o(h) ± o(h) = o(h), o(h) x o(h) = o(h), c x o(h) = o(h).
Before we talk about the setting of a multiple state model, let us review the single decrement
model and multiple decrement model in Chapters 1 - 7 and Chapters 8 - 9.
( ·Actex 2013 I Johnny Li and Andrew Ng I SoA Exam MLCChapter 1O: Multiple State Models
The Single Decrement Model
In the ordinary single decrement model, there are two states, alive (0) and dead ( l):
~~o_._A_li_ve~__.1--~~µ_x+_'~~~~~1~~l._D_e_a_d~---'
Model l: The Single Decrement Model
The single decrement model is characterized by the force of mortality:
d
-Fx(t) l P(t T < h)
Ji = dt = --lim t+hqx - ,qx = lim < x - t + =Jim hqx+t
x+t SJt) Sx(t) h-;O h h-;O hPr(Tr > f) h-;O h '
and this means
"qx+t = µx+th + o(h) .
So you can see why the approximation" di q_w ~ µx+idt for small dt" holds.
For this model, the probability for (x) to stay in state 0 (alive) for a period of length tis
1 Px =exp(- J~µx+sds).
The probability for (x) to enter state l (dead) on or before time tis ,qx = J~ ,pxµx+sds.
The Double Decrement Model
In multiple decrements, what we need are the various forces of decrement. In the double
decrement model, they are 2 modes of decrements, say, accidental death (1) and death due to
other causes (2).
1. Accidental
0. Alive
2. Others
Model 2: The Double Decrement Model
(: Actex 2013 I Johnny Li and Andrew Ng I SoA Exam MLC
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