Actuarial Models Coursework

Project description
There are two questions for this coursework which is from a actuarial course module (CT4 Actuarial Models). The questions are in intermediate level and the first one consists of three parts and the second one consists of four parts.
The Questions are attached which are exercise 3 and 4

Actuarial Models: example sheet 5
*=easy, **=intermediate, ***=difficult
Exercises 3 and 4 are part of the coursework (for MATH39511 and
MATH69511). Submit your answers to the reception on the ground
floor before noon on Wednesday 5 November. Do not forget to fill
in a cover sheet.
** Exercise 1
(In Exercise 1(a) from last week you were asked to solve the question using the analytical
approach with the forward equations. In this exercise you are asked to solve the same question
but now via a probabilistic method.) Consider the time homogeneous MJP{Xt
: t ?0}
generated by the Q-matrix associated to the following model:
h:healthy i:ill
d:dead
-?
@
@
@
@
@
@R
µ
¡
¡
¡
¡
¡
¡ª
?
Prove, without making use of the Kolmogorov forward equations, that
p
id
(t) := Pr(Xt =d|X0=i) = 1?e
??t
, t ?0.
(Hint: draw a sample path of the MJP.)
** Exercise 2
LetX={Xt
: t?0}be an MJP and letCt
be the current holding time att, i.e. Ct
is the
length of time between t and the last jump beforet (if there is no jump before t, then we
defineCt =t). Show that fors?tandw?t?s,
Pr(Ct > w, Xt =k|Xs =i) =p
ik
(s, t?w)p
kk
(t?w, t).
(Hint: draw a sample path of the MJP.)
** Exercise 3
Consider the following graphical representation of a Q-matrix of a time inhomogeneous MJP,
which is used for critical illness insurance. Heretrepresents the age of a critically ill person.
1
1:ill
2:dead due to illness
3:dead by other causes
©
©
©
©
©
©
©
©
©*
H
H
H
H
H
H
H
H
Hj
µ(t)
?(t)
(a) Show that the transition probabilities from state 1 to respectively state 1, 2 and 3 are
given by
p
11
(s, t) =e
?
R
t
s
(µ(u)+?(u))du
p
12
(s, t) =
Z
t
s
µ(u)e
?
R
u
s
(µ(v)+?(v))dv
du
p
13
(s, t) =
Z
t
s
?(u)e
?
R
u
s
(µ(v)+?(v))dv
du,
where0?s?t.
(b) In order to determine the transition rate ?(t), the insurance company uses the ex-perience from a group of policyholders who have (or have had) a standard (instead
of a critical illness) life insurance policy with the company. It is assumed that the
mortality rate (i.e. the hazard function) of these policyholders, denoted bye?(t) is
constant over each age interval [k, k+ 1), wherek?N, i.e.
e?(t) =e?
k+1/2
, k ?t < k+ 1,
where for eachk ?N, e?
k+1/2
is a strictly positive number. From past data, the
company knows thatp
k
, the probability that such a policyholder of exact age klives
at least until agek+ 1 is given by
p
55
= 0.996, p
56
= 0.995, p
57
= 0.994.
Determinee?
55.5
, e?
56.5
ande?
57.5
.
(c) The company assumes thatµ(t) = 0.1 and?(t) = e?(t) for all ages t. Compute the
probability that a person who is critically ill and of exact age 55 dies within two years
because of the illness.
2
** Exercise 4
The following four state MJP with constant transition rates represents a model for the status
of an employee of a law firm. Here timetrepresents time in years.
1:junior
associate
2:senior associate
with possibility
of becoming partner
3:senior associate
with no possibility
of becoming partner
4:partner
-©
©
©
©
©
©*
H
H
H
H
H
Hj
0.25
0.14
0.01
(a) Find expressions for the transition probabilitiesp
11
(t),p
12
(t),p
13
(t) andp
14
(t) for all
t?0.
(b) What is the probability of a junior associate becoming partner?
(c) What is the probability of a junior associate becoming partner within 5 years?
(d) What is the probability that it takes a junior associate at least 3 years to become a
senior associate?
** Exercise 5
(The formulas appearing in this exercise are very important for estimating the transition
rates of a time homogeneous MJP and we will make frequently use of it in Chapter 9.)
LetX={Xt
: t ?0} be a time homogeneous MJP with transition ratesµ
ik
and write
µ
i =
Pd
j=1,j6=i
µ
ij
. For simplicity we assume µ
i >0 for alli. Let for n?1,J(n) and H(n)
be thenth jump, respectively holding time of the MJP.
(a) Denote byf
H(1)|X0
(h
1
|i) the density function of H(1) conditional on X0 =i. Show
that
f
H(1)|X0
(h
1
|i) =µ
i
e
?µih1
, h
1>0.
(b) Denote byf
H(1),X
J(1)
|X0
(h
1
, k|i) the joint density/mass function of H(1) and XJ(1)
conditional onX0=i. Show that
f
H(1),X
J(1)
|X0
(h
1
, k|i) =µ
ik
e
?µi
h1
, h
1>0, k 6=i.
(c) Denote byf
H(1),X
J(1),H(2)|X0
(h
1
, k, h
2
|i) the joint density/mass function ofH(1),H(2)
andXJ(1)
conditional onX0=i. Show that
f
H(1),X
J(1)
,H(2)|X0
(h
1
, k, h
2
|i) =µ
ik
e
?µih1
µ
k
e
?µ
k
h2
, h
1
, h
2>0, k 6=i.
3
(d) Denote byf
H(1),X
J(1)
,H(2),X
J(2)
|X0
(h
1
, k, h
2
, l|i) the joint density/mass function ofH(1),
H(2) and XJ(1)
, XJ(2)
conditional onX0=i. Show that
f
H(1),X
J(1)
,H(2),X
J(2)
|X0
(h
1
, k, h
2
, l|i) =µ
ik
e
?µih1
µ
kl
e
?µ
k
h2
, h
1
, h
2>0, k 6=i, l 6=k.
(e) Write down an expression for (i) the joint density/mass function ofH(1),H(2),H(3)
andXJ(1)
, XJ(2)
conditional on X0 =i and (ii) the joint density/mass function of
H(1),H(2),H(3) and XJ(1)
, XJ(2)
, XJ(3)
conditional onX0=i.
*** Exercise 6 (not so important)
(This exercise introduces the Poisson process, which is a process that we will encounter a lot
in the course unit Risk Theory.) Fix ? >0. LetX={Xt
: t?0}be the time homogeneous
MJP with (infinite) state space{0,1,2,3, . . . ,}generated (as in the beginning of Chapter 5
in the notes) by the Q-matrix with transition rates given by
µ
ik =
8
>
<
>
:
?? if k=i,
? if k=i+ 1,
0 otherwise.
One can assume that the transition probabilitiesp
ik
(t) satisfy the forward equations
p
0
ik
(t) =
? X
j=1
p
ij
(t)µ
jk
with boundary condition
p
ik(0) =
(
1 ifi =k
0 ifi 6=k.
(a) Show by induction or otherwise that the (unique) solution to the forward equations
is given by
p
ik
(t) = e
??t
(?t)
k?i
(k?i)!
, k ?i, i = 0,1,2, . . . .
Note that in particular,p
0k
(t) is the probability mass function of a Poisson distribution
with parameter?t.
(b) In this part, you are asked again to prove that p
0k
(t) is the probability mass function
of a Poisson distribution with parameter?t, but now via a probabilistic method.
(i) Assuming thatX0
= 0, write the event{Xt ?k}in terms of the jump times
J(1), J(2), . . . of the MJP. (Hint: draw a sample path of the MJP).
(ii) Recall that the sum ofn?1 independent exponentially distributed (with the
same parameter??0) random variables has a gamma distribution with param-etersnand?, i.e. the probability density function of the sum is given by
f(x) =?
n
e
??x
x
n?1
(n?1)!
, x > 0.
4
Using this and part (a), show that
Pr(Xt =k|X0
= 0) = e
??t
(?t)
k
k!
.
Here you can make use of the following identity:
?
k
Z
?
t
e
??x
x
k?1
(k?1)!
·
?x
k
?1
¸
dx= e
??t
(?t)
k
k!
.
(c) Show that givenX0
= 0, the incrementsXt ?Xs
, 0 ?s < t, are Poisson distributed
as well with parameter?(t?s), i.e.
Pr(Xt ?Xs =k|X0
= 0) = e
??(t?s)
(?(t?s))
k
k!
.
(Hint: use the law of total probability by conditioning on Xs
.)
5

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