Subject CT6 CMP Upgrade 2014/15

Transcription

CT6: CMP Upgrade 2014/15
Page 1
Subject CT6
CMP Upgrade 2014/15
CMP Upgrade
This CMP Upgrade lists all significant changes to the Core Reading and the ActEd
material since last year so that you can manually amend your 2014 study material to
make it suitable for study for the 2015 exams. It includes replacement pages and
additional pages where appropriate.
Alternatively, you can buy a full replacement set of up-to-date Course Notes at a
significantly reduced price if you have previously bought the full price Course Notes in
this subject. Please see our 2015 Student Brochure for more details.
This CMP Upgrade contains:

all changes to the Syllabus objectives and Core Reading.

changes to the ActEd Course Notes, Series X Assignments and Question and
Answer Bank that will make them suitable for study for the 2015 exams.
The Actuarial Education Company
© IFE: 2015 Examinations
Page 2
1
Changes to the Syllabus objectives and Core Reading
1.1
Syllabus objectives
1.2
Core Reading
There have been no changes to the Core Reading.
2
Page 3
Changes to the ActEd Course Notes
Chapter 4
Page 7
One of the Core Reading equations wasn’t in a bold font as it should have been. A
replacement page is provided.
Page 8
Some extra explanatory Core Reading has been added to the derivation of P W  w  .
A replacement page is provided.
Page 4
3
Changes to the Q&A Bank
There have been no changes to the Q&A Bank.
4
Page 5
Changes to the X Assignments
There have been no changes to the X Assignments.
Page 6
5
Other tuition services
In addition to this CMP Upgrade you might find the following services helpful with
your study.
5.1
Study material
We offer the following study material in Subject CT6:
●
Online Classroom
●
Flashcards
●
Sound Revision
●
MyTest
●
Revision Notes
●
ASET (ActEd Solutions with Exam Technique) and Mini-ASET
●
Mock Exam
●
Additional Mock Pack.
For further details on ActEd’s study materials, please refer to the 2015 Student
Brochure, which is available from the ActEd website at www.ActEd.co.uk.
5.2
Tutorials
We offer the following tutorials in Subject CT6:

a set of Regular Tutorials (usually lasting three full days)

a Block Tutorial (usually lasting three full days)

a Revision Tutorial (usually lasting one full day)

Live Online Tutorials (usually lasting three full days)

Live Online Revision Tutorials (usually lasting half a day).
For further details on ActEd’s tutorials, please refer to our latest Tuition Bulletin, which
is available from the ActEd website at www.ActEd.co.uk.
5.3
Page 7
Marking
You can have your attempts at any of our assignments or mock exams marked by
ActEd. When marking your scripts, we aim to provide specific advice to improve your
chances of success in the exam and to return your scripts as quickly as possible.
For further details on ActEd’s marking services, please refer to the 2015 Student
Brochure, which is available from the ActEd website at www.ActEd.co.uk.
5.4
Feedback on the study material
ActEd is always pleased to get feedback from students about any aspect of our study
programmes. Please let us know if you have any specific comments (eg about certain
sections of the notes or particular questions) or general suggestions about how we can
improve the study material. We will incorporate as many of your suggestions as we can
when we update the course material each year.
If you have any comments on this course please send them by email to [email protected]
or by fax to 01235 550085.
All study material produced by ActEd is copyright and is sold
for the exclusive use of the purchaser. The copyright is owned
by Institute and Faculty Education Limited, a subsidiary of
the Institute and Faculty of Actuaries.
Unless prior authority is granted by ActEd, you may not hire
out, lend, give out, sell, store or transmit electronically or
photocopy any part of the study material.
You must take care of your study material to ensure that it is
not used or copied by anybody else.
Legal action will be taken if these terms are infringed. In
addition, we may seek to take disciplinary action through the
profession or through your employer.
These conditions remain in force after you have finished using
the course.
CT6-04: Reinsurance
Page 7
Question 4.2
Find E (Y ) when X has a Pareto distribution with parameters   200 and   6 , and
M  80 .
Under excess of loss reinsurance, the reinsurer will pay Z where:
Ï0
Z =Ì
ÓX - M
if X £ M
if X > M
With a retention level of M , the mean amount paid by the reinsurer is:
•
E (Z ) =
Ú ( x - M ) f ( x ) dx
(1.3)
M
( )
Similarly, we can calculate E Z 2 using:
•
( ) = Ú (x - M )
E Z
2
2
f ( x)dx
M
( )
2
Then var ( Z ) = E Z 2 - ÈÎ E ( Z )˘˚ .
More generally, the moment generating function of Z , the amount paid by the
reinsurer, is:
( )
M t0
e
0
M Z (t ) = E etZ = Ú
1.2
• t x-M )
f
M
f ( x) dx + Ú e (
( x) dx
The reinsurer’s conditional claims distribution
Now consider reinsurance (once again) from the point of view of the reinsurer.
The reinsurer may have a record only of claims that are greater than M. If a claim
is for less than M the reinsurer may not even know a claim has occurred. The
reinsurer thus has the problem of estimating the underlying claims distribution
when only those claims greater than M are observed. The statistical terminology
is to say that the reinsurer observes claims from a truncated distribution.
In this case the values observed by the reinsurer relate to a conditional distribution,
since the numbers are conditional on the original claim amount exceeding the retention
limit.
 IFE: 2015 Examinations
Page 8
CT6-04: Reinsurance
Let W be the random variable with this truncated distribution. Then:
W = X -M|X >M
Suppose that the underlying claim amounts have PDF f ( x ) and CDF F ( x ) .
Suppose that the reinsurer is only informed of claims greater than the retention
M and has a record of w = x  M. What is the PDF g(w) of the amount, w, paid by
the reinsurer?
The argument goes as follows:
P (W < w ) = P ( X < w + M | X > M )
=
P ( X < w + M and X > M )
P(X > M)
=
P (M < X < w + M )
P(X > M)
=
ÚM
=
F (w + M ) - F (M )
1 - F (M )
w +M
f (x )
dx
1 - F (M )
(using Bayes' Formula)
(since F (M ) = P(X < M ))
This derivation also uses the result:
b
P (a < X < b) = Ú f ( x) dx = F (b) - F (a)
a
Differentiating w.r.t. w, the PDF of the reinsurer’s claims is
g (w ) =
f (w + M )
, w > 0.
1 - F (M )
(1.4)
Note that this is just the original PDF applied to the gross amount w + M , divided by
the probability that the claim exceeds M .
Question 4.3
Using the notation above, if X is Exp(  ) , find the distribution of W.
 IFE: 2015 Examinations

Subject CT6 CMP Upgrade 2014/15

Transcription

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