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 2015 study material to make it suitable for study for the 2016 exams. 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 2016 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 2016 exams.
1 Changes to the Syllabus objectives and Core Reading
1.1 Syllabus objectives There have been no changes to the syllabus objectives.
1.2 Core Reading Chapter 6 Page 35, Section 6.4 There was a typo in the fourth line of the second paragraph on this page: the word “underling” should be changed to “underlying”. Chapter 10 Page 29 The same typo appears in the third line of the Core Reading paragraph on this page: again, the word “underling” should be changed to “underlying”.
2 Changes to the ActEd Course Notes Chapter 5 Pages 60-61, Solution 5.17 The end-of-year reserve was incorrectly used in place of the DSAR in the calculation of the EDS and the ADS in this solution. Please replace page 59-60 with the replacement pages that you’ll find at the end of this upgrade document. On page 61 the answer for the mortality profit at the top of the page should be changed from £159,964 to £702,037. Chapter 6 Page 12, Section 4.1 In the section on whole life assurance policies with compound increases, the sentence in the middle of the page should be changed to read as follows (change shown in bold): “With this contract, the sum assured increases (by multiplying the previous sum assured by (1 )b+ ) at the end of each year.”
Chapter 10 Page 32, Section 5.1, Example In the integral expression shown in the solution on this page, the age subscript of the
( )dam function should be changed from 48 to 48 t+ . The corrected expression should
now read as follows:
22
48 480
50,000 ( ) ( )t dt
t
EPV v ap a dtm=
= Ú +t
Page 46, Solution 10.7 In the first line of this solution, the second age given at the end of the line should be changed from 1x + to x t+ .
Chapter 12 Pages 23-26, Section 2.3, Single premium unitised with-profits example If you purchased your 2015 materials prior to 27th January 2015, this section and associated solutions 12.15, 12.16 and 12.17) would have contained some errors. On that date ActEd sent an email to everyone who had purchased the 2015 CMP or Course Notes giving details of, and sending replacement pages, for this section and solutions. If you purchased your 2015 materials on or after 27th January 2015 then your notes would have included the correct, updated, versions. We advise that you check that you have the correct updated versions. If the first sentence at the top of page 24 (under “Solution”) begins with “The first thing we need to do...”, then you have the correct versions and you need take no further action. However, if instead this first sentence begins with “We will need the dependent probabilities...”, then you have the old versions and you will need to replace them. Please discard the whole of pages 23-26 and pages 43-46, and replace them with the attached pages 23-26a (plus copyright page) and pages 43-46, which are attached at the end of this upgrade document.
3 Changes to the Q&A Bank Part 2 Solutions Page 5, Solution 2.7 In the last paragraph (starting “Since the...”), the word “quarterly” should be changed to “annual”. We have made further changes to improve the solution, so you might prefer to replace the whole of pages 5-6 with the replacements pages attached at the end of this upgrade document. Part 3 Solutions Page 27, Solution 3.28(c)
In the second line of formula (in the definition of s dxM ) the subscript of the s dC
symbol in the summation should be changed from x to y . The corrected expression
should now read as follows:
64
s d s dx
y x
M C=
= Â y
Part 4 Solutions Page 13, Solution 4.14 The last line of formulae is incorrect, because it shows the net income, rather than the outgo. Replace the expression by the following: (3,000 1,981.09) 0.0007447 (10 5) 0.98825 £4.18- ¥ - - ¥ = -
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 CT5:
Mock Exam
Additional Mock Pack
ASET (ActEd Solutions with Exam Technique) and Mini-ASET
Revision Notes
Flashcards. For further details on ActEd’s study materials, please refer to the 2016 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.
5.2 Tutorials We offer the following tutorials in Subject CT5:
a set of Regular Tutorials (usually lasting two or three full days)
a Block Tutorial (lasting two or three full days)
a Revision Day (lasting one full 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.
The Online Classroom is an exciting approach to studying for the actuarial exams. It acts as either a valuable add-on to a face-to-face tutorial or a great alternative to a tutorial, particularly if you're not based in the UK or near a tutorial venue. At the heart of the Online Classroom in each subject is a comprehensive, easily-searched collection of over 100 tutorial units. These are a mix of:
teaching units, helping you to really get to grips with the course material, and
guided questions, enabling you to learn the most efficient ways to answer questions and avoid common exam pitfalls.
The best way to discover the Online Classroom is to see it in action. You can watch a sample of the Online Classroom tutorial units on the ActEd website at www.ActEd.co.uk.
5.4 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 2016 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.
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.
and: : 1: 11 x tx t n t x t n ta v p a++ - + + - -= +
So:
( )
( )
1 1 1 1 1:: : : : :
1 1 11: 11: 1 : :
1 11: 11: 1 :
11 :
1
t x t n tx n x n x t n t x n x n
x t x t x t n tx t n t x n x n
x t x t n tx t n t x n
x t t x n
V P A P a P
v p A P v p a P
vp A P a
vp V
+ -+ -
+ + + + - -+ + - -
+ + + - -+ + - -
+ +
+ = - +
= - + +
= -
=
Multiplying through by 1 i+ then gives:
( )( )1 1 11: : :
1t x t tx n x n x nV P i p V+ ++ + =
Solution 5.16
The death strain formula would change to:
( )4
121
0 if the life survives to 1
1 if the life dies in the year 1t
tDS
S i V t t+
+ÏÔ= Ì+ - Æ +ÔÓ
This is because (assuming deaths occur half way through the year on average) the payment of the sum assured occurs 8/12 through the year, on average. So, in order to revalue the payment to the end of the year of death, we need to accumulate the payment with 4 months interest.
where 6V is the reserve held at the policy anniversary in 2013 (which is at exact
duration 6). Now:
6 71 7160,000V A P a= -
where:
½
65
65
60,000 60,000 1.04 0.527862,631.05
12.276
AP
a
¥ ¥= =
using ½(1 )x xA i A+ .
So:
½6 71 71
½
60,000 1.04 2,631.05
60,000 1.04 0.61548 2,631.05 9.998 11,354.90
V A a= ¥ ¥ -
= ¥ ¥ - ¥ =
and the death strain at risk is:
½60,000 1.04 11,354.90 49,833.33DSAR = ¥ - =
The expected death strain for this policy was:
701,900 1,900 0.024783 49,833.33
2,346,537
EDS q DSAR= ¥ ¥ = ¥ ¥
=
During the policy year, 33 people died, so the actual death strain was: 33 33 49,833.33 1,644,500ADS DSAR= ¥ = ¥ =
CT5-12: Profit testing Page 23
The Actuarial Education Company IFE: 2016 Examinations
2.3 Example 3: Single premium unitised with-profits contract
For this example we assume the contract has a five-year term, paid for by a single premium. The allocation rate is 100% and there is no bid-offer spread. The only charges under the contract are an admin fee equal to 0.5% of the bid value of the unit fund, deducted at the end of each year by cancelling units, and a penalty on surrender calculated as a percentage of the unit fund value depending on policy duration as follows:
After 1 year: 2.4%
After 2 years: 1.8%
After 3 years: 1.2%
After 4 years: 0.6% The unit price increases each year according to the insurer’s declared bonus interest rate. The full value of the unit fund, plus any discretionary terminal bonus, is paid out on maturity or at the end of the year of earlier death. Surrender is permitted only at the end of each year, when the fund value less the above surrender penalty would be paid out. You can assume that, over the long term, all profits earned from investment returns are fully distributed to policyholders through the bonus payments (we explain the significance of this assumption after the solution, below). Calculate the projected profit for each policy year based on the following assumptions: Age at entry: 45 exact Single premium: £20,000 Mortality: 0.002xq = for 45, 46, ..., 49x =
Surrender rate: 5% of all policies in force at the end of each of Years 1-4 Bonus interest rate: 3.5% pa Non-unit interest: 2% pa Initial expenses: £300 Renewal expenses: £25 at the start of Year 2, and thereafter at the start of each
subsequent year, inflating at 2.5% pa Claim expenses: £100 per death, surrender, or maturity Non-unit reserves: £50 per policy in force at the start of Year 5
Page 24 CT5-12: Profit testing
IFE: 2016 Examinations The Actuarial Education Company
Solution The first thing we need to do is to project the unit fund values and the policy fees each year. At the end of the first year, the fund value (before deduction of the policy fee) is the single premium plus one year’s bonus: 20,000 1.035 20,700¥ =
The policy fee at the end of the year is: 20,700 0.005 103.50¥ =
which means that the fund remaining at the end of the year is: 20,700 103.50 20,596.50- =
Working through the other years in a similar way, we obtain the following table of projected values:
Year t Fund at end of year before deduction of
policy fee
(A)
Policy fee
(B)
Fund at end of year after deduction of
policy fee
(C)
1 20,700 103.50 20,596.50
2 21,317.38 106.59 21,210.79
3 21,953.17 109.77 21,843.40
4 22,607.92 113.04 22,494.88
5 23,282.20 116.41 23,165.79
where, for years 2, 3, 4, 5t = :
1( ) ( ) 1.035
( ) ( ) 0.005
( ) ( ) ( )
t tA C
B A
C A B
-= ¥
= ¥
= -
CT5-12: Profit testing Page 25
The Actuarial Education Company IFE: 2016 Examinations
Next we need to work out the surrender penalties. These are equal to the end-year fund values (after charges) multiplied by the appropriate percentage rates:
Year t Fund at end of year Surrender penalty
1 20,596.50 494.32
2 21,210.79 381.79
3 21,843.40 262.12
4 22,494.88 134.97
5 23,165.79 0
We will also need the dependent probabilities of decrement by death and surrender. Because surrenders take place at the end of each year, the dependent probability of dying is the same as the independent probability, ie equal to 0.002 each year. The dependent surrender probability is found as:
( )1 0.002 0.05 0.0499- ¥ =
as we are told that 5% of the end of year in-force policies surrender each year. The dependent probability of a policy staying in force over any particular year is then: 1 0.002 0.0499 0.9481- - =
(or alternatively this can be calculated as ( ) ( )1 0.002 1 0.05 0.9481- ¥ - = ).
The non-unit cashflows can now be calculated, which for the first four years of the contract are as follows:
Year t Initial and renewal
Expenses
(1)
Interest
(2)
Policy fee
(3)
Expected surrender
profit
(4)
Expected claim
expenses
(5)
Expected non-unit Cashflow
(6)
1 – 300 – 6 103.5 24.67 – 5.19 – 183.02
2 – 25 – 0.5 106.59 19.05 – 5.19 94.95
3 – 25.62 – 0.51 109.77 13.08 – 5.19 91.53
4 – 26.27 – 0.53 113.04 6.73 – 5.19 87.78
5
Page 26 CT5-12: Profit testing
IFE: 2016 Examinations The Actuarial Education Company
where:
( )
{ }{ }
( ) ( )
1(1) (1) 1.025 3 5
(2) (1) 0.02
(3) fund value 0.005
(4) surrender penalty 0.0499
(5) 100 0.002 0.0499 1 4
(6) (1) (2) (3) (4) (5)
t t t
t
-= ¥ £ £
= ¥
= ¥
= ¥
= - ¥ + £ £
= + + + +
Question 12.15
Calculate the table entries for Year 5.
Having worked out the expected cashflows, we now need to calculate the expected profit for each year, taking into account the effect of the non-unit reserves. For this policy, we are told that a non-unit reserve of 50 is required per policy in force at the start of Year 5 (and also therefore per policy in force at the end of Year 4); at all other times no non-unit reserve is to be held. How will this affect the expected profit in Year 4? For a policy in force at the start of Year 4, the insurer expects a cashflow of 87.78 by the end of the year. However, the insurer now has to set aside reserves of 50 for each policy that’s remained in force through to the end of the year. The probability of doing this is 0.9481 (from above). So the expected profit for Year 4 (per policy in force at the start of the year) reduces to: 87.78 50 0.9481 40.38- ¥ =
Question 12.16
Calculate the expected profit that will now be earned from a policy in force at the start of Year 5, including the reserve of 50 held at the start of the year.
As there are no reserves held at any time in Years 1-3, the expected profits for these years are all equal to the expected cashflows shown in the table above.
CT5-12: Profit testing Page 26a
The Actuarial Education Company IFE: 2016 Examinations
Non-unit reserves, such as in this example, are required for unit-linked and unitised with-profits policies wherever a future negative cashflow is expected.
Question 12.17
Given that there was an expected negative cashflow of 11.06, it might have seemed logical to hold the smallest reserve possible to cover this expected cost, ie 11.06 rather than the 50 actually held. Suggest a reason why the insurer might have used the higher figure.
What’s the point of profit-testing for unitised with-profits? As explained in an earlier chapter, the idea of a with-profits contract is to return to the policyholder the profits earned by the insurer on the policy over the policy term. One approach for unitised with-profits is to earmark the investment profits for policyholders (via the unit fund), and deduct explicit charges to cover the insurer’s non-unit expenses and other costs, as in the example we have just looked at. By projecting the future non-unit profits, the insurer can check that its charges are on track to cover these outgoes. This was why in the above example we assumed that all the investment profits were distributed to policyholders. An exam question may or may not state this assumption explicitly, but it generally would be implied, as the main point of profit testing these contracts in this way would be to “test” the adequacy of the charges in covering the non-unit liabilities.
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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
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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.
CT5-12: Profit testing Page 43
The Actuarial Education Company IFE: 2016 Examinations
Solution 12.11
The bid value of premiums allocated will be 2,000 0.6 0.95 1,140¥ ¥ = .
The fund at end of year, before management charge, is 1,140 1.06 1,208.40¥ = .
The management charge is 1% 1,208.40 12.08¥ = .
The fund after deduction of management charge will be 1,208.40 12.08 1,196.32- = (or
do as 99% of fund value). Solution 12.12
The bid value of premiums allocated will be 2,000 0.98 0.95 1,862¥ ¥ = .
The fund after allocation is 1,862 3,209.40 5,071.40+ = .
The fund at end of year, before management charge, is 5,071.40 1.06 5,375.68¥ = .
The management charge is 1% of this = 53.76. The fund after deduction of this management charge will be:
5,375.68 53.76 5,321.92- =
Solution 12.13
Income to the non-unit fund comes from:
● at the start of the year, the margin between premium and cost of premiums allocated, ie 2,000 1,140 860- = , and
● at the end of the year the fund management charge of 12.08 (from the unit fund calculations).
Note that interest on the non-unit fund is negative in the first year due to the effect of expenses:
( )860 1,150 6% 17.40- ¥ = -
The total of all these elements is 854.68.
Page 44 CT5-12: Profit testing
IFE: 2016 Examinations The Actuarial Education Company
Solution 12.14
Non-unit profit/loss is income less outgo, ie:
854.68 1,150 9.54 304.86- - = -
Solution 12.15
The expenses in Year 5 are the Year 4 expenses increased by inflation at 2.5%. The amount is therefore: 26.27 1.025 26.93¥ = The interest is equal to the interest lost over the year due to the expenses incurred at the start of the year. Its amount is: 26.93 0.02 0.54- ¥ = - The policy fee income is read from the table, and for Year 5 this is 116.41. As there is no surrender penalty (and no-one surrenders!) then the expected surrender profit from the surrender penalty is zero. The claim expenses of 100 are paid out on all policies that become claims during the year. All policies in force at the start of year 5 will ultimately claim at the end of the year – either by surviving and receiving the maturity benefit, or by dying during the year and receiving the death benefit. The expected amount of claim expenses is then equal to:
( )49 49 100 100q p+ ¥ =
The expected cashflow for year 5 is then calculated as:
The Actuarial Education Company IFE: 2016 Examinations
The completed cashflow table now reads as follows:
Year t Initial and renewal
Expenses
(1)
Interest
(2)
Policy fee
(3)
Expected surrender
profit
(4)
Expected claim
expenses
(5)
Expected non-unit Cashflow
(6)
1 – 300 – 6 103.5 24.67 – 5.19 – 183.02
2 – 25 – 0.5 106.59 19.05 – 5.19 94.95
3 – 25.62 – 0.51 109.77 13.08 – 5.19 91.53
4 – 26.27 – 0.53 113.04 6.73 – 5.19 87.78
5 – 26.93 – 0.54 116.41 0 – 100 – 11.06
Solution 12.16
The expected profit for Year 5 will equal:
{ } { } { }
{ }
expected cashflow reserve at start of year interest on reserve
expected cost of reserve at end of year
+ +
-
Putting in the values (and noting that the reserve at the end of the year is zero) we obtain: 11.06 50 0.02 50 0 39.94- + + ¥ - = Solution 12.17
All reserves need to be prudent, so that there will be a high probability that the liabilities (future outgo) will be covered. The insurer would therefore have projected its cashflows on more cautious assumptions, leading to a somewhat higher negative cashflow than 11.06 in Year 5, and decided that a reserve of 50 was necessary to cover this. Solution 12.18
£186.97 will be the profit expected by the end of the coming year (Year 2), from a policy that was in force at exact time 1.
Page 46 CT5-12: Profit testing
IFE: 2016 Examinations The Actuarial Education Company
Solution 12.19
From the definition given in Section 2.1, the required probability is:
1 55 55 55( ) 1 ( ) ( ) 1 0.005 0.1 0.895d wap aq aq
So:
2 1 55186.97 ( ) 186.97 0.895 167.34E P ap
Solution 12.20
The first-year element of the profit signature is unchanged from the profit vector, because the amount ( 803.99 ) is already the amount per policy in force at the beginning of year 1, ie at inception. For year 3 we need: 3( )PS 3 2 55( ) ( )PRO ap
and in general we will need: 1 55( ) ( ) ( )t t tPS PRO ap
We calculate 55 56( ) , ( ) ,ap ap etc from:
( ) 1 ( ) ( )d wx x xap aq aq
(using the dependent probabilities given in Section 2.1); and 2 55( )ap (for example)
from the cumulative probability: 2 55 55 56( ) ( ) ( )ap ap ap
[Total 5] In practice, the values of these two functions are quite similar, and an annuity certain for the expected lifetime can be used as a reasonableness check on the value of a life annuity. Solution 2.7
The prospective reserve is:
1 |61:93,000proV a= ¥ [1]
The retrospective reserve is the accumulated value of the premium less benefits:
60 601 |60:1
61 613,000retro D D
V P aD D
= ¥ - ¥ [½]
where P is the single premium. Now:
|60:103,000P a= [½]
so:
60 601 | |60:10 60:1
61 61
60| |60:10 60:1
61
3,000 3,000
3,000 ( )
retro D DV a a
D D
Da a
D
= ¥ - ¥
= - [1]
Since the difference between the annuities in the brackets represents the value at age 60 of annual payments made in arrears for the 9 years between ages 61 and 70, this can be written as:
