ct52010-2013
TRANSCRIPT
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Faculty of Actuaries Institute of Actuaries
EXAMINATION
26 April 2010 (am)
Subject CT5 Contingencies
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 14 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
Faculty of Actuaries
CT5 A2010 Institute of Actuaries
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CT5 A20102
1 Explain what the following represent:
(a) [ ]x rl +
(b) |n m xq
(c)x
d
[3]
2 Define spurious selection, giving two distinct examples. [3]
3 Calculate the standardised mortality ratio for the population of Urbania using thefollowing data:
Standard Population Urbania
Age Population Deaths Population Deaths
60 2,500,000 26,170 10,000 130
61 2,400,000 29,531 12,000 145
62 2,200,000 32,542 11,000 173
[3]
4 A life insurance company offers an increasing term assurance that provides a benefitpayable at the end of the year of death of 10,000 in the first year, increasing by 100 on
each policy anniversary.
Calculate the single premium for a five year policy issued to a life aged 50 exact.
Basis:
Rate of interest 4% per annum
Mortality AM92 Select
Expenses Nil
[4]
5 A population is subject to the force of mortality x= e0.0002x1.
Calculate the probability that a life now aged 20 exact:
(i) survives to age 70 exact [2]
(ii) dies between ages 60 exact and 70 exact [3]
[Total 5]
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CT5 A20103 PLEASE TURN OVER
6 You are provided with the following extract from a life table:
x lx
50 99,813
51 97,702
52 95,046
Calculate 0.75p50.5using two different methods. [5]
7 A company is about to establish a pension scheme that will provide an age retirementbenefit of n/60ths of final pensionable salary where n is total number of years of
service. Final pensionable salary is the average salary in the three years before
retirement.
An employee who will become a member of the pension scheme is currently aged 55
exact has and will be granted exactly 20 years of past service. The employees salary
in the year before the valuation date was 40,000.
(i) Calculate the present value of benefits for this member (including future
service). [3]
(ii) Calculate the contribution required to fund this benefit as a percentage of
future salaries. [3]
Basis:
Pension Scheme from the Formulae and Tables for Actuarial Examinations
[Total 6]
8 100 graduates aged 21 exact decide to place the sum of 1 per week into a fund to beshared on their retirement at age 66 exact.
(i) Show that each surviving member can expect to receive on retirement a fund
of approximately 7,240. [4]
Basis:
Rate of interest 4% per annum
Mortality AM92 Ultimate
One of the survivors uses the accumulated fund to buy a weekly annuity payable for
10 years certain. After 10 years the annuity is payable at two-thirds of the initial level
for the rest of life.
(ii) Calculate the weekly amount of the annuity on the basis used in part (i). [2]
[Total 6]
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CT5 A20104
Active(A) Retired (R)
Dead (D)
xx
x
9 A life insurance company models the experience of its pension scheme contractsusing the following three-state model:
(i) Derive the dependent probability of a life currently Active and agedxretiring
in the year of agexto (x + 1) in terms of the transition intensities. [2]
(ii) Derive a formula for the independent probability of a life currently Active and
agedxretiring in the year of agexto (x + 1) using the dependent probabilities.
[4]
[Total 6]
10 The decrement table extract below is based on the historical experience of a very largemultinational companys workforce.
Age (x) Number of employees
( )xal
Deaths
( )dxad
Withdrawals
( )wxad
40 10,000 25 120
41 9,855 27 144
42 9,684
Recent changes in working conditions have resulted in an estimate that the annual
independent rate of withdrawal is now 75% of that previously used.
Calculate a revised table assuming no changes to the independent death rates, stating
your results to one decimal place. [7]
11 Thieles differential equation for the policy value at duration t (t> 0), t xV , of an
immediate life annuity payable continuously at a rate of 1 per annum from agexis:
1t x t x t xx tV V Vt
+
= +
(i) Derive this result algebraically showing all the steps in your working. [5]
(ii) Explain this result by general reasoning. [3]
[Total 8]
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CT5 A20105 PLEASE TURN OVER
12 On 1 January 2005, a life insurance company issued 1,000 10-year term assurancepolicies to lives aged 55 exact. For each policy, the sum assured is 50,000 for the
first five years and 25,000 thereafter. The sum assured is payable immediately on
death and level annual premiums are payable in advance throughout the term of this
policy or until earlier death.
The company uses the following basis for calculating premiums and reserves:
Mortality AM92 Select
Interest 4% per annum
Expenses Nil
(i) Calculate the net premium retrospective reserve per policy as at 31 December
2009. [6]
(ii) (a) Give an explanation of your numerical answer to (i) above.
(b) Describe the main disadvantage to the insurance company of issuing
this policy.
(c) Give examples of how the terms of the policy could be altered so as to
remove this disadvantage.
[3]
There were, in total, 20 deaths during the years 2005 to 2008 inclusive and a further 8
deaths in 2009.
(iii) Calculate the total mortality profit or loss to the company during 2009. [3]
[Total 12]
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CT5 A20106
13 A life insurance company issues a 3-year unit-linked endowment assurance policy to amale life aged 45 exact.
Level premiums of 4,000 per annum are payable yearly in advance throughout the
term of the policy or until earlier death. 95% of the premium is allocated to units in
the first policy year, 100% in the second and 105% in the third. A policy fee of 50 is
deducted from the bid value of units at the start of each year. The units are subject to
a bid-offer spread of 5% on purchase. An annual management charge of 1.75% of thebid value of units is deducted at the end of each policy year.
Management charges are deducted from the unit fund before death, surrender and
maturity benefits are paid.
If the policyholder dies during the term of the policy, the death benefit of 125% of the
bid value of the units is payable at the end of the policy year of death. On maturity,
100% of the bid value of the units is payable.
The policyholder may surrender the policy only at the end of the first and second
policy years. On surrender, the bid value of the units less a surrender penalty ispayable at the end of the policy year of exit. The surrender penalty is 1,000 at the
end of the first policy year and 500 at the end of the second policy year.
The company uses the following assumptions in carrying out profit tests of this
contract:
Rate of growth on assets in the unit fund 5.5% per annum in year 1
5.25% per annum in year 2
5.0% per annum in year 3
Rate of interest on non-unit fund cash flows 4.0% per annum
Mortality AM92 SelectInitial expenses 200
Renewal expenses 50 per annum on the second and third
premium dates
Initial commission 15% of first premium
Renewal commission 2.0% of the second and third years
premiums
Rate of expense inflation 2.0% per annum
Risk discount rate 7.0% per annum
For renewal expenses, the amount quoted is at outset and the increases due to inflation
start immediately. In addition, you should assume that at the end of the first and
second policy years, 12% and 6% respectively of all policies still in force then
surrender immediately.
(i) Calculate the profit margin for the policy. [13]
(ii) Calculate the expected present value of profit for the policy if the company
assumed that there were no surrenders at the end of each of the first and
second policy years. [3]
[Total 16]
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CT5 A20107
14 A life insurance company issues a 30-year with profits endowment assurance policyto a life aged 35 exact. The sum assured of 100,000 plus declared reversionary
bonuses are payable on survival to the end of the term or immediately on death if
earlier.
(i) Show that the quarterly premium payable in advance throughout the term of
the policy or until earlier death is approximately 616.
Pricing basis:
Mortality: AM92 Select
Interest: 6% per annum
Initial commission: 100% of the first quarterly premium
Initial expenses: 250 paid at policy commencement date
Renewal commission: 2.5% of each quarterly premium from the start of the
second policy year
Renewal expenses: 45 at the start of the second and subsequent policy
years
Claim expense: 500 on death; 250 on maturityFuture reversionary bonus: 1.92308% of the sum assured, compounded and vesting
at the end of each policy year (i.e. the death benefit does
not include any bonus relating to the policy year of
death)
[10]
At the end of the 25thpolicy year, the actual past bonus additions to the policy have
been 145,000.
(ii) Calculate the gross prospective policy reserve at the end of that policy year
immediately before the premium then due.
Policy reserving basis:
Mortality: AM92 Ultimate
Interest: 4% per annum
Bonus loading: 4% of the sum assured and attaching bonuses,
compounded and vesting at the end of each policy year
Renewal commission: 2.5% of each quarterly premium
Renewal expenses: 90 at the start of each policy year
Claim expense: 1,000 on death; 500 on maturity
[6][Total 16]
END OF PAPER
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Faculty of Actuaries Institute of Actuaries
EXAMINERS REPORT
April 2010 Examinations
Subject CT5 Contingencies
Core Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as the
interpretation of the syllabus to which the examiners are working. They have however given
credit for any alternative approach or interpretation which they consider to be reasonable.
R D Muckart
Chairman of the Board of Examiners
July 2010
Comments
These are given in italics at the end of each question.
Faculty of ActuariesInstitute of Actuaries
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 2
1 (i) The number of lives still alive at agex + rout of lxlives alive at agexsubjectto select mortality.
(ii) The probability that a life agex will die between agex + nandx + n + m.
(iii) The number of lives that die betweenxand (x + 1) out of xl lives alive atx.
Question generally answered well.
2 Spurious selection occurs when mortality differences ascribed to groups are formedby factors which are not the true causes of these differences.
For example mortality differences by region may be put down to the actual class
structure of the region itself whereas a differing varying mix of occupations region by
region could be having a major effect. So Region is spurious and being confoundedwith occupation.
Another example might be in a company pension scheme which might be showing a
significant change in mortality experience which could be viewed as change over
time. However withdrawers from the scheme may be having an effect as their
mortality could be different. To that degree Time Selection may be spurious.
Question generally answered well. Credit was given for a wide range of valid examples.
3 The Standardised mortality ratio is the ratio of actual deaths in the population dividedby the expected number of deaths in the population if the population experienced
standard mortality.
Actual number of deaths for Urbania = 130+145+173 = 448
Mortality rates in standard population are:
Age 60: 26,170 / 2,500,000 = 0.0104680
Age 61: 29,531 / 2,400,000 = 0.0123046
Age 62: 32,542 / 2,200,000 = 0.0147918
Expected number of deaths for Urbania
= 0.010468 10,000 + 0.0123046 12,000 + 0.0147918 11,000 = 415
SMR = 448/415 = 107.95%
Question generally answered well.
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 3
4 1 1[50]:5 [50]:5
(10,000 100) 100( )EPV A IA= +
5 5[50] 5 [50] 55 [50] 5 [50] 55 559,900( ) 100(( ) (5 ( ) ))A v p A IA v p A IA= + +
5 9557.81799,900(0.32868 *0.38950)9706.0977
v=
5 9557.8179100* 8.5639 (5*0.38950 8.57976)9706.0977
v + +
132.96 4.34= +
137.30=
Many students answered the question well. The most common error was the use of 10,000 asthe multiplier before the temporary assurance function rather then 9,900.
5 exp( )x t
t x sx
p ds+
=
0.0002exp( ( 1) )x t S
xe ds
+=
0.0002exp( )x t x tSx x
e ds ds+ += +
0.0002( ) 0.0002
exp0.0002
x t xe e
t
+ = +
(i) Probability =
0.0002 70 0.0002 20
exp 500.0002
x xe e
= +
0.6362=
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 4
(ii) This is the probability that the life survives to 60 and then dies between 60 and
70
40 20 10 60Probability (1 )p p=
0.0002 60 0.0002 20 0.0002 70 0.0002 60
exp (60 20) . 1 exp (70 60)0.0002 0.0002
x x x xe e e e = + +
0.725 (1 0.8773)x=
0.0889=
This question was answered poorly overall. It was an unusual representation of the x
function but other than that was a straight forward probability and integration question.
6 p50= 97,702 / 99,813 = 0.978850p51= 95,046 / 97,702 = 0.972815
Uniform distribution of deaths
50 0.25 51 50 51
0.5 50 50
(1 0.25(1 )) 0.978850* (1 0.25*(1 0.972815))0.982588
(1 0.5(1 )) (1 0.5*(1 0.978850))
p p p p
p p
= = =
Constant force of mortality
t= ln(pt)
50 = ln(0.978850) = 0.021377
51= ln(0.972815) = 0.027561
0.5*0.021377 0.25*0.0275610.5 50 0.25 51* * 0.989368*0.993133 0.982574p p e e
= = =
Generally answered well. A limited number of students used the Balducci Assumption as oneof their answers. This is not in the CT5 Course whilst the above 2 methods clearly are. This
method was however credited solution not published as not in CT5
7 (i) Age retirement benefit
5555
54 55
(20 )140,000
60
z raz raM R
s D
+
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 5
1 (20*128,026 963,869)40,000
60 9.745*1,389
+=
173,584=
(ii) Contributions
55
54 55
40,000.
sN
Ks D
88,615.40,000
9.745*1,389K x=
261,868K=
Therefore K= 173,584 / 261,868 i.e. 66.3%
Most students answered reasonably well. Most common error was the wrong sxfunction.
Also some students included early retirement calculations which were not asked for.
Also students often did not include the past service benefits in the final contribution rate
believing the final result would have been too high (the question however was quite specific
on providing past benefits).
8 (i)(66 21)
21:45
45 21
1.04Fund 52*
a
p
=
4566 2121:45 21:45 21:45
1 1 8695.6199*(1 * / ) * 1 0.17120*
2 2 9976.3909a a v l l a
= =
21:45 0.42539a=
44 65 2121:45 21:44
8821.2612* / 21.045 .17805* 21.2029976.3909a a v l l= + = + =
21:4520.777a =
( )
4552*1.04 (20.777)therefore fund 7,240
8695.61999976.3909
= =
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 6
(ii) Let annuity be P per week. Then EPV of annuity at 66 is
1010 66 7610
252 ( * . )3
P a v p a+
10(1 ) 6589.9258252 *0.675564* (8.169 0.5)3ln(1.04) 8695.6199
vP = +
[ ]52 8.272 2.618]P= +
566.26P=
Therefore pension is given by
7,240 566.26P=
12.79P=
Many students struggled with this question and indeed a large number did not attempt it. As
will be seen from the solution above the actuarial mathematics involved are relatively
straightforward.
Note that 52.18 (i.e. 365.25/7) would have been an acceptable alternative to 52 as the
multiplier which will of course have adjusted the answer slightly.
9 (i) We are looking to derive ( )rxaq in terms of xand x
Use the Kolmogorov equations (assuming the transition intensities are
constant across a year age):
( )
( )
( )
( ) (1 )( )
r tt x
rx
aq et
aq e
+
+
=
= +
(ii) Similarly
( )( ) (1 )( )
dxaq e
+= +
Note that:
( )1 (( ) ( ) )r dx xaq aq e + + =
log(1 (( ) ( ) ))r dx xaq aq + = +
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 7
So
( ) (( ) ( ) )( log(1 (( ) ( ) )))
r r dx x xr d
x x
aq aq aqaq aq
= +
+
this can be rearranged to show
( )log(1 (( ) ( ) ))
( ) ( )
rr dxx xr d
x x
aqaq aq
aq aq = +
+
Given that:
1 ,rxq e=
then
( )
(( ) ( ) )1 1 (( ) ( ) )
rx
r dx x
aqr r d aq aq
x x xq aq aq + = +
In general this was poorly answered with most students making a limited inroad to the
question.
However, the question did not specify that constant forces must be assumed. So, a valid
alternative to part (i) is:
( )1 1
0 0 0
( ) ( ) exp
trx t x x t x r x r x taq ap dt dr dt + + + +
= = +
This makes no assumptions and provides an answer in the form asked for in the question, and
so would merit full marks. If constant forces are assumed, the above expression will turn into
the answer in the above solution.
For part (ii) a solution is only possible if some assumption is made. The following
alternatives could be valid:
(1) Assume dependent decrements are uniformly distributed over the year of age
With this assumption, deaths occur on average at age x + , so:
12 1
2 12
( ) ( ) ( )( ) ( )
( ) 1 ( )
r d r rx x xr r d r r x
x x x x x dx x
ad ad q aqq aq aq q q
al aq
+ = = + =
(This is covered by the Core Reading in Unit 8 Section 10.1.3.)
(2) Assume independent decrements are uniformly distributed over the year of age
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 8
This leads to two simultaneous equations:
( )
1
dd x
x rx
aqq
q=
and
( )
1
rr x
x dx
aqq
q=
which results in a quadratic equation inr
xq . (This is covered by the Core Reading Unit 8
Section 10.1.6.)
Whilst a full description has been given above to assist students, in reality those who
successfully attempted this question did assume constant forces.
10 First calculate ( ) and ( )d wx xaq aq
Age (x) Number of
employees( )xal
( )dxaq ( )wxaq
40 10,000 .00250 .01200
41 9,855 .00274 .01461
42 9,684
From this table and relationship
1 1( ) / (1 *( ) ) and ( ) / (1 *( ) )
2 2
d d w w w d x x x x x xq aq aq q aq aq= =
Calculate andd wx xq q
40d
q =.00250/(1.006) = .00252 and 41d
q = .00274/(1.00731) = .00276
40w
q =.01200/(1.00125) = .01201 and 41w
q = .01461/(1.00137) = .01463
Adjusting for the 75% multiplier of independent withdrawal decrements:
40
41
40
41
1 3( ) .00252* 1 * *.01201 .00251
2 4
1 3( ) .00276* 1 * *.01463 .00274
2 4
3 1( ) .01201* * 1 *.00252 .00900
4 2
3 1( ) .01463* * 1 *.00276 .01096
4 2
d
d
w
w
aq
aq
aq
aq
= = = =
= =
= =
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 9
Using the above data the Table can now be reconstructed
Age (x) Number of
employees
( )xal
Deaths
( )dxad
Withdrawals
( )wxad
40 10,000 (10000*.00251)=25.1 10000*.00900=90.0
41 9,884.9 (9,884.9*.00274)=27.1 9,884.9*.01096=108.3
42 9749.5
It should be noted that if more decimal places are used in the aq factors then the deaths at 40
become 25.0 so full credit was given for this answer also.
Because of the limited effect on the answer from the original table students were asked to
show the result to 1 decimal place. Many failed to do so and were penalised accordingly.
11 (i) Policy value at duration t of an immediate annuity payable continuously at arate of 1 per annum and secured by a single premium at age x is given by:
0
st x x t s x t V a e p ds
+ += =
0 0
s s
t x x t s x t s x t
V a e p ds e p dst t t t
+ + +
= = =
1ln( ) (ln ln )s x t s x t x t s x t x t s x t
s x t
p p l lp t t t
+ + + + + + + ++
= = = +
( )s x t s x t x t s x t p pt
+ + + + +
= +
0
( )s
t x s x t x t x t s
V e p dst
+ + + +
=
0
sx t x t s x t x t sa e p ds
+ + + + +=
00
s sx t x t s x t s x ta e p e p ds
+ + + +
=
1x t x t x ta a+ + += +
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 10
1x t t x t xV V+= +
(ii) Consider a short time interval (t, t + dt) then equation implies:
1 ( )t dt t x t t x t xV V V dt dt V dt o dt + + = + +
where
x t t xV dt+ = reserve released as a result of deaths in time interval
(t, t + dt)
1 dt = annuity payments made in time interval (t, t + dt)
t xV dt = interest earned on reserve over time interval (t, t + dt)
In general very poorly answered on what was a standard bookwork question.
12 (i) Annual premium Pfor the term assurance policy is given by:
1 1
[55]:10 [55]:5
[55]:10
25,000 25,000A AP
a
+=
where
1 1
[55]:10 [55]:525,000 25,000A A+
( )1/2 10 5[55] 10 [55] 65 [55] 5 [55] 6025,000 (1 ) ( ) ( )i A v p A A v p A= + +
8821.2612(0.38879 0.67556 0.52786)
9545.992925,000 1.019804
9287.2164(0.38879 0.82193 0.4564)9545.9929
=
+
( )25,495.10 (0.38879 0.32953) (0.38879 0.36496) 2118.39= + =
Therefore
2118.39257.46
8.228P = =
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 11
Net Premium Retrospective Reserves at the end of the fifth policy year is
given by:
[55]5 1
[55]:5 [55]:5
60
(1 ) 50,000l
i Pa Al
+
[ ]9545.9929
1.21665 257.46 4.59 50,000 1.019804 (0.38879 0.36496)9287.2164
=
41.71=
(ii) Explanation more cover provided in the first 5 years than is paid for by the
premiums in those years. Hence policyholder in debt at time 5, with size of
debt equal to negative reserve.
Disadvantage if policy lapsed during the first 5 years (and possibly longer),the company will suffer a loss which is not possible to recover from the
policyholder.
Possible alterations to policy structure
Collect premiums more quickly by shortening premium payment term or make
premiums larger in earlier years, smaller in later years
Change the pattern of benefits to reduce benefits in first 5 years and increase
them in last 5 years.
(iii) Mortality Profit = EDS ADS
Death strain at risk = 50,000 (42) = 50,042
59(1000 20) 50,042EDS q= 980 0.00714 50,042 350,154= =
8 50, 042 400, 336ADS= =
Total Mortality Profit = 350,154 400,336 = -50,182 (i.e. a mortality loss)
Quite reasonably answered by the well prepared student.
In (i) it should be noted that in this case the retrospective and prospective reserves are equal.
If the student recognised this, explicitly stated so and then did the easier prospective
calculation full marks were given. No credit was given for a prospective calculation without
explanation.
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 12
13
Annual premium 4000.00 Allocation % (1st yr) 95.0%
Risk discount rate 7.0% Allocation % (2nd yr) 100.0%
Interest on investments (1st yr) 5.5% Allocation % (3rd yr) 105.0%Interest on investments (2nd yr) 5.25% B/O spread 5.0%
Interest on investments (3rd yr) 5.0% Management charge 1.75%
Interest on non-unit funds 4.0% Surrender penalty (1st yr) 1000
Death benefit (% of bid value of units) 125% Surrender penalty (2nd yr) 500
Policy Fee 50
% prem
Initial expense 200 15.0%Renewal expense 50 2.0%
Expense inflation 2.0%
(i) Multiple decrement table:
xd
xq s
xq
45 0.001201 0.12
46 0.001557 0.06
47 0.001802 0.00
x ( )dxaq ( )
sxaq ( )ap 1( )t ap
45 0.001201 0.11986 0.878943 1.000000
46 0.001557 0.05991 0.938536 0.878943
47 0.001802 0.00000 0.998198 0.824920
Unit fund (per policy at start of year)
yr 1 yr 2 yr 3
value of units at start of year 0.000 3690.074 7693.641
Alloc 3800.000 4000.000 4200.000B/O 190.000 200.000 210.000
policy fee 50.000 50.000 50.000
Interest 195.800 390.604 581.682
management charge 65.727 137.037 213.768
value of units at year end 3690.074 7693.641 12001.554
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 13
Cash flows (per policy at start of year)
yr 1 yr 2 yr 3
unallocated premium + pol fee 250.000 50.000 150.000
B/O spread 190.000 200.000 210.000
expenses 800.000 131.000 132.020Interest 14.400 4.760 2.881
man charge 65.727 137.037 213.768
extra death benefit 1.108 2.995 5.407
surrender penalty 119.856 29.953 0.000
end of year cashflow 189.926 287.755 133.461
probability in force 1 0.878943 0.824920
discount factor 0.934579 0.873439 0.816298
expected p.v. of profit 133.280
premium signature 4000.000 3285.769 2882.069
expected p.v. of premiums 10167.837
profit
Margin 1.31%
(ii) Revised profit vector (309.781, 257.802, 133.461)
Revised profit signature (309.781, 257.492, 133.093)
Revised PVFNP = 289.515 + 224.904+ 108.643 = 44.032
Again most well prepared students made a good attempt at this question. The most common
error was to ignore dependent decrements.
Substantial credit was given to students who showed how they would tackle this question even
if they did not complete all the arithmetical calculations involved.
14
(i) Let Pbe the quarterly premium. Then:
EPV of premiums:
(4)
[35]:304 @6% 56.1408Pa P=
where
( )(4) 3030 [35][35]:30 [35]:30 3 18a a p v=
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 14
3 8821.261214.352 1 0.17411
8 9892.9151
=
14.0352=
EPV of benefits:
0.5 1.5 29 29.5[35] [35] [35]1 29
100,000( (1 ) ... (1 ) )q v q b v q b v+ + + + +
30 3030 [35]100,000 (1 )b v p+ +
where b= 0.0192308
( )
0.52 2 30 30
[35] [35] [35]0.5 0.5 1 29
100,000 (1.06)
(1 ) (1 ) ... (1 )(1 ) (1 ) q b v q b v q b vb b= + + + + + ++ +
30 30
30 [35]100, 000(1 )b v p+ +
0.5 1 30
30 [35][35]:30
100,000(1.06) @ 100,000 @
(1 )A i v p i
b = +
+
0.5100,000 (1.06) 8821.26120.32187 0.30832
(1 ) 9892.9151b
= +
8821.2612100,000 0.30832
9892.9151+
4,742.594 27,492.112 32,234.706= + =
where
1.061 0.04
1i
b= =
+
EPV of expenses (at 6%)
(4) (4)
[35]:30 [35]:1 [35]:30250 0.025 4 0.025 4 45 1P Pa Pa a = + + +
1 30
30 [35][35]:30500 250A v p+ +
250 0.025 56.1408 0.025 4 0.97857 45 13.352P P P= + + +
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 15
0.5 8821.2612 8821.2612500 1.06 0.18763 0.17411 250 0.174119892.9151 9892.9151
+ +
2.30566 906.322P= +
where
( )(4) [35][35]:1[35]:13
18
a a p v=
3 9887.20691 1 0.9434 0.97857
8 9892.9151
= =
Equation of value gives:
56.1408 32, 234.706 2.30566 906.322P P= + +
33,141.028615.60
53.8351P = =
(ii) Gross prospective policy value (calculated at 4%) is given by:
1prospective 1/2 5 (4) (4)
5 60 60:560:5 60:560:5
245,000(1 ) @ 245,000 @ 0.025 4 90 4
(1 )V i A i v p i Pa a Pa
b = + + + +
+
1 5
5 6060:51000 500A v p+ +
(4)1 5 6560:5 60:50.5 60:5
60
245,000@ 245,000 @ 90 0.975 4
(1.04)
lA i v i a Pa
l = + +
0.5 5 565 6560:5
60 60
1000 1.04 500l l
A v vl l
+ +
(4) 5 65
60:560:5 60
3 3 8821.2612where 1 4.55 1 0.82193 4.4678
8 8 9287.2164
la a v
l
= = =
4
601 060:5
60
1.04 465.9551and 1 0 @ 0.05017
1.04 9287.2164
td
i A il
+
= = = = =
0.5
245,0000.05017 245, 000 0.94983 90 4.55 0.975 4 615.60 4.4678
(1.04)= + +
( )
0.51000 1.04 0.82499 0.78069 500 0.78069+ +
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Subject CT5 (Financial Mathematics Core Technical) April 2010 Examiners Report
Page 16
12,052.954 232,708.35 409.5 10,726.473 45.177 390.345 234,880= + + + + =
Part (i) answered reasonably well. Students had more problems with (ii)
END OF EXAMINERS REPORT
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Faculty of Actuaries Institute of Actuaries
EXAMINATION
6 October 2010 (am)
Subject CT5 Contingencies
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 14 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
Faculty of ActuariesCT5 S2010 Institute of Actuaries
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CT5 S20102
1 Calculate:
(a) 20|10 [45]q
(b) 30 [45]:[50]p
Basis: AM92 Select[3]
2 Calculate 0.5p45.75using the Uniform Distribution of Deaths assumption.
Basis: AM92 Ultimate[3]
3 Calculate the single premium payable for a temporary reversionary annuity of12,000 per annum payable monthly in arrear to a female life currently aged 55 exacton the death of a male life currently aged 50 exact. No payment is made after 20years from the date of purchase.
Basis:
Rate of interest 4% per annumMortality of male life PMA92C20Mortality of female life PFA92C20Expenses Nil
[4]
4 A gymnasium offers membership for a three-year period at a fixed fee of 240 perannum payable monthly in advance. The contract may only be cancelled at a renewalanniversary. Monthly premiums cease immediately on the death of the member.
Calculate the expected present value of membership fees if the gymnasium sells 120memberships:
Basis:
Rate of interest 6% per annumRate of mortality 1% per annumProbability of renewal 80% at each anniversaryExpenses Nil
[5]
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CT5 S20103 PLEASE TURN OVER
5 A pension scheme provides an age retirement benefit of n/80ths of final pensionablesalary where n is total number of years of service. Final pensionable salary is theaverage salary in the three years before retirement. Normal retirement age is 65 andage retirement is only permitted between ages 60 and 65 exact.
A member of the pension scheme currently aged 45 exact has 12 years of service andtheir salary in the year before the valuation date was 25,000.
Give a formula for the expected cashflows between the 66thand 67thbirthdays as aresult of entitlement from this past service. [5]
6 Calculate:
(a)30:40
A
(b)30:40:20
a
Basis:
= 0.01 throughout for the life aged 30 now
= 0.02 throughout for the life aged 40 now
= 4% per annum[6]
7 A life insurance company issues a 10-year term assurance policy to a life aged 55
exact. The sum assured which is payable immediately on death is given by theformula:
50,000 (1 0.1 ) 0,1,2........,9t t + =
where tdenotes the curtate duration in years since the inception of the policy.
Level premiums are payable monthly in advance throughout the term of the policy oruntil earlier death.
Calculate the monthly premium for this policy using the following basis:
Mortality AM92 SelectInterest 4% per annumExpenses Nil
[6]
8 Describe the causal factors that explain observed differences in mortality andmorbidity. [6]
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CT5 S20104
9 The actuary advising a pension scheme has decided that the independent mortality inthe standard table for pension schemes (PEN) from page 142 of the Formulae andTables for Actuarial Examinations is no longer appropriate for that pension scheme.
Calculate the revised row of the service table for age 61, assuming that the revisedindependent mortality rate at that age is 80% of the previous independent mortalityrate.
[7]
10 Define the following terms, giving formulae and defining all notation used:
(a) Crude mortality rate(b) Indirectly standardised mortality rate
[7]
11 A life insurance company issues a four-year unit-linked policy to a male life. Thefollowing non-unit cash flows, tNUCF(t= 1,2,3,4), are obtained at the end of each
year tper policy in force at the start of the year t:
Year t 1 2 3 4
tNUCF 50.2 43.1 32.1 145.5
Assume that the annual mortality rate for the male life is constant at 1% at all ages.
(i) Show that the annual internal rate of return is 6%. [3]
The company sets up reserves in order to zeroise future negative cash flows. The rateof interest earned on non-unit reserves is 2.5% per annum.
(ii) Calculate the net present value of the profits after zeroisation using a riskdiscount rate of 6% per annum. [3]
(iii) Comment on the results obtained in (i) and (ii) above. [1][Total 7]
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CT5 S20105 PLEASE TURN OVER
12 A life insurance company issued a with profits whole life policy to a life aged 40exact on 1 January 2000. Under the policy, the basic sum assured of 50,000 andattaching bonuses are payable immediately on death. Level premiums are payableannually in advance under the policy until age 65 or earlier death.
The company declares simple reversionary bonuses at the start of each year includingthe first year and the bonus entitlement on the policy is earned immediately the bonus
is declared.
(i) Give an expression for the gross future loss random variable under the policyat the outset, defining symbols where necessary. [4]
(ii) Calculate the annual premium using the following assumptions:
Mortality AM92 SelectInterest 6% per annumBonus loading 2.5% per annum simpleInitial expenses 300
Renewal expenses 25 at the start of the second and subsequent policyyears while the policy is in force
Claim expenses 250[4]
On 31 December 2009, the policy is still in force. Bonuses declared to date total13,750.
(iii) Calculate the gross premium prospective reserve for the policy as at31 December 2009 using the following assumptions:
Mortality AM92 UltimateInterest 4% per annumBonus loading 3% per annum simpleRenewal expenses 35 at the start of each policy year while the policy is in
forceClaim expenses 250
[4][Total 12]
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CT5 S20106
13 On 1 January 2009, a life insurance company issued 10,000 joint life whole lifeassurance policies to couples. Each couple comprised one male life aged 60 exact andone female life aged 55 exact when the policy commenced. Under each policy, a sumassured of 100,000 is payable immediately on the death of the second of the lives todie.
Premiums under each policy are payable annually in advance while at least one of the
lives is alive.
The life insurance company uses the following basis for calculating premiums and netpremium reserves:
Mortality PMA92C20 for the malePFA92C20 for the female
Interest 4% per annumExpenses Nil
(i) Calculate the annual premium payable under each policy. [4]
During the calendar year 2009, there was one claim for death benefit, in respect of apolicy where both the male and the female life died during the year. In addition, therewere 20 males and 10 females who died during the year.
(ii) Calculate the mortality profit or loss for the group of 10,000 policies for thecalendar year 2009. [10]
[Total 14]
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CT5 S20107
14 A life insurance company issues four-year without profits endowment assurancepolicies to male lives aged 56 exact. The sum assured is 21,500 payable on maturityor at the end of the year of death if earlier. Premiums of 5,000 are payable annuallyin advance throughout the term of the policy.
The company holds net premium reserves for these policies, calculated using AM92Ultimate mortality and interest of 4% per annum.
Surrenders occur only at the end of a year immediately before a premium is paid. Thesurrender value is 70% of the net premium reserve calculated at the time the surrendervalue is payable.
The company uses the following assumptions in carrying out profit tests of thiscontract:
Rate of interest on cash flows 4% per annumMortality AM92 SelectSurrenders 10% of all policies still in force at the end of each of
the first, second and third policy yearsInitial expenses 600Renewal expenses 45 per annum on the second and subsequent
premium datesRisk discount rate 6% per annum
Calculate the expected profit margin for this contract. [15]
END OF PAPER
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INSTITUTE AND FACULTY OF ACTUARIES
EXAMINERS REPORT
September 2010 examinations
Subject CT5 Contingencies
Core Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim of
helping candidates. The questions and comments are based around Core Reading as theinterpretation of the syllabus to which the examiners are working. They have however givencredit for any alternative approach or interpretation which they consider to be reasonable.
T J BirseChairman of the Board of Examiners
December 2010
Institute and Faculty of Actuaries
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 2
1 (a) 20|10 [45] 65 75 [45]( ) /
(8,821.2612 6,879.1673) / 9,798.0837 0.198212
q l l l=
= =
(b) 75 8030 [45]:[50][45] [50]
6,879.1673 5,266.46040.380951
9,798.0837 9,706.0977
l lp
l l
= = =
Question generally done well.
2 .5 45.75 .25 45.75 .25 46*p p p=
.25 45.75 45 45.25* / (1 .75* ) .25*.001465 / (1 .75*.001465)q q q= =
.000367 by UDD=
.25 46 46.25* .25*.001622 .000406q q= = =
.5 45.75Hence (1 .000367)*(1 .000406) .999227p = =
In general question done well. However many students did not appreciate the split in line 1
above and attempted to apply formula directly.
3 Value of Single Premium is:
( )( ) ( ) ( ) ( )( )
( ) ( )
( )
(12) (12)
55:20 50:55:20
20 2055 20 55 75 50:55 20 50:55 70:75
20
20
12 1,000
13 13 13 1312,000 24 24 24 24
8784.95513 1312,000 18.210 10.93324 249917.623
8784.955 921316.909 24 9917.623
a a
v p v p
v
v
=
=
( )38.134 138.792 249941.92312,000((17.668 4.201) (16.367 3.099))
2,388
=
=Many students struggled with how to break down the monthly annuity functions into those
which could then utilise the Tables. However question generally done well by well prepared
students.
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 3
4 The value of 1 per annum payable monthly for 1 year is
(12) (12)(12)1 :1:1
. 11/ 24(1 . )x x xx xx v p v p+= =
:1Where 1
x =
Therefore
(12)
:11 11 /24(1 0.99 /1.06) 0.96973
x = =
The probability of reaching the beginning of each year is :
Year 1=1
Year 2 = 0.99*0.8 = 0.792 Year 3 = 0.792 * 0.792 = 0.6273
The value is therefore
2120 240 0.96973 (1 0.792 /1.06 0.6273 / (1.06) ) + + 64,388=
This question was overall done very poorly with few students realising that the key
element to the calculation involved a one year annuity due payable monthly.
5 The formula is:
1945 0.5 45 66 65 65 66
44 45 45 0.5 44 45 6515
12 ( ) 12 ( )25000 25000
80 ( ) 80 ( )t t t
tt
z r rl z r rl
s l rl s l rl
+ + + +
+ +=
+
Question done very poorly. Many students attempted to use annuity functions
whereas the question sought was a pure cash flow one.
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 4
6 (a)
______.04 .01 .02 .02 .01
030:40
.05 .07 .06 .07
0
.05 .06 .07
0
.05
{ (1 )*.01 (1 )*.02}
{.01*( ) .02*( )}
(.01* .02* .03* )
.01 .0 *
.05
t t t t t
t t t t
t t t
t
A e e e e e dt
e e e e dt
e e e dt
e
= +
= +
= +
=
.06 .07
0
2 .03* *
.06 .07
(1/ 5 1/ 3 3 / 7) .10476
t te e
+
= + =
(b)20 .04 .01 .02
30:40:20 0
20 .070
20.07
0
1.4
* *
1
.07
(1/ .07) / .07) 10.763
t t t
t
t
a e e e dt
e dt
e
e
=
=
=
= =
Question generally done well.
7 Let Pbe the monthly premium. Then equating expected present value of premiumsand benefits gives:
(12) 1 1[55]:10 [55]:10[55]:10
12 45000 5000( )Pa A IA= +
where
( )
( ) ( )( )
(12) 1010 [55][55]:10[55]:10
1 0.5 10 0.510 [55][55]:10 [55]:10
1 0.5 1010 [55][55]:10 [55]
11 8821.26121 8.228 0.458 1 .67556 8.056
24 9545.9929
1.04 1.04 0.68354 0.62427 0.06044
( ) 1.04
a a v p
A A v p
IA IA v p
= = =
= = =
=
( )( )10 10 [55] 65650.5
10
1.04 (8.58908 0.62427 7.89442 10 0.62427 0.52786) 0.3728
45000 0.06044 5000 0.372812 568.99
8.056
47.42
IA v p A
P
P
= =
+ = =
=
In general question done well by well prepared students.
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 5
8 Occupation either because of environmental or lifestyle factors mortality may bedirectly affected. Occupations may also have health barriers to entry, e.g. airlinepilots
Nutrition poor quality nutrition increases morbidity and hence mortality
Housing standard of housing (reflecting poverty) increases morbidity
Climate climate can influence morbidity and may also be linked to natural disaster
Education linked to occupation but better education can reduce morbidity, e.g. byreducing smoking
Genetics there is genetic evidence of a predisposition to contracting certainillnesses, even if this has no predictive capability
A straightforward bookwork question generally done well although not all students captured
the full range. All valid examples not shown above were credited.
Students who misunderstood the question and tried to answer using Class, Time, Temporary
Initial Selection were given no credit.
9 Use the formula
( )
(1 0.5(( ) ))
x
xx
aq
q aq
=
to derive the independent probabilities:
( ) (50 / 6548)0.00809
(1 0.5*((219 516) / 6548))(1 0.5(( ) ))
dd x
x dx
aqq
aq = = =
+
( ) (219 / 6548)0.03496
(1 0.5*((50 516) / 6548))(1 0.5(( ) ))
ii xx i
x
aqq
aq = = =
+
( ) (516 / 6548)0.080455
(1 0.5*((50 219) / 6548))(1 0.5(( ) ))
rr x
x rx
aqq
aq = = =
+
Then the revised 80%*0.00809 0.006472dxq = = then use the formula
1 1( ) (1 ( ...) ( . ...) ...)
2 3x x x x xaq q q q q = + + +
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 6
to derive dependent probabilities:
1 1( ) (1 ( ) ( . )) 0.0061046
2 3d d i r i r x x x x x xaq q q q q q= + + =
1 1( ) (1 ( ) ( . )) 0.0334465
2 3i i d r d r x x x x x xaq q q q q q= + + =
1 1( ) (1 ( ) ( . )) 0.0787948
2 3r r d i d ix x x x x xaq q q q q q= + + =
The resulting service table is:
lx dx ix rx
6,548 40 219 516
This question was done poorly. Many students appeared not to remember the derivation
process for multiple decrements etc. Some students wrote down the final table without
showing intermediate working. This gained only a proportion of the marks.
10 (a) Crude mortality rate = actual deaths / total exposed to risk
, ,
,
cx t x t
xc
x t
x
E m
E=
where
,c
x tE is central exposed to risk in population between agexandx+t
mx,tis central rate of mortality in population between agexandx+t
(b) Indirectly standardised mortality rate
, ,
,
, ,
, ,
s c sx t x t
xs c
x t
xc s
x t x t
xc
x t x t
x
E m
E
E m
E m
=
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 7
,s c
x tE is central exposed to risk in standard population between agexandx+t
,s
x tm is central rate of mortality in standard population between agexandx+t
This question generally done well. Other symbol notation was accepted provided it wasconsistent and properly defined.
11
Year t xq xp 1t xp tNUCF Profit Signature
1 0.01 0.99 1 50.2 50.22 0.01 0.99 0.99 43.1 42.7
3 0.01 0.99 0.9801 32.1 31.54 0.01 0.99 0.9703 145.5 141.2
(i) PV of profit @ 6%
2 3 450.2 42.7 31.5 141.2
47.4 38.0 26.4 111.8
0.0 6%
v v v v
IRR
= +
= +
= =
(ii) 2
32.1
31.31.025V= =
1 2 11.025 43.1 72.3xV p V V = =
revised cash flow in year 1 150.2 50.2 71.6 121.8xp V= = =
andNPVof profit = 121.8/1.06 + 111.8 = -3.1
(iii) As expected, theNPVafter zeroisation is smaller because the emergence ofthe non- unit cash flow losses have been accelerated and the risk discount rate
is greater than the accumulation rate.
Parts (i) and (iii) done well generally. In Part (ii) many students failed to develop the
formulae properly although they realised the effect in (iii).
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 8
12 (i) The gross future loss random variable is
( ) 40 4040 40
40 1 min( 1,25)50,000 1 1 ( )T T
K Kb K v I e ea fv Pa+ + + + + + +
Note: select functions also acceptable
where b is the annual rate of bonusI is the initial expensee is the annual renewal expense payable in the 2ndand
subsequent yearsf is the claim expenseP is the gross annual premiumK40(T40) is the curtate (complete) random future lifetime of a life
currently aged 40
(ii) The annual premium Pis given by
[ ] ( )[ ] [40]40[40]:25 4050,250 1,250 300 25( 1)Pa A IA a= + + +
0.5 0.513.29 50, 250 1.06 0.12296 1, 250 1.06 3.85489
300 25(15.494 1)
P = +
+ +
13.29 6361.402 4961.065 300 362.35P = + + +
901.79P =
(iii) The required reserve is
( )50 50 50:155064,000 1,500 35 901.79A IA a a+ +
0.5 0.564,000 1.04 0.32907 1,500 1.04 8.55929
35 17.444 901.79 11.253
= +
+
21, 477.560 13,093.196 610.54 10,147.84= + +
25,033.32=
In general question done well by well prepared students. In (i) credit also given if the
formulae included a limited term on the expense element although in reality this is unlikely.
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 9
13 (i) Let Pbe the annual premium. Then equating expected present value ofpremiums and benefits gives:
60 :55 60 :55100000m f m f Pa A=
where60 55 60 :5560 :55
15.632 18.210 14.756 19.086m f m f m fa a a a= + = + =
0.5 0.5
60 :55 60 :55 60 :551.04 1.04 (1 )
m f m f m f A A d a= =
0.51.04 (1 0.038462 19.086) 0.2711804= =
19.086 100000*0.2711804P =
1,420.83P = .
(ii) Reserves at the end of the first policy year:
Where both lives are alive:
0.5 61 :56
60 :55
100000 1.04 1m f
m f
a
a
0.5 15.254 17.917 14.356100000 1.04 1 1448.0115.632 18.210 14.756
+ = = +
Where the male life is alive only:
61 61
0.5
100000
0.04100000 1.04 1 15.254 1420.83 15.254 20475.94
1.04
m mA Pa
=
Where the female life is alive only:
56 56
0.5
100000
0.04100000 1.04 1 17.917 1420.83 17.917 6247.12
1.04
f fA Pa
=
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 10
Mortality Profit = Expected Death Strain Actual Death Strain
(a) Both lives die during 2009 = 1 actual claim.
Mortality Profit
( ) ( )( ) ( )
0.560 55
10,000 1 100000 1.04 1448.01
10, 000 0.002451 0.001046 1 100532.38 97954.99
m fq q=
= =
(b) Males only die during 2009 = 20 actual deaths (and therefore we needto change reserve from joint life to female only surviving).
Mortality Profit
( ) ( )( ) ( )
55 6010,000 20 6247.12 1448.01
10, 000 0.998954 0.002451 20 4799.11 21520.95
f mp q= = =
(c) Females only die during 2009 = 10 actual deaths (and therefore weneed to change reserve from joint life to male only surviving).
Mortality Profit
( ) ( )
( ) ( )
60 5510,000 10 20475.94 1448.01
10, 000 0.997549 0.001046 10 19027.93 8265.02
m fp q=
= =
Hence overall total mortality profit
97954.99 21520.95 8265.02 68,169.01= + + =
i.e. a mortality loss
Part (i) generally done well. Part (ii) was challenging and few students realised the full
implications of reserve change on 1st
death. Only limited partial credit was given ifstudents used only joint life situations.
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 11
14 Reserves required on the policy per unit sum assured are:
56:40 56:4
56:4
57:31 56:4
56:4
58:22 56:4
56:4
59:13 56:4
56:4
1 0
2.8701 1 0.23364
3.745
1.9551 1 0.47797
3.745
1.01 1 0.73298
3.745
aV
a
aV
a
aV
a
aV
a
= =
= = =
= = =
= = =
Multiple decrement table:
T[56] 1t
dq
+
[56] 1t
sq
+
[56] 1( )
t
daq
+
[56] 1( )
t
saq
+ [56] 1( ) tap + 1 [56]( )t ap
1 0.003742 0.1 0.003742 0.09963 0.896632 1.0000002 0.005507 0.1 0.005507 0.09945 0.895044 0.8966323 0.006352 0.1 0.006352 0.09936 0.894283 0.8025254 0.007140 0.0 0.007140 0.0 0.992860 0.717685
Probability in force [56] 1 [56] 1 [56] 1( ) (1 ) (1 )d s
t t tap q q+ + + =
The calculations of the profit vector, profit signature and NPV are set out in the tablebelow:
Policy
year Premium Expenses Interest
Death
claim
Maturity
claim
Surrender
claim
In force
cash flow
1 5000 600.00 176.00 80.45 0.00 350.31 4145.232 5000 45.00 198.20 118.40 0.00 715.38 4319.423 5000 45.00 198.20 136.57 0.00 1096.1613 3920.4754 5000 45.00 198.20 153.51 21346.49 0.00 16346.80
Policy
year
Increase in
reserves
Interest on
reserves Profit vector
Cum probability
of survival
Discount
factor
NPV
profit
1 4504.02 0.00 358.78 1.00000 0.943396 338.472 4174.53 200.93 345.82 0.89663 0.890000 275.963 3816.72 411.05 514.84 0.80253 0.839619 346.914 15759.07 630.36 42.63 0.71768 0.792094 24.24
Total NPV= 308.63
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Subject CT5 (Contingencies Core Technical) September 2010 Examiners Report
Page 12
The calculations of the premium signature and profit margin are set out in the tablebelow:
Policy year 1 2 3 4
Premium 5000.00 5000.00 5000.00 5000.00probability in force 1.00000 0.89663 0.80253 0.71768discount factor 1.00000 0.943396 0.890000 0.839619p.v. of premium signature 5000.000 4229.40 3571.22 3012.91=> expected p.v. of premiums 15813.53
profit margin = 2.0%
Many well prepared students were able to outline the process required without being totally
accurate on the calculation. Significant credit was awarded in such situation.
Many students failed to appreciate the multiple decrement element.
END OF EXAMINERS REPORT
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INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
26 April 2011 (am)
Subject CT5 Contingencies
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 13 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
CT5 A2011 Institute and Faculty of Actuaries
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CT5 A20112
1 Give a different example of selection shown by each of the following mortality tables:
(a) ELT15
(b) PMA92
(c) AM92
[3]
2 Calculate:
(a) 23 65p
(b) 10|5 60q
(c)65:10
s
Basis:
Mortality PMA92C20
Rate of interest 4% per annum [4]
3 Calculate ( )xIa
Basis: x = 0.02 for allx
= 4% per annum
[4]
4 Outline the benefits that are usually provided by a pension scheme on retirement dueto ill health. [5]
5 A pension scheme uses the following model to calculate probabilities, where thetransition intensities are = 0.05 and = 0.08.
Calculate:
(a) the dependent probability of retirement
(b) the independent probability of death from active service
using the Kolmogorov equations. [5]
Active Retired
Dead
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CT5 A20113 PLEASE TURN OVER
6 (i) Define uniform distribution of deaths [2]
(ii) Using the method in (i) above calculate 1.25q65.5 [4]
Basis:
Mortality ELT15(Males) [Total 6]
7 Explain how education influences morbidity. [6]
8 A life insurance company issues a with profits whole life assurance policy to a lifeaged 40 exact. The sum assured of 100,000 plus declared reversionary bonuses are
payable immediately on death. Level premiums are payable annually in advance to
age 65 or until earlier death.
A simple bonus, expressed as a percentage of the sum assured, is added to the policy
at the start of each year (i.e. the death benefit includes the bonus relating to the policy
year of death).
The following basis is used to price this policy:
Mortality AM92 Select
Rate of Interest 4% per annum
Initial expenses 300 plus 50% of the first annual premium, incurred at the
policy commencement date
Renewal commission 2.5% of each premium from the start of the second policy
year
Claim expense 350 at termination of the contract
Using the principle of equivalence, calculate the level simple bonus rate that can be
supported each year on this policy if the annual premium is 3,212. [6]
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CT5 A20114
9 A male life aged 52 exact and a female life aged 50 exact take out a whole lifeassurance policy. The policy pays a sum assured of 100,000 immediately on first
death. Premiums are payable for a period of five years, monthly in advance.
Calculate the monthly premium payable.
Basis:
Mortality PMA92C20 (male life), PFA92C20 (female life)Rate of interest 4% per annum
Expenses Nil [7]
10 Calculate the expected present value and variance of the present value of anendowment assurance of 1 payable at the end of the year of death for a life aged 40
exact, with a term of 15 years.
Basis:
Mortality AM92 Select
Rate of interest 4% per annumExpenses Nil [8]
11 A life insurance company issues a 4-year unit-linked endowment policy to a life aged61 exact under which level premiums of 2,500 are payable yearly in advance
throughout the term of the policy or until earlier death. In the first policy year 40% of
the premium is allocated to units, while in the second and subsequent policy years
110% of the premium is allocated to units. The unit prices are subject to a bid-offer
spread of 5%.
If the policyholder dies during the term of the policy, the death benefit of 10,000 or
the bid value of the units, whichever is higher, is payable at the end of the policy year
of death.
The policyholder may surrender the policy, in which case a value equal to a fixed
percentage of the total premiums paid on the policy is payable at the end of the policy
year of surrender. The percentage is based on the policy year of surrender as follows:
Policy year % of total premiums payable
as a surrender value
1 0
2 25
3 50
4 75
On maturity, 105% of the bid value of units is payable.
An annual management charge of 0.5% of the bid value of units is deducted at the end
of each policy year before death, surrender and maturity benefits are paid.
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CT5 A20115 PLEASE TURN OVER
The company uses the following assumptions in carrying out profit tests of this
contract:
Rate of growth on assets in the unit fund 4.25% per annum
Rate of interest on non-unit fund cash-flows 3.5% per annum
Independent rate of mortality AM92 Select
Independent rate of surrender 6% per annum
Initial expenses 325
Renewal expenses 74 per annum on the second
and subsequent premium dates
Initial commission 10% of first premium
Renewal commission 2.5% of the second andsubsequent years premiums
Risk discount rate 5.5% per annum
(i) Construct a multiple decrement table for this policy assuming that there is a
uniform distribution of both decrements over each year of age in the single
decrement table. [3]
(ii) Construct tables showing the growth of the unit fund and the non-unit fund.
Include all commissions in the non-unit fund. [7]
(iii) Calculate the profit margin for this policy on the assumption that the company
does not zeroise future expected negative cashflows. [3]
[Total 13]
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CT5 A20116
12 On 1April 1988, a life insurance company issued a 25-year term assurance policy to alife then aged 40 exact. The initial sum assured was 75,000 which increased by 4%
per annum compound at the beginning of the second and each subsequent policy year.
The sum assured is payable immediately on death and level monthly premiums are
payable in advance throughout the term of the policy or until earlier death.
The company uses the following basis for calculating premiums and reserves:
Mortality AM92 Select
Rate of interest 4% per annum
Initial commission 50% of the total premium payable in the first policy year
Initial expenses 400 paid at the policy commencement date
Renewal commission 2.5% of each premium from the start of the second policy
year
Renewal expenses 75 per annum, inflating at 4% per annum compound, at the
start of the second and subsequent policy years (the renewal
expense quoted is as at the start of the policy and the
increases due to inflation start immediately)
Claim expense 300 on termination (the claim expense is fixed over the
duration of the policy)
(i) Show that the monthly premium for the policy is approximately 56. [10]
(ii) Calculate the gross premium prospective reserve as at 31 March 2011. [6][Total 16]
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CT5 A20117
13 (i) Explain, including formulae, the following expressions assuming that the sumassured is payable at the end of the year of death:
death strain at risk
expected death strain
actual death strain [6]
(ii) A life insurance company issues the following policies:
25-year term assurances with a sum assured of 200,000
25-year endowment assurances with a sum assured of 100,000
The death benefit under each type of policy is payable at the end of year of
death.
On 1 January 2000, the company sold 10,000 term assurance policies to male
lives then aged 40 exact and 20,000 endowment assurance policies to male
lives then aged 35 exact. For each type of policy, premiums are payable
annually in advance.
During the first ten years, there were 145 actual deaths from the term
assurance policies written and 232 actual deaths from the endowment
assurance policies written.
(a) Calculate the death strain at risk for each type of policy during 2010.
During 2010, there were 22 actual deaths from the term assurance policies and
36 actual deaths from the endowment assurance policies.
Assume that there were no lapses/withdrawals on each type of policy during
the first eleven years.
(b) Calculate the total mortality profit or loss to the office in the year 2010.
(c) Comment on the results obtained in (b) above.
Basis:
Mortality AM92 Ultimate
Rate of interest 4% per annum
Expenses Nil [11]
[Total 17]
END OF PAPER
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INSTITUTE AND FACULTY OF ACTUARIES
EXAMINERS REPORT
April 2011 examinations
Subject CT5 ContingenciesCore Technical
Introduction
The attached subject report has been written by the Principal Examiner with the aim ofhelping candidates. The questions and comments are based around Core Reading as theinterpretation of the syllabus to which the examiners are working. They have however givencredit for any alternative approach or interpretation which they consider to be reasonable.
T J BirseChairman of the Board of Examiners
July 2011
Institute and Faculty of Actuaries
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 2
1 (a) Time selection because it is based on a period of three calendar years(b) Class selection applies only to male pensioners(c) Temporary initial selection as there are select rates
Other valid answers acceptable
This question was generally done well. However some students did not supply different
selection types for each part and this was penalised.
2 (a) 8823 6565
3534.0540.366307
9647.797
lp
l= = =
(b) 70 7510|5 6060
( ) (9238.134 8405.160)0.084771
9826.131
l lq
l
= = =
(c)
( )( )
10 10 1065:10 65 10 65 75
65:1010 65 10 65
10 10
(1 ) (1 ) ( )
8,405.160(1.04) (13.666 (1.04 ) 9.456)9,647.797
8,405.1609,647.797
1.48024 (13.666 0.67556 0.87120 9.456) / 0.87120
13.764
i a i a v p as
p p
+ + = =
=
=
=
This question was generally done well for parts (a) and (b) but students struggled more withpart (c).
31 2 3
0 1 2
.04 .02 .06
0.06 .06 2 .06 31
.06 2 .06
( ) 2 3 .......
Now * throughout.
Hence
( ) (1 2 3( ) 4( ) .........) at force of interest 6%
(1/(1 )) ((1 ) / .06)
294.
t t tx t x t x t x
x
x
Ia v p dt v p dt v p dt
vp e e e
Ia e e e a
e e
= + + +
= =
= + + + +
=
=
8662 0.970591
286.19
=
This question was not done well. The majority of students failed to realise that the increasing
function I was not continuous, although the payment is continuous. Instead most attempted
to compute0
( ) tx t xIa tv p dt
= . Only minimal credit was given for this.
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 3
4 Schemes usually allow members to retire on grounds of ill-health and receive apension benefit after a minimum length of scheme service.
Benefits are usually related to salary at the date of ill-health retirement in similar waysto age retirement benefits.
However, pensionable service is usually more generous than under age retirementwith years beyond those served in the scheme being credited to the member e.g. actualpensionable service subject to a minimum of 20 years, or pensionable service thatwould have been completed by normal retirement age.
A lump sum may be payable on retirement and a spouse pension on death afterretirement.
Other valid points were credited. Generally this bookwork question was done well.
5 The Kolmogorov equations in this case are:
( )
( )
( )
( )
r tt x
d tt x
aq et
aq et
+
+
=
=
For the case where t= 1 the solution for the dependent probability of retirement is:
( )( ) (1 )rxaq e +=
+
Hence the dependent probability of retirement is
(0.05 0.08)0.08( ) (1 )0.08 0.05
0.07502
rxaq e
+= +
=
The formula for the independent probability of death is
1dxq e=
Hence the independent probability of death is:
0.051 0.04877dxq e= =
Generally this question was completed satisfactorily by well prepared students.
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 4
6 (i) The definition of the uniform distribution of deaths (UDD) is .s x xq s q=
(alternatively t x x t p + is constant).
(ii) We have
1.25 65.5 0.5 65.5 0.75 66 p p p=
0.5 65.5 0.5 65.5 65 65(1 ) (1 (0.5 / (1 0.5 ))) by UDDp q q q= = = (1 ((0.5 0.02447) / (1 0.5 0.02447))) = 0.98761
0.75 66 0.75 66 661 1 0.75 1 0.75 0.02711p q q= = = = 0.97967
Hence
1.25 65.5
1.25 65.5 1.25 65.5
= 0.98761 0.97967 0.96753
1 1 0.96753
= 0.03247
p
q p
=
= =
A straightforward question that was generally done well.
7 Education influences the awareness of a healthy lifestyle, which reduces morbidity.
Education includes formal and informal processes, such as public health awarenesscampaigns.
Shows in:
Increased income Better diet Increased exercise Better health care Reduced alcohol and tobacco consumption
Lower levels of illicit drug use Safer sexual practices
Some effects are direct (e.g. drug use); some are indirect (e.g. exercise)
Students generally scored on a range of points but in most cases did not write enough of them
to gain all the marks.
Students who mentioned over indulgence risks for the better educated were given credit.
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 5
8 Let bbe the simple bonus rate (expressed as a percentage of the sum assured). Thenthe equation of value at 4% p.a. interest is (whereP = 3,212):
( ) ( )
[40] [40][40]:25
0.5 0.5
(.975 0.025) (100,000 350) 1,000 ( ) 300 0.5
(.975 15.887 0.025)
(100,000 350) 1.04 0.23041 1,000 1.04 7.95835 300 0.5
49,833.6179 23,579.5423 8,115.9564 1,906
24,348.
P a A b IA P
P
b P
b
b
+ = + + + +
+ =+ + + +
= + +
=
07563.00
8,115.9564=
i.e. a simple bonus rate of 3% per annum
Generally done well although some students treated b as not vesting in the first year.
9 Value of benefits using premium conversion
1/252:50 52:50
1/252:50
100,000 100,000 (1.04)
100,000 (1.04) (1 (0.04 /1.04) )
101,980.4 (1 0.0384615 17.295)
34,143.89
A A
a
=
=
=
=
Value of monthly premium ofP
(12) (12) (12)5 57:5557:5552:5052:50:5 52:50
(12)52:5052:50
(12)57:5557:55
5 57:55
52:50
12 12
11 / 24 17.295 0.458 16.837
11 / 24 15.558 0.458 15.100
(0.82193 9,880.196 9,917
lP P v
l
lv
l
=
= = =
= = =
=
.623) / (9,930.244 9,952.697)
0.81491
=
Hence (12)52:50:5
12 12 (16.837 0.81491 15.100) 54.3823P P P= =
Therefore:
34,143.89 / 54.3823 627.85P= =
There was an anomaly in this question in that it was not fully clear that the premium paying
period ceased on 1stdeath within the 5 year period. Even though the vast majority of students
who completed this question used the above solution a small minority used
(12)
512Pa
i.e.ignoring the joint life contingency. This was credited.
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 6
None the less many students struggled with this question
10 Expected present value is[40]:15
A where
1 1[40]:[40]:15 [40]:15 15
A A A= +
15 15[40] 15 [40] 55 15 [40]A v p A v p= +
9,557.8179 9,557.81790.23041 0.55526 0.38950 0.55526
9,854.3036 9,854.3036
= +
0.23041 0.20977 0.53855= +
0.55919=
Variance
2 2[40]:15 [40]:15
( )A A=
2 2 1 2 1[40]:[40]:15 [40]:15 15
A A A= +
2 2 15 2 2 15[40][40] 15 55 15 [40]( ) ( )A v p A v p= +
9,557.8179 9,557.81790.06775 0.30832 0.17785 0.308329,854.3036 9,854.3036
= +
0.06775 0.05318 0.29904
0.31361
= +
=
2So variance 0.31361 0.55919 0.000917= =
Note answers are sensitive to number of decimal places used.
Question done well by well prepared students. Many students failed to realise that the
endowment function needed to be split into the term and pure endowment portions.
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 7
11 Summary of assumptions:
Annual premium 2,500.00 Allocation % (1st yr) 40%Risk discount rate 5.5% Allocation % (2nd yr +) 110%Interest on investments 4.25% Man charge 0.5%Interest on sterling provisions 3.5% B/O spread 5.0%
Minimum death benefit 10,000.00 Maturity benefit 105%
% prm TotalInitial expense 325 10.0% 575Renewal expense 74 2.5% 136.5
(i) Multiple decrement table:
x dxq sxq
61 0.006433 0.0662 0.009696 0.0663 0.011344 0.0664 0.012716 0.06
x ( )dxaq ( )
sxaq ( )ap 1( )t ap
61 0.006240 0.05981 0.933953 1.000000
62 0.009405 0.05971 0.930886 0.93395363 0.011004 0.05966 0.929337 0.86940464 0.012335 0.05962 0.928047 0.807969
(ii) Unit fund (per policy at start of year)
yr 1 yr 2 yr 3 yr 4
value of units at start of year 0.00 985.42 3,732.08 6,581.15alloc 1,000.00 2,750.00 2,750.00 2,750.00B/O 50.00 137.50 137.50 137.50
interest 40.37 152.91 269.65 390.73management charge 4.95 18.75 33.07 47.92value of units at year end 985.42 3,732.08 6,581.15 9,536.46
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 8
Non-unit fund (per policy at start of year)
yr 1 yr 2 yr 3 yr 4
unallocated premium 1,500.00 250.00 250.00 250.00
B/O spread 50.00 137.50 137.50 137.50expenses/commission 575.00 136.50 136.50 136.50interest 34.12 8.72 8.72 8.72man charge 4.95 18.75 33.07 47.92extra death benefit 56.25 58.95 37.62 5.72extra surrender benefit 58.94 148.20 168.91 121.41extra maturity benefit 0.00 0.00 0.00 442.51end of year cashflow 1,016.76 149.71 93.36 536.62
(iii)
probability in force 1 0.933953 0.869404 0.807969discount factor 0.947867 0.898452 0.851614 0.807217
expected p.v. of profit 419.03
premium signature 2,500.00 2,213.16 1,952.79 1,720.19expected p.v. of
premiums 8,386.15 profitmargin 5.00%
Credit was given to students who showed good understanding of the processes involved even
if the calculations were not correct. Generally well prepared students did this question quite
well.
12 (i) LetPbe the monthly premium. Then:
EPV of premiums:
(12)[40]:25
12 @ 4% 186.996Pa P=
where
( )(12) 2525 [40][40]:25 [40]:2511
124
a a p v=
11 8821.261215.887 1 0.37512
24 9854.3036
=
15.583=
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 9
EPV of benefits:
0.5 1.5 24 24.5[40] [40] [40]1 24
75,000( (1 ) ... (1 ) )q v q b v q b v+ + + + +
where b= 0.040.5 0.5
1 / 25 /[40] 25 [40] 65[40]:25
75,000 (1 ) 75,000 (1 )@ @
(1 ) (1 )
i iA i A p v A i
b b
+ + = = + +
0.5
75,000 8821.26121 1 1
(1.04) 9854.3036
=
7709.6880=
where
/ /1.04 1 0.00 i.e. 0%1
i ib
= = =+
EPV of expenses (at 4% unless otherwise stated
(12) (12) @0%
[40]:25 [40]:1 [40]:25
1
[40]:25
0.5 12 400 0.025 12 0.025 12 75 1
300
6 400 0.025 12 15.583 0.025 12 0.982025 75 23.27542300 0.05422
6 400 4.6749 0.2946 1745.6558 16.26
P Pa Pa a
A
P P P
P P P
= + + +
+
= + + + +
= + + + +
6
10.3803 2161.9218P= +
where
( )(12) [40][40]:1[40]:111
124
11 9846.53841 1 0.96154 0.98202524 9854.3036
a a p v=
= =
( )@0% 64[40] 1 64 [40] 64[40]:25[40] [40]
11 ....
8934.877139.071 17.421 23.27541
9854.3036
la l l e e
l l+ = + + =
= =
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 10
1 0.5 1 0.5 25 65[40]:25 [40]:25 [40]:25
[40]
0.5
1.04 1.04
8821.26121.04 0.38896 0.37512 0.05422
9854.3036
lA A A v
l
= =
= =
Equation of value gives:
186.996 7709.6880 10.3803 2161.9218
9871.609855.89
176.6157
P P
P
= + +
= =
(ii) Gross prospective policy value at t= 23 (calculated at 4%) is given by:
[ ] [ ]
[ ]
[ ]
(12)prospective 23 0.5 0.563 63 64 63 63 64 63:2
(12)2363 63:2
75,000 (1.04) (1.04) 300 0.025 12
75 (1.04) 1 (1.04) 12
184,853.66 0.98058 0.011344 (1.04) 0.988656 0.012716 0.96154
300 0.9
V v q p q v v q p q v Pa
p v Pa
= + + + +
+ +
= +
+
[ ]
[ ]
(12) 2 65
63:263:2 63
8058 0.011344 0.988656 0.012716 0.96154 0.025 12 55.89 1.90629
184.854 1 (1.04) 0.988656 0.96154 12 55.89 1.90629
11 11 8821.2612
where 1 1.951 1 0.9245624 24 9037.3973
la a v
l
+ +
+ +
= =
1.90629
4,335.0628 6.8932 31.9628 367.6104 1,278.5106
3,463.02
=
= + + +
=
This question was generally not done well especially part (ii). In part (i) although it was
commonly recognised that a resultant rate of interest of 0% emerged students did not often
seem to know how to progress from there.
13 (i) The death strain at risk for a policy for year t + 1 (t = 0, 1, 2) is the excess ofthe sum assured (i.e. the present value at time t + 1 of all benefits payable ondeath during the year t + 1) over the end of year provision.
i.e. DSAR for year t + 1 1tS V+=
The expected death strain for year t + 1 (t = 0, 1, 2) is the amount that thelife insurance company expects to pay extra to the end of year provision forthe policy.
i.e. EDS for year t + 1 1( )tq S V+=
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 11
The actual death strain for year t+ 1 (t= 0, 1, 2) is the observed value att+1 of the death strain random variable
i.e. ADS for year t + 1 1( )tS V+= if the life died in the year tto t+1= 0 if the life survived to t + 1
Note: Full credit given if definition of death strain is given for a block ofpolicies rather than for a single policy as per above.
(ii) (a) Annual premium for endowment assurance with 100,000 sum assuredgiven by:
35:2535:25
100,000 100,0000.38359 2,393.40
16.027EAP A
a= = =
Annual premium for term assurance with 200,000 sum assured givenby:
140:25
40:25
1 2525 4040:25 40:25
200,000
where
8,821.26120.38907 0.37512 0.38907 0.33573 0.05334
9,856.2863
200,000 0.05334671.62
15.884
TA
TA
AP
a
A A v p
P
=
=
= = =
= =
Reserves at the end of the 11th year:
for endowment assurance with 100,000 sum assured given by:
11 46:14 46:14100,000
100,000 0.58393 2,393.40 10.818
58,393.0 25,891.8 32,501.2
EA EAV A P a=
=
= =
for term assurance with 200,000 sum assured given by:
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Subject CT5 (Contingencies Core Technical) Examiners Report, April 2011
Page 12
111 51:14 51:14
1 1414 5151:14 51:14
11
200,000
where
8,821.26120.58884 0.57748 0.58884 0.52583 0.063019,687.7149
200,000 0.06301 671.62 10.69
12,602.0 7,179.6 5,422.4
TA TA
TA
V A P a
A A v p
V
=
=
= = =
=
= =
Therefore, sums at risk are:
Endowment assurance: DSAR = 100,000 32,501.2 = 67,498.8
Term assurance: DSAR = 200,000 5,422.4 = 194,577.6
(b) Mortality profit = EDS ADS
For endowment assurance
4519768 67, 498.8 19768 0.001465 67, 498.8 1, 954, 773.3
36 67,498.8 2,429,956.8
EDS q
ADS
= = =
= =
mortality profit = 475,183.5 (i.e. a loss)
For term assurance
509,855 194,577.6 9,855 .002508 194,577.6 4,809, 246.1
22 194,577.6 4,280,707.2
EDS q
ADS
= = =
= =
mortality profit = 528,538.9Hence, total mortality profit = 528,538.9 475,183.5 = 53,355.4
(c) Although there is an overall mortality profit in 2010, the actual numberof deaths for the endowment assurances is approximately 25% higherthan expected, which is a concern. Further investigation would berequired to determine reasons for poor mortality experience for theendowment assurances, e.g. there may have been limited underwritingrequirements applied to this type of contract when they were written.
Generally (a) was done well. The most common error in (b) was to assume reserves at 10
years rather than 11. On the whole well prepared students coped with (b) well. Many
students did not attempt (c) or at best gave a somewhat sketchy answer.
END OF EXAMINERS REPORT
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INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
4 October 2011 (am)
Subject CT5 Contingencies
Core Technical
Time allowed: Three hours
INSTRUCTIONS TO THE CANDIDATE
1. Enter all the candidate and examination details as requested on the front of your answer
booklet.
2. You must not start writing your answers in the booklet until instructed to do so by the
supervisor.
3. Mark allocations are shown in brackets.
4. Attempt all 14 questions, beginning your answer to each question on a separate sheet.
5. Candidates should show calculations where this is appropriate.
Graph paper is NOT required for this paper.
AT THE END OF THE EXAMINATION
Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this
question paper.
In addition to this paper you should have available the 2002 edition of the Formulae
and Tables and your own electronic calculator from the approved list.
CT5 S2011 Institute and Faculty of Actuaries
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CT5 S20112
1 Calculate:
(a) 10|1 [50]q
(b) 10 [60] 1p +
(c) (12)[40]:20
a
Basis:Mortality AM92Rate of interest 6% per annum [3]
2 Calculate 0.5 75.25q using the assumption of a constant force of mortality.
Basis:Mortality AM92 [3]
3 In a special mortality table with a select period of one year, the followingrelationships are true for all ages:
0.5 [ ]
0.5 [ ] 0.5
0.25
0.45
x x
x x
q q
q q+
=
=
Expressp[x] in terms ofpx . [3]
4 A term assurance contract with a term of 20 years pays a sum assured of 1immediately on death to a life now aged 30 exact.
Calculate the expected value and variance of this contract.
Basis:Mortality AM92 UltimateRate of interest 4% per annum [4]
5 (a) Write down the random variable form of 1:x yA .
(b) Calculate 1:x yA on the following assumptions: