CT5 CMP upgrade 2014 - Actuarial Education Company Upgrade/CT5-PU-14.pdf · CT5: CMP Upgrade 2013/14 Page 1 ... The company assumes that mortality will ... An alternative solution

CT5: CMP Upgrade 2013/14 Page 1

The Actuarial Education Company © IFE: 2014 Examinations

Subject CT5

CMP Upgrade 2013/14

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 2013 study material to make it suitable for study for the 2014 exams. It includes replacement pages and additional pages where appropriate. Alternatively, you can buy a full replacement set of up-to-date Course Notes at a significantly reduced price if you have previously bought the full price Course Notes in this subject. Please see our 2014 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 2014 exams.

1 Changes to the Syllabus objectives and Core Reading

1.1 Syllabus objectives No changes have been made to the syllabus objectives.

1.2 Core Reading No changes have been made to the Core Reading.

Page 2 CT5: CMP Upgrade 2013/14

© IFE: 2014 Examinations The Actuarial Education Company

2 Changes to the ActEd Course Notes Chapter 5 P40 In the example, the wording of parts (i) and (ii) has been changed so that it reads “portfolio” instead of “policy”. This now reads: Example A life insurance company has a portfolio of 10,000 single premium one-year term assurances. For each policy, there is a sum assured of $50,000 payable at the end of the year if the policyholder dies during the year. The company assumes that mortality will be 1% pa. (i) Calculate the expected death strain for this portfolio. (ii) Given that 89 people die during the year, calculate the actual death strain and

hence the mortality profit or loss for this portfolio. Chapter 7 P19 In the first line of the expression for the expected present value of the expenses, 2 xp should be 2 [60]p . So the expression should read:

( )2 2[60] 2 [60]EPV expenses 0.015 120 1 1.019231 1.019231P v p v p= + + + +

Chapter 9 P25 Self-assessment question 9.20 has been reworded and the solution to this question has been changed. Replacement pages are provided.



Chapter 10 P43 In the solution to Question 10.5, the calculation of the cost of death benefits features an incorrect figure. It should read: ( )1

12 40,000 1,390.35 0.58 0.000937 £1.75¥ - ¥ ¥ = As additional clarification, we should state that this calculation assumes that the death payments are made at the end of the month, after all charges have been deducted.



3 Changes to the Q&A Bank Solution 1.9 P6 Part (ii) has been corrected to read: (ii) is true. Since mortality is very light over the age range 25 to 35, |25:10A will be very

slightly more than 10 0.386v = . Question 2.21 P7 This question has been replaced. Replacement pages for the question and the solution are provided. Question 2.32 P13 This question is new. Replacement pages for the question and the solution are provided.



Solution 4.8 P5 This solution has been enhanced. It now reads: Salaries change only on the 1 July of each year, so a rate of £30,000 at 1 January 2005 must mean that the person is paid an amount of £30,000 from 1 July 2004 to 1 July 2005. This corresponds to 30.75s . Earnings from 1 July 2005 to 1 July 2006 correspond to 31.75s . So the answer is:

31.75

30.75

30,000 12

ss

È ˘+Í ˙

Î ˚ [2]

Alternative solution We are given a salary rate, which is no use: we need to convert it into a salary amount. Salaries change only on the 1 July of each year, so a rate of £30,000 at 1 January 2005 must mean that the person is paid an amount of £30,000 from 1 July 2004 to 1 July 2005. This corresponds to 30.75s . Earnings over the year 2005 correspond

to 31.25s . So the answer is 31.25

30.7530,000

ss

. [2]



4 Changes to the X assignments Solution X1.7 An alternative solution using life table functions has been added. Replacement pages are provided.

Solution X1.12 An alternative solution using life table functions has been added. Replacement pages are provided.

Question X2.8 This question has been replaced. Replacement pages are provided. Question X3.12 This question has been replaced. Replacement pages are provided. Question X3.13 This question has been replaced. Replacement pages are provided.



Solution X4.1 This solution has been enhanced. Replacement pages are provided. Solution X4.2 This solution has been updated. Replacement pages are provided.



5 Other tuition services

In addition to this CMP Upgrade you might find the following services helpful with your study.

5.1 Study material We offer the following study material in Subject 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 2014 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)

• CT5 online classroom. For further details on ActEd’s tutorials, please refer to our latest Tuition Bulletin, which is available from the ActEd website at www.ActEd.co.uk.

5.3 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 2014 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.



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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.


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Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the

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These conditions remain in force after you have finished using the course.

CT5-09: Contingent and reversionary benefits Page 25


Question 9.20

Can you find a way to derive the formula on the right-hand side of this expression, ie n

n yx y x y na v p a , by considering payments made at time t (using a similar approach

as used for the Type 2 annuity above)?

Example Ralph and Betty are both aged 70 exact. Upon Betty’s death, Ralph will receive £20,000 pa payable annually in advance starting from the end of the year of Betty’s death and ceasing on Ralph’s death. Ralph may receive a maximum of 20 payments. Ralph’s mortality follows PMA92C20, Betty’s mortality follows PFA92C20 and the interest rate for all future years is 4%i = pa. Calculate the EPV of this benefit to Ralph. Solution We have:

2020 70 70:7070:9070:20

20 202,675.203 2,675.20311.562 1.04 4.527 1.04 4.339 9.766

9,238.134 9,238.134

1.771154

f mm ma v p a a

- -

+ -

= - ¥ ¥ + ¥ ¥ -

=

So, the EPV to Ralph is: £20,000 1.77115 £35,423¥ =

Type 5 – an annuity payable to (y) on the death of (x) and guaranteed for n years The expected present value of this benefit is:

1: | ++ n

x y n y x y nnA a v p a

Page 26 CT5-09: Contingent and reversionary benefits


Question 9.21

Prove this result.

The first term in the above expression is the expected present value of the guaranteed benefit, which is paid to ( )y following the death of ( )x . The second term is the

expected present value of the benefit paid to ( )y once the n -year guarantee period is

up. This second term can also be found by considering payments made at time t that are n or more years after (x)’s death. So, payments will be made at time t if:

(y) is alive (probability t yp )

(x) died before time t n (probability t n xq )

So we have:

tt y t n x

n

EPV v p q dt

noting that there can be no payment when t n . Substituting s t n :

0

n sn s y s xEPV v p q ds

Taking out constant factors involving n we get:

0

n sn y s y n s x

nn y x y n

EPV v p v p q ds

v p a

as before.

CT5-09: Contingent and reversionary benefits Page 51


Pulling a factor of nn xyv p outside the integral, we get:

( )

:0 0

:

•

+ + + + + + + += =

+ + +

= - -

= - - -

Ú Ún

t n sy t t xy x t y xy n xy y s n s x n y n x s n

t s

ny xy n xy y n x n y n

v a p dt a a v p v a p ds

a a v p a a

m m

Solution 9.18 Left hand side The only restriction from the normal whole life version is that payments following the death of (x) after n years will not be made. We can therefore calculate the payments we need by simply integrating to n rather than to infinity. Right hand side We need to deduct all annuity payments, currently included in x ya , that relate to the

death of (x) after n years with (y) alive at the time of (x)’s death. (Note that the death of (x) after n years, with (y) dead at the time of (x)’s death, is already excluded from x ya ).

The required deduction therefore equates to a reversionary annuity from time n, with both (x) and (y) alive at that point, appropriately discounted to the start of the contract. Solution 9.19 The annuity begins at time t, which is the instant of (x)’s death. Hence the factor

t x x tp dt .

The annuity is payable to life (y) for life thereafter, but for a maximum of n years. Hence:

(y) has to be alive at time t (probability t yp )

the expected present value of the temporary annuity from that time is :y t n

a

and we need to discount this value to time zero (using tv ).

Page 52 CT5-09: Contingent and reversionary benefits


Solution 9.20 First, consider an annuity that starts n years after the death of (x) and is payable to (y). This annuity will be in payment at time t + n if:

(x) has died before time t and

(y) is alive at time t + n. So, the expected present value of this annuity is:

0

t nt x t n yv q p dt

Now, the Type 4 annuity is equal to a reversionary annuity less this one above. So, the expected present value of the Type 4 annuity is:

0 0

0

t t nt x t y t x t n y

n tn y t x t y nx y

nn yx y x y n

v q p dt v q p dt

a v p v q p dt

a v p a

Solution 9.21 We can write the expected present value of this benefit in terms of an integral as follows:

( )0

•+ + + ++Ú t n

t x x t t y n y t y t nnv p p a v p a dtm

Here we’re thinking about ( )x dying at time t , with ( )y still alive. Then ( )y receives

an annuity that is guaranteed for n years. Furthermore, if ( )y is still alive at time

+t n , he then receives an annuity throughout his remaining lifetime. If we multiply out the brackets, then the first term is just:

1:0

•+ =Ú t

t x x t t y x yn na v p p dt a Am

CT5: Q&A Bank Part 2 – Questions Page 7


2 Exam-style Questions

Question 2.20

On 1 January 2009, a life insurance company sold a number of 10-year pure endowment policies, each with a benefit amount of £40,000, to lives then aged 30. Level premiums are payable annually in advance. (i) Calculate the annual premium. [3] (ii) On 1 January 2010, there were 50 of these policies still in force. During 2010,

one policyholder died. Calculate the company’s mortality profit for 2010. [5] Assume AM92 Select mortality and 4% pa interest. Ignore expenses. [Total 8] Question 2.21

An insurer issues a combined term assurance and annuity contract to a life aged 35. Level premiums are payable monthly in advance for a maximum of 30 years. On death before age 65 a benefit is paid immediately. The benefit is £200,000 on death in the first year of the contract, £195,000 on death in the second year, £190,000 on death in the third year, etc, with the benefit decreasing by £5,000 each year until age 65. No benefit is payable on death after age 65. On attaining age 65 the life receives an annuity of £10,000 pa payable monthly in arrears. Calculate the monthly premium on the basis of: Mortality: up to age 65: AM92 Select over age 65: PFA92C20 Interest: 4% pa Expenses: none [8]

Page 8 CT5: Q&A Bank Part 2 – Questions


Question 2.22

A life office sold a portfolio of 10,000 term assurance policies on 1/1/2003. The policies had a term of 2 years, with premiums paid annually in advance and were sold to a group of males aged 60 exactly at that date. Each policy had a sum assured of £50,000, which is payable at the end of the year of death. The company prices the product assuming AM92 Ultimate mortality. (i) The same premium was charged for each year. The premium was calculated by

setting the expected present value of the premiums equal to the expected present value of the benefit payments plus 10% of the standard deviation of this present value. Calculate the premium assuming 4% pa interest. [7]

(ii) During the first policy year 75 policyholders died. Calculate the net premium

reserve at the end of 2003 and hence the mortality profit for the portfolio for calendar year 2003. [5]

(iii) A director of the company has calculated the profit of the business as premiums

received less sum assured paid on death less the net premium reserve. He calculates the profit as “just under £3.5m” and writes to ask why this conflicts with the mortality profit set out above. Show that the director’s figures are numerically correct and then explain why the two figures differ. [6]

[Total 18] Question 2.23

On 1 January 2010 a pension scheme had 100 members aged 75 exact, each eligible for a pension of £10,000 pa, payable annually in advance. In addition, the members were entitled to a death benefit of £20,000 payable at the end of the year of death. No premiums were being paid in respect of these contracts after January 2010. Given that 4 of the lives died during 2010, calculate the mortality profit for these contracts for calendar year 2010 using the following basis: Mortality: PFA92C20 Interest: 4% pa Expenses: none [5]

CT5: Q&A Bank Part 2 – Questions Page 13


Question 2.32

On 1 January 1998 a life insurance company issued a number of 20-year pure endowment policies to a group of lives aged 40 exact. In each case, the sum assured was £75,000 and premiums were payable annually in advance. On 1 January 2012, 500 policies were still in force. During 2012, 3 policyholders died, and no policy lapsed for any other reason. The office calculates net premiums and net premium reserves on the following basis: Interest: 4% per annum Mortality: AM92 Select (i) Calculate the profit or loss from mortality for this group for the year ending

31 December 2012. [6] (ii) Explain why the mortality profit or loss has arisen. [2] (iii) Calculate the probability of there being a mortality loss (as opposed to a

mortality profit) in 2013. [3] [Total 11]










CT5: Q&A Bank Part 2 – Solutions Page 15


Solution 2.21

If the monthly premium is P , the premium equation is:

1(12) (12)1 65| 65| |[35]:30[35]:30 [35]:30

[35]12 205,000 5,000( ) 10,000

DPa A IA a

D = - + [2]

The factors (calculated using the appropriate mortality and interest rates) are:

(12) 65|| [35]:30[35]:30

[35]

11 111 17.631 0.72508 17.299

24 24

Da a

D

Ê ˆ= - - = - ¥ =Á ˜

Ë ¯ [1]

( )1 1| | |[35]:30 [35]:30 [35]:30

689.231.02 1.02 0.32187 0.04789

2,507.02A A A

Ê ˆ= - = - =Á ˜Ë ¯ [1]

( )

( )

1 65[35] 65 65|[35]:30

[35]( ) 1.02 ( ) ( ) 30

689.231.02 7.47005 7.89442 30 0.52786

2,507.02

0.96506

Ê ˆ= - +Á ˜

Ë ¯

Ê ˆ= - + ¥Á ˜Ë ¯

=

DIA IA IA A

D

[2]

(12) 11 116565 24 2413.871 14.329a a= + = + = [1]

So the premium equation becomes:

12 17.299 205,000 0.04789 5,000 0.96506

10,000 0.27492 14.329

¥ = ¥ - ¥

+ ¥ ¥

P

So:

44,385.4

£213.8212 17.299

P = =¥

per month [1]

[Total 8]

Page 16 CT5: Q&A Bank Part 2 – Solutions


Solution 2.22

(i) Premium calculation The expected benefit outgo per policy is:

1 6260 62|60:2

6050,000 50,000

802.4050,000 0.45640 0.48458

882.85

50,000 0.0159775

£798.88

DA A A

D

Ê ˆ¥ = -Á ˜Ë ¯

Ê ˆ= - ¥Á ˜Ë ¯

= ¥

= [1]

The variance per unit sum assured is equal to 2 1 1 2| |60:2 60:2( )A A .

From first principles, the variance is:

2 260 60 61 (0.0159775)vq v p q [1]

where v is calculated at 21.04 1 8.16% . So the variance is:

( )1 2 21.0816 0.008022 1.0816 0.991978 0.009009 (0.0159775)

0.014801

- -¥ + ¥ ¥ -

= [2] Therefore, assuming that the lives are independent, then the variance is:

250,000 0.014801 37,002,178

and the standard deviation is 6,082.9 . [1]

The expected present value of the premiums is:

( )6060:2

0.9919871 1 1.953825

1.04Pa P v p P P

Ê ˆ= + = + =Á ˜Ë ¯ [1]



Solution 2.32

This question is CT5 September 2007 Question 12 (extended and with the dates changed). (i) Mortality profit 2012 is the 15th policy year so we need to calculate the death strain at risk (DSAR) in this year. This will require the reserve at time 15, which in turn requires that we first calculate the net premium from policy commencement. The net premium can be found from the equation:

[ ] [ ]1

40 :2040 :20£75,000A Pa=

¤

[ ] [ ]

2060

4040 :20

20

£75,000

£75,000 1.04 9,287.2164

13.930 9,854.3036

£2,315.81

v l

Pa l

-

=

¥ ¥=¥

= [1½]

Or, you could use[ ]60

40

D

D to calculate [ ]

140 :20

A , leading to the same answer.

Page 32 CT5: Q&A Bank Part 2 – Solutions


So, the reserve per policy at time 15 is:

115 55:555:5

5

£75,000 £2,315.81

9,287.2164£75,000 1.04 £2,315.81 4.585

9,557.8179

£49,281.26

V A a

-

= -

= ¥ ¥ - ¥

= [1]

So, the DSAR per policy in the 15th year is:

0 49,281.26

£49,281.26

DSAR = -

= - [½]

The expected death strain (EDS) across the 500 policies is:

( )54500

500 0.003976 £49,281.26

£97,971.15

EDS q DSAR¥=

= ¥ ¥ -

= - [1]

The actual death strain (ADS) is:

( )3

3 £49,281.26

£147,843.78

ADS DSAR= ¥

= ¥ -

= - [1]

So, the mortality profit (MP) is:

97,971.15 147,843.78

£49,873

MP EDS ADS= -

= - +

= [1]

[Total 6]



(ii) Why the mortality profit has arisen When asked to explain why mortality profit or loss has arisen you should think about the type of policy, in this case a pure endowment. Ask yourself whether, to maximise profits for the life office, you would want more or fewer deaths than expected. In this case, the expected number of deaths is: 54500 500 0.003976 1.988 3q = ¥ = < [1]

and so we have more actual deaths than expected. However, this is a pure endowment so no benefit is paid upon death. This means that if more lives die than expected, the life office will make a mortality profit. [1] [Total 2] (iii) Probability of there being a mortality loss in 2013 This will occur if we have less deaths than expected. The expected number of deaths in 2013 is: 55497 497 0.004469 2.2211q = ¥ =

So, there will be a mortality loss if there are 2 or fewer deaths. [1] The number of deaths has a binomial distribution:

( )497,0.004469D Bin [½]

So, the probability is:

( ) ( )

( )

497 496

4952

1 0.004469 497 0.004469 1 0.004469

4970.004469 1 0.004469

2

0.6169

- + ¥ ¥ -

Ê ˆ+ ¥ ¥ -Á ˜Ë ¯

= [1½]

[Total 3]










CT5: Assignment X2 Questions Page 3


Question X2.6

A man aged 65 exact buys a whole life annuity that provides payments at the end of each policy year. The first payment is £10,000, and subsequent payments increase by 3% pa compound. Let X denote the present value random variable for this annuity. Interest is assumed to be 3% pa and mortality is assumed to follow the AM92 Ultimate table. (i) Derive an expression for X and simplify this as much as possible. [2] (ii) Calculate ( )E X . [1]

(iii) Calculate the prospective reserve for this policy at time 5, assuming it is still in

force. [2] (iv) Calculate the probability that the present value of the benefit received by the

policyholder is greater than £250,000. [2] [Total 7] Question X2.7

On 1 January 2008 an insurer issued a block of 25-year annual premium endowment policies that pay £120,000 at maturity, or £60,000 at the end of the year of earlier death to lives aged exactly 65. The premium basis assumed 4% interest, AM92 Select mortality and allowed for an initial expense of £200 and renewal expenses of 1% of each subsequent premium. Reserves are calculated on the same basis as the premiums. (i) Calculate the premium. [3] (ii) Calculate the reserve required per policy at 31 December 2012. [3] (iii) There were 197 policies in force on 1 January 2012. During 2012 there were 9

deaths, interest was earned at twice the rate expected and expenses were incurred at twice the rate expected. By considering the total reserve required at the start and end of the year, and all the cashflows during the year, calculate the profit or loss made by the insurer from all sources (not just from mortality) in respect of these policies for the 2012 calendar year. [6]

[Total 12]

Page 4 CT5: Assignment X2 Questions


Question X2.8

A life insurance company issued a with profits whole life policy to a life aged 20 exact, on 1 July 2002. Under the policy, the basic sum assured of £100,000 and attaching bonuses are payable immediately on death. The company declares simple reversionary bonuses at the start of each year. Level premiums are payable annually in advance under the policy. (i) Give an expression for the gross future loss random variable under the policy at

the outset. Define symbols where necessary. [3] (ii) Calculate the annual premium, using the equivalence principle.

Basis: Mortality AM92 Select Interest 6% per annum Bonus loading 3% simple per annum Expenses Initial £200 Renewal 5% of each premium payable in the second and

subsequent years Assume bonus entitlement is earned immediately on payment of premium. [4]

(iii) On 30 June 2005 the policy is still in force. A total of £10,000 has been declared

as a simple bonus to date on the policy.

The company calculates reserves for the policy using a gross premium prospective basis, with the following assumptions:

Mortality AM92 Ultimate Interest 4% Bonus loading 4% per annum simple Renewal expenses 5% of each premium

Calculate the reserve for the policy as at 30 June 2005. [4] [Total 11]



Question X3.11

The following table shows (in £’s) a profit testing calculation with some of the entries missing for a three-year endowment assurance contract issued to a group of lives aged exactly 57 with a sum assured of £5,000 payable at the end of the year of death. Outgo terms are shown as negative entries.

Year Premium Expenses Interest Expected

cost of claims

Expected cost of

increasing reserves

Profit vector

1 1,530 –50 ? ? ? –51

2 1,530 ? ? ? ? 21

3 1,530 ? ? ? ? 45

The initial rate of mortality at each age is 1%. The rate of accumulation used is 6%. Reserves are calculated using an interest rate of 4%. The reserves are zero at the start and end of the contract. The interest earned on the reserve in the third year is £195. (i) Complete the table. [7] (ii) Calculate the internal rate of return. [2] (iii) Explain the effect that changing to a weaker reserving basis would have on the

internal rate of return. [2] (iv) Calculate the net present value using a risk discount rate of 7%. [2] (v) Explain the effect that changing to a weaker reserving basis would have on the

net present value. [2] [Total 15]



Question X3.12

A life insurance company issues a 5-year with profits endowment assurance policy to a life aged 60 exact. The policy has a basic sum assured of £10,000. Simple reversionary bonuses are added at the start of each year, including the first. The sum assured (together with any bonuses attaching) is payable at maturity or at the end of year of death, if earlier. Level premiums are payable annually in advance throughout the term of the policy. (i) Show that the annual premium is approximately £2,476. Basis: Mortality: AM92 Select Interest: 6% per annum Initial expenses: 60% of the first premium Renewal expenses: 5% of the second and subsequent premiums Bonus Rates: A simple reversionary bonus will be declared each

year at a rate of 4% per annum [5] The office holds net premium reserves using a rate of interest of 4% per annum and AM92 Ultimate mortality. In order to profit test this policy, the company assumes that it will earn interest at 7% per annum on its funds, mortality follows the AM92 Ultimate table and expenses and bonuses will follow the premium basis. (ii) Calculate the expected profit margin on this policy using a risk discount rate of

9% per annum. [13] [Total 18]



Question X3.13

A life insurance company issues a 3-year unit-linked endowment assurance contract to a female life aged 60 exact under which level premiums of £5,000 per annum are payable in advance. In the first year, 85% of the premium is allocated to units and 104% in the second and third years. The units are subject to a bid-offer spread of 5%, and an annual management charge of 0.75% of the bid value of the units is deducted at the end of each year. If the policyholder dies during the term of the policy, the death benefit of £20,000 or the bid value of the units after the deduction of the management charge, whichever is higher, is payable at the end of the year of death. On survival to the end of the term, the bid value of the units is payable. The company holds unit reserves equal to the full bid value of the units but does not set up non-unit reserves. It uses the following assumptions in carrying out profit tests of this contract: Mortality: AM92 Ultimate Surrenders: None Expenses: Initial: 600 Renewal: 100 at the start of each of the second and third policy years Unit fund growth rate: 6% per annum Non-unit fund interest rate: 4% per annum Risk discount rate: 10% per annum (i) Calculate the expected net present value of the profit on this contract. [10] (ii) State, with a reason, what the effect would be on the profit if the insurance

company did hold non-unit reserves to zeroise negative cashflows, assuming it used a discount rate of 4% per annum for calculating those reserves. (You do not need to perform any further calculations.) [2]

[Total 12]

End of paper












Question X4.1

A pension scheme provides a pension of 1/35 of career average salary in respect of each full year of service, on age retirement between the ages of 60 and 65. A proportionate amount is provided in respect of an incomplete year of service. At the valuation date of the scheme, a new member aged exactly 45 has an annual rate of salary of £63,000. Calculate the expected present value of the future service pension on age retirement in respect of this member, using the Pension Fund Tables in the Tables and assuming that salaries increase continuously. [3] Question X4.2

A pension scheme provides a pension on ill-health retirement of 1/80th of Final Pensionable Salary for each year of pensionable service subject to a minimum pension of 20/80ths of Final Pensionable Salary. Final Pensionable Salary is defined as the average salary earned in the three years before retirement. Normal retirement age is 65 exact. Derive a formula for the present value of the ill-health retirement benefit for a member currently aged 35 exact with exactly 10 years past service and salary for the year before the calculation date of £20,000. [5]



Question X4.3

A three-state transition model is shown in the following diagram:

Alive Sick

Dead

v

Assume that the transition probabilities are constant at all ages with 2%=m , 4%=n ,

1%=r and 5%=s .

Calculate the present value of a sickness benefit of £2,000 pa paid continuously to a life now aged 40 exact and sick, during this period of sickness, discounted at 4% pa and payable to a maximum age of 60 exact. [4] Question X4.4

You are given the following statistics in respect of the population of Urbania:

Males

Females

Age band Exposed to risk

Observed Mortality rate

Exposed to risk

Observed Mortality rate

20–29 125,000 0.00356 100,000 0.00125 30–39 200,000 0.00689 250,000 0.00265 40–49 100,000 0.00989 200,000 0.00465 50–59 90,000 0.01233 150,000 0.00685

Calculate the directly and indirectly standardised mortality rates for the female lives, using the combined population as the standard population. [6]

Question X4.5

Explain how geographical location can affect mortality. [5]

CT5: Assignment X1 Solutions Page 7


Solution X1.7

The survival probability can be written as: 2 63.25 0.75 63.25 64 0.25 65p p p p= ¥ ¥ [1]

From the Tables: 64 641 0.97801p q= - =

You could also have calculated 64p as 65

64

l

l.

(a) UDD Under the uniform distribution of deaths assumption: 0.25 65 651 0.25 1 0.25 0.02447 0.99388p q= - = - ¥ = [1]

and:

63 630.75 63.25

0.25 63 63

0.980350.98519

1 0.25 1 0.25 0.01965

p pp

p q= = = =

- - ¥ [1]

So: 2 63.25 0.98519 0.97801 0.99388 0.95763p = ¥ ¥ = [½ ]

Alternative solutions

You could also have used the UDD formula ( )1- +-

=-

xt s x s

x

t s qq

s q to say that:

630.75 63.25

63

0.75 0.75 0.019650.01481

1 0.25 1 0.25 0.01965

¥= = =- - ¥

qq

q

and hence: 0.75 63.25 1 0.01481 0.98519= - =p

Page 8 CT5: Assignment X1 Solutions


Or, another way to do this is to use:

65.252 63.25

63.25

lp

l=

and interpolate between the life table values from ELT15 (Males). We have:

63.25 64 630.25 0.75

0.25 81,076 0.75 82,701

82, 294.75

l l l= ¥ + ¥= ¥ + ¥=

and 65.25 66 650.25 0.75

0.25 77,353 0.75 79,293

78,808

l l l= ¥ + ¥= ¥ + ¥=

So, 2 63.2578,808

0.9576382,294.75

p = =

Markers: please give credit for correct alternatives. (b) CFM Under the constant force of mortality assumption:

( )0.25 0.250.25 65 65 0.97553 0.99383p p= = = [1]

and:

( ) ( )0.75 0.750.75 63.25 63 0.98035 0.98523p p= = = [1]

So: 2 63.25 0.98523 0.97801 0.99383 0.95761p = ¥ ¥ = [½ ]

[Total 6]



The variance of the present value random variable is:

( ) { }

{ }

{ }

2min 1,10

min 1,102

2min 1,10

2

var 10,000 var

110,000 var

10,000var

K

K

K

W a

v

d

vd

+

+

+

È ˘= Í ˙Î ˚

È ˘-Í ˙=Í ˙Î ˚

È ˘= Í ˙Î ˚ [1]

Now { }min 1,10Kv + is the present value of a benefit of 1 paid at time 10 or at the end of the year of death of ( )x , whichever is sooner. So it is the present value of a 10-year

endowment assurance on ( )x , and:

{ } ( )2min 1,10 2:10 :10var K

x xv A A+È ˘ = -Í ˙Î ˚ [1]

We can calculate the values of these assurances using premium conversion:

( )0.04:10 :101 1 1 7.747656 0.696210x xA d a e-= - = - - ¥ = [1]

Since 2:10xA is equal to :10xA evaluated using a force of interest of 2 0.08d = , and:

( )9 0.1 10

0.08 0.02:10 0.1

0

1@ 0.08 6.642533

1k k

xk

ea e e

ed

- ¥- -

-=

-= = = =-

Â [1]

it follows that:

( )2 0.08:10 1 1 6.642533 0.489298xA e-= - - ¥ = [1]

So:

{ }min 1,10 2var 0.489298 0.696210 0.004589Kv +È ˘ = - =Í ˙Î ˚



The standard deviation of the present value random variable is therefore:

( )0.04

10,000 10,0000.004589 0.004589 17, 277

1d e-¥ = ¥ =

- [1]

[Total 8] Solution X1.12

This question is CT5 April 2005 Question 6. Uniform distribution of deaths This assumption says that t x xq t q= for integer values of x and 0 1t£ £ . Since we are

starting at age 45½ , which is not an integer, we must first write ½ 45½p in terms of

45t p . So we begin by writing:

45½ 45½

½ 45

pp

p= [½]

This can be easily seen from the following diagram:

Then under the UDD assumption:

45 45½ 45½

½ 45 45

1 1 1 0.002660.99867

1 1 ½ 1 ½ 0.00266

- - -= = = =- - - ¥

q qp

q q [1]

45 45½ 46

½ 45p ½ 45½p

45p



Alternative solutions

Alternatively, you could use the UDD formula ( )1- +-

=-

xt s x s

x

t s qq

s q to say that:

45½ 45½

45

½ ½ 0.002660.00133

1 ½ 1 ½ 0.00266

¥= = =

- - ¥q

qq

and hence ½ 45½ 1 0.00133 0.99867= - =p .

Also:

6014 46

46

86,7140.91023

95, 266

lp

l= = = [½]

So: 14½ 45½ ½ 45½ 14 46 0.99867 0.91023 0.90902p p p= ¥ = ¥ = [1]

Or, another way to do this is to use:

6014½ 45½

45½

lp

l=

and interpolate between the life table values from ELT15 (Males). We have:

45½ 45 460.5 0.5

0.5 95,521 0.5 95, 266

95,393.5

l l l= ¥ + ¥= ¥ + ¥=

So: 14½ 45½86,714

0.9090195,393.5

p = =

Markers: please give credit for correct alternatives.



Constant force of mortality We now assume that m is constant between integer ages.

In this case:

½ ½½ 45½ 45½0

½ ½45

½

exp

1 0.00266 0.99867

tp dt e

e p

[1]

So: 14½ 45½ ½ 45½ 14 46 0.99867 0.91023 0.90902p p p= ¥ = ¥ = [1]

Note that the two assumptions give the same answer correct to 4dp. Another possible assumption that you could have used here is the Balducci assumption, which you met in Subject CT4 or Subject 104. The Balducci formula is given on Page

33 of the Tables. It states that ( )1 1t x t xq t q- + = - for integer values of x and 0 1t£ £ .

In this case: ½ 45½ ½ 45½ 451 1 ½ 1 ½ 0.00266 0.99867p q q= - = - = - ¥ = [1]

So: 14½ 45½ ½ 45½ 14 46 0.99867 0.91023 0.90902p p p= ¥ = ¥ = [1]

as before. [Maximum 5] Markers: please award the full five marks for correct solutions using either method.



There were 9 deaths during 2012. So the reserves required on 31 December 2012 (using the prospective reserve figure of £12,215.36) total: (197 9) 12,215.36 £2,296,487.68- ¥ = [½]

So the profit earned in 2012 was:

2012 1,916,152.02 605,065.80 12,101.32 200,729.32

9 60,000 2, 296, 487.68

£126,642

Profit = + - +

- ¥ -

= -

ie a loss of approximately £127,000. [1] [Total 6] Solution X2.8

This question is CT5 September 2005 Question 10. (i) Gross future loss random variable Suppose that:

b is the level of simple bonus, expressed as a percentage of the sum assured

I is the initial expense

e is the renewal expense, payable at the start of each year, including the first

f is the termination expense, payable at the time a claim is made

P is the annual premium

K is the curtate future lifetime of the policyholder

T is the complete future lifetime of the policyholder [1 for all notation defined, ½ mark if up to three items missing] Then the gross future loss random variable at the outset is:

( ) 1 1100,000 1 1 + +È ˘= + + + + + -Î ˚ T TK KL b K v I e a f v P a [2]

[Total 3]



(ii) Annual premium Note that the rate of interest is 6% pa in this part of the question. The expected present value of the premiums is: [20] 16.877P a P= [½]

The expected present value of the expenses is:

( )[20]200 0.05 1 200 0.79385P a P+ - = + [½]

The expected present value of the benefits is:

[20] [20]100,000 3,000( )A IA+ [½]

Now:

½ ½[20] [20]1.06 1.06 0.04472 0.04604ª ¥ = ¥ =A A [½]

and:

½ ½[20] [20]( ) 1.06 ( ) 1.06 2.00874 2.06812ª ¥ = ¥ =IA IA [½]

So: EPV benefits 10,808.58= [½]

Using the principle of equivalence we have:

16.877 10,808.58 200 0.79385

£684.48

P P

P

= + +

fi = [1] [Total 4] Award full marks for the correct final premium here.



(iii) Reserve at time 3 Note that the rate of interest is 4% pa in this part of the question. The reserve at time 3 is:

3 23 23 23110,000 4,000( ) 0.95 V A IA Pa= + - [2]

½ ½110,000 1.04 0.12469 4,000 1.04 6.09644

0.95 684.48 22.758

£24,057.70

= ¥ ¥ + ¥ ¥

- ¥ ¥

=

or £24,058 to the nearest £1. [2] [Total 4] Markers : award full marks for the correct final answer here. Award method marks appropriately. Solution X2.9

This question is CT5 April 2005 Question 14 (with the dates changed). (i) Definitions The death strain at risk for a policy issued t years ago when the policyholder was aged x , which provides a sum assured of S payable at the end of the year of death and provides no benefit on survival to time 1t + is given by: 1tDSAR S V+= - [1]

It is the amount of money, over and above the reserve at time 1t + , that has to be paid

in respect of each death during the policy year ( ), 1t t + .

The expected death strain for such a policy is:

( )1x t tEDS q S V+ += - [1]



This is the amount that the life insurance company expects to pay extra to the year-end reserve for the policy. For a group of identical policies, the expected death strain is given by: expected number of deaths ¥ DSAR [1]

The actual death strain is:

( )1

0 if the policyholder survives to time 1

if the policyholder dies in the year , 1t

tADS

S V t t+

+ÏÔ= Ì - +ÔÓ [1]

So it is the observed value of the indicator random variable:

( )0 if the policyholder survives to time 1

1 if the policyholder dies in the year , 1

tD

t t

+Ï= Ì +Ó

[1]

multiplied by the death strain at risk. For a group of identical policies, the actual death strain is given by: actual number of deaths ¥ DSAR [1] [Total 6] (ii)(a) Death strain at risk for each type of policy for calendar year 2004 The end of calendar year 2012 is time 3, when time is measured in years from the start of the policies. Term assurance To calculate the reserve at time 3, we first need to calculate the annual premium for the policy. If we denote this by P , then:

145:15 45:15

150,000Pa A= [½]

From the Tables: 45:15 11.386a = [½]

Also:

1 6045:1545:15

45

882.850.56206 0.03592

1,677.97

DA A

D= - = - = [½]



Alternatively, you can calculate this as:

1 6045 6045:15

45

882.850.27605 0.45640 0.03592

1,677.97

DA A A

D= - = - ¥ =

So:

150,000 0.03592

£473.2111.386

P¥= = [½]

The reserve at time 3 is:

13 48:1248:12

150,000 473.21

882.85150,000 0.63025 473.21 9.613

1,484.43

£777.52

V A a= -

Ê ˆ= - - ¥Á ˜Ë ¯

=

[1] Or, you could calculate the term assurance using:

1 6048 6048:12

48

882.850.30695 0.45640

1, 484.43

0.035511

DA A A

D= - ¥

= - ¥

=

which leads to the same answer. The death strain at risk for each term assurance policy is then: 3 150,000 777.52 £149, 222DSAR S V= - = - = [½]



Pure endowment Let the annual premium for the pure endowment be P ¢ . Then:

6045:15

45

882.8575,000 11.386 75,000 £3, 465.71

1,677.97

DP a P P

D= fi = ¥ fi =¢ ¢ ¢ [1]

The reserve at time 3 is:

13 48:1248:12

75,000 3,465.71

882.8575,000 3,465.71 9.613

1,484.43

£11,289.63

V A a= -

= ¥ - ¥

=

[1]

There is no death benefit if the policyholder dies during calendar year 2012, so the death strain at risk for each pure endowment policy is: 30 £11, 290DSAR V= - = - [½]

Temporary annuity Watch out here – these policyholders are aged 55 at entry and have PMA92C20 mortality. The reserve at time 3 for the temporary annuity is:

( )( )( )

3 58:2

258 2 58

2

25,000

25,000

1 0.001814 1 0.0021101 0.00181425,000

1.04 1.04

£47,018.15

V a

v p v p

=

= +

Ê ˆ- --= +Á ˜Ë ¯

= [1]

There is no death benefit for this policy. However, if the policyholder survives to time 3, there is a survival benefit of £25,000, which is not included in the reserve at time 3. [½]



So the death strain at risk for each temporary annuity is:

( )30 25,000 £72,018DSAR V= - + = - [½]

This calculation is quite sensitive to rounding and to the method of calculation used. For example, if you had calculated the annuity as:

2 258 2 58 6058:2

9,826.13115.356 1.04 14.632 1.881

9,864.803-= - = - ¥ ¥ =a a v p a

then you would get: 3 47,023.16=V and 72,023= -DSAR

(ii)(b) Total mortality profit or loss Term assurance policies There are 5,000 15 4,985- = term assurance policies in force on 1 January 2012.

The expected death strain for this group of policies is: 474,985 149,222 4,985 0.001802 149,222 £1,340,457EDS q= ¥ = ¥ ¥ = [½]

The actual death strain for this group of policies is: 8 149,222 £1,193,776ADS = ¥ = [½]

So the mortality profit from this group of policies is: £146,681MP EDS ADS= - = [½]



Pure endowment policies There are 2,000 5 1,995- = pure endowment policies in force on 1 January 2012.

The expected death strain for this group of policies is:

( ) ( )471,995 11,290 1,995 0.001802 11,290 £40,587EDS q= ¥ - = ¥ ¥ - = - [½]

The actual death strain for this group of policies is:

( )1 11,290 £11,290ADS = ¥ - = - [½]

So the mortality profit from this group of policies is: £29,297MP EDS ADS= - = - [½]

Temporary annuity policies There are 1,000 5 995- = temporary annuity policies in force on 1 January 2012.

The expected death strain for this group of policies is:

( ) ( )57995 72,018 995 0.001558 72,018 £111,643EDS q= ¥ - = ¥ ¥ - = - [½]

The actual death strain for this group of policies is:

( )1 72,018 £72,018ADS = ¥ - = - [½]

So the mortality profit from this group of policies is: £39,625MP EDS ADS= - = - [½]

Total mortality profit The total mortality profit is then: 146,681 29,297 39,625 £77,759- - = [½]

[Total 13]



(iv) Net present value The net present value calculated using a risk discount rate of 7% is:

2 351 20.79 44.10 47.66 18.16 36.00 £6.50NPV v v v= - + + = - + + = [Total 2] (v) Effect on net present value The net present value will increase. [½] This is because the profits in the later years will be released sooner, which will increase their present value. [1½] [Total 2] Solution X3.12

This question is CT5 April 2009 Question 14. (i) Annual premium Let the annual premium be P . The expected present value of the premiums is:

[60]:5(premiums) 4.398EPV Pa P [½]

The expected present value of the expenses is:

[60]:5(expenses) 0.6 0.05 1

0.6 0.05 3.398

0.7699

EPV P P a

P P

P

[½] Turning our attention to the benefits, simple bonuses of 4% pa are declared at the start of each year. This means that the sum assured increases by £10,000 0.04 £400 each

time a bonus is awarded. [½] Since bonuses are added at the start of the year, we must allow for a sum assured of £10,400 on death during the first year, £10,800 on death during the second year, and so on. The payment at the end of the fifth year will be £12,000, irrespective of whether the policyholder dies during the final year or survives to the end of the contract.



An expression for the expected present value of the benefits is:

[60]:5 [60]:5(benefits) 10,000 400( )EPV A IA [½]

Alternatively, if you split off the maturity benefit and treat the contract as a term assurance plus a pure endowment, you can write:

1 1 1

[60]:5 [60]:5 [60]:5( ) 10,000 400( ) 12,000

10,000 0.039428 400 0.126806 12,000 0.711612

EPV benefits A IA A

To evaluate the increasing assurance, we use the formula:

5 5[60] 5 [60] 65 65 5 [60][60]:5

( ) ( ) ( ) 5 5 IA IA v p IA A v p [1]

Looking up values in the Tables at 6% interest:

5[60]:5

5

8,821.2612( ) 5.4772 (1.06) 5.50985 5 0.40177

9,263.1422

8,821.26125(1.06)

9,263.1422

IA

giving

[60]:5( ) 3.68486IA . [½]

Also, at 6% interest:

[60]:50.75104A [½]

This gives:

(benefits) 10,000 0.75104 400 3.68486

£8,984.35

EPV

[½]

The premium is found by solving:

4.398 8,984.35 0.7699

£2,476.32

P P

P



So, the premium is £2,476, as stated. [½] [Total 5] (ii) Profit margin To calculate a net premium reserve for a with-profits policy we need to use the following method: 1. Calculate the net premium for the contract ignoring bonuses. 2. Calculate the net premium reserve. The expected present value of future benefits

should include allowance for bonuses declared so far, but no allowance for future bonuses. The premium used should be the net premium calculated in 1.

In order to carry out the profit test, we need to calculate the net premium reserve held at each duration. The net premium, NP , for the contract, using 4% interest and AM92 Ultimate mortality is:

60:5 60:5

10,000 0.8249910,000 £1,813.16

4.550

NP a A NP [1]

By time 1, one bonus of £400 has been awarded, so the sum assured is £10,400. The net premium reserve at time 1 is therefore:

1 61:4 61:410,400 1,813.16

10,400 0.85685 1,813.16 3.722

£2,162.66

V A a

[½]

By time 2, two bonuses of £400 have been awarded, so the sum assured is £10,800. The net premium reserve at time 2 is therefore:

2 62:3 62:310,800 1,813.16

10,800 0.89013 1,813.16 2.857

£4,433.21

V A a

[½]



By time 3, three bonuses of £400 have been awarded, so the sum assured is £11,200. The net premium reserve at time 3 is therefore:

3 63:2 63:211,200 1,813.16

11,200 0.92498 1,813.16 1.951

£6,822.30

V A a

[½]

By time 4, four bonuses of £400 have been awarded, so the sum assured is £11,600. The net premium reserve at time 4 is therefore:

4 64:1 64:111,600 1,813.16

11,600 0.96154 1,813.16 1

£9,340.70

V A a

[½]

At time 5, the contract expires, so no reserves will be held at that point. Alternatively, the net premium reserves may be calculated as follows:

1 61:4 61:4

61:4 61:4 61:4

1 60:5 61:4

61:461:4

60:5

10, 400 1,813.16

10,000 1,813.16 400

10,000 400

10,000 1 400

V A a

A a A

V A

aA

a

where the final equality uses the formula for the net premium reserve of an endowment assurance given on page 37 of the Tables. (You needn’t show the first two lines of the above in the exam.) The net premium reserve at time 2 is:

62:32 2 60:5 62:3 62:3

60:5

10,000 800 10,000 1 800

aV V A A

a

with other reserves calculated similarly.



Traditionally, profit tests are set out with the years as row headings and the cashflows as column headings. It can be easier to set out these questions in an exam if you transpose the table, as shown below. Markers, please award full credit either way if the actual numbers are correct. Carrying out the profit test: Note Year 1 2 3 4 5 Premium ( P ) 2,476.32 2,476.32 2,476.32 2,476.32 2,476.32 (1) Expenses –1,485.79 –123.82 –123.82 –123.82 –123.82 (2) Interest 69.34 164.68 164.68 164.68 164.68 (3) Benefit –83.43 –97.30 –113.25 –131.59 –12,000 (4) Reserves at

start of year 0 2,162.66 4,433.21 6,822.30 9,340.70

(5) Interest on reserves

0 151.39 310.32 477.56 653.85

(6) Expected reserves at end of year

–2,145.31 –4,393.27 –6,753.31 –9,234.74 0

(7) Profit vector –1,168.87 340.66 394.15 450.71 511.73 (8) Probability of

survival to start of year

1 0.991978 0.9830412 0.9731007 0.9620619

(9) Profit signature

–1,168.87 337.93 387.47 438.59 492.32

[7, lose ½ for each incorrect value, subject to minimum of 0] Markers, award full method marks here if the method has been followed through correctly, ie if one incorrect value early in the calculation has led to many incorrect values but the method is otherwise perfect, award 6.5 marks out of 7. Notes: (1) 0.6 P for year 1; 0.05 P for years 2 to 5 (2) 0.07( (1))P

(3) 60 1(10,000 400 ) kk q for years 1,2,3,4k

12,000 in year 5, as the benefit is paid irrespective of whether the policyholder

dies in the final year or survives to the end of the term (4) 0 for year 1; 1k V for years 2,3, 4,5k

(5) 0.07 (4)

(6) 60 1 k kV p for years 1,2,3,4k ; 0 for year 5

(7) (1) (2) (3) (4) (5) (6) P

(8) 1 60k p for years 1,2,3,4,5k



(9) (7) (8)

The profit margin is defined as:

( )

NPV

EPV premiums

where the calculations are performed at the risk discount rate. Using the risk discount rate of 9% pa, the net present value of the profit signature is:

2 3 4 51,168.87 337.93 387.47 438.59 492.32

£141.95

NPV v v v v v

[1] The expected present value of the premiums is:

2 3 460 2 60 3 60 4 60( ) 2,476.32 1

2,476.32 4.17044

£10,327.34

EPV premiums vp v p v p v p

[1]

The profit margin is:

141.95

0.013710,327.34

ie 1.37%. [1] [Total 13]



Solution X3.13

This question is CT5 September 2006 Question 9. (i) Expected net present value of the profit Unit fund The expected cashflows in the unit fund are given in the table below. Cashflows out of the fund are shown as negative entries.

Year Premium Cost of allocation

Fund at start of

year

End fund before charge

Management charge

Fund at end of year

1 5,000 4,037.50 4,037.50 4,279.75 - 32.10 4,247.65

2 5,000 4,940 9,187.65 9,738.91 - 73.04 9,665.87

3 5,000 4,940 14,605.87 15,482.22 - 116.12 15,366.10

[4, lose ½ for each incorrect value, subject to minimum of 0] Markers, award full method marks here if the method has been followed through correctly, ie if one incorrect value early in the calculation has led to many incorrect values but the method is otherwise perfect, award 3.5 marks out of 4. Non-unit fund The expected cashflows in the non-unit fund are:

Year Premium less cost

of allocation

Expenses Interest Expected benefit

cost

Management charge

Profit vector

1 962.50 –600 14.50 –126.37 32.10 282.73

2 60 –100 –1.60 –93.10 73.04 –61.66

3 60 –100 –1.60 –46.86 116.12 27.66

[3, lose ½ for each incorrect value, subject to minimum of 0] Markers: the marks for the expected benefit cost column are awarded separately below.



Markers, award full method marks here if the method has been followed through correctly, ie if one incorrect value early in the calculation has led to many incorrect values but the method is otherwise perfect, award 2.5 marks out of 3. The expected benefit cost figures are calculated as follows. If the policyholder dies in Year 1, a death benefit of £20,000 is payable at the end of the first year. The sum of £4,247.65 comes from the unit fund, and the remainder comes from the non-unit fund. The expected death cost in Year 1 is:

( ) 6020,000 4,247.65 15,752.35 0.008022 126.37q- = ¥ = [1]

The entries for Years 2 and 3 are calculated in a similar way:

( ) 6120,000 9,665.87 10,334.13 0.009009 93.10q- = ¥ = [½]

and ( ) 6220,000 15,366.10 4,633.90 0.010112 46.86q- = ¥ = [½]

If the policyholder survives to the end of Year 3, she receives the full bid value of the units. This comes from the unit fund, so there is no cashflow from the non-unit fund. The expected present value of the profit, discounted at the risk discount rate, is:

( )

60 2 602 3

2 3

282.73 61.66 27.66EPV profit

1.10 1.10 1.10

282.73 61.66 9,129.717 27.661 0.008022

1.10 9, 287.21641.10 1.10

226.91

p p-= + +

-= + - + ¥

= [1]

[Total 10] (ii) Effect of holding non-unit reserves to zeroise negative cashflows Holding non-unit reserves to offset the negative cashflow at time 2 defers the emergence of profit at time 1. Because we are discounting the profit flows at a higher rate (10%) than the interest earned on the reserves (4%), deferring the emergence of profit will reduce its expected present value. [2]



Assignment X4 – Solutions Markers: This document does not necessarily give every possible approach to solving each of the questions. Please give credit for other valid approaches. Solution X4.1

Throughout all the symbols used are as defined for the Formulae and Tables. This also means that values of xr for all 60x < are zero. We develop a general

solution that assumes retirement is possible at any age, but we realise that all the values representing the cost of retirement at ages before 60 will be computed as zero. This will then meet the requirement in the question that retirement can only occur at ages older than 60. Once you have practised lots of questions of this type, you should be able just to write down the correct commutation function formula. In this solution we show where the formula comes from. But you only have to write down the last line to get full marks. Assuming that salaries increase continuously, the amount of salary earned in the year of age [45 ,45 1]t t , 0,1,2,t is:

45

44.563,000 ts

s

noting that £63,000 will be the average salary over the year of age [44.5, 45.5]. [½ ] The career average salary on retirement in the year of age [45 ,45 1]u u+ + + , u = 0, 1,

2, … is, on average:

( )

( )1

45 46 452

144.52

63,000us s s

u s

++ + +

+

[½ ]

assuming a uniform distribution of retirements over each year of age. The annual pension starting on retirement in that year of age is then:

( ) ( )

( )( )1 1 1

45 46 45 45 46 452 2 2

144.544.52

63,000 63,00035 35

u uu s s s s s s

su s

+ ++ + + + + + +=

+

[½ ]



The career average salary on retirement at age 65 exact is:

( )45 46 64

44.563,000

20

s s s

s

+ + +

and the annual pension starting on retirement at age 65 exact is:

( ) ( )45 46 64 45 46 64

44.5 44.5

2063,000 63,000

35 20 35

s s s s s s

s s

+ + + + + +=

[½ ]

The expected present value of the retirement pension is then:

{ ( )

( ) ( ) }

45.5 46.51 145 45 45.5 45 46 46 46.52 245

44.5 45

64.5 65145 46 64 64 64.5 45 46 64 65 652

63,000

35

r r

r r

s v r a s s v r as v l

s s s v r a s s s v r a

+ + +

+ + + + + + + + [½ ]

Using the symbols as defined in the Tables and rearranging, the value becomes:

( ){ 145 45 46 47 64 652

44.5 45

63,000

35ra ra ra ra ras C C C C C

s D+ + + + +

( ) ( )}1 146 46 47 48 64 65 64 64 652 2

ra ra ra ra ra ra ras C C C C C s C C+ + + + + + + + +

( )45 46 6444.5 45

63,000

35s ra s ra s raM M M

s D= + + + [½ ]

45

44.5 45

63,000 63,000 2,013,657£184, 437

35 35 8.438 2,329

s raR

s D

¥= = =¥ ¥

[1]

[Maximum 3] Markers: award full marks for the correct final answer.



Alternative solution Assuming that the member’s salary grows continuously, and that retirements are uniformly distributed over each year of age, the expected present value of benefits earned from the year of service (45,46) is:

0.5 1.545 45 45 46145.5 46.52

44.5 45 44.5 45

19.5 2045 64 45 6564.5 65

44.5 45 44.5 45

63,000

35

r r

r r

s r s rv a v a

s l s l

s r s rv a v a

s l s l

Ï+Ì

Ó

¸+ + + ˝

˛

Note that the first term in the brackets contains a factor of ½, but the others do not. The reason for this is that if retirement occurs in the year (45,46), then assuming this occurs on average in the middle of the year, half the year’s salary will contribute to the pension, but if retirement occurs after the end of that year, the full year’s salary will contribute to the pension. Using the symbols as defined in the Tables and rearranging, this becomes:

( )45 145 46 47 64 652

45 44.5

4545 45

45 44.5 45 44.5

63,000

35

63,000 63,000 1

35 35

ra ra ra ra ra

ra s ra

sC C C C C

D s

sM M

D s D s

+ + + + +

= =

Summing similar terms over all future years of service from (45,46) to (64, 65), the expected present value of the benefits is:

( )45 46 64 45

45 44.5 45 44.5

63,000 1 63,000 1

35 35s ra s ra s ra s raM M M R

D s D s+ + + = [3]

Markers: award full marks for the correct final answer and method marks appropriately.



Solution X4.2

This question is Subject CT5, April 2005, Question 8. Define: j = the valuation rate of interest [¼]

1

1v

j=

+ [¼]

xi = the number of ill-health retirements between x and 1x + , 64x £ [¼]

xl = the number of active lives at age x exact [¼]

Both xi and xl must come from a suitable service table. [¼]

ixa = the expected present value at age x of a pension of 1 pa payable on ill-health

retirement at age x , and payable in accordance with the scheme rules [¼]

{ }xs is a salary scale such that:

( )

( )expected salary earned in year of age , 1

expected salary earned in year of age , 1x t

x

x t x ts

s x x+ + + +

=+

[¼]

1 2 3

3x x x

xs s s

z - - -+ += [¼]

Assume that ill-health retirements occur uniformly over each year of age and part years of service count proportionately. [¼]



Past service benefit The member has 10 years of past service, so is already entitled to 10/80ths of final

pensionable salary when he retires. If he retires in the year of age ( ), 1y y + , we are

assuming it occurs at age ½y + , so his FPS will be:

½

3420,000

syz +

[¼]

Note that we have 34s in the denominator since he earned £20,000 between age 34 and

35. The expected present value of the past service benefit is:

½ 1½ 29½35 35½ 36 36½ 64 64½35½ 36½ 64½

35 34 35 34 35 34

35½ 64½35 35½ 35½ 64 64½ 64½

3534 35

1020,000

80

1020,000

80

i i i

i i

i z i z i zv a v a v a

l s l s l s

i v z a i v z a

s v l

È ˘¥ + + +Í ˙

Î ˚

È ˘+ += ¥ Í ˙

Í ˙Î ˚ [¼]

Note that we will deal with the guarantee in the future service benefit. Now define:

xx xD v l= [¼]

½½ ½

z ia x ix x x xC i v z a+

+ += [¼]

and: 1 64z ia z ia z ia z ia

x x xM C C C+= + + + [¼]

Then the expected present value of the past service benefit is:

35 36 64 35

34 35 34 35

10 1020,000 20,000

80 80

Ê ˆ+ + +¥ = ¥ ¥Á ˜

Ë ¯

z ia z ia z ia z iaC C C M

s D s D

[¼]



Future service benefit – years of service from age 35 to 45 On retirement in ill-health, whenever that occurs, the member will receive the minimum pension of 20/80ths of FPS. 10/80ths have already been accounted for in the past service benefit, so the 10 years of service from age 35 to 45 are covered by the remaining 10/80ths. [¼] If we follow the same method as for the past service liability, we get:

½ 1½ 29½35 35½ 36 36½ 64 64½35½ 36½ 64½

35 34 35 34 35 34

35½ 64½35 35½ 35½ 64 64½ 64½

3534 35

1020,000

80

1020,000

80

i i i

i i


l s l s l s

i v z a i v z a

s v l

È ˘¥ + + +Í ˙

Î ˚

È ˘+ += ¥ Í ˙Í ˙Î ˚

35 36 64 35

34 35 34 35

10 1020,000 20,000

80 80


s D s D

Ê ˆ+ + += ¥ = ¥ ¥Á ˜Ë ¯

[¼]

If the guaranteed 20 years of service is valued altogether (rather than as past/future) then the expected present value of the guarantee is the sum of the past service expression and the future service expression from age 35 to 45:

35 36 64 35

34 35 34 35

20 2020,000 20,000

80 80


s D s D

Ê ˆ+ + +¥ = ¥ ¥Á ˜Ë ¯

Future service benefit – years of service from age 45 Once age 45 is reached, the minimum service of 20 years is attained and any additional service beyond this age will give extra years of service for the member’s ill health pension. [¼] Consider the year of service from age 45 to age 46. If we follow the same method as for the past service liability, the expected present value of the benefit in respect of the year of future service from age 45 to age 46 is:

10½ 11½ 29½45 45½ 46 46½ 64 64½45½ 46½ 64½

35 34 35 34 35 34

45½ 64½45 45½ 45½ 64 64½ 64½

3534 35

120,000 0.5

80

1 0.520,000

80

i i i

i i


l s l s l s

i v z a i v z a

s v l

È ˘¥ + + +Í ˙

Î ˚

È ˘+ += ¥ Í ˙Í ˙Î ˚

[¼]



Notice the main difference here; that the first term has 0.5 as the coefficient instead of 1 because, remember, if the member retires between 45 and 46, we assume it will occur halfway through the year. Here, we define:

½z ia z ia z iax x xM M C= - [¼]

So, the expected present value of the benefit in respect of the year of future service from age 45 to age 46 is:

45 46 64

34 35

45

34 35

1 0.520,000

80

120,000

80

z ia z ia z ia

z ia

C C C

s D

M

s D

È ˘+ + +¥ Í ˙Í ˙Î ˚

= ¥

[¼]

Similarly, for other ages x between 46 and 64, the expected present value of the benefit in respect of the year of future service from age x to age x + 1 is:

34 35

120,000

80

z iaxM

s D¥ [¼]

Here, we define:

1 64z ia z ia z ia z ia

x x xR M M M+= + + + [¼]

So, the expected present value of the benefit in respect of all the years of future service from age 45 is:

45 46 64

34 35

45

34 35

120,000

80

120,000

80

z ia z ia z ia

z ia

M M M

s D

R

s D

Ê ˆ+ + +¥ Á ˜Ë ¯

= ¥

[¼]



Putting these two elements together, the expected present value of the future service benefit is:

35 45

34 35

10120,000

80

z ia z iaM R

s D

Ê ˆ+¥ Á ˜

Ë ¯ [¼]

Combining this with the past service benefit gives a total expected present value of:

35 45

34 35

20120,000

80

z ia z iaM R

s D

Ê ˆ+¥ Á ˜

Ë ¯ [¼]

[Maximum 5] Alternative solution for the future service liability Some people may prefer to view the future service liability as one component rather than the breakdown we have given above into that before age 45 and that after age 45. If the member retires through ill health before age 45, he will receive the minimum pension of 20/80ths of FPS. 10/80ths have been accounted for in the past service benefit, so the remaining 10/80ths will form part of the future service benefit. [¼] If the member retires between 45 and 46, we assume it will occur halfway through the year, so his future service pension would be 10½ /80ths of FPS. Similarly, if he retires between 46 and 47, his future pension would be 11½ /80ths of FPS, and so on. [¼] So the future service benefit is:

½ 1½35 35½ 36 36½35½ 36½

35 34 35 34

9½ 10½44½ 45 45½4444½ 45½

35 34 35 34

11½ 29½46 46½ 64 64½46½ 64½

35 34 35 34

120,000 10 10

80

10 10½

11½ 29½

i i

i i

i i

i z i zv a v a

l s l s

z i ziv a v a

l s l s

i z i zv a v a

l s l s

È¥ + +Í

Î

+ +

˘+ + + ˙

˚

[½]



This can be written as:

35½ 44½35 35½ 35½ 44 44½ 44½

34 35

45½ 46½45 45½ 45½ 46 46½ 46½

34 35 34 35

64½64 64½ 64½

34 35

120,000 10

80

10½ 11½

29½

i i

i i

i

i v z a i v z a

s D

i v z a i v z a

s D s D

i v z a

s D

È Ê ˆ+ += ¥ Í Á ˜Ë ¯ÍÎ

+ + +

˘+ ˙

˙̊

[¼]

( )35 44 45 46 64

34 35

10 10½ 11½ 29½ 120,000

80

z ia z ia z ia z ia z iaC C C C C

s D

È ˘+ + + + + +Í ˙= ¥Í ˙Î ˚

[¼] The numerator in the bracketed term above is:

( )( )

( )( )

( )

35 44 45 46 64

35 64 45 46 64

35 45 46 64

46 47 64

64

35 45 45 46

10 10½ 11½ 29½

10 ½ 1½ 19½

10 ½

½

½

10 ½ ½

+ + + + + +

= + + + + + +

= + + + +

+ + + +

+

+

= + - + -

z ia z ia z ia z ia z ia

z ia z ia z ia z ia z ia

z ia z ia z ia z ia

z ia z ia z ia

z ia

z ia z ia z ia z ia z

C C C C C

C C C C C

M C C C

C C C

C

M M C M C( ) ( )46 64 64½+ + -ia z ia z iaM C [½]

Defining z iaxM and z ia

xR in the same way as before ... [¼]

... the above expression is:

35 45 46 64 35 4510 10z ia z ia z ia z ia z ia z iaM M M M M R+ + + + = + [¼]



This gives the same answer as before for the expected present value of the future service benefit, ie:

35 45

34 35

10120,000

80

z ia z iaM R

s D

Ê ˆ+¥ Á ˜

Ë ¯ [¼]

Solution X4.3

This question is CT5 April 2005 Question 5. The probability of a life aged x , who is currently sick, staying in the sick state for t years is given by:

40 0exp

tSSt x s x sp ds

.

Since the transition intensities are assumed to be constant, the expression simplifies to:

40

tSSt p e . [1]

The expected present value of the sickness benefit is then:

( )20 20400 0

2,000 2,000 tt SSte p dt e dtd r nd - + +- =Ú Ú [1]

( )20

0

2,000 te d r nd r n

- + +È ˘= -Í ˙+ +Î ˚

[1]

( )20 ln1.04 0.052,0001

ln1.04 0.05e- +È ˘= -Í ˙Î ˚+

£18,652.72= [1]

[Total 4]

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