current

Faculty of Actuaries Institute of Actuaries

EXAMINATIONS

18 April 2000 (am)

Subject 105 — Actuarial Mathematics 1

Time allowed: Three hours

INSTRUCTIONS TO THE CANDIDATE

1. Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.

2. Mark allocations are shown in brackets.

3. Attempt all 17 questions, beginning your answer to each question on aseparate sheet.

Graph paper is not required for this paper.

AT THE END OF THE EXAMINATION

Hand in BOTH your answer booklet and this question paper.

In addition to this paper you should have availableActuarial Tables and an electronic calculator.

Faculty of Actuaries105—A2000 Institute of Actuaries

105—2

1 In the context of a pension scheme, explain the term “prospective servicebenefit” and state one example. [2]

2 In a select mortality investigation, θx,r corresponds to the number of deathsaged x next birthday at entry with duration r at the policy anniversaryfollowing death. θx,r divided by the appropriate central exposed to risk givesan estimate of µ[y]+t .

Derive the values of y and t to which this estimate applies. State clearly anyassumptions used. [2]

3 Mortality levels for a certain country have been studied at national andregional level. Explain the circumstances under which a particular regionmay have an Area Comparability Factor of 0.5. [2]

4 A 25 year annual premium endowment assurance policy was sold to a lifeaged 40 exact at outset. Death benefits are payable at the end of the year ofdeath. Calculate the Zillmerised net premium reserve at the end of the tenthyear per unit sum assured.

Basis: Mortality: A1967–70 SelectInterest: 3% per annumInitial expense: 2.5% of the sum assured [3]

5 A life insurance company sells an annual premium whole life assurance policywith benefits payable at the end of the year of death. Expenses are incurred atthe start of each year, and claim expenses are nil.

(a) Write down a recursive relationship between the gross premiumreserves at successive durations, calculated on the premium basis.Define all symbols used.

(b) Explain the meaning of this formula. [3]

6 Calculate A30 30 301

: : using A1967–70 mortality and interest of 4% per annum. [3]

105—3 PLEASE TURN OVER

7 A pension scheme provides a pension of 145 of final pensionable salary for each

year of service, with a maximum of 23 of final pensionable salary, upon

retirement at age 65.

Final pensionable salary is defined as average annual salary over the 3 yearsimmediately preceding retirement.

A member is now aged exactly 47 and has 14 years of past service. He earned£40,000 in the previous 12 months.

Calculate the expected present value now of this member’s total pension onretirement, using the symbols defined in, and assumptions underlying, theFormulae and Tables for Actuarial Examinations. [3]

8 The random variables Tx and Ty represent the exact future lifetimes of twolives aged x and y respectively.

Let the random variable g(T) take the following values:

g(T) = a T T n

a T T nn x y

T T x yx y

if

if

max{ , }

max{ , }max{ , }

≤

>

RS|T|

(i) Describe the benefit which has present value equal to g(T). [2]

(ii) Express E[g(T)] as concisely as possible in the form of an annuityfunction. [1]

[Total 3]

9 Define the term “asset share” in the context of a with-profit policy. [3]

10 The number of people sick with a new disease is expected to increaseaccording to the logistic model. The initial number sick is 100,000 and it isbelieved that the number sick with the disease will never exceed 250,000. Atthe outset, sickness is assumed to grow at 5% per annum.

Calculate the number of people who are sick after exactly 10 years. [3]

11 A multiple decrement table is subject to two forces of decrement α and β.Under the assumption of a uniform distribution of the independentdecrements over each year of age, ( )aq x

α = 0.2 and ( )aq xβ = 0.05.

Calculate qxα and qx

β . [3]

105—4

12 An insurer sells combined death and sickness policies to healthy lives aged 35.The policies, which are for a term of 30 years, pay a lump sum of £20,000immediately on death, with an additional £10,000 if the deceased is sick at thetime of death. There is also a benefit of £3,000 per annum payablecontinuously to sick policyholders. There is no waiting period before benefitsare payable. Annual premiums of £500 are payable continuously by healthypolicyholders.

The mortality and sickness of the policyholders are described by the followingmultiple state model, in which the forces of transition depend on age.

px tgh, is defined as the probability that a life aged x who is in state g(g = H, S

or D) is in state h at age x + t (t ≥ 0 and h = H, S or D). The force of interest isδ.

Express in integral form, using the probabilities and the various forces oftransition, the expected present value of one such policy at its commencement.

[4]

13 A pension scheme provides the following benefit to the spouse of a member,following the death of the member in retirement:

A pension of £10,000 per annum payable during the lifetime of the spouse, butceasing 30 years after the death of the member if that is earlier. All paymentsare made on the anniversary of the member’s retirement.

Calculate the expected present value of the spouse’s benefit in the case of afemale member retiring now on her 60th birthday, who has a husband agedexactly 64.

Basis: a(55) Ultimate mortality at 8% per annum interest [8]

H = healthy S = sick

D = dead

σx

νxµx

ρx


14 (i) Discuss the suitability of the crude death rate, the standardisedmortality rate and the standardised mortality ratio for comparing

(a) the mortality, at different times, of the population of a givencountry

(b) the mortality, at a certain time, of two different occupationalgroups in the same population [6]

(ii) The following table gives a summary of mortality for one of theoccupational groups and for the country as a whole.

Occupation A Whole Country

Exposed ExposedAge group to risk Deaths to risk Deaths

20–34 15,000 52 960,000 3,10035–49 12,000 74 1,400,000 7,50050–64 10,000 109 740,000 7,100

37,000 235 3,100,000 17,700

Calculate the crude death rate, the standardised mortality rate and thestandardised mortality ratio for Occupation A. [4]

[Total 10]

15 An insurer issues 15 year term assurance policies to lives aged exactly 50 whohave provided satisfactory answers on a basic medical questionnaire. The sumassured of £100,000 is payable at the end of the year of death during the policyterm. The policy includes an option at the end of the term which allowspolicyholders to convert their policy to a whole life policy for the same sumassured (payable at the end of the year of death). The premiums payable forthis whole life policy are the office’s standard premium rates, irrespective ofthe health of the policyholder effecting the option.

The insurer calculates annual premiums for all products using A1967–70Select mortality and 4% per annum interest, with an expense allowance of 5%of all premiums.

(i) Describe:

(a) the North American method and(b) the conventional method

for pricing mortality options. [5]

(ii) Using the conventional method calculate the extra annual premium theinsurer should charge above that for a term assurance policy with nooption. [5]

(iii) Without performing any further calculations, describe what otherconsiderations would arise if the option were such that the policy couldbe converted on the 10th or the 15th anniversary. [3]

[Total 13]

105—6

16 A life insurance company issues a 4 year unit-linked policy with a levelpremium of £1,000 payable annually in advance to a life aged exactly 61. Thedeath benefit at the end of the year of death is £4,000, or the bid value of theunits if greater. The maturity value is the bid value of the units.

95% of each premium is invested in units at the offer price. The bid price is95% of the offer price. Premiums payable in the first two years are invested incapital units which are subject to a management charge of 6% per annum.Subsequent premiums are invested in accumulation units for which themanagement charge is 1% per annum. Management charges are deducted atthe end of each year from the bid value of units before benefits are paid.

Capital units are actuarially funded using factors of 61 :4t tA + − calculated using

A1967–70 Ultimate with 5% per annum interest for t = 0, 1, 2 and 3.

The company uses the following assumptions to profit test this contract:

Rate of interest on unit investments: 8% per annumRate of interest on sterling fund: 4% per annumMortality: A1967–70 UltimateInitial expenses: £100 plus 20% of the first premiumRenewal expenses: £20 on the first policy anniversary, and

increasing with inflation at 5% perannum on each subsequent anniversary

(i) Using a risk discount rate of 12% per annum calculate the expected netpresent value of the profit on this contract. [12]

(ii) Without performing any further calculations, state with reasonswhether your answer in (i) would be higher or lower for each of thefollowing, if

(a) the risk discount rate were 10% per annum(b) the policyholder were aged 50 exactly(c) capital units were actuarially funded at 4% per annum [5]

[Total 17]

105—7

17 A man aged exactly 30 effected a 35 year with profit endowment assurance fora sum assured of £50,000. Level annual premiums are payable throughout thepolicy term, ceasing on earlier death. The sum assured, with attachingbonuses, is payable at the end of the year of death, or on maturity. Compoundreversionary bonuses vest at the end of each policy year.

(i) Show that the premium (to the nearest £1) is £990 per annum using thefollowing basis:

Mortality: A1967–70 UltimateInterest: 6% per annumExpenses: Initial: £250 plus 60% of the annual premium

Renewal: 2.5% of second and subsequent premiumsBonuses: 1.923% per annum [7]

(ii) The random variables Tx and Kx represent the exact future lifetime andthe curtate future lifetime of a life aged x, respectively. Using Tx , Kx orboth, express, in stochastic form, the gross future loss random variablefor this policy at duration t, where t is an integer and 0 < t < 35. Usethose elements of the basis set out in part (i) as needed. Assume bonusdeclarations have been in line with the original bonus loadings. [3]

(iii) Immediately before the 11th premium is due, and just after the 10thbonus has brought the sum assured plus accumulated bonuses to£60,000, the policyholder wishes to convert the policy to a non-profitwhole life policy, with premiums of an unchanged amount payable untildeath.

Using the mortality and interest elements of the premium basis set outin part (i), and allowing for renewal expenses of 2.5% of all futurepremiums as well as an alteration expense of £100, calculate the revisedsum assured. [6]

(iv) State one other consideration, if any, that the office should take intoaccount before completing the alteration in (iii), and explain why theyshould do so. [2]

[Total 18]


EXAMINATIONS

April 2000


EXAMINERS’ REPORT

� Faculty of Actuaries

� Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) — April 2000 — Examiners’ Report

Page 2

1 “Prospective service benefit” means a benefit not dependent on either past or

future service explicitly, although it may depend on total expected service.

Examples include — lump sum death benefit of 4 � salary or spouse’s pension

death in service of 120

n

� final salary where n is based on deceased member’s

total potential service to NPA, including any past service.

Many candidates confused “prospective” with “future”.

2 x next birthday at entry � x � ½ on average at entry assuming birthdays

uniformly distributed over policy year.

r at policy anniversary after death means exact duration r � 1 at theanniversary before death (the start of the policy year rate interval for

duration) and hence r � ½ mid-year when the force of mortality isestimated. No assumptions are necessary.

The force estimated is �[x�½]+r�½ , so y = x � ½, t = r � ½.

3 If its age/sex profile is such that if it experienced the same age/sex specific

mortality rates as the country, then its crude death rate would be twice that of the country, i.e. the region has a much older age structure (and/or higher maleproportion) than the country.

4tV Zillmer =

[ ] : [ ] :

[ ]: [ ]:

1x t n t x t n t

x n x n

a aI

a a

� � � �

� �

��

��

Here 10V = 50:15 50:15

[40]:25 [40]:25

1 (.025)a a

a a

� ��

��

��

= 11.671 11.671

1 (.025)17.180 17.180

� ��

� �

= 0.30368


Page 3

5 (a) ( tV � + GP � et) (1 + i) = q

x+t (S) + px+t (t+1V �)

wheretV � = gross premium reserve @ time t

GP = office premium

et

= expenses incurred at time t

i = interest rate in premium/valuation basis

S = Sum Assured

px+t (qx+t) probability life aged x + t survives (dies within) one year on

premium/valuation mortality basis.

(b) Income (opening reserve plus interest on excess of premium over expense,and reserve) equals outgo (death claims and closing reserve for survivors)if assumptions are borne out.

6 1

30:30:30A = �1

30:30 : 30½A = 60:60

30:30 60:60

30:30

½D

A AD

� ��

� �

= 60:60

30:30 60:60

30:30

.04 .04½ 1 1

1.04 1.04

Da a

D

� ��

��

= .04 2487.2117 .04

½ 1 (19.701) 1 (9.943)1.04 10236.789 1.04

� ��

= ½[1 � .75773 ��(.24297)(.61758)]

= .0461

7 Future service = 18 + 14 past � total = 32 > max of 30.

� Value of benefit = 47 65

46 47

2(40,000)

3

z ra

s

s C

s D

� ��

= 2 4.28 35846

(40,000)3 4.18 15778

� � � � � ��

= £62,033

Most candidates allowed for retirement at any age, not just 65, and many failed to noticethat service exceeded 30 years so the maximum of 2/3rds applied.


Page 4

8 (i) A continuous annuity of £1 p.a. payable for a minimum of n years and

continuing thereafter until the death of the survivor of x and y.

(ii) E[g(T)] = :xy n

a .

Rather than defining asset share, some candidates discussed bonuses and policy payouts.

9 The asset share for a with-profit policy is the accumulated value of premiums

less deductions plus an allocation of profits from non-profit business. Theaccumulation is at actual earned rates of return.

The deductions include expenses, cost of benefits, tax, transfers to shareholders,cost of capital and contribution to free assets.

Rather than defining asset share, some candidates discussed bonuses and policy payouts.

10 In logistic model P(t) =

1

t KCe

�

��

or t

C e K��

� ��

� ��

As t � P(t) � K

� � K =

.05

250,000

P(0) =

1

1

250,000C

�

� ��

� � = 100,000 � C = 0.000006

� P(10) =

1

(.05)(10) 1(.000006)

250,000e

�

��

��

= 130,904

Only a minority of candidates seemed familiar with the logistic model.

11 Under UDD in single decrement table

( )x

aq� = (1 ½ )

x xq q� �

� = ½x x x

q q q� � �� = 0.2

( )x

aq� = (1 ½ )

x xq q� �

� = ½x x x

q q q� � �� = 0.05

� x x

q q� �� = 0.15 � =

x xq q� � + 0.15

( 0.15) ½( 0.15)x x x

q q q� � �� = 0.2

� 2½( ) .925 0.15x x

q q� �

� � � = 0.2


Page 5

OR 2( ) 1.85 0.1x x

q q� �

� � = 0

Roots are 2

1.85 1.85 0.4

2

�� 0.05573 (and q > 1 is invalid)

xq� = 0.05573 and

xq� = 0.20573

Alternatively, x

q� = � �( ) 1 ½

x xaq q

� �� and

xq� = � �( ) 1 ½

x xaq q

� ��

Using iteration, and taking starting values in denominators of ( )x x

q aq� �

� etc.

1st iterationx

q� = 0.2 ÷ [1 � (.5)(.05)] = .205128

x

q� = 0.05 ÷ [1 ��(.5)(.2)] = .055556

Similarly, 2nd iterationx

q� = .20571,

xq� = .05571

3rd iterationx

q� = .20573,

xq� = .05573

4th iterationx

q� = .20573,

xq� = .05573

Hence x

q� = .20573,

xq� = .05573

A large number of candidates used formulae appropriate when decrements are uniform in

the multiple decrement table, but the question specified that independent decrements were

uniform in the single decrement tables.

12 EPV = 30

0 35,500t hh

te p dt�� (premiums)

30

0 35, 3520,000 t hh

t te p dt��

�� (death from healthy)

30

0 35, 3530,000 t hs

t te p dt��

�� (death from sick)

30

0 35,3,000 t hs

te p dt�� (sickness income)

13 EPV = 30

60 64 30 60 60 641 31

10,000 (1 ) ( )f m t f f m tt t t t t

t tp p v p p p v

�

�

� �

� ��

�

= 30

64 30 60 64 60 641 31 1

10,000m t f m t f m t

t t t t tt t t

p v p p v p p v

� �

�

� � �

� ��

�

= � �30

64:30 30 64 60:94 60:6410,000

m m f m f ma p v a a� �


Page 6

64:30

m

a = 94

64 94

64

Da a

D� = 7.616 �

16.4(1.707)

5844.0

� ��

= 7.611

30 64 30

m

p v = 94

64

D

D =

16.4

5844.0 = 0.002806297

:

60:94

f ma = 1.666

60:64

f ma = 6.854

� EPV = 10,000{7.611 + (.002806297)(1.666) � 6.854} = £7,617

Very few candidates provided a satisfactory answer. Many did not attempt to deal withthe term aspect of the question, and most of those who did assumed the annuity ended 30years after retirement rather than 30 years after the pensioner’s death.

14 (i) Crude death rate is heavily influenced by mortality at older ages

(a) OK if population structures by age and sex are reasonablystable. Therefore beware large scale emigration/immigration. Easy andpractical.

(b) Not suitable — age and sex distributions in occupational groupslikely to vary significantly.

Standardised Mortality Rate

Again influenced by mortality at older ages.

(a) OK to use but need age specific mortality rates at each timepoint.

Changing population structure has no effect.

(b) Copes well with age/sex variations provided age specific rates areavailable for occupational groups.

But use of a fixed age structure may be unrepresentative of givenoccupation.

Standardised Mortality Ratio

Heavily influenced by relative mortality at older ages.

(a) Fine but ensure standard rates used are same each time.


Page 7

(b) Good except for possible problems gathering the data on agedistributions.

Use of occupational age structure maintains relevance.

(ii) Occupational A

Crude Rate

= 235 / 37,000 = 0.00635

Standardised Mortality Rate

= (960,000 � 52

15000

+ 1,400,000 � 74

12000

+ 740,000 � 109

)10,000

� 3,100,000

= (3,328 + 8,633.33 + 8,066) ÷ 3,100,000 = 0.00646

Standardised Mortality Ratio

= 235 �

3,10015,000

960,000

7,50012,000

1,400,000

7,10010,000

740,000

� ��

� ��

� ��

�

= 235 � 48.44 64.29

95.95

��

= 235 � 208.68

= 1.126

Answered quite well in general, although some students tended to describe the variousmeasures in general rather than relate them to the specific situations described.

15 (i) (a) North American Method

Relies on double decrement table with explicit proportions whochoose to exercise option and a special mortality table for thosepeople post option. While theoretically accurate, it is often difficult to obtain sufficientdata to estimate experience.


Page 8

(b) Conventional Method

Assumes all eligible lives actually take up option, and that theyare subject to Ultimate mortality as opposed to Select if normalunderwriting carried out. If there are many option dates etc., thenthe most costly from the insurers point of view is assumed.

(ii) Insurer charges 1

.95

� ��

(P[65]) (100,000) per annum for whole life policy

i.e. (.05254)(100,000) � .95 = £5,530.53 p.a.

At option date (age 65), the value of benefits provided is

100,000 A65

= (100,000)(.58705) = £58,705

The insurers net liability at option date present value of benefits – (present value of

premiums less expenses)

= 100,000 A65

– (.95)(5,530.53) 65

a��

= 58,705 – (.95)(5,530.53)(10.737)

= 58,705 – 56,412.20 = £2,292.80

Extra premium, P’, spread over term assurance policy term, is from:-

.95P � [50]:15a�� = 2,292.80 65

[50]

D

D

� P � = (2,292.80) 2,144.1713

4,581.3224

� ��

� (.95) (11.028)

� P � = £102.43 per annum

(iii) The office needs to decide which option is costlier, not just in the value ofthe option benefit, but its impact on the overall premium required overthe period to the option exercise date. In this case, it needs to compare the above option cost in premium terms

plus the 15 term assurance premium to the similarly calculated extra

premium for the 10 year option combined with a 10 year term insurancepremium. It should then charge the higher combined premium, thereby havingoption cost at any date more than covered.

Part (i) was well answered, but (ii) and (iii) were very poorly answered. Many candidatestreated the contract as a whole life from the start making the option cost the differencebetween a term assurance and a whole life policy for the life aged 50.


Page 9

16 (i) q61 = .016 013 56 p61 = .983 986 44 0 p61 = 1.0 5%

61:4A = .82703

q62 = .017 749 72 p62 = .982 250 28 1 p61 = 0.983 986 62:3A = .86624

q63 = .019 654 64 p63 = .980 345 36 2 p61 = 0.966 521 63 2:A = .90792

q64 = .021 743 10 p64 = .978 256 90 3 p61 = 0.947 524 64:1A = .95238

Capital unit fund — fully funded

Y/e fund Management FundYear Cost of alloc. Fund b/f after 8% growth Charge 6% c/f

1 902.50 – 974.70 58.48 916.222 902.50 916.22 1,964.21 117.85 1,846.363 – 1,846.36 1,994.07 119.64 1,874.434 – 1,874.43 2,024.38 121.46 1,902.92

Capital unit fund — a-funded

Available Needed at Extra death ManagementYear Cost of alloc. Fund b/f @ y/e after 8% year end cost charge

1 746.39 – 806.10 793.67 1.96 10.472 781.78 793.67 1,701.49 1,676.35 3.02 22.123 – 1,676.35 1,810.46 1,785.17 1.75 23.544 – 1,785.17 1,927.98 1,902.92 – 25.06

Premium unit fund

1%Fund Management Fund

Year Cost of alloc. Fund b/f @ year end charge c/f

3 902.50 – 974.70 9.75 964.954 902.50 964.95 2,016.85 20.17 1,996.68

Death cost (using full Cap. Units) Yr 1 � q61 (4000 – 916.22) = 49.38

Yr 2 � q62 (4000 � 1846.36) = 38.23

Yr 3 � q63 (4000 � 1874.43 � 964.95) = 22.81

Yr 4 � q64 (4000 � 1902.92 � 1996.68) = 2.18


Page 10

Sterling fund

(4%)Premium less Sterling Death Management Profit Profit

Year cost of alloc. Expense interest cost charge vector signature

1 253.61 300.00 (1.86) 49.38 10.47 �87.16 �87.162 218.22 20.00 7.93 38.23 22.12 190.04 187.003 97.50 21.00 3.06 22.81 33.29 90.04 87.034 97.50 22.05 3.02 2.18 45.23 121.52 115.14

NPV = �87.16v + 187v2 + 87.03v3 + 115.14v4 = 206.37

Alternative approach whereby entire death cost is charged to sterling fund is also valid,providing a-funded capital unit management charge is correspondingly increased.

(ii) (a) Given the shape of the cash flows, with the positives after thenegatives, a discount rate of 10% would mean larger NPV.

(b) Death cost would reduce, probability of being in force and hencepremium income would increase, causing NPV to increase. A-funding factors would also decrease, accelerating the cash flows.Given risk discount rate (12%) > sterling fund rate this willincrease NPV.

(c) At 4%, factors will be bigger, unit reserves increase and profit isdeferred. Because risk discount rate exceeds sterling fund rate,NPV decreases.

Generally well answered, although candidates often failed to give reasons for their correctconclusions in (ii).

17 (i) 6%

30:35Pa�� = 4%4%

1 1

30:35 30:35

150,000 250

1.01923A A

� ��

� ��

6%

30:35.025 .575Pa P� ��

Because bonuses vest at year end, maturities get an extra bonuscompared to deaths in last year, and so the death benefit function isdivided by (1 + bonus loading).

� �6%

30:35.975 .575P a �� = 250 + 50,000 65 65

30:35

30 30

1

1.01923

D DA

D D

� ��

� ��

6%

30:35a�� = 15.019

4%

30:35A = .27483


Page 11

4%

65

4%

30

D

D=

2144.1713

10433.31 = .20551

� P(14.0685) = 250 + 50,000{.06801 + .20551} � P = 989.87 = £990 p.a.

(ii) Gross future loss = PV future outgo � PV future income

= PV future benefit payment + PV future expenses

� PV future premiums

= G(K30+t) + (.025)(990) 30min[ 1, 35 ]

tK t

a��

��

� (990) 30min[ 1, 35 ]

tK t

a��

��

where G(K30+t) = 30 30 1

.06 30

35 35

.06 30

50,000 (1.01923) 35

50,000 (1.01923) 35

t tt K K

t

t

t

v K t

v K t

� ��

�

�

�

� � ��

� ��

(iii) Reserve before alteration = reserve after alteration + cost of alteration

Before

10V =

4%

4% 6%65 65

40:25 40:25

40 40

160,000 (.975)(990)( )

1.01923

D DA a

D D

� ��

� � ��

��

= 1

60,000 (.40005 .30690) .306901.01923

� ��

� � � (.975)(990)(13.081)

= 23,897.55 � 12,626.44 = 11,271.11 say £11,271

After

10V = x 6% 6%

40 40(.975)(990)A a� ��

= x (.15807) � (.975)(990)(14.874) = (.15807)(x) � 14,357

� 11,271 = (.15807)(x) � 14,357 + 100 � x = 161,498 say £161,500

(iv) The amount at risk is immediately significantly increased (by £100,000)and the term for which there is a death strain has been extended. There is a grave risk of adverse selection against the office unless itunderwrites the alteration as effectively a new business case. A simpledeclaration of health will not suffice in this case given the size of thechange of the immediate risk.

Parts (i), (iii) and if attempted (iv) were well answered although most students missed thedifferent bonus treatment needed for death benefits compared with the maturity benefit.Few candidates seemed familiar with the concept of the gross future loss as a randomvariable and answers to part (ii) were weak.


EXAMINATIONS

19 September 2000 (am)




1. Write your surname in full, the initials of your other names and yourCandidate’s Number on the front of the answer booklet.



Graph paper is not required for this paper



In addition to this paper you should have available, ActuarialTables and an electronic calculator.

� Faculty of Actuaries105—S2000 � Institute of Actuaries

105—2

1 Two lives, each aged x, are subject to the same mortality table. According to themortality table and a certain rate of interest, Ax = 0.5 and Ax x = 0.8.

Calculate 2x xA , using the same mortality table and interest rate. [2]

2 The following data are available in relation to a particular country and one of itsregions:

Region A Country

Age groupPopulation at30 June 2000

(000s)

Deaths in2000

Population at30 June 2000

(000s)

Deaths in2000

0–39 645 350 13,580 8,34740–59 450 2,295 8,100 45,36060+ 385 27,500 6,290 489,860

Calculate the standardised mortality ratio for region A by reference to thecountry as a whole. [2]

3 (i) A life insurance policy provides a benefit of £10,000 payable immediatelyon the death of a life (x), if (x) dies after a life (y). Express in integral formthe expected present value of the benefit under this policy. [1]

(ii) Set out, giving a reason, the most appropriate annuity factor to valueannual premiums payable under the policy. [1]

[Total 2]

4 A healthy life aged exactly 35 has a policy providing an income benefit of £50 perweek payable during sickness. The benefit is not payable beyond age 60. Thereis no deferred or waiting period.

Calculate the present value of this benefit.

Basis: Mortality: English Life Table No. 12-MalesSickness: Manchester Unity Sickness Experience 1893/97

Occupation Group AHJInterest: 4% per annum [3]

5 An annuity of 1 is payable annually in arrears while at least one of two lives, (x)and (y), is alive.

Derive an expression in terms of joint-life and single life functions for thevariance of the present value of the annuity. [3]

6 Describe three types of bonus that may be given to a with profits contract. [3]


7 In the context of a life insurance contract, explain how an asset share may bebuilt up using a recursive formula. [3]

8 (i) On 1 January 1990 a life insurance company issued a 20-year annualpremium without profits endowment assurance policy to a life then agedexactly 40, which is still in force. The sum assured of £100,000 is payableat the end of the year of death within the term of the policy, or onsurvival. The company values the policy using a modified net premiummethod, with a Zillmer adjustment.

Calculate the reserve for the policy on 31 December 1999.

Basis: Mortality: A1967–70 SelectInterest: 4% per annumZillmer adjustment: 2% of the sum assured [3]

(ii) Without carrying out any further calculations, explain how the value ofthe policy would differ if the company used a Zillmer adjustment of 1% ofthe sum assured, with the same mortality and interest assumptions. [2]

[Total 5]

9 A life insurance company issues a special reversionary annuity contract. Underthe contract an annuity of £10,000 per annum is payable monthly for life, to afemale life now aged exactly 60, on the death of a male life now aged exactly 65,provided the male life dies within 10 years of the start date of the policy.Payments commence on the first monthly policy anniversary after the date ofdeath.

Calculate the single premium required for the contract.

Basis: Mortality: a(55) Ultimate mortality, male or female as appropriateInterest: 6% per annumExpenses: none [5]

10 A pension scheme provides an ill-health retirement pension of 1/60 of FinalPensionable Salary for each year of company service, with fractions of a year tocount proportionately, subject to a maximum pension of 40/60 of FinalPensionable Salary. Retirement due to ill-health may take place at any agebefore age 65. Final Pensionable Salary is defined as the average annual salaryover the three-year period preceding retirement.

Derive commutation functions to value the ill-health retirement pension for amember aged exactly 25, who has completed exactly 5 years company service todate. Define carefully all the symbols that you use. [7]

11 Describe the component method of population projection used for British OfficialProjections, stating carefully any assumptions that you make and defining all thesymbols that you use. [7]

105—4

12 A life insurance company issues only single premium without profit termassurance policies.

The premium is to be calculated for a special 3-year term assurance for lives agedexactly 60 where the basic sum assured is £100,000, payable at the end of theyear of death.

This special policy carries a “guaranteed insurability” option that may be selectedat the outset of the 3-year policy in return for the payment of an additional singlepremium.

This option provides a guarantee to the policyholder that a further £100,000 ofsum assured may be purchased, at a subsequent policy anniversary, on normalpremium rates and without evidence of health.

The further sum assured purchased will not itself carry any further options, andwill expire at the end of the 3-year term of the original policy.

A policyholder who has paid the additional single premium can subsequentlydecide whether or not to effect the increase in sum assured and then at whichpolicy anniversary — the first or second, but not both.

The company uses the “North American experience” method for pricing theoption.

Calculate the additional single premium payable at outset for a policyholderchoosing the option.

Basis: Mortality: A1967–70 Select, except in the case of policyholders whodecide to exercise their option to increase the sum assured.For these policyholders, the mortality basis assumed toapply, from the point of increase in sum assured, is 150% ofA1967–70 Ultimate.

Interest: 5.5% per annum

Proportion of policyholders at the first anniversary who decide toincrease their sum assured at that point: 20%

Proportion of policyholders at the second anniversary who decide toincrease their sum assured at that point: 20%

Expenses: none [7]


13 A life insurance company uses the following 3-state model, to estimate the profitin respect of a 2-year combined death benefit and sickness policy issued to ahealthy policyholder aged exactly 55 at inception.

In return for a single premium of £6,000 payable at the outset the company willpay the following benefits:

£16,000 if the policyholder dies within 2 years, payable at the end of the year ofdeath;

£8,000 at the end of each of the 2 years if the policyholder is sick at those times.

Let St represent the state of the policyholder at age 55 + t, so that S0 = H and fort = 1 and 2, St = H, S or D.

The company uses transition probabilities defined as follows:

55ij

tp + = P(St+1 = jSt = i)

For t = 0 and 1 the transition probabilities are:

55HD

tp + = 0.08 55SD

tp + = 0.15 55SH

tp + = 0.75 55HS

tp + = 0.12

The transitions in the multiple state model are the only sources of randomness.

(i) One possible outcome for this policy is that the policyholder is healthy attimes 0, 1 and 2. List all the possible outcomes and the associated cashflows. [3]

(ii) Calculate the probability that each outcome occurs. [5]

(iii) Assuming a rate of interest of 8% per annum, calculate the net presentvalue at time 0 of the profit for each outcome. [2]

(iv) Calculate the mean and standard deviation of the net present value of theprofit at time 0 for the policy. [5]

[Total 15]

Healthy (H) Sick (S)

Dead (D)

105—6

14 On 1 September 1992, a life insurance company issued a whole life with profitspolicy to a life then aged exactly 45. The basic sum assured was £100,000. Thesum assured and attaching bonuses are payable immediately on death. Levelmonthly premiums are payable in advance to age 85 or until earlier death. Thecompany calculated the premium on the following basis:

Mortality: A1967–70 SelectInterest: 4% per annumBonus loading: 0.97087% per annum compound, vesting at the

beginning of each policy yearExpenses: initial: 50% of the first year’s premiums, incurred at the outset

renewal: 5% of the second and each subsequent year’s premiums,incurred at the beginning of the respective policy years.

(i) Show that the monthly premium is £229, to the nearest £. [7]

(ii) Immediately before payment of the premium due on 1 September 2000, atthe request of the policyholder, the insurance company alters the policy toa paid-up policy, with no future premiums payable. The sum assuredunder the policy is reduced, with no further bonuses payable.

The insurance company calculates the reduced sum assured afteralteration by equating prospective gross premium policy reservesimmediately before and after alteration, allowing for an expense ofalteration of £100.

Bonuses have vested at the rate of 4% per annum compound at thebeginning of each policy year from the date of issue of the policy. Thecompany calculates prospective gross premium policy reserves for thepurpose of the alteration using the following assumptions:

Mortality: A1967–70 UltimateInterest: 4% per annumExpenses: noneAllowance for future bonuses: none

Calculate the sum assured after alteration. [6][Total 13]

15 A life insurance company issues a 3-year unit-linked endowment assurancecontract to a male life aged exactly 62 under which level annual premiums of£4,000 are payable in advance throughout the term of the policy or until earlierdeath. 101% of each year’s premium is invested in units at the offer price.

The premium in the first year is used to buy capital units, with subsequent years’premiums being used to buy accumulation units. There is a bid-offer spread inunit values, with the bid price being 95% of the offer price.

The annual management charges are 5.25% on capital units and 1.25% onaccumulation units. Management charges are deducted at the end of each year,before death, surrender or maturity benefits are paid.

105—7

On the death of the policyholder during the term of the policy, there is a benefitpayable at the end of the year of death of £10,000 or the bid value of the unitsallocated to the policy, if greater. On maturity, the full bid value of the units ispayable.

A policyholder may surrender the policy only at the end of each year. Onsurrender, the bid value of the accumulation units plus a proportion of the capitalunits is payable. The proportion of the capital units payable on surrender isdetermined by the year of surrender, as follows:

Year of surrender Proportion of capitalunits paid out

1 0.852 0.903 1

The life insurance company uses the following assumptions in carrying out profittests of this contract:

Mortality: A1967–70 UltimateExpenses: initial: £300

renewal: £60 at the start of each of the second and third policyyears

Unit fund growth rate: 9% per annumSterling fund interest rate: 4.5% per annumRisk discount rate: 15% per annumSurrender rates: 15% of all policies still in force at the end of each of

the first and second years

(i) The company holds unit reserves equal to the full bid value of theaccumulation units and a proportion 62 :3t tA + − (calculated at 4%), of the full

bid value of the capital units, calculated just after the payment of thepremium due at time t (t = 0, 1 and 2). The company holds no sterlingreserves.

Calculate the profit margin on the contract. [17]

(ii) Assume instead that the company holds unit reserves equal to the full bidvalue of both the accumulation and capital units and that the companyalso holds sterling reserves, at the start of each policy year, equal to 10%of the annual premium. Calculate the revised profit margin on thecontract. [6]

[Total 23]


EXAMINATIONS

September 2000


EXAMINERS’ REPORT

� Faculty of Actuaries� Institute of Actuaries

Subject 105 (Actuarial Mathematics 1) — September 2000 — Examiners’ Report

Page 2

1 Ax = 0.5

Axx = 0.8

12 xxA = Axx

2xxA = Ax − 1

xxA = Ax − ½ xxA = 0.5 − ½ × 0.8 = 0.1.

2 The standardised mortality ratio (SMR) =, ,

, ,

cx t x t

xsc

x t x tx

E m

E m

�

�

=350 2,295 27,500

8,347 45,360 489,860645 * 450 * 385 *

13,580 8,100 6,290

+ +� �+ +� ��

=30,145

32,899.93

= 0.9163.

3 (i) £0

10,000 (1 )tt y t x x tv p p dt

∞

+− µ�

where x = age of (x)

y = age of (y)

(ii) The premium should be payable as long as (x) is alive, while the benefit isstill payable. It does not matter whether (y) is alive. The mostappropriate annuity factor is, therefore:

( )mxa�� , where m denotes frequency of payment.

4 Value = 35 60

35

50( )K KD

−

K35 = 13 13 /13 26 / 26 52 /52 104 / all35 35 35 35 35K K K K K+ + + +

= 462592 + 143625 + 154161 + 179711 + 716291 = 1656380

K60 = 970852.7

D35 = 23986


Page 3

Value = £1429.02

Alternative

60

35

50 69.056 129.405DD

� �−� �

� �, based on value of 1 p.w. all periods, whole of life

= £1428.99

5 Required: ( )xyK

Var a

( )xyK

Var a =1

( 1)xyK

Var a + −��

=1

1 xyKv

Vard

+� �−� ��

=1

2

1( )xyK

Var vd

+

= 2 22

1( ) )xy xyA A

d−

= 2 2 2 22

1( ( ) )x y xy x y xyA A A A A A

d+ − − + −

where “2” denotes evaluation at rate of interest i2 + 2i. Other functions areevaluated at rate of interest i.

6 The following are three types of guaranteed reversionary bonuses. The bonusesare usually allocated annually in arrears, following a valuation.

Simple — the rate of bonus each year is a percentage of the initial basic sumassured under a policy. The effect is that the sum assured increases linearly overthe term of the policy.

Compound — the rate of bonus each year is a percentage of the basic sumassured and the bonuses previously added. The effect is that the sum assuredincreases exponentially over the term of the policy.

Super compound — two compound bonus rates are declared each year. The firstrate (usually the lowest) is applied to the basic sum assured. The second rate isapplied to the bonuses previously added. The sum assured increasesexponentially over the term of the policy. The sum assured usually increasesmore slowly than under a compound allocation in the earlier years and faster inthe later years.


Page 4

7 An asset share is evaluated for an individual policy or for a block of policies,usually for non-unit linked policies.

The asset share is the accumulation of premiums less deductions associated withthe contract plus an allocation of profits on non-profit business, all accumulatedat the actual rate of return earned on investments. Deductions include allexpenditure associated with the contract or contracts.

The asset share may be built up recursively on a year-to-year basis. Initially, theasset share is zero. Each year, the cash flows including premiums received,deductions made to cover actual costs and provisions made to cover otherliabilities and provision for profits allocated to the policy or group of policies arerecorded. A suitable rate of return is used to accumulate the asset shares pluspremiums less deductions plus profit allocations to the year-end to determine theasset share. The process is repeated for subsequent years.

8 (i) Reserve = 50:10 50:10

[40]:20 [40]:20

100,000 1 2,000a a

a a

� �� − −� ��

��

��

50:10a�� = 8.207

[40]:20a�� = 13.772

Reserve = £39,216.24

(ii) Using a Zillmer adjustment has the effect of reducing the policy value.Changing the Zillmer adjustment from 2% of the sum assured to 1% of thesum assured has the effect of reducing the amount of the Zillmeradjustment and hence increasing the policy value, as at 31 December1999.

9 Premium = (12) (12)1065 60 65 6010,000

m f m f

a a� �� −� ��

��

(12)65 60m f

a�� = 60 60:65f f m

a a− = 10.996 − ½(7.753 + 7.335)

= 3.452


Page 5

(12)10 65 60

m f

a�� = 75 7070 70:75

65 60

m f

f f m

D la a

D l

� �� −� ��

=6809 780683

17857 897001× × (8.328 − ½(4.9 + 4.525))

= 0.381307 × 0.870326 × 3.616

= 1.200

∴ Premium = 10,000(3.452 − 1.2) = £22,520.

10 Define a service table:

lx+t = no. of members aged x + t last birthday

ix+t = no. of members who retire due to ill-health age x + t last birthday

sx+t / sx = ratio of earnings in the year of age x + t to x + t + 1 to the earnings inthe year of age x to x + 1

Define zx+t = 13 (sx−3 + sx−2 + sx−1);

ixa = value of annuity of 1 p.a. to an ill-health

retiree aged exactly x + t.

Let (AS) be the member’s expected salary earnings in the year of age 25 to 26.

Assume that ill-health retirements take place uniformly over the year of age.

Consider ill-health retirement between ages 25 + t and 25 + t + 1, t < 35.

The present value of the retirement benefits related to future service:

( ) 25 ½25 ½ 25 25

25 ½2525 25 25

½ ( ) ( ½)( )60 60

z iatit t t

t s

t AS z i Cv t ASa

s lv D

+ ++ + + +

+ ++ +

=

where 25z ia

tC + = z25+t+½ v25+t+½ i25+t 25 ½i

ta + +

and 25s D = s25 v25 l25

Similarly it may be shown that the present value of the benefits is, in total:

25 26 60 61 6425

( )½ 1½ ... 35 35 ... 35

60z ia z ia z ia z ia z ia

s

ASC C C C C

D� �+ + + + + +� �


Page 6

=

( )25 26 60 61 64

25

60 64

( )½ 1½ ... 35½ 36½ ... 39½

60

½ ... 4½

z ia z ia z ia z ia z ias

z ia z ia

ASC C C C C

D

C C

� + + + + + +�

�− +�

= ( ) ( )25 26 64 60 61 6425

( )... ...

60z ia z ia z ia z ia z ia z ia

s

ASM M M M M M

D� �+ + − + + +� �

[where z iaxM =

64

0½ ]

xz ia z ia

x t xtC C

−

+=Σ −

= 25 6025

( )60

z ia z ias

ASR R

D� �−� �

where z iaxR =

64

0

xz ia

x ttM

−

+=Σ

Similarly it may be shown that the present value of benefits related to pastservice is:

2525

5( )60

z ias

ASM

D

where 25z iaM =

30

250

.z iat

t

C +=�

11 Px(n) = Survivors to n of Px−1(n −1) + migrants during (n − 1, n) whosurvived to be age x at n (net migrants are considered, i.e.migrants less emigrants).

P0(n) = Births during (n − 1, n) + migrants during (n − 1, n) who survivedto be age 0 at n

where Px(n) is the population age x last birthday at n, where n refers tomid-year n.

Let

B(n) = births during (n − 1, n)

Mx(n) = net migrants during (n − 1, n) who survive to be age x lastbirthday at n

qx−½(n − 1) = probability that a life aged last birthday x − 1 at n − 1 dies in(n − 1, n), assuming those aged x − 1 last birthday at n − 1 havebirthdays uniformly distributed over the calendar year.

½q0(n − 1) = probability that a life born in (n − 1, n) dies in (n − 1, n).


Page 7

Then we have:

Px(n) = Px−1(n − 1) × (1 − qx−½(n − 1)) + Mx(n)

P0(n) = B(n) × (1 − ½q0(n − 1)) + M0(n)

Projections are carried out separately for each sex to give values P0(n),P1(n), ..., Px(n), ....

B(n) and Mn(x) are determined using separate models. Total births in (n − 1, n),B(n), are projected using

B(n) = { }½ ( 1) ( ) ( )f fx x xP n P n f nΣ − +

where ( )fxP n is the number of females aged x last birthday at n.

{ }½ ( 1) ( )f fx xP x P n− + gives the average female population aged x last birthday

over the year (n − 1, n).

fx(n) is the fertility rate over (n − 1, n) for women aged x last birthday at the dateof birth.

The summation is taken over all ages where fx(n) > 0.

The sex ratio at birth has been estimated empirically to be 1.06 : 1 (males :females). This ratio is used to obtain male and female births, as follows:

Bm(n) =1.06 ( )

2.06B n

Bf(n) =( )

2.06B n

Migration numbers are estimated directly from the International PassengerSurvey.


Page 8

12 Construct multiple decrement tables

For those not exercising the option:

Age No. alive No. ofdeaths

60 100,000 669.9061 79,464.08 770.9462 62,954.51 1,117.42

For those exercising the option:

Age No. alive No. ofdeaths

61 19,866.02 477.1962 35,147.46 935.79

Premiums payable:

67.987,2]01774972.0*)00970168.1(*)006699.1(

00970168.0*)006699.1(0066990.0[000,1003

20

=−−

+−+=

v

vvP

P1 = 100,000[0.00970168v + (1 − .00970168) * 0.01774972v2] = 2,498.85

P2 = 100,000[0.01774972v] = 1682.44

where 0P is the premium payable at the outset, 1P is the premium payable at the

first anniversary for additional cover and 2P is the premium payable at thesecond anniversary for additional cover.

Cost of benefits =

[ ]32 )79.935*242.117,1()19.477*294.770(90.669000,100

000,100vvv ++++

=4,730.567

Value of premiums=

1100,000

[2987.67*100,000+2,498.85*19,866.02v+1,682.44*15,738.63v2]

=3,696.12.

Option premium=

4,730.57-3,696.12 = £1,034.45


Page 9

13 (i)

Outcome Cashflow

(1) HHH 6,000, 0, 0(2) HHS 6,000, 0, −8,000(3) HHD 6,000, 0, −16,000(4) HSH 6,000, 8,000, 0(5) HSS 6,000, −8,000, −8,000(6) HSD 6,000, −8,000, −16,000(7) HD 6,000, −16,000

“−” indicates cashflow to policyholder

(ii) Complete the set of transition probabilities:

55HH

tp + = 0.8, 55SS

tp + = 0.1

The probability that each outcome occurs is:

Outcome Probability

(1) 0.64(2) 0.096(3) 0.064(4) 0.09(5) 0.012(6) 0.018(7) 0.08

1.000

(iii) The net present value of each outcome is:

Outcome NPV of Profit

(1) 6,000(2) −858.711(3) −7,717.421(4) −1,407.407(5) −8,266.118(6) −15,124.829(7) −8,814.815


Page 10

(iv) Mean = Σ NPV × liability= 6000 × 0.64 − 858.711 × 0.096 + ...= £2,060.36

Variance = ΣNPV2 × Probability − (Mean)2

= 34,009,436.35

Standard deviation = £5,831.76

14 (i) 1.04 / 1.0097087 = 1.03

� death benefits evaluated at 3% p.a.

Value of death benefits = 100,000 [45]A

= 100,000 × 1.04½ × 3%[45]A

= 100,000 × 1.04½ × 0.42060

= 42,892.952

Value of premiums = (12) 4%[45]:40

Pa��

(12)[45]:40

a�� = [45] 85 85

[45] [45]

11124

N N DD D

� �− −− � ��

� �

=99744.168 1095.1562 11 241.28824

15680.3705 24 5680.3705

− � �− −� ��

= 17.366651 − 0.438864

= 16.92779

Value of premiums = 16.92779P

Value of expenses = 0.45P + 0.05P [45]:20a��

= 0.45P + 0.05P × 17.366651

= 1.31833P

∴ 16.92779P = 42892.952 + 1.31833P

P = 2747.882

Monthly premium = £228.99


Page 11

(ii) Sum assured = 1.048 × 100,000 = 136,856.91

Value of reserves before alteration =

136,856.91 (12)53 53:32

228.99 12A a− × ��

53A = ½ 4%531.04 A = 1.04½ × 0.4226 = 0.430969

(12)53:32

a�� = 53 85 85

53 53

111

24N N D

D D� �− − −� ��

=60363.851 1095.1562 11 241.28824

14020.9326 24 4020.9326

− � �− −� ��

= 14.74004 − 0.43083

= 14.30921

Value of reserves = 19,661.094.

= 53SA = S × 0.430969

∴ S × 0.430969 + 100 = 19,661.094

∴ S = £45,388.63

15 (i) Multiple decrement table

age (x) dxq s

xq (al)x ( )dxad ( )s

xad

62 0.01774972 0.15 100,000 1774.972 14733.75463 0.01965464 0.15 83491.274 1640.991 12277.54264 0.02174310 0 69572.741 1512.727 065 68060.014


Page 12

Unit Funds (ignoring actuarial funding)

Year, t 1 2 3

Value of Capital Units at start 0 3963.790 4093.703Premium to CUs 3838 0 0Interest on CUs 345.42 356.741 368.433Management charge on CUs 219.630 226.828 234.262Value of CUs at end 3963.790 4093.703 4227.874

Value of Accumulation Units at start 0 0 4131.127Premium to AUs 0 3838 3838Interest on AUs 0 345.42 717.221Management charge on AUs 0 52.293 108.579Value of AUs at end 0 4131.127 8577.769

Surrender value of units 3369.222 7815.460 12805.643

Capital Unit Fund (allowing for actuarial funding)

Year, t 1 2 3

Actuarial funding factor 0.89097 0.92528 0.96154Value of CUs at start 0 3667.616 3936.259Premium to CUs 3419.543 0 0Interest on CUs 307.759 330.085 354.263Management charge on CUs 195.683 209.879 225.252Value of CUs at end 3531.618 3787.822 4065.270

Sterling Fund

Year, t 1 2 3

Unallocated premium 580.457 162 162Expenses 300 60 60Interest 12.621 4.59 4.59MC on Capital Units 195.683 209.879 225.252MC on Accumulation Units 0 52.293 108.579Surrender profit 23.927 15.218 0Extra death benefit 114.813 40.902 0Cost of extra allocation 113.546 123.692 162.604End of year cashflow 284.329 219.297 277.817

Probability in force 1 0.834913 0.695727Discount factor 0.869565 0.756144 0.657516Expected present value 247.243 138.445 127.088

Expected p.v. of profit = 512.776

Expected p.v. of premiums = 4000 × 2.25208 = 9008.323

Profit margin = 5.69%


Page 13

(ii) Revised Sterling Fund (ignoring reserves)

Year, t 1 2 3

Unallocated premium 162 162 162Expenses 300 60 60Interest −6.21 4.59 4.59MC on Capital Units 219.630 226.828 234.262MC on Accumulation Units 0 52.283 108.579Surrender profit 87.602 60.199 0Extra death benefit 107.141 34.89 0End of year cash flow 55.881 411.01 449.449

Reserves at start of year 400 400 400Interest on reserves 18 18 18Change in reserves at year end −66.035 −66.683 −400

Revised cashflow −260.084 95.693 467.449

Expected present value −226.160 60.412 213.835

Expected present value of profit = 48.087



EXAMINATIONS

11 April 2001 (am)

Subject 105 � Actuarial Mathematics 1



1. Write your surname in full, the initials of your other names and yourCandidate�s Number on the front of the answer booklet.







� Faculty of Actuaries105 A2001 � Institute of Actuaries

105 A2001�2

1 In the context of a unit-linked contract, state a key reason for the use of actuarialfunding of capital units. [2]

2 In the context of a pension fund, state what is meant by a transfer value. [2]

3 Under the Manchester Unity model of sickness, you are given the followingvalues:

t px = 1 − .05t2 (0 ≤ t ≤ 1)

x tz + = 0.1 (0 ≤ t ≤ 1)

Calculate sx . [3]

4 Some time ago, a life office issued an assurance policy to a life now aged exactly55. Premiums are payable annually in advance, and death benefits are paid atthe end of the year of death. The office calculates reserves using gross premiumpolicy values. The following information gives the reserve assumptions for thepolicy year just completed. Expenses are assumed to be incurred at the start ofthe policy year.

Reserve brought forward at the start of the policy year £12,500Annual premium £1,150Annual expenses £75Death benefit £50,000Mortality A1967�70 ultimateInterest 5.5% per annum

Calculate the reserve at the end of the policy year. [3]

5 Life insurance company A calculates paid-up policy values for endowmentassurance policies by applying the net premium reserve as a single premium atthe time of the alteration. It holds net premium reserves based on A1967�70ultimate mortality and 3% per annum interest.

Life insurance company B calculates its corresponding paid-up values byreducing the sum assured to (t/n) times the original sum assured, where n is theoriginal term of the policy and t is the number of premiums which have been paidat the time of the calculation of the paid-up sum assured.

The sum assured is paid at the end of the term or the end of the year of death, ifearlier. Premiums are payable annually in advance.

Identify, showing your calculations, which company pays the higher paid-up sumassured after 15 years, immediately before payment of the 16th premium, for a25-year endowment assurance policy originally taken out by a life then agedexactly 40. [4]

105 A2001�3 PLEASE TURN OVER

6 A life office prices sickness insurance contracts using the following three statemodel in which the forces of transition depend on age:

Level premiums are payable continuously. Benefits are payable continuouslyduring periods of sickness. There is no death benefit, and the contracts have adeferred period of three months and include a waiver of premiums during periodsof benefit payment. Reserves are always positive under the normal premiumbasis.

State briefly, with reasons, what effect the following changes will have on thepremium (certain increase, certain decrease, not certain), if the same net presentvalue of profit is to be achieved:

(a) an increase in the death rate from the sick state together with an increasein the rate of transition from the healthy state to the sick state

(b) a fall in the death rate from the sick state together with a fall in the rateof transition from the sick state to the healthy state. [4]

7 A population is subject to two modes of decrement, α and β, between ages x andx + 1. In the single decrement tables

t xpα =

2x

x t� �� +� �

and

t xpβ = 3

xx t

� �� +� �

where 0 ≤ t ≤ 1.

Write down an integral expression for( )xaq α . Hence or otherwise obtain anexpression for this probability in terms of x only. [6]

healthy sick

dead

ρx

σx

µx υx

105 A2001�4

8 The following data relate to a population projection being carried out using thecomponent method, and specifically give information about the female populationof the country.

P(x, t) = population at 1 January 2001+ t aged x last birthday, x, t = 0, 1, 2, ..

M(x, t) = estimated net number of emigrants from the population during theyear 2001 + t, aged x last birthday at 1 January 2001 + t

q(x, t) = independent probability that a life who attains exact age x during theyear 2001 + t, dies during that year

The following is a selection from the available data.

x P(x,0) q(x,0) q(x,1) M(x,0) M(x,1)

54 728,610 .0121 .0115 37,013 31,46155 700,369 .0136 .0129 35,868 30,12656 678,123 .0152 .0144 34,312 28,99457 620,975 .0170 .0161 31,179 24,943

Calculate, from the information given, the projected number of females aged 57last birthday at 1 January 2003. State any assumptions you make. [6]

9 Define each of the following terms and give one example of each:

(a) class selection(b) selective decrement(c) spurious selection [6]

10 A life office has just sold a single premium deferred pension policy to a lady agedexactly 45. This policy guarantees to pay a cash sum of £200,000 on her 60th

birthday, which must be used to buy a whole of life annuity at that time. Thepolicy also carries an annuity option whereby the policyholder can elect to receivea pension of £15,000 per annum payable monthly in advance from the same date,until her death.

The office invests the single premium such that the value of related assets on thepolicyholder�s 60th birthday will be normally distributed with a mean value of£250,000 and a standard deviation of £50,000. It also believes that the annualinterest rate, i, which will be available on the policyholder�s 60th birthday is arandom variable where i =

.04 with probability .25



The distribution of the value of assets on the policyholder�s 60th birthday isindependent of this annual interest rate, i.


Calculate the probability that the value of assets on the policyholder�s 60th

birthday is less than the cost of providing the annuity benefit, assuming thepolicyholder is alive at age 60.

Basis for annuity rates: Mortality: a(55) female ultimateInterest: i per annumExpenses: Nil [8]

11 An annuity of £40,000 per annum is payable annually in arrear in respect of twolives both aged 40.

• The first payment is deferred until the end of the year in which the first ofthe two lives dies, and

• Payments continue until 5 years after the death of the survivor.

Assume the two lives are independent with respect to mortality.

Calculate the expected present value of this annuity.

Basis: A1967�70 ultimate mortality4% per annum interest [10]

12 Two life offices operating in the same economy have maintained the followingrecords in respect of their male assured lives data:

deaths, subdivided by age last birthday at the preceding policyanniversary, and duration at the preceding policy anniversary

in force, each 1 January, subdivided by age next birthday at issue, andcalendar year of policy issue

(i) Describe how you would calculate select forces of mortality, definingcarefully all symbols you use and stating necessary assumptions. Stateclearly the ages and durations to which the resultant rates would apply.

[8]

(ii) Discuss briefly the advantages and disadvantages of pooling the data ofthe two companies to form one mortality rate estimate for eachcombination of age and duration. [3]

[Total 11]

105 A2001�6

13 A life insurance company sells with-profit whole life policies, with the sumassured payable immediately on the death of the life assured and with premiumspayable annually in advance ceasing with the policyholder�s death or on reachingage 65 if earlier.

The company markets two versions of this policy, one with simple reversionarybonuses and the other with compound reversionary bonuses. In both cases thebonuses are added at the end of the policy year.

The company prices the products using the following basis:

Mortality A1967�70 selectInterest 4% per annumExpenses initial £250

renewal 2% of second and subsequent premiumsclaim £150 at termination of contract

Bonuses simple 6% of basic sum assured per annumcompound 4% of accumulated sum assured and bonuses per

annum

(i) Write down an expression for the gross future loss at the point of sale foreach of these policies, assuming they are sold to a life aged x exact (x < 65)at outset. Write the expression in terms of functions of the randomvariables Tx and Kx , which represent the exact future lifetime and thecurtate future lifetime of (x), respectively. [5]

(ii) Calculate the gross premium required for each of the two policies for asum assured of £200,000 and a life aged 40 exact at outset, using theequivalence principle. [8]

(iii) After 10 years, bonuses totalling £90,000 have been declared for thecompound reversionary bonus contract. Calculate the net premiumreserve for that policy at that time, using A1967�70 ultimate mortalityand interest of 4% per annum. [4]

[Total 17]

105 A2001�7

14 A life office issues an endowment assurance with a term of five years to a lifeaged exactly 55. The sum assured is £100,000, payable at the end of the fiveyears or at the end of the year of death if earlier. Premiums are payableannually in advance throughout the term of the policy.

The office assumes that initial expenses will be £300, and renewal expenses,which are incurred at the beginning of the second and subsequent years of thepolicy, will be £30 plus 2.5% of the premium. The funds invested for the policyare expected to earn 7.5% per annum, and mortality is expected to follow theA1967�70 select life table. The office holds net premium reserves, usingA1967�70 ultimate mortality and interest of 4% per annum.

The office sets premiums so that the net present value of the profit on thecontract is 15% of the annual premium, using a risk discount rate of 12% perannum.

(i) Calculate the premium. [12]

(ii) Without carrying out any further calculations, state with brief reasonswhat the effect on the premium would be in each of the following cases:

(a) The reserves are calculated using a lower rate of interest.(b) The office uses a risk discount rate of 15%.(c) Mortality is assumed to be A1967�70 ultimate.

[6][Total 18]


EXAMINATIONS

April 2001


EXAMINERS� REPORT


Subject 105 (Actuarial Mathematics 1) � April 2001 � Examiners� Report

Page 2

Examiners� Comments

The overall standard of scripts was somewhat disappointing, especially with regard toanswers offered for Questions 10, 11 and 12 where candidates were required to applyprinciples from the syllabus to a problem that they may not have seen before. It is alsoobvious that many candidates did not read the question asked, or at least did notaddress the specifics of the question in their solution. Comments on the individualquestions follow.

Question 3

A surprisingly high number of candidates used the half-year approximation to work outthe integral without any justification, and a number omitted the factor of 52.18

Question 4

Many students used q55 and not q54 to calculate the death cost

Question 7

The majority of candidates could not quote any of the acceptable integrals and a furthernumber did not know how to obtain an expression for the force of the decrement fromthe available data.

Question 8

Common errors included adding rather than subtracting the net number of emigrantsand not applying a survival factor to these emigrants. Clearly students did not read thequestion correctly.

Question 9

Generally answered well, although a number of candidates seemed unaware of selectivedecrements.

Question 10

This question was answered very poorly. Far too many candidates averaged the interestrate as a first step and calculated the cost of annuity benefits at the expected interestrate (6%). A further number then overlooked the option when finalising their solutions.


Page 3

Question 11

This question was answered poorly by many candidates. Most candidates who madeprogress favoured the alternative answer shown below. While a number of studentsidentified the reversionary element correctly, few were able to correctly handle the 5-year element.

Question 12

While many candidates had a general understanding of the construction of selectmortality rates, most had trouble applying it with the data given. Only a minority ofcandidates appreciated that the in force data would have to be manipulated in order toget a central exposed to risk which corresponded with the data for deaths.

Question 13

Part (ii) was generally well done. The commonest errors were to limit the benefit term to25 years and to ignore that it was paid immediately on death. Part (i) was not answeredwell as students seemed to confuse random variables and their expected values. Part(iii)proved the most difficult. Many students calculated a gross premium reserve, otherscorrectly omitted allowance for expenses and future bonuses but used the grosspremium from (ii) and finally most of those who could correctly compute a net premiumdid so using select mortality.

Question 14

This was generally answered quite well. A significant minority failed to recognise it as aprofit test question, and others constructed the reserves using the gross premium. Inpart (ii), most students identified the correct effect on the NPV of the profit ( areduction) but wrongly assumed that the premium must also reduce.


Page 4

1 The office can more closely match income and outgo. The initial strain caused byhigh initial expenses is reduced by capitalising the higher management chargesfrom the capital unit fund.

2 When a member leaves a pension scheme with an entitlement to deferredbenefits, they may elect in lieu to have a cash payment made by the scheme toeither a new scheme or an individual pension policy.

3 sx = 1

052.18 t x x tp z dt+�

= 1 2

052.18 (1 .05 ) (0.1)t dt−� = 5.218

13

0

.053t

t� �

−� ��

= (5.218) .05

13

� �−� ��

= 5.131

4 In general: (tV + P − E) (1 + i) = (qx+t) (S) + (px+t) (t+1V)

Here (12,500 + 1,150 − 75) (1.055) = (q54) (50,000) + (p54) (t+1V)

q54 = .00755572

� p54 = .99244428

� t+1V = 1

.99244428{(13,575)(1.055) − (.00755572)(50000)}

= 1

.99244428{14,321.62 − 377.79} = 14,050


Page 5

5 Company A: 15 40:25V = 55:10

40:25

1a

a−��

�� = 1 −

8.37117.169

= 0.5124

PUP SA = 15 40:25

55:10

V

A =

0.51240.75619

= 0.6777

Company B: t = 15 n = 25 � PUP SA = tn

= 1525

= 35

= 0.6

Therefore Company A provides the higher PUP SA.

6 (a) Uncertain → healthy to sick transition rate increase causesreduced premium income and higher claims butincreased sickness death rate leads to reducedclaims.Depends on interaction of both effects.

(b) Definite increase → Both effects mean people are sick for longer.Therefore higher premiums will be needed tomeet higher claims.

7 ( )xaq α = 1

0( ) ( )t x x tap a dtα

+µ� = 1

0( )( )( )t x t x x tp p dtα β α

+µ�

= 1

0( )t x x t t xp p dtα α β

+µ�

But t x x tpα α+µ = ( )t x

dp

dtα−

� t x x tpα α+µ =

2d xdt x t

� �− � �+� �

= (−1) (−2) (x2) (x + t)−3 = 2

3

2( )

xx t+

∴ ( )xaq α = 2 31

3 30

2.

( ) ( )x x

dtx t x t+ +� = 2x5

1

60

1( )

dtx t+�

= 2x5

1

50

1 15 ( )x t

� �� −

+� ��

= 5

5

21

5 ( 1)x

x

� �−� �+� �


Page 6

8 We need P(57, 2) so we need to project P(55, 0) for 2 years, allowing appropriatelyfor emigrants.

2001 P(55, 0) = 700369 all have their 56th birthday during 2001 � use q(56, 0)

P(56, 1) = (700,369) (1 − .0152) − (35,868) [1 − (½) (.0152)] assuming netemigration is spread uniformly across year

= 689723.4 − 35,595.4 = 654,128.0

2002 P(57, 2) = (654,128) (1 − .0161) − (28,994) [1 − (½) (.0161)] using (q57, 1)

= 643,596.5 −28,760.6 = 614,835.9 = 614,836

9 (a) Class selection: Refers to a factor affecting relative mortality whichis a permanent feature, e.g. age, sex, smokingstatus etc.

(b) Selective decrement: When lives grouped by one decrement experiencedifferent levels of another decrement, e.g. ill healthretirers usually experience heavier mortality thanother scheme members or retired members ofsimilar age/sex

or

marriage/mortality.

(c) Spurious selection: An investigation wrongly suggests that a certainselection is present when it is not.

It usually results from unrecognised heterogeneityin the data, with perhaps changing proportions oflives subject to different underlying mortalityrates, e.g. occupational differences being theunderlying cause of �regional� mortality effects

or

changing sex mix leading to wrongly attributingmortality rate progressions to temporary initialselection.


Page 7

10 (12)60a�� = a60 +

1324

� @ 4% = .542 + 13.294 = 13.836

@ 6% = .542 + 10.996 = 11.538

@ 8% = .542 + 9.294 = 9.836

� annuity per annum available is @ 4%: (200,000 ÷ 13.836) = 14,455 p.a.@ 6%: (200,000 ÷ 11.538) = 17,334 p.a.@ 8%: (200,000 ÷ 9.836) = 20,333 p.a.

so the guaranteed minimum annuity option will only be chosen if i = .04

� if i = .06 or .08 (occurs with probability 0.75), the office will only make a loss ifassets at age 60 < £200,000.

The guarantee has value at vesting of (15,000) (12)60a��

= (15,000) (13.836) = 207,540

so the insurer makes a loss if i = .04 (with prob = .25) if assets at age60 < 207,540

� Probability of loss = (.75) [Prob (Assets < 200,000)] + (.25) [Prob(Assets < 207,540)]

= (.75) 250,000 200,000 250,000

50,000 50,000A

P� �− −� �<� ��

� � where A ~ N(250,000, 50,0002)

+ (.25)250,000 207,540 250,000

50,000 50,000A

P� �− −� �<� ��

� �

= (.75) [P(z < − 1)] + (.25) [P(z < − .8492)]

= (.75) [1 − .84134] + (.25) [1 − .80234] = (.75) (.15866) + (.25) (.19766)

= .118995 + .049415 = .16841 = 0.17


Page 8

11 From considering only the second condition, the first five payments are certainto occur:

� EPV = 5

1

40,000 t

t

v=� = 540,000a

Payments thereafter will be made if either life was alive five years earlier, withprobability 5 40:40t p−

� EPV = 5 40:406

40,000 tt

t

v p∞

−=�

Setting n = t − 5 (∴ t = n + 5) we get

EPV = 540:40

1

40,000 nn

n

v p∞

+

=� = 5

40:401

40,000 nn

n

v v p∞

=�

= 540:4040,000 v a

However, the first condition means that no payment occurs on any date whenboth lives are still alive, which occurs with probability 40:40t p .

The previous values are too high by 40:401

40,000 tn

t

v p∞

=� = 40,000 a40:40 .

Therefore the total expected present value is given by:

540:4040:40540,000[ ( ) ]a v a a+ −

= 540 40 40:40 40:40540,000[ ( ) ]a v a a a a+ + − −

= � ��

+ − −� ��

5 41:41 41:41415

40 40:40 40:40

240,000

N NNa v

D D D

= (2)(125,015.43) 109,071.05 109,071.05

40,000 4.45182 (.82193)6,986.4959 6,794.7238 6,794.7238

� �� + − −� ��

= 40,000[4.45182 + (.82193) {(2)(17.89387) − (16.05232)} − 16.05232]

= 40,000[4.45182 + (.82193) (19.73542) − 16.05232]

= (40,000)(4.62063) = 184,825


Page 9

Alternative solution:

The first condition is the sum of two reversionary annuities:

(40,000)(2)( 40 40a ) = (80,000)( 40 40:40a a− )

or

is the difference between a last survivor annuity and joint life annuity:

(40,000)( 40:40a − 40:40a ) = (80,000)( 40 40:40a a− )

The second condition pays a 5 year annuity certain, with the first payment at theend of the year of death of the survivor:

(40,000)(5

a�� ) 40:40A

Using 40:40A = 1 − 40:40da�� = 1− d(2 40 40:40a a−�� )

evaluating and summing the 2 elements leads to the same solution as above.

12 (i) Force of mortality � Deaths ÷ Corresponding central exposed to risk

In this case let θy,r = number of deaths where

y = age last birthday at previous policy anniversary

r = duration at previous policy anniversary

� both are policy year rate intervals

,

,

y rcy rE

θ estimates µ[y+½−r]+r+½

At the start of the policy year rate interval lives are aged y last birthday,with duration in force of exactly r years. Assuming birthdays are spreaduniformly over the policy year, this gives an average exact age at the startof the rate interval of y + ½, or y + ½ − r at the start of the policy.

The duration in force half-way through the rate interval (appropriate forforce of mortality) is clearly r + ½.

For the central exposed to risk, initially we define

n Px,t = number of lives in force on 1.1.n where

x = age next birthday at issue

t = calendar year of issue


Page 10

� x + n − t = age next birthday at following policy anniversary

� x + n − t − 1 = age last birthday at following policy anniversary

� x + n − t − 2 = age last birthday at previous policy anniversary

and also n − t − 1 = duration at previous policy anniversary

For correspondence with deaths we need

n Py+2+t−n, n−r−1 and n+1 Py+1+t−n, n−r

and the appropriate central exposed to risk for calendar year n is:

,c

n y rE = ½(n Py+2+t−n, n−r−1 + n+1 Py+1+t−n, n−r)

assuming all movements (new business, deaths, lapses etc.) are spreadevenly throughout the calendar year.

Then µ[y+½−r]+r+½ = ,

,

y rc

n y rn

E

θ

�

summing the central exposed to risk over the years of the study.

Alternative method for (i)

It is probably easier to actually restructure in force as follows:

At each census date, calculate r = census year − calendar year of issue

Let y = x + r

� age y − 2 last birthday at preceding policy anniversary

� duration r − 1 at preceding policy anniversary

Redefine in force as n Py,r = no. of lives in force on 1.1.n where

y = age next birthday at following policyanniversary

r = duration at following policy anniversary


Page 11

� Appropriate central exposed to risk for calendar year n is

,c

n y rE = ½(n Py+2, r+1 + n+1Py+2, r+1)

assuming n Py,r varies linearly over the calendar year

(rest of part (i) solution is as above)

(ii) Pooling the date will give rise to more credible estimates of trueunderlying mortality rates, since greater exposure means lower variance.

However, one must be wary of heterogeneity in the data from the twooffices:

e.g. differing geographical coveragediffering underwriting standardsdifferent distribution or target market etc.

13 (i) Simple bonus version:

L = 250 + (S[1 + (.06)Kx] + 150) { }min[1 ,65(.98) .02x

x

TK x

v P a P+ −− +��

Compound bonus version:

L = 250 + { }min[1 ,65( [(1.04) 150) (.98) .02x x

x

K TK x

S v P a P+ −+ − +��

(ii) Equivalence principle � E[L] = 0

Also we shall assume E[T] ��

E[K] + ½

Simple bonus:

� [ ] 1[ ] [ ] 1

[ ]

[ ] [ ]

250 ( 150) (.06 ) ( )

more easily valued as

(.94 150) .06 ( )

xx x

x

x x

DS A S IA

D

S A S IA

+++ + +

+ +

��

= [ ]:65(.98) .02x x

P a −� �+� �

��


Page 12

In this case:

250 + (1.04)½ [40][40]

[40]

{(.94) (200,000) 150} (.06) (200,000)R

AD

� �+ +� �

� ��

= [40]:25[(.98) .02]P a +��

� ½ 57,705.359250 (1.04) 188,150 (.27284) (12,000)

6,981.5977� �+ +� ��

= P[(.98) (15.609) + .02]

� 250 + (1.04)½ [51,334.85 + 99,184.22] = P[15.31682]

� P = 153,749.94 ÷ 15.31682 = £10,038 p.a.

Compound bonus:

[40]½[ ]

*250 (1.04) 200,000 150

1.04 x

AA

� �� + +� ��

= 15.31682P

* at 1i b

b−+

i.e. 0%

� ½ 200,000(250) (1.04) (150) (.27284)

1.04� �+ +� ��

= 15.31682P

� 250 + (1.04)½ [192,307.69 + 40.926] = 15.31682P

� P = 196,407.87/15.31682 = £12,823p.a.

(iii) Net Premium Reserve for WP policies

(i) allows for accrued bonuses only(ii) net premium ignoring any bonuses

� 10V = 50 50:15290,000 (NP)A a− ��

where NP = 200,000 40

40:25

Aa��

= ½(200,00) (1.04) (.27331)

15.599

= 3,573.60 p.a.


Page 13

� 10V = (290,000) (1.04)½ (.38450) − (3,573.60) (10.995)

= 113,713.23 − 39,291.73 = 74,421.50

14 (i) Reserves 1 55:5V = 1 − 56:4

55:5

a

a

��

�� = 1 −

3.7204.547

= 0.18188

Similarly 2V = .37189

3V = .57115

4V = .78007

5V = 0 assuming all claims paid in cash flow outgo

Also q[55] = .00447362 p[55] = .99552638 � 0 p[55] = 1q[55]+1 = .00625190 p[55]+1 = .99374810 1 p[55] = .995526q57 = .01049742 p57 = .98950358 2 p[55] = .989302q58 = .01168566 p58 = .98831434 3 p[55] = .978917q59 = .01299373 p59 = .98700627 4 p[55] = .967478

Opening Closing ProfitYear Premium Expense Reserve Interest Claim Reserve Vector

1 P 300 0 .075P−22.5 447.36 18106.63 1.075P−18876.492 P .025P+30 18188 .073125P+1361.8 625.19 36956.50 1.048125P−18061.893 P .025P+30 37189 .073125P+2786.9 1049.74 56515.44 1.048125P−17619.284 P .025P+30 57115 .073125P+4281.4 1168.57 77095.44 1.048125P−16897.615 P .025P+30 78007 .073125p+5848.3 100000 0 1.048125P−16174.70

� Profit Signature NPV of Profit Signature

1 1.075P − 18,876.49 × v = .95982P − 16,854.012 1.043446P − 17,981.08 × v 2 = .83183P − 14,334.413 1.036912P − 17,430.79 × v 3 = .73805P − 12,406.894 1.026027P − 16,541.36 × v 4 = .65206P − 10,512.335 1.014038P − 15,648.67 × v 5 = .57539P − 8,879.48

3.75715P − 62,987.12

NPV = .15P = 3.75715P − 62,987.12

� P = 62,987.12 ÷ (3.75715 − .15) = £17,462


Page 14

(ii) (a) Lower interest rate � larger reservesLarger reserves � profit deferredProfit deferred � NPV reduced

(since risk rate (12%) > earned (7.5%))� profit falls below 15% premium� need to increase premium to re-establish 15% margin

(b) Higher risk rate � lower NPV etc.� need higher premium to meet profit requirement

(c) Ultimate mortality � more death claims� earlier claims (NPV claims increases) and reduced

premium income� NPV falls� need to increase premium


EXAMINATIONS





1. Write your surname in full, the initials of your other names and yourCandidate�s Number on the front of the answer booklet.







� Faculty of Actuaries105 S2001 � Institute of Actuaries

105 S2001�2

1 Under the Manchester Unity model of sickness, you are given the followingvalues:

= 5xs

1

0= 0.9t xp dt�

Calculate the value of xz . [2]

2 Give a formula for 21(2003)P in terms of 20(2002)P , based on the componentmethod of population projection. ( )xP n denotes the population aged x lastbirthday at mid-year n.

State all the assumptions that you make and define carefully all the symbols thatyou use. [3]

3 A life insurance company issues a policy under which sickness benefit of £100 perweek is payable during all periods of sickness. There is a waiting period of 1 yearunder the policy.

You have been asked to calculate the premium for a life aged exactly 30, who isin good health, using the Manchester Unity model of sickness.

Describe how you would allow for the waiting period in your calculation, giving areason for your choice of method. [3]

105 S2001�3 PLEASE TURN OVER

4 An employer recruits lives aged exactly 20, all of whom are healthy whenrecruited. On entry, the lives join a scheme that pays a lump sum of £50,000immediately on death, with an additional £25,000 if the deceased was sick at thetime of death.

The mortality and sickness of the scheme members are described by the followingmultiple-state model, in which the forces of transition depend on age only.

All surviving members retire at age 65 and leave the scheme regardless of theirstate of health.

,abx tp is defined as the probability that a life who is in state a at age x (a = H, S, D)

is in state b at age x + t ( 0 and , , )t b H S D≥ = .

Write down an integral expression for the expected present value, at force ofinterest δ , of the death benefit in respect of a single new recruit. [3]

5 A pension scheme provides a pension of 1/60 of career average salary in respect ofeach full year of service, on age retirement between the ages of 60 and 65. Aproportionate amount is provided in respect of an incomplete year of service.

At the valuation date of the scheme, a new member aged exactly 40 has anannual rate of salary of £40,000.

Calculate the expected present value of the future service pension on ageretirement in respect of this member, using the Pension Fund Tables in theFormulae and Tables for Actuarial Examinations. [3]


ρx

σx

µx νx

Dead (D)

105 S2001�4

6 A life insurance company issues a special annuity contract to a male life agedexactly 70 and a female life aged exactly 60.

Under the contract, an annuity of £10,000 per annum is payable monthly to thefemale life, provided that she survives at least 10 years longer than the male life.The annuity commences on the monthly policy anniversary next following thetenth anniversary of the death of the male life and is payable for the balance ofthe female�s lifetime.

Calculate the single premium required for the contract.

Basis: Mortality: a(55) Ultimate, males or females as appropriateInterest: 8% per annumExpenses: none [4]

7 The staff of a company are subject to two modes of decrement, death and withdrawal from employment.

Decrements due to death take place uniformly over the year of age in theassociated single-decrement table: 50% of the decrements due to withdrawaloccur uniformly over the year of age and the balance occurs at the end of the yearof age, in the associated single-decrement table.

You are given that the independent rate of mortality is 0.001 per year of age andthe independent rate of withdrawal is 0.1 per year of age.

Calculate the probability that a new employee aged exactly 20 will die as anemployee at age 21 last birthday. [4]

8 The following data are available from a life insurance company relating to themortality experience of its temporary assurance policyholders.

,x dθ The number of deaths over the period 1 January 1998 to 30 June 2001,

aged x nearest birthday at entry and having duration d at the policyanniversary next following the date of death.

, ( )y eP n The number of policyholders with policies in force at time n, aged y

nearest birthday at entry and having curtate duration e at time n, wheren = 1.1.1998, 30.6.1998, 30.6.2000 and 30.6.2001.

Develop formulae for the calculation of the crude central select rates of mortalitycorresponding to the ,x dθ deaths and derive the age and duration to which these

rates apply. State all the assumptions that you make. [6]


9 (i) State the conditions necessary for gross premium retrospective and prospective reserves to be equal. [3]

(ii) Demonstrate the equality of gross premium retrospective and prospectivereserves for a whole life policy, given the conditions necessary for equality.

[4] [Total 7]

10 A life insurance company issues a special term assurance policy to two lives agedexactly 50 at the issue date, in return for the payment of a single premium. Thefollowing benefits are payable under the contract:

(i) In the event of either of the lives dying within 10 years, a sum assured of£100,000 is payable immediately on this death.

(ii) In the event of the second death within 10 years, a further sum assured of£200,000 is payable immediately on the second death.

Calculate the single premium.

Basis: Mortality: A1967�70 UltimateInterest: 4% per annumExpenses: None [8]

105 S2001�6

11 A life insurance company sells term assurance policies with terms of either 10 or20 years.

As an actuary in the life office, you have been asked to carry out the first reviewof the mortality experience of these policies. The following table shows thestatistical summary of the mortality investigation. In all cases, the central ratesof mortality are expressed as rates per 1,000 lives.

All policies 10-year policies 20-year policies

Age Numberin force

Centralmortality

rate

Numberin force

Centralmortality

rate

Numberin force

Centralmortality

rate

�24 6,991 1.08 6,013 0.86 978 2.1225�44 6,462 2.05 5,438 1.74 1,024 3.6845�64 5,815 13.26 4,942 11.55 873 22.9465� 3,051 75.70 2,570 71.53 481 97.70

Total 22,319 18,963 3,356

(i) Calculate the directly standardised mortality rate and the standardisedmortality ratio separately in respect of the 10-year and 20-year policies.In each case, use the �all policies� population as the standard population.

[6]

(ii) You have been asked to recommend which of these two summarymortality measures should be monitored on a regular basis.

Give your recommendation, explaining the reasons for your choice. [3] [Total 9]


12 A life insurance company offers an option on its 10-year without profit termassurance policies to effect a whole life without profits policy, at the expiry of the10-year term, for the then existing sum assured, without evidence of health.Premiums under the whole life policy are payable annually in advance for thewhole of life, or until earlier death.

(i) Describe the conventional method of pricing the mortality option, statingclearly the data and assumptions required. Formulae are not required.

[3]

(ii) A policyholder aged exactly 30 wishes to effect a 10-year without profitsterm assurance policy, for a sum assured of £100,000.

Calculate the additional single premium, payable at the outset, for theoption, using the conventional method.

The following basis is used to calculate the basic premiums for the termassurance policies.

Basis: Mortality: A1967�70 SelectInterest: 6% per annumExpenses: none [4]

(iii) Describe how you would calculate the option single premium for the policydescribed in part (ii) above using the North American method, statingclearly what additional data you would require and what assumptions youwould make. [4]

(iv) State, with reasons, whether it would be preferable to use theconventional method or the North American method for pricing themortality option under the policy described in part (ii) above. [3]

[Total 14]

105 S2001�8

13 (i) On 1 September 1996, a life aged exactly 50 purchased a deferred annuitypolicy, under which yearly benefit payments are to be made. The firstpayment, being £10,000, is to be made at age 60 exact if he is then alive.The payments will continue yearly during his lifetime, increasing by1.923% per annum compound.

Premiums under the policy are payable annually in advance for 10 yearsor until earlier death.

If death occurs before age 60, the total premiums paid under the policy,accumulated to the end of the year of death at a rate of interest of 1.923%per annum compound, are payable at the end of the year of death.Calculate the annual premium.

Basis: Mortality: before age 60: A1967�70 Ultimate

after age 60: a(55) Males Ultimate

Interest: 6% per annum

Expenses: initial: 10% of the initial premium, incurredat the outset

renewal: 5% of each of the second andsubsequent premiums, payable at thetime of premium payment

claim: £100, incurred at the time of paymentof the death benefit [9]

(ii) On 1 September 2001, immediately before payment of the premium thendue, the policyholder requests that the policy be altered so that there is nobenefit payable on death and the rate of increase of the annuity inpayment is to be altered. The premium under the policy is to remainunaltered as is the amount of the initial annuity payment.

The life insurance company calculates the revised terms of the policy byequating gross premium prospective reserves immediately before andafter the alteration, calculated on the original pricing basis, allowing foran expense of alteration of £100.

Calculate the revised rate of increase in payment of the annuity. [7] [Total 16]

105 S2001�9

14 A life insurance company issues a 3-year unit-linked endowment assurancecontract to a male life aged exactly 60 under which level annual premiums of£5,000 are payable in advance throughout the term of the policy or until earlierdeath. 102% of each year�s premium is invested in units at the offer price.

The premium in the first year is used to buy capital units, with subsequent years�premiums being used to buy accumulation units. There is a bid-offer spread inunit values, with the bid price being 95% of the offer price.

The annual management charges are 5% on capital units and 1% onaccumulation units. Management charges are deducted at the end of each year,before death, surrender or maturity benefits are paid.

On the death of the policyholder during the term of the policy, there is a benefitpayable at the end of the year of death of £12,000 or the bid value of the unitsallocated to the policy, if greater. On maturity, the full bid value of the units ispayable.

The policy may be surrendered only at the end of the first or the second policyyear. On surrender, the life insurance company pays the full bid value of theaccumulation units and 80% of the nominal bid value of the capital units,calculated at the time of surrender.

The company holds unit reserves equal to the full bid value of the accumulationunits and a proportion,

60 :3t tA + − (calculated at 4% interest and A1967-70 Ultimate

mortality), of the full bid value of the capital units, calculated just after thepayment of the premium due at time t (t = 0,1 and 2). The company holds nosterling reserves.

The life insurance company uses the following assumptions in carrying out profittests of this contract:

Mortality: A1967�70 Ultimate

Expenses: initial: £400

renewal: £80 at the start of each of the second and thirdpolicy years

Unit fund growth rate: 8% per annum

Sterling fund interest rate: 5% per annum

Risk discount rate: 15% per annum

Surrender rates: 20% of all policies still in force at the end of each ofthe first and second years

Calculate the profit margin on the contract. [18]


EXAMINATIONS

September 2001


EXAMINERS� REPORT


Subject 105 (Actuarial Mathematics 1) � September 2001 � Examiners� Report

Page 2

Examiners� Comments

Overall the standard of attempts was lower than the examiners would have expected. There wasevidence that many candidates spent too much time on the earlier questions, with consequent timeproblems later on.

Questions 1,2,3,4,9 and 11 were well answered. In question 5, career average salary was not dealtwith well. Candidates had difficulty with year-end decrements in question 7 and with the duration inquestion 8. Question 9(i) was poorly answered, although it was a standard question.

Question 10 was the most poorly attempted, with few candidates scoring more than half marks. Therewas an ambiguity in this question: the benefit payable on the second death could have beeninterpreted as £200,000 or £300,000. Candidates were given credit for either approach. Manycandidates did not give sufficient detail in their answers to question 12(i). Question 13(ii) was poorlyattempted and the answers to question 14 were not as strong as one would have expected for a fairlystandard question.


Page 3

11

052.18x t x x ts p z dt

��

� �

� �

� � � � �

�

� � �

1

01 1 1

0 0 0

52.18 * 55.56

0.9

x t x t x x xx

x t x t t x

l z dt s l sz

l dt l dt p dt

2 The required formula is:

� � �1221 20 2120(2003) (2002)(1 (2002)) (2003)P P q M

1220 (2002)q is the probability that a life aged 20 last birthday at mid-year 2002

dies between mid-year 2002 and mid-year 2003, assuming those aged 20 last birthday at mid-year 2002 have birthdays uniformly distributed over the

calendar year.

21(2003)M denotes the number of migrants entering the population during mid- year 2002 and mid-year 2003 who survive to be aged 21 last birthday at mid-year 2003.

The formula is applied separately to males and females.

3 To allow for the fact that benefit cannot be paid for at least one year, the sicknessbenefit could be valued using the factor

31

30

100 xK KD�

, where x is the ceasing age for benefits.

However, the factor 31K is not accurate as it takes into account sickness of alldurations, whereas a new policyholder aged 30 cannot experience sickness of alldurations from age 31. For this reason and because the numerical effect is notsignificant, I would use the factor 30K rather than 31K in the above formula.

4 The required expression is

45

20, 20 20, 200

25,000 {2 . 3 . }t HH HSt t t te p p dt��

� ��


Page 4

5 The value of the benefits is

12

401

24039

40,000 40,000 1,758,471£63,816.35

60 60 (3.48 3.58).5,204

s raRs D

� �

�

6 The required single premium is given by

� � � �10 (12) 7010 60 70 70 :7070 |70

60

10000 f m fm f

ff D

p v a a aD

� ��

� �3571.2

10000 7.308 5.1068858.7

� �

= £8,876.90

7 Construct a multiple decrement table.

Age No. alive No. deaths No. withdrawalsover year

No. withdrawalsat year end

20 100000 97.50 4997.5 4745.25

21 90159.75 87.9058

At age 20, no. of deaths = 100000*0.001(1-0.5*0.05) = 97.50

no. of withdrawals over year = 100000*0.05*(1-0.5*.001) = 4997.5

no. of withdrawals at year end = 100000*(1-0.05)*(1-0.001)*0.05 = 4745.25

Required probability = 87.9058/100000 = 0.00087906.

8 Define a census taken at time t after the start of the period of investigation(1.1.98), '

, ( )x dP t , of those lives having a policy in force at time t, who were x

nearest birthday at entry and will be duration d on the policy anniversary nextfollowing time t.

The central exposed to risk is then given by

3.5 ', ,0

( )tc

x d x dt

E P t dt�

�

� �


Page 5

Assuming that ', ( )x dP t varies linearly between the census dates the integral can

be approximated by

� � � �� ' ' ' ' ' '1 1 1 1 12 2 2 2 2, , , , , ,

1 1 1 1* 0 * 2 2 *1 2 32 2 2 2x d x d x d x d x d x dP P P P P P� � � � �

However, the census data have been recorded according to age x nearest birthdayat entry and curtate duration d at time t. The following formula may be written:

', , 1( ) ( )x d x dP t P t

�

� .

Substituting this into the equation above gives

,cx dE � � � � ��

� � � ��

1 1 12 2 2, 1 , 1 , 1 , 1

1 12 2, 1 , 1

1 1 1* 0 * 2 22 2 21 *1 2 32

x d x d x d x d

x d x d

P P P P

P P

� � � �

� �

� � �

� �

,,

,

x dx d c

x d

mE

��

� estimates � � 1x dm� �

because the average age at entry is x assuming

birthdays are uniformly distributed over the policy year and the exact durationat the start of the rate year of death is d � 1 for all lives (no assumptions arenecessary).

9 (i) Gross premium retrospective and prospective reserves will be equal if:

� The mortality and interest rate basis is the same for the retrospectiveand prospective reserves and is the same as that used to determine thegross premium at the date of issue of the policy.

� The same expenses (excluding the initial expenses) are valued in theretrospective and prospective reserves and also the expenses valued inthe retrospective reserves are the same as those used to determine theoriginal gross premium.

� The gross premium valued in the retrospective and prospectivereserves is that determined on the original basis using the equivalenceprinciple.

(ii) The prospective reserves at time t are given by

( ) ( )m mx t x t x t x tSA ea fA Ga� � � �� (a)

where S is the sum assurede is the annual rate of renewal expensesf is the claim expenseG is the annual rate of gross premium


Page 6

The retrospective reserve at time t is given by

� �( )1 1: | : | : |: |

mxx t x t x tx t

x t

DGa SA I ea fA

D�

� � � �� (b)

where I is the additional initial expense.

The original gross premium is given by

( ) ( ) 0m mx x x xGa SA I ea fA� � � � �� .. (c)

Add � �( ) ( )m mxx x x x

x t

DGa SA I ea fA

D�

� � � �� , which is identically 0, to (a).

Combining terms, e.g. ( ) ( ) ( ): |

m m mx xx x t x t

x t x t

D DGa Ga Ga

D D�

� �

� �� gives (b), the

expression for the retrospective reserve.

10 The value of benefit (a) and (b) is

��

�

�

��

�

��

|10:50:501

|10:5012100000 ��AA

� ��

�

�

��

�

��

50:50

60:60

50

60|10:50:50|10:50:50 2104.1 2

11

DD

DDadA ��

��

|10:50:50|10:50|10:50:50 2 aaa ��

98781.75857.4354

51.24729103.59513

50:50

60:6050:50|10:50:50 �

�

�

�

�

DNNa��

207.8|10:50 �a��

621178.00607.45975942.2855

50

60��

DD

57117.05857.43542117.2487

50:50

60:60��

DD

064438.00607.4597

0842.14775555.176704.104.1 21

21

50

60501|10:50 �

�

�

�

�

DMMA

� the required premium is


Page 7

� � � �� 57117.0621178.0*298781.7207.8*2038462.0104.1064438.0*2100000 21

��

= £13,369.55

Alternative solution

The benefits payable may be regarded as a sum of £200,000 payable on eitherdeath, less a sum of £100,000 payable on the first death.

� the value of the benefits is:

|10:50:501

|10:5011000002*200000 ��AA �

064438.01|10:50 �A

��

��

��

50:50

60:60|10:50:50|10:50:50

104.1 211

DDadA ��

� � 124011.057117.098781.7*038462.0104.1 21

��

� the required premium is

200000*2*0.064438-100000*0.124011

=£13,374.10

(The difference in the two answers is due to rounding)


Page 8

11 (i) The directly standardised rate (DSR) is given by

, ,

,

csx t x t

xcsx t

x

E mDSR

E�

�

�

The standardised mortality ratio (SMR) is given by

, ,

, ,

cx t x t

xc sx t x t

x

E mSMR

E m�

�

�

For 10-year policies:

6.991 * 0.86 ...13.56053

6.991DSR

�

� �

�

6.013 * 0.86 ...0.920149

6.013 *1.08 ...SMR

�

� �

�

For 20-year policies:

6.991 * 2.12 ...21.06187

6.991DSR

�

� �

�

0.978 * 2.12 ...1.424669

0.978 *1.08 ...SMR

�

� �

�

Note: In each of the above the DSR is expressed as the number of deathsper 1,000.

(ii) I would favour the standardised mortality ratio. The directly standardisedmortality rate requires ,x tm to be recorded for each age group, for the 10-

year and 20-year policies separately. The data may not be readilyavailable. The SMR requires the number of deaths in each age and policygroup only to be recorded: these data should be easily recorded.

12 (i) In pricing the mortality option using the conventional method, the actuarypricing the option assumes:

� that all lives eligible to take up the option will do so, and

� that the mortality experience of those who take up the option will bethe Ultimate experience which corresponds to the Select experiencethat would have been used as a basis if underwriting had beencompleted as normal when the option was exercised

The mortality basis used is not usually assumed to change over time, sothat the only data required are the Select and Ultimate mortality tablesused in the original pricing basis.

In pricing the mortality option, the actuary values the premium incomeassuming that the premium payable at the end of the ten years iscalculated using Select rates according to the original premium basis and


Page 9

values the premiums assuming Select rates apply only from the date ofissue of the original policy. The actuary values the liabilities similarly.The difference in the present value of the premium income and benefitliability per policy originally issued gives the additional option singlepremium, per policy issued.

(ii) Whole life premium payable = � �40100000P

Whole life premium which should be paid according to the actuary�s basis= 40100000P

Option premium = present value of the difference in premiums =

� ��

� �

10 4040 4040

30

10

100000

1 33542.311100000 0.01063 0.01058 *14.874 £41.18

33828.7641.06

lP P v a

l

� ��

� ��

� � � � �

� �

��

(iii) I would require the following data:

� an estimate of the probability of those reaching age 40 aspolicyholders, who exercise the option

� a multiple decrement table to describe the mortality and otherrelevant decrements (such as surrender) of those who exercise theoption, commencing at age 40

� the basis on which the whole life premium payable is to be calculated:this would normally be assumed to based on the 1967-70 Selectmortality, similar to the premium basis set out

I would calculate the present value of the additional liability, using themultiple decrement table from age 40 and allowing for the probability ofexercise of the option and A1967-70 Select mortality before age 40.

I would then calculate the whole life premium payable and also thepresent value of the whole life premiums payable, similarly to the methodused to calculate the additional liability.

The difference between the two values, per term assurance policy issued,would be the option premium.


Page 10

(iv) The more accurate method is the North American method. However, Iwould favour the conventional method for the following reasons.

� There may not be sufficient data available to apply the NorthAmerican method.

� If the policy were being sold in a market where the conventionalmethod was generally used for pricing, then there would be adequateexperience of the use of the method in the market.

� Even if sufficient data were available in respect of the North Americanmethod, they might not be appropriate for pricing the portfolioconcerned, particularly if the pricing were being done when thebusiness was first issued.

13 (i) Present value of annuity payments:

1.01923 1.06 1.04� � annuity payments are valued at 4% p.a.

Value = � �1625.1106.11100000100000

50

6010

%460

50

%660

��

lla

DD

��

10

1 30039.78710000 12.625 64,821.99

32669.8551.06� �

Present value of death benefits:

Present value of death benefit at age t�50

tt vsP %6

%923.1|�� , where P is the annual premium.

� � � �ttt

t vvPP %6%401923.001923.1

06.11101923.1

01923.001923.1

��

� the present value of the death benefits is

� �1%6

|10:501%4

|10:5001923.001923.1 AAP �

4% 60150:10| 50:10|

50

2855.59420.68436 0.0631824597.0607

DA AD

� � � � �

6% 601 50 601050:10|50

1 30039.7870.25736 0.55839* *0.3913632669.8551.06

lA A Al

� � � �

=0.056421


Page 11

� the present value of the death benefits is

� � PP 358347.0056421.0063182.001923.001923.1

��

Present value of premiums less expenses:

150:10| 50:10|0.95 0.05 100

0.95 *7.599 0.05 100*0.056421

7.16905 5.6421

Pa P A

P P

P

� � �

� � �

� �

��

PP 358347.099.648216421.516905.7 ��

49.518,9£�P

(ii) At the date of alteration:

Present value of annuity payments before alteration

� � � �

�

��

6%4%60 6060 5

55 55

110000 10000 (11.625 1) 10000 * 0.708453 *12.625

1.06

89442.19

D la

D l

Present value of death benefit before alteration

� �

� �

4% 4% 6%1.923% 1 1 15| 55:5| 55:5| 55:5|

1.019230.01923

1.01923*5.295953*0.045886 0.045886 0.0432490.01923

0.382777

3643.459

Ps A P A A

P P

P

� �

� � �

�

�

��

Present value of annuity payments after alteration

=89442.19+3643.459-100 = 92985.649

Present value of annuity payments before alteration, based on a rate ofinterest of 6% after age 60

10.813* 89442.19 76605.02

12.625� �


Page 12

Estimated interest rate underlying annuity payments after alteration

034479.002.0*02.7660519.8944219.89442649.9298504.0 �

�

�

�

Revised rate of escalation

%467.21034479.1

06.1��

14 Multiple decrement table

Agedxq ( )d

xal ( )dxad ( )s

xad60 0.014432 100000 1443.246 19711.3561 0.016014 78845.4 1262.596 15516.5662 0.01775 62066.25 1101.658 12192.92

Probabilities of survival

Age t xp60 161 0.78845462 0.620662

Unit fund (ignoring actuarial funding)

Year 1 2 3Fund brought forward 0 4970.97 5100.215Premiums allocated to CU 4845 0 0Interest 387.6 397.6776 408.0172Management charge 261.63 268.4324 275.4116Fund carried forward 4970.97 5100.215 5232.821

Fund brought forward 0 0 5180.274Premiums allocated to AU 0 4845 4845Interest 0 387.6 802.0219Management charge 0 52.326 108.273Fund carried forward 0 5180.274 10719.02

Surrender values 3976.776 9260.446 15951.84

Unit fund (with actuarial funding)

Year 1 2 3Actuarial funding factor 0.890605 0.925148 0.961538Fund brought forward 0 4598.885 4904.053Premiums allocated to CU 4314.979 0 0Interest 345.1983 367.9108 392.3242Management charge 233.0089 248.3398 264.8189


Page 13

Fund carried forward 4427.169 4718.456 5031.558

Sterling fund

Year 1 2 3Unallocated premium 685.021 155 155Expenses 400 80 80Interest 14.251 3.75 3.75Management charge 233.0089 300.6658 373.0918Mortality charge 109.2946 33.64881 0Surrender profit 88.7785 125.6126 0Additional allocation 135.3906 146.0999 201.2624Fund at year end 376.3742 325.2797 250.5794Present value of profit = 623.4689

Present value of premiums = 10774.61



EXAMINATIONS

9 April 2002 (am)




1. Enter all the candidate and examination details as requested on the front of your answerbooklet.

2. You must not start writing your answers in the booklet until instructed to do so by thesupervisor.


4. Attempt all 14 questions, beginning your answer to each question on a separate sheet.



Hand in BOTH your answer booklet, with any additional sheets firmly attached, and thisquestion paper.

In addition to this paper you should have available Actuarial Tables andyour own electronic calculator.

� Faculty of Actuaries105 A2002 � Institute of Actuaries

105 A2002�2

1 (i) Explain what is meant by the following expression:

3|4 q [40] 1� .

(ii) Calculate its value using A1967�70 mortality. [3]

2 Two lives, each aged exactly 35, are independent with respect to mortality and areeach subject to a constant force of mortality of 0.02.

Calculate the value of the following expression for these lives:

135:3510|20 q [4]

3 (i) Define the term Total Fertility Rate and explain the difference between ratescalculated on a cohort basis and a period basis. [3]

(ii) In the context of population projection, state, with a reason, which basis ispreferable. [2]

[Total 5]

4 Due to a downturn in the economy, the numbers unemployed in a certain country areexpected to increase. The current number unemployed is 100,000, and this isexpected to rise towards but not exceed 300,000 following the logistic growth model.The initial rate of increase in unemployment will be 25% per annum. Calculate howlong it will take for the unemployed population to reach 200,000. [5]

5 A life insurance company sold a number of 4-year single premium policies with aguaranteed amount payable at maturity.

The closest matching investment available was a 5-year zero-coupon bond. Interestrates at the time of the insurance company selling the policies and investing themoney in the bonds were 5.25% effective per annum.

The office invested all the premiums received in these assets. The insurance companyguaranteed a return of 5.0% per annum at maturity. On death, the return was notguaranteed but the company promised to pay out the full market value of the relatedasset immediately at the date of death.

If the distribution of 1 + i is log-normal with parameters � = 0.05 and � = 0.01, andmortality follows English Life Table No. 12 � Males, calculate the probability that theoffice makes a loss on a policy sold to someone aged exactly 56. You should assumethat the company sells the matching asset at the time of any claim.

(Remember that if X~lognormal with parameters � and �, then loge(X) is normallydistributed with mean � and standard deviation �.) [5]


6 A retiring employee aged exactly 60 is given a choice between the following twopensions:

Pension A is payable annually in arrear throughout the pensioner�s lifetime, with atleast 4 payments guaranteed to be made. The first payment is £20,000 and paymentsincrease by 0.9709% per annum compound thereafter.

Pension B is payable annually in arrear, with an initial payment of £13,000. Eachsubsequent payment increases by £1,000 and payments cease immediately on death.

Calculate the expected present value of each pension using the following basis:

Mortality: A1967�70 SelectInterest: 4% per annum [6]

7 Describe how nutrition and education affect mortality. [6]

8 A mortality investigation of pensioners who retired due to ill health is beingundertaken to investigate if there are any initial temporary select effects. Data inrespect of deceased pensioners are categorised as follows:

x: age last birthday at deathr: curtate number of years between retirement and death

(i) Estimates of �[y]+t are made by dividing the death data by its correspondingcentral exposed to risk. Derive the values of y and t in terms of x and r,stating clearly any assumptions you need to make. [3]

(ii) The following data are also available in respect of one pensioner:

Date of birth 1 August 1936Date of retirement 1 November 1998Date of death 1 July 2001

Calculate the contribution of this individual to each of the appropriate centralexposed to risk figures corresponding to the available summary data. [3]

[Total 6]

105 A2002�4

9 A life insurance company sells 4-year unit-linked endowment assurance contracts tomales aged 61 exact. Premiums of £1,000 are payable annually in advance.

Capital units are bought by the premium in the first year, and accumulation units arebought thereafter. 102% of each year�s premium is invested in units at the offer price.There is a bid-offer spread in unit values, with the bid price being 95% of the offerprice.

Capital units bear a management charge of 5% per annum of the bid value of the fund,and this charge is deducted at the end of every year.

The death benefit under the policy is paid at the end of the year of death, and is thefull bid value of units under the policy, after deduction of relevant managementcharges.

The pricing actuary assumes that fund growth will be 7.5% per annum and thatmortality experience will follow A1967�70 Select. He is contemplating using part ofthe management charge for actuarial funding of capital units. The actuarial factor atduration t would be [61] :4t tA

� � at a rate of interest of 4% per annum, with mortality as

above.

Assuming non-unit fund growth is 5% per annum, and ignoring expenses, calculatethe non-unit fund cash flow for the first year of the policy if:

(a) the full value of capital units is held(b) only the actuarially funded value of capital units is held [6]

10 The following 3 state model is used to price various sickness policies. The forces oftransition �� , � and � depend only on age.

The following probabilities are defined:

,ijx tp is the probability that a life aged x in state i will be in state j at age x + t;

,iix tp is the probability that a life aged x in state i will remain in state i until age x + t;

,ij

t x zp is the probability that a life aged x in state i will be in state j at age x + t, havingbeen in state j for period z.

H: healthy S: sick

D: dead

�x

�x

�x

�x


Using these probabilities and / or forces of transition, write down an expression forthe expected present value of each of the following sickness benefits for a lifecurrently aged 35 and healthy. The constant force of interest is �.

(a) £1,000 per annum payable continuously while sick, but all benefits cease atage 65

(b) £1,000 per annum payable continuously while in the sick state for anycontinuous period in excess of a year. However, any benefit period is limitedto 5 years payments, but the number of possible benefit periods is unlimited

(c) £1,000 per annum payable continuously throughout the first period of sicknessonly [6]

11 (i) A life insurance company prices endowment assurance policies allowing formortality, expenses and interest. For surrenders, it wants to base the values itis prepared to pay on gross premium reserves, using the premium basisunchanged except for the interest rate. If it is to make a profit on surrenders,state in what direction it should change the interest rate element of the basis, ifthe reserves are:

(a) retrospective(b) prospective

Give reasons for your answers. You should assume that experience is thesame as the premium basis. [4]

(ii) A policyholder aged exactly 60 has 5 years remaining on his endowmentpolicy which has a sum assured of £100,000 payable immediately on death, ormaturity, whichever occurs first. He can no longer afford to pay any furtherpremiums. He is offered a choice of:

(a) a surrender value of £41,000

(b) a paid up sum assured of £54,000

(c) a whole life policy, without future premiums, with a death benefit,payable immediately on death, of £100,000

Show which he should choose, assuming he wants the one with the highestexpected present value of benefits.

Basis: Mortality: A1967�70 UltimateInterest: 6% per annum [4]

[Total 8]

105 A2002�6

12 A small employer decides to set up a pension scheme for his 2 employees, who aredescribed by the following details:

Age(exact)

Past service(years)

Expected salaryin next year (£)

30 5 25,00035 6 20,000

The scheme will provide a pension of 1/60th of pensionable salary for each year ofservice (fractions of a year counting proportionally) on retirement for any reason.Pensionable salary is the average annual salary earned in the final 36 months ofemployment.

The employer meets the full cost of the scheme. The contribution rate is determinedby equating the expected present value of the total scheme liabilities to the expectedpresent value of contributions. Contributions are calculated to be a constantpercentage of the total salaries of the members at any time.

(i) Using the symbols defined in, and assumptions underlying, the Formulae andTables for Actuarial Examinations, calculate the contribution rate required forthe scheme. Ignore the possibility of new members joining the scheme. [8]

(ii) Immediately after the scheme is set up, a new employee joins the companyand pension scheme. She is aged exactly 40, and will earn £30,000 in the nextyear. The employer decides to maintain the contribution rate determined inpart (i) and to apply it to the new total salaries. Determine whether the fundingrate is sufficient to meet the liabilities of the extended membership. [3]

[Total 11]

13 100 people aged exactly 50 are each sold a 15-year endowment assurance policy withsum assured £100,000. The premiums are paid annually in advance, and the sumassured is paid on maturity or at the end of the year of earlier death.

The life insurance company�s assumptions are:

Mortality: A1967�70 Ultimate, and the lives are independent with respect tomortality


Expenses: Initial: £300Renewal: 2.5% of each premium, including the first

Let P be the gross annual premium.

(i) State the gross future loss random variable for one policy at the outset. [3]

105 A2002�7

(ii) Using your answer to part (i) or otherwise, evaluate, in terms of P,

(a) the mean and variance of the loss (in present value terms) for a singlepolicy at outset

(b) the mean and variance of the loss (in present value terms) for the entireportfolio at outset. [7]

Note: 50:15A at 12.36% per annum = 0.20426

(iii) Show what values the gross annual premium P can take if the companyrequires that the probability it incurs a loss (in present value terms) on theentire portfolio has to be less than 2.5%. Use the Normal approximation. [4]

[Total 14]

14 A life insurance company issues a number of 3-year term assurance contracts to livesaged exactly 60. The sum assured under each contract is £200,000, payable at the endof the year of death. Premiums are payable annually in advance for the term of thepolicy, ceasing on earlier death.

The company carries out profit tests for these contracts using the followingassumptions:

Initial expenses: £200 plus 35% of the first year�s premium

Renewal expenses: £25 plus 3% of the annual premium, incurred at the beginning ofthe second and subsequent years

Mortality: A1967�70 Ultimate

Investment return: 7% per annum

Risk discount rate: 15% per annum

Reserves: One year�s office premium

(i) Show that the office premium, to the nearest pound, is £4,396, if the netpresent value of the profit is 25% of the office premium. [10]

(ii) Calculate the cash flows if the company held zero reserves throughout thecontract, using the premium calculated in part (i). [2]

(iii) Explain why the company might not hold reserves for this contract and theimpact on profit if they did not hold any reserves. [3]

[Total 15]

Page 1


REPORT OF THE BOARD OF EXAMINERS ONTHE EXAMINATIONS HELD IN

April 2002


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 howevergiven credit for any alternative approach or interpretation which they consider to bereasonable.

K FormanChairman of the Board of Examiners

11 June 2002



Page 2

EXAMNINER�S COMMENTS

The overall standard of scripts was better than in recent sittings. However answers werevery disappointing for questions 5, 10 and 13(iii) in particular, where the question posed aproblem not seen in recent examinations. It is also clear that many candidates� statisticalknowledge or understanding is not up to the standard required. Finally candidates are urgedto read the questions carefully. In many cases the answers for questions 5, 8, 9, 10 and 12omitted elements asked for or added details not required for the question.

Comments on individual questions follow after the solution to each question.


Page 3

1 3|4 q [40] 1� is the probability that a life now aged 41, who entered the population ofinterest a year ago subject to select mortality at that time, will survive for 3 moreyears, and die during the following 4, when aged between 44 and 48.

= ( 3 p [40] 1� )( 4 q 44 ) for a 2 year select table

= 44

[40] 1

ll

�

. 44 48

44

l ll� = 44 48

[40] 1

l ll �

� = (33,309.271 � 32,934.221) / 33,484.739

= 0.0112006

Comment on Question 1Well answered, with only a small minority of candidates mixing up the survival and deathperiods.

2 135:3510|20 q = (.5)( 10|20 q 35:35 ) = .5[10 p 35:35 {1 � 20 p 45:45 }]

= .5[10 p 352 � 30 p 35

2] = .5[(e�.2)2 � (e�.6)2] = .5(.67032 ��.30119) = .1846

Comment on Question 2Answers were generally of a reasonable standard. The commonest errors related to thefactor of .5, and a range of errors in evaluating the required integrals.

3 Total fertility rates summarise the age specific fertility rates fx (i.e. the ratio of birthsto population of women aged x generating them). The summation is over all ages forwhich fx > 0, often taken as 15-49.

Cohort: fertility rates are summed (over a period of time) for women born in aspecified period e.g. all those born in the same calendar year

Period: fertility rates are summed at a point of time (e.g. the rates experienced in onecalendar year) for women of different ages

Cohort rates are generally preferred for their greater stability and their smooth rate ofchange over timeorPeriod rates are quicker and easier to obtain, and therefore suitable for immediate use

Other sensible reasons also gained credit.

Comment on Question 3

Good standard, although some candidates mixed up cohort and period rates while othersprovided formulae that dealt with numbers of births rather than fertility rates.


Page 4

4 Logistic model

{1/P(t)}{dP(t)/dt} = � � kP(t) or also P(t) = [Ce��t + (k/�)]�1 or = �/[C�e��t + k]

Current rate: � � k100,000 = .25Limiting population: � � k300,000 = 0leading to k = (.25/200,000) = 1/800,000 and � = .375 (=3/8)

We want t such that P(t) = 200,000

From t = 0 (now) 100,000 = [C + (1/300,000)] �1 so C = 2/300,000 or 0.00000666667

200,000 = [(2/300,000) e�.375t + (1/300,000)] �1

(1/200,000) � (1/300,000) = (2/300,000) e�.375t

e�.375t = (1/4) so using logs t = 3.70 years

Comment on Question 4Overall standard was quite good, although a surprising number of candidates did not seem toknow the logistic model at all. The commonest error was to use � = 0.25.

5 Insurance company received P so guaranteed maturity benefit= [(1.05)4 ] * P = 1.21550625P

The company invests P @5.25% so is due to receive 1.2915479P in 5 years.

On death, the office breaks even because it pays out exactly the value of assetavailable. This occurs with probability 4q56 = (1 � [l60/l56]) = 0.0690

At maturity (t = 4) office loses money only if yields at the time are j such that{1.2915479P / (1 + j)} < 1.21550625P i.e (1 + j) > 1.06256

Prob (1 + j > 1.06256) for lognormal (1 + j)= Prob (z > [Ln 1.06256 � 0.05] / 0.01) from standard normal= Prob (z >1.07) = 1 � .85769 = 0.14231

Maturity occurs with probability 1 � .0690 = .9310 so the overall probability of a lossis 0.9310 * 0.14231 = 0.1325 = 0.13

Comment on Question 5Very poor standard of answers. Many made no reasonable attempt. Of those who did, sometried to calculate a surrender profit or loss, even though this was clearly zero. Many tried tocalculate the value of the zero coupon bond at the end of 4 years (one year short ofredemption) by considering the distribution of (1+i)4 and accumulating rather than using thedistribution of 1+i directly and discounting.


Page 5

6 Pension AEPV = (20,000/1.009709) 4(a + v4

4 p[60] a64) @ (1.04/1.009709) = 3% = (20,000/1.009709)[3.7171 + (.88849)(.947214)(11.962 �1)] = 256,363

Pension BEPV = 12,000 a [60] + 1,000(Ia) [60] = 12,000 a [60] + (1,000S[60]+1/D[60])= 12,000[12.710 ��1] +1,000 [307,254.58/2,815.3028] = 249,657

Comment on Question 6Well answered. Common mistakes in A were not getting initial level correct (missing divisorof 1.009709) and using 4% interest for the deferred period until life annuity commences. InB, many evaluated (Ia)[60] using S[60].

7 Nutrition influences morbidity and (in longer term) mortality.Lack of nutrition leads to general weakening of bodyPoor quality increases risk of disease / hinders recoveries.Excessive / inappropriate can lead to obesity and associated diseases (e.g. hypertension,heart disease). This can arise from social factors e.g. ready processed food / fast food etc.Poor / lack of nutrition can arise from adverse economic circumstances.

Education (covering formal and also general awareness from public healthcampaigns).It influences awareness of elements of healthy lifestyle. This can affect behaviour inmany areas e.g. nutrition / diet; personal health and hygiene; awareness of effects oftobacco, alcohol, drugs;Education level will also have a bearing on income level, occupation , standard ofhousing and general lifestyle, all of which are themselves regarded as influencers ofmortality..

Other reasonable points also received credit.

Comment on Question 7Well answered overall. Some candidates were inclined to repeat the same point rather thanidentifying distinct influences on mortality.

8 (i) Age last birthday = x at start of rate interval in which diesCurtate duration = r at start of rate interval in which diesNo assumptions needed

x + .5 at mid-point of interval, with duration r +.5 so we are estimating �[x+.5�(r+.5)]+r+.5 = �[x�r]+r+.5

This does require an assumption of an even spread of retirements over the yearof age, because we have no other information about ages at entry (we can onlydeduce that they can range from x ��r + 1 to x � r �1).


Page 6

(ii) Retired aged 62 years and 3 months. Dies aged 64 years and 11 months sototal exposure is 2 years 8 months.

(62,0) 9 months(63,0) 3 months(63,1) 9 months(64,1) 3 months(64,2) 8 months

Comment on Question 8Very poorly answered, especially part (ii). Some otherwise correct answers omittedassumptions completely while others gave �standard� assumptions e.g. policy anniversariesspread evenly over the year of age when there are no policies (only retirements). Overall, theunderstanding of the different rate intervals and the associated assumptions seems confused.In part(ii), many candidates calculated the total exposure incorrectly, including in somecases not even calculating the age at death correctly.

9 (a)

Capital units no actuarial funding

Year Cost ofinvestment

Fund at endbefore m.c.

ManagementCharge

Fund at end

1 969 1,041.67 52.08 989.59

Non-unit fund

Year Premiumless cost ofallocation

Interest Death cost Managementcharge

Cashflow

1 31 1.55 0 52.08 84.63

(b)

A funding factors

[61]:4A 0.85697

[61] 1:3A�

0.89045 from (M[61]+1 ��M65 + D65) / D[61]+1= (1,337.8829 ��1,258.7316 + 2,144.1713) / 2,541.7641


Page 7

A funded capital unit fund

Year Cost ofinvestment

Fund availableat end

Fund neededat end

ManagementCharge

1 830.40 892.68 881.18 11.50

Non-unit fund

Year Premiumless cost ofallocation

Interest Death cost Managementcharge

Cashflow

1 169.60 8.48 0.78 11.50 188.80

The death cost is q[61]*(full capital unit fund � A funded capital unit fund @ t = 1)i.e. 0.00723057*(989.59-881.21)

Comment on Question 9Handled very well. Errors, where they occurred, were to include a death cost in (a), use ofthe wrong funding factor at t=1 and incorrect calculation of the death cost in (b). Somecandidates completed a full profit test for each year of the contract, wasting valuable time.

10 (a) EPV =30

35,01,000 t HS

te p dt��

�

(b) EPV = 1,0006

0 1 35,

HSt

tz

e p dzdt�

��

� �or

��

�

�

��

6

1 ,3535,350p000,1 drdtpee SS

rtr

t

HH

tt ��

�

(c) EPV = 1,000 35 35 ,0 035,

HHt r SS

t t rt

e p e p drdt� �

��

� �� or

��

��

��

�0 3535,35|35,350

)(p000,1 drdtpae rtrtSS

rtrt

HH

tt

��

Comment on Question 10This was a testing question that was not answered well at all. Most attempted (a) but oftengot it wrong, while very few candidates made any real attempt at (b) and (c), even though (b)in particular just required direct use of a formula given in the appropriate Core Reading.Where an attempt was made at (b) or (c), candidates often used the benefit ceasing age for(a), although none applied in (b) or (c).


Page 8

11 (i) The retrospective and prospective reserves equal each other on the premiumbasis. We want the SV calculation to result in a lower reserve.

Retrospective reserve needs to be done at a smaller interest rate, as it isaccumulating past excess premiums over claims/ expenses.

Prospectively, the interest rate needs to be higher than the premium basis, sothat the discounting of the excess of future outgo (claims / expenses) overpremium income results in a lower answer.

(ii) SV = 41,000

PUPSA = 54,000 60:5A = 54,000((1.06).5 1 160:5 60:5 60:5

{ }A A A� � )

where 160:5

A = v5 5 p60 = (.747258)(27,442.681/30,039.787) = .68265

EPV of PUPSA = 54,000(1.02956{.75477 � .68265} + .68265) = 40,873

Whole Life option 100,000 60A = (1.06).5 (.39136) = 40,293

So SV is best

Comment on Question 11Well answered. Those who got (i) wrong often wrestled with reserving formulae rather thanconsidering the underlying concept needed. In part (ii), a surprisingly large number ofcandidates overvalued the paid up option by multiplying the entire endowment factor by1.060.5 rather than just the death element.


Page 9

12 EPV of past pensions: (n/60)(Sal)( z iaxM + z ra

xM )/sDx

EPV future pensions: (1/60)(Sal)( z ia

xR + z ra

xR )/sDx

EPV of contributions @ 1% of salary: (.01)(Sal)(s

xN )/sDx

age salary past service sDx z iaxM z ra

xM z iaxR

z raxR

sxN

30 25,000 5 28,043 8,636 88,345 231,941 2,915,486 540,02035 20,000 6 22,276 8,513 88,345 188,977 2,473,760 417,224

EPV past pension EPV future pension EPV cont. 1%sal

7,204.78 46,764.89 4,814.218,696.18 39,844.63 3,745.95

Total 15,900.96 86,609.52 8,560.16

Total Liability = 15,900.96+86,609.52 = 102,510.48

Contribution rate needed = 102,510.48/8,560.16 = 11.98% of salary

New employee

Age salary pastservice

sDx z iaxR

z raxR

sxN

40 30,000 0 18,629 147,045 2,032,033 317,121

EPVpast

pension

EPVfuture

pension

EPVcont.

1%sal

0.00 58,486.18 5,106.89

Contribution needed = 58,486.18 / 5,106.89 = 11.45%.

Therefore the contribution rate of 11.98% established for the original 2 members ismore than that required to meet the costs of the new entrant, and the scheme is insurplus.

Comment on Question 12Answered very well, but a disappointing number of candidates overlooked the ill-healthretirement benefits.


Page 10

13 All values at t = 0

(i) Future loss random variable =50

50

min( 1,15)min( 1,15)100,000 300 .975K

Kv Pa�

��

= 50

50min( 1,15)

min( 1,15) 1100,000 300 .975K

K vv Pd

�

� ��

(ii) (a) Mean for single policy just take expected value of random variable

X = EV one policy = 100,000 50:15A + 300 � .975 P 50:15a��

= (100,000)(.44395) + 300 �(.975)P(9.823)

= 44,695 � 9.577425P

Y = Variance one policy

Variance = (using 2nd form of loss r.v.)

= [100,000 + (.975P/d)]2Var ( 50min( 1,15)Kv � )

= [100,000 + (.975P/d)]2 2 250:15 50:15( [ ] )A A�

where the 2 superscript denotes at i2 + 2i

= [100,000 + (17.225P)]2(.007168397)

or Standard Deviation = (100,000 + 17.225P)(.084666)

(b) 100 policies

Mean = 100X

Variance = 100Y assuming the lives are independent or

Standard Deviation = 10 Std Dev above


Page 11

(iii) Using Central Limit Theorem (n = 100) we can assume normality of portfolioloss.

We want Prob (loss > 0) < .025

Prob ([loss � mean]/ Std Dev > [0 � mean]/std Dev) < .025

Prob (z > � mean/Std Dev) < .025

This means that (� mean / Std Dev) > 1.96 or (mean/ std Dev) < �1.96

(100)(44,695 � 9.577425 P) < (�1.96) (84,666 + 14.5837P)

P � 463,5445/929.158448 = 4,988.86 say 4,989

Comment on Question 13A very mixed standard. It is clear some candidates do not have a good understanding of thedifference between a random variable and its expectation, at least in this context. Commonerrors in (i) were to use assurance or life annuity functions, to miss the +1 in the Kx+1 termsor to give a profit (rather than loss) random variable.

In (ii), very few got the variance correct for a single policy, usually not making theconversion of the annuity into (1-vn)/d format used in the model solution, and then missingthe cross-product or covariance term between the benefit and premium random variables. Asurprising number of candidates missed the independence of lives within the portfolio andtherefore concluded that the variance of the portfolio was 1002 times the variance of onepolicy.

In (iii), of the few candidates who attempted this part, many started with considering aloss < 0, when the loss has to be > 0 to be a loss.


Page 12

14 (i)

age qx px t�1px

60 0.01443246 0.98556754 161 0.01601356 0.98398644 0.9855675462 0.01774972 0.98225028 0.96978510

Year Prem. Expense Opening

reserveInterest Death

claimClosingreserve

Profit vector Profitsignature

NPV

1 P 0.35P+200

0 0.0455P�14

2,886.49 0.985568P �0.290068P�3,100.49

�0.290068P�3,100.49

�0.252233P�2,696.08

2 P 0.03P+25

P 0.1379P�1.75

3,202.71 0.983986P 1.123914P�3,229.46

1.107693P�3,182.85

0.837575P�2,406.69

3 P 0.03P+25

P 0.1379P�1.75

3,549.94 0 2.107900P�3,576.69

2.044210P�3,468.62

1.344101P�2,280.67

1.929443P�7,383.44

Therefore 1.929443P � 7383.44 = .25P � Premium = 4,396.36 = 4,396.

(ii) If we use this premium, and ignore reserves, the cash-flows per policy in forceat the start of each year are (�43, 1,334, 986).

(iii) As the cash flows in years 2 and 3 are all positive, there is no need to establishreserves at the end of any year.

In such a scenario, the profits emerge earlier and because the discount rateexceeds the earned rate of interest, the NPV increases.

Comment on Question 14Answered well overall. The most common error was mishandling of reserves. Adisappointing number of students started from a commutation function approach when acashflow model was needed. In (iii), a number of candidates made the general statement thatit was not necessary to hold reserves for term assurance contracts because the probability ofdeath was low, without any reference to the specifics of the cashflows in this case.


EXAMINATIONS













� Faculty of Actuaries105�S2002 � Institute of Actuaries

105 S2002�2

1 Explain in words what is meant by the logistic model for projecting the size of apopulation.

Write down a differential equation whose solution gives a formula for the size of apopulation based on the logistic model.

Define carefully all symbols that you use. [3]

2 Describe how option pricing techniques may be used to determine the value of theguarantee under a deferred annuity policy with a guaranteed minimum annuity. [3]

3 Define each of the following terms and give an example of each in life assurancebusiness:

(a) class selection(b) spurious selection(c) adverse selection [3]

4 Explain what is meant by the following, in the context of with profit life insurancecontracts:

(a) earned asset share(b) retrospective valuation reserve [4]

5 A life insurance company issues a whole life assurance policy to a life aged exactly60, paying a sum assured S, together with attaching bonuses, immediately on death.Compound bonuses are added annually in advance. Premiums under the policy arepayable annually in advance, ceasing at exact age 85 (the last premium is payable atage 84 exact) or on earlier death.

Write down an expression for the net future loss random variable at outset for thispolicy. Define carefully all the symbols that you use. [4]

6 A life insurance company issues a temporary annuity policy to two independent lives,each aged exactly 60. The annuity of £10,000 per annum is payable quarterly inarrears, while at least one of the lives is alive. The annuity is payable for a maximumof 10 years.


Basis: mortality: A1967�70 Ultimateinterest: 4% per annumexpenses: ignore [4]


7 Two lives, (x) and (y), are assumed to be independent with respect to mortality andare each assumed to be subject to a constant force of mortality of 0.01. Calculate theprobability that (x) dies more than 10 years after (y). [4]

8 Calculate

2[20]:[20]A

using A1967�70 mortality and interest of 4% per annum. [4]

9 Members of a pension scheme are subject to two modes of decrement namely death(d) and withdrawal (w). The following assumptions are made in respect of the twodecrements:

Independent rate dxq is A1967�70 Ultimate;

Independent rate wxq is 0.05 per annum at age 20 last birthday and increases by

5% at each successive age attained. (For example, the annual rate ofwithdrawal at age 20 + t last birthday is � � � �0.05 1.05 t

� );

the decrements are statistically independent;

each decrement is uniformly distributed in its single decrement table.

Calculate the probability that a new entrant aged exactly 20 will withdraw from thescheme at age 22 last birthday. [5]

10 (i) Define, giving a formula, the term �Standardised Mortality Ratio�. Define allthe symbols that you use. [2]

(ii) Show how the Standardised Mortality Ratio may be expressed as a weightedaverage, setting out clearly what function is averaged and what the weightsare. [3]

[Total 5]

105 S2002�4

11 A pension scheme provides the following deferred benefits for a member aged 55exact, who leaves service before Normal Pension Age (NPA), which is age 65 exact.

(a) A deferred pension of £10,000 per annum, payable monthly in advance fromNPA. The pension is payable for a minimum of 60 monthly payments. Thepension increases monthly in deferment and payment at the effective rate of3.846% per annum compound.

(b) On the death of the member after NPA, a dependant�s pension of 50% of themember�s pension entitlement at the date of death. The pension is payablemonthly in advance beginning on the first day of the month following the dateof the member�s death, or the fifth anniversary of the member�s NPA, if laterand increases monthly in payment at the effective rate of 3.846% per annumcompound.

Calculate the expected present value of the deferred benefits.

Basis: Mortality: A1967�70 UltimateInterest: 8% per annumProportion withdependants: 90% of members have dependants at the date of

retirement Age difference: members are the same age as their dependants

(assume that females are treated exactly the same asmales)

Expenses: none [11]

12 A life insurance company issues a disability insurance contract to a healthy life agedexactly 30. Under the contract, a benefit of £20,000 per annum is payable weekly inthe event of disability. The benefit continues to be payable during disability, until thepolicyholder recovers or reaches age 65. The benefit increases continuously inpayment at the rate of 3% per annum compound.

There is no waiting period or deferred period. Premiums continue to be payableduring periods of disability.

Disability benefit payments are valued using rates of claim inception and termination.

(i) Describe the method of valuing disability benefit payments under thiscontract, setting out the data required. [6]

(ii) Derive commutation functions for valuing the benefits payable under thecontract, stating clearly any assumptions that you make and defining carefullyall the symbols that you use. [7]

[Total 13]


13 A life insurance company issues a long-term care contract to a healthy life aged 50exact. Under the contract, the life insurance company will pay the costs of long-termcare while the policyholder satisfies the conditions for payment.

The conditions for payment are assessed each year on the policy anniversary, justbefore payment of the premium then due. If the policyholder satisfies the conditions,the full annual amount of the benefit payable is paid immediately.

Regular premiums are payable annually in advance under the policy until death andare waived during periods of benefit payment.

For those lives needing care at 100% of maximum, the current payment on the policyanniversary is £50,000. The company uses the following data in respect of theexpected proportions of lives at each age needing care at different expected costlevels, for pricing the long-term care contract.

Exact age Proportion needingcare at 50% of

maximum

Proportion needingcare at 100% of

maximum

51�70 0.01 0.0171�85 0.04 0.0686+ 0.08 0.10

Basis: Mortality: A1967�70 UltimateInterest: 6% per annumBenefit inflation: Maximum payment at 100% care level at policy

anniversary t(t = 1, 2,..) = £50,000 � (1.019231)t

Expenses: 10% of each premium

(i) Write down an expression for the expected present value of benefits (includingthe waiver of premium benefit) at outset for the contract. Define carefully allthe symbols that you use. [4]

(ii) Calculate the annual premium payable under the contract. [10] [Total 14]

105 S2002�6

14 A life insurance company issues a 2-year unit-linked endowment assurance contractto a male life aged exactly 63, under which level annual premiums of £6,000 arepayable in advance throughout the term of the policy, or until earlier death.

102% of each year�s premium is invested in units at the offer price.

The premium in the first year is used to buy capital units, with the second year�spremium being used to buy accumulation units. There is a bid-offer spread in unitvalues, with the bid price being 95% of the offer price.

The annual management charges are 5% of the bid value of capital units and 1% ofthe bid value of accumulation units. Management charges are deducted at the end ofeach year, before death, surrender or maturity benefits are paid.

On the death of the policyholder during the term of the policy, there is a benefitpayable at the end of the year of death of £10,000, or the full bid value of the unitsallocated to the policy, if greater. On maturity, the full bid value of the units ispayable.

The policyholder may surrender the policy only at the end of the first policy year.The surrender value is equal to 87% of the bid value of the capital units.

The life insurance company uses the following assumptions in carrying out profit testsof this contract:

Mortality: A1967�70 UltimateSurrender: 10% of policies then in force, occurring at the end of the

first policy yearExpenses: Initial: £500

Renewal: £100 at the start of the second policy yearUnit fund growth rate: 8% per annumNon unit fund interest rate: 4% per annumRisk discount rate: 15% per annum

(i) Calculate the net present value on this contract, assuming that the companyholds unit reserves equal to the full bid value of the accumulation units andcapital units. [12]

(ii) Assume that the company holds unit reserves equal to the full bid value of theaccumulation units and a proportion, 63 :2t tA

� �(calculated at 4% and

A1967�70 Ultimate mortality), of the full bid value of the capital units(t = 0, 1).

Calculate the net present value on the contract. [9]

(iii) Explain what the effect would be on the answers in parts (i) and (ii) if themortality assumption were changed to mortality of A1967�70 Select. [2]

[Total 23]


EXAMINATIONS

September 2002


EXAMINERS� REPORT

Introduction


K G FormanChairman of the Board of Examiners12 November 2002



Page 2

EXAMINERS� COMMENTS

The overall standard of attempts by candidates was high. A number of questions wereanswered very well. The more challenging questions were less well answered, such asQuestions 2,6,7, 12 and 13, with evidence of lack of proper preparation. A common mistake,which was also a feature of previous examinations, was to misread some of the questions.

Detailed comments are given after the solution to each question.


Page 3

1 A logistic model for projecting the size of a population is a model under which aninitial rate of growth for the population is assumed to decrease over time in proportionto the size of the population.

The model may be expressed in the form

� �

� ��

1 dP tkP t

P t dt� �� , where

P(t) is the size of the population at time t� is the rate of growth, a constant, >0k is a constant, >0.

Full credit was given for the solution set out. However, a fuller treatment would be to assumethat the initial rate of growth, � , may also be negative, in which case � may be assumed toeither increase or decrease over time in proportion to the size of the population. Fewcandidates gave this fuller treatment.

The question was answered well in general. A number of candidates were vague in thedefinition, omitting the point that the initial rate of growth decreased over time in proportionto the size of the population.

2 A guaranteed annuity rate corresponds to a call option on the bonds that would benecessary to ensure that the guarantee was met, i.e. at an exercise price that generatedthe required fixed rate of return. Alternatively, it can be modelled by an option toswap floating rate returns at the option date for fixed rate returns sufficient to meet theguaranteed option.

It is difficult to ensure that the whole investment fund corresponds to a single optiontraded in the market. An approximation is possible by using options written onindices.

At the date of policy issue, all guarantees will be out of the money, i.e. they will haveno intrinsic value because current market rates are more than sufficient to meet theguarantees, but will have a time value that is the result of the views of many investors(�the market�) of the present value of the likely future costs of the option.

Thus the market price of a suitable option produces a way of pricing the guaranteedannuity rates.

This question was poorly answered. Many candidates described other techniques rather thanoption pricing techniques.

3 (a) Class selection is the process whereby lives are divided into separate groups,within which mortality or morbidity is homogenous, where each group is


Page 4

specified by a category or class of a particular characteristic of the population.An example in life assurance business is the use of individual rating factorswhich produce mortality differences, e.g. smoking status.

(b) Spurious selection is the process whereby lives are divided into separategroups, within which mortality or morbidity is homogenous, where thedifferences in mortality or morbidity are due to factors other than those used toform the groups. An example is a change in underwriting over time leading tomortality improvements, where such improvements are ascribed to the passageof time.

(c) Adverse selection is the process whereby lives are divided into groups thattend to act against a controlled selection process imposed on the groups, inrespect of mortality or morbidity. An example is where smokers will tend toselect policies from a life office that does not use smoking status as a ratingfactor.

This question was generally well answered. Credit was given for all reasonable examples.Some candidates confused self-selection with adverse selection in part (c).

4 (a) The asset share for a with profit contract is the accumulation of premiums lessdeductions associated with the contract plus an allocation of profits on non-profits business, all accumulated at the actual rate of return earned oninvestments. The deductions include expenses, claims, cost of capital andtransfer to shareholder funds, if relevant and are based on actual experience.

(b) The retrospective valuation reserve is the expected accumulation of pastpremiums received, less expected expenses and benefits including anyreversionary or interim and terminal bonuses included in past claims.

This question was well answered. Some candidates omitted the allocation from non-profitbusiness in part (a).

5 The net future loss random variable is given by

� � 60 6060

1min( 1,25)1 K T

KS b v Pa�

�� .

b is the annual rate of future bonus

60 60,K T are the curtate and complete future lifetimes of a life aged 60

P is the annual premium

This question was well answered. A common error was the inclusion of 60K rather than160 �K .


Page 5

6 The required single premium is given by

� � � � � ��

4 4 460:60:10| 60:10| 60:60:10|

10000 10000 2a a a� �

� � � �

� �

4 703 360 708 860:10|

60

3 38 8

1516.997211.551 7.9572855.5942

7.500

Da a aD

� � � �

� � � �

�

� ��

� �

4 70:703 360:60 70:708 860:60:10|

60:60

3 38 8

1039.01728.943 5.4982487.2117

6.865

Da a aD

� ��

�

� ��

�

�

� the single premium is � � 350,81£865.6500.7*2*10000 ��

This question was relatively poorly answered. Many candidates struggled with the correctevaluation of the annuity factors.

7 The required probability is

100 0

0.01 0.1 0.010

0.020.1

00.1

, where exp 0.01

*0.01*

0.01* *0.02

0.5*0.45242

tt y y t t x t x

t t

t

p p dt p ds

e e dt

ee

e

�

� �

� � � �

��

�

�

� ��

�

�

��

� � �

�

�

� �

�

This question was answered less well than the examiners had expected, with many candidatessetting out the initial integral expression incorrectly.


Page 6

8 � ��

2 1[20]:[20] [20]:[20][20]

1[20] [20]:[20]2

A A A

A A

� �

� �

� �

� � � � � � � �20:2020:20

20

113312.0

adAA

��

�

= 1 � 0.038462*21.509= 0.17272

2[20]:[20] 0.13312 0.5*0.17272A� � �

= 0.04676

This question was well answered by most candidates.

9 Independent rates of withdrawal:

Age attained Rate20 0.0521 0.052522 0.055125

Probability of survival to age 22 =

22

20

34029.2830.95*0.9475* 0.95*0.9475* 0.89856834088.257

ll

� �

� � � � � � 055103.000079739.0*1*055125.01 21

2221

2222 ��dww qqaq

Required probability = 0.898568*0.055103 = 0.049514

This question was well answered. Some candidates used wq22 in place of � �waq 22 .

10 (i) The Standardised Mortality Ratio is the ratio of the actual deaths in apopulation compared with the expected deaths, based on standard mortalityrates.

The formula is

, ,

, ,

cx t x t

xc sx t x t

x

E m

E m

�

�, where

,cx tE is the central exposed to risk in the population between ages x and x + t


Page 7

,x tm is the central rate of mortality for the population between ages x and x +t

,s

x tm is the central rate of mortality for a standard population between ages xand x + t

(ii) The Ratio may be written in the form

,, ,

,

, ,

x tc sx t x t s

x txc sx t x t

x

mE m

m

E m

�

�

which is the weighted average of the age-specific mortality differentialsbetween the population being studied and the standard population.

i.e. ,

,

x ts

x t

m

m

weighted by the expected deaths in the population being studied based onstandard mortality.

i.e. , ,c sx t x tE m

Part (i) was well answered in general. A common error was not basing expected deaths onthe mortality of a standard population.

Part (ii) was very poorly answered, with few candidates obtaining full marks.

11 (a) The net rate of interest is 4% per annum in deferment and payment.

The value of the deferred pension in payment is

� � � �12 1265 70705|

55 6510000* * *D Da a

D D� �

��


Page 8

� �

� �

65

55

125(12)5|

70

6512

70

2144.1713 0.5851093664.5684

* 1.021537*4.4518 4.5477

1516.9972 0.7074982144.1713

8.957 0.458 8.499

DD

ia ad

DD

a

� �

� � �

� �

� �

��

��

Required value = 10000*0.585109*(4.5477+(0.707498*8.499)) = 61792

(b) The value of the pension on death in retirement is

� �

8607797.0681.27442102.23622

459.2498.5957.7

*1****10000*5.0*9.0

65

70

70:707070|70)12(

70|70

1270

65

70)12(70|70

65

70

65

70

55

65

��

��

��

�

�

��

�

�

��

��

��

��

��

ll

aaaa

alla

ll

DD

DD

��

��

Required value =

� �� 0.9*0.5*10000*0.585109* 0.707498* 0.8607797*2.459 1 0.8607797 *8.499� �

= 6147

Total value = £67,939

Attempts at valuing the benefits in part (a) were reasonable in general. Attempts at valuingthe benefits in part (b) were very poor.

12 (i) The approach uses two double decrement tables. One table relates to healthypolicyholders and decrements of falling sick and dying. Recovery andsubsequent rates of sickness are allowed for in the table. The table is used tocalculate probabilities of surviving to be a healthy policyholder at age 30+t,� �� 30

30

alal t� , 350 �� t . The table is also used to calculate the dependent initial

rate of falling sick at age 30+t, � � taq�30 , 350 �� t . The rate � � taq

�30 is calledthe inception rate for disability.

The second table relates to policyholders receiving disability benefits and hasdecrements of recovery from disability and dying (while disabled). Thesurvival probabilities from this double decrement table are used, together withan appropriate interest rate, to determine the present value at the date ofbecoming disabled of a disability annuity of £20,000 per annum, increasing inpayment continuously at the rate of 3% per annum compound and payable


Page 9

according to the policy conditions until the policyholder dies while disabled,recovers or reaches age 65.

The probabilities of surviving as a healthy policyholder to age 30+t, theinception rates for disability at age 30+t and the disability annuity payablefrom age 30+t are calculated for each value of t and are integrated or summedover the range 350 �� t .

The data required are those set out in the two multiple decrement tables above,for ages from 30 to 65.

(ii) The value of a disability benefit of £1 p.a. payable weekly to a healthy lifenow aged 30 exact is

� �

� ��

3530

3030300

20000*t

i t ittt

t

ala v a dt

al

�

��

��

�

� , where

� � � �30 30, it tal a

� � are based on a double decrement table for healthy

policyholders described in part (i) above and gives the number surviving toage 30 + t while healthy and the force of inception of disability at age 30 + t.

30i

ta �

� is a continuous annuity based on the second double decrement table

described above, evaluated at rate of interest i� , where

1 11.03

ii ��

Assume that lives becoming disabled in � �30 ,30 1t t� � � do so on average at

age 1230 t� � and the integral is approximated by

� � � ��

12

12

3035 3030

300 30

20000*ti it tt

t

ad v a

al v

� � ��

� ��

�

� , where

� �30i

tad�

is the number of lives becoming disabled at age 30 + t last birthdayin the first decrement table described in part (i) above.


Page 10

Define commutation functions as follows

� �12

12

3030 3030

iia it ttC ad v a �

� � ��

� � 3030 30D al v�

30 300

tia ia

tt

M C��

�

�

� ��

So the value is approximated by

30

3020000*

iaMD�

Poorly answered in general. The evidence was that many candidates were not well preparedfor this topic, with some candidates not attempting the question and others giving very pooranswers. For a well-prepared candidate, the question should have been relativelystraightforward and a small number of candidates did achieve high marks.

13 (i) The expected present value of benefits at outset is given by

� �4% 2 6% 250 50

50 5050 501 1 1 1

c c ct tt t

t c t c

D DI L P ID D

� �

� �

� �

� � � �

� � � ��

� � � �� ,

using A1967-70 Ultimate

50c

tI�

is the proportion of policyholders needing care at exact age 50+t, atbenefit level 1, 2c �

Benefit levels: 1c � is the benefit level at 50% of maximum 2c � is the benefit level at 100% of maximum

1 2£25,000, £50,000L L� �

P is the annual premium


Page 11

(ii)Exact age Proportion

needing careat 50% ofmaximum

Proportionneeding careat 100% ofmaximum

Totalproportion

Expectedpayment

per current life

51-70 0.01 0.01 0.02 75071-85 0.04 0.06 0.10 400086+ 0.08 0.10 0.18 7000

Present Value of Long Term Care Benefits (all calculated at 4% interest):

D50 = 4597.0607

Exact age(x to x+ t)

Expectedpayment

per current lifeA

1 50( ) /x x tN N D� ��

B

Value

A*B51-70 750 12.37730 9282.9871-85 4000 2.44004 9760.1686+ 7000 0.18574 1300.18

Total Present Value 20343.32

Present value of Waiver Payments (all calculated at 6% interest)

Present value is = P*{(0.02*(D51+D52+�.+D70)+0.1*(D71+D72+�.+D85)+0.18*(D86+D87+��)}/D50

This can be rewritten as:

PV = P*{0.02*a50+0.08*v20*(l70/l50)*a70+0.08*v35*(l85/l50)*a85}

= P*(0.02*12.120+0.08*0.31180*0.723055*7.018+0.08*0.13011*0.20712*3.297)

= 0.376084*P

So the final premium equation allowing for expenses is:

0.9*P* 50a�� (at 6%) = 20343.32+0.376084*P

i.e. P = 20343.32/(0.9*13.120�0.376084)

i.e. P = £1,779.52

The examiners expected this to be a challenging question. It required the application of basicactuarial techniques to pricing a product that was probably relatively unfamiliar to mostcandidates. A number of candidates performed very well, achieving full, or near to full,


Page 12

marks. However, the majority of candidates performed poorly, with part (i) being betteranswered than part (ii).

14 (i)

Multiple decrement table

x dxq s

xq � �xal � �dxad � �

sxad

63 0.01965464 0.1 100000 1965.464 9803.45464 0.0217431 0 88231.08 1918.417 065 86312.67

Unit Fund

Year, t 1 2

Value of Capital units at start 0 5965.164Premium to Capital units 5814 0Interest on Capital units 465.120 477.213Management charge on CUs 313.956 322.119Value of Capital units at end 5965.164 6120.258

Value of Accumulation units at start 0 0Premium to Accumulation units 0 5814Interest on Accumulation units 0 465.120Management charge on Aus 0 62.791Value of Accumulation Units at end 0 6216.329

Total value of units 5965.164 12336.587Surrender value 5189.693 0

Non-unit Fund

Unallocated premium 186 186Expenses 500 100Interest -12.56 3.44MC on Capital units 313.956 322.119MC on Accumulation units 0 62.791Surrender profit 76.023 0Extra death benefit 79.303 0End of year cash flow -15.884 474.350

Probability in force 1 0.882311Profit signature -15.884 418.52Discount factor 0.869565 0.756144Expected present value -13.812 316.465Net present value 302.65


Page 13

(ii)

Unit FundYear, t 1 2

Actuarial funding factor 0.92528 0.96154Value of Capital units at start 0 5735.744Premium to Capital units 5379.578 0Interest on Capital units 430.366 458.860Management charge on CUs 290.497 309.730Value of Capital units at end 5519.447 5884.873

Total value of units 5519.447 12101.202*Surrender value 5189.693 0*including accumulation units

Non-unit FundUnallocated premium 620.422 186Expenses 500 100Interest 4.817 3.440Surrender profit 32.327 0Extra death benefit 88.064 5.118MC on Capital units less cost of additionalallocation 99.656 79.463

MC on Accumulation units 0 62.791End of year cash flow 169.158 226.576

Probability in force 1 0.882311Profit signature 169.158 199.91Discount factor 0.869565 0.756144Expected present value 147.094 151.161Net present value £298.25

(iii) If A1967-70 Select mortality were used in the profit tests instead of A1967-70Ultimate mortality, the cost of the extra death benefit would decrease and, separately,the profit signature would increase. The effect of these two factors would be toincrease the net present value of profit in part (i) and part (ii).

Part (i) was generally well answered. A surprising number of candidates calculated theprobability of being in force for year 2 incorrectly.

A number of candidates achieved full marks for part (ii). However, in general this part wasless well answered than part (i). Common errors were the incorrect calculation of thesurrender profit, extra death benefit and management charge on capital units less the chargeof additional allocation.


Page 14

Part (iii) was well answered. A number of candidates described the effect of changing themortality basis from Ultimate to Select in the actuarial funding factors, in addition to the twofactors given in the solution. Full credit was also given for this approach.


EXAMINATIONS

9 April 2003 (am)












� Faculty of Actuaries105�A2003 � Institute of Actuaries

105 A2003�2

1 In the context of Manchester Unity sickness functions, state the relationship between26|2626s and 26|26

26z . [2]

2 (i) In the context of with profit policies, describe the super compound method ofadding bonuses. [2]

(ii) Give a reason why a life insurance company might use the super compoundmethod of adding bonuses as opposed to the compound method. [1]

[Total 3]

3 Under a policy issued by a life insurance company, the benefit payable on death, atthe end of the year of death, is a return of premiums paid without interest. A levelpremium of £1,500 is payable annually in advance, throughout the term of the policy.

For a policy in force at the start of the tenth year, you are given the followinginformation:

Reserve at the start of the year, 9V: £11,300Reserve at the end of the year per survivor, 10V: £13,200Probability of death during the year: 0.04Rate of interest earned: 5% p.a.

Calculate the profit expected to emerge at the end of the tenth year per policy in forceat the start of that year. Ignore expenses and all decrements other than death. [3]

4 Compare the use of the component method and the logistic mathematical modellingmethod for projecting the size of the population in a certain country. [4]

5 A researcher into international mortality experience is interested in comparing deathrates in different countries by cause of death (cancer, heart disease, accidents etc.).An initial study compares crude death rates by cause of death for each country, andindicates a wide range of experience among the different countries.

(i) Comment on the approach of using crude rates for this comparison, indicatingany advantages and disadvantages of this method. [2]

(ii) Suggest an alternative approach which addresses any shortcomings identifiedin (i). You should assume any data required are available. [2]

[Total 4]


6 A life insurance company sells disability insurance contracts, under which the benefitis £100 per week, payable while a life insured is alive, disabled and aged not morethan 65. It calculates premiums and reserves using the inception rate / disabilityannuity methodology.

Calculate the expected present value of future benefit payments for the following twopolicyholders:

(a) A 45 year old who is healthy at the valuation date, and whose policy has adeferred period of one year. The value should take into account all possiblefuture periods of sickness claims.

(b) A 55 year old who has been receiving benefit payments for the last two years.The value should allow only for the remaining payments under the currentsickness claim.

Basis: Interest: 6% per annumMorbidity & Mortality: S(ID) in the Actuarial Tables [4]

7 A life insurance company issues a reversionary annuity policy to a male and a female,both of whom are aged exactly 60.

The annuity commences immediately on the death of the first of the lives to die and ispayable subsequently while the second life is alive, for a maximum period of 20 yearsafter the commencement date of the policy.

The annual amount of the annuity is £10,000 and is payable continuously.

Calculate the single premium for the policy.

Basis: Mortality: PMA92C20 for the male life and PFA92C20 for the female life.The lives are independent with respect to mortality.


Expenses: Initial: £300 incurred at the outset

Annuity: 2% per annum of the annuity payment, incurredcontinuously while the annuity is being paid

[7]

105 A2003�4

8 An insurer sells a special 3-year single premium non-profit term assurance policy foran initial sum assured of £250,000. This policy includes an option such that thepolicyholder can double the sum assured at the end of the second year of the policy bypaying an additional premium at that time, based on normal mortality rates, withoutevidence of health.

All death benefits are payable immediately on death.

The company uses the North American method for pricing this policy.

Calculate the premiums payable by a female life aged exactly 55 at the outset whodoes take up the option.

Basis: Normal mortality: ELT15 (Females)

Mortality of those whoexercise the option: 300% of ELT15 (Females)


Expenses: None

Proportion of policyholders 40% of those alive on the secondwho exercise the option: policy anniversary

[8]

9 (i) Explain why a life insurance company might need to set up non-unit reservesin relation to a unit-linked assurance contract. [3]

(ii) A ten-year contract has the following profit signature before non-unit reservesare set up:

(�1, 0, +1, -2, +1, +1, 0, �1, 0, +1)

If positive non-unit reserves are set up to zeroise negative cash flows, writedown the revised profit signature. You should ignore interest. [2]

(iii) State the advantages of cash flow techniques for product pricing comparedwith traditional commutation functions. [3]

[Total 8]


10 A member of a pension scheme is aged exactly 40, having joined the scheme at ageexactly 22. He earned £30,000 in the immediately preceding 12 months. Finalpensionable salary is defined as the annual average earnings over the three yearsimmediately prior to retirement. Normal Retirement Age is a member�s 65thbirthday.

Using the functions and symbols defined in, and assumptions underlying, theExample Pension Scheme Table in the Actuarial Tables, calculate the expectedpresent value of each of the following:

(i) A pension on ill-health retirement of two-thirds of final pensionable salary. [3]

(ii) A pension on retirement at any stage on grounds other than ill-health of one-eightieth of final pensionable salary for each year of service (fractions of ayear counting proportionately), subject to a maximum of 40 years. [3]

(iii) A lump sum on retirement at any age for any reason of £50,000. [3][Total 9]

11 In a select mortality investigation, tabulations of in force populations are available fora certain class of business, in the following 2 ways:

On each of 1 January 2000, 2001 and 2002, Px,t is available where x and t are definedas:

Method x tA Age last birthday Curtate durationB Age next birthday at

issue plus calendar yearof census minuscalendar year of issue

Duration at policyanniversary during year ofcensus

Two different tabulations of deaths in each of the years 2000�2002 are also available,�y,r where y and r are defined as:

Method y r1 Age last birthday at

policy anniversary priorto death

Duration at policyanniversary followingdeath

2 Age last birthday atdeath

Curtate duration at death

These data are to be used to estimate select forces of mortality. For each tabulation ofdeaths:

(i) Determine the ages and durations to which these estimates apply, stating allassumptions you make. [6]

(ii) Indicate which of the tabulations of census data gives the best match to each ofthe tabulations of deaths and write down an appropriate approximation to therequired exposed to risk. State all assumptions you make. [4]

[Total 10]

105 A2003�6

12 You are the actuary of a life insurance company which issued 5,000 with-profitendowment assurance policies to lives then aged exactly 40 on 1 January 2002. Eachpolicy had an original sum assured of £100,000 and a term of 20 years, with annualpremiums of £4,300 payable in advance throughout the term, ceasing on earlier deathor discontinuance.

You are given the following information, most but not all of which is needed tocalculate asset shares:

� The office holds net premium prospective reserves for in force policies basedon AM92 Ultimate mortality and 4% per annum interest.

� On death, policies receive the original sum assured plus previously declaredreversionary bonuses and any applicable terminal bonuses. The claimpayment is made at the end of the calendar year of death.

� On discontinuance within the first two years, policies receive a surrender valueequal to 25% of premiums paid. The surrender value is payable at the end ofthe calendar year of discontinuance.

� On 31 December 2002, the office declared a reversionary bonus of 2% of theoriginal sum assured for all policies fully in force on that date (i.e. notincluding any policies terminating during 2002 for reason of death orsurrender).

� On 31 December 2002, the office also declared a terminal bonus for deathclaims which arose in the previous 12 months whereby the total death benefitpayable is 125% of the original sum assured plus 125% of any attachingreversionary bonuses.

� Expenses incurred were £15.0 million on 1 January 2002.

� During 2002, 4 policyholders died and 200 discontinued.

� The office earned interest of 6.5% on its assets during 2002.

The company uses actual death claims when calculating asset shares and ignores allother factors affecting profit or expenses not given above.

(i) Calculate the asset share per in force policy on 31 December 2002. [7]

(ii) State with reasons which information given is not required for your calculationin (i). [3]

[Total 10]


13 (i) Describe the benefit whose present value is shown below. Tx and Ty are thecomplete future lifetimes of two lives aged x and y respectively:

( ) £100,000 if

( ) 0 otherwise

yTx yg T v T T

g T

� � ��

[2]

(ii) The policy in (i) was originally paid for by a single premium at outset. Thepolicyholders, who are both still alive, now request that the benefit bemodified immediately to be paid on the earlier death of either life.

Calculate the level premium payable annually in advance from now until thefirst death of either life if the policy is amended in the manner requested.

Basis: Mortality: (x) subject to force of mortality of 0.02(y) subject to force of mortality of 0.03(x), (y) independent with respect to mortality

Interest: force of interest of 0.04

Renewal expenses: 2.5% of all premiums payable from thealteration date

Alteration expenses: £100 [8]

(iii) State, with reasons, any actions the life insurance company should undertakebefore proceeding with the alteration described in (ii). [2]

[Total 12]

105 A2003�8

14 A life insurance company sells 4-year decreasing term assurance policies, with levelpremiums payable annually in advance for the term of the policy, but ceasing onearlier death. The initial sum assured is £200,000 decreasing by £50,000 at eachpolicy anniversary and the death benefit is payable at the end of the year of death.

The company allows for the following when calculating premiums:

Initial expenses: £300 plus 25% of the annual premium

Renewal expenses: £30 per annum plus 2.5% of annual premium, incurred at thetime of payment of the second and subsequent premiums

Mortality: AM92 Select

Interest: 4% per annum (for all rates needed)

For a male aged exactly 60 at outset:

(i) Write down the gross future loss random variable at the outset of the policy.[3]

(ii) Calculate the office premium using commutation functions, setting theexpected value of the gross future loss random variable to zero. [4]

(iii) Calculate the office premium using a discounted cash flow projection,assuming no withdrawals, ignoring reserves and using the same profit criterionas in (ii). [6]

(iv) Without further calculation explain the effect of:

(a) allowing for the setting up of reserves in the calculation in part (iii)

(b) having set up the reserves in (iv)(a), increasing the discount rate to10% per annum

[3][Total 16]


REPORT OF THE BOARD OF EXAMINERS

April 2003


EXAMINERS� REPORT

Introduction


J CurtisChairman of the Board of Examiners

3 June 2003



Page 2

Overall Comments

The standard this year was generally good, slightly improved from last year. Candidatesseemed to cope well with the new areas that were examined this time (mainly questions 6 and12). However, the following areas continue to prove the most difficult for candidates:-estimation of select forces of mortality (question 11), mortality options (question 8), andcontingent assurances / reversionary annuities (question 7 and 13(ii)), despite the questionsasked being very standard for these topics.

Comments for individual questions follow after each question which we hope will assiststudents.

1 126|26 26|26 26|2626 0.5 2626 26 260

( )ts z p dt z p� ��Well answered. The main error, if one was present, was to confuse the exact andapproximate relationships.

2 (i) The super compound bonus method is a method of allocating annual bonusesunder which two bonus rates are declared each year. The first rate, usually thelower, is applied to the basic sum assured and the second rate is applied to thebonuses added in the past.

(ii) The sum assured and bonuses increase more slowly than under other methodsfor the same ultimate benefit, enabling the office to retain surplus for longerand thereby providing greater investment freedom.

This method also rewards longer standing policyholders and discouragessurrenders, relative to other methods.

Very well answered overall. In part (ii), other reasons, where valid, were accepted.

3 The death benefit in year 10 is £15,000

Profit emerging per policy in force at the start of the year is:

([9V + P]*1.05) � (15,000*0.04) � ([1 ��0.04]*10V) =([11,300 + 1,500]*1.05) � (15,000*0.04) � (0.96*13,200) = £168

Well answered. Two common errors recurred, using a wrong death benefit (usually ninetimes the premium) and omitting the survival probability of 0.96 for closing reserves.

4 The component method builds up recursively year on year, allowing explicitly foreach of the 3 key elements: births, deaths and net emigration. Each of these can bemodelled separately to incorporate changing trends, although to do so relies ondetailed data and / or assumptions, usually split by age and sex.


Page 3

The logistic model is easy to apply, but is restricted in the variation it can allow for apopulation, relying on 2 parameters which give a limiting population and an initialgrowth rate, which reduces as population increases. The model does not lend itself tounderstanding mechanisms of population changes. In reality, growth varies over timein a different manner and most recent projections using the logistic and similarmodels have tended to overestimate the population.

Also well answered. Occasionally candidates gave extremely lengthy and detaileddescriptions of the two methods, too much for the marks available, while at the same timeoverlooked the comparison of the two approaches, which was the main thrust of the questionasked.

5 (i) Crude rates are easily calculated, relying only on total population at risk andtotal deaths for each cause of death in this case.

However, the relative results for different countries can vary widely if thedeath rate for a certain cause of death (a) varies by age � as most do � and(b) population structures vary by age between countries. Differences in thecrude rates for a cause of death would then be confounded with differences inpopulation structures.

(ii) The rates could be standardised. Direct standardisation is best, whereby eachcountries actual age-specific death rates are applied to a common population.

Any reasonable standard population could be chosen, but where possible itshould have some relevance to the study e.g. a European study couldstandardise according to the population in Europe sorted by age as follows:

Directly standardised death rate for cause A for a given country =

s c Ax x

xs c

xx

E m

E

�

�

Where s cxE is the central exposed to risk at age x in the standard population

and Axm is the central mortality rate from cause A at age x in the country in

question.

Generally very well answered, especially part (i). While many candidates did not relate theiranswers to the specific question which concerned a cause of death study and wrote aboutmortality rates generally, this was accepted by the examiners. In part (ii), alternativesuggestions were also accepted, where justified.

6 (a) (0.242488)(100)(52.18) using (1/ )

45HS all

a = £1,265.30

(b) (5.4952)(100)(52.18) using 55,2SS

a = £28,673.95

Very well answered. The only common error was the omission of the 52.18 factor.Candidates seemed clearly familiar with the new examination tables.


Page 4

7 Value 10,000*1.02 300I� � where

60:20 60:20 60:60:202*f m fmI a a a� � �

� �20 801 160 802 260:20

60

6,953.53615.132 (0.456387) (7.006) 12.8699,826.131

m la a v al

� � � � � � ��

� �20 801 160 802 260:20

60

7,724.73716.152 (0.456387) (8.489) 13.1139,848.431

f la a v al

� � � � � � ��

� �20 80 801 160:60 80:802 260:60:20

60 60

6,953.536 7,724.73713.590 (0.456387) (5.357) 12.2339,826.131 9,848.431

f mm f m fl la a v a

l l� � � � � � ��

I = 12.869+13.113-2*12.233=1.516

Value =10000*1.02*1.516+300

� Premium = £15,763

This question caused considerable problems to candidates . Common errors were to onlyallow for one reversion (usually on death of male), omit the factor of 2 for joint life annuity,use a factor of 0.98 instead of 1.02 for expenses, or assuming that the annuity ran for 20years from the first death. A surprisingly high proportion of candidates used erroneousformulae to convert annuities from annually in advance to continuous, often dividing by1.040.5. This is a basic actuarial function which is given in the examination tables.

8 Let the full single premium at commencement = P

The premium (based on normal mortality) payable at the time of exercising the optionon the 2nd anniversary =

30.444,1583,93

554000,250000,250000,250)05.1( 5.0

57

575.057

5.0�� v

ldvvq

Therefore the premium required at duration 2, if the option is exercised, is £1,444.30

Thus equating the expected present value of all premium income with the expected present valueof all claims, we get:


Page 5

79.759,5 toleading 54.278,675.518

))554(3499450(532,94000,250)05.1()30.444,1(

532,94583,93)4.0(

)3(000,250)05.1()30.444,1()4.0(

)])3)(2)(4(.)6[(.(000,250)05.1()30.444,1()4.0(

325.02

357

25655

55

5.02

55

57

357

357552

255|155

5.02552

��

��

��

��

PP

vvvvP

vdvdvdl

vllP

vqvqpvqvqvpP

Alternative approach based on non-option policy

If the policy were a simple 3-year term assurance without any options, the singlepremium at commencement would be:

80.684,3)554499450(532,94000,250)05.1(

)(000,250)05.1(

][000,250)05.1(A000,250

325.0

357

25655

55

5.0

357552

2565555

5.01|3:55

��

��

��

vvv

vdvdvdl

vqpvqpvq

To allow for the option, the initial single premium needs to be increased by:]})(000,250)(000,250[05.1{4.0 57

*5757

*57

5.02552 vqqvqqvp ��

*57q represents the mortality of optioners post-option = 3q57

(The 1st term in square brackets represents the extra mortality of optioners on the original SA,and the 2nd term represents the extra mortality on the additional SA over and above that paidfor by the normal rates premium paid at the time of exercising option, t=2)

00.075,2532,94554*4000,250)05.1)(4.0(

4000,250)05.1)(4.0()4(000,250)05.1())(4.0(

35.0

3

55

575.057

5.03552

��

��

v

vldqvp

The total single premium at outset = 3,684.80+2,075.00 = 5,759.80 (same as above, allowingfor rounding)

The premium payable by policyholders at t=2 when exercising their option is (unchangedfrom original solution):

30.444,1583,93

554000,250000,250000,250)05.1( 5.0

57

575.057

5.0�� v

ldvvq

This proved the most difficult question for students, with few fully correct answers. A numberof candidates seemed to misread the question and tried to calculate the cost of the option(instead of the premiums) while others treated the policy as annual premium. Many studentscalculated the basic premium for a policy with no option and tried to calculate the additionalpremium required for the option so the examiners have provided an alternative solutionalong these lines.


Page 6

9 (i) To zeroise future negative cash flows.

The office must meet all future outgo (additional to unit liabilities) e.g. deathclaims in excess of units, expenses, maturity guarantees. It can take credit forfuture income to the non-unit fund but cannot assume recourse to futurecapital.

If there are negative cash flows, we cannot assume that they will be met fromsubsequent positive cash flows or future capital (lapse risk, regulations). Theyare future losses which we need to reserve for now. With adequate non-unitreserves established, the minimum expected cash flow in future years,allowing for release of reserves, is zero, hence the �zeroisation� of cash flows.

(ii) (�2, 0, 0, 0, +1, 0, 0, 0, 0, +1)

(iii) Cash flow approach is more flexible in general and allows for clarity ofthought and ease of presentation of resultsAllows for complex policies (varying benefits, options)Permits variable or stochastic premium basis e.g. interest basisBest (often only) approach for multiple state model situationsAllows amount and timing of cash flows to be observedProvides net cash flows useful for investment strategyAllows for explicit amount of profit to be calculated.Makes explicit allowance for cost of capitalOnly way to calculate non-unit reservesFacilitates repeating with altered basis for sensitivity testing (once spreadsheetor program set up)

Generally well answered, especially part (ii). Some candidates only gave examples of outgoin part (i), without considering offsetting income while in part (iii) some candidates tended toconcentrate on only one reason.

10 (i) 40 40

39 40

2 2 7.814 58,094(30,000) (30,000) £47,527.513 3 7.623 25,059

z ia

ss Ms D

� �

(ii) 40 6240 40

39 40

1830,00080

30,000 7.814 (18)(128,026) 2,884, 260 159,03080 7.623 25,059

£77,153.73

z ra z raz ra

ss M R Rs D

� ��

� �


Page 7

(iii) 40 40

40

369 78250,000 50,000 £17,945.123, 207

i rM MD

� ��

Well answered throughout. The commonest mistakes related to omitting the salaryadjustment, treating (i) as service-related, omitting the factor for age 62 in the future servicepart of (ii). In (iii), some candidates used annuity functions and / or omitted one of the typesof retirement.

11 (i) We are estimating �[x]+t

From 1�y,r y is policy year rate interval and lives are aged between y and y + 1at the start of the interval in which death occurs, giving an average age at thepolicy anniversary before death of y + .5, assuming an even spread ofbirthdays over the policy year.

r is also a policy year rate interval, and is the same as a duration of r � 1 yearsexact at the policy anniversary before death, without assumption.

The age at entry is y + .5 � (r �1) = y � r + 1.5 and the duration midwaythrough the rate interval (needed for the duration when estimating forces ofmortality) is r ��+ .5 = r � .5 so we are estimating �[y�r+1.5]+r�.5. No furtherassumptions are required.

From 2�y,r y is age last birthday at death giving a life year rate interval, withlives y exact at the start of the interval without assumptions needed.

r is again a policy year rate interval, giving duration r years exact at the policyanniversary before death, without assumption.

The average age at entry is y � r, but we must assume an even spread ofbirthdays over the policy year because the 2 rate intervals are not the same (theage at entry could range from y � r �1 to y � r + 1 based on the information wehave) and the duration midway through the rate interval is r + .5 so we areestimating �[y�r]+r+.5.

(ii) Census A gives a life year for age, with y last birthday, and a policy year forduration with r curtate.

Census B gives y next birthday at next policy anniversary, which is also y � 2last birthday at previous policy anniversary. It also gives duration r at policyanniversary following census or r �� curtate at census.

For the 1�y,r deaths, census B fits perfectly but we just need to be careful withage labels. To get y last birthday at previous policy anniversary, and r ��curtate, we need Py+2,r.


Page 8

The approximate exposed to risk is 2000 2001 20021 12, 2, 2,2 2

B B By r y r y rP P P� � �

� ��

for

estimating �[y�r+1.5]+r�.5

For the 2�y,r deaths, census A fits perfectly. To get y last birthday, andr curtate, we need Py,r.

The approximate exposed to risk is 2000 2001 20021 1, , ,2 2

A A Ay r y r y rP P P� ��

� �for

estimating �[y�r]+r+.5

We assume that Px,t.varies linearly between census dates.

Generally not well answered , especially part (ii). In (i), some candidates based their answeron the census data rather than on the death data, listed standard assumptions regardless of ifthey applied here. Others, having defined the age and duration labels correctly did notdefine the force of mortality at all or incorrectly.

In part (ii), while many students correctly matched the censuses to the death tabulations,almost none got the correct age / duration labels for census B matched with deaths method 1.

There was a slight discrepancy in the question between the number of years of death data (3)and the time period spanned by the censuses (2). This was not central to any of the answersrequired, but the examiners accepted all valid interpretations / assumptions made by studentsin this regard.

12 (i) Death claims in 2002 get SA, no reversionary bonus, and terminal bonus =125,000

Discontinuances in 2002 get 0.25*4,300 = 1,075

2002 money flows:

Premium income: 5000*4,300 = 21,500,000Expenses: 15,000,000

Balance: 6,500,000

Interest during 2002 @ 6.5%: 422,500

Balance @ 31/12/2002 before claims: 6,922,500

Death claims 2002: 4*125,000 500,000Surrender claims: 200*1,075 215,000

Total funds 31/12/2002: 6,207,500

No. of policies in force 31/12/2002: 5000 ��4 ��200 4,796


Page 9

Asset share per policy in force at 31/12/2002 = 6, 207,5004,796

= £1,294

(ii) The basis for net premium reserves and the 2002 reversionary bonusdeclaration were the unnecessary items.

Neither affected the cash flows during 2002 nor therefore the year end assetshare.

Well answered, especially as this was the first time an asset share calculation had appeared.The main error was to allow for reserves in some way. Some students tried to do thecalculation per policy sold but this usually led to errors.

13 (i) A (contingent) whole life assurance with benefit of £100,000 paidimmediately on the death of (y) providing it occurs after (x)�s death

(ii) Reserve before alteration

V = 0

100,000 (1 )tt x t y y te p p dt

�

��

��

= .04 .02 .03 .07 .09

0 0 0

100,000 (1 ).03 3,000t t t t te e e dt e dt e dt� � �

� � � � �

� ��

� ��

= 1 13,000 9,523.81.07 .09

� ��

� � � � �

Reserve post alteration:

xyxy aPA ��)975.0(000,100 �

0

.04 .02 .03 .09

0 0

( )

(0.05) 0.05

1.05 0.555556.09

txy t x t y x t y t

t t t t

A e p p dt

e e e dt e dt

�

��

� �

� �

� � �

� � ��

� �

� ��

�

� �

0.09.04 .02 .03 .09

0.090 0 0

11.61861

t t t t txy t x t y

ea e p p e e e ee

� � ��

� � � � �

�

� � ��


Page 10

or alternatively

.09

0

te�

�

�.09 1 1 (at 1 0.094174) 11.6186ia i e

d i�

��

Reserve before = Reserve after + alteration expense

9,523.81 = (100,000)(.555556) ��P(.975)(11.6186) + 100

so P = £4,072.32 p.a.

(iii) Both lives should be underwritten at this time.The proposed change increases the probability of claim payout by the insurersubstantially with regard to life y. Previously if y was worse than assumedmortality, it was a margin for the office, but now the office is at immediaterisk in relation to y. The risk with regard to x is similar to that before thealteration as regards the likelihood of a claim arising, but because the claimwould now be paid immediately on x�s death, the present value could increasesignificantly.

Part (i) was well answered. In part (ii), many candidates made a good effort but manyomitted or could not calculate the pre-alteration reserve. In part (iii), many candidates madegeneral comments about underwriting without explaining why in the context of this particularalteration.

14 (i) Gross future loss random variable (GFL r.v.) =

[60]

[60] [60]

1[60] 1( ){(200,000 (50,000)( )} 300 30 (.975 .225)K

K Kv K a P a�

��

for K[60] < 4

or 3 4300 30 (.975 .225)a P a� � �� for K[60] � 4

(ii) E(GFL r.v.) = 0

�1 1

[60]:4 [60]:4 [60]:4[60]:4250,000 50,000( ) 300 30( 1) (.975 .225)A IA a P a� � � � � ��

[60] 641[60]:4

[60]

400.74 372.69 0.0318547880.56

M MA

D� �

� � �

[60] 64 641[60]:4

[60]

4 7380.21 5813.76 4(372.69)( ) 0.08595666880.56

R R MIA

D� � � �

� � �


Page 11

[60] 64[60]:4

[60]

12475.24 9186.74 3.734555880.56

N Na

D� �

� � ��

leading to 7,963.68 � 4,297.83 + 300 + 82.04 = 3.41619P P = £1,184.91

(iii)

q[60] 0.005774 p[60] 0.994226 0p[60] 1q[60]+1 0.00868 p[60]+1 0.99132 1p[60] 0.994226q62 0.010112 p62 0.989888 2p[60] 0.985596q63 0.011344 p63 0.988656 3p[60] 0.97563

Year Prem Expense Interest Claim Cash flow Profit Signature NPV

1 P 0.25P�300 0.03P�12 1154.8 0.78P�1466.8 0.78P�1466.8 0.75P�1410.382 P 0.025P�30 0.039P�1.2 1302 1.014P�1333.2 1.008145P�1325.5 0.932087P�1225.53 P 0.025P�30 0.039P�1.2 1011.2 1.014P�1042.4 0.999394P�1027.39 0.888458P�913.3424 P 0.025P�30 0.039P�1.2 567.2 1.014P�598.4 0.989289P�583.817 0.845648P�499.049

Total NPV = 3.416193P ��4,048.28

So P = £1,185.03 (same as above except for rounding due to use of commutation functions)

(iv) (a) Profit is deferred but as earned interest and risk discount rate are equal,there is no impact on NPV or premium.

(b) Profit is deferred but because the discount rate exceeds earned rate,NPV falls and premium would have to increase to satisfy the sameprofit criterion.

Parts (ii) and (iii) were handled well throughout, with only the death benefit element of part(ii) causing any difficulty. In part (i), a number of students gave the expectation of therandom variable, and among those who did give a random variable many omitted the selectnotation and / or struggled with the benefit element. In part (iv), many gave correct answersfor (b), but in (a) very few students recognised that there would be no impact on the premiumbecause the earned interest rate equalled the discount rate.


EXAMINATIONS













� Faculty of Actuaries105—S2003 � Institute of Actuaries

105 S2003—2

1 A life insurance company issues a number of 3-year unit-linked policies to lives eachaged 40 exact. The year-end non-unit fund cash flows ( )tNUCF , per policy in force atthe start of policy year t, are as follows (in £’s):

Year (t) 1 2 3( )tNUCF 100 100 �150

Non-unit fund reserves are to be set up at each year end for each policy then in forceto zeroise future negative cash flows. Calculate the adjusted value of ( )tNUCF at theend of year 1, assuming that interest is earned on reserves at the rate of 5% per annumand that the mortality basis is AM92 Select. [3]

2 Describe four benefit options that may be available to an individual member of apension scheme who leaves the scheme before normal pension age. [4]

3 A life insurance company issues a disability insurance policy to a healthy life agedexactly 45.

The benefits under the policy are as follows. There is no waiting period, but there is adeferred period of one year. A benefit of £10,000 per annum is payable continuouslywhile the policyholder is sick, after the completion of the deferred period. The benefitis payable until the policyholder reaches age 65, dies or recovers. Premiums arewaived while the policyholder is in receipt of benefit payment.

Level annual premiums are payable continuously under the policy until age 65 or thepolicyholder’s earlier death.

Calculate the annual premium.

Basis: Sickness: S(ID) Tables

Mortality: ELT(15) Males


Expenses: Initial: 60% of the annual rate of premium

Regular: £50 per annum, assumed incurred continuously in allyears of the policy, including periods of sickness

Claim: 1.5% of sickness benefit payments made to thepolicyholder

[4]

105 S2003—3 PLEASE TURN OVER

4 Describe the calculation of a surrender value for a without-profit endowmentassurance policy, under which level annual premiums are payable monthly in advanceand cease on earlier death or surrender and the sum assured is payable immediately ondeath. Give formulae, defining carefully all the symbols that you use. [5]

5 A life insurance company issues a term assurance policy to a life aged 55 exact for aterm of 10 years. The sum assured is payable immediately on death. The sum assuredis given by

£100,000 (1 0.05 ) = 0,1, 2....,9.t t� �

where t denotes the curtate duration in years since the inception of the policy.

Level premiums are payable monthly in advance for a period of 10 years or untilearlier death. The life insurance company calculates the premium using theequivalence principle.

Calculate the annual premium.

Basis: Mortality: AM92 SelectInterest: 4% per annumExpenses: None [5]

6 A pension scheme provides a pension of 1/60 of Final Pensionable Salary for eachyear of scheme service upon retirement for any reason. Fractional years of servicecount proportionately. Final Pensionable Salary is defined as the average annualsalary in the three years immediately prior to retirement. Members are required tocontribute continuously at the rate of 5% of salary.

You are given the following data in respect of Member A as at 1 January 2003:

Age: 50 exactAnnual rate of salary: £50,000

Using the data in the Actuarial Tables, calculate, in respect of Member A:

(i) The expected present value of future contributions payable. [3]

(ii) The expected present value of the pension benefits on retirement for anyreason based on future service. [2]

[Total 5]

105 S2003—4

7 A life insurance company uses the following 3-state model, to calculate premiums fora 3-year sickness policy issued to healthy policyholders age 50 exact at inception.

In return for a single premium of P payable at the outset the company will pay abenefit of £10,000 at the end of each of the 3 years if the policyholder is sick at thattime.

Let tS represent the state of the policyholder at age 50 t� , so that 0 =S H and for1, 2 and 3, = , or .tt S H S D�

The life insurance company uses transition probabilities defined as follows:

150 = ( = | = )ijt ttp P S j S i��

For 0, 1 and 2t � the transition probabilities are:

50 = 0.05HDtp � 50 = 0.15SD

tp � 50 = 0.80SHtp � 50 = 0.1HS

tp�

The life insurance company calculates P as the expected present value of the benefitpayments, assuming interest at 6% per annum and expenses of 5% of P.

Calculate P. [5]

8 A life insurance company issues 10-year unit linked policies to lives aged exactly 50.Premiums paid in the first two years of the policies are applied to purchase capitalunits, with premiums in subsequent years being applied to purchase accumulationunits.

The management charge on the capital unit fund is 5% of the bid value of the units,deducted at the end of each policy year. The management charge on the accumulationunit fund is 1% of the bid value of the units, deducted at the end of each policy year.

The life insurance company wishes to use actuarial funding assuming a rate of interestof 3% per annum. In calculating the actuarial funding factors, the life insurancecompany assumes that mortality is constant, with

= 0.001 for 50 60xq x� � .

The life insurance company ignores surrenders.


Dead (D)


(i) Calculate the actuarial funding factor to be applied at the end of the third yearof a policy. [4]

(ii) The life insurance company is considering using a higher rate of interest foractuarial funding factors. It wishes to assume the same mortality basis and toignore surrenders in calculating the revised actuarial funding factors.

Describe how you would determine the maximum rate of interest it would beprudent to use in calculating the actuarial funding factor to be applied at theend of the third year of the policy. Set out the considerations you would takeinto account. [5]

[Total 9]

9 You are a consulting actuary to a client who wishes to invest £1m now to provide animmediate income for his partner and himself in retirement. Both the client and hispartner are aged 60 exact.

The client wishes to provide a payment annually in advance each year while either heor his partner is alive. He wishes the amount of the payment to be

£ (1.05) = 0,1, 2.......tI t�

where I denotes the amount of the initial payment and t denotes the curtate duration inyears since the inception of the policy.

The client further requests that he wishes the amount of the initial payment I to besuch that the capital of £1m is at least 95% likely to be sufficient to provide therequired payments and he asks you to advise what the maximum value of the initialpayment I should be.

In carrying out the calculations, you assume that the only source of random variationis the future mortality of the client and his partner.

Calculate the required value of I based on the following assumptions.

Mortality: The client and his partner are independent withrespect to mortality and are each subject to themortality of PMA92C20.

Rate of future investment returns: 6% per annum

Expenses: none [9]

105 S2003—6

10 A life insurance company offers an option on its 10-year level term assurance policiesto effect a whole life without profits policy, for the sum assured, without evidence ofhealth. The option may be exercised once only, either on the fifth anniversary of thepolicy or at the expiry of the 10-year term. If the option is exercised on the fifthpolicy anniversary, the term assurance policy ceases immediately.

The sums assured under the 10-year term assurance policy and under the whole lifepolicy are both payable immediately on death. A single premium, inclusive of theoption premium, is payable at the outset under the term assurance policy and levelpremiums under the whole life policy are payable annually in advance until death.The premiums under the whole life policy are calculated using the company’s normalannual premium basis.

(i) Describe the conventional method of pricing the mortality option, statingclearly the data and assumptions required. [4]

(ii) A policyholder aged exactly 45 wishes to effect a 10-year without profits termassurance policy, for a sum assured of £200,000.

Calculate the total single premium payable under the term assurance policy,using the conventional method to calculate the option premium.

The following basis is used to calculate the basic term assurance premium:

Basis: Mortality: AM92 Select Interest: 4% per annum

Expenses: none [5][Total 9]

11 On 1 January 2000, a life insurance company issued an endowment assurance policyto a life aged exactly 50 for a term of 10 years.

Under the policy, a sum assured of £100,000 is payable on survival to age 60 exact orat the end of the year of death on earlier death. Level premiums are payable annuallyin advance for 10 years or until earlier death.

On 1 January 2003, the policy is still in force and the life insurance companycalculates on a prospective basis both the gross premium reserve and the net premiumreserve for the policy at this date, using the assumptions shown below. The sameassumptions were used to calculate the gross premium at inception as follows:

Mortality: AM92 UltimateInterest: 4% per annumExpenses: Initial: £300 incurred at the outset

Renewal: 5% of each premium

(i) Calculate the gross premium reserve as at 1 January 2003. [3]

(ii) Calculate the net premium reserve, with Zillmer adjustment, as at 1 January2003. Identify clearly the Zillmer adjustment. [2]


(iii) Explain why the net premium reserve with Zillmer adjustment calculated inpart (ii) might be used in preference to the net premium reserve with noZillmer adjustment, calculated as at 1 January 2003, using the sameassumptions. [2]

(iv) Assume instead that the life insurance company calculated the gross premiumreserve as at 1 January 2003 using a rate of interest of 3.5% per annumfollowing a general fall in market interest rates, with all other assumptionsunchanged. Assume also that the net premium reserve with a Zillmeradjustment, calculated in part (ii), is unchanged.

State, giving a reason, whether you consider it appropriate to use thisunchanged net premium reserve with a Zillmer adjustment for reservingpurposes. [2]

[Total 9]

12 A life insurance company issues a two-year without-profit policy to a member, agedexactly 50, of a certain club. The policy provides the following benefits:

(a) on death as a member during two years, a sum of £10,000

(b) on withdrawal from the club within two years, a return of 75% of premiumspaid without interest

(c) on survival as a member to the end of two years, the sum of £5,000

Death and withdrawal benefits are payable at the end of the year of death orwithdrawal respectively and the survival benefit is payable on the maturity date of thepolicy. There are no decrements from membership of the club other than death orwithdrawal.

A premium of £3,000 is payable annually in advance under the policy for 2 years oruntil earlier death or withdrawal.

Calculate the net present value of the profit under the policy to the life insurancecompany.

Basis: Mortality: the independent rate of mortality is that ofAM92 Select

Withdrawal: the independent rate of withdrawal is 5% per annum

Rate of decrements: Mortality and withdrawal occur uniformly throughouteach policy year in the respective associated singledecrement tables.

Expenses: £150 incurred at outset

Rate of interest: 5% per annum

Reserves: Ignore

Risk discount rate: 15% per annum [9]

105 S2003—8

13 A life insurance company issues a special annuity policy to a male and a female life,both aged exactly 60.

Under the policy, an annuity is payable annually in arrear for a maximum of 4 years,ceasing on the first death of the two lives. The first payment under the policy is£10,000 and subsequent payments increase by 1.9231% per annum compound.

(i) Calculate the standard deviation of the present value of benefits under theannuity policy.

Basis: Mortality: The male and the female lives are independent withrespect to mortality and are subject to the mortality ofPMA92C20 and PFA92C20 respectively.

Interest: 6% per annum[8]

(ii) State, with reasons, whether the standard deviation would be higher, lower orthe same if the annuity were to cease on the second death of the two lives,other conditions remaining unchanged. [2]

[Total 10]

14 A life insurance company uses the following multiple-state model for pricing andvaluing annual premium long-term care contracts, which are sold to lives that arehealthy at outset.

Under each contract, the life company will pay the costs of long-term care while thepolicyholder satisfies the conditions for payment. These conditions are assessed everyyear on the policy anniversary, just before payment of the premium then due. If thepolicyholder satisfies the conditions, the annual amount of the benefit payable is paidimmediately. A maximum of four benefit payments may be made under the policy,after which time the policy expires. The policy also expires on earlier death.

Premiums are payable annually in advance under the policy until expiry, and arewaived if a benefit is being paid at a policy anniversary.

1: Claim level 1 2: Claim level 20: Healthy

3: Dead

105 S2003—9

For lives at claim level 1, benefits of 60% of the maximum level are paid, while livesat claim level 2 receive 100% of the maximum level. The current maximum level is£50,000 per annum and is expected to increase by 6% per annum compound in thefuture.

ijxp is the probability that a life aged x in state i will be in state j at age x+1 and the

insurer uses the following probabilities for all values of x:

00 0.87xp � 01 0.1xp �02 0.0xp �

11 0.6xp �12 0.3xp �

22 0.6xp �

(i) Calculate the annual premium under the contract.

Basis: Interest: 6% per annumExpenses: 7.5% of each premium

[9]

(ii) A policyholder has already received two benefit payments at level 1, and isabout to receive a third benefit instalment. Calculate the reserves the officeshould hold for this policy immediately after the benefit payment is made, ifthe policyholder is assessed as entitled to either:

(a) benefit at level 1 = £42,000 per annum(b) benefit at level 2 = £70,000 per annum

Reserve basis: Transition probabilities: as given


Benefit inflation: Inflation of the maximum benefitlevel of 7% per annum.

[5] [Total 14]


REPORT OF THE BOARD OF EXAMINERS

September 2003


EXAMINERS’ REPORT

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. J Curtis Chairman of the Board of Examiners 11 November 2003 © Faculty of Actuaries © Institute of Actuaries


EXAMINATIONS

September 2003


EXAMINERS’ REPORT

© Faculty of Actuaries © Institute of Actuaries


Page 3

Overall Comments The standard of answering overall was at a lower level than the examiners expected. Candidates found particular difficulty with questions 4, 8, 9, 13 and 14. Attempts at questions 9 and 13 in particular were generally unsatisfactory. In relation to the other questions many candidates performed well. Individual comments follow after each question and we hope that these will be of assistance to students. 1 [ ]

[ ]

40

40 1

0.999212

0.999038

p

p +

=

=

The reserve required per policy in force at the end of year 2, 857.14205.1

1502 ==V

The cost of this, at the end of year 2, per policy in force at the start of year 2 = [ ] 720.142*2140 =+ Vp The adjusted value of ( ) 720.42720.1421002 −=−=NUCF

The reserve required per policy in force at the end of year 1, 686.4005.1720.42

1 ==V

The cost of this, at the end of year 1, per policy in force at the start of year 1 = [ ] 654.40*140 =Vp The adjusted value of ( ) 35.59654.401001 =−=NUCF

The question was well answered in general. A number of candidates used incorrect mortality rates. 2 Return of the member’s contributions Under this option, the total of the member’s contributions are returned, with or

without interest. This option is available normally only after a short period of service. There is likely to be a tax charge on the sum paid to the member.

A deferred pension payable from normal pension age This option provides for the member to receive, from the scheme the member is

leaving, a pension payable from normal pension age. The pension is normally based on the number of years’ service to the date of leaving and final pensionable salary at


Page 4

the date of leaving. The basic amount of the deferred pension is increased each year, from the date of leaving to normal pension age, by a revaluation rate.

An immediate pension from the date of leaving This option provides an immediate pension payable from the scheme, from the date of

leaving. This option is normally restricted to members close to normal pension age. The pension can be calculated in a number of ways: a common method is to determine the pension amount as that which is actuarially equivalent to the deferred pension the member would otherwise have received.

A transfer cash equivalent The transfer cash equivalent is an amount determined by the scheme actuary as a fair

assessment of the present value of the deferred pension and other benefits given up by the member leaving the scheme. The transfer cash equivalent may be paid to a new scheme that the member is joining, or to a special individual policy that a member can effect for this purpose with a life insurance company.

This question was well answered in general. Some candidates just listed the benefit options, whereas use of the word “Describe” required a fuller treatment. 3 Let P be the annual premium. P is given by

( )( ) ( )1/ 1/45:20 45:2045:20 45:20

10000*1.015* 0.6 50HS all HS allP a a a P a− = + +

( )11.299 0.242488 10150*0.242488 0.6 50*11.299

£289.41

P P

P

∴ − = + +

∴ =

Overall this question was answered well. Some candidate had difficulty with valuing the waiver benefit. 4 The retrospective policy value is determined, using a basis that reflects the experience

of the policy and takes account of the cost of surrender. The formula for the policy value is as follows:

( ) ( ){ }12 121 1: :: :

xx t x tx t x t

x t

D Ga SA I ea fA CD +

− − − − − , where

x is the age of policyholder at inception


Page 5

t is the policy duration at which the surrender value is being calculated G is the annual office premium S is the sum assured I are the initial expenses, in excess of the regular expenses occurring each year e are the regular annual expenses f are the additional expenses that occur when the contract terminates C are the surrender expenses The prospective policy value is calculated using a basis that reflects the future

expected investment earnings, future expected expenses and future expected mortality experience of the surrendering policyholders, less the cost of surrender. The formula is as follows:

( ) ( )12 12

: :: :x t n t x t n tx t n t x t n tSA ea fA Ga C+ − + −+ − + −

+ + − −

Additional definition: n is the original term of the policy. A table of surrender values by policy duration is produced. The surrender value at a

particular duration is usually a blend of the retrospective and prospective policy values, subject to a minimum of zero. Generally, the retrospective policy value is given a greater weighting at earlier durations and the prospective value is given a greater weighting at later durations. Other considerations, such as the asset share and marketing influences, are also generally taken into account. Where possible, the surrender value should be less than the asset share. Marketing considerations may mean adjusting surrender values upwards.

Most candidates did not answer this question well. The examiners’ view was that this was a standard theoretical question and well-prepared candidates should have scored reasonably. Very few candidates mentioned both prospective and retrospective reserves; most formulae given were not fully correct; and very few candidates dealt with the considerations set out in the final part of the solution.


Page 6

5 Let P be the annual premium. P is given by 1 1(12)

[55]:10 [55]:10 [55]:1095000 5000( )Pa A IA= +

[ ]

(12) 65[55]:10 [55]:10

55

689.230.458 1 8.228 0.458 1 8.0561104.05

Da aD

⎛ ⎞ ⎛ ⎞⎜ ⎟= − − = − − =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

( )

1 0.5 65[55]10 [55]10

[55]

0.5

[55] 65 651 0.5[55]:10

[55]

0.5

= 1.04

= 1.04 0.68354 0.624274 0.060439

10*( ) = 1.04 *

9482.75 5441.07 10*363.82= 1.04 * 0.3726921104.05

DA AD

R R MIA

D

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠

− =

− −

− −=

95000*0.060439 5000*0.372692= = £944.048.056

P +

Candidates attempted this question well in general. There were some minor errors in the formulae and numerical calculations.


Page 7

6 (i) Salary at age 50 exact ⇒ salary earned between age 49.5 and 50.5, assuming that the salary increase was given at age 49.5.

49.5 0.5*(9.031 9.165) 9.098s = + = Value of future contributions

= 50

49.5 500.05*50000*

*

sNs D

1636382500* £25,036.40.9.098*1796

= =

(ii) Value of future retirement benefits

50 50

49.5 50

50000 50000 1604000 363963* * £100,365.26.60 60 9.098*1796

z ra z iaR Rs D

+ += = =

The solution given is based on the assumption that Member A’s salary was increased 6 months before the valuation date. The examiners gave full credit for any other sensible assumption so long as the assumption was stated. For example, assuming that the salary had just been increased, 50s would be used in place of 5.49s . Candidates answered the question well, in general. 7 The remaining transition probabilities are: 50 500.85 0.05HH SS

t tp p+ += = Probability of being sick at 1t = = 0.1 Probability of being sick at 2t =

= 50 51 50 51 0.85*0.1 0.1*0.05 0.09HH HS HS SSp p p p+ = + = Probability of being sick at 3=t = HSSHHSSSSSHSSSHSHHHSHHHH pppppppppppp 525150525150525150525150 +++ = 0.85*0.85*0.1+0.85*0.1*0.05+0.1*0.05*0.05+0.1*0.8*0.1 = 0.08475


Page 8

P is given by ( )32 08475.009.01.0*1000095.0 vvvP ++= 23.585,2£=P Most candidates scored well on this question, with many getting full marks. 8 (i) The actuarial funding factor is given by 53:7A at a rate of interest of 3% and mortality given by 0.001 53 60xq x= ≤ ≤ . 53:7 53:71A da= − 2 6

53:7 1 0.999 (0.999 ) ... (0.999 )a v v v= + + + +

=71 (0.999 ) 6.39873

1 0.999vv

−=

−

53:70.031 *6.39873 0.813631.03

A = − =

(ii) In assessing the maximum rate of interest, I would make a prudent

estimate of the level of the company’s future renewal expenses (including renewal commissions) and express this as a regular percentage of the projected bid values of the funded capital and accumulation units, say i%. I would use discounted cash flow techniques to calculate i. Conventionally, the rate i% tends to be the management charge used for accumulation units, 1% in this case. In practice, we might tend to increase the 3% interest rate to 4% (5%-1%). Mathematically, however, the maximum rate of interest is ( ) ( )%5%100%%5 −− i . In this case, assuming %1=i , this would give a maximum theoretical rate of 4.21%. In assessing whether this would be prudent to use, I would compare the funded value of capital units at the end of the third year using the revised actuarial funding factor with the surrender value of capital units at that time. The funded value should not be less than the surrender value. A further check should be made to ensure that this remains the case at all subsequent policy durations.


Page 9

I would also consider whether the mortality assumption was appropriate for calculating the actuarial funding factor. The assumed level of mortality should not be lighter than that prudently expected for the group of policyholders. Otherwise the company would be anticipating future management charges it might not receive.

Part (i) was not well answered. Many candidates did not show that the actuarial funding factor as the present value of an endowment benefit. Credit was given for variations from the solution set out: if a candidate assumed that the amount of the management charge being pre-funded was 3% per annum and used a rate of

interest of 95.003.0 for the present value of the endowment benefit, credit was given; if a

candidate assumed that the death benefit was payable immediately on death rather than at the end of the year of death in the calculation of the present value of the endowment benefit, credit was also given. Part (ii) caused particular difficulties. Few candidates mentioned the use of discounted cash flow techniques or the considerations set out in the final two paragraphs of the solution. 9 Let t be the future lifetime of the joint status. For the payments to be exactly 95%

likely to be sufficient, since the lives are independent with respect to mortality, the value of t is given by

60:60 0.05t p =

60 60:60

260 60

60

60

2 0.05

( ) 2 0.05 0

2 4 4*0.05 0.02532 or 1.97472

0.02532

t t

t t

t

t

p p

p p

p

p

⇒ − =

⇒ − + =

± −⇒ = =

⇒ =

60

600.02532 39 40tl t

l+⇒ = ⇒ < <

Therefore, for the payments to be at least 95% likely to be sufficient, there must be at

least 40 payments. Alternative derivation that there must be at least 40 payments


Page 10

For payments to be at least 95% likely to be sufficient, t is given by

( ) 95.0

95.0

260

60:60

≥⇒

≥

q

q

t

t

97468.060 ≥⇒ qt

02532.060 ≤⇒ pt

798.248131.9826 6060 ≤⇒= +tll

40≥⇒ t

I is given by

401.061000000 1 0.9524%1.05

Ia i= = − =

40 33.44892a = £29,896I = This was the most poorly answered of all the questions, with few candidates gaining many marks. The question was based on a practical application of standard joint life mortality and the examiners would have expected candidates to have performed much better. 10 (i) Under the conventional method, the premiums that should be charged and the

premiums that will be charged for the new policy or policies that the policyholder can opt to take are determined. The present value of the differences between the premiums is then calculated and this is the present value of the cost of the option. Where there is more than one option, the present value of one option only is taken into account: the option chosen is the one that gives the highest present value of the differences in premiums.

In carrying out the calculations, the following assumptions are made:

all lives eligible to take up the option will do so; the mortality experience of those who take up the option will be the Ultimate experience which corresponds to the Select experience that would have been used as a basis if underwriting had been completed as normal when the option had been exercised.


Page 11

The mortality basis used is not usually assumed to change over time, so the

only data required are the Select and Ultimate mortality rates used in the original pricing basis.

(ii) The present value of the differences in premiums are as follows: Option exercised at the fifth anniversary

Present value = [ ]

[ ]

[ ]

5050 5050

5045 50200000

AD A aD a a

⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

= 0.51366.61 0.32907 0.32868200000* *1.04 *17.4441677.42 17.444 17.454

⎛ ⎞−⎜ ⎟⎝ ⎠

= 96.10 Option exercised at the tenth anniversary

Present value = [ ]

[ ]

[ ]

5555 5555

5545 55200000

AD A aD a a

⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

= 200000* 0.51105.41 0.38950 0.38879*1.04 *15.8731677.42 15.873 15.891

⎛ ⎞−⎜ ⎟⎝ ⎠

= 154.62 The cost of the option is the greater value, i.e., £154.62 The basic single premium is given by

[ ]

[ ]

554510.5 0.5[45]:10

45

0.5

200000*1.04 200000*1.04 *

462.68 430.55200000*1.04 £3,906.751677.42

M MP A

D

−= =

−= =

∴ The total single premium = £3,906.75 + £154.62 = £4,061.37. Candidates performed well on this question in general. In part (ii) there is a subtle point that if the 5 year option is taken then a release of the Term Assurance reserve would take place. The Examiners did not expect students to cover this and the solution is based on this assumption. A few candidates did point this out and due credit was allowed within the total marks in these cases.


Page 12

11 (i) The original gross premium is given by 50:10 50:100.95 300 100000Pa A= +

50:10

50:10

8.314

0.68024

a

A

=

=

£8,650.47P = The gross premium reserve = 53:7 53:7100000 0.95*8650.47*A a−

53:7

53:7

6.166

0.76286

a

A

=

=

Gross premium reserve = £25,614.14 (ii) Net premium reserve with Zillmer adjustment

= 53:7 53:7

50:10 50:10100000 1 300*

a aa a

⎛ ⎞− −⎜ ⎟⎜ ⎟

⎝ ⎠

= 25,835.94 − 222.49 = £25,613.45 £222.49 is the Zillmer adjustment.

(iii) The net premium reserve with Zillmer adjustment equals the gross premium reserve calculated in part (i) (subject to rounding errors). If the insurance company actuary is satisfied that there are sufficient margins in the gross premium reserve then the net premium reserve with Zillmer adjustment would be adequate. In addition, the use of the net premium reserve with Zillmer adjustment compared with the use of the reserve without adjustment would reduce the company’s funding requirements.

(iv) If the life insurance company’s actuary decided that the gross premium reserve using 4% interest was no longer adequate given the fall in market interest rates and that 3.5% interest should be used, this would give a higher value for the gross premium reserve. The net premium reserve calculated in part (ii) was equal to the gross premium reserve using 4% interest and this net premium reserve would not be adequate.


Page 13

Many of the well prepared performed well on this question. A surprising number of candidates showed a lack of understanding of a Zillmer adjustment. 12 The multiple decrement table is as follows.

( )Age x

( )xal ( )dxad ( )w

xad

50 100000 192.17 4995.07 51 94812.76 252.55 4734.16 52 89826.04

Values for the multiple decrement table are calculated from formulas of the following

type:

( )12( ) 1

( ) ( ) *( )

d d wx x x

d dx x x

aq q q

ad al ad

= −

=

1( ) ( ) ( ) ( )d w

x x x xal al ad ad+ = − −

The profit test is set out as follows.

Year 1 2 Premium 3000 3000Expenses 150Interest 142.5 150Death benefit 19.217 26.637Withdrawal benefit 112.389 224.692Survival benefit 4737.02Cash flow 2860.894 −1838.349 Probability in force 1 0.94813Discounted cash flow 2487.734 −1317.954 Net present value £1,169.78

Candidates performed well on this question in general. Where errors occurred, they were mostly in respect of the multiple decrement table. A number of candidates did not use a cash flow approach which is what the Examiners were expecting.


Page 14

13 (i) With 0.06i = and payments increasing at the rate of 1.9231% per annum, we can value at 4%, but we must make the initial payment = 10000/1.019231.

age xl (male) xl (female)

k kpxy Pr(Kxy = k)

60 9826.131 9848.431 0 1 0.004504 61 9802.048 9828.163 1 0.995496 0.005364 62 9773.083 9804.173 2 0.990132 0.006361 63 9738.388 9775.888 3 0.98377 0.007514 64 9696.99 9742.64 ≥4 0.976257 0.976257

k min( ,4)|ka min( ,4) 2ka E[x] 2E[x ] 0 0 0 0 0 1 0.961538 0.924556 0.005158 0.00496 2 1.886095 3.557353 0.011998 0.02263 3 2.775091 7.70113 0.020851 0.057863 ≥4 3.629895 13.17614 3.543709 12.86329

3.581717 12.94875

Variance = 12.94875 − (3.581717)2 = 0.120052 Std Dev: (0.120052)0.5 = 0.346485 ∴ Std Dev for this annuity is (10000/1.019231)*0.346485=3399.48 Alternative solution With 0.06i = and payments increasing at the rate of 1.9231% per annum, we

can value at 4%, but we must make the initial payment = 10000/1.019231. We require

( ) ( )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=−=

+

+ dvVaraVaraVar

xy

xyxy

K

KK

5,1min

|5,1min|4,min

1)1()(

( )( )2|5:60:60|5:60:60

22

1 AAd

−=

age xl (male) xl (female)

k Pr(Kxy = k)

60 9826.131 9848.431 0 0.004504 61 9802.048 9828.163 1 0.005364 62 9773.083 9804.173 2 0.006361 63 9738.388 9775.888 3 0.007514


Page 15

64 9696.99 9742.64 ≥4 0.976257

k ( )kKv xyk =+ Pr*1%4 ( )kKv xy

k =+ Pr*1%16.8

0 0.0043308 0.0041642 1 0.0049593 0.0045852 2 0.0056549 0.0050272 3 0.0064230 0.0054904 ≥4 0.8024121 0.6595242

0.8237801 0.6787912

( ) == 22

|5:60:60 82378.0A 0.6786136

Variance = ( ) 12005.06786136.06787912.012%4

=−d

Std Dev: (0.12005)0.5 = 0.34648 ∴ Std Dev for this annuity is (10000/1.019231)*0.34648=3399.43 (ii) If the annuity were a last survivor annuity, the standard deviation would be

smaller. The chances of both lives dying during the 4 years would be much lower, so more annuities would be payable for 4 years, with a consequent reduction in the deviation from the average present value of the annuity payments.

This question was very poorly answered in general. Many candidates were unable to make any reasonable attempt. The examiners had expected the question to be challenging, but not to the extent experienced. 2 alternative solutions are given which the Examiners hope will assist.


Page 16

14 (i) With no recovery to the healthy state, premiums are payable only until the first claim.

00 00 (0.87)t

t x t xp p= = ∴EPV premiums P{1 + 0.87v+(0.87v)2 + (0.87v)3 +...} = 5.578947P Valuing the benefit from the point when the first claim arises, we get the

following probabilities:

the first claim payment will be at level 1; the second claim payment will be at level 1 with probability 0.6 and

level 2 with probability 0.3; the third claim payment will be at level 1 with probability

20.6 0.36= and at level 2 with probability 0.6*0.3 0.3*0.6 0.36+ = ;

the fourth claim payment will be at level 1 with probability 216.06.0 3 = and at level 2 with probability

324.06.0*6.0*3.03.0*6.0*6.06.0*3.0*6.0 =++ . If the first claim is in n years time, the expected present value will be

50000*0.6*1.06 *n nv . With v at 6%, this is 30,000 for all n. Similarly the present value of any level 2 claim will be 50,000, so we can ignore interest in valuing claims.

The EPV of all claims at the point of the first claim payment arising is

therefore: 30,000*(1 0.6 0.36 0.216) 50,000*(0 0.3 0.36 0.324) 114,480+ + + + + + + = Finally the probability that the first claim occurs at the end of year 1 is 0.1, at

the end of year 2 is (0.87)*(0.1), at the end year 3 is (0.87)2*(0.1) and in general at the end of year n is (0.87)n−1*(0.1).

The probability of a claim is therefore

2 0.10.1*(1 0.87 0.87 ...) 0.769230.13

+ + + = =

The EPV of all claims = (0.76923)*(114,480) = 88,061.45


Page 17

The equation of value is: (1 − 0.075)*5.578947*P = 88,061.45 ⇒ P = £17,064.43 (a) If the third instalment is at level 1, then the fourth claim will be at level

1 with probability 0.6, or at level 2 with probability 0.3. However, interest and claim inflation no longer cancel, so the reserve

immediately after paying the third claim is:

1.07 1.0742,000* *(0.6) 70,000* *(0.3) £47,0801.05 1.05

V ⎛ ⎞ ⎛ ⎞= + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(b) If the third instalment is at level 2, then the fourth can only be at level

2, and will occur with probability 0.6. This gives the following reserve value

1.0770,000* *(0.6) £42,8001.05

V ⎛ ⎞= =⎜ ⎟⎝ ⎠

This question was also not answered well. Many candidates valued the policies as four-year policies only and many also failed to appreciate that interest could be ignored in valuing claims in part (i) after which the question became much easier to complete. Few candidates made reasonable attempts at part (ii).


EXAMINATIONS

21 April 2004 (am)

Subject 105 Actuarial Mathematics 1



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.





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 Actuarial Tables and your own electronic calculator.

Faculty of Actuaries 105 A2004 Institute of Actuaries

105 A2004 2

1 (a) Give a formula for the Area Comparability Factor, defining all terms you use.

(b) Explain the role of this Factor in calculating standardised mortality rates, indicating any advantages it has over other available methods.

[3]

2 A life insurance company uses the following model for pricing and valuing sickness and other contracts.

,abx tp is the probability that a life now aged x and in state a will be in state b when aged

x + t

,aax tp is the probability that a life now aged x and in state a will remain continuously in

state a until age x + t

Explain what is represented by each of the following integrals:

(a) 65 12

,012,000

x tx te p dt

(b) 30 30 ( ) 11 22

35, 35 ,0 010,000

t t rt x t t re p p drdt

[3]

3 Explain the main differences in approach between the conventional and North American methods for pricing mortality options in life assurance contracts. [4]

1: Healthy

3: Dead

2: Sick

x

x

x x

105 A2004 3 PLEASE TURN OVER

4 In a certain country, the population has reached a stationary size, and there is no immigration or emigration. Women between the ages of 20 39 inclusive are regarded as being of childbearing age and mortality in this age range is zero. In the past every woman had a new baby on each of her 21st, 26th, 31st and 36th birthdays. From 1 January 2004, a change in birth patterns means that every woman is expected to have a new baby on each of her 23rd, 28th, 33rd and 38th birthdays. During the transition from one pattern to the other, it is expected that every woman will still have 4 babies, with a gap of at least 5 years between consecutive births.

Calculate the Total Fertility Rate for:

(a) the calendar year 2003 (b) the calendar year 2004 (c) women born in 1962 (d) women born in 1982

[4]

5 (a) Explain what is meant by

2[ ][ ]n x yq

(b) Evaluate 225 [40][40]q assuming both lives are subject to AM92 mortality.

[4]

6 In a select mortality investigation, x,r corresponds to the number of deaths aged x

nearest birthday at death with duration r at the policy anniversary preceding death. x,r

divided by the appropriate central exposed to risk gives an estimate of [y]+t.

Derive the values of y and t to which this estimate applies, stating any assumptions used.

[5]

7 The future lifetimes of two individuals aged x and y are independent, and subject to constant forces of mortality of 0.02 and 0.03 respectively.

(i) Calculate the probability that their first death occurs after 3 years and before 8 years from now. [3]

(ii) Calculate the probability that their second death occurs after 3 years and before 8 years from now. [3]

[Total 6]

105 A2004 4

8 A company issues a block of 5-year single premium investment policies to lives each aged 60 exact at commencement of a policy. It guarantees simple annual reversionary bonuses of 8% per annum of the single premium, with the possibility of a terminal bonus at maturity. The death benefit is 5 times the single premium.

All premiums received under this policy are invested in an asset class where 5-year returns have a normal distribution with a mean of 50% and standard deviation of 25%. The company intends to declare terminal bonuses on maturity such that the proceeds of the policy are the greater of the guaranteed amount and 90% of the underlying asset value.

Calculate the probability that:

(a) the insurer makes a loss on a particular policy.

(b) a policyholder receives a terminal bonus.

Basis: Mortality: ELT15 (Females) Expenses: Ignore

[6]

9 A retirement benefits scheme provides a lump sum retirement benefit equal to 3/80ths of the salary rate at retirement for each completed year of service in the scheme. Fractions of a year do not get credit. Retirement can occur at any age after attaining age 60 but not later than a member s 65th birthday.

Calculate the total service liability for the lump sum benefit in respect of a member aged 63 exact on the valuation date who has exactly 30 years of past service and is earning £40,000 per annum.

Basis: Interest: 6% per annum Salary increases: Nil Independent mortality rates: PMA92Base Independent retirement rates: Age 63 last birthday 10%

Age 64 last birthday 6%

State any other assumptions you rely on. [6]

10 List the main categories of costs incurred by life insurance companies, giving an example of each, and indicating the manner in which they are usually allowed for in calculating premiums. [8]


11 (i) In the context of Manchester Unity Sickness Tables, state the meaning of:

(a) the force of sickness xz

(b) the annual rate of sickness xs

[2]

(ii) An insurance sickness policy provides combined endowment and sickness benefits. The sickness benefit is £200 per week for the first 26 weeks of sickness, £150 per week for the next 26 weeks and £100 per week thereafter while sickness lasts. All sickness payments cease on a policyholder s 65th

birthday. There are no waiting or deferred periods.

The endowment part of the policy pays £10,000 immediately on the death of the policyholder or on survival to age 65.

Premiums are waived during periods of sickness.

Calculate the level premium per annum payable continuously by a new policyholder aged 35. Premiums are payable to age 65 but cease on earlier death.

Basis: Sickness: S(MU) Mortality: ELT 15 (Males) Interest: 4% per annum Expenses: Nil

[7] [Total 9]

105 A2004 6

12 On 1 January 1993, a life insurance company issued a number of 25-year without profit endowment assurance policies to lives then aged 35 exact. Level premiums were payable annually in advance throughout the term of the policy, ceasing on the earlier death of the life assured. The sum assured was payable on survival to the end of the term, or at the end of the year of death, if earlier.

Premiums and reserves were calculated on the following basis:

Mortality: AM92 Select Interest: 6% per annum Expenses: 60% of the first premium

5% of each premium excluding the first

Calculate, as at 31 December 2003, the profit or loss for the calendar year 2003 in respect of these policies, given the following information:

The total sums assured in force on 1 January 2003 were £50,000,000.

The total death claims occurring during 2003 and paid on 31 December 2003 were £200,000.

During 2003, policies with sums assured of £2,500,000 were surrendered. Surrender values, paid on 31 December 2003, were calculated as the retrospective reserve using the above basis, but with interest at 4% per annum.

During 2003, policies with sums assured of £1,000,000 (before alteration) were made paid up with effect from 31 December 2003. Paid-up sums assured were calculated on a proportionate basis, namely the original sum assured * t/25 where t is the number of premiums actually paid.

The company incurred expenses of £100,000 on 1 January 2003.

The company earned a total return of 7% on its assets during 2003.

Ignore tax, and assume that reserves for paid-up policies ignore future expenses. [10]


13 A life insurance company issues a policy to male lives aged 45 exact, providing the following benefits:

A decreasing term assurance with a death benefit, which is payable immediately on death, of £200,000 in the first year, £190,000 in the second year thereafter reducing by £10,000 each year until the benefit is £10,000 in the 20th year, with cover ceasing at age 65.

An annuity of £25,000 per annum, increasing by £2,000 each year, where the first payment is made on the policyholder s 65th birthday, and continues annually for life thereafter.

The policy is paid for by level quarterly premiums payable in advance for 20 years, ceasing on earlier death.

Calculate the premium, using the equivalence principle.

Basis:

Mortality: AM92 Select


Expenses: Initial: £200 plus 35% of the premiums paid in the first year

Renewal: 5% of all subsequent premiums and £40 per annum, increasing by 4% per annum compound, on each policy anniversary

Claim: Death: £250*(1.04)t where t is the exact duration of the policy at death, measured in years with fractions counting

Annuity: 2% of annuity payments [14]

105 A2004 8

14 (i) Under a 4-year unit-linked policy issued to a male aged 60 exact, the following non-unit cash flows, NUCFt, (t = 1,2,3,4) are obtained at the end of year t per policy in force at the start of year t.

Year t 1 2 3 4 NUCFt 400 210 190 450

Mortality follows AM92 Select.

(a) Show that the annual internal rate of return lies between 5% and 6%.

(b) If the rate of interest earned on non-unit reserves is 7.5% per annum, calculate the reserves required at times t = 1, 2 and 3 in order to zeroise future negative cash flows.

(c) Without doing any further calculations, explain what effect the zeroisation of future negative cash flows in part (b) above will have on the internal rate of return relative to that in (a) above. [7]

(ii) A unit-linked endowment policy with an annual premium of £5,000 and a term of 2 years is to be issued to a male life aged 60 exact. 97.5% of each premium will be allocated to units at the offer price. The units will be subject to a bid-offer spread of 4%.

At the end of each year a management charge of 1% of the bid value of the units will be deducted from the unit fund.

If the policyholder dies during the term of the contract the office will pay out the greater of £40,000 and the bid value of the units at the end of the year of death (after the deduction of the management charge).

The company carries out all profit test calculations on the contract using the following basis:

Mortality: AM92 Select Rate of growth on assets in the unit fund: 9% per annum Rate of interest on non-unit fund cash flows: 6% per annum Expenses: £250 at time 0; £50 at time 1 Risk discount rate: 12% per annum

(a) If the policyholder dies in the second year of the contract, calculate the amounts of the non-unit fund cash flows in both of the years of the contract.

(b) Hence calculate the net present value of the profit assuming that the policyholder dies during the second year of the contract.

(c) The policyholder could also die in the first year, or survive to the end of the term of the contract. Calculate the net present value of the profit for each of these two events.

(d) Hence or otherwise, calculate the expected net present value of the profit under this contract.

[11] [Total 18]

END OF PAPER


EXAMINATIONS

April 2004


EXAMINERS REPORT


Subject 105 (Actuarial Mathematics) April 2004

Examiners Report

Page 2

In general terms this was a relatively straightforward paper of standard questions with the possible exceptions of Questions 4 and 8. It was well done by the well prepared students. The Examiners noted, however, that many students appeared unprepared for this examination and often their marks were well short of the required pass mark resulting in overall a disappointing pass ratio.

1 (a) , , , ,

, ,

ACF

s c s c sx t x t x t x t

x xs c c

x t x tx x

E m E m

E E

where

,cx tE : Central exposed to risk in population being studied between ages x and x + t

,s c

x tE : Central exposed to risk in standard population between ages x and x +t

,s

x tm : central rate of mortality either observed or from a life table in standard

population for ages x to x + t

(b) When multiplied by the crude death rate for the population or area under consideration, the ACF provides a standardised mortality rate ( the indirectly standardised rate ). This approach is often favoured when data required by other methods, usually local age-specific mortality rates, are unavailable.

Question 1 was generally well done although clearly many students could not remember the standard formula.

2 (a) Expected present value of a benefit of 12,000 p.a. payable continuously to a life now aged x and healthy whenever x is sick, with the benefit ceasing at age 65

(b) EPV of a benefit of 10,000 p.a. payable continuously to a life now aged 35 and healthy throughout their first period of sickness, ceasing at age 65 in any event

This question was done reasonably well. In part (b) of the question there was an erroneous symbol x in the formula which should have been 35. The examiners gave full credit for using either x or 35 in the answer above.


Examiners Report

Page 3

3 Conventional assumes all eligible lives exercise the option, experience ultimate mortality according to some table and pay premiums on the option policy based on select mortality from the same table as if underwriting took place at the time of commencement of the option policy.

The North American approach assumes that only a certain proportion of eligible lives exercise the option. Opters and non-opters are subject to different mortality levels. This is normally achieved by having a double decrement table of mortality / exercise of option for original policyholders and also a mortality table for post-option mortality for those who exercise the option.

Question done well. Credit was given for other appropriate comments.

4 Total Fertility Rate = xx

f where fx is age specific fertility rate at age x.

For calendar years, we use the period rate approach where we sum the fx s observed in that year.

For women born in a calendar year, we sum across the fx s observed over their lifetime, each x coming from the rate observed in the calendar year in which they were aged x. (None of this is required from the student, it is just explanation for the following results).

Up to the end of 2003, fx = 1 for x = 21, 26, 31, 36 and fx = 0 otherwise.

Therefore, the answer to (a) and (c) = 4, seeing as all relevant births occur before the change at the end of 2003.

From 1 January 2004, fx = 1 for x = 23, 28, 33, 38 and fx = 0 otherwise.

The answer to (d) is also 4. They will have babies when they are aged 21 (in 2003), 28, (in 2010), 33 (in 2015) and 38 (in 2020).

The answer to (b) is zero. There are no women who will have babies in 2004. Those aged 23, 28, 33, and 38 all had babies during 2002 when aged 21, 26, 31 and 36, and therefore will not have their next baby until 2009 when they are aged 28, 33 and 38 respectively.

This question was not done well and many students failed to understand the concept of a Total Fertility Rate attempting often to construct probabilities.

The solution above is a full one. One mark was awarded for each part if the student just wrote down the correct numerical answer.


Examiners Report

Page 4

5 (a) 2[ ][ ]n x yq represents the probability that a select life, now aged y, will die within

n years, having been predeceased by a select life now aged x

(b) 2 2 2651 1 125 [40][40] 25 25 [40][40][40]2 2 2

[40]* *{( ) } *{(1 ) }

lq q q

l

= 8,821.2612 212 9,854.3036*{(1 ) } 0.005495

Question was done reasonably well.

6 x nearest birthday at death

x ½ at start of rate interval (life year from x ½ to x + ½) during which life dies or x at mid-point when the force of mortality is estimated. No assumption necessary.

r at policy anniversary preceding death means exact duration r at the anniversary before death (the start of the policy year rate interval for duration) and hence r + ½ mid-year. No assumption necessary

The average age at entry [y] is therefore [(x ½)

r], but we must assume an even spread of birthdays over the policy year because the two rate intervals are not the same type and therefore not coincident. (Based on the information we have the age at entry could range from (x ½) (r + 1) to (x + ½) (r) i.e. x

r 1½ to x

r + ½, on average x

r ½.)

Therefore we get an estimate of [x r ½]+r+½

Well prepared students scored well on this question. For full marks all comments regarding assumptions needed to be stated.

7 (i)

3|5 3 8 3 3 8 8

3 3 8 8

0 0 0 0

3 8 0.15 0.4

0 0

* *

exp[ 0.02 ]*exp[ 0.03 ] exp[ 0.02 ]*exp[ 0.03 ]

exp[ 0.05 ] exp[ 0.05 ] .8607 .6703 0.1904

xy xy xy x y x yq p p p p p p

dt dt dt dt

dt dt e e

Alternatively, the joint life status has constant hazard rate 0.02+0.03 = 0.05 giving a probability of the first death occurring between time 3 and 8:

8 8 0.05 0.15 0.43|5 :3 3

0.05 0.1904txy t xy x t y tq p dt e dt e e


Examiners Report

Page 5

(ii)

3|5 3 8 3 3 3 3 8 8 8 8

.06 .09 0.15 .16 .24 0.40

( * ) ( * )

( ) ( )

(.9418 .9139 .8607) (.8521 .7866 .6703) .9950 .9685 0.0265

x y x y x y x yxy xy xyq p p p p p p p p p p

e e e e e e

Alternatively,

8 0.02 0.03 0.053|5 3|5 3|5 3|5 3

(0.02 0.03 0.05 ) 0.0265t t tx y xyxyq q q q e e e dt

Although this question was a simple application of probabilities it was surprisingly not done well overall.

8 (a) Insurer makes a loss if either the policyholder dies or the asset value s 5-year return is less than 40% for survivors

Probability of loss = 40 505 60 5 60 25* [ ]q p = 5 60 5 60 * [ 0.4]q p

655 60

60

87,0930.94942877

91,732

[ 0.4] 1 [0.4] 1 0.65542 0.34458

lp

l

Probability of loss = 0.05057123 (0.94942877)*(0.34458) 0.3777

(b) Terminal bonus is received if both the policyholder is alive and the asset value exceeds 155.556% of single premium

Probability of terminal bonus = 55.556 505 60 25

*(1 [ ])p = 5 60 *(1 [.2222])p

[0.2222] 0.58792 interpolating linearly between values for 0.22 and 0.23

Probability of terminal bonus = 0.94942877*(1-.58792)=0.3912

This question was done very poorly overall. Even though the question defined a Normal Distribution very few students appreciated how to apply this in this case.

9 Dependent decrement rates from independent using:

12

( ) (1 )r r dx x xaq q q etc. assuming a uniform distribution of decrements in the single

decrement tables.


Examiners Report

Page 6

Age qr qd (aq)r (aq)d (ap)

63 0.1 0.009189 0.099541 0.00873 0.89173 64 0.06 0.010604 0.059682 0.010286 0.930032

Probability of retiring at 64 last birthday = (.89173)*(.059682) = 0.05322

Probability of retiring at 65 = (.89173)*(.930032) = 0.829338

Assuming that those retiring age 63 or 64 last birthday can be represented as retiring on average half-way through the year then we get

Age Benefit Discount Probability EPV

63 45,000 0.971286 0.099541 4,350.73 64 46,500 0.916307 0.05322 2,267.61 65 48,000 0.889996 0.829338 35,429.16

Total 42,047.50

Despite the definitions given in the question, many students failed to appreciate that they needed to use dependent decrements and produced an answer based merely on independent decrements. Limited credit was given for a solution based on independent decrements and to score well the dependent approach was necessary.

10

Category Example Pricing

Initial Commission Allow for directly, usually premium related

Marketing, promotional Per policy on estimated volumes

Underwriting / Processing proposal / Issue of policy documentation

Usually per policy, although some elements might be tied to other driver (e.g. medical expenses might be sum assured related)

Renewal commission Allow for directly, usually premium related

administration Per policy per annum, allow for inflation

Claim Calculation and payment of benefit

Per policy, allow for inflation


Examiners Report

Page 7

Overhead Central services e.g. IT, legal Per policy per annum

The solution above are the main items the Examiners were seeking. The question was open to wide interpretation as the word Costs was used as opposed to Expenses. Thus the Examiners gave full credit within the total marks for other valid references. Allowing for this the question was well done.

11 (i) (a) the force of sickness xz is the probability that a person aged exactly x

is sick at that moment

(b) the annual rate of sickness xs is the expected number of weeks

sickness that a life aged exactly x will experience in the year of age x to x + 1

(ii)(0 / 26) (26 / 26) (52 / ) (0 / )

35:30|35:30| 35 35 35 3510,000 200 150 10052.18

HS HS HS all HS allPPa A a a a a

using S(ID) notation and where all sickness benefit functions are understood to terminate at age 65.

Using values from S(MU) tables, and noting that 35:30| 35:30|1A a we get

*16.979 10,000{(1 (0.039221)(16.979)}

200(10.813 1.931) 150(2.203) 100(2.972 11.859) (29.778)52.18

P

P

P(16.979-0.571) = 3,340.72+(2,548.80+330.45+1,483.10) = 7,703.07

P = 469.47 per annum

(Theoretically, some adjustment to the age should be made to reflect the fact that a healthy 35 year-old cannot receive the 2nd / 3rd levels of benefits immediately, but the usual adjustments are approximate and have only minor influence on the result, so are ignored here.)

Question done well by well prepared students.


Examiners Report

Page 8

12 Premium per 100,000 given by:

[35]:25| [35]:25|100,000 0.95 .55

100,000*(.24198) *[(0.95)(13.392) 0.55] 1,987.94

A Pa P

P P

Reserve per 100,000 for fully in force policy at 31 12 2002 given by:

10 45:15| 45:15|

10

100,000 0.95

(100,000)(.42556) (0.95)(1,987.94)(10.149) 23,389.18

V A Pa

V

Reserve per 100,000 for fully in force policy at 31 12 2003 given by:

11 46:14| 46:14|

11

100,000 0.95

(100,000)(.45028) (0.95)(1,987.94)(9.712) 26,686.47

V A Pa

V

SV at 31 12 2003 per 100,000 SA given by

1[35]11 [35]11|[35]:11| :

46(0.95 .55 100,000 )

DSV Pa P A

D @4%

[35] 46[35]:11|

[35]

52,662.65 29,905.969.0772

2,507.02

N Na

D

[35] 461[35]:11|

[35]

481.53 460.840.0082528

2,507.02

M MA

D

112,507.02

[{1,987.94}*{(.95*9.0772) 0.55} {100,000*0.0082528}] 23,690.441,611.07

SV

Cost of PUPs at 31 12 2003 per 100,000 SA given by:

1111 46:14|25( )100,000 44,000*0.45028 19,812.32PUPV A

Total funds available at 31 12 2003 before paying any claims or setting up reserves:

[{500 * (P+10V)} 100,000] * (1.07) = 13,469,759.20 = A

Total claims paid on 31 12 2003 = (200,000 + 25 * 11SV) = 792,261 = B

Total closing reserves required:

(500 2 25 10)11V + 10 * 11PUPV = 12,553,958.81 = C

Profit for 2003 = A B C = 123,539.39


Examiners Report

Page 9

Question not done well and very few complete answers were presented

13 Equivalence principle EPV premiums = EPV benefits + EPV expenses

Let P = quarterly premium

EPV premiums: (4) 653[45]:20| 8[45]:20|

[45]

3 689.238 1,677.42

4 4 [ (1 )]

4 [13.785 (1 )] 54.2563

DPa P a

D

P P

EPV death benefit:

1 1[45]:20|[45]:20|

[45] 65 [45] 65 650.5 0.5

[45] [45]

0.5 0.5

210,000 10,000( )

20210,000*(1.04) [ ] 10,000*(1.04) [ ]

462.68 363.82 13,987.39 5,441.07 20*363.82210,000*(1.04) [ ] 10,000*(1.04) [

1,677.42 1,6

A I A

M M R R M

D D

]77.42

12,621,61 7,720.60 4,901.01

EPV annuity: 6565 65

[45](23,000 2,000( )

Da Ia

D)

(0.410887)({23,000*12.276} {2,000*113.911})

209,622.20

EPV expenses: Death claim: 250 * 20q[45] = 250 * (1 0.90030) = 24.92 Annuity .02 * EPV annuity = 4,192.44

Premium related:

[45] 1

[45]

(4) (4)[45]:20| [45]:1|

38

(0.05)(4 ) (0.30)(4 )

(0.05)(54.2563 ) (0.3)(4 )(1 [1 ])

2.712815 1.182171 3.8950

D

D

Pa Pa

P P

P P P

Other:

65[45] 65[45]:20|

[45]160 40 @ 0% 160 40({1 } {1 })

160 40[35.282 (0.90030)(17.645)] 935.85

la e e

l


Examiners Report

Page 10

54.2563P = 4,901.01 + 209,622.20 + 24.92 + 4,192.44 + 3.8950P + 935.85

50.3613P = 219,676.42

Hence P = 4,362.01

Quarterly Premium is 4,362 to nearer whole unit.

Very few complete answers were presented but many well prepared students did successfully complete a number of parts.

For the item of Renewal Expenses, credit was also given if the student took the alternative approaches of:

1. Assuming that this particular expense applied throughout life.

2. If the inflation escalator of 4% applied from year 3 rather than year 2

14 (i) (a)

Year t q[60]+t 1 p[60]+t 1 t 1p[60] NUCFt Profit signature

NPV @ 5% NPV @ 6%

1 0.005774 0.994226

1 400 400.00

380.95 377.36 2 0.008680 0.991320

0.99423

210 208.79

189.38 185.82 3 0.010112 0.989888

0.98560

190 187.26

161.77 157.23 4 0.011344 0.988656

0.97563

450 439.03

361.19 347.76

Total 7.85 1.01

Because there is a change in sign in NPV between 5% and 6%, there must be a solution to NPV = 0 for an interest rate between 5% and 6%.

(b) 3V = 0 since policy has positive cash flow in year 4.

2V = 190 / 1.075 = 176.74

Clearly 1V = 0 since 210 (NUCF2) > p[60]+1*2V

(i) (c) It will increase it. The rate on non-unit reserves exceeds the IRR so in this case the deferral of profits, by introducing reserves, will increase NPV and IRR.

(Usually the discount rate exceeds the non-unit rate of return and allowing for reserves would then reduce NPV and IRR)


Examiners Report

Page 11

(ii) These preliminary calculations, while also an alternative way to get the answer required in (ii)(d), are presented here as background calculations for (ii)(a), (b) and (c).

Unit fund

Fund

Year Cost of allocation brought forward Interest Mgmt. charge

Fund at end

t at bt ct dt et

5000*.975*.96 et-1 .09*(at+bt)

.01*(at+bt+ct)

at+bt-ct-dt

1 4,680.00 0.00 421.20 51.01 5,050.19 2 4,680.00 5050.19 875.72 106.06 10,499.85

Non-unit fund

Year Unallocated

premium Expense

Interest Death cost Mgmt. charge Cash flow Prof Sig.

NPV t ft gt ht it jt kt lt mt

5,000 at .06*(ft gt)

q[60]+t 1*(40,000 et)

dt ft gt+ht it+ jt t 1p[60]*kt

1.12 t* lt

leading to

1 320.00 250.00 4.20 201.80 51.01 76.59 76.59 68.38

2 320.00 50.00 16.20 256.06 106.06 136.20 135.41 107.95

Total 39.57

(a) Yr 1: 320 250 +4.20 + 51.01 = 125.21 Yr 2: 320 50 +16.20 + 106.06 (40,000 10,499.85) = 29,107.89

(b) NPV = 125.21v 29,107.89v2 = 23,092.84 (@12%) (c) Die Yr 1 Cash flow: 320 250+4.20+51.01 (40,000 5,050.19) = 34,824.60

NPV = 34,824.6v = 31,093.39 (@12%)

Survive: Yr 1 = 125.21 Yr 2 = 320 50 + 16.20 + 106.06 = 392.26

NPV = 125.21v + 392.26v2 = 424.50 (@12%)

(d) NPV for contract = 31,093.39q[60] 23,092.84p[60]q[60]+1 + 424.502p[60]

= 39.57

(same NPV as in preliminary calculations above in non-unit cash flows)

A very straightforward question done very well by well prepared students many of whom scored virtually full marks.

END OF EXAMINERS REPORT


EXAMINATIONS





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.





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 Actuarial Tables and your own electronic calculator.

Faculty of Actuaries 105 S2004 Institute of Actuaries

105 S2004 2

1 A life insurance company issues an annuity policy to two lives aged 65 and 62 exact in return for a single premium. Under the policy an annuity of £10,000 per annum is payable monthly in advance while at least one of the lives is alive.


Basis: Mortality: PMA92C20 in respect of the life aged 65 exact

PFA92C20 in respect of the life aged 62 exact Interest: 4% per annum Expenses: none

[3]

2 A member of a pension scheme is aged 50 exact, having joined the scheme at age 30 exact. His current salary is £50,000 per annum. Final pensionable salary is defined as the annual average earnings over the three years immediately prior to retirement. Normal Retirement Age is a member s 65th birthday. Salary increases take place six months before the member s birthday.

Using the functions and symbols defined in, and the assumptions underlying, the Example Pension Scheme Table in the Actuarial Tables, calculate the expected present value of the following:

A pension on retirement at any stage on grounds of ill health of one-sixtieth of final pensionable salary for each year of service, with fractions of a year counting proportionately. [4]

3 A life insurance company issues a policy to a life aged 50 exact. The policy provides the following sickness benefit:

£100 per week for the first two years of sickness, reducing to £50 per week thereafter during sickness. Sickness benefit ceases at age 65, or on earlier recovery or death. There is no waiting or deferred period.

Level premiums under the policy are payable weekly in advance until age 65 or until earlier death. Any premiums falling due during periods of sickness are waived.

Calculate the weekly premium.

Basis: Mortality: ELT 15 (Males) Sickness Table: S(ID) in the Actuarial Tables Interest: 6% per annum Expenses: 5% of each premium

(Expenses continue even when premiums are waived) [5]

105 S2004 3 PLEASE TURN OVER

4 (i) Describe the use of risk classification by life insurance companies in underwriting life assurance policies. [2]

(ii) State two limitations to the use of risk classification and explain how life insurance companies deal with these limitations. [3]

[Total 5]

5 A life insurance company issued a non-profit term assurance policy to a life aged x exact at the outset, with a term of 20 years. Under the policy, the sum assured of £100,000 is payable at the end of the year of death. Premiums under the policy are level and payable monthly in advance for 20 years, or until earlier death.

The company values the policy at duration t years using a gross premium prospective policy value, tV .

Derive algebraically the relationship between tV and 1t V . Define all the symbols

that you use, where necessary. [6]

6 On 1 January 2001, a life insurance company issued a 10-year joint life non-profit term assurance policy to two lives aged 50 exact. Under the policy, the sum assured of £500,000 is payable immediately on the death of the first of the lives to die. Premiums of £1,000 per annum are payable annually in advance for 10 years, or until the first death of the lives assured.

On 31 December 2003 the policy is still in force. Calculate the gross premium prospective policy value at this date, using the following valuation assumptions:

Mortality: PMA92C20 for the first life and PFA92C20 for the second life Interest: 4% per annum Expenses: Renewal: 3% of each premium

Claim: £200 on payment of a claim [6]

105 S2004 4

7 A double decrement table is to be constructed from two single decrement tables. The modes of decrement are and . For each of the single decrement tables you are given

2= .x t x xl l t d and 3= .x t x xl l t d for 0 1t

where i

xl

= the number of lives in the single decrement table i at age x exact

(i = , )

ixd

= the number of decrements over [x, x + 1] in the single decrement

table i (i = , )

(i) Show that

. = 2t x x t xp tq for 0 1t

where

it xp

= the probability that a life aged x exact survives t years

ix t

= the force of decrement by cause i at age x + t

ixq

= the probability that a life aged x exact becomes a decrement

by cause i over [x, x + 1]

in the single decrement table for cause i (i = , ). [3]

(ii) Hence or otherwise show that the dependent initial rate of decrement at age x exact due to cause is:

2= 1

5x xxaq q q

[3] [Total 6]


8 A life insurance company issues 10-year non-profit term assurance policies, for a sum assured S, to lives aged x exact. It offers an option on the policies to effect, either on the fifth policy anniversary or at the expiry of the 10-year term, a whole life non-profit policy for the same sum assured, without evidence of health. Premiums under the term assurance policies are payable annually in advance for 10 years, or until earlier death, or until the fifth policy anniversary, if the option is then exercised. Premiums under the whole life policy are payable annually in advance for the whole of life. The sums assured under the term assurance and whole life policies are payable at the end of the year of death.

An additional single premium is charged at the outset under the term assurance policy for the mortality option. The company uses the North American method for pricing options.

Give formulae for calculating the additional single premium charged at outset for the mortality option. You may ignore expenses. Define all the symbols that you use, where necessary.

[8]

9 A life insurance company sells with profit whole life policies, with the sum assured and attaching bonuses payable immediately on the death of the life assured and with level premiums payable annually in advance ceasing with the policyholder s death or on reaching age 65 if earlier.

Simple reversionary bonuses vest under the policies at the end of each year.

The company prices the product using the following basis:

Mortality: AM92 Select Interest: 4% per annum Expenses: Initial: £250

Renewal: 2% of second and subsequent years premiums Claim: £150 at termination of contract

Bonuses: Simple: 6% of basic sum assured per annum

(i) Write down an expression for the gross future loss at the point of sale for one of these policies, assuming it is sold to a life aged x exact (x < 65) at the outset. Write the expression in terms of functions of the random variables T[x]

and K[x], which represent the exact future lifetime and the curtate future lifetime of (x) respectively. [3]

(ii) Calculate the gross premium required for one of these policies for a sum assured of £200,000 and issued to a life aged 40 exact at the outset, using the equivalence principle. State any assumptions you make. [6]

[Total 9]

105 S2004 6

10 The following data are available from a life insurance company, relating to the mortality experience of its term assurance policyholders.

,x d

The number of deaths over the period 1 January 2000 to 30 September 2003,

aged x nearest birthday at entry and having exact duration d at the next policy anniversary following the date of death.

, ( )y eP n

The number of policyholders with policies in force at time n, aged y nearest

birthday at entry and having curtate duration e at time n, where n = 1.1.2000, 30.9.2000, 30.9.2002 and 30.9.2003.

(a) Develop formulae for the calculation of the crude select forces of mortality corresponding to the ,x d deaths.

(b) Derive the age and duration to which these estimates apply.

Assume that all months are of equal length. State all other assumptions that you make.

[11]

11 A special 3-year endowment assurance policy provides that the death benefit payable at the end of year of death is £10,000 plus the endowment assurance net premium reserve for that year that would have been held had death not occurred. £10,000 is payable on survival to the end of the 3 years.

On the basis set out below, use a discounted cash flow method to calculate the level annual premium payable in advance for a life aged 57 exact. The requirement is that at the discount rate defined below the value of the annual emerging surpluses should sum to zero.

Basis: Mortality: AM92 Select for experience and reserves Expenses: 20% of the first annual premium 5% of subsequent premiums Reserves: Value as a normal endowment assurance for a 3-year term

on a net premium basis using a valuation rate of interest of 4% per annum. Ignore the effect on reserving of the extra death benefit defined above.

Interest earnings: 7% per annum on cash flow Discount rate: 10% per annum

Ignore tax and any other items. [12]


12 A pension scheme provides the following benefits in respect of a former male member of the scheme who has just left service:

(a) A pension to him for life of £10,000 per annum if he survives to age 65: the pension commences on his 65th birthday and is guaranteed payable for five years in any event.

(b) A spouse s pension of £5,000 per annum, commencing immediately on the death of the former member before his 65th birthday and payable for life to the spouse.

A spouse s pension is payable on death in deferment if the former member is married at the date of death.

Pensions are payable monthly in advance.

Pensions in payment and deferment are increased monthly in arrears at the effective rate of 2.8846% per annum.

The former member is now aged 62 exact. You are not given any information as to whether he has a spouse.

Calculate the expected present value of these benefits using the following basis:

Basis: Valuation rate of interest: 7% per annum

Mortality in deferment and in retirement: PMA92C20 for the former member and

PFA92C20 for his spouse

Proportion of former members with a spouse at each age up to age 65: 90%

Age difference of spouses: Females are exactly 3 years younger than their husbands

Assume that death before retirement occurs at the mid-point of the year of age in respect of each year of age.

[12]

105 S2004 8

13 You are a member of a committee responsible for monitoring the trend in assured lives mortality rates. You have been presented with the following ratios of actual to expected mortality rates on the basis of a standard table constructed twenty years ago ( Standard Table A ) and the total expected deaths over the period 2000 2003 based on this table.

Age Ratio of Actual to Expected Mortality Rates

Total Expected Deaths (000 s)

2000 2001 2002 2003 2000 2001 2002 2003

15 44 1.80 2.00 10 10 45+ 0.90 0.80 20 20

You have also been given details of the exposed to risk data in the two age groups 15 44 and 45+ corresponding to Standard Table A. The exposed to risk data are described as Standard Population A .

(i) Define, giving a formula, the term Standardised Mortality Ratio . Define all the symbols that you use. [2]

(ii) Show how the Standardised Mortality Ratio may be expressed as a weighted average. Describe the function averaged and the weights. [3]

(iii) Calculate the Standardised Mortality Ratios for the periods 2000 2001 and 2002 2003 with reference to Standard Table A, using the data presented. [2]

(iv) The committee measured the change in mortality between the periods 2000 2001 and 2002 2003 by calculating a Comparative Mortality Factor (CMF) for each period. This factor was calculated as

12

rr , where

1r

was the expected number of deaths for the period obtained by applying

the observed mortality rates to Standard Population A

2r

is the expected number of deaths in Standard Population A over a

two-year period based on Standard Table A

The CMF was 0.95 for the period 2001 2001 and 0.99 for the period 2002 2003, which led the committee to conclude that mortality was deteriorating.

(a) Explain the difference between the results of your calculation of the Standardised Mortality Ratios in part (iii) and these CMF figures. (Hint: Express the CMF figures as weighted averages.)

(b) State, giving a reason, which set of figures you think provides the better results.

105 S2004 9

(c) Comment on the conclusion of the committee that mortality was deteriorating. [6]

[Total 13]

END OF PAPER


EXAMINATIONS

September 2004


EXAMINERS REPORT


Subject 105 (Actuarial Mathematics 1) September 2004

Examiners Report

Page 2

In general, well prepared candidates did well on this examination which contained reasonably standard questions. Indeed some students scored high marks testifying to the fairly straightforward nature of the paper. The Examiners noted however that there were many candidates who were just not well prepared for the examination and this resulted in a large number being quite a few marks below the required pass level.

Questions without further comment below were those that were in general done well by candidates.

1 The premium is given by:

(12)65:62

10000P a

12 1165 62 65:62 2465:62

a a a a

= 13.666 + 15.963 12.427 0.458

= 16.744

P = £167,440

2 The expected present value is given by:

50 5049 50 50

5000020

0.5*60* *

5000020*45392 363963

0.5*60* 9.031 9.165 *1796

£64,861

z ia z iaM Rs s D

3 Let P be the weekly premium. P is given by

0/ 2 /50 50 50:15

52.18* 100 50 0.95*52.18 *HS all HS all

P a a P a

100 *0.456447 50*0.184025 0.95* *9.516P P

£4.25P


Examiners Report

Page 3

4 (i) Risk classification is used as an underwriting tool by life insurance companies. The company divides policyholders into different risk groups according to factors that affect mortality. The company s expectation is that policyholders in the same risk group are homogeneous with respect to mortality risk. The groups are defined by the use of rating factors, e.g., age, sex, smoking habit.

(ii) In theory the company should add rating factors to its underwriting system until the all mortality differences are fully accounted for, apart from random variation. In reality, the ability of prospective policyholders to provide accurate responses to questions and the cost of collecting information limit the extent to which rating factors can be used. In addition, from a marketing point of view, proposers are anxious that the process of underwriting should be straightforward and speedy.

In setting underwriting terms, companies compromise between the conflicting requirements of risk classification and marketing and use a limited number of rating factors. It is important for a company not to omit a significant rating factor that is used by other companies in the market: otherwise, there would be a risk of selection against the company.

Credit was given for other suitable points and description.

5 Let P =monthly premium G = annual equivalent premium (=12P)

e = annual regular expenses f = claim expenses

121:20 :20

' 100000t x t t x t tV f A G e a

1 1:20 1:20 1x t x tx t t x t t

A vq vp A

and 12 12 12

:20 :1 1:20 1x tx t t x t x t ta a vp a

12 1211:20 1 :1 1:20 1

' 100000t x t x t x tx t t x t x t tV f vq vp A G e a vp a

12 1

1:20 1:1100000 100000x t x t x t tx t

f vq G e a vp f A

12

1:20 1x t tG e a

= 12

1:1100000 'x t x t tx t

f vq G e a vp V

12

:1' 1 100000t x tx t

V G e a i q f


Examiners Report

Page 4

= 1* 'x t tp V

Many students attempted to just write down the relationship which was not satisfactory. To score well the relationship had to be derived from 1st principles and the nature of the monthly premium effect clearly brought out.

6 The gross premium prospective policy value is given by:

1 53:53:753:53:7

500, 200 1,000*0.97*A a

760 6053:53 60:6053:53:7

53 53

* * *fm

m f

lla a v a

l l

79826.131 9848.431 1

16.716 * * *14.0909922.995 9934.574 1.04

16.716 0.745975*14.09

= 6.205

11/ 2 760 60

53:53:753:53:7 53 53

(1.04) * 1 * *fm

m f

llA da v

l l

= (1.04)1/2*(1 0.038462 * 6.205 0.745975)

= 0.015676

the gross premium policy value is:

500,200 * 0.015676 1,000 * 0.97 * 6.205 = £1822 to nearer £


Examiners Report

Page 5

7 (i) Let 2x t x xl l t d and 3

x t x xl l t d

21t x xp t q

and 2

t x xq t q

2t x

t x x t xp

p tqt

(ii) Therefore 1

0r x r x x rx

aq p p dr

= 1

3

0

1 2x xr q rq dr

= 1

4

0

2 2x xq r r q dr

=

15

2

0

2

5x xr

q r q

= 2

15x xq q

This question was done very poorly and few candidates derived satisfactory answers.

8 The expected present value of the benefits is given by

9 21 55 1

50 1

d wt tx t x t

x tt tx x

ad adS v v A

al al, (I) where

1

0

tx t x x t

t

A S p q v

The expected present value of the premium income is given by

255 1

5 5:101

wtx t

x x t x txt x

adP a P v a

al, (II) where

9

:100

tx t x

t

a ap v and 0

tx t x

t

a p v


Examiners Report

Page 6

xP is the premium for the term assurance and 5xP or 10xP is the premium for the

whole life assurance at the date on which the option is effected.

The additional single premium is given by (I) (II).

A double decrement table is constructed for all lives that effect the term assurance policy, with decrements of death and exercising the option, with the following definitions:

dx

ad , the number of decrements due to death aged x last birthday;

4wx

ad and 9

wx

ad , the number of decrements due to exercise of the option at the

fifth policy anniversary and at the expiry of the 10-year term respectively; and

xal , the number of lives aged exactly x in the double decrement table.

x tt x

x

alap

al

The dashed functions represent the mortality of those who have exercised the option.

The above solution is just one of a number of possible approaches and credit was given to candidates whose chosen method showed clear definitions. It was not totally necessary to adopt a multiple decrement approach as movements took place at discrete points and again credit was given for other methods.

9 (i) [ ]

[ ][ ] min[1 ,65 ]

250 1 0.06 150 0.98 0.02x

x

Tx K x

L S K v Pa P

(ii) Equivalence principle 0E L

Assume 12E T E K

40 40250 0.94 150 0.06S A S IA =

40 :250.98 0.02Pa P

12

40 [40]250 1.04 0.94 200,000 150 0.06 200,000A IA

= 40 :250.98 0.02Pa P

12250 1.04 188,150* 0.23041 12,000 *7.95835


Examiners Report

Page 7

= 0.98*15.887 0.02P

1

2250 1.04 43351.64 95500.2 15.58926P

£9,099.32P

10 ,x d is classified as x nearest birthday at entry and duration d at policy anniversary

following death. Define a census taken at time t after the start of the period of investigation (1.1.2000), ,'x dP t , of those lives having an in force policy at time t,

who were aged x nearest birthday at entry and will be duration d on the policy anniversary following time t.

The Central Exposed to Risk is then given by

3.75

, ,0

' .t

cx d x d

t

E P t dt

Then assuming that ,'x dP t varies linearly between the census dates (1.1.2000,

30.9.2000, 30.9.2002, 30.9.2003) the integral can be approximated by

, ,31 * ' 0 ' 0.752 4 x d x dP P

, ,1 *2 ' 0.75 ' 2.752 x d x dP P

, ,1 *1 ' 2.75 ' 3.752 x d x dP P

However the censuses ,'x dP t have not been recorded. The recorded censuses

,x dP t have lives classified by x nearest birthday at entry and curtate duration d at

time t. We can write

, , 1'x d x dP t P t

Substituting into the previous formula gives an expression for the required Central Exposed to Risk.

Then: ,,

,

x dx d c

x dE estimates 0.5x d

because the average age at entry is x assuming birthdays are uniformly distributed over the policy year, and the exact duration at the mid-point of the rate year (policy year) of deaths is d 0.5 for all lives (no assumptions are necessary).

This question was generally done well by well prepared students but many did not appreciate the relatively straightforward triangulation method.


Examiners Report

Page 8

11 If St is surplus in year t per policy in force at begin year t then:

(t 1V+P

Et)*1.07 = qt(10000 + tV) + (1

qt)*tV + St

Where tV etc is relevant reserve, P the required premium, Et is expenses for year t and qt the relevant mortality for year t

So St = (P

Et)*1.07 + t 1V *1.07

tV 10000* qt

We need to sum t-1 p[x]*St*vt at 10% for t = 1, 2, 3 and set to zero.

1V = 10000*(1

[57] 1:2 [57]:3/a a ) = 10000*(1 (1+v*l59/l[57]+1)/2.873)

= 10000*(1 1.956/2.873) = 3191.79

2V=10000*(1 1/2.873) = 6519.32 and 3V=10000 using 4% interest.

The following table can now be completed:

Year end t 1 2 3

Prem-Expense 0.8*P 0.95*P 0.95*P

t 1V[57] 0 3191.79 6519.32 10000*q[57]+t 1 41.71 61.80 71.40 Interest 0.056*P 0.0665*P+223.43 0.0665*P+456.35

tV[57] 3191.79 6519.32 10000.00

St 0.856P 3233.50 1.0165*P 3165.90 1.0165*P 3095.73

t-1 p[57] 1.00000 0.99583 0.98967

t-1 p[57]*St 0.856*p 3233.50 1.0123*P 3152.70 1.006*P 3063.75

Therefore:

(0.856*P 3233.5)*v+(1.0123*P 3152.70)*v2+(1.006*P 3063.75)*v3=0 at 10%

i.e. 2.3706*P=7846.92

P = £3,310.10

Very few students produced a full answer here. Although most solutions attempted were as above, it was also acceptable to take the 3rd year reserve as zero i.e. assuming the £10000 maturity value had been paid. This approach would have given a numerical answer of £3287.7


Examiners Report

Page 9

12 The inflation rate of 2.8846% p.a. combined with the valuation rate of 7% p.a. means that all benefits can be valued at 4% per annum effective.

The expected present value of the member s pension is given by

Benefit a

12653 65:562

110000* * *

1.04

la

l

12 12 1257070565:5 65

* *l

a a v al

at 4%i

125125

* 1.021537*4.4518 4.5477i

a ad

70

65

9238.1340.957538

9647.797

l

l

5 0.82193v

1270 11.562 0.458 11.104a

12

65:513.287a

the expected present value is given by

9647.797

10000* *0.889*13.2879773.083

116,607

Benefit b

The expected present value of the spouse s pension on death before retirement is given by

21262

62 0.5 59 0.50.5620

15000 * * *

1.04t

t ttt

dh a

l

1259.5 0.5* 16.982 16.652 0.458 16.359a


Examiners Report

Page 10

Similarly,

(12)60..5 16.024a

(12)61.5 15.679a

the value is given by

34.694 41.3985000* 0.980581*0.9* *16.359 0.942866*0.9* *16.024

9773.083 9773.083

49.1930.906602*0.9* *15.679

9773.083

= 866

the total expected present value is

116,607+866 = £117,473.

13 (i) The Standardised Mortality Ratio is the ratio of the actual deaths in a population compared with the expected deaths, based on standard mortality rates.

The formula is

, ,

, ,

cx t x t

xc sx t x t

x

E m

E m, where

,cx tE

is the central exposed to risk in the population between ages x and

x t

,x tm

is the central rate of mortality for the population between ages x and

x t

,s

x tm

is the central rate of mortality for a standard population between ages x

and x t


Examiners Report

Page 11

(ii) The Ratio may be written in the form

,, ,

,

, ,

x tc sx t x t s

x txc sx t x t

x

mE m

m

E m

which is the weighted average of the age-specific mortality differentials between the population being studied and the standard population.

i.e. ,

,

x ts

x t

m

m,

weighted by the expected deaths in the population being studied based on standard mortality.

i.e. , ,c sx t x tE m

(iii) The SMR for 2000 2001 is 1.8*10 0.9*20

1.230

The SMR for 2002 2003 is 2*10 0.8*20

1.230

(iv) (a) A formula for the CMF is

, ,

, ,

s cx t x t

xs c s

x t x tx

E m

E m

which may be written in the form

,, ,

,

, ,

x ts c sx t x t s

x txs c s

x t x tx

mE m

m

E m.

This is simply a weighted average of

,

,

x ts

x t

m

m,


Examiners Report

Page 12

weighted by

, ,s c s

x t x tE m .

The differences between the SMR and CMF figures indicates that the Standard Population A and the observed population have different proportions in the two age ranges.

As the CMF<SMR, this indicates that Standard Population A is more heavily weighted to the older age group.

(b) In my opinion, use of the SMR gives better results for comparing the population in each of the two periods. The mortality experience in the two periods is compared using Standard Population A exposed to risk in the CMF calculations and the observed population exposed to risk in the SMR calculations. Standard Population A appears to have a significantly different composition from the observed population. Therefore, using the Standard Population A exposed to risk in the weight calculations could introduce differences in the results which have nothing to do with underlying mortality differences. Use of the observed population exposed to risk removes this difficulty and results should be more reliable.

(c) I disagree with the committee s conclusion. The SMR figures indicate that the mortality experience has not changed between 2000 2001 and 2002 2003.

In part (iv) other acceptable comments were given credit.

END OF EXAMINERS

REPORT

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healthy x x d

life aged x

year of death

lives aged x

number of deaths aged

pension of year

dead x s

sick gh p x