МАТЕМАТИКА СТРАХУВАННЯ ЖИТТЯ · fifth year, ie just before the sixth...

МАТЕМАТИКА

СТРАХУВАННЯ

ЖИТТЯ

ЗАДАЧІ

до розділу 1 .......................................................................................................................................... 1

до розділу 2 .......................................................................................................................................... 7

до розділу 3 ........................................................................................................................................ 10

до розділу 4 ........................................................................................................................................ 20

до розділу 5 ........................................................................................................................................ 24

до розділу 6 ........................................................................................................................................ 36

до розділу 7 ........................................................................................................................................ 40

до розділу 8 ........................................................................................................................................ 46

до розділу 9 ........................................................................................................................................ 68

Задачі до розділу 1

1

1.

2.

3.

4.

5. (CT1, April 2005)

6.

7.

8.


2

9.

10.

11.

Claire, aged exactly 30, buys a whole life assurance with a sum assured of

₤50,000 payable at the end of the year of her death. Calculate the standard deviation

of the present value of this benefit using AM92 Ultimate mortality and 6% pa

interest. 12.

13.

14.


3

15. (CT5, April 2005)

16.

Evaluate A40 and 40A based on AM92 Ultimate mortality at 4% pa interest.

17.

18.

19.

20.


4

21. (Subject 104, April 2000, Question 7) (СT5, April 2007)

22. (CT5, April 2005)

23. (CT5, April 2005)

24.


5

25.

26. (CT1, April 2005)

27. (CT1, April 2005)

28. (CT5, April 2006)


6

29. (CT5, May 2004)

30. (СT5, April 2007)


7

1. У розглянутій популяції сила смертності дорівнює 0.02 для всіх вікових категорій.

Обчислити

(і) Ймовірність того, що новонароджений помре протягом 5 років.

(іі) Ймовірність того, що особа, яка дожила до 90 років, помре до досягнення 95 років.

(ііі) Ймовірність того, що особа, яка має вік 5 років помре в інтервалі від 10 до 20

років.

(iv) Повне математичне сподівання тривалості життя новонародженого.

(v) Усічене математичне сподівання тривалості життя новонародженого.

2. Усічене математичне сподівання тривалості життя пенсіонерів, що вийшли на пенсію

у 60 років, складає 12 років. Знайти ймовірність того, що пенсіонер віку 65 років доживе до

70 років, припускаючи, що сила смертності є сталою.

3. Довести, що

)(stxtxtxstxs

ppt

4.

Якщо виконується припущення UDD, то 21

0

xx ee .

5.

Нехай xx TT * коли nTx 0 і nTx * коли nx . Довести, що

n

xtnx dtpe0

|:

0

6. Особа у віці 50 років підлягає додатковому випадковому ризику у віці від 50 до 51

року. Стандартна ймовірність смерті у віці від 50 до 51 року 0.006. Додатковий ризик дає

добавку до стандартної сили смертності, що рівномірно спадає від 0.03 на початку року до 0

наприкінці року. Обчислити ймовірність того, що особа доживе до 51 року.

7. За таблицями ELT15(M) знайти 3.286.5 q та 3.286.5 p використовуючи припущення

(і) UDD

(іі) стала сила смертності

(ііі) Бальдуччі.

8. Прямий ануїтет у розмірі 60 тис. фунтів стерлінгів виплачується протягом 20 років

особі, якій на момент укладення договору виповнилось 40 років. Розрахувати очікувану

поточну вартість цього ануїтету використовуючи таблиці AM92, припущення про селективну

смертність при 6% річних.

9. Довести, що

][

][1

|:][

x

nxx

nxD

MMA

10. (N1 April 2008)


8

11. (N8 april 2008)

12 (N6 april 2008)

13. (N1 april 2007)

14. (N7 april 2007)


9

15. (N5 april 2006)

16. (N2 april 2011)

17. (N6 april 2011)

18. (N3 September 2011)


10

1.

2.

3.

An annual premium endowment policy is surrendered with one year to go.

Explain why the reserve is a sensible surrender value for the insurer to pay.

4.

5.

6.

A 10-year term assurance with a sum assured of ₤500,000 is issued to a male

aged 30. Calculate the prospective and retrospective reserves at the end of the

fifth year, ie just before the sixth premium has been paid, assuming AM92

Ultimate mortality and 4% pa interest.

7.

8.

On 1 January 1997 a life insurance company issued a number of 30-year pure

endowment contracts to lives then aged 35. Level premiums are payable

annually in advance throughout the term of the contract or until earlier death.


11

In each case, the only benefit is a sum assured of ₤20,000, payable on survival

to the end of the term.

On 1 January 2005, 600 policies were still in force. During 2005, 3

policyholders died. Assuming that the company uses net premium policy

reserves, calculate the profit or loss from mortality for calendar year 2005 in

respect of this group of policies.

Basis:

mortality: AM92 Ultimate

interest: 4% pa

9.

10.

At the start of a particular year a life insurance company had a portfolio of

5,000 female pensioners, all aged exactly 60, who each receive an income of

₤10,000 per annum, paid annually in arrears.

The company holds net premium reserves, calculated using PFA92C20

mortality and 4% pa interest.

During that year, 9 pensioners died. Calculate the mortality profit or loss.

11. A life insurance company issues 20-year temporary assurance policies to lives

aged 45. The sum assured, which is payable immediately on death, is ₤400,000

for the first 10 years, and ₤100,000 thereafter. Level annual premiums are

payable in advance for 20 years, or until earlier death.

The premium basis is:

Mortality: AM92 Ultimate

Interest: 4% per annum

Expenses: nil.


12

12. (СT5 April 2006)

13. (СT5 April 2006)


13

14. (СT5 April 2006)


14

15. (СT5 April 2005)


15

16. (СT5 September 2005)

17. (СT5 Specimen May 2004)


16


19. (105 April 2003)


17

20. (105 April 2003)

21. (105 September 2003)


18

22. (N4 April 2007)

23. (N12 April 2007)

24. (N3 April 2008)

25. (N10 April 2011)


19

26. (N12 April 2008)


20

1.

2.

3.

4.

5.

6.

7. (N2, April 2008)

8. (N13, April 2007)


21


22

9. (N11, April 2008)

10. (СT5, N4 September 2011)


23

(N13, April 2008)

11. (N3, April 2011)


24

1.

2.

3.

4.

5. (СT5 April 2006)


25

6. (СT5 April 2005)

7.


26




27

10.

11.


28

12. (105 April 2004)

13.


29

14. (105 September 2004)

15.


30

16.


31

17. (105 April 2003)

18. (105 September 2003)


32

19.


33

20. (N13, April 2007)


34

21. (N11, April 2008)


35

22. (N8, April 2011)


36

1.

2.

3.

4.

5.

6.

7.

8.


37

9.

Jack, aged 60, wants to buy a reversionary annuity. If he dies before age 65 and

before his wife Vera, who is also now aged 60, she will receive an income of

₤10,000pa. The income will be paid annually in arrears (from the end of the year of

Jack's death) until Vera's 75th birthday or until her earlier death.

Calculate the single premium payable assuming PA92C20 mortality and 4% pa

interest. 10.

Jim and Dot, both aged 60, buy an annuity payable monthly in advance for at most

20 years, which is payable while at least one of them is alive. Calculate the

expected present value of the annuity assuming 4% pa interest and PA92C20

mortality. 11.

Marge and Homer, both aged exactly 55, take out a policy that provides a lump

sum of ₤50,000 payable immediately on the second death. Premiums are payable

annually in advance while at least one of Marge and Homer is alive.

Calculate the annual premium for the policy assuming PA92C20 mortality, 4% pa

interest, and no expenses. 12.

13. (CT5 April 2007)


38

14. (N5 April 2008)

15. (N10 April 2008)

16. (N9 April 2006)


39

17. (N9 April 2006)

18. (Subject 105, April 2000)

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 pa payable during the lifetime of the spouse, but ceasing 15 years

after the death of the member if that is earlier. All payments are made on the anniversary

of the member's retirement.

Calculate the expected present value of the spouse's benefit in the case of a female

member retiring now on her 60th birthday, who has a husband aged exactly 55.

Basis: PA92C20 mortality, 4%pa interest

19. (N9 April 2011)


40

1. (СT5 April 2006)

2. (СT5 April 2005)


41


4. (105 April 2004)


42

5. (105 September 2003)


43

6. (105 September 2003)


44

7. (105 April 2002)


45

8. (СT5 April 2011)


46

1. (СT5 April 2006)

2. (СT5 April 2006)


47

3. (СT5 April 2005)

4. (СT5 April 2005)


48


49



50


7.


51

8.

9.


52

10. (105 April 2004)

11.


53


54

12. (105 September 2004)


55

13.

14. (105 April 2003)


56

15.


57

16. (105 September 2003)

17. (105 September 2003)


58

18. (105 April 2002)

19.


59

20.

21. (105 September 2002)


60

22.

23.


61

24.


62

25.


63

26.

27.

28.

29.

30.


64

31.

32.

33.

34.


65

35.

36.

37.

38. (N4, April 2011)


66

39. (N11, April 2011)


67

Розв’язки задач до розділу 9

68

1. (N2, April 2007)

2. (N8, April 2007)

3. (N9 April 2008)


69

4. (N16 April 2006)


70


71

5. Why would you expect mortality rates to decrease over time and when would this not be the

case?

6. How would you expect the mortality rates for judges and divers to compare?

7. Fat people in the UK tend to come from a "rich" or a "poor" background, rather than an

"average wealth" background. Suggest possible reasons for this.

8. List four factors that could adversely affect the mortality of a homeless person in a

developed country.

9. What problem must you be careful to avoid before you can make the claim that a small

intake of alcohol is good for you?

10. An actuarial student is preparing a mortality table based on data obtained by recording the

ages at death on graves in a local cemetery. Suggest how each of the forms of selection

described above might be present.

11. Criticise the following statements from the viewpoint of how selection might affect the

assumed reality:

(a) "There were 1 million reported crimes in 1980 and 2 million in 1990. So the

numbers of minor and serious crimes committed have doubled over the last decade."

(b) "1% of the HIV tests carried out in a London clinic proved positive. So there are

500,000 HIV positive people in this country."

(c) "People who take out life assurance policies experience higher mortality rates than

the general population because they believe they are likely to die sooner than other people."

12. Explain why it might be necessary to subdivide policies by method of selling in order to

obtain homogeneous lapse rates.

13. Why will voluntary resignation tend to select those with lighter mortality and ill health-

retirement rates?

14. What is the risk to a life company of using one mortality table for all classes of lives

together?

15. What is the problem with producing tables for different classes of lives?

16. Explain what would be the likely effect on a company's mortality experience if it issued

assurance policies for the same rates of premium to smokers and non-smokers, when most

of the other companies in the market place charged different rates for the two groups.

17. (N1 April 2011)

18. (N7 April 2011)

МАТЕМАТИКА СТРАХУВАННЯ ЖИТТЯ · fifth year, ie just before the sixth...

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