1. Chapter 8 LIFE ANNUITIES
Basic Concepts
Commutation Functions
Annuities Payable mthly
Varying Life Annuities
Annual Premiums and Premium Reserves

2. 8.1 Basic Concepts
We know how to compute present value of contingent payments
Life tables are sources of probabilities of surviving
We can use data from life tables to compute present values of payments which are contingent on either survival or death

3. Example (pure endowment), p. 155
Yuanlin is 38 years old. If he reaches age 65, he will receive a single payment of 50,000. If i = .12, find an expression for the value of this payment to Yuanlin today. Use the following entries in the life table: l38 = 8327, l65 = 5411

4. t Ex = (t px ) (1 + t) – t = v t t px
Pure Endowment
Pure endowment: 1 is paid t years from now to an individual currently aged x if the individual survives
Probability of surviving is t px
Therefore the present value of this payment is the net single premium for the pure endowment, which is:
t Ex = (t px ) (1 + t) – t = v t t px

5. Example (life annuity), p. 156
Aretha is 27 years old. Beginning one year from today, she will receive 10,000 annually for as long as she is alive. Find an expression for the present value of this series of payments assuming i = .09
Find numerical value of this expression if px = .95 for each x

6. Series of payments of 1 unit as long as individual is alive
Life annuity
Series of payments of 1 unit as long as individual is alive
present value (net single premium) of annuity ax 1 1 1
…..
…..
age x
x + 1
x + 2
x + n px
2px
npx
probability

7. Temporary life annuity
Series of n payments of 1 unit (contingent on survival)
present value ax:n|
last payment 1 1 1
…..
age x
x + 1
x + 2
x + n px
2px
npx
probability

8. n - years deferred life annuity
Series of payments of 1 unit as long as individual is alive in which the first payment is at x + n + 1
present value n|ax
first payment 1 1 … …
age x
x + 1
x + 2
x + n
x + n +1
x + n + 2
n+1px
n+2px
probability
Note:

9. äx ….. … äx:n| ….. n|äx … … Life annuities-due 1 1 1 1 px 2px npx 1 1

10. Note
but

11. 8.2 Commutation Functions
Recall: present value of a pure endowment of 1 to be paid n years hence to a life currently aged x
Denote Dx = vxlx
Then nEx = Dx+n / Dx

12. Life annuity and commutation functions
Since nEx = Dx+n / Dx we have
Define commutation function Nx as follows:
Then:

13. Identities for other types of life annuities
temporary life annuity
n-years delayed l. a.
temporary l. a.-due

14. Accumulated values of life annuities
temporary life annuity
since
and
we have
similarly for temporary life annuity-due:
and

15. Examples (p. 162 – p. 164)
(life annuities and commutation functions) Marvin, aged 38, purchases a life annuity of 1000 per year. From tables, we learn that N38 = 5600 and N39 = Find the net single premium Marvin should pay for this annuity
if the first 1000 payment occurs in one year
if the first 1000 payment occurs now
Stay verbally the meaning of (N35 – N55) / D20
(unknown rate of interest) Given Nx = 5000, Nx+1=4900, Nx+2 = 4810 and qx = .005, find i

16. Select group
Select group of population is a group with the probability of survival different from the probability given in the standard life tables
Such groups can have higher than average probability of survival (e.g. due to excellent health) or, conversely, higher mortality rate (e.g. due to dangerous working conditions)

17. Notations
Suppose that a person aged x is in the first year of being in the select group
Then p[x] denotes the probability of survival for 1 year and q[x] = 1 – p[x] denotes the probability of dying during 1 year for such a person
If the person stays within this group for subsequent years, the corresponding probabilities of survival for 1 more year are denoted by p[x]+1, p[x]+2, and so on
Similar notations are used for life annuities: a[x] denotes the net single premium for a life annuity of 1 (with the first payment in one year) to a person aged x in his first year as a member of the select group
A life table which involves a select group is called a select-and-ultimate table

18. Examples (p. 165 – p. 166)
(select group) Margaret, aged 65, purchases a life annuity which will provide annual payments of 1000 commencing at age 66. For the next year only, Margaret’s probability of survival is higher than that predicted by the life tables and, in fact, is equal to p , where p65 is taken from the standard life table. Based on that standard life table, we have the values D65 = 300, D66 = 260 and N67 = If i = .09, find the net single premium for this annuity
(select-and-ultimate table) A select-and-ultimate table has a select period of two years. Select probabilities are related to ultimate probabilities by the relationships p[x] = (11/10) px and p[x]+1 = (21/20) px+1. An ultimate table shows D60 = 1900, D61 = 1500, and ä 60:20| = 11, when i = .08. Find the select temporary life annuity ä[60]:20|

19. The following values are based on a unisex life table: N38 = 5600, N39 = 5350, N40 = 5105, N41 = 4865, N42 = It is assumed that this table needs to be set forward one year for males and set back two years for females. If Michael and Brenda are both age 40, find the net single premium that each should pay for a life annuity of 1000 per year, if the first payment occurs immediately.

20. 8.3 Annuities Payable mthly
Payments every mth part of the year
Problem: commutation functions reflect annual probabilities of survival
First, we obtain an approximate formula for present value
Assume for a moment that the values Dy are also given for non-integer values of y

21. Annuity payable every 1/m part of the year
Usual life annuity ax 1 1 1
…..
…..
age x
x + 1
x + 2
x + n
Annuity payable every 1/m part of the year
a(m)x
1/m
1/m
1/m
1/m
…..
…..
age x
x + 1/m
x + 2/m
x + (m-1)/m
x + 1

22. Annuity payable every 1/m part of the year
a(m)x
1/m
1/m
1/m
1/m
…..
…..
age x
x + 1/m
x + 2/m
x + (m-1)/m
x + 1

23. Annuity payable every 1/m part of the year
a(m)x
1/m
1/m
1/m
1/m
…..
…..
age x
x + 1/m
x + 2/m
x + (m-1)/m
x + 1

24. Using linear interpolation for Dx+i+j/m

25. Using linear interpolation for Dx+i+j/m

26. Continuous life annuity

27. Annuity payable m-thly, deferred
n|a(m)x
a(m)x+n
1/m
1/m
1/m
1/m …
…..
…..
age x
x + n
x + n+1/m
x + n+2/m
x +n+ (m-1)/m
x + n+1

28. Annuity payable m-thly, temporary
a(m)x:n|
1/m
1/m
1/m
1/m
…..
age x
x + 1/m
x + 2/m
x +n+ (m-1)/m
x + n

29. Examples
Page 168, 8.10

30. 8.4 Varying Life Annuities
Arithmetic increasing annuities
It is sufficient to look at the sequence 1,2,3,….
Temporary decreasing annuities

31. Example
Ernest, aged 50, purchases a life annuity, which pays 5,000 for 5 years, 3,000 for 5 subsequent years, and 8,000 each year after. If the first payment occurs in exactly 1 year, find the price in terms of commutation functions.

32. Arithmetic increasing annuity
(Ia)x 1 2 n
…..
…..
age x
x + 1
x + 2
x + n px
2px
npx
probability

33. Arithmetic increasing annuity, temporary
(Ia)x:n| 1 2 n
…..
age x
x + 1
x + 2
x + n
x + n+1 px
2px
npx
probability

34. Arithmetic decreasing annuity, temporary
(Da)x:n| n
n-1 1
….. x
x + 1
x + 2
x + n
x + n+1

35. Arithmetic decreasing annuity, temporary
(Da)x:n| n
n-1 1
…..
(Ia)x:n| x
x + 1
x + 2
x + n
x + n+1 1 2 n
….. x
x + 1
x + 2
x + n
x + n+1

36. Arithmetic decreasing annuity, temporary
(Da)x:n| n
n-1 1
…..
(Ia)x:n| x
x + 1
x + 2
x + n
x + n+1 1 2 n
….. x
x + 1
x + 2
x + n
x + n+1
(n+1)ax:n|
n+1
n+1
n+1
….. x
x + 1
x + 2
x + n

37. Arithmetic decreasing annuity, temporary
(Da)x:n| n
n-1 1
…..
age x
x + 1
x + 2
x + n
x + n+1 px
2px
npx
probability

38. Examples
Georgina, aged 50, purchases a life annuity which will pay her 5000 in one year, 5500 in two years, continuing to increase by 500 per year thereafter. Find the price if S51 = 5000, N51 = 450, and D50 = 60
Redo the previous example if the payments reach a maximum level of 8000, and then remain constant for life. Assume S58 = 2100
Two annuities are of equal value to Jim, aged 25. The first is guaranteed and pays him 4000 per year for 10 years, with the first payment in 6 years. The second is a life annuity with the first payment of X in one year. Subsequent payments are annual, increasing by each year. If i = .09, and from the 7% -interest table, N26=930 and D25= 30, find X.

39. 8.5 Annual Premiums and Premium Reserves
Paying for deferred life annuity with a series of payments instead of a single payment
Premium reserve is an analog of outstanding principal
Premiums often include additional expenses and administrative costs
In such cases, the total payment is called gross premium
Loading = gross premium – net premium
General approach: actuarial present values of two sequences of payments must be the same (equation of value)

40. Annual premiums P = tP(n|äx)1 1 1 … … …
age x
x + 1
x + t-1
x +t
x + n
x + n +1
x + n + 2
t is the number of premium payments
Present value of premiums is P äx:t|
Present value of benefits is n|äx
Therefore P äx:t| = n|äx

41. Example
Arabella, aged 25, purchases a deferred life annuity of 500 per month, with the first benefit coming in exactly 20 years. She intends to pay for this annuity with a series of annual payments at the beginning of each year for the next 20 years. Find her net annual premium if D25 = 9000, D 45 = 5000, ä25 = 15 and ä45 = 11.5

42. … … … Reserves P P P P 1 1 1 ntV (n|äx)
age x
x + 1
x + t -1
x +n-1
x + n
x + n +1
x + n + 2
ntV (n|äx)
Analog of outstanding principal immediately after premium t has been paid
Assume that the number of premium payments is n
Reserve ntV (n|äx) = PV of all future benefits – PV of all future premiums

43. Loading and Gross premiums
Arabella, aged 25, purchases a deferred life annuity of 500 per month, with the first benefit coming in exactly 20 years. She intends to pay for this annuity with a series of annual payments at the beginning of each year for the next 20 years. Assume that 50% of her first premium is required for initial underwriting expenses, and 10% of all subsequent premiums are needed for administration costs. In addition, 100 must be paid for issue expenses. Find Arabella’s annual gross premium, if D25 = 9000, D 45 = 5000, ä25 = 15, and ä45 = 11.5

44. Chapter 9 LIFE INSURANCE
Basic Concepts
Commutation Functions and Basic Identities
Insurance Payable at The Moment of Death
Varying Insurance
Annual Premiums and Premium Reserves

45. 9.1 Basic Concepts Benefits are paid upon the death of the insured
Types of insurance
Whole life policy
Term insurance
Deferred insurance
Endowment insurance

46. Whole life policy
Benefit (the face value) is paid to the beneficiary at the end of the year of death of inured person
If the face value is 1 and insurance is sold to a person aged x, the premium is denoted by Ax x
x + 1
x + 2
x + t+1
….. Ax
age
probability px
2px
tqx
x + t 1

47. Whole life policy ….. Ax px 2px qx+t 1 x x + 1 x + 2 x + t+1 age
probability px
2px
qx+t
x + t 1

48. Term insurance
Benefit (the face value) is paid to the beneficiary at the end of the year of death of inured person, only if the death occurs within n years
If the face value is 1 and insurance is sold to a person aged x, the premium is denoted by A1x

49. Deferred insurance Does not come into force until age x+n
If the face value is 1 and insurance is sold to a person aged x, the premium is denoted by A1x:n|

50. n-year endowment insurance
Benefit (the face value) is paid to the beneficiary at the end of the year of death of inured person, if the death occurs within n years
If the insured is still alive at the age x+n, the face value is paid at that time
If the face value is 1 and insurance is sold to a person aged x, the premium is denoted by Ax:n|
Exercise:

51. Examples
Rose is 38 years old. She wishes to purchase a life insurance policy which will pay her estate 50,000 at the end of the year of her death. If i=.12, find an expression for the actuarial present value of this benefit and compute it, assuming px = .94 for all x.
Michael is 50 years old and purchases a whole life policy with face value 100,000. If lx= 1000(1-x/105) and i=.08, find the price of this policy.
Calculate the price of Rose’s and Michael’s policies if both policies are in force for a term of only 30 years.
Calculate the price of Rose’s and Michael’s policies if both policies are to be 30 years endowment insurance.

52. 9.2 Commutation Functions
Recall:

53. Commutation Functions
Recall:
So we need:

54. Whole life insurance

55. Term insurance

56. n-year endowment insurance

57. Note
We can represent insurance premiums in terms of actuarial present values of annuities, e.g. Ax = 1 – d äx
Hence they also can be found using “old” commutation functions

58. Examples
Juan, aged 40, purchases an insurance policy paying 50,000 if death occurs within the next 20 years, 100,000 if death occurs between ages 60 and 70, and 30,000 if death occurs after that. Find the net single premium for this policy in terms of commutation functions.
Phyllis, aged 40, purchases a whole life policy of 50,000. If N40 = 5000, N41 = 4500, and i = .08, find the price.

59. 9.3 Insurance Payable at the Moment of Death
We consider scenario when the benefit is paid at the end of the year of death
Alternatively, the benefit can be paid at the moment of death

60. Divide each year in m parts

62. Taking the limit as m→∞ we get:

63. Premium for insurance payable at the moment of death
Whole life policy:
Term policy:

64. Examples
Find the net single premium for a 100,000 life insurance policy, payable at the moment of death, purchased by a person aged 30 if i = .06 and tp30 = (.98)t for all t
Solve the previous example if it is 20 years endowment insurance, force of interest is .06 and lx = 105 – x, 0 ≤ x ≤ 105.

65. Remarks Using integration by parts, we can get Āx = 1 – δ āx
Approximate formula: Āx ≈ (i/δ) Ax
To obtain it, use linear interpolation in the following expression: