Topic 2 - Life Assurance and Annuity Functions

Question 1

Topic 2 –Life and Annuity Functions
Topic overview

• Define simple assurance and annuity functions
• Determine mean and variance functions
  • For assurance functions
  • For annuity functions
• Derive relationship between assurance and
  annuity functions

Answer 1

Topic overview

Question 2

What is a life insurance contract?

Answer 2

• Agreement between Life co and policyholder
• Policyholder agrees to make payment(s)
• Life co agrees to pay lump sum or series of
  payments on occurrence of insured event
  (Insured event normally on survival or on
  death)

Question 3

What are the different types of life assurance contract?

Answer 3

Whole life (WL)
Term assurance (TA)
Pure endowment (PE)
Endowment assurance (EA)

Question 4

What is a whole life contract?

Answer 4

Payout on death

Question 5

What is a term assurance contract?

Answer 5

Payout on death during specific term

Question 6

What is a pure endowment contract?

Answer 6

Payout on survival beyond a specific term

Question 7

What is an endowment assurance contract?

Answer 7

• Payout on survival beyond a term or death if earlier
• So combination of a TA and PE

Question 8

In most contracts, what is the payout referred to as?

Answer 8

Sum Assured

Question 9

Determine mean & variance functions

What does the standard notation Tx refer to?

Answer 9

• Complete future lifetime of a life aged x
• How much longer we expect a life aged x to live

Question 10

State the formula for calculating the complete expectation of life for an individual aged x

Answer 10

E[Tx] = ∫0,∞ 𝑡𝑝𝑥 dt = 𝑜ex

Note: in all cases where 𝑜ex is stated, the 𝑜 symbol should be placed above the e

Question 11

Determine mean & variance functions

What does the standard notation Kx refer to?

Answer 11

• Curtate random future lifetime of a life aged x
• Integer of Tx

Question 12

State the formula for calculating the curtate expectation of life

Answer 12

E[Kx] = Σ𝑘=1,∞ 𝑘𝑝𝑥 = ex

ex ≈ 𝑜ex - 0.5

• Assume that benefits are paid at the END of year of death

Question 13

Give the present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid at the end of year of death

Answer 13

Whole Life Assurance
- Pays sum assured on policyholder’s death

PV of a sum assured of £1 payable at end of YOD
PV = 𝑣^(𝐾𝑥 + 1)

Question 14

Determine mean & variance functions

State the standard notation for the complete future lifetime of a life aged x (ie how much longer we expect a life aged x to live)

Answer 14

Tx

Question 15

What does the standard notation E[Tx] refer to?

Answer 15

The complete expectation of life for an individual aged x

Question 16

Determine mean & variance functions

Give the standard notation for the curtate random future lifetime of a life aged x (integer of Tx)

Answer 16

Kx

Question 17

Give the expected present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid at the end of year of death

Answer 17

Whole Life Assurance
- Pays sum assured on policyholder’s death

EPV of a sum assured of £1 payable at end of YOD
EPV = E[𝑣^(𝐾𝑥 + 1)] = Σ𝑘=0,∞ 𝑣^(𝐾𝑥 + 1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘
EPV = A,x

Question 18

Give the expected present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid at the end of year of death

Whole Life Assurance
- Pays sum assured on policyholder’s death

EPV of a sum assured of £1 payable at end of YOD
EPV = E[𝑣^(𝐾𝑥 + 1)] = Σ𝑘=1,∞ 𝑣^(𝐾𝑥 + 1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘
EPV = A,x

Find A35 – basis AM92 Ultimate@ 4% pa

Answer 18

A35 = 0.19219 (pg 96)

Question 19

Give the variance of the present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid at the end of year of death

Answer 19

• Whole Life Assurance
• Variance
• Use Var[X] = E[X^2] – [E(X)]^2

var[𝑣^(𝐾𝑥 + 1)] = E[𝑣^(𝐾𝑥 + 1)(2)] - E[𝑣^(𝐾𝑥 + 1)]^2
= Σ𝑘=0,∞ [𝑣^(𝐾𝑥 + 1)]^2 𝑘𝑝𝑥 𝑞,𝑥+𝑘 - (Ax)^2
= Σ𝑘=0,∞ [𝑣^2]^(𝐾𝑥 + 1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘 - (Ax)^2
= 2^Ax - (Ax)^2

2^Ax based on v^2 i.e. (1/1+i)^2 and is tabulated

Question 20

Determine mean & variance functions

Whole Life Assurance

So far our expectations and variances of sum assured are only based on sum assured (S) of £1, what if S>1?

Answer 20

• E(S𝑣^[𝐾𝑥+1]) = SE(𝑣^[𝐾𝑥+1])
• Var[S𝑣^(𝐾𝑥+1)] = S^(2)Var[𝑣^(𝐾𝑥+1)]

Question 21

Determine mean & variance functions

Example
• Calculate the EPV and Variance of a whole life assurance with a sum assured of £100,000 for a life currently aged 40. Payable at the EOYD.
• Basis AM92 Ultimate @ 6% pa

Answer 21

• 100000A40@6% = 100000x0.12313 = £12,313
• Var = 1000002[(2^A40) – (A40)^2]
• 100000^(2)[0.02707 – 0.12313^(2)] = £119,090,031

Question 22

Determine mean & variance functions

Question
• Calculate the EPV and standard deviation of a whole life assurance with a sum assured of £20,000 for a life currently aged 65. Payable at the EOYD.
• Basis AM92 Ultimate @ 4% pa

Answer 22

• 20000A65@4% = 20000x0.52786 = £10,557
• SD = (20000^2[(2^A65) – (A65)^2])^0.5
• (20000^2[0.30855 – 0.52786^2])^0.5 = £3,459

Question 23

What is term assurance?

Answer 23

Pays sum assured on policyholder’s death during a specific period

Question 24

Give the formula for the present value of a sum assured of £1 on policyholder’s death during a specific period payable at the end of YOD

Answer 24

PV = F = { 𝑣^(𝐾𝑥 + 1) 𝑖𝑓 𝐾𝑥 < ⴖ
={ 0 𝑖𝑓 𝐾𝑥 ≥ ⴖ

Question 25

Give the formula for the expected present value of a sum assured of £1 on policyholder’s death during a specific period payable at the end of YOD

Answer 25

EPV = E(F) = Σ𝑘=0,ⴖ-1 𝑣^(𝐾𝑥+1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘 + 0.𝑛𝑝𝑥
EPV = E(F) = Σ𝑘=0,ⴖ-1 𝑣^(𝐾𝑥+1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘 = 𝐴1^𝑥:𝑛¬

Question 26

What is a pure endowment?

Answer 26

Pays sum assured on policyholder’s survival beyond a specific period

Question 27

Give the formula for the present value of a sum assured of £1 on policyholder’s survival beyond a specific period

Answer 27

PV = G = { 0 𝑖𝑓 𝐾𝑥 < ⴖ
= { v^ⴖ 𝑖𝑓 𝐾𝑥 ≥ ⴖ

Question 28

Pure Endowment

Give the formula for the expected present value of a sum assured of £1 on policyholder’s survival beyond a specific period payable at the end of the period

Answer 28

EPV = E(G) = v^(n) npx + 0.nqx
EPV = E(G) = v^(n) npx = 𝐴𝑥:n^1¬

Question 29

Give the formula for the variance of a term assurance

Answer 29

Var(F) = 2^𝐴1^𝑥:𝑛¬ - (𝐴1^𝑥:𝑛¬)2

based on v^2 i.e. (1/1+i)^2

Question 30

Give the formula for the variance of a pure endowment

Answer 30

Var(G) = 2^𝐴𝑥:𝑛^1¬ - (𝐴𝑥:𝑛^1¬)2

based on v^2 i.e. (1/1+i)^2

Question 31

Determine mean & variance functions

Example
• Calculate the EPV of a 20 year pure endowment with a sum assured of £100,000 payable on survival to age 65, for a life currently aged 45.
• Basis AM92 Ultimate @ 4% pa

Answer 31

• 100,000𝐴45:20^1¬@4%
• 100,000v^(20)20p45
• 100,000 (1.04^-20) x 8821.2612/9801.3123
• =100,000 (1.04^-20)x0.9 = £41,075

Question 32

Determine mean & variance functions

Question
• Calculate the SD of a 20 year pure endowment with a sum assured of £100,000 payable on survival to age 65, for a life currently aged 45.
• Basis AM92 Ultimate @ 4% pa
• Hint use fact 2^𝐴𝑥:𝑛^1¬ based on v^2 i.e. (1/1+i)^2

Answer 32

• [100,000^2(2^𝐴45:20^1¬ - (𝐴45:20^1¬)2)]^0.5
• [100,000^2(v^[2(20)]20p45 – (v^(20)20p45)^2)]^0.5
• [100,000^2(1.04^[-2(20)] x 0.9 – (1.04^-20 x 0.9)^2)]^0.5
• [100,000^2(0.18746 – (0.41075)^2)]^0.5
• £13,691

Question 33

What is endowment assurance?

Answer 33

Pays sum assured on policyholder’s survival beyond a specific period or death if earlier

Question 34

Give the formula for the present value of a sum assured of £1 on policyholder’s survival beyond a specific period, or death if earlier, payable at the end of the period

Answer 34

PV = H = { v^(𝐾𝑥 +1) 𝑖𝑓 𝐾𝑥 < ⴖ = 𝑣^min(𝐾𝑥 + 1, 𝑛)
= { v^ⴖ 𝑖𝑓 𝐾𝑥 ≥ ⴖ

Question 35

Give the formula for the expected present value of a sum assured of £1 on policyholder’s survival beyond a specific period, or death if earlier, payable at the end of the period

Answer 35

EPV = E(H) = E(F) + E(G)
EPV = E(H) = 𝐴𝑥:𝑛¬ = 𝐴1^𝑥:𝑛¬ + 𝐴𝑥:𝑛^1¬

Question 36

Give the formula for the variance of an endowment assurance

Answer 36

Var(H) = 2^𝐴𝑥:𝑛¬ - (𝐴𝑥:𝑛¬)^2
• based on v^2 i.e. (1/1+i)^2
• Note - Var(H) ≠ Var(F) + Var(G)

Question 37

Determine mean & variance functions

Example
• Calculate the EPV of a 25 year endowment assurance with a sum assured of £50,000 payable on survival to age 65 or at the end of the year of earlier death, for a life currently aged 40.
• Basis AM92 Ultimate @ 6% pa

Answer 37

• 50,000𝐴40:25¬ @6% = 50,000x0.24787 = £12,394
• NB 𝐴𝑥:𝑛¬ tabulated for x+n = 60 & 65

Question 38

Determine mean & variance functions

Question
• Calculate the EPV of a 15 year endowment assurance with a sum assured of £75,000 payable on survival to age 60 or at the end of the year of earlier death, for a life currently aged 45.
• Basis AM92 Ultimate @ 6% pa

Answer 38

75,000𝐴45:15¬ @6% = 75,000x0.42556 = £31,917

Question 39

Question
• Calculate the EPV of 10 year term assurance with a sum assured of £90,000 payable at the end of the year of death, for a life currently aged 55.
• Basis AM92 Ultimate @ 4% pa
• Use EA = TA + PE

Answer 39

• 90,000𝐴1^55:10¬ = 90000(𝐴55:10¬ - 𝐴55:10^1¬)
• 90,000(0.68388–8821.2612/9557.8179x1.04^-10)
• £5,434

Question 40

What is a deferred assurance contract?

Answer 40

WL (whole life) assurance pays sum assured on policyholder’s death deferred for n years

Question 41

Deferred assurance contract

Give the formula for the present value of a sum assured of £1 on policyholder’s death deferred for a period of n years payable at the end of year of death

Answer 41

PV = J = { 0 𝑖𝑓 𝐾𝑥 < 𝑛
{ 𝑣^(𝐾𝑥 + 1) 𝑖𝑓 𝐾𝑥 ≥ 𝑛

Question 42

Deferred assurance contract

Give the formula for the expected present value of a sum assured of £1 on policyholder’s death deferred for a period of n years payable at the end of year of death

Answer 42

• EPV = n|Ax
• n|Ax = Ax - 𝐴𝑥:𝑛^1¬ = v^n npx Ax+n

Question 43

Determine mean & variance functions

Example
• Calculate the EPV of 10 year term assurance with a sum assured of £80,000 payable at the end of the year of death, for a life currently aged 45.
• Basis AM92 Ultimate @ 4% pa

Answer 43

• 80,000𝐴1^45:10¬= 80000(A45 – v^10 10p45 A55)
• 80,000(0.27605–0.3895x9557.8179/9801.3123x1.04^-10)
• 80,000x0.019455 = £1,556

Question 44

Determine mean & variance functions

Question
• Calculate the EPV of 10 year endowment assurance with a sum assured of £80,000 payable at the end of the year of death, for a life currently aged 45.
• Basis AM92 Ultimate @ 4% pa

Answer 44

• 80,000𝐴45:10¬ = 𝐴1^45:10¬ + 𝐴45:10^1¬
• 80,000(0.019455 + 9557.8179/9801.3123x1.04^-10)
• £54,259

Question 45

Give the present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid immediately on death.

Answer 45

• So far only discussed death benefits payable at the end of year of death i.e. Kx+1
• Can evaluate benefits payable immediately on death
• Replace Kx + 1 with Tx (a continuous function)
• Whole Life Assurance
• PV = 𝑣^(𝑇x)

Question 46

Give the formula for the expected present value of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid immediately on death.

Answer 46

EPV = E[𝑣^(𝑇x)] = ʃ 0,∞ 𝑣^(𝑡) 𝑡𝑝𝑥 μ,𝑥+𝑡 dt = Ǡx

Question 47

Give the formula for the variance of a £1 sum assured payout on a policyholder’s death from a whole life assurance contract. Assume benefits are paid immediately on death.

Answer 47

Var[𝑣^(𝑇x)] = 2^Ǡx - (Ǡx)^2

Question 48

Give the formula for the present value of a sum assured of £1 on policyholder’s death during a specific period payable immediately at death

Answer 48

• Term Assurance
• Pays sum assured on policyholder’s death during a specific period

PV = Ꞙ = { 𝑣^(Tx) 𝑖𝑓 T𝑥 < ⴖ
={ 0 𝑖𝑓 T𝑥 ≥ ⴖ

Note: Ꞙ represents the letter F with a bar on top

Question 49

Give the formula for the expected present value of a sum assured of £1 on policyholder’s death during a specific period payable immediately at death

Answer 49

EPV = E(Ꞙ) = Ǡ1^𝑥:ⴖ¬

Note: Ꞙ represents the letter F with a bar on top

Question 50

Give the formula for the variance of a sum assured of £1 on policyholder’s death during a specific period payable immediately at death

Answer 50

Var(Ꞙ) = 2^Ǡ1^𝑥:ⴖ¬ - [Ǡ1^𝑥:ⴖ¬]^2

Note: Ꞙ represents the letter F with a bar on top

Question 51

Give the formula for the present value of a sum assured of £1 on policyholder’s survival beyond a specific period or death if earlier, payable immediately at death

Answer 51

• Endowment Assurance
• Pays sum assured on policyholder’s survival beyond a specific period or death if earlier

PV = Ĥ = { 𝑣^(Tx) 𝑖𝑓 T𝑥 < ⴖ
= { 𝑣^(ⴖ) 𝑖𝑓 T𝑥 ≥ ⴖ

Note: Ĥ represents the letter H with a bar on top

Question 52

Give the formula for the expected present value of a sum assured of £1 on policyholder’s survival beyond a specific period or death if earlier, payable immediately at death

Answer 52

EPV = E(Ĥ) = E(Ꞙ) + E(G)
EPV = E(Ĥ) = Ǡ𝑥:ⴖ¬ = Ǡ1^𝑥:ⴖ¬ + Ǡ𝑥:ⴖ^1¬

Note: Ĥ represents the letter H with a bar on top
Note: Ꞙ represents the letter F with a bar on top

Question 53

State some of the common relationships between benefits paid
immediately and at EOY of death

Answer 53

• Ǡ𝑥 ≈ (1+i)^0.5 A𝑥 ≈ i/δ A𝑥
• Ǡ1^𝑥:𝑛¬ ≈ (1+i)^0.5 𝐴1^𝑥:𝑛¬ ≈ i/δ 𝐴1^𝑥:𝑛¬
• Ǡ𝑥:𝑛¬ ≈ (1+i)^0.5 𝐴1^𝑥:𝑛¬ + 𝐴𝑥:𝑛^1¬

Question 54

Summary
•Introduced to assurance functions
- Whole life
- Term assurance
- Pure endowment
- Endowment assurance
•Derived EPV and variance for functions
•Looked at benefits paid at EOY of death & immediately
•Looked at relationships between both

Answer 54

Summary