Topic 2 - Life Assurance and Annuity Functions Flashcards
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
Topic overview
What is a life insurance contract?
- 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)
What are the different types of life assurance contract?
Whole life (WL)
Term assurance (TA)
Pure endowment (PE)
Endowment assurance (EA)
What is a whole life contract?
Payout on death
What is a term assurance contract?
Payout on death during specific term
What is a pure endowment contract?
Payout on survival beyond a specific term
What is an endowment assurance contract?
• Payout on survival beyond a term or death if earlier
• So combination of a TA and PE
In most contracts, what is the payout referred to as?
Sum Assured
Determine mean & variance functions
What does the standard notation Tx refer to?
- Complete future lifetime of a life aged x
- How much longer we expect a life aged x to live
State the formula for calculating the complete expectation of life for an individual aged x
E[Tx] = ∫0,∞ 𝑡𝑝𝑥 dt = 𝑜ex
Note: in all cases where 𝑜ex is stated, the 𝑜 symbol should be placed above the e
Determine mean & variance functions
What does the standard notation Kx refer to?
- Curtate random future lifetime of a life aged x
- Integer of Tx
State the formula for calculating the curtate expectation of life
E[Kx] = Σ𝑘=1,∞ 𝑘𝑝𝑥 = ex
ex ≈ 𝑜ex - 0.5
- Assume that benefits are paid at the END of year of death
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
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)
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)
Tx
What does the standard notation E[Tx] refer to?
The complete expectation of life for an individual aged x
Determine mean & variance functions
Give the standard notation for the curtate random future lifetime of a life aged x (integer of Tx)
Kx
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=0,∞ 𝑣^(𝐾𝑥 + 1) 𝑘𝑝𝑥 𝑞,𝑥+𝑘
EPV = A,x
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
A35 = 0.19219 (pg 96)
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
- 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
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?
- E(S𝑣^[𝐾𝑥+1]) = SE(𝑣^[𝐾𝑥+1])
- Var[S𝑣^(𝐾𝑥+1)] = S^(2)Var[𝑣^(𝐾𝑥+1)]
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
• 100000A40@6% = 100000x0.12313 = £12,313
• Var = 1000002[(2^A40) – (A40)^2]
• 100000^(2)[0.02707 – 0.12313^(2)] = £119,090,031
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
• 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
What is term assurance?
Pays sum assured on policyholder’s death during a specific period
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
PV = F = { 𝑣^(𝐾𝑥 + 1) 𝑖𝑓 𝐾𝑥 < ⴖ
={ 0 𝑖𝑓 𝐾𝑥 ≥ ⴖ
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
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¬
What is a pure endowment?
Pays sum assured on policyholder’s survival beyond a specific period
Give the formula for the present value of a sum assured of £1 on policyholder’s survival beyond a specific period
PV = G = { 0 𝑖𝑓 𝐾𝑥 < ⴖ
= { v^ⴖ 𝑖𝑓 𝐾𝑥 ≥ ⴖ
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
EPV = E(G) = v^(n) npx + 0.nqx
EPV = E(G) = v^(n) npx = 𝐴𝑥:n^1¬
Give the formula for the variance of a term assurance
Var(F) = 2^𝐴1^𝑥:𝑛¬ - (𝐴1^𝑥:𝑛¬)2
based on v^2 i.e. (1/1+i)^2
Give the formula for the variance of a pure endowment
Var(G) = 2^𝐴𝑥:𝑛^1¬ - (𝐴𝑥:𝑛^1¬)2
based on v^2 i.e. (1/1+i)^2
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
• 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
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
• [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
What is endowment assurance?
Pays sum assured on policyholder’s survival beyond a specific period or death if earlier
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
PV = H = { v^(𝐾𝑥 +1) 𝑖𝑓 𝐾𝑥 < ⴖ = 𝑣^min(𝐾𝑥 + 1, 𝑛)
= { v^ⴖ 𝑖𝑓 𝐾𝑥 ≥ ⴖ
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
EPV = E(H) = E(F) + E(G)
EPV = E(H) = 𝐴𝑥:𝑛¬ = 𝐴1^𝑥:𝑛¬ + 𝐴𝑥:𝑛^1¬
Give the formula for the variance of an endowment assurance
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)
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
• 50,000𝐴40:25¬ @6% = 50,000x0.24787 = £12,394
• NB 𝐴𝑥:𝑛¬ tabulated for x+n = 60 & 65
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
75,000𝐴45:15¬ @6% = 75,000x0.42556 = £31,917
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
• 90,000𝐴1^55:10¬ = 90000(𝐴55:10¬ - 𝐴55:10^1¬)
• 90,000(0.68388–8821.2612/9557.8179x1.04^-10)
• £5,434
What is a deferred assurance contract?
WL (whole life) assurance pays sum assured on policyholder’s death deferred for n years
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
PV = J = { 0 𝑖𝑓 𝐾𝑥 < 𝑛
{ 𝑣^(𝐾𝑥 + 1) 𝑖𝑓 𝐾𝑥 ≥ 𝑛
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
• EPV = n|Ax
• n|Ax = Ax - 𝐴𝑥:𝑛^1¬ = v^n npx Ax+n
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
• 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
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
• 80,000𝐴45:10¬ = 𝐴1^45:10¬ + 𝐴45:10^1¬
• 80,000(0.019455 + 9557.8179/9801.3123x1.04^-10)
• £54,259
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.
• 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)
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.
EPV = E[𝑣^(𝑇x)] = ʃ 0,∞ 𝑣^(𝑡) 𝑡𝑝𝑥 μ,𝑥+𝑡 dt = Ǡx
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.
Var[𝑣^(𝑇x)] = 2^Ǡx - (Ǡx)^2
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
• 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
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
EPV = E(Ꞙ) = Ǡ1^𝑥:ⴖ¬
Note: Ꞙ represents the letter F with a bar on top
Give the formula for the variance of a sum assured of £1 on policyholder’s death during a specific period payable immediately at death
Var(Ꞙ) = 2^Ǡ1^𝑥:ⴖ¬ - [Ǡ1^𝑥:ⴖ¬]^2
Note: Ꞙ represents the letter F with a bar on top
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
• 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
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
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
State some of the common relationships between benefits paid
immediately and at EOY of death
• 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¬
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
Summary
