Tutorial 3 Flashcards
Question 1
- Calculate the net premium for a 30 year term assurance for a life currently aged 25 with sum assured of £125,000, with the following conditions:
- Basis AM92 @ 4%
a) Death benefits paid at EOY, premiums annually in advance
b) Death benefits paid at EOY, premiums quarterly in advance
c) Death benefits paid immediately, premiums monthly in advance
- How much would the premium in a) change by if the
contract was an endowment assurance? - What does this change in premium represent?
- Calculate the net premium for a 30 year term assurance for a life currently aged 25 with sum assured of £125,000, with the following conditions:
- Basis AM92 @ 4%
• Death benefits paid at EOY, premiums annually in advance
• 𝑃1^25:30 = 125000𝐴1^25:30/ ἂ25:30
• 𝐴1^25:30=A25 – D55/D25A55 =0.13386-1105.41/3733.77x0.3895=0.01855
• ἂ25:30=ἂ25 – D55/D25 ἂ55=22.52-1105.41/3733.77x15.873 = 17.821
• 𝑃1^25:30 = 125000x0.01855/17.821 = £130.11 pa
• Death benefits paid at EOY, premiums quarterly in advance
• 𝑃1^25:30 (4)= 125000𝐴1^25:30/ ἂ25:30(4)
• ἂ25:30(4)= ἂ25:30 - ([4−1]/2𝑥4)(1 – D55/D25) = 17.821-3/8x(1-1105.41/3733.77)
• = 17.557
• 𝑃1^25:30 (4)= 125000x0.01855/17.557 = £132.07 pa
• Death benefits paid at immediately, premiums monthly in advance
• Ṕ1^25:30 (12)= 125000Ǡ1^25:30/ ἂ25:30(12)
• ἂ25:30(12)= ἂ25:30 - ([12−1]/2𝑥12)(1 – D55/D25) = 17.821-11/24x(1 - 1105.41/3733.77)
• =17.4984
• Ǡ1^25:30 = 1.04^(0.5)𝐴1^25:30 = 1.04^(0.5)x0.01855 = 0.01892
• Ṕ1^25:30 (12) = 125000x0.01892/17.4984 = £135.16 pa
• How much would the premium in a) change by if the contract was an endowment assurance, what does it represent?
• 𝑃25:30 = 125000𝐴25:30/ ἂ25:30
• 𝐴25:30=1 - d ἂ25:30 = 1 - (0.04/1.04)x17.821 = 0.31458
• 𝑃25:30 = 125000x0.31458/17.821 = £2,206.53 pa
• So change is 2206.53 - 130.11 = £2,076.42
• Premium of pure endowment – 125000x1105.41/3733.77/17.821 =
£2,076.60
[Note that Ṕ = P with bar on top]
Question 2
• Calculate the net premium payable for a whole life
assurance with sum assured of £80,000 for a life
currently aged 50. Assume death benefits are paid at the end of the year of death. Premiums are payable annually in advance and are guaranteed for two years (i.e. payable by the member or their family irrespective of death in that period) and payable for the duration of the policy thereafter.
• Basis - Mortality PFA92C20, interest 4% pa
• P50 = 80000A50/[ἂ2 +v^(2)l52/l50ἂ52]
• 80000A50 = 80000(1 - d ἂ50)
• 80000(1 - 0.04/1.04x19.539) = £19,880
• [ἂ2 +v^(2)l52/l50ἂ52]
• = 1.04/0.04[1 - 1.04^(-2)] + 1.04^(-2)x9941.454/9952.697x19.034
• =1.96154 + 17.5781 =19.53964
• P50 = 19880/19.53964 = £1,017.42
Question 3
• A life currently aged exactly 30 purchases a 25 year
endowment assurance with sum assured of £120,000.
Assume death benefits are paid immediately on death and premiums are paid continuously for the duration of the contract. Basis AM92 Ult, Interest 6% pa
a) Calculate the net premium reserve at t = 10 using the prospective method.
b) Calculate the net premium reserve at t = 12 using the retrospective method.
c) Derive a formula for the net premium reserve in terms of annuities and use it to verify your answers in a and b.
a) Calculate the net premium reserve at t = 10 using the
prospective method.
• Ṕ30:25 = 120000 Ǡ30:25 / ᾱ30:25
• ᾱ30:25 ≈ ἂ30:25 - ½ (1 – v^(25) 25p30)
• ἂ30:25 = ἂ30- v^(25) 25p30 ἂ55
• 16.372 – 13.057 x 9557.8179/9925.2094x1.06^-25
• 16.372 - 13.057x0.22437 = 13.4424
• ᾱ30:25 ≈ 13.4424 – 0.5(1 - 0.22437) = 13.0546
• Ṕ30:25 = 120000(1 - ln1.06x13.0546)/13.0546 = £2,199.89 pa
• 120000Ǡ40:15 - 2199.89ᾱ40:15
• ᾱ40:15 ≈ ἂ40:15 - ½ (1 – v^(15) 15p40)
• ἂ40:15 = ἂ40- v^(15) 15p40 ἂ55
• 15.491 – 13.057 x 9557.8179/9856.2863x1.06^-15
• 15.491 - 13.057x0.40463 = 10.20775
• ᾱ40:15 ≈ 10.20775 – 0.5(1 - 0.40463) = 9.9101
•10,Ṽ = 120000(1 - ln1.06x9.9101) – 2199.89x9.9101 = £28,905
b) Calculate the net premium reserve at t = 12 using the retrospective method.
• 2199.89x1.06^(12)xl30/l42x ᾱ30:12- 120000 x1.06^(12)xl30/l42xǠ1^30:12
• ᾱ30:12 ≈ ἂ30:12 - ½ (1 – v^(12) 12p30)
• ἂ30:12 = ἂ30- v^(12) 12p30 ἂ42
• 16.372– 15.253 x 9837.0661/9925.2094x1.06^-12
• 16.372-15.253x0.492556 = 8.85904
• ᾱ30:12 ≈ 8.85904 – 0.5(1 - 0.492556) = 8.60532
• Ǡ1^30:12 = (1 - ln1.06x8.60532 – 0.492556) = 0.0060214
• 12,Ṽ = 2199.89x8.60532/0.492556 – 120000x0.0060214/0.492556
• = £36,967
c) Derive a formula for the net premium reserve in terms of annuities and use it to verify your answers in a and b.
• Ǡ,𝑥+𝑡:𝑛−𝑡 - Ṕ,𝑥:𝑛𝑎 ᾱ,𝑥+𝑡:𝑛−𝑡
• Ǡ,𝑥+𝑡:𝑛−𝑡 - (Ǡ,𝑥:𝑛/ᾱ,𝑥:𝑛) ᾱ,𝑥+𝑡:𝑛−𝑡
• (1 - 𝛿 ᾱ,𝑥+𝑡:𝑛−𝑡) – (1- 𝛿 ᾱ,𝑥:𝑛) ᾱ,𝑥+𝑡:𝑛−𝑡/ᾱ,𝑥:𝑛
• (1 - 𝛿 ᾱ,𝑥+𝑡:𝑛−𝑡) – ᾱ,𝑥+𝑡:𝑛−𝑡/ᾱ,𝑥:𝑛 + 𝛿ᾱ,𝑥+𝑡:𝑛−𝑡
• 1 – ᾱ,𝑥+𝑡:𝑛−𝑡/ᾱ,𝑥:𝑛
• 10,Ṽ = 120000(1 – ᾱ,40:15/ᾱ,30:25)
• 10,Ṽ = 120000(1 - 9.9101/13.0546) = £28,905
•12,Ṽ = 120000(1 – ᾱ,42:13/ᾱ,30:25)
• ᾱ,42:13 ≈ ἂ42:13 - ½ (1 – v^13 13p42)
• ἂ,42:13 = ἂ,42 - v^(13) 13p42 ἂ,55
• 15.253– 13.057 x 9557.8179/9837.0661x1.06^-13
• 15.253 – 13.057x0.45553 = 9.30514
• ᾱ, 42:13 ≈ 9.30514 – 0.5(1-0.45553) = 9.03291
• 12,Ṽ = 120000(1 – 9.03291/13.0546) = £36,968 vs £36,967
Question 4
• A life assurance company issues a 10 year endowment assurance and a whole life assurance to two identical lives currently aged 55.
• If the net premium reserve is identical at t=5, calculate the sum assured for the endowment assurance.
Assume the following
• Death benefits are paid at the end of the year of death
• The sum assured for the whole life assurance is £50,000
• Basis Mortality AM92 Ult , interest 4%
• 5,V,WL = 50000(1 - ἂ,60/ἂ,55)
• 5,V,WL = 50000(1 - 14.134/15.873) = £5,477.86
• 5,V,EA = SA(1 - ἂ,60:5/ ἂ,55:10)
• 5477.86 = SA(1 - 4.55/8.219)
• SA = 5477.86/(0.446405) = £12,27
