Tutorial 4 Flashcards
Question 1
• A life currently aged 50 exactly has received a quote for a whole life assurance with the following properties:
• Initial sum assured is £100,000
• Sum assured increases by 1.9231% at start of each year excluding the first year
• Benefits paid at the end of the year of death
• Basis AM92 Ultimate , Interest 6% pa
a) Calculate the EPV of this benefit.
b) Calculate the monthly premium if premiums are paid monthly in advance for 15 years.
• The life has received a revised whole life assurance quote whereby the sum assured will increase as per above until the 5th year and remain level thereafter.
c) Calculate the EPV of this benefit and the revised monthly premium.
a) Calculate the EPV of this benefit.
=100000(vq50 + 1.019231v^2 1|q50 + 1.019231^(2) v^(3) 2|q50
• 100000/1.019231(1.019231vq50 +1.019231^(2)v^(2)1|q50 + 1.019231^(3) v^(3)2|q50)
• 1.06/1.019231 = 1.04
• (100000/1.019231)A50 @ 4%pa
• (100000/1.019231) x 0.32907
• £32,286.11
b) Calculate the monthly premium if premiums are paid monthly in advance for 15 years.
• 12P ἂ50:15¬(12)@6% = 100000/1.019231 A50 @ 4%pa
• ἂ50:15¬(12) = ἂ50:15¬ - [12−1]/2𝑥12
(1 – v^(15)l65/l50)
• ἂ50:15¬(12)= 10.038 -11/24
(1 – 1.06^(-15) x8821.2612/9712.0728) = 9.7534
• 12Px9.7534 = 32286.11
• P = £275.85 pm
• The life has received a revised whole life assurance quote whereby the sum assured will increase as per above until the 5th year and remain level thereafter.
c) Calculate the EPV of this benefit and the revised monthly premium.
• = 100000(vq50 + 1.019231v^(2)1|q50 + 1.019231^(2) v^(3)2|q50 + 1.019231^(3) v^(4)3|q50 + 1.019231^(4) v^(5)4|q50 + 1.019231^(4) v^(6)5|q50 + 1.019231^(4) v^(7)6|q50 …………)
• 100000/1.019231(1.019231vq50 + 1.019231^(2)v^(2)1|q50 + 1.019231^(3) v^(3)2|q50 + 1.019231^(4)v^(4)3|q50 + 1.019231^(5)v^(5)4|q50)+100,000(1.019231^(4)v^(6)5|q50 + 1.019231^(4) v^(7)6|q50 …………)
• (100000/1.019231)𝐴1^50:5¬ @4% pa
• 𝐴1^50:5¬ = (A50 – [D55/D50]A55)
• 𝐴1^50:5¬ = (0.32907 - 0.3895x1105.41/1366.61) = 0.014015
• [100000/1.019231]𝐴1^50:5¬ @4% pa = 100000/1.019231x0.014015 = £1,375.06
• +100,000(1.019231^(4) v^(6)5|q50 + 1.019231^(4) v^(7) 6|q50 …………)
• +100,000(1.019231^(4) v^(6)5p50q55 + 1.019231^(4) v^(7)6p50q56 …………)
• +100,000x1.019231^(4) v^(5)5p50 (vq55 + v^(2) p55q56 …………)
• +100,000x1.019231^(4) v^(5)5p50 A55@6%
• +100,000x1.019231^(4)1.06-^(5)x9557.8179/9712.0728x0.26092
• =£20,706.91
• EPV = 1375.06 + 20706.91 = £22,081.97
• Monthly premium = 22081.97/(12x9.7534) = £188.67 pm
Question 2
• A life currently aged 60 exactly has been offered a whole life annuity of £15,000 pa. If the annuity is assumed to decrease by 1.9231% pa of each year excluding the first, calculate the following:
a) The EPV if the annuity is paid annually in advance
b) The EPV if the annuity is paid annually in arrears
• Basis - Mortality AM92 Ultimate, interest 4% pa
a) Calculate the EPV if the pension is paid annually in advance
• 15000(1 + 1.019231^(-1)v p60 + 1.019231^(-2) v^(2)2p60 +…….)
• 15000(1+ v’p60 + v’^(2)2p60 +…….)
• v’ = 1/1.019231 x 1/1.04
• 15000 ἂ60@6%
• 15000x11.891 = £178,365
b) Calculate the EPV if the pension is paid annually in arrears
• 15000(vp60 + 1.019231^(-1)v^(2)2p60 +…….)
• 15000x1.019231(1.019231^(-1)vp60 + 1.019231^(-2) v^(2)2p60 +..)
• 15000x1.019231(v’p60 + v’^(2)2p60 +..)
• v’ = 1/1.019231 x 1/1.04
• 15000 x 1.019231 x a60 @6% pa
• 15000x 1.019231 (ἂ60 - 1) @6% pa
• 15000 x 1.019231 (11.891 - 1) = £166,507
Question 3
• A life currently aged 50 exactly purchases a decreasing term assurance with a duration of 20 years. The initial sum assured is £120,000 and will decrease each subsequent year by £5,000.
• Basis AM92 Ultimate, Interest 6% pa
a) Calculate the EPV of this benefit.
b) Calculate the EPV if the policy was a decreasing endowment assurance.
c) Calculate the initial premium for each contract if premiums are expected to increase by £5 pa from year 2.
Assume premiums are paid annually in arrears for the duration of the contract
a) Calculate the EPV of this benefit.
125000 𝐴1^50:20¬ - 5000(𝐼𝐴)1^50:20¬
𝐴1^50:20¬ = A50 – A70xl70/l50x1.06^-20
= 0.20508 – 0.48265 x 8054.0544/9712.0728 x 1.06^-20
= 0.20508 – 0.48265x0.258574 = 0.080279 (𝐼𝐴)1^50:20¬ = (IA)50 –v^(20)20p50[20A,70 + (IA),70]
= 4.84555 – 0.258574[20x0.48265 + 5.33628] = 0.96971
EPV = 125000x0.080279 – 5000x0.96971 = £5,186.33
b) Calculate the EPV if the policy was a decreasing endowment assurance.
EPV = 125000𝐴1^50:20¬ - 5000(𝐼𝐴)1^50:20¬ + (125000 – 20x5000) 𝐴50:20^1¬
EPV = 5186.33 + 25000x0.258574 = £11,650.68
EPV = 125000𝐴50:20 - 5000(𝐼𝐴)50:20
125000𝐴50:20 = 125000(𝐴1^50:20 + 𝐴50:20^1)
5000(𝐼𝐴)50:20 = 5000[(𝐼𝐴)1^50:20 + 20x𝐴50:20^1]
c) Calculate the initial premium for each contract if premiums are expected to increase by £5 pa from year 2.
• P - 5 + 5(Ia)50:20¬
• a50:20¬ = a50 – v^(20)20p50a70 = (ἂ50 - 1) – v^(20)20p50(ἂ70 - 1)
• a50:20¬ = (14.044 - 1) – 0.258574(9.14 - 1) = 10.9392
• (Ia)50:20¬ =(Ia)50 –v^(20)20p50[20a70 + (Ia)70]
= [(Iἂ)50 - ἂ50) - 0.258574[20(ἂ70 - 1) + (Iἂ)70 - ἂ70]
=162.497 - 14.044 – 0.258574[20x8.14 + 67.198 - 9.14] = 91.34
• a) 5186.33 = (P - 5)x[10.9392]+5x91.34
P = £437.36
• b) 11650.68 = (P - 5)x[10.9392]+5x91.34
P = £1,028.29
Question 4
• A life currently aged 60 exactly purchases a 10 year endowment assurance with initial sum assured of £50,000. The policy is subject to the following simple reversionary bonuses excluding the first year:
• Bonus in years 2 and 3 = 5%
• Bonus in years 4+ = 1.5%
• Calculate the net premium reserve after 6 years.
Assume the following
• Death benefits are paid at the end of the year of death
• Basis Mortality PFA92C20 , interest 4% pa
• P = 50000 𝐴60:10¬/ἂ60:10¬
• ἂ60:10¬ = ἂ60 - v^(10)10p60 ἂ70
• ἂ60:10¬ = 16.652 - 12.934 x 9392.621/9848.431 x 1.04^(-10)
• ἂ60:10¬ = 16.652 - 12.934 x 0.6443 = 8.31863
• P = 50000 (1 - 0.04/1.04 x 8.31863)/8.31863
• P = £4,087.53
• Bonus = 50000(2 x 5%+3 x 1.5%) = £7,250
• 6,V = 57250 𝐴66:4¬ - 4,087.53ἂ66:4¬
• ἂ66:4¬ = ἂ66- v^(4)4p66 ἂ70
• ἂ66:4¬ = 14.494 – 12.934 x 9392.621/9658.285 x 1.04^(-4)
• ἂ66:4¬ = 14.494 - 12.934 x 0.83129 = 3.7421
• 6,V = 57250 (1 - 0.04/1.04 x 3.7421) - 4087.53 x 3.7421
• 6,V = £33,714.2
