Acturial Science - CM1 A - CH 7 Accumulating and Discounting - Question Bank

Ch 7 Discounting and Accumulating
Question Bank




Tip : In this chapter, calculations is often the major problem. Keep it in mind and always double check your
answers as formulas are limited in this chapter.




Q1


Question




Answer
𝑏 𝑏
𝛿(𝑡).𝑑𝑡
𝐴𝑉𝑎𝑡 𝑡=𝑏 = ∫ 𝜌(𝑡) ∗ 𝑒 ∫𝑡 . 𝑑𝑡 (𝐹𝑜𝑟𝑚𝑢𝑙𝑎)
𝑎
6 6
6 6
0.06.𝑑𝑡 6
𝐴𝑉𝑎𝑡 𝑡=6 = ∫ 100 ∗ 𝑒 ∫𝑡 . 𝑑𝑡 = ∫ 100 ∗ 𝑒 (0.06𝑡)𝑡 . 𝑑𝑡 = ∫ 100 ∗ 𝑒 (0.36−0.06𝑡) . 𝑑𝑡
0 0 0

6 6
(0.36−0.06𝑡)
𝑒 (0.36−0.06𝑡) 100
= 100 ∫ 𝑒 . 𝑑𝑡 = 100 ∗ ( ) = ∗ (𝑒 0.36−0.36 − 𝑒 0.36−0 )
0 −0.06 0
−0.06

= 722.2157


12 12
𝛿(𝑡).𝑑𝑡 (0.05+0.0002𝑡 2 ).𝑑𝑡
𝐴𝑉𝑎𝑡 𝑡=12 = 𝐴𝑉𝑎𝑡 𝑡=6 ∗ 𝑒 ∫6 = 722.2157 ∗ 𝑒 ∫6
0.0002𝑡 3 12
(0.05𝑡+ )6
= 722.2157 ∗ 𝑒 3 = 722.2157 ∗ 𝑒 0.7152−0.3144 = 722.2157 ∗ 𝑒 0.4008
= €1078.2815

, 𝑁𝑜𝑡𝑒 ∶ 𝐼𝑛 𝑡ℎ𝑖𝑠 𝑞𝑢𝑒𝑠, 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑠 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑠𝑡𝑟𝑒𝑎𝑚 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡, 𝑏𝑢𝑡 𝑎

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑎𝑠𝑒 100), 𝑏𝑢𝑡 𝑠𝑡𝑖𝑙𝑙 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑠𝑡𝑟𝑒𝑎𝑚 𝑛𝑜𝑡 𝑎
6
𝑠𝑖𝑛𝑔𝑙𝑒 𝑝𝑎𝑦𝑚𝑒𝑛𝑡. 𝑆𝑜 𝑑𝑜 𝑛𝑜𝑡 𝑡ℎ𝑖𝑛𝑘 (100 ∗ 𝑒∫0 0.06.𝑑𝑡 ) 𝑤𝑜𝑢𝑙𝑑 𝑤𝑜𝑟𝑘.




Q2


Question




Answer

a) 𝐴𝑉 𝑡=10 = 𝑃𝑉𝑡=0 ∗ 𝐴(0,10) + 𝑃𝑉𝑡=5 ∗ 𝐴(5,10)
10 10
𝛿(𝑡).𝑑𝑡 𝛿(𝑡).𝑑𝑡
= 100 ∗ 𝑒 ∫0 + 100 ∗ 𝑒 ∫5
10 10
(0.05+0.001𝑡+0.0001𝑡 2 ).𝑑𝑡 (0.05+0.001𝑡+0.0001𝑡 2 ).𝑑𝑡
= 100 ∗ 𝑒 ∫0 + 100 ∗ 𝑒 ∫5
10 10
0.001𝑡2 0.0001𝑡3 0.001𝑡2 0.0001𝑡3
[(0.05𝑡+ + ) ] [(0.05𝑡+ + ) ]
2 3 2 3
= 100 ∗ 𝑒 0 + 100 ∗ 𝑒 5


= 100 ∗ [𝑒 0.58333−0 + 𝑒 0.58333−0.26667 ] = 100 ∗ (1.792 + 1.3725)
= €316.4536


b) For variable force of interest to be equivalent to constant force of interest,

316.4536 = 𝑃𝑉𝑡=0 ∗ 𝑒 𝛿∗10 + 𝑃𝑉𝑡=5 ∗ 𝑒 𝛿∗5

316.4536 = 100 ∗ 𝑒 𝛿∗10 + 100 ∗ 𝑒 𝛿∗5