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Question
An annuity immediate is to be paid for some years at 12% p.a. The present value of the annuity is ₹ 10,000 and the accumulated value is ₹ 20,000. Find the amount of each annuity payment
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Solution
Given, P = ₹ 10,000, r = 12% p.a., A = ₹ 20,000
∴ i = `"r"/(100) = (12)/(100)` = 0.12
Now, `(1)/"P" - (1)/"A" = "i"/"C"`
∴ `(1)/(10,000) - (1)/(20,000) = (0.12)/"C"`
∴ `(2 - 1)/(20,000) = (012)/"C"`
∴ `(1)/(20,000) = (0.12)/"C"`
∴ C = (0.12)(20,000)
∴ C = 2,400
∴ The amount of each annuity payment is ₹ 2,400.
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For annuity due,
C = ₹ 20,000, n = 3, I = 0.1, (1.1)–3 = 0.7513
Therefore, P = `square/0.1 xx [1 - (1 + 0.1)^square]`
= 2,00,000 [1 – 0.7513]
= ₹ `square`
The future amount, A = ₹ 10,00,000
Period, n = 20, r = 5%, (1.025)20 = 1.675
A = `"C"/"I" [(1 + "i")^"n" - 1]`
I = `5/200` = `square` as interest is calculated semi-annually
A = 10,00,000 = `"C"/"I" [(1 + "i")^"n" - 1]`
10,00,000 = `"C"/0.025 [(1 + 0.025)^square - 1]`
= `"C"/0.025 [1.675 - 1]`
10,00,000 = `("C" xx 0.675)/0.025`
C = ₹ `square`
