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Question
Solve the following :
Find the present value of an annuity immediate of ₹20,000 per annum for 3 years at 10% p.a. compounded annually. [(1.1)–3 = 0.7513]
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Solution
Given, C = ₹20,000, n = 3 years, r = 10 % p.a.
∴ i = `"r"/(100) = (10)/(100)` = 0.1
Since, P = `"C"/"i"[1 - (1 + "i")^"n"]`
∴ P= `(20,000)/(0.1)[1 - (1 + 0.1)^-3]`
= 2,00,000[1 – (1.1)–3]
= 2,00,000[1 – 0.7513]
= 2,00,000(0.2487)
= ₹49,740
∴ Present value of an annuity immediate is ₹49,740.
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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`
For an annuity due, C = ₹ 2000, rate = 16% p.a. compounded quarterly for 1 year
∴ Rate of interest per quarter = `square/4` = 4
⇒ r = 4%
⇒ i = `square/100 = 4/100` = 0.04
n = Number of quarters
= 4 × 1
= `square`
⇒ P' = `(C(1 + i))/i [1 - (1 + i)^-n]`
⇒ P' = `(square(1 + square))/0.04 [1 - (square + 0.04)^-square]`
= `(2000(square))/square [1 - (square)^-4]`
= 50,000`(square)`[1 – 0.8548]
= ₹ 7,550.40
