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प्रश्न
Find the present value of an annuity due of ₹ 600 to be paid quarterly at 32% p.a. compounded quarterly. [Given (1.08)−4 = 0.7350]
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उत्तर
Given: C = ₹ 600,
Amount is invested every quarter for one year.
∴ n = 4
Rate of interest is 32% p.a.
∴ r = `(32)/(4)` = 8%
i = `"r"/(100) = (8)/(100)` = 0.08
Now, P' = `("C"(1 + "i"))/"i"[1 - (1 + "i")^-"n"]`
∴ P' = `(600(1 + 0.08))/(0.08)[1 - (1 + 0.08)^-4]`
= `(600(1.08))/(0.08)[1 - (1.08)^-4]`
= (7,500)(1.08)[1 – 0.7350]
= 8,100 × 0.2650
P' = 2,146.5
∴ Present value of annuity due is ₹ 2,146.5.
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Therefore, P = `square/0.1 xx [1 - (1 + 0.1)^square]`
= 2,00,000 [1 – 0.7513]
= ₹ `square`
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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]`
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= `"C"/0.025 [1.675 - 1]`
10,00,000 = `("C" xx 0.675)/0.025`
C = ₹ `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
