SINKING FUND & PERPETUITY

1. Sinking Fund

Definition: A sinking fund is an annuity created for accumulating money over a period of time, which can be used for paying off a loan or for replacement of some asset or other financial obligations.

Formula

S = R × [(1+i)^n − 1] / i Where: S = Total amount accumulated (Future Value) R = Periodic payment (money set aside each period) i = Interest rate per period (annual rate ÷ number of compoundings per year) n = Total number of payment periods To find R (periodic payment): R = iS / [(1+i)^n − 1]

📝 NOTE: If compounded half-yearly: i = rate/(2×100), n = years × 2. If quarterly: i = rate/(4×100), n = years × 4.

SOLVED EXAMPLE: Ex.1 — Company Sinking Fund

Problem: A company sets aside ₹5000 annually for 10 years in a sinking fund at 5% p.a. compound interest. Find the surplus after paying debenture of ₹60,000.

Given: R = ₹5000, n = 10, i = 5/100 = 0.05

Using: S = R × [(1+i)^n − 1] / i

S = 5000 × [(1.05)^10 − 1] / 0.05

Let x = (1.05)^10 → log x = 10 × log(1.05) = 10 × 0.0212 = 0.2120

x = antilog(0.2120) = 1.629

S = 5000 × [1.629 − 1] / 0.05 = 5000 × 0.629/0.05 = 5000 × 12.58 = ₹62,900

Surplus = ₹62,900 − ₹60,000 = ₹2,900

Answer: Surplus amount = ₹2,900

SOLVED EXAMPLE: Ex.2 — Sinking Fund for Education

Problem: A sinking fund is created to accumulate ₹10,00,000 for child’s education at end of 25 years. Interest = 4% p.a. Find annual payment. [Given (1.04)^25 = 2.6658]

S = ₹10,00,000, n = 25, i = 4/100 = 0.04

R = iS / [(1+i)^n − 1] = (0.04 × 1000000) / [(1.04)^25 − 1]

= 40000 / [2.6658 − 1] = 40000 / 1.6658 = ₹24,012.50

Answer: Annual payment = ₹24,012.50

SOLVED EXAMPLE: Ex.3 — Half Yearly Sinking Fund [CBSE 2023]

Problem: Accumulate ₹1,00,000 for daughter’s education after 10 years. Invest at 5% p.a. compounded half-yearly. [Given (1.025)^20 = 1.6386]

S = ₹1,00,000, rate = 5% half-yearly → i = 5/(2×100) = 0.025

n = 10 × 2 = 20 periods

R = iS / [(1+i)^n − 1] = (0.025 × 100000) / [1.6386 − 1]

= 2500 / 0.6386 = ₹3,914.81

Answer: Each half-yearly payment = ₹3,914.81

THINGS TO REMEMBER

a) Sinking Fund = money set aside periodically to accumulate a future amount

b) Use S = R[(1+i)^n − 1]/i when you know R and want S

c) Use R = iS/[(1+i)^n − 1] when you know target amount S and want to find R

d) Always convert annual rate to per-period rate: i = annual rate / (100 × compoundings per year)

e) Always convert years to number of periods: n = years × compoundings per year

f) Surplus = Amount accumulated − Amount to be paid

2. PERPETUITY

Definition: An annuity in which payments continue forever (infinite term). Example: Property rent received forever.

Two Types of Perpetuity

(1) Perpetuity payable at END of each period
(2) Perpetuity payable at BEGINNING of each period

Type 1 (End): P = R / i Type 2 (Beginning): P = R + R/i = R(1 + 1/i) Where: P = Present value of perpetuity R = Periodic payment amount i = Interest rate per period

NOTE: When interest is calculated only at the end of a year, nominal rate = effective rate.