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Quantitative Aptitude · Mathematics of Finance

Perpetuity and Sinking Fund for CA Foundation

Updated 1 October 2026 · Fact-checked

A perpetuity is a series of equal payments that never ends, so its present value is C ÷ i. A sinking fund is built by equal periodic deposits that grow with interest to a target sum. Find the target, then use A = S × i ÷ [(1+i)^n − 1].

Understand Perpetuity and Sinking Fund

An annuity is a series of equal payments for a fixed number of periods. A perpetuity is the same thing with no end date. The payments go on forever. Examples are a scholarship paid every year from a fixed deposit, or a fixed dividend on a preference share that never matures.

Why does a never-ending stream have a finite present value? Because far-away payments are discounted so heavily that they add almost nothing. If you deposit a sum P at rate i, it earns P × i every period. If you withdraw only that interest, the deposit stays intact forever. So P × i = C, which gives P = C ÷ i.

A sinking fund works the other way round. You know a sum you will need on a future date. It may be to repay a loan or debenture, or to replace a worn-out machine. You deposit equal instalments at regular intervals. Each deposit earns compound interest. By the target date the deposits plus interest equal the sum needed.

So a sinking fund is the future value of an annuity, solved backwards for the instalment. You are given the future amount S, the rate i and the number of periods n. You find the annuity A.

For an asset, the sum needed is usually the replacement cost minus the scrap value you expect from the old asset. Read the question for this adjustment.

Key formulas to remember

Present value of perpetuity (immediate)
PV = C ÷ i
C is the payment per period. The first payment comes at the end of the first period. i is the rate per period as a decimal.
Present value of perpetuity due
PV = C + C ÷ i = C × (1 + i) ÷ i
Use when the first payment is made immediately, at time 0.
Rate per period
i = annual rate ÷ number of compounding periods per year
For quarterly payments with quarterly compounding, divide the annual rate by 4.
Future value of an annuity
S = A × [(1+i)^n − 1] ÷ i
A is the instalment at the end of each period. n is the number of periods.
Sinking fund instalment
A = S × i ÷ [(1+i)^n − 1]
Gives the equal end-of-period deposit needed to reach the target sum S.
Amount to be accumulated for replacement
S = cost of new asset − scrap value of old asset
Use when the question asks for a fund to replace an asset and gives scrap value.

How to solve Perpetuity and Sinking Fund questions

Use this method for any perpetuity or sinking fund question. First decide which of the two it is. Then keep the rate and the period in the same unit.

  1. 1Decide the type. If payments continue forever, it is a perpetuity. If a fixed sum is needed on a future date, it is a sinking fund.
  2. 2Convert the rate to a rate per payment period. Divide the annual rate by the number of compounding periods in a year. Convert n the same way.
  3. 3For a perpetuity, check the timing. If the first payment is one period from now, use C ÷ i. If it is paid today, add one extra C.
  4. 4For a sinking fund, find the target sum S. Subtract scrap value from the replacement cost if both are given.
  5. 5Find (1+i)^n from the data given in the question, or compute it step by step for small n.
  6. 6Apply A = S × i ÷ [(1+i)^n − 1] and calculate carefully.
  7. 7Check your answer against the options. A sinking fund instalment must be less than S ÷ n, because interest also helps build the fund. A perpetuity value must be more than one payment C whenever i is below 100%.

Quickest way: Option elimination and reverse check

When to use it: Use in the 2-hour MCQ paper when you want the answer in under a minute and want to avoid negative marks.

  1. For a perpetuity, compute C ÷ i mentally. Divide by 8% as 12.5 times C, by 10% as 10 times C, by 5% as 20 times C.
  2. Look for the trap option. It is often C ÷ (annual rate) when payments are quarterly, or C ÷ i without the extra C for a due.
  3. For a sinking fund, calculate S ÷ n first. This check can eliminate any option that is equal to or above it. Options below S ÷ n can still be wrong, so do not pick one without further checking.
  4. Keep the small n values in mind. At 10%, the factor [(1+i)^n − 1] ÷ i is 2.1 for n = 2 and 3.31 for n = 3. So A = S ÷ 2.1 or S ÷ 3.31.
  5. Reverse check the survivor. Multiply the instalment by the factor. It should come back to S.
  6. If the data needed for (1+i)^n is missing or the calculation is long, skip and return later.

Common mistakes in Perpetuity and Sinking Fund

  • Using the annual rate for a perpetuity of quarterly or monthly payments.

    Students see 12% and divide directly, forgetting the payment is made every quarter.

    Fix: Convert the rate to the period rate first. For quarterly payments at 12% compounded quarterly, use i = 3%.

  • Using C ÷ i for a perpetuity due.

    Students memorise one formula and ignore the words 'at the beginning of each period' or 'first payment today'.

    Fix: For a due, add one payment: PV = C + C ÷ i.

  • Dividing the target sum by n and calling it the sinking fund instalment.

    It feels natural to split the amount equally, and students ignore the interest earned.

    Fix: Use A = S × i ÷ [(1+i)^n − 1]. The correct instalment is always below S ÷ n when i is positive.

  • Forgetting to subtract scrap value in an asset replacement question.

    Students read only the cost of the new asset and rush to the formula.

    Fix: Underline the cost and scrap value words. The fund required is cost minus scrap, unless the question says otherwise.

  • Mixing up the annuity factor and the sinking fund factor.

    Both use (1+i)^n − 1 and i, but one is the reciprocal of the other.

    Fix: Remember: the annuity gives S from A, so S = A × factor. The sinking fund gives A from S, so A = S ÷ factor.

  • Treating a perpetuity as having a future value.

    Students apply annuity ideas to a stream that has no last payment.

    Fix: Only present value is defined for a perpetuity. Never try to compute its future value.

Worked examples

Example 1

A trust pays a scholarship of ₹3,000 at the end of every quarter forever. Money earns 12% per annum compounded quarterly. What is the present value of the scholarship? (A) ₹25,000 (B) ₹1,00,000 (C) ₹36,000 (D) ₹3,00,000

Show the solution
  1. Payments are quarterly, so use the quarterly rate: i = 12% ÷ 4 = 3% = 0.03.
  2. The first payment is at the end of the first quarter, so it is a perpetuity immediate.
  3. PV = C ÷ i = 3,000 ÷ 0.03.
  4. 3,000 ÷ 0.03 = 1,00,000.
  5. Option (A) comes from using 12% instead of 3%, which is the trap.

Answer: (B) ₹1,00,000

Example 2

A person wants to receive ₹5,000 at the beginning of every year forever, starting today. The rate of interest is 10% per annum. What is the present value? (A) ₹50,000 (B) ₹55,000 (C) ₹45,000 (D) ₹60,000

Show the solution
  1. The first payment is made today, so this is a perpetuity due.
  2. PV = C + C ÷ i.
  3. C ÷ i = 5,000 ÷ 0.10 = 50,000.
  4. PV = 5,000 + 50,000 = 55,000.
  5. Check with the other form: C × (1+i) ÷ i = 5,000 × 1.10 ÷ 0.10 = 55,000.

Answer: (B) ₹55,000

Example 3

A new machine will cost ₹2,50,000 after 2 years. The old machine will fetch a scrap value of ₹40,000 then. A sinking fund is created by equal deposits at the end of each year at 10% per annum compound interest. What is the annual deposit? (A) ₹1,05,000 (B) ₹1,25,000 (C) ₹1,00,000 (D) ₹90,000

Show the solution
  1. Amount to be accumulated: S = 2,50,000 − 40,000 = 2,10,000.
  2. i = 0.10 and n = 2.
  3. (1+i)^n = 1.1² = 1.21, so (1+i)^n − 1 = 0.21.
  4. A = S × i ÷ [(1+i)^n − 1] = 2,10,000 × 0.10 ÷ 0.21.
  5. 2,10,000 × 0.10 = 21,000, and 21,000 ÷ 0.21 = 1,00,000.
  6. Reverse check: deposits of 1,00,000 and 1,00,000. The first earns 10,000 interest. Total = 1,00,000 × 1.10 + 1,00,000 = 2,10,000. Correct.
  7. S ÷ n = 1,05,000 is the trap option. The true instalment must be lower.

Answer: (C) ₹1,00,000

Exam tips

  • Read the timing words carefully: 'end of each year' means immediate, 'beginning' or 'starting today' means due.
  • Check that the rate and the payment period match before you touch any formula.
  • In sinking fund questions, look for scrap value, and for whether the amount needed is the cost or the cost less scrap.
  • Expect small values of n such as 2, 3 or 4, and rates like 5% or 10%. Learn the factors for these so you can finish fast.
  • If a sinking fund option equals S ÷ n or is larger than it, eliminate it at once. This check can remove options at or above S ÷ n, but it cannot confirm an option below it, so still verify the survivor.

Practice questions from Mathematics of Finance

Perpetuity and Sinking Fund: frequently asked questions

What is the perpetuity present value formula in CA Foundation?

For a payment C at the end of each period and a rate i per period, PV = C ÷ i. If the first payment is made immediately, it is a perpetuity due, and PV = C + C ÷ i. Always use the rate per payment period.

What is the difference between an annuity and a perpetuity?

An annuity is a series of equal payments for a fixed number of periods, so it has a last payment. A perpetuity has no end and goes on forever. An annuity has both present and future values. A perpetuity has only a present value.

How do I calculate a sinking fund instalment?

First find the sum S needed on the future date. Then use A = S × i ÷ [(1+i)^n − 1], where i is the rate per period and n is the number of deposits. This assumes deposits are made at the end of each period.

Why is the sinking fund instalment less than the target divided by the number of years?

Each deposit earns compound interest until the target date. So the deposits themselves need to cover only part of the target. The interest covers the rest.