Suppose a machine costing ₹50,000 is to be replaced at the end of 10 years, at that time it will have a salvage value of ₹5,000. In order to provide money at that time for a machine costing the same amount, a sinking fund is set up. The amount in the fund at that time is to be th

Question

Suppose a machine costing ₹50,000 is to be replaced at the end of 10 years, at that time it will have a salvage value of ₹5,000. In order to provide money at that time for a machine costing the same amount, a sinking fund is set up. The amount in the fund at that time is to be the difference between the replacement cost and salvage value. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly. What should each payment be?

Accepted Answer

> [!TLDR]
> The problem asks for the quarterly payment into a sinking fund that must accumulate to ₹45,000 (replacement cost minus salvage) in 10 years, with interest at 8% compounded quarterly. The answer is found using the ordinary annuity future value formula: each payment is approximately ₹745.50.

The core idea here is a **sinking fund** — a way to set aside money regularly so that, with compound interest, it grows to a specific target amount by a future date. You are not saving the entire ₹50,000 because you will get ₹5,000 back from the old machine at replacement time. So the net amount you need to accumulate is:

$$
\text{Target amount} = 50000 - 5000 = 45000
$$

This ₹45,000 is the future value of an ordinary annuity: equal payments made at the end of each quarter, earning 8% per year compounded quarterly. The key is to match the compounding frequency (quarterly) with the payment frequency (quarterly). That means the interest rate per quarter is:

$$
i = \frac{8\%}{4} = 2\% = 0.02
$$

And the total number of quarterly payments over 10 years is:

$$
n = 10 \times 4 = 40
$$

Now, the future value $F$ of an ordinary annuity (payments at the end of each period) is given by:

$$
F = P \cdot \frac{(1+i)^n - 1}{i}
$$

where $P$ is the payment per period. We know $F = 45000$, $i = 0.02$, $n = 40$. We solve for $P$:

$$
P = F \cdot \frac{i}{(1+i)^n - 1}
$$

Let’s work through the calculation step by step.

1. **Compute $(1+i)^n$**  
   $$
   (1.02)^{40}
   $$  
   You can compute this using a calculator or by logarithms.  
   $$
   (1.02)^{40} \approx 2.208039
   $$

2. **Subtract 1**  
   $$
   (1.02)^{40} - 1 \approx 2.208039 - 1 = 1.208039
   $$

3. **Divide $i$ by that result**  
   $$
   \frac{0.02}{1.208039} \approx 0.016555
   $$

4. **Multiply by the target amount**  
   $$
   P = 45000 \times 0.016555 \approx 745.00
   $$

Let’s be more precise:  
   \[ …

Concept understanding — Present Value Perpetuity

Present Value of a Perpetuity: The Intuition

The Precise Statement

Where Does This Formula Come From?

The Critical Assumption

Real-World Examples …

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