Mr. Arya wants to know the amount he should pay for a gold mine expected to yield an annual return of Rs 4 lakh for the next 10 years, after which it will be worthless. Find the amount he should pay for the mine, if he wants to yield 18% annual return on his investment and also set up a sinking fund to replace the purchase price. Assume that the sinking fund earns 10% annually. [Use $(1.1)^{10} = 2.5937$ ]

Q: Mr. Arya wants to know the amount he should pay for a gold mine expected to yield an annual return of Rs 4 lakh for the next 10 years, after which it will be worthless. Find the amount he should pay for the mine, if he wants to yield 18% annual return on his investment and also set up a sinking fund to replace the purchase price. Assume that the sinking fund earns 10% annually. [Use $(1.1)^{10} = 2.5937$]

Let P be the amount Mr. Arya should pay for the mine. **The annual income of Rs 4,00,000 must do two jobs:** - **Provide the desired return:** 18% of the price = $0.18 P$. - **Fund the sinking fund** that must accumulate to P in 10 years (to replace the purchase price) at 10% per annum. **Sinking-fund deposit factor** (annual deposit that grows to Re 1 in 10 years at 10%): - Factor = $i / ((1 + i)^{10} - 1) = 0.10 / (2.5937 - 1) = 0.10 / 1.5937 = 0.062747$. **Balance equation:** $4,00,000 = 0.18 P + 0.062747 P$ $4,00,000 = 0.242747 P$ **$P = 4,00,000 / 0.242747$ ≈ Rs 16,47,806.** **Mr. Arya should pay approximately Rs 16,47,806 (about Rs 16.48 lakh) for the gold mine.** Check: return part = $0.18 \times 16,47,806$ ≈ Rs 2,96,605; sinking-fund deposit ≈ Rs 1,03,395, which at 10% over 10 years grows to about $1,03,395 \times 15.937$ ≈ Rs 16.48 lakh = P. ✓

Financial Mathematics

Valuation with Sinking Fund (Capital Recovery)

Official Answer

Let P be the amount Mr. Arya should pay for the mine.

The annual income of Rs 4,00,000 must do two jobs:

Provide the desired return: 18% of the price = $0.18 P$ .
Fund the sinking fund that must accumulate to P in 10 years (to replace the purchase price) at 10% per annum.

Sinking-fund deposit factor (annual deposit that grows to Re 1 in 10 years at 10%):

Factor = $i / ((1 + i)^{10} - 1) = 0.10 / (2.5937 - 1) = 0.10 / 1.5937 = 0.062747$ .

Balance equation:

$4,00,000 = 0.18 P + 0.062747 P$

$4,00,000 = 0.242747 P$

$P = 4,00,000 / 0.242747$ ≈ Rs 16,47,806.

Mr. Arya should pay approximately Rs 16,47,806 (about Rs 16.48 lakh) for the gold mine.

Check: return part = $0.18 \times 16,47,806$ ≈ Rs 2,96,605; sinking-fund deposit ≈ Rs 1,03,395, which at 10% over 10 years grows to about $1,03,395 \times 15.937$ ≈ Rs 16.48 lakh = P. ✓

sinking fundcapital recoveryannual returngold mine valuationwasting assetdeposit factorfuture value annuitypurchase price

Marking Scheme

11 mark: recognising the annual income splits into desired return ( $0.18P$ ) and sinking-fund contribution.
21 mark: correct sinking-fund deposit formula $i/((1+i)^{10} - 1)$ .
31 mark: evaluating the factor $0.10/1.5937 = 0.062747$ .
41 mark: forming the equation $4,00,000 = 0.242747 P$ .
51 mark: final answer P ≈ Rs 16,47,806 (accept Rs 16.47–16.48 lakh).

Hint

The Rs 4 lakh annual income must cover TWO things: (1) the 18% desired return on the price P, i.e. 0.18P, and (2) the yearly sinking-fund deposit that grows to P in 10 years at 10%. Set $4,00,000 = 0.18P + P \cdot i/((1+i)^{10} - 1)$ and solve for P.

Quick Oral Answer

The Rs 4 lakh yearly income has to earn Mr. Arya his 18% return and also rebuild his capital through a sinking fund earning 10%, so $4,00,000 = 0.18P + 0.062747P$ , giving a fair price of about Rs 16.48 lakh.

Analysis & Explanation

This is a classic capital-recovery-with-sinking-fund valuation, a signature Financial Mathematics problem.

Concept — Two competing demands on income

When a wasting asset (a mine that becomes worthless) is bought, the buyer wants an ordinary return on the money invested AND wants to get the original capital back by the end of the asset's life.
The return demand is met by $0.18 P$ each year; the capital-replacement demand is met by depositing a fixed amount annually into a sinking fund that compounds at 10% to reach P.

Key formula — sinking fund deposit

The annuity that accumulates to a future value F in n years at rate i requires an annual deposit $F \cdot i / ((1 + i)^n - 1)$ . Here the accumulated future value must equal P, so the deposit is $P \cdot 0.062747$ .

Exam trap

The most common error is using the 18% rate for BOTH the return and the sinking fund. The sinking fund earns a DIFFERENT rate (10%), which is why the deposit factor uses 0.10, not 0.18.
Another trap is treating the whole Rs 4 lakh as pure return (dividing $4,00,000 / 0.18$ ), which ignores capital replacement.

Real-world

Mining, oil, and equipment-leasing firms use exactly this method to price wasting assets, ensuring investors recover their capital while earning a market return.

Common Mistakes

1Using the 18% return rate for the sinking fund too, instead of the given 10% rate that the fund actually earns.
2Dividing Rs 4,00,000 by 0.18 alone, ignoring the need to replace the purchase price via the sinking fund.
3Mis-computing the sinking-fund factor as $((1+i)^{10} - 1)/i$ (the accumulation factor) instead of its reciprocal $i/((1+i)^{10} - 1)$ .

Interesting Facts

Sinking funds date back to 18th-century Britain, where Prime Minister William Pitt the Younger formalised a national sinking fund in 1786 to pay down public debt.

The idea of separating 'return of capital' from 'return on capital' is the foundation of modern depreciation accounting and asset valuation.

Because the sinking fund here earns only 10% while the investor demands 18%, the two-rate structure (dual-rate valuation) is sometimes called the 'Hoskold' method after engineer H. E. Hoskold, who introduced it for mine valuation in 1877.

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Frequently Asked Questions

Why is the sinking fund calculated at 10% but the return demanded at 18%?

The 18% is the return Mr. Arya wants ON his investment (his required yield). The 10% is the interest rate his sinking-fund deposits actually EARN when reinvested. These are two independent rates, and using both correctly (a dual-rate valuation) is the whole point of the problem.

What does 'set up a sinking fund to replace the purchase price' mean here?

Since the mine becomes worthless after 10 years, Mr. Arya would lose his capital P. To avoid this, he sets aside a fixed sum each year into a fund that grows at 10% to exactly P by year 10, so his original capital is recovered even though the asset is exhausted.