Perpetuity
A perpetuity is a stream of equal payments that continues for ever, such as the interest from a permanent endowment that funds a yearly scholarship. Although the payments never end, their present value is finite because distant payments are worth very little today.
If a payment R is made at the end of every period and the interest rate per period is i, the present value is PV = R/i. Example: ₹5,000 every year for ever at 8% per annum has PV = 5000/0.08 = ₹62,500. This is the amount that, invested at 8%, earns exactly ₹5,000 each year without the principal ever being used up.
If the first payment is made immediately (a perpetuity due), add one payment: PV = R + R/i. With the same figures, PV = 5000 + 62,500 = ₹67,500.
Rearranging, i = R/PV: an investment of ₹1,00,000 paying ₹6,000 a year for ever gives a rate of 6%.
Sinking fund
A sinking fund is a fund into which equal amounts are set aside regularly so that, with interest, it accumulates to a required amount on a future date. Firms use sinking funds to replace machinery or repay a loan or debenture at maturity.
If payments R are made at the end of each period for n periods at rate i per period, the amount accumulated is that of an ordinary annuity: A = R[(1 + i)ⁿ − 1]/i, so R = A·i/[(1 + i)ⁿ − 1]. Example: to have ₹1,00,000 after 5 years at 10% compounded annually, R = 100000 × 0.1/(1.1⁵ − 1) = 10000/0.61051 ≈ ₹16,379.75 per year.
Key points
- Perpetuity (end of period): PV = R/i.
- Perpetuity due: PV = R + R/i = R(1 + i)/i.
- Sinking fund payment: R = A·i/[(1 + i)ⁿ − 1].
- A sinking fund looks ahead to a future amount; a perpetuity's PV looks at today's value.
Common mistakes
- Using a percentage (8) instead of a decimal (0.08) in R/i.
- Using the present-value annuity formula for a sinking fund instead of the amount (future value) formula.