Simple and compound interest

Interest, principal and rate

Banks and post offices pay interest, extra money, on money kept with them, and people pay interest when they borrow. A notice such as ‘FD @ 10% p.a.’ gives the rate of interest; p.a. stands for per annum, every year. The money deposited is the principal. If you keep ₹6000 in a fixed deposit at 10% p.a., after one year you have 6000 + 0.10 × 6000 = ₹6600. In other words the deposit becomes 110% of itself: 6000 × 1.1 = 6600. In a fixed deposit (FD) the money stays locked for the chosen period, and taking it out early usually means paying a penalty.

Without compounding

Suppose the bank pays the interest out at the end of every year and returns the principal at the end. Then ₹6000 at 10% p.a. earns ₹600 each year. Over 3 years that is ₹1800 of interest, so ₹7800 is received in all, 130% of the deposit, a gain of 30%. In general, with principal p and yearly rate r (as a decimal), the interest each year is p × r, the interest over t years is prt, and the total is p + prt = p(1 + rt). This is often called simple interest.

With compounding

Now suppose each year’s interest is added back into the deposit. The first year still earns ₹600, making ₹6600. The second year earns 10% of ₹6600, which is ₹660, making ₹7260. The third earns ₹726, making ₹7986. This is compounding: every year’s interest also earns interest. Each year the amount is multiplied by 1.1, so after 3 years it is 6000 × 1.1 × 1.1 × 1.1 = 6000 × 1.331, a gain of 33.1%. After t years the amount is p(1 + r)ᵗ, and the compound interest is p(1 + r)ᵗ − p.

Why compounding wins in the long run

At first the two methods are close. ₹20,000 at 10% p.a. for 2 years gives ₹24,000 without compounding and ₹24,200 with it, a difference of only ₹200. But the gap widens with time and with more periods. Without compounding, ₹1000 at 10% grows by the same ₹100 every year, a straight line, and doubles in 10 years. With compounding, it is multiplied by 1.1 every year, a curve that bends upwards, and doubles in just over 7 years. The first is linear growth; the second is exponential growth. The same rule describes a city growing 3% a year or bacteria multiplying 2.5% an hour.

Comparing loans

Interest on loans works the same way. Giridhar borrows ₹12,500 at 12% p.a. for 3 years without compounding, and pays 12,500 × 0.12 × 3 = ₹4500 in interest. Raghava borrows the same amount at 10% p.a. compounded yearly, and pays 12,500 × (1.1³ − 1) = ₹4137.50. Even though compounding grows faster, the lower rate keeps Raghava’s interest ₹362.50 smaller over these 3 years.

Decline and depreciation

Compounding can also run downwards. Many things lose value with age and use; this is depreciation. A TV bought for ₹21,000 that loses 5% of its value in a year is worth 0.95 × 21,000 = ₹19,950. A village of 1250 people that shrinks by about 10% every decade has 1250 × 0.9 × 0.9 × 0.9 ≈ 911 people after three decades, or around 910. Each decade multiplies by 0.9 instead of 1.1. For the rest of the chapter, see the Chapter 1 notes, percentage increase and decrease, and the NCERT solutions.