Simple and compound interest, and why the gap gets absurd

Simple interest is charged on the original amount, forever. Compound interest is charged on the original amount plus all the interest already added. Over a year the two barely differ. Over decades the difference is the whole story.

The two formulas

Simple: A = P(1 + rt)

Compound: A = P(1 + r/n)^nt

P is the principal, r the annual rate as a decimal, t the years, and n the number of times interest is added per year.

The difference in the maths is small: one multiplies, the other raises to a power. The difference in outcome is not.

₹1,00,000 at 10%

• 1 year — simple ₹1,10,000 · compound (annual) ₹1,10,000 · difference ₹0
• 5 years — simple ₹1,50,000 · compound ₹1,61,051 · difference ₹11,051
• 10 years — simple ₹2,00,000 · compound ₹2,59,374 · difference ₹59,374
• 20 years — simple ₹3,00,000 · compound ₹6,72,750 · difference ₹3,72,750
• 30 years — simple ₹4,00,000 · compound ₹17,44,940 · difference ₹13,44,940

At thirty years compound returns more than four times what simple does. Nothing changed except that the interest was allowed to earn interest.

Notice how the gap behaves: negligible at one year, modest at five, then it runs away. Compounding is not slow-then-fast because the rate changes — the rate is constant throughout. It is because the base it applies to keeps growing.

Compounding frequency

The same 10% on ₹1,00,000 for 10 years, compounded at different intervals:

• Annually — ₹2,59,374
• Quarterly — ₹2,68,506
• Monthly — ₹2,70,704
• Daily — ₹2,71,791
• Continuously — ₹2,71,828

Frequency matters, and it matters less than people expect. The jump from annual to quarterly is real; from monthly to daily is small; and there is a hard ceiling — the continuous limit, P × e^rt. No bank can compound its way past that, however often it advertises "daily compounding".

This is what APR and APY exist to disentangle. APR is the nominal annual rate ignoring compounding. APY (or effective annual rate) includes it. 12% compounded monthly is 12% APR and 12.68% APY. When comparing two products, compare APY to APY — the headline rate on its own is not comparable.

The rule of 72

A useful piece of mental arithmetic: divide 72 by the annual rate and you get roughly the years to double.

• At 6% — about 12 years
• At 8% — about 9 years
• At 12% — about 6 years
• At 18% (a typical credit card) — about 4 years

It is accurate to within a few per cent for rates between about 5% and 20%, which covers nearly everything you will meet. It is also the fastest way to sanity-check an investment claim: anything promising to double your money in two years is claiming a 36% annual return, and you should want to know how.

Time beats amount

The most consequential thing about compounding is how much it rewards starting early. Two people, both saving ₹10,000 a month at 10%:

• A saves from 25 to 35, then stops entirely. Ten years of contributions, ₹12 lakh in.
• B saves from 35 to 60. Twenty-five years, ₹30 lakh in.

At 60, A has about ₹2.1 crore. B has about ₹1.3 crore. A contributed two and a half times less and finishes well ahead, because A's money had twenty-five extra years to compound.

This is why "start now with a small amount" is better advice than "wait until you can afford a serious amount". The variable that dominates is time, and it is the only one you cannot go back and get more of.

The same maths, pointed at you

Credit cards compound too. A typical card at 3% per month is 42.6% effective annual, not 36%.

Carry ₹1,00,000 and pay only the 5% minimum each month, and you will be paying for roughly ten years and repay around ₹2,50,000. The minimum payment is calculated so that it barely exceeds the monthly interest, which means the balance falls very slowly by design.

Two specific traps worth knowing. Most cards charge interest from the transaction date — not the statement date — the moment you carry any balance at all, so the interest-free period disappears entirely once you miss a full payment. And cash advances usually have no grace period whatsoever: interest starts at the ATM.

Which yields a decision rule that is simply arithmetic: paying off an 18% debt is a guaranteed 18% return. No investment offers a guaranteed 18%. Clear high-interest debt before investing, every time.

Inflation runs the same formula

At 6% inflation, prices double roughly every 12 years — the rule of 72 again. Money in a savings account at 3% while inflation is 6% is losing about 3% of its purchasing power annually, compounding.

So the number that matters is the real return, not the nominal one: subtract inflation from your rate before deciding whether something is growing. A 7% fixed deposit against 6% inflation is a 1% real return, which is much less impressive than 7% sounds.

Working it out

Our compound interest calculator handles the frequency variations and regular monthly contributions, which is the case most people actually have and the one the textbook formula does not cover directly. The simple interest calculator is there for the products that genuinely use it — most short-term consumer loans and many fixed deposits.

One last check worth doing on anything you are offered: work out the total paid or received over the full term, not the annual rate. Rates are designed to be compared; totals are designed to be understood, and they are the number that tells you what actually happened.