The Maths Behind Loans and Investments

Everything rests on compounding

One idea underlies all of it: money earning a return earns a return on the return.

A = P × (1 + r)ⁿ

Simple interest would give P × (1 + r×n) - linear growth. Compounding is exponential, and the difference is invisible over a few years and enormous over a few decades.

₹1,00,000 at 8%: after five years, simple interest gives ₹1,40,000 and compounding gives ₹1,46,933 - a difference of 5%. After thirty years, simple gives ₹3,40,000 and compounding gives ₹10,06,266 - a difference of 196%.

This single fact explains why starting to save at 25 rather than 35 matters more than any decision about which fund to choose. The first decade of contributions has the longest to compound, so it contributes disproportionately to the final total. Someone who invests for ten years and then stops often ends up ahead of someone who starts ten years later and never stops.

The rule of 72

Divide 72 by the annual percentage return for the approximate number of years to double. At 6%, twelve years. At 12%, six years. At 3%, twenty-four.

It is accurate enough between roughly 4% and 15%, and it is the fastest sanity check available. Used in reverse it is a useful filter: an investment promising to double in two years is claiming a 41% annual return, which should prompt hard questions about what risk is being taken - or whether the offer is genuine at all.

Applied to inflation, it is sobering. At 6% inflation, prices double every twelve years. A salary that has not doubled in twelve years has fallen in real terms.

Regular contributions: the annuity formula

A monthly investment is not one sum compounding - it is many sums, each compounding for a different length of time. The first instalment compounds for the full period; the last, for one month. Summing that series gives:

FV = P × [ ((1 + i)ⁿ − 1) / i ] × (1 + i)

P = monthly amount
i = monthly rate (annual ÷ 12)
n = number of months

₹10,000 a month for twenty years at 12%: total contributed ₹24 lakh, projected value roughly ₹99.9 lakh. Growth is about three quarters of the total.

Now shift the period. The same ₹10,000 a month for ten years produces about ₹23.2 lakh, of which ₹12 lakh is contribution. Doubling the duration did not double the outcome - it more than quadrupled it. That non-linearity is the argument for starting early stated mathematically.

The loan formula is the same idea inverted

A loan asks the mirror question: what constant payment, discounted at the interest rate, has a present value equal to the amount borrowed?

EMI = P × r × (1 + r)ⁿ / ((1 + r)ⁿ − 1)

₹50,00,000 at 8.5% over twenty years gives an EMI of about ₹43,391. Over 240 months that totals roughly ₹1.04 crore, meaning ₹54 lakh of interest on a ₹50 lakh loan.

The payment is constant but its composition is not. Interest is charged on the outstanding balance, so the first payment is roughly ₹35,400 interest and ₹8,000 principal. By the final year almost all of it is principal.

Two conclusions follow directly. Selling or refinancing early means you have repaid far less than the payments suggest. And an extra payment made in year two removes vastly more total interest than the same payment in year eighteen, because it eliminates eighteen years of compounding on that amount.

Why long tenures cost so much

Going from twenty to thirty years reduces the monthly payment by about ₹4,900 - roughly 11% - and adds about ₹34 lakh in interest. Each additional year of tenure buys progressively less affordability at progressively higher cost, because the extra years are the ones where the balance has already been substantially reduced.

That is not an argument against long tenures, which sometimes make the difference between buying and not buying. It is an argument for seeing the total before choosing.

The number that makes all the others honest

Every figure above is nominal. Inflation determines what those rupees will actually buy.

Real return ≈ nominal return − inflation
Exactly:      ((1 + nominal) / (1 + inflation)) − 1

A fixed deposit at 7% with inflation at 6% earns a real return of about 0.9%, and after tax on the interest it is very likely negative. The account balance grows and the purchasing power shrinks.

This reframes what "safe" means. Cash carries no nominal risk and near-certain real loss over decades. Equities carry substantial short-term risk and have historically been the more reliable preserver of purchasing power over long periods. Neither is safe in every sense - they are exposed to different risks, on different timescales.

Any projection over more than a few years should be converted to today's purchasing power before you react to it. A projected corpus of ₹5 crore in thirty years at 6% inflation is worth about ₹87 lakh in today's money.

What every calculator quietly assumes

These are the assumptions that make the arithmetic tractable and the results optimistic.

• A constant rate of return. The most consequential simplification. Real returns arrive unevenly, and the order matters enormously - poor years early in accumulation are survivable, poor years early in retirement drawdown can be ruinous. This is sequence-of-returns risk, and no simple formula captures it.
• No fees. A 1% annual expense ratio over thirty years typically consumes a fifth or more of the final value. It compounds against you exactly as returns compound for you.
• No taxes. Capital gains on redemption, tax on interest income, and their interaction with holding periods all reduce what you actually receive.
• Perfect discipline. Every instalment paid, nothing withdrawn early, no panic selling in a downturn. Investor behaviour during market falls is the largest single source of underperformance in the real data.
• Stable rules. Tax rates, interest rates and regulations all change over a thirty-year horizon.

None of this makes the calculators useless. It makes them a lower bound on uncertainty rather than a forecast. Run them at several rates, plan around the pessimistic case, and revisit annually.

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