Compounding: How Returns Build on Returns, and Simple Interest

Compounding is interest that goes on to earn interest. Simple interest pays on the original amount alone, so five years at 12.00 per cent adds 60.00 per cent. Compounding adds 76.23 per cent. On the Rs 96,46,25,655 that Sankalp Industrial Systems Limited, invented, carries as a Year 5 present value, the difference over those five years is Rs 15,65,98,952, and discounting is this arithmetic run backwards.

A tailor's shop already holds the whole of this guide. He buys cloth, stitches it, sells the shirts, and at the end of the month he has a surplus of Rs 3,000 over what the cloth cost him. He can do one of two things with that Rs 3,000. He can take it home, a perfectly reasonable thing to do, and next month he will buy exactly the same amount of cloth as he bought this month. Or he can leave it in the shop and buy a little more cloth next month than he bought this month.

Both tailors are earning the same margin on the same trade. Neither is cleverer than the other and neither has found a better supplier. The only difference between them is where the surplus sits at the start of the next month, and after one month there is no difference at all. Ask again after five years and the two shops are not the same size. The decision about whether the surplus rejoins the working amount or leaves it is the entire difference between simple interest and compounding; nothing that follows adds a second idea to it.

What is actually different, and where does that difference live?

The difference lives in one place only, and it is worth being blunt about how small that place is. Both methods charge the same rate. Both charge it once a period. Both run for the same number of periods. They disagree about one thing only: the amount the rate is charged on. One of them charges it on a number that is frozen at its opening size for the whole run. The other charges it on a number that is refreshed at the end of every period. Last period's interest is standing inside this period's base.

There is genuinely nothing more to it. There is no second mechanism hiding underneath, no extra assumption smuggled in, and nothing about markets or risk or how anybody behaves. The gap measured in every figure below is produced entirely by a number being refreshed rather than frozen. One sentence rebuilds every figure below, including the ones that look implausible.

Grey stands for simple interest in every drawing below, green stands for compounding, and the pale shading between them is the gap the two produce. Red appears once only, in the drawing of the mistake near the end. Simple interest is not the wrong answer to anything; it is the base case, and it carries a colour of its own for that reason.

Simple Interest: what does it do, and why does it come first?

Simple interest charges the rate on the amount at the start, for every period, no matter what has happened since. Beginning with Rs 1,00,00,00,000 at 12.00 per cent, the first year adds Rs 12,00,00,000, and so does the second, and so does the eleventh. The rupees added are identical every time because the number they are a percentage of never moves. Plotted, the total is a straight line. A constant addition never looks like anything else.

Simple interest comes first rather than as a footnote at the end, for a practical reason. The compound case is only legible against something, and the something is this. To a reader who has never watched a straight line being drawn, the curve that sits beside it later does not read as a curve; it reads as just another line that happens to be higher. Simple interest is the base case, and a reader who has not held it clearly cannot see what compounding actually adds.

There is a second reason. Simple interest is not a relic. Simple interest is the arithmetic behind a great many ordinary arrangements, and a reader who files it under history will misread those arrangements when they meet them.

Where does simple interest still turn up, outside a textbook?

In more places than most readers expect, and the common thread is short periods and interest that is settled rather than left standing. A hand loan between neighbours repaid in three months with a flat charge is simple interest. A supplier who agrees a fixed late charge on an overdue bill is applying simple interest to the original invoice, not to the invoice plus last month's late charge. Interest that is calculated on a daily balance and swept out every quarter leaves nothing in for the next quarter to work on, so over that quarter it behaves as simple interest.

There is a structural version of the same thing at Sankalp Industrial Systems Limited, invented. Its term loanBorrowing for a fixed period with agreed repayment dates, as against a facility that is drawn and repaid as the business needs it., Tranche 1, is Rs 3,00,00,00,000 carrying 7.80 per cent charged quarterly, and the company pays that interest across rather than letting it stand. Interest that is paid across never becomes part of the amount interest is charged on. The debt at the start of the next quarter is the same Rs 3,00,00,00,000 it was at the start of this one, so from the lender's side the charge on that tranche behaves in a straight line, whatever the borrower does with its own cash afterwards.

The test is never the name of the arrangement; it is whether the interest is settled or left standing. The question classifies almost any arrangement, whatever it happens to be called.

What does compounding do that simple interest does not?

Compounding refreshes the base. At the end of each period the interest earned is added to the amount, and the next period charges the rate on that larger amount. Nothing else changes. The rate does not rise, the periods do not shorten, and no extra money is put in from outside. The only new thing in year two is that year one's interest is now standing in the queue with the original amount, earning at the same rate as everything else.

Refreshing the base is why compound growth is so often described badly. People reach for words like explosive and exponential, and those words make it sound as though something dramatic is happening in each period. Nothing dramatic happens in any period. Each individual period is dull, and the result is startling only because the dullness is applied to a base that has been quietly getting larger the whole time.

The useful way to see it is as layers. Take five years at 12.00 per cent and ask where the total growth of 76.23 per cent actually comes from. Exactly 60.00 percentage pointsThe unit for the distance between two percentages, kept separate so that a move from 60 to 76 is not misread as a rise of 16 per cent. of it is interest charged on the original amount, the part simple interest would also have paid. The remaining 16.23 points is interest charged on interest. The second layer is empty in year one, is worth 1.44 points by the end of year two, and has reached 16.23 points by the end of year five.

Can the whole difference be written as one line of arithmetic?

The whole difference can be written as one line, and writing it as arithmetic rather than as words removes most of the confusion at a stroke. Simple interest multiplies the rate by the number of periods and then applies that once. Compounding multiplies one plus the rate by itself, once for each period.

Everything that follows, the widening gap, the doubling times and the link back to discounting, is a consequence of the difference between adding a rate to itself and multiplying by it repeatedly. Notice what the compound expression does not contain: it has no term for how the money was earned, no term for who is holding it, and no term for what happens in the world. The expression is a multiplication carried out n times.

Why can nobody see the difference in the first year?

Because in the first year there is no difference. At that point the starting amount is the only amount there is, so both methods take the rate on it. Rs 1,00,00,00,000 at 12.00 per cent becomes Rs 1,12,00,00,000 under either method, and no amount of care will find a rupee between them. The equality holds at every rate and over every kind of period, and it is not a coincidence: interest on interest requires interest to exist first.

Year two is where the two part company, and the parting is small. Simple interest ignores the first year's earnings and charges 12.00 per cent on the original amount again, so the total reaches 124.00 on a base of 100.00. Compounding charges 12.00 per cent on the whole 112.00 and reaches 125.44. The two are 1.44 points apart. On a chart drawn at any sensible scale that is a couple of pixels. By year five they are 16.23 points apart, an easy distance to see. The difference is not merely small early on; it is invisible early on, and that is what allows a wrong sentence about it to survive for years.

How large is the gap on the invented company's own rupees?

Large enough to be worth an argument, and the figures are already on the table from the discounting work, so nothing new has to be assumed. Sankalp Industrial Systems Limited, invented, is forecast to produce free cash flow to the firmThe cash a business has left over for everyone who put money into it, after tax has been paid and after it has bought what it needs to keep going. of Rs 1,70,00,00,000 in Year 5, and at the company's assumed 12.00 per cent that is worth Rs 96,46,25,655 today. Take that Rs 96,46,25,655 and run it forward five years, simply the same journey in the other direction.

Under simple interest it earns 12.00 per cent of the original amount five times over. The five charges come to 60.00 per cent in total, or Rs 57,87,75,393 of interest, and the amount finishes at Rs 1,54,34,01,048. Under compounding it multiplies by 1.762341683, adds Rs 73,53,74,345 of interest, and finishes at Rs 1,70,00,00,000, exactly the amount discounted back from in the first place. Landing back on the face amount is not luck. Landing there is the test that shows the two operations undo one another.

Five years at a rate nobody would call remarkable produces Rs 15,65,98,952 of interest that simple interest would never have paid, and that is 27.06 per cent more interest than simple interest pays in the whole period. Every figure in that table is an assumption about an invented company, and the 12.00 per cent is that company's own assumed rate rather than an observation about anything.

What does thirty years do that five years gives no warning of?

Thirty years changes the shape of the answer so completely that readers who accepted the five year figure without blinking reject the thirty year one. The arithmetic is identical; only the exponent has moved. The rejection is worth watching for in yourself.

At this length the point is the shape, and a tidy starting number keeps the arithmetic within reach of the head, so the illustration runs on a round Rs 1,00,00,00,000 rather than on the company's awkward figure. The rate holds still at 12.00 per cent and both methods run out to thirty years.

The second row carries the whole idea in one comparison, so look at it before anything else. The thirtieth year of simple interest is charged on a number that has not moved since the beginning, so it adds the same Rs 12,00,00,000 the first year added. The thirtieth year of compounding adds Rs 3,20,99,91,656, or 26.75 times as much, and the rate did not change to produce it.

Read that again if it looks like a typing error. The rate did not change. The starting amount did not change. Nothing was added from outside. Six and a half times as much money comes out of the same rate over the same thirty years, purely because the base was refreshed each year rather than frozen. The gap alone, Rs 25,35,99,22,121, is just over seven times all the interest that simple interest pays in the whole period.

How long does money take to double at 12.00 per cent?

Doubling time collapses everything into one number that a reader can carry around, and no comparison between the two methods is cleaner. Under simple interest it is easy and exact. Doubling requires the original amount added on again, and the rate adds 12.00 per cent of the original amount each year, so it takes one divided by 0.12 years, or 8.33 years and not a day less.

Under compounding the answer needs a natural logarithmThe power to which 2.718281828 must be raised in order to reach a given number. It is the tool that gets time out of an exponent and onto its own.. The number of periods is sitting in the exponent and has to be brought down. The logarithm of 2 divided by the logarithm of 1.12 is 6.1163, printed here as 6.12 years. Compounding does the same job two and a fifth years sooner, at the identical rate, purely because it is working on a base that keeps refreshing.

There is a shortcut worth knowing, for checking figures without a calculator. Dividing 72 by the rate written as a whole number gives an approximate doubling time. At 12.00 per cent that gives 6.00 years against a true 6.12, so the shortcut is 0.12 years light, about six weeks over a six year run. The shortcut is not exact and it drifts at very high and very low rates, but as a way of catching a doubling claim that is badly wrong it does the job.

What has any of this got to do with discounting?

Everything, and the connection is close enough that the two belong side by side. Compounding forward and discounting back are not two techniques. Both are one operation pointed in two directions, and the two numbers involved are reciprocalsOne divided by a number. Multiplying by a number and then by its reciprocal lands back exactly where it began. of each other.

Here is the check, using figures that are already settled. Five years at 12.00 per cent gives a compounding factor of 1.762341683. One divided by that is 0.567426856, and that is precisely the Year 5 discount factorThe multiplier that restates a later amount at today's date. It is always smaller than one, and it shrinks the further away the amount is. used throughout the valuation work on this invented company. Nothing was re-derived and no second convention was introduced. The factor that carries an amount five years forward and the factor that carries it five years back are the same object, written once as a multiplier greater than one and once as its reciprocal.

The reciprocal relationship is also why both sides of the journey have to use compounding. Money sent out by adding a fixed number of rupees each year and brought home by dividing has its two halves running different rules, so it arrives somewhere other than where it set off from. The circle closes: Rs 96,46,25,655 multiplied by 1.762341683 is Rs 1,70,00,00,000, and Rs 1,70,00,00,000 multiplied by 0.567426856 is Rs 96,46,25,655. The convention used throughout is year-end discountingA convention under which each cash flow is dated to the closing day of its year, not spread evenly across the twelve months., and both directions use it.

Where does this picture stop being useful?

Sooner than the drawings suggest, and it is worth saying plainly because a thirty year curve is a seductive object. Compounding describes what a multiplication does when it is repeated. Compounding promises nothing. The arithmetic does not say that any rate is available, that any rate will hold for thirty years, that anybody will still be there to pay it, or that the rupees at the end will buy what the rupees at the start would have bought. Every one of those is a separate question, and each is settled somewhere else.

A few neighbouring matters are covered separately. The nominal rateA rate as it is quoted, before anything has been done about how often it is charged or about prices rising. of 12.00 per cent used throughout is charged once a year here, and everything changes when it is charged more often than that.