Compound Interest: How a Balance Grows on Its Own Interest

Compound Interest works out what a balance becomes when each charge is added to it and then charged again. Enter the balance, the rate with the span it covers, and how many spans to run, and the tool returns the figure at every step. Nothing is paid against it at any setting, and an untouched balance is the clearest way to see the growth.

Rates themselves, and what a month or a year does to one, are covered separately. The arithmetic below runs out to the rupee on one invented household's own balance, and the shape that balance makes is the point.

A balance that nobody touches is a drawing convention rather than a description of anybody. The Bhosale household, invented, did not leave its card alone. The household paid the required minimum in full and on the due date in every single month, and the balance rose anyway. An untouched run is the cleanest way to see the curve, in the same way a diagram of a falling stone leaves the air out.

The market lane where Ashok Bhosale runs his tailoring counter was dug up for drainage work and stayed dug up for five months. The counter took Rs 52,800/- across the year against Rs 96,000/- the year before. Meghna Bhosale's salary from Sahyadri Freight Services Private Limited did not move, nothing about the household's spending changed, and money in fell by Rs 43,200/- while money out stayed where it was. Nobody took a decision that produced the Rs 48,594/- this tool starts from.

What does this working tool compute?

The question is simple to state. If a balance sits at a rate, and each charge is added to the balance before the next charge is worked out, what is the balance after a stated number of spans? The adding step is the whole subject. CompoundingInterest being charged on interest that has already been added to the balance, so the amount being charged on keeps rising even when the rate never moves. is interest being charged on interest that was itself already charged.

Think about a jar of grain in a household kitchen where a tenth of what is in the jar has to be scooped out and put back on top of the jar as a heap, every month. In month one the scoop comes out of the original grain. In month two the scoop is taken from the original grain plus last month's heap, so it is bigger. Nobody changed the size of the scoop. The scoop is a proportion, and the thing it is a proportion of got larger. A rate never grows; the amount it is applied to does, and there is nothing else to the mechanism.

How a lender computes a charge day by day is covered separately, as is the difference between one kind of borrowing and another. Growth on money a household sets aside is covered separately too, for a reason that turns on the order in which these subjects are read.

What goes in, and where does each figure come from?

Three inputs, and each one has a document behind it. A tool whose inputs are remembered rather than read produces an answer that is exact to the rupee and wrong. Each of the three figures below is traced to the document it is read from.

Input one, the balance

The balanceWhat is owed on the account at a stated moment. On a card it is the figure the statement carries as the total amount due for that cycle. is read off the statement balanceThe figure the card statement carries as at its statement date, being what was owed before plus anything charged plus anything spent, less anything paid. line of the statement, not recalled from the last time anybody looked. For the Bhosale household on 31 March of year two that figure is Rs 48,594/-. Every later figure is that number grown. An error in it is an error in every row and every point on every curve below. Walk to the drawer and read it.

Input two, the rate and the span it covers

A rate on its own is not a number the tool can use. Three and a half per cent of what, over how long? The rate must arrive with the periodThe span of time a rate covers. A rate quoted for a month and a rate quoted for a year are different instructions even when the number printed beside them is the same. it is quoted for, and the count of spans entered below must be measured in that same unit. The household's contracted figure is 3.5 per cent a month, invented, and it applies to the whole balance once the card is not cleared in full.

The rate and the count of spans are a matched pair, and the tool cannot see when they have been mismatched. Both inputs are perfectly valid numbers on their own. No other input error in this calculator passes without a complaint from the software or an odd-looking figure at the point of entry. The answer comes back clean, confident and enormous, and that is worse than an error message.

Input three, how many spans to run

The last input is a count, and it is the one input with no document at all behind it. The count is a question rather than a fact: how far out to look. Six statements, twelve, sixty. Nothing about the household fixes the number. The count does fix the honesty of the reading. A balance looked at over three months and the same balance looked at over five years produce two feelings that are very hard to hold at the same time.

How does a balance grow when nothing is paid against it?

Here is the run, to the rupee, on the household's own closing figure. The starting balance is Rs 48,594/-, the rate is the contracted 3.5 per cent a month, and the run is untouchedNo payment of any size is made against the balance and nothing new is spent on it, so the only thing happening in each span is the charge being added.: nothing paid, nothing spent. Money is held in whole rupees at every step and each charge is rounded to the nearer rupee, upward at a half.

The check runs in both directions on every row. Rs 48,594/- plus Rs 24,835/- of interest is Rs 73,429/-, and Rs 48,594/- plus Rs 3,34,234/- is Rs 3,82,828/-. Twelve months of untouched interest on this balance is Rs 24,835/-, more than half the balance again, added by nothing happening at all. Sit with that one for a moment before reading on, because it is the figure most people would not guess within Rs 5,000/-.

In plain figures, the readings run as follows. At six spans the balance is Rs 59,734/-. At twelve it is Rs 73,429/-, with Rs 24,835/- of charge. At twenty-four it is Rs 1,10,956/-, at thirty-six Rs 1,67,662/-, and at sixty Rs 3,82,828/-, of which Rs 3,34,234/- is charge. At sixty spans the straight line reads Rs 1,50,641/- against the curve at Rs 3,82,828/-. The two are identical at one span and Rs 2,32,187/- apart at sixty, and everything between those two facts is what compounding does.

Why does the growth speed up rather than hold steady?

Look back at the third column of the table. The charge in the first month is Rs 1,701/-. The charge in the sixtieth month is Rs 12,946/-. Same debt, same contract, same rate, and nothing in the agreement changed on any date in between. The charge is more than seven times larger because the balance it is taken from is more than seven times larger.

The rate is fixed and the base is not, so a fixed proportion of a rising base is a rising amount, and that is the whole of what people mean when they say compounding accelerates. There is no second mechanism hiding underneath it. The accelerationThe growth getting faster over time. The rate never moves. The amount it is applied to keeps rising, and each period adds more than the last. is arithmetic rather than intent, which is also why nobody sends a letter when it starts.

Acceleration is also why doubling timeHow long a balance takes to become twice its starting size at a given rate, with nothing paid against it. is a more useful handle than a percentage for most people. At this household's contracted 3.5 per cent a month, Rs 48,594/- stands at Rs 96,692/- after twenty months and Rs 1,00,076/- after twenty-one, so it passes twice its starting size inside the twenty-first month. A household that hears three and a half per cent hears a small number. A household that hears twice over in under two years hears something quite different, and both sentences describe the same contract.

What does the period do to the answer?

Everything, and it does it silently. One input pair can be entered in a way that is individually correct and jointly meaningless: a rate quoted for one span and a count measured in another. Nothing about either figure looks wrong. The output is a clean number with no warning attached.

Run the household's 3.5 per cent a month over twelve monthly spans and the balance is Rs 73,429/-. Enter the same rate but count in years. Twelve years goes in as one hundred and forty-four monthly spans, and the tool returns Rs 68,86,412/-. A count in the wrong unit does not produce a slightly wrong answer; it produces an answer with no relationship at all to the question, and it produces it confidently. Copy the words a month or a year off the agreement along with the number, every time.

What happens if the rate is simply multiplied by the number of spans?

Multiplying the rate by the number of spans is the reading almost everybody starts from, and it is not stupid. Multiplication is how simple interestInterest charged only on the original amount, so each period adds exactly the same rupee figure. A carried card balance does not work that way. works, it is how most people were taught percentages at school, and for one span it is exactly right. Three and a half per cent times twelve is forty-two per cent, and forty-two per cent of Rs 48,594/- is Rs 20,409/-, giving a balance of Rs 69,003/-. Clean, quick, and low.

Who else runs this same arithmetic, and what for?

Three parties run this computation on the same balance, and they are not asking the same question of it. Knowing that makes the output easier to read.

A lender's system runs it to produce a charge line. The lender's version carries the highest stakes on being exactly right, and it is the only one of the three that produces a figure anybody has to pay. An assessor looking at an application runs it forwards to see what an existing balance becomes over the life of a new borrowing. A balance that is rising on its own changes what a household can carry alongside it. An analyst reading a lender's own books runs it across a portfolio to work out what a stock of revolving balances turns into, the same curve drawn from the other side of the counter.

The household is the only one of the four parties for whom this arithmetic is a question about time rather than a question about money, and time is the input nobody else can supply. A lender knows the rate. A system knows the balance. Only the household knows how long this is likely to sit there, and that is the input that changes the answer most. Time is also why the count of spans is left as a dial rather than fixed at some sensible default. There is no sensible default. There is only how long.

What does this tool deliberately not do?

The same arithmetic that runs a debt forwards runs money set aside forwards too. One formula covers both, and the formula does not know which side of a sheet it is standing on. So the obvious next move, and the one nearly every treatment of this subject makes, is to swing the tool around and show what the same rate would do to money a household puts away.

The calculator runs forwards only on a balance owed. Growth on money set aside sits much later in the reading order, after the buffer and after protection, and putting it here would place the most risk-carrying content in this whole subject in front of a reader who may be in debt today. A household holding a balance at 3.5 per cent a month does not need a curve showing what could have been.

The Bhosale household, Meghna Bhosale, Ashok Bhosale, Ira Bhosale and Sahyadri Freight Services Private Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.