SIP Future Value: Arithmetic on an Assumption You Choose

A systematic investment plan (SIP) is a standing instruction that moves the same amount into the same place on the same date each month, and this calculator multiplies that amount by a period at a rate the reader chooses. Given what would go in each month, for how many months, and a rate the reader supplies, it shows the arithmetic across three assumptions at once rather than one. The output is arithmetic on a chosen rate, not a projection, a forecast or a promise.

The worked default in plain text, so it survives with the panel closed. Rs 2,000/- a month for 120 months puts in Rs 2,40,000/-, and the arithmetic gives Rs 2,94,500/- at an assumed 4 per cent a year, Rs 3,65,892/- at an assumed 8 per cent, and Rs 4,60,077/- at an assumed 12 per cent. Each reconciles by addition: Rs 2,40,000/- plus Rs 54,500/- of growth, plus Rs 1,25,892/-, plus Rs 2,20,077/- respectively. The width between the outer two is Rs 1,65,577/-, or 0.69 times everything put in. All three rates were typed in for the illustration, and none of them is suggested here.

Two settings are worth trying before reading on. At 36 months the band almost closes: the width falls to Rs 9,791/- against Rs 72,000/- put in, 0.14 times, so over three years the choice of rate barely matters. At 150 months the width reaches Rs 3,01,274/- against Rs 3,00,000/- put in, and the assumption begins moving the answer by more than every rupee the household handed over. The longer the period runs, the less the answer is about the money and the more it is about the assumption. A long period is usually presented the other way round.

A calculator of this shape is usually built to make one number look attractive, and it does that in two quiet moves. The calculator fills the rate in, so nobody has to decide anything, and prints a single large figure at the bottom, so there is nothing to compare it with. The calculator above refuses both moves. The distance between two assumptions that are not far apart in any dramatic way is what is left, and that distance is the honest content of the calculation. The width is the thing to read, rather than the number.

What does this working tool compute?

The tool runs the same short calculation three times over and puts the three results beside each other. The calculator is given a monthly amountWhat would go in each month, taken from what a household actually has spare. No calculator can produce it., a periodA count of months, not a length of time anybody is promising anything will be held for. in months, and three assumed ratesFigures the reader chooses. The arithmetic is run at each one, and nobody is claiming any of them will happen.. The panel then shows all three answers together, the amount that would have been put in, the build-up instalment by instalment, and the gap between the highest and the lowest result. The gap is the figure worth carrying away. The output of this tool is a width, not an amount, and a household that reads it as an amount has read the opposite of what the arithmetic says.

The everyday version runs like this. Somebody asks a tailor how much cloth a set of curtains will take. The tailor measures the window, does the arithmetic exactly right, and then says it depends on how much drop is wanted at the bottom, so it is between four metres and seven metres. The arithmetic was never the difficult part. The unknown input was, and an honest tailor hands back the range rather than picking a drop for the customer. A calculator that hands back the range is that tailor.

What goes in, and where is each figure found?

Two of the three inputs can be read off something. The third cannot be read off anything at all, and that difference shapes everything that follows. The field note beside each box names the document and stops there. Meaning comes later.

Input one: what would go in each month

The monthly amount is a figure about a household’s own money, and it is found in that household’s own cash flow. Not in a brochure, not in a minimum amount somebody quotes, and not in what anybody else is putting in. The surplus is what actually arrives in a month less what actually leaves, and whatever part of that surplus a household decides is spare goes here. If the subtraction has never been done, that is the work to do first, and it is a different job from this one.

The worked example uses Rs 2,000/- a month because the invented Bhosale household ran a recurring deposit that took exactly that, paying in Rs 64,000/- of deposits out of the Rs 42,770/- that leaves in an ordinary month. The figure is used here because this household has already demonstrated it can find that sum, not because it is right for anybody.

Input two: for how many months

A count, and nothing more interesting than a count. The date the payments would stop, less today, counted in months. Ten years is 120 months. Twenty five years is 300. The field runs from 12 to 360 because that is the width of the drawing area, not because anything is being recommended about either end of it.

One warning about this input, and it is about arithmetic rather than conduct. Because compoundingGrowth applied to growth already added, rather than only to the original amount. works on the months that have already passed, the period makes the answer look most impressive. The shape of the curve is a fact, not a reason to type a larger number into the field. A household that stretches the period because the result got bigger has changed nothing about its own life; it has only changed a number on a screen.

Input three: the rate, which the reader supplies

There is no document anywhere that holds this figure. No statement to read it off, no line on a bank record, no printed number that settles it, and it is not a fact about the household or about anything else. The rate is the only input in this calculator that cannot be looked up, and it is also the input that moves the answer more than the other two put together.

So the field opens blank and says so on its face. A figure typed into it sets the arithmetic running. Left blank, the panel shows nothing except an explanation of why. A blank panel is not a broken calculator. A blank panel is the calculator working exactly as intended.

Why does the rate field carry no default?

Because a default is a suggestion. Nobody experiences a prefilled box as a neutral starting point. A number sitting in a field on a screen that looks like it knows what it is doing reads as a view about what belongs there, and once it has been read that way, everything after it is a forecast wearing the clothes of a calculation.

Consider what would happen if the field came filled in. The tool would print a result, and the reader would be holding a sentence of the form: this much for this long ends with that. Nobody wrote that sentence. The prefilled field wrote it, and a claim would have been made about a household's money without a word of the claim appearing in the text. A default rate turns arithmetic into a forecast silently. The refusal has to sit on the tool’s own face rather than in a note at the bottom.

The second reason is about how people arrive at a rate when asked to choose one. Daniel Kahneman and Amos Tversky established that people take a small run of recent outcomes as evidence of an underlying rate, and are markedly more confident in that reading than the evidence supports. A prefilled field feeds it directly: the number gets anchored, and the reader's own judgement never gets exercised at all. A blank field forces the choice into the open, where the reader at least knows a choice was made and who made it.

The third reason is the plainest. There is no basis for a figure. Any default would have to rest on a stated view about what happens to money in markets over a period, and arithmetic supplies no such view.

What does the computation actually do, month by month?

The computation does one thing repeatedly. The balance grows by one month of the assumed rate, and this month's amount is added. Grow, then add: the whole operation repeats once per month for as many months as the period runs. Nothing else happens. There are no fees in it, no taxes in it, no charges of any kind in it, and no adjustment anybody has slipped in.

The monthly rate is the annual rate divided by twelve, so an assumed 8 per cent a year is 0.6667 per cent a month. Start at nil. There was nothing there to grow in the first month, so the first month adds Rs 2,000/- and grows nothing. The table below is that one step, run six times, and 120 repeats of it reach Rs 3,65,892/-.

The bottom row is a reconciliation and it works in both directions. Rs 2,40,000/- went in, Rs 3,65,892/- came out at this one assumption, and the Rs 1,25,892/- between them is everything the assumed rate added across the ten years. The check is a subtraction, and it needs no trust in the tool at all. The whole computation is short enough for a household to check by hand, and that is the only reason to trust any figure here.

The panel at the top splits that same total a second way, instalment by instalment. Each Rs 2,000/- is a separate deposit compounding for however many months are left after it goes in. The first has 119 months behind it and grows to Rs 6,535/- at a 12 per cent assumption; the sixtieth has 60 months and grows to Rs 3,633/-; the hundred and twentieth has none at all and is still Rs 2,000/-, at every assumption anybody could type. The first sixty instalments carry between three quarters and four fifths of all the growth, at either outer assumption. Both curves in the build-up therefore fall steadily from left to right and land on the flat line at the last instalment.

Two notes on the arithmetic itself. The running balance carries its paise and only the figure shown is rounded, once, to the nearest rupee. Rounding every month instead puts the ten year result at an assumed 8 per cent two rupees lower. The two rupee gap is worth knowing when the check is done on paper. And every result here assumes the amount goes in at the end of each month. Putting it in at the start instead gives each amount one extra month of growth, so every figure rises a little. Neither convention is more correct. The convention used has to be stated, so a figure from somewhere else can be compared against it fairly.

Why does the tool show a range and never one figure?

Because one figure would be a claim and a rangeSeveral results shown together, so that no single one of them reads as the answer. is a description. The difference between a claim and a description is the whole honest basis of this calculator, and it is worth unpacking slowly.

The same Rs 2,000/- a month for the same 120 months at three different assumed rates gives three completely different amounts: Rs 2,94,500/- at an assumed 4 per cent, Rs 3,65,892/- at an assumed 8 per cent, and Rs 4,60,077/- at an assumed 12 per cent. The three rates are round numbers, evenly spaced, chosen because they are easy to check by hand. Nothing makes any one of them more likely than the others, and no probability attaches to any of the three.

Had only the middle one been shown, a reader would have left holding Rs 3,65,892/-, a number accurate to the rupee. The arithmetic is exact and its input was a guess, so the precision is entirely real and entirely misleading. Exact arithmetic on a guess produces an exact answer to a question nobody can answer, and the decimal places make it look like something it is not.

What does the spread between two assumptions tell a household?

Now the figure this whole calculation exists to produce. The result at the 12 per cent assumption, Rs 4,60,077/-, less the result at the 4 per cent assumption, Rs 2,94,500/-, gives a difference of Rs 1,65,577/-. Set against the Rs 2,40,000/- contributedThe monthly amount multiplied by the number of months. It needs no assumption at all. over the ten years and the ratio is 0.69.

The ratio of 0.69 is easy to skate over, so read it again. The choice of rate moves the answer by 0.69 times every rupee the household put in, across the whole ten years, and the household controls every rupee it put in while controlling none of the rate. All those months of finding Rs 2,000/-, all that discipline, all of it real and all of it the household's own doing, and a swing between two unremarkable assumptions is nearly as large.

The spreadThe distance in rupees between the results at two different assumptions. is not a measure of anything going wrong. The spread measures how much of the printed answer was never arithmetic at all. Rs 2,40,000/- of the Rs 4,60,077/- figure is money. The rest is a consequence of a number somebody typed into a box.

What does the spread say about the calculation as a whole?

The width says the calculation is dominated by its least reliable input, and it says so in rupees rather than in a warning sentence. A caution asks for care. A width states exactly how much room there was for the answer to be somewhere else.

A builder quoting a house works the same way. If the quotation comes back at Rs 9,00,000/- and the builder adds that the figure moves by six lakh depending on what happens to steel, that is not a price. The quotation is arithmetic with the part nobody controls named alongside it. Naming that part settles the question to ask next, and that is far more use than a price. A calculation whose width is nearly as large as everything put in is not a calculation that produces an amount, and knowing that is worth more than any of the three figures inside the width.

The width also puts the two parts of the answer in their proper sizes. The part a household earns, saves and carries in month after month is Rs 2,40,000/-, entirely real and entirely to the household's credit. The part that varies with the assumption is Rs 1,65,577/- wide, and none of it is under anybody's control. Any presentation that shows the second part as a single figure has quietly transferred credit from the first to the second.

What is this tool not?

The tool is not a projectionA statement about what will happen. A projection needs a rate somebody is prepared to stand behind., it is not a forecast, and it is not a promise. The refusal sits on the panel itself rather than in a footnote. A footnote is read after the number, and by then the reading has already happened.

The practical version of the distinction matters more than the definition. The arithmetic in this calculator and the arithmetic on a screen somebody else is holding can be identical, line for line, and mean two entirely different things. Three rates typed into three boxes by a household are that household's own assumptions, with all three answers visible side by side. One rate filled in by somebody else, with one figure printed at the bottom, turns the same arithmetic into that person's statement about what is going to happen to the household's money. The calculation is identical and the meaning is opposite, and the only two things that changed are who chose the rate and whether more than one answer is visible.

With somebody else's screen in front of a household and a person waiting for an answer, two questions are worth asking out loud, and neither is confrontational. Who chose the rate in that box, and what does the same arithmetic show at two other rates. A calculation that can only survive at one rate has said something about itself, and a person happy to run it at three has said something too. Asking for the other two figures is not a refusal of anything.

What can no arithmetic of this kind tell a household?

No arithmetic of this kind can say what will happen. Not approximately, not on average, not within a range anybody could name. Every figure here is conditional on a rate somebody chose, and the rate is the one part of the calculation that nobody controls, nobody can look up and nobody can settle. Run perfectly, the arithmetic is still exact arithmetic on a guess.

The arithmetic also cannot tell a household whether to put money anywhere at all. Whether to put money anywhere sits upstream of all of this, and the arithmetic has no view on it. A household with an outstanding card balance costing 3.5 per cent a month has a contracted cost in front of it that behaves in a completely different way from anything here, and comparing the two is covered separately. The invented Bhosale household holds no arrangement of the kind computed here, has a buffer covering 0.73 months of what leaves, and owes Rs 71,594/-. The Bhosale figures are the truth about the household the Rs 2,000/- was borrowed from, not a comment on anybody reading.

And no product enters the calculation, so it cannot tell a household anything about a particular product. The calculation is arithmetic in the abstract with two figures from one invented household dropped into it. The arithmetic is a way of seeing how much of a printed answer was ever arithmetic in the first place, and that is the entire claim being made.

What is worth writing down once the panel is closed?

Three things, and the largest of the three figures is not one of them.

The first is the amount and the period. Both belong to the household and both are facts. Rs 2,000/- a month for 120 months is Rs 2,40,000/-. The Rs 2,40,000/- does not move when anybody changes an assumption and it does not depend on anything happening in a market. If only one number survives the reading, this is the one worth keeping.

The second is the width, written with both of its rates attached. Rs 1,65,577/- between a 4 per cent and a 12 per cent assumption, over 120 months, at Rs 2,000/- a month. A width without its two rates attached is not a fact about anything, in the same way that a distance is not a distance until it is said between what and what. The width is worth writing out in full or not writing at all.

The third is a habit rather than a figure, and it is what all of this is really for. Whenever a single number appears at the bottom of somebody’s calculation, the question to ask is what it becomes at two other assumptions. The width that comes back shows how much of that number was arithmetic and how much of it was a choice. The question costs nothing to ask, it needs no expertise to understand the answer, and nobody being straight about the figures will mind being asked.

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