Top-Down vs Bottom-Up Market Sizing: Which Route Can Be Run

Top-down starts from a total somebody published and takes a share of it. Bottom-up starts from a count and a price and multiplies up. The useful question is not which is more accurate but which one the available evidence can actually support. A bottom-up build stops dead when a number is missing. A top-down run returns a clean figure for any pair it is fed.

What are the two routes, and what does each one need?

Both routes are set out one at a time under how a market total is estimated. A run from above starts with a total that somebody else has already put a number on, and takes a portion of it. A build from below starts with a count of things and a figure for each thing, and multiplies the two together. The two pairs of inputs are the whole of the difference in shape.

The question everybody asks next is which one is more accurate, and it is the wrong question. Accuracy is the distance between an estimate and the true answer. Accuracy cannot be judged without the true answer, and in a real sizing nobody has it. If somebody had it, no estimate would be needed. The comparison worth running instead is one that can be settled today, on any pair of routes, without knowing the answer at all: what does each route do when a number it needs turns out not to exist?

The shape is the shape of counting the people at a wedding. One way is to ask the caterer how many plates went out and take the share thought to be guests rather than staff. The other is to count the tables and count the chairs at a table and multiply. If nobody knows how many chairs a table holds, the second method simply stops, and the stop is visible. The first method never stops. The caterer's method gives a number even when the share has been quietly guessed, and the number looks exactly like the number a known share would have produced.

Can three builds made of different parts land on the same published figures?

They can. Three builds follow. Each one is assembled out of components that have nothing in common with the components of the other two, and each one finishes exactly on a figure the business itself published. The exactness is the whole point of them.

Build one is units times price. Anjani Stationers Private Limited, invented for teaching, sold 2,50,000 registers in the year at a realised priceWhat buyers actually paid on average once every discount and allowance had come off, as against whatever price was printed on a list. of Rs 108.00/- each. The two multiplied together come to Rs 2,70,00,000/-, the revenue that business published for the year. Not a rupee out.

Build two is the customer book, and it shares not one component with build one. There is no count of registers in it and no price of a register in it. The same business sold to 36 accounts. The Sunrise Public School group took Rs 81,00,000/- of the year. The other thirty five accounts took Rs 5,40,000/- each, Rs 1,89,00,000/- between them. Rs 81,00,000/- added to Rs 1,89,00,000/- comes to Rs 2,70,00,000/-. The same published figure, reached from a completely different direction, and the name count of one plus thirty five is the published thirty six.

Build three moves to a different business and a different kind of thing entirely. Setu Bazaar, an invented marketplace, reaches 50,000 buyers. Its heaviest 5,000 hold 40.00 per cent of what it kept, and the other 45,000 hold the remaining 60.00 per cent. Carry those two shares across to the gross merchandise valueThe total value of the goods that crossed a marketplace in a period, a much larger quantity than the marketplace's own revenue, and one that belongs to the sellers rather than to it. that crossed the marketplace. The heavy band then carries Rs 2,00,00,00,000/- of goods, or Rs 4,00,000/- a buyer. The other band carries Rs 3,00,00,00,000/-, or Rs 66,666.67/- a buyer. The two add to Rs 5,00,00,00,000/-, the flow that marketplace published. Carrying a split of what the marketplace kept across to what crossed it holds the same 4.00 per cent charge on both bands, and that step is the shared input in the failure below.

One honest wrinkle sits inside build three, and it is worth naming rather than smoothing over. The figure for a buyer in the second band is a recurring decimalA division whose answer never stops repeating, such as two hundred thousand divided by three. Writing it down at any number of places rounds it. Multiplying the written figure back then does not return exactly the starting figure.: two hundred thousand divided by three, which runs on forever. Rounded to the paisa it reads Rs 66,666.67/-, and 45,000 of those comes to Rs 3,00,00,00,150/-, Rs 150/- more than the band actually carries. Carried unrounded the band lands exactly. Three builds, three landings on the published figures, and the only gap anywhere is Rs 150/- in five hundred crore that appears solely because a figure was written down to the paisa before it was multiplied back.

What happens when the total belongs to the business itself?

Now the other route, run once, in full. Setu Bazaar handled Rs 5,00,00,00,000/- of goods in the year and kept a take rateWhat proportion of everything traded across a marketplace that marketplace itself ends up keeping, found by dividing its own revenue by the value that crossed. of 4.00 per cent of that flow. Four per cent of Rs 5,00,00,00,000/- is Rs 20,00,00,000/-, the recognised revenueThe revenue a business is entitled to report for a period under the accounting rules it follows. The rules for what may go into that line, and when, are settled separately. that marketplace published. The subtraction gives the same answer from the other side: Rs 5,00,00,00,000/- of goods crossed, Rs 4,80,00,00,000/- reached the sellers, and the difference is Rs 20,00,00,000/-.

The run on Setu Bazaar has the shape of a run from above. A total at the top, a share applied to it, a figure at the bottom. Its total is one business's own flow rather than a market, so the run is top-down in form only. Every rupee of that Rs 5,00,00,00,000/- crossed one marketplace and is knowable to that marketplace down to the last order. Nobody had to find it, nobody had to draw a boundary around it, and nobody had to decide whose sales counted. The run on that marketplace is a run from above with the hard part, finding the total, taken out.

The four cells below make the harder half plain. The same run cannot be performed on Anjani Stationers at all, in any form. There is no total anywhere to take a share of. Nobody publishes what schools in that district spend on things to write in. Nobody publishes what the register makers of that city sell between them. The route is not difficult on that business; it is unavailable. Three of the four cells can be filled and one cannot, and the reason is an absence of evidence rather than a difference of skill.

Where did each per-unit figure actually come from?

Now take the builds apart, and this is the centre of the matter. Each of the three builds raises one question, taken one at a time. Not whether the arithmetic is right. The arithmetic is right. The question is where the figure for each single thing came from before it was multiplied.

Rs 108.00/- a register is recoverable by dividing the published revenue of Rs 2,70,00,000/- by the published count of 2,50,000 registers. Rs 5,40,000/- an account is recoverable by dividing the published remainder of Rs 1,89,00,000/- by the published thirty five names. Rs 4,00,000/- of goods a heavy buyer is recoverable by taking 40.00 per cent of the published flow of Rs 5,00,00,00,000/- and dividing it by the published 5,000 buyers. Three of the three per-unit figures are divisions of the totals they later rebuild, and zero of the three are independent measurements.

So each build multiplied a count by that total over a count, and returned the total. The build could never have done anything else. The arithmetic is correct and the arithmetic is empty. Those two facts sit together comfortably, and the comfort is exactly what makes the emptiness hard to hold on to. The household version takes four seconds: checking a month's spending by taking the total from a bank statement, dividing it by thirty, and multiplying the answer by thirty. Nobody would call that a check on their spending. Dressed in a count of registers and a price a register, though, the same operation starts to look like a check.

The ruling is not new. Under Revenue Growth vs Monetisation Improvement, revenue is activity times yield, and the identity returns revenue by construction, so it could never have finished anywhere but on revenue. The general form of the same ruling is this: a reconciliation that was constructed cannot confirm anything. Not the level, not the components, not the total it landed on. The exact landing shows only that arithmetic works, and arithmetic was never in doubt.

If the arithmetic confirms nothing, what is it good for?

There is a useful half to all this, and it is genuinely useful. Splitting a total into a count and a figure for each thing gives attribution: which of the two terms a movement came out of. Attribution is a different claim from confirming a level, and it survives even when the split was constructed.

Work it on a bare construction with no name, trade, country or year attached. Arithmetic can demonstrate the property without describing anybody's business. A total of 100 in one period and 110 in the next, with the count held at 10 in both. The figure for each thing therefore went from 10 to 11. A constructed split cannot confirm a level and can still locate a movement. The split says nothing about whether the level of 100 was right. The split does say that whatever the level was, the change did not come out of the count. The count did not move.

Locating the movement is worth having, and it is checkable by somebody else in a way the level never was. If a total rose while the count stood still, anybody can look at the two counts and see for themselves that they match. The claim about where the movement came from rests on an observation rather than on a division. The claim about the level rests on nothing.

How a change in a product splits into two effects, what the cross termThe part of a change in a product that belongs to neither factor on its own. Both factors moved at the same time, and the factor the part is charged to is a convention rather than a finding. between them is, and which convention decides that split are all covered separately under Revenue Growth vs Monetisation Improvement, where Laspeyres 1871 and Paasche 1874 are named for the choice.

What would a real sizing look like, and why does none of this carry over?

The difference fits in one sentence, and then it can be worked. For Anjani Stationers and Setu Bazaar both sides were published; in a real sizing neither side is. The gap between those two conditions is the whole difference, and it is very wide.

A real build from below, for the register trade of one city, would need the count of makers, and nobody publishes the count. The build would need what each maker sells, and nobody publishes that either. The build would need a price for each maker's output, and no firm holds anybody's price but its own. A real run from above would need a total for that trade, and there is no such total anywhere, drawn on any convention, by anybody. Two routes agreeing in this example is a property of the example and never a validation of a method.

Every landing above rested on a figure and its own total both being already printed. A real sizing never meets that condition. The professional version of that is the sentence actually used in a room: when two sizing methods are presented as agreeing, the question is whether either of them used a number derived from the other. Most of the time one number is shared, and the agreement is arithmetic rather than evidence.

The denominatorThe number underneath in a division, being the thing a share is a share of. Which one is the right one for a share of a field, and why it is so rarely available, is settled separately. problem sits underneath both routes and is worked separately under Market Fragmentation: What a Share Is Worth in a Crowded Field. The narrower matter is behaviour: on published components, both routes land; on real evidence, one of them stops and the other does not.

Which route fails loudly, and which one fails quietly?

The comparison was set up on failure behaviour rather than on accuracy for one reason: failure behaviour is what actually separates the two routes.

A build from below fails loudly. Take away a count or take away a price and there is nothing to multiply. The build stops, and the person running it knows it has stopped, in the same second. Work it on the case at hand. Bhavani Register Works turns out 1,50,000 registers a year. The count is published. Its price is not published. Its cost of paper is not published. Its works cost is not published. So a build from below covering the makers on that lane reaches the first multiplication and stops there, with the missing number named. The output of that build is a sentence rather than a figure, and the sentence is true.

A run from above fails quietly. Hand it a total and hand it a share, and out comes a clean number, whichever pair it was handed. Feed it a total drawn on the wrong convention and it returns a number. Feed it a share nobody measured and it returns a number. Feed it a total for one country and a share observed in another and it returns a number, correct to two decimal places, with no complaint anywhere. A route that always returns a number cannot signal when it should have refused. The finding is not new either. The concentration index reaches it about a boundary, under Market Concentration vs Market Share: How Each Measure Fails, and a route that cannot refuse is the same finding wearing a new hat.

The household version is a calculator. A calculator gives an answer for every input, including the inputs typed by mistake. A clean answer says nothing about whether the keys were the right keys. The tool actually wanted, and the one almost no estimating method provides, is a tool that stops.

So which route should be reached for, and what should be written beside it?

The straight answer is about evidence rather than preference: the route to reach for is the route the evidence supports. Where the analyst holds a count that can be defended and a figure for each thing that came from somewhere other than the total, the build runs from below. Where the only thing available is somebody else's total, the run goes from above, followed by the thing almost nobody does.

Print the total and the share as separate lines rather than only the product. The product hides both inputs and the two lines hide nothing. A reader who sees Rs 40 crore cannot check anything. A reader who sees a total, a share, and the product of the two can go and argue with either one. Printing both costs a line of space and converts a figure into a claim.

Then the harder instruction, and it is the one that separates a useful document from a confident one. Where neither route is supported, write that, and name which number is missing. A note saying that a build from below stops at the second maker's price, with no price published for that maker anywhere, is a more useful document than a figure. The note points the next reader straight at the missing figure. The note is checkable by anybody in a way a figure never is. And the note will still be true in a year, a claim very few estimates can make.

None of this makes sizing worthless. The claim is about what a sized figure has to carry with it before it can be read. A figure with its route named and its inputs separated is ordinary professional work. A figure standing alone with the word triangulated beside it is the thing to be careful about.

How does an analyst or a lender actually use any of this?