Price, Return and Level Series: What Each One Answers

Read one record three ways. The price series says what one unit cost on a date, in rupees. The return series says what changed between one month and the next, in per cent. A rebased index says how far the record has travelled from a month somebody picked, and carries no units at all. The six year record is all three at once, and only the middle one compares with anything else.

Before a single figure is read, where it came from. The six year record was written to a recipe and then worked out again from that recipe, month by month, so anybody who repeats the arithmetic lands on the same closing figure rather than having to take it on trust. Each of the three readings is a division, and a division can be redone by hand.

The six year record has appeared already. The record is 72 monthly observations of the Nakshatra unit, an invented object written for teaching, running from the month ending 31 January 2019 to the month ending 31 December 2024 against an opening markThe figure a record is measured out from, fixed before its first observation. Here it is Rs 100.00/- on 31 December 2018, a chosen reference point rather than something that was observed happening. of Rs 100.00/- on 31 December 2018. Earlier notes settled that its order is part of its data, then ran three kinds of moving average across its prices. None of that is built again here.

One thing has not been said yet, and it quietly causes most of the confusion in this subject. When somebody puts a column of numbers in front of a reader and calls it the record, a choice has already been made out of sight, and the same 72 observations can arrive as three completely different looking columns. One is in rupees. One is in per cent. One is in numbers with no units at all. The three columns are not three datasets but three readings, and telling them apart before anything else is done is the cheapest habit in this subject.

Here is the everyday version. A household describes its electricity in three ways without thinking about it. A bill of Rs 2,400/- in December is a level. Up 8 per cent on November is a change. Running at about 130 against the summer month the household treats as normal is an index. Nobody in that kitchen is confused. Each sentence carries its own units. On a spreadsheet the units come off, and the same three sentences turn into three columns of bare numbers that look interchangeable and are not.

What are the three ways to read one record?

Take the record as it stands and write it out three times. Every figure in all three columns comes out of the same 72 observations, and no information is added or removed by moving between them.

The first column is the price series: Rs 100.00/- at the opening mark, Rs 96.8923/- at its lowest in August 2019, Rs 140.8315/- at December 2021, Rs 238.6196/- at its highest in November 2023, Rs 187.4539/- at the close. Every entry answers the question what did one unit cost on that date.

The second column is the return series: 2.00 per cent in January 2019, minus 1.00 per cent in August 2019, 16.00 per cent in October 2022 which is its largest month, minus 13.00 per cent in July 2024 which is its smallest, and 14.00 per cent in December 2024. Every entry answers a different question: what changed over that month.

The third column is a rebased index, and the word rebased is doing real work in that phrase. Divide every price by the price in one chosen month and multiply by 100.00. Choose December 2021 and the column reads 71.0068 at the opening mark, 100.0000 at December 2021 itself by construction, 169.4363 at November 2023 and 133.1051 at the close. Every entry answers a third question: how far the record has travelled from the month somebody picked.

Read the bottom row across. Three numbers, and two of them look nothing like each other: Rs 187.4539/-, 14.00 per cent, 133.1051. A reader handed any one of them alone would have no way of producing the other two. The month's change is 14.0000 per cent computed off the prices and 14.0000 per cent computed off the index, to four decimal places, so the December 2024 row is also the proof that the three columns agree.

What does the price series answer, and what are its units?

The price series is the reading people picture when they hear the word data. A price series is a column of rupee amounts against dates, and each amount answers one narrow question: what did a single unit cost on that date. Rs 96.8923/- in August 2019 says a unit cost that much then. The figure says nothing whatever about what happened during August, and nothing about whether that was a good month or a bad one.

The prices are carried forward rather than looked up one at a time. Start at the opening mark, apply January 2019's change of 2.00 per cent to get Rs 102.00/-, apply February's to that, and keep going for 72 steps. Each month works on the price the previous month left behind, and the word compoundingApplying each period's change to the amount the previous period finished with, rather than to the amount everything started with. The rupee value of a fixed percentage move keeps changing as a record travels. describes exactly that. The same percentage move is worth a different number of rupees in different months. October 2022 rose 16.00 per cent and that was worth Rs 25.4252/-. July 2024 fell 13.00 per cent and that was worth minus Rs 25.9550/-. The price underneath it was higher, so the smaller percentage moved more rupees.

Here is the first thing a price series cannot do. A rupee move is not comparable even against another rupee move inside the same record, let alone across two different objects. Two units priced at Rs 40/- and Rs 4,000/- can be given the identical month and will report rupee moves a hundred times apart, and nobody reading those two rupee figures side by side learns anything at all about which unit had the bigger month.

The second thing it cannot do is sit still. Watch where the line goes: out from Rs 100.00/-, down to Rs 96.8923/- by August 2019, up past Rs 140.8315/- at the end of 2021, on to Rs 238.6196/- in November 2023, and back to Rs 187.4539/- at the close. The record spent exactly 5 of its 72 months below the opening mark, all of them in 2019. The record is quoted to the paisaThe hundredth part of a rupee. Carrying a figure that far, or to four decimal places as these notes do, says how precisely the arithmetic was kept and not that anybody could deal at exactly that figure. throughout, and every one of those figures is a level rather than a movement.

What does the return series answer, and why does it travel when a price does not?

Each month's return is that month's price divided by the previous month's, less one, written as a per cent. December 2024 is Rs 187.4539/- divided by Rs 164.4332/-, or 1.14. Less one, the return is 14.00 per cent. Done 72 times, that gives the second column. The conversion costs one observation. 72 prices plus an opening mark give 72 returns, and the opening mark itself never gets one. There is no month before it to divide by.

Across the whole six year record this column has a centre of 1.00 per cent and a spread of 5.00 per cent. The centre and the spread are the mean and the standard deviation already familiar from elsewhere, applied to a column of 72 numbers, and no new machinery is needed to read them. Its largest month is 16.00 per cent and its smallest is minus 13.00 per cent.

The reason this column matters is that it is a pure number, and a pure number travels. Divide rupees by rupees and the rupees cancel, leaving something that can be set beside any other percentage in the world. The price series can never have that single property. A unit priced in the tens and a unit priced in the thousands both produce returns on the same scale, so which one had the bigger December can be answered. Being a pure number also means the column can be averaged, lined up against a second record month by month, or checked against an assumption about how wide a typical month should be.

Look at the shape of it too. The bars in the figure below go up and down around a fixed line and show no sign of drifting off anywhere over six years. The price path in the previous figure ends up more than twice as high as it started. The difference between drifting and staying put is not a small stylistic matter, and the comparison below is about nothing else.

Price Series vs Return Series: what does each one keep, and what does each one lose?

The comparison between price and return is the one the rest of this guide hangs off. Same record, same 72 observations, two columns, and each one keeps something the other throws away.

The price series keeps the level and loses comparability. It records that a unit cost Rs 238.6196/- in November 2023, which the return column can never give. A price cannot stand beside anything measured in different money, or beside anything at all whose starting figure was different.

The return series keeps the movement and loses the level. It records that December 2024 was a 14.00 per cent month, and sets that beside any other 14.00 per cent month anywhere. A return cannot say what anything cost. Handed the return column alone, nobody can rebuild a single price without being told one price to start from. 72 percentages describe a journey with no starting point attached to it.

Then there is the difference most readers have never had pointed out to them, and it is the reason so much of this subject runs on returns. Look at where each column sits in its first three years against its last three, and the two behave nothing alike. Averaged over January 2019 to December 2021, the price is Rs 115.0525/-. Averaged over January 2022 to December 2024, it is Rs 190.4591/-, a gap of Rs 75.4066/-. Nothing about the record pulls that price back towards any particular figure, so the second stretch simply sits somewhere else than the first.

Now do the same with the return column. The first three years average 1.00 per cent and the last three years average 1.00 per cent, agreeing to the last decimal place. One of these two columns has a centre that stays put and the other does not, and that is the whole of the practical difference. A figure computed on the return column, such as its average or its spread, describes the record rather than describing the stretch of months that happened to be picked. The same figure on the price column mostly describes the choice of dates.

The same point arrives from a second direction, without any new arithmetic in this guide. Lined up against its own reading from the month before, the price column agrees with its own past almost perfectly, at 0.9650 on a scale where 1.0000 would be a perfect match. The return column manages only 0.2011. A price is mostly a restatement of last month's price. A return is mostly news. Both of those readings, and the reason a level series wanders in the first place, belong to the separate notes on whether a level series sits still, where the idea is called stationarityThe property of a series whose behaviour does not depend on when it is looked at: no drift in its centre and no change in its spread over time. Whether the six year record has it, and the test for it, are settled in their own notes., and to the notes on lagLining a column up against its own earlier readings, so that each month sits beside the month before it. The reading a lag gives, and how far back it is worth going, are handled separately.. Stationarity and lag name where that machinery lives.

One more angle, on a single month. The units question gets sharpest when the record is stripped down to one step. December 2024 moved from Rs 164.4332/- to Rs 187.4539/-. The movement can be written three ways: Rs 23.0207/- in rupees, 14.00 per cent in per cent, and 16.3462 index points on the December 2021 base. All three are correct. The other two both require knowing where the movement started from. Only the middle one can be handed to somebody who knows nothing else about this record and still mean something. Which of the several ways of turning a level into a change is the one to reach for, including the one called a log differenceA way of writing a change that uses logarithms instead of a plain division. Where it helps, what it costs and why some work prefers it are set out in the separate notes on turning levels into changes., is a separate question settled in the notes on turning levels into changes.

What is a rebased index, and what does choosing a base change?

The third reading is the one people find slipperiest, and the arithmetic behind it is a single division. A month is chosen. Every price in the record is divided by that month's price and multiplied by 100.00. The division is the whole recipe, and the month chosen is called the base.

Base the six year record at December 2021, where the price is Rs 140.8315/-, and the column reads 71.0068 at the opening mark, exactly 100.0000 at December 2021, 169.4363 at the November 2023 high and 133.1051 at the close. None of those four numbers carries a unit: an index reading is not rupees and it is not per cent, and writing it with either attached is the commonest formatting mistake made with this reading. A reading of 133.1051 is a statement that the record stands where it stands relative to one particular month, and nothing else.

Now the part worth pausing on. Move the base to a different month and every single number in that column changes. Base it at the November 2023 high instead and the close reads 78.5576 rather than 133.1051. The column looks like a completely different record and is not one. Here is what does not change. Dividing every price by the same number leaves the ratio between neighbouring prices untouched, so the monthly changes computed off the index are identical at every base that could be picked. December 2024 is 14.0000 per cent off the prices and 14.0000 per cent off the index at any base at all, to four decimals. The base changes every level and no return.

What happens if the same record starts a thousand times higher?

Here is the same point with the volume turned up, and it is worth doing because it settles an objection that sits in the back of most readers' minds. Take the identical 72 monthly changes and run them from an opening mark of Rs 1,00,000.00/- instead of Rs 100.00/-. The record now closes at Rs 1,87,453.89/- rather than Rs 187.4539/-.

Every single price is a thousand times larger. Not one monthly change differs, at any decimal place worth checking. A return series does not know what the level underneath it was, and that is a feature rather than a shortcoming. It is why two objects that will never be priced anything alike can still be laid side by side and asked which one had the better month, and it is why a percentage figure survives being moved between records, currencies and reporting conventions when a rupee figure does not.

Which of the three readings can be averaged?

Averaging is where the three readings separate most sharply, and the test is simple: after the division, the question is what the units of the answer are and whether that describes anything.

Averaging the return column gives 1.00 per cent, a real statement: the typical month of this record moved about one per cent. Averaging the price column gives Rs 152.7558/-. The figure is arithmetically correct and describes nothing. The record opened at Rs 100.00/-, dipped to Rs 96.8923/-, climbed to Rs 238.6196/- and closed at Rs 187.4539/-. Only 2 of its 72 months landed within Rs 5/- of that average, and only 7 landed within Rs 10/- of it. The line crosses the path exactly once, in February 2022. An average price is the centre of a journey rather than a description of any part of it, and the record spent almost none of its time anywhere near the figure.

The index column sits in between and inherits the same problem the price column has, with an extra one on top: its average depends on the base, so two people averaging the same record can report different figures while both being right. Based at December 2021 the average index reading is 108.4671, and based anywhere else it is something different.

How is the reading a question needs decided?

No rule with clauses is needed. Hearing what the question asks for is what matters, and a question comes in only four shapes.

Ask what it cost, and the answer is the price series. How much did it move, and the answer is the return series. Did it move more than something else did, and the answer is the return series again, on both objects. How far has it come since a date everybody has agreed to treat as the starting line, and the answer is a rebased index with that date as its base.

Here is the line worth keeping: if the question can be answered without knowing the units, it is a question about returns. Applied to the four shapes above, it sorts them immediately. Most of the damage in this subject is done by columns that arrived without a label, so the reading in use belongs in the sentence itself, said out loud.

Who reaches for which reading at work?

The choice is not academic, and three quite different people make it every week without calling it anything.

A lender looking at a borrower is usually after levels. The balance today, the balance at the last review, the shortfall in rupees: those questions have to be answered in money, because money is what gets recovered. A percentage change in a facility that nobody can size in rupees is not a usable answer, and this is the one setting where reaching for the return column is the mistake.

An analyst comparing two things is always after returns, for the reason this guide has laboured. Two objects that will never be priced alike can only be set beside each other once both have been converted, and the conversion is the first thing done rather than the last. A comparison table built in rupees is a table nobody can read across.

A household watching its own costs almost always ends up building an index without knowing that is the name for it. Somebody says the monthly grocery bill is running at about 130 compared with two years ago, and that sentence is a rebased index with a base month chosen for convenience. The moment a second household says its own figure is 115, the two are only comparable if both chose the same base month, and they almost never have. That is the whole hazard of the third reading in one everyday sentence, and it is the reason a published index is always accompanied by its base.

The habit that serves all three is the same one: the reading being quoted is named in the sentence, every time. The naming costs four words. A figure recomputed as its window slides forward, covered in the notes on the rolling windowA figure worked out over a fixed number of recent months, then worked out again as the months move on, so it is a series of readings rather than one. The reading it gives and how late it reacts are covered on their own., and a record with a repeating calendar pattern taken out of it, a seasonally adjustedA record with its repeating calendar pattern removed, so that the months can be read against each other without a January effect or a June effect sitting inside them. The removal, and what it does and does not change, are handled separately. record, both still have to say which of the three readings they were built on before anybody can use them.