Inflation Impact: Converting an Amount Between Two Years

To move an amount between two years, multiply it by the ratio of the two index values, destination over source. The ratio is the whole computation. Everything difficult sits in the inputs: whether both values came from one series, what base that series is stated on, and whether the answer says which year's money it is in.

Underneath that sentence is a single idea. A price index is a running record of how much a fixed shopping list costs, written as a number rather than as a bill. If the list cost 100.00 in one year and 108.83 in another, then the second year's rupee is smaller, and the ratio of those two numbers is exactly how much smaller. Converting an amount between the two years is nothing more than applying that ratio. There is no second formula and no special case.

The computation runs forwards, runs backwards, and then runs on a pair of amounts where the rupee answer and the purchasing answer point in opposite directions. Two input faults make the arithmetic produce a confident number about nothing: index values pulled from two different series, and index values pulled from two different base years.

Left on the figures it opens with, the panel runs the Republic of Sankhya, invented and illustrative throughout. An index of 100.00 for the earlier year and 108.83 for the later one, an amount of Rs 50,000/- recorded in the earlier year and Rs 53,000/- recorded in the later one. The multiplier is 1.0883, the multiplier adds Rs 4,415/-, and Rs 50,000/- of the earlier year's money is Rs 54,415/- of the later year's. Carried straight back at 100.00 over 108.83 it returns Rs 50,000/- with nothing left over. Against that, the Rs 53,000/- actually recorded is short by Rs 1,415/- in later year money and by Rs 1,300/- in earlier year money. The pair reads as a rise of 6.00 per cent in rupees and a fall of 2.60 per cent in what those rupees buy. The identity closes on the fourth decimal. The purchasing ratio of 0.9740 multiplied by the index ratio of 1.0883 gives 1.0600, the rupee rise. Every number the sections below work through is one of those.

The saved settings underneath the fields each move one thing. Setting the recorded amount to Rs 54,415/- is the year where the payment exactly stood still: up 8.83 per cent in rupees and nil in what it buys. Setting it to Rs 47,000/- puts it 13.63 per cent behind. Running the conversion the other way, from 108.83 into 100.00 on Rs 54,415/-, returns Rs 50,000/- and reads as down 8.11 per cent in rupees with the purchasing line unmoved. The third button under the second release drives the central fault. Two index values that were never on one ruler get divided anyway.

Where do the three index levels come from?

The levels are not handed down. Sankhya has a consumer basket with three parts and a set of price movements attached to each part, and the whole index is built by weighting those movements. Food carries a weight of 40 per cent and its prices rose 12.0 per cent, contributing 4.80 points. Energy carries 10 per cent and rose 9.0 per cent, contributing 0.90 points. Everything else, the part that moves slowly, carries 50 per cent and rose 2.0 per cent, contributing 1.00 point. Add the three contributions and the basket cost 6.70 per cent more. The second index level comes from that addition and from nowhere else.

A series has to start somewhere, and the starting point is a decision rather than a measurement. Set the first year at 100.00 by convention. The second year is then 100.00 multiplied by 1.0670, or 106.70. In the year after that the slow moving part of the basket set the pace at 2.00 per cent. The third year is 106.70 multiplied by 1.0200. That works out at 108.834 and is published to two decimals as 108.83.

The last row is worth pausing on. Prices rose in both years, and yet the pace fell hard, from 6.70 per cent to 2.00 per cent. A reader who watches only the rate would say prices were calming; a reader who watches only the level would say prices were still climbing. Both are looking at the same three numbers. A value published to two decimals is the value everybody else will use, and rebuilding it from scratch is how two people get different answers from the same series. Every conversion below uses the published 108.83 rather than the unrounded 108.834.

What are the two inputs the conversion starts from?

Two numbers, and nothing else. The conversion needs the index valueA single number in a price series, showing what the basket cost in that period compared with whatever period the series treats as its starting point. for the year the amount is stated in, and the index value for the year it is to be stated in. The first is the source, the second is the destination. Everything else that might be tempting to gather, the inflation rate for each year, the weights, the sub components, is already inside those two numbers.

A ratio between two different indices is not a conversion of anything, so both values must be lifted from the same series and the same base. This is easier to break than it sounds. Documents quote whatever series suited their author, and two of them sitting on the same desk will often be measuring different baskets or counting from different starting years. Taking one value from each and dividing produces a number that looks exactly like a multiplier and is not one.

Think of two rulers. One is marked in centimetres, the other in inches, and both have the number 30 printed on them somewhere. Nobody would measure a table with one and a doorway with the other and subtract. The moment the marks come from two different rulers the arithmetic stops describing the room. Index values are marks on a ruler, and the ruler is the series.

How is the multiplier worked out?

Divide the destination year's index value by the source year's. Nothing else happens at this step. Converting Rs 50,000/- from the first Sankhya year into the third means dividing 108.83 by 100.00. The division gives a multiplierThe single number an amount is multiplied by to restate it in another year's money. The multiplier is one index value divided by another, and it carries no unit. of 1.0883.

Two things are worth noticing about that number. The multiplier has no unit. Rupees have already cancelled out of the top and the bottom, and that is what lets one multiplier be applied to a salary, a rent, a loan balance or a turnover line without any adjustment. And it is bigger than one only because the destination year is the later of the two. The multiplier is greater than one when the conversion runs into a year where prices are higher, and smaller than one when it runs into a year where prices are lower, and the direction is decided entirely by which value goes on top.

What does the multiplier do to the amount?

The multiplier multiplies the amount, and that is the last step. Rs 50,000/- multiplied by 1.0883 is Rs 54,415/-. The units are the whole point, so they have to be carried through the sentence. Rs 50,000/- of first year money is Rs 54,415/- of third year money. The amount of stuff has not changed. The size of the rupee it is counted in changed.

How is the same conversion run the other way?

The ratio turns upside down. Going from the third year back to the first, the destination is now 100.00 and the source is 108.83. The multiplier is 100.00 divided by 108.83, or 0.9189 to four decimals. Applied to Rs 54,415/- it returns Rs 50,000/- exactly, with nothing left over.

The backward multiplier is not a second rule to remember. The same ratio with the two values swapped does the whole job, and that makes converting between years a relationship rather than a direction. A reader who has only ever converted forwards has memorised a procedure and will freeze the first time a document asks for an old figure restated in old money. A reader who has seen the ratio invert knows there is only one thing to decide: which year goes on top. The year on top is always the year the answer is to be stated in.

DeflatingRestating a later amount in an earlier year's money, by multiplying it by the earlier year's index value divided by the later year's. The arithmetic is the same as converting forward, with the ratio inverted. is the name that usually gets attached to the backward direction, because the multiplier is below one and the amount comes down. There is no separate arithmetic hiding behind the word. Deflating is this same division with the smaller number on top.

How does an amount rise in rupees and fall in what it buys?

The step most readers actually came for is this one, where the conversion stops being bookkeeping and starts saying something. Consider a payment of Rs 50,000/- in the first Sankhya year that had become Rs 53,000/- by the third. Read in rupees it went up: Rs 3,000/- more, a rise of 6.00 per cent. The rise is the nominal amountAn amount exactly as it was recorded at the time, in the money of its own year, with no adjustment for what prices did afterwards. change, a fact about the rupee figures on the two documents.

Now put both amounts in the same year's money before comparing them. Rs 50,000/- of first year money is Rs 54,415/- of third year money, worked out above. The payment actually received in the third year was Rs 53,000/-. Set against what standing still would have taken, the payment falls short by Rs 1,415/-, and that shortfall is 2.60 per cent of it. The same pair of amounts rose 6.00 per cent in rupees and fell 2.60 per cent in what those rupees buy, and both statements are arithmetically correct.

Run it in the other year's money and the story does not change. Rs 53,000/- of third year money deflated to the first year is Rs 53,000/- multiplied by 100.00 over 108.83. That comes to Rs 48,699.81 and rounds to Rs 48,700/-. Against Rs 50,000/- that is short by Rs 1,300/-, and 1,300 against 50,000 is the same 2.60 per cent fall. The percentage is the same in either year's money, but the rupee size of the shortfall is not. A real figure carrying no year cannot be used at all. Rs 1,415/- and Rs 1,300/- are the same event measured with two different rulers.

An amount stated in one chosen year's money is called a real amountAn amount restated into the money of one chosen year, ready to be compared with other amounts stated in that same year's money., and a run of them all stated in that same year's money is often labelled constant rupeesA run of figures all restated into the money of one named year, making the differences between them differences in quantity rather than in the size of the rupee.. The label is only useful when it names the year. Constant rupees of which year is not a pedantic question; without the answer, the number cannot be compared with anything.

Why does the choice of index change the answer?

Because each index is a different shopping list, and the lists did not move together. Sankhya's consumer basket rose 6.70 per cent over its first step. Sankhya's wholesale basket carries no services at all and weights manufactured goods at 63 per cent. The wholesale basket rose 4.62 per cent over the same step, worked out the same way: 2.64 points from primary goods, 1.35 from fuel and 0.63 from manufactured goods.

Run the same Rs 50,000/- across that one step on each. On the consumer series the multiplier is 106.70 over 100.00 and the answer is Rs 53,350/-. On the wholesale series the multiplier is 104.62 over 100.00 and the answer is Rs 52,310/-. Same amount, same two years, two answers Rs 1,040/- apart, and neither series has made an error.

Set the recorded Rs 53,000/- against each of those. Against the consumer answer of Rs 53,350/- the payment is down 0.66 per cent in what it buys; against the wholesale answer of Rs 52,310/- it is up 1.32 per cent. One choice of basket says the payment fell behind and the other says it got ahead, on the same two amounts across the same two years.

So which is right? The one whose basket matches what the amount stands for. A household's monthly outgo is a consumer basket question. A factory's bill for inputs before anything reaches a shelf is a wholesale basket question. The answer depends on whose prices the amount represents, and that is a question about the amount rather than about the index.

What does an index value of 108.83 actually mean on its own?

On its own, nothing at all. An index value is never an absolute measurement of anything. An index value is a comparison with one particular period that the compiler chose and set to 100.00, and that period is the base yearThe period a price series has been set to 100.00. Every other value in the series reads as a comparison with that period and nothing else.. Read 108.83 without knowing which year the series counts from and it is a number with no referent, like a temperature quoted without its scale.

Something else happens when the same Sankhya prices are written on a second year base. Dividing every value by 106.70 and multiplying by 100 turns the series that read 100.00, 106.70 and 108.83 into 93.72, 100.00 and 102.00. Not one price in Sankhya changed. The prices are identical and the arithmetic is identical, and the printed numbers are completely different. The restatement is called rebasingRestating a whole price series to make a different period read 100.00, without changing any of the underlying prices or any of the changes between them..

Here is the trap, and it is quiet. Suppose one document quotes the third year as 108.83 and another quotes it as 102.00, and a reader who has not checked either base divides one by the other. The ratio comes out at 1.0670, reading as a 6.70 per cent rise. Both numbers describe the very same year in the very same country, so the true change is zero. The division recovered the distance between the two bases. That distance is a fact about the two documents and not a fact about prices.

Two index series stated on different bases cannot be compared until one of them is restated onto the other's base, and skipping that step is the commonest silent error in this whole computation. Silent is the word that matters. Nothing goes wrong on screen. The division works, the answer looks plausible, the units look right, and there is no warning anywhere. Run it through the panel above and the damage has a size. Dividing 102.00 by 100.00 gives a multiplier of 1.0200 and converts Rs 50,000/- to Rs 51,000/-, a figure Rs 3,415/- below the Rs 54,415/- the same amount is genuinely worth. The recorded Rs 53,000/- then reads as up 3.92 per cent in what it buys when it is down 2.60 per cent. The fix is one multiplication. Take 102.00 on the second year base, multiply by 106.70 over 100.00, and it returns 108.834, the published 108.83 to within 0.004 of an index point. All of that gap is the rounding in 102.00 rather than any movement in prices. At that point both numbers are on one ruler and can be divided.

Try two settings before moving on. Set the base selector to a second year base and watch the index labels change from 100.00, 106.70 and 108.83 to 93.72, 100.00 and 102.00 while the converted amount barely moves, the two rupee difference being nothing but the two decimal rounding in the restated values. Then set the recorded amount to Rs 54,415/- exactly and both changes settle: up 8.83 per cent in rupees and 0.00 per cent in what it buys. Standing still is defined exactly there, and it takes an 8.83 per cent rise in rupees to achieve it across those two Sankhya years.

Where would a reader find the numbers this computation needs?

Sankhya offers a tidy series because it was built to. Outside it, every input to this arithmetic has to be fetched from somewhere specific. The table below answers exactly one question about each of them: where it sits. Meanings were settled higher up. Fetching is what is left, and it decides whether the answer is worth anything at all.

Why does a lender run this before reading a revenue line?

Consider a shopkeeper walking into a branch with three years of figures. Turnover was Rs 50,00,000/- in the first year and Rs 53,00,000/- in the third. Written down it is a rising line, and a rising line is what a lender wants to see. The same arithmetic as above at a hundred times the size makes Rs 53,00,000/- of third year money Rs 48,69,981/- of first year money, so the shop is taking 2.60 per cent less in real terms than it was.

The two readings support completely different conversations. A shop growing 6.00 per cent is expanding and may be able to carry more debt. A shop whose takings are shrinking 2.60 per cent while its prices rise with everybody else's is losing customers or losing volume, and the rising rupee line was hiding it. A lender converts before comparing because a business that has only kept pace with prices and a business that has actually grown look identical on a rupee line and mean opposite things about the borrower.

The same test runs on a household. A salary that went from Rs 50,000/- to Rs 53,000/- a month over those two Sankhya years bought less at the end than at the start, even though every payslip in between showed a bigger number than the last. Nobody in that household did anything wrong and nothing on any payslip was incorrect. The rupee simply moved underneath them, and the only way to see it was to put both figures on one ruler.

The Republic of Sankhya, with its consumer basket and its wholesale basket, are invented.
Educational material. Not advice on any investment, tax, budget or market position.