Purchasing Power Parity and Interest Rate Parity Compared

Purchasing power parity says the exchange rate should settle where one basket of goods costs the same in either currency, and trade is what pushes it there. The interest condition says that where money earns more, that currency should be expected to weaken by roughly the gap, or holding it would be a gain with nothing given up. One arbitrages goods, the other arbitrages money.

Every price, return and exchange rate below belongs to the Republic of Sankhya and its trading partner Marut, two invented economies whose figures are illustrative.

Both conditions are trying to answer the same question. Where would an exchange rate come to rest if nobody could make money out of moving it? Where they differ is in what is allowed to move. Purchasing power parity moves goods, so it is built entirely out of what things cost. The interest condition moves money, so it is built entirely out of what money earns. Neither one is derived from the other, and that turns out to be the most useful thing about the pair.

The two ingredients are already in place. The first is arbitrageBuying something where it is cheap and selling it where it is dear, until the two prices are close enough that the trip is no longer worth taking. and it closes a price gap in any market. The second is the Sankhya policy rateThe rate a central bank fixes as the anchor for short term lending, which drags most other short dated rates in that economy up or down with it. , locked at 6.00 per cent.

What is purchasing power parity, and what does it compare?

Purchasing power parity starts from a very ordinary thought. If a bag of groceries costs one amount in Sankhya rupees and another amount in Marut units, and somebody could buy it cheaply in one place and sell it dearly in the other, then the exchange rate between the two currencies has a job to do. The rate has to be the number that makes both prices describe the same thing.

So a basketA fixed list of goods and services in fixed quantities, priced repeatedly so that a change in the total is a change in prices and not a change in what was bought. is priced twice. The same list of things, the same quantities, once in each currency. In Sankhya that basket costs Rs 8,000/-. In Marut the identical basket costs 100 Marut units. Dividing one by the other gives 80.00 Sankhya rupees to the Marut unit. **Purchasing power parity is that division and nothing more: the price of the basket in one currency over its price in the other.**

Look at what happens if the market rate sits somewhere else. Suppose the market rate were 100.00 instead of 80.00. A trader with 100 Marut units converts them into Rs 10,000/-, buys the Sankhya basket for Rs 8,000/-, ships it to Marut and sells it for 100 Marut units. The trader is 25 Marut units better off and has done nothing clever. As long as that is true, goods keep moving in that direction. The movement lifts Sankhya prices and pushes Marut prices down, and the rate is dragged towards the level where the trip stops being worth taking. **The mechanism inside purchasing power parity is trade, so the condition has nothing at all to say about interest rates.**

Here is the part worth sitting with. The basket matches at 80.00, but look at the items inside it and not one of them does. Grain works out at 85.71, cloth at 72.73, a month of rent at 77.78, a haircut at 100.00 and bus fares at 80.00. Only the total lands on 80.00. Purchasing power parity is a statement about a basket, not about any item in it, and a reader who tests it on a single product is testing something the condition never claimed.

What is the interest rate parity idea, and what does it compare?

Now set the basket aside completely. Forget that goods exist. The interest condition is built from a different raw material, and it deserves the same amount of room, so work it from the beginning.

Money in Sankhya earns 6.00 per cent over a year. Money in Marut earns 2.00 per cent. Suppose an investor holds Rs 1,00,000/- and money can cross the border freely in either direction. There are two ways to spend the year.

Route one, the money stays in Sankhya. Rs 1,00,000/- at 6.00 per cent becomes Rs 1,06,000/- after a year. Route two, the money moves to Marut. At today's rate of 80.00 the Rs 1,00,000/- buys 1,250 Marut units, those earn 2.00 per cent and become 1,275 Marut units, and at the end of the year they come home at whatever the rate is then.

Ask what that end rate would have to be for the two routes to land level. The end rate is Rs 1,06,000/- divided by 1,275 Marut units, or 83.14. A rate of 83.14 sits 3.92 per cent above 80.00. **So the interest condition is not a claim about goods at all: it says that a currency paying the higher returnWhat a sum of money grows to over a stated period, expressed as a percentage of what was put in at the start. should be expected to weaken by roughly that gap. Otherwise route one beats route two, and the higher paying currency is a gain with nothing given up.**

Two details are worth pinning down. First, the 3.92 per cent is not the simple difference of 6.00 and 2.00, a subtraction that would give 4.00. The comparison is between two grown sums and not two rates, so the figure is one plus the Sankhya return divided by one plus the Marut return, less one. Second, the word doing all the work in that sentence is expected. Nobody has fixed a rate for the end of the year and nobody has signed anything. The condition describes what the market would have to be anticipating for the two routes to look equally attractive today.

What does each condition actually arbitrage?

Both conditions are described as closing a gap, and both do. The difference lies in what has to physically move for the gap to close, and that single difference explains most of the distance that ever opens between the two.

Purchasing power parity closes its gap by moving goods. Somebody has to buy grain in one country, put it on a ship or a truck, wait, pay freight, pay whatever the border charges, and sell it in the other. The trip is slow, it costs money, and there is a floor below which it does not pay for itself. Worse, a great deal of what a household actually spends on cannot make the trip at all. Look back at the Sankhya basket: rent for a month, a haircut and bus fares add up to Rs 4,000/- of the Rs 8,000/- total. **Exactly half the basket by value is non-tradableA good or service that cannot practically be bought in one country and delivered in another, such as a haircut, a bus ride or a month of rent., so no amount of arbitrage in goods can touch it.**

The interest condition closes its gap by moving money, and money has none of those problems. A deposit can be converted and placed in another currency in an afternoon at a cost that rounds to very little. Nothing is loaded, nothing waits at a port, and nothing is unshippable. So when a gap opens on the money side, it tends to close quickly, and when a gap opens on the goods side, it can sit there for years.

Think of a street vendor and a bank branch on the same road. The vendor who spots that onions are cheaper two districts away still has to hire a tempo, drive there, load, drive back and sell before the onions turn. The branch manager who spots that a deposit pays more across the road moves the money before lunch. Same instinct, completely different friction.

Why can the two conditions disagree, and why is that the point?

The comparison stops being a definition exercise and starts earning its keep at the point where the two conditions disagree.

Write down what feeds each condition. Purchasing power parity takes in exactly two numbers: the price of the basket in Sankhya and the price of the same basket in Marut. The interest condition takes in exactly two numbers as well: the return on money in Sankhya and the return on money in Marut. Now look for an input that appears on both lists. There is not one. **The two conditions share no input at all, neither can be rearranged into the other, and that is precisely why they are allowed to point at different levels.**

Compare that with something that looks similar and is not. The balance of payments records what a country earned across its border and what claims it handed over, and the current account plus the capital account equals the change in reserves. Minus Rs 14,000 crore set beside plus Rs 19,000 crore gives plus Rs 5,000 crore, and always will. Whatever is not paid for out of earnings is paid for by handing over a claim, and any residual is booked as errors and omissions. The equality between the two sides is an identityA relationship that holds by definition rather than by discovery, because one side is constructed out of the other and the two can never come out different.. An identity cannot disagree. So when it agrees, it has confirmed nothing.

Purchasing power parity and the interest condition are the opposite case. The two conditions were never obliged to agree, so on any occasion where they do, the agreement carries information, and on any occasion where they do not, the disagreement carries information too. **A route that can disagree is informative precisely because it can, and a rearrangement is never informative however neatly it closes.** Before a second calculation is called a cross-check, the question to settle is whether it could possibly have come out differently. If it could not, it is not a check, it is the same statement wearing different clothes.

What do both conditions give on the same set of Sankhya figures?

Put the two side by side on one set of numbers. Purchasing power parity takes the Sankhya basket at Rs 8,000/- and the Marut basket at 100 units and returns 80.00. The interest condition takes 6.00 per cent against 2.00 per cent and returns an expected weakening of 3.92 per cent. Starting from the observed rate of 80.00, that points at about 83.14 a year out. The observed market rate today is 80.00.

One thing has to be settled before anything is read into the agreement. Purchasing power parity landing exactly on the observed rate of 80.00 here was chosen when these teaching figures were written. The coincidence was not discovered, and it is not a finding about anything. A worked instance is easier to follow when one of the two conditions starts flush, so the Sankhya basket prices were picked to make the division come out at the observed rate. In any real set of numbers there is no reason at all for a parity level and a market rate to coincide, and everything that follows rests on that.

Now move one input at a time and watch which side responds. Every row below holds everything except the named input at its published Sankhya level.

The last row carries the whole comparison. Two of the rows change the goods answer and leave the money answer untouched. Two of them do the reverse. And the final row is worth pausing on: when Marut pays more than Sankhya, the sign flips and the condition implies the rupee should be expected to strengthen rather than weaken, the same rule read from the other end.

What does it mean when the observed rate sits far from either condition?

Sooner or later a market rate turns up a long way from a parity level, and the temptation is to reach for a verdict. The arithmetic supports something narrower.

Carry the Sankhya case on a step. The rate moves from 80.00 to 84.00 Sankhya rupees to the Marut unit, and nothing at all happens to the basket prices, so purchasing power parity still says 80.00. There is now a gap of 4.00, or 5.00 per cent of 80.00. The gap establishes that the arbitrage described by purchasing power parity has not closed. The gap establishes nothing beyond that. A gap stays open for two very different reasons, and the claim is smaller than it looks.

The first is that it has not closed yet. Goods take time to move and the trade that would narrow the gap may be under way. The second, and the one people forget, is that it cannot close. Half the Sankhya basket by value is rent, haircuts and bus fares, and no shipment of anything closes a gap in the price of a haircut. **A gap between a market rate and a parity level is a question about which of those two situations holds, and it is never on its own a signal.**

So a gap of this kind does not establish that a currency is priced too high or too low, and the words usually reached for here import a judgement that the division of one basket price by another simply does not carry. A gap does support three questions: how much of the basket could physically move, how long a move of that kind takes, and whether anything is stopping it. All three are answerable. The verdict is not.

What does neither condition promise?

Both conditions describe a level a rate would settle at if the arbitrage they name were allowed to run to the end. Neither one says when that will happen, and neither says the market has to get there at all.

Put the timing question plainly. Purchasing power parity does not carry a clock. Purchasing power parity says where trade would leave the rate if goods could move without cost. Goods cannot move without cost, and half the basket cannot move at any price, so the level it names may be approached over many years or never reached. The interest condition is faster because money is faster, but it is a statement about what the market appears to be anticipating, and an anticipation is not an outcome. If everyone expects a 3.92 per cent weakening and something else entirely happens, the condition was never wrong about anything. No such claim was ever made.

Both also carry the phrase other things equalA working assumption that everything not being discussed is being held still, so a single cause can be traced. Real conditions never oblige. underneath every line of the arithmetic, and other things are never equal. Prices move while goods are in transit. Returns move while money is placed. A basket bought in one country is not perfectly the same basket as one bought in another. **The level either condition names is a resting point rather than a forecast, and treating a resting point as a prediction of next quarter is the most common thing that goes wrong with both.**

What does an analyst actually do with a gap between the two?

The practical habit worth taking away is smaller than it might seem. When somebody presents a number and calls it the currency's gap, the first thing to ask is which benchmarkThe reference level a measurement is taken against. Change the reference and the same measurement produces a different number without anything real having changed. that gap was measured against. The same observed rate produces two completely different gaps depending on which of the two conditions it is held up to.

Worked on the Sankhya figures, the observed rate is 84.00. Measured against purchasing power parity at 80.00, that is 5.00 per cent above. Measured against 83.14, the level the interest condition implies for a year on from 80.00, the same rate is 1.04 per cent above. **The same rate of 84.00 is five per cent away from one benchmark and about one per cent away from the other, so a gap quoted without naming its benchmark says nothing whatsoever.**

A loan repayable in a foreign currency is money, and the arbitrage that touches money is the money arbitrage, so a lender pricing that loan wants the money-side reading. An analyst looking at a manufacturer whose costs are domestic and whose sales are exported wants the goods-side reading. The question there is whether the firm's prices stay competitive when goods actually move. Same currency, same day, two different questions, two different benchmarks. Asking which one is on the table is most of the work.