International economics · Parity conditions
PPP and interest rate parity: which formula, which way, and where the numericals bite
Purchasing power parity and interest parity look like four formulas to memorise. They are one idea, that arbitrage removes easy profit, applied to goods, to interest rates with a forward contract, and to interest rates without one. This page derives each, gives you a lab for each, and walks through the numeric traps. 23 past-paper questions at the end.
A question to hold on to
India is the home country and the US the foreign country. Interest rates are 3% in India and 1% in the US. The rate is ₹80 per dollar spot and ₹81 in the 3-month forward market. Which are true of the resulting covered interest arbitrage flows: the rupee depreciates in the spot market, the rupee depreciates in the forward market, interest rates fall in India, interest rates fall in the US?
UGC NET Dec 2023Solved at the end ↓
The idea
One idea behind four formulas
An arbitrage is a profit with no risk and no money of your own: buy something where it is cheap and sell it where it is dear. If one were available, everyone would do it, and their trading would erase it. All four parity conditions are what is left once that has happened. Each applies the idea to a different kind of price.
| Condition | What arbitrage equalises | Says | Section |
|---|---|---|---|
| Absolute PPP | The price of a basket in two countries | S = P ÷ P* | Absolute PPP |
| Relative PPP | The same, as changes | %ΔS ≈ π − π* | Relative PPP |
| Covered interest parity | Returns at home and abroad, with a forward contract | F ÷ S = (1 + i) ÷ (1 + i*) | Covered parity |
| Uncovered interest parity | The same, using the spot rate you expect | Expected %ΔS ≈ i − i* | Uncovered parity |
Throughout, S is the spot exchange rate in rupees per dollar (the home currency per unit of foreign currency), F the forward rate, i the Indian interest rate, i* the American one, P and P* the Indian and American price levels, and π and π* the two inflation rates. If you have not met the quoting conventions, the page on exchange rates takes five minutes.
Interactive
Absolute PPP and the law of one price
The law of one price says an identical, tradable good costs the same everywhere when measured in one currency. If a tonne of steel were cheaper in Mumbai than in Rotterdam, traders would buy in Mumbai and sell in Rotterdam until prices matched. Absolute purchasing power parity (PPP) applies that to a whole basket:
Absolute PPP
S in rupees per dollar, P the basket’s rupee price, P* its dollar price. Equivalent to saying the real exchange rate equals 1.
The best-known test is the Big Mac, and JAM 2022 gives its data. A burger costs 178 rupees in India and $5.58 in the US. At the market rate of ₹69.69 per dollar the Indian one costs $2.55, which is 54% cheaper. The exchange rate that would equalise the two is 178 ÷ 5.58 = ₹31.90 per dollar. Click through the table.
The data are those printed in JAM 2022 Q50 (January 2019). Pick a country to compare its burger with the American one at the market exchange rate and at the PPP rate.
India
A burger costs 178 in local currency. At the market rate (69.69 per dollar) that is $2.55, which is 54% cheaper than in the US. The PPP rate that would make it cost $5.58 is 31.9 per dollar, against the market's 69.69.
JAM 2022 · moderate (numerical)
Big Mac prices and market exchange rates are given for seven countries. The cheapest price (in US dollars) of a Big Mac is ______ (two decimals).
- Convert each price to dollarsDivide the local price by the rate: Norway 50 ÷ 8.53 = 5.86; Japan 390 ÷ 108.44 = 3.60; Mexico 49 ÷ 17.31 = 2.83; China 20.90 ÷ 6.85 = 3.05; India 178 ÷ 69.69 = 2.55; Russia 110.17 ÷ 66.69 = 1.65.
- Pick the smallestRussia, at $1.65. The US price is already in dollars at 5.58.
Answer
$1.65.
Why PPP is only roughly true
Only tradable goods can be arbitraged. A haircut or a flat cannot be shipped abroad, and these are most of what households buy. Transport costs, tariffs, taxes and different brands add further wedges. This leaves a pattern known as the Balassa-Samuelson effect: in poorer countries wages in tradables are low, and since the same wages apply to the non-tradable sectors, services are cheap, so overall price levels are low compared with the market exchange rate. That is why a rupee buys more in India than its dollar value suggests, and why economists compare incomes at PPP rather than market rates.
Pause and predict
A basket costs ₹3,000 in India and $60 in the US. The market rate is ₹80 per dollar. Compared with absolute PPP, the rupee is
Derivation
Relative PPP: inflation and the exchange rate
Absolute PPP asks for price levels, which are hard to compare. Relative PPP asks only for changes: the exchange rate moves by the difference in the two inflation rates. It can hold when levels differ permanently because of transport costs, since those costs cancel out of a change. The derivation is three steps, and the lab inside it draws the path of the rate year by year.
From the price of a basket to relative PPP
We want to show
If the exchange rate always buys the same basket in both countries, then its percentage change is about the inflation differential: .
- Write absolute PPP at two dates.
Absolute PPP says a basket costs the same in both countries when measured in one currency: S = P ÷ P*, where S is rupees per dollar, P the rupee price of the basket in India and P* its dollar price in the US. Suppose it holds today and a year from now:
- Divide one by the other.
Prices have grown by π in India and π* in the US, so P1 = P0(1 + π) and P1* = P0*(1 + π*). The starting prices cancel and only growth rates are left:
This is relative PPP, exact. It needs no assumption about the levels, only that the same relation holds at both dates.
- Subtract one and simplify.
CUET 2023: India 10%, US 4%. Exact: 1.10 ÷ 1.04 − 1 = 5.77%. Shortcut: 10 − 4 = 6%. The question’s options are 6%, 3%, 14% and 2.5%, so the shortcut is what it wants. When options are close together, use the exact form.
So
A country with higher inflation than its partners must expect its currency to depreciate by about the difference. The explanation is the one the lab shows: if the rate did not move, its goods would get steadily dearer in world markets, and relative PPP says the exchange rate offsets that.
That one result answers a group of questions. CUET 2019: which causes depreciation, according to PPP? An increase in the nation’s own inflation (the other options are the opposite or unrelated). Hyderabad 2015 and 2018 test the same direction: if A’s inflation exceeds B’s, A’s currency weakens. CUET 2024 asks what holding PPP implies, and the true statements are the three in which the exchange rate equals the ratio of price levels, the law of one price holds, and price levels are equal in a common currency.
Worked examples
Comparing incomes at PPP
Market exchange rates exaggerate how rich rich countries are, so economists convert incomes at PPP. Two JNU questions test the method, and both can be done in one line once you see that the PPP rate comes from the basket.
JNU 2019 · moderate
An Indian professional earns ₹2,50,000 a month, and a counterpart in the UK earns £5,000. The nominal rate is £1 = ₹100. A burger costs £3 in the UK and ₹120 in India. Who is better off in effect?
- Get the PPP rate from the basket₹120 ÷ £3 = ₹40 per pound. By purchasing power, £1 buys what ₹40 buys, not ₹100.
- Convert the British salary£5,000 × 40 = ₹2,00,000 in Indian purchasing power.
- Compare₹2,50,000 against ₹2,00,000.
Answer
The Indian professional is better off. At the market rate, £5,000 would look like ₹5,00,000, which is exactly the trap.
JNU 2013 · hard
A$50 = US$1 and B$1 = US$1.5. Petrol costs A$75 a litre in A and B$1 in B. In PPP terms petrol is three times dearer in A than in B. Compare someone earning A$50,000 a month with someone earning B$2,000 a month, in PPP terms.
- Find the PPP rate between the two currenciesPetrol is three times dearer in A at the PPP rate x (A$ per B$): 75 ÷ x = 3 × 1, so x = 25.
- ConvertA$50,000 ÷ 25 = B$2,000.
- CompareB$2,000 against B$2,000.
Answer
They are at par. The market rates (A$75 = US$1.50 and B$1 = US$1.50) show equal petrol prices; the PPP rate is what separates them.
Vocabulary
The forward market, in plain words
The spot rate is for exchange now (two business days is standard). A forward contract fixes today the rate at which currency will be exchanged on a later date, say in three months. The forward premium on the foreign currency is the percentage by which the forward rate exceeds the spot rate, and a negative premium is a forward discount:
Forward premium on the foreign currency
Both rates in home currency per foreign unit. Annualise a premium for a short period by multiplying by the number of such periods in a year.
GATE 2024: the forward rate is 20 and the spot rate 8, so the premium is (20 − 8) ÷ 8 = 1.5, which is 150%. With S = 80 and a three-month F = 81, it is 1.25% for the quarter and 5% annualised.
Four words the papers like to match to their meanings:
- Hedging: avoiding or covering an exchange-rate risk, by taking a forward contract, for instance (UoH 2020: forward rates protect against the risk that the rate changes before a deal is complete).
- Speculation: deliberately accepting exchange-rate risk in the hope of profit. A forward contract can also be used to gamble that a currency will rise or fall, so UoH 2020’s answer is “all of the above”.
- Arbitrage: buying a currency where it is cheaper and selling where it is dearer, with no risk (UGC NET June 2024).
- A swap: a spot sale of a currency combined with a forward repurchase of the same currency, in one transaction (UGC NET June 2023). CUET 2025 matches the others: a forward transaction is an agreement today to buy or sell at a rate agreed today, and futures are the same on an organised market with standardised amounts.
Interactive and derivation
Covered interest parity
Suppose you hold rupees for three months. You can lend them in India, or sell them for dollars, lend in the US, and lock in today, with a forward contract, the rate at which to convert back. Both routes are riskless. If one pays more, money floods into it until prices change to remove the difference. The derivation walks through it in four steps.
Where covered interest parity comes from
We want to show
With no riskless profit available, the forward rate satisfies , so the forward premium on the foreign currency is about .
- Two ways to hold one rupee for a period.
Take ₹1 and a period of, say, three months. Route 1: lend it at home at the period rate i. Route 2: sell the rupee for dollars at the spot rate S, lend the dollars at the American rate i*, and fix today, with a forward contract, the rate F at which those dollars will be turned back into rupees.
Both routes are riskless, because the forward contract removes all exchange-rate uncertainty. Both start with ₹1 and end with rupees.
- Follow each rupee through.
Route 1 ends with 1 + i rupees. Route 2 takes three steps:
Put in UGC NET Dec 2023’s numbers: S = 80, F = 81, and quarterly rates i = 0.75% and i* = 0.25% (3% and 1% a year, divided by four). Route 1 gives 1.0075. Route 2 gives 1.0025 × 81 ÷ 80 = 1.0150. Route 2 wins by 0.75% of the sum invested every three months.
- A difference is an invitation to arbitrage.
Anyone can borrow rupees at i, take route 2 and keep the difference, with no risk and no money of their own. So they do, in huge amounts, and the trades move four prices. Buying dollars spot pushes S up (the rupee weakens on the spot market). Selling dollars forward pushes F down. Money leaving India pushes Indian interest rates up, and money arriving in America pushes American rates down. All four moves narrow the gap, and they stop only when the two routes pay the same.
- Set the routes equal.
The left side is the forward premium on the dollar. For small rates the denominator 1 + i* is close to 1, which gives the exam form: forward premium ≈ i − i*. In the example, the no-profit forward rate is 80 × 1.0075 ÷ 1.0025 = ₹80.40, well below the market’s ₹81, which is why the arbitrage pays.
So
The country with the higher interest rate has a forward discount on its currency, just big enough to cancel its higher rate. Interest rates and the forward rate are two prices of the same thing, and arbitrage ties them together. (Hyderabad 2021 writes it as r = r* + d, where d is the forward discount on the home currency.)
Now play with it. The lab starts with the UGC NET example. Move the forward rate and see where money flows, then use the presets for the other covered question, and the third for the uncovered case.
Route 1: stay in India
₹1lend at home: × (1 + i) = × 1.0075₹1.0075
Route 2: go through dollars, covered by a forward contract
₹1÷ 80$0.01250× 1.0025$0.01253× F = 81₹1.015
Home return, per quarter0.75%
Covered return abroad1.503%
Forward premium on $, per year5%
Forward rate with no profit₹80.399
Gain per ₹100, per quarter₹0.753
Which way does the money go?
Lending in the US pays more in rupees than lending at home (6.01% against 3% a year), so money leaves India. Arbitrageurs buy dollars spot (the rupee weakens in the spot market), sell dollars forward (pulling the forward rate down), Indian interest rates are pushed up and American ones down, until the gap closes.
The parity condition
Covered parity needs F = S(1 + i)/(1 + i*) = 80.399. The market forward rate is 81. The shortcut is the premium equals the interest differential: 5% against 2%.
Reading the exam versions
- Hyderabad 2021 writes covered parity as r = r* + d, where d is the forward discount on the home currency. So a higher rate at home goes with a forward discount on the home currency of exactly the size of the gap. It is the same condition as a forward premium on the foreign currency equal to i − i*.
- CUET 2023: the US interest rate is 3% a year, spot ₹80, three-month forward ₹81. The return on a US bond in rupee terms is 3% plus the forward premium of 1/80 × 4 = 5%, which is 8%.
- DSE 2008 (Delhi School level): if covered interest parity holds, expected currency depreciation equals the interest differential plus the risk premium. Covered parity fixes the forward premium at i − i*. Expected depreciation differs from the forward premium by whatever extra reward investors need for bearing the risk, so expected depreciation = (i − i*) + risk premium.
Expectations
Uncovered interest parity
Skip the forward contract and use the rate you expect. Now the foreign return is risky, so the condition holds only if investors do not care about that risk (otherwise a risk premium enters, as in the DSE question). Under that assumption, the currency with the higher interest rate must be expected to depreciate, by about the interest differential:
Uncovered interest parity
Expected change in the rupee price of a dollar over the period, approximately.
Why? If India paid more and the rupee were not expected to weaken, everyone would hold rupees, pushing the rate up now until holding rupees looks no better than holding dollars. JNU 2021 puts it in words: if the interest rate is higher in X than in Y, the currency of Y is expected to appreciate relative to X.
UGC NET Dec 2023 · hard
India’s rate is 6%, the US 4%. The dollar is ₹80 today and is expected to rise to ₹84 in a year. According to uncovered interest arbitrage parity, which is true?
- Return at home6%.
- Expected return in rupees on a US deposit1.04 × 84 ÷ 80 − 1 = 9.2%. The dollar is expected to rise 5%, on top of the 4% interest.
- Compare9.2% against 6%: the US deposit is expected to pay more, so capital flows out of India. Parity would need an expected rise of only about 2%.
Answer
Capital will flow out of the Indian economy.
A rate rise: the jump and the drift
It seems to contradict an earlier statement. Raising an interest rate attracts capital and makes the currency appreciate at once (Hyderabad 2019: rupee appreciates against the yen when Indian real interest rates rise relative to Japan’s). Yet uncovered parity says the currency with the higher rate is expected to depreciate afterwards. Both are true, one after the other: the currency jumps up when the news arrives, then drifts down, and the drift is what offsets the higher rate for investors. A question that says “expected to appreciate” after a rate rise (Hyderabad 2018) contradicts JNU 2021’s key, so we have left it out of the practice set.
Exam focus
Numeric traps and the mistakes examiners are fishing for
Watch out for
The mistakes examiners are fishing for
- Annualising the wrong way. A premium or a rate quoted for three months is multiplied by four, not by three, to compare with an annual interest rate (CUET 2023, UGC NET Dec 2023).
- Putting the exchange rate the wrong way up. Every formula here takes S as home currency per foreign unit. If the question quotes dollars per rupee, invert first.
- Using the premium on the wrong currency. A premium on the dollar is a discount on the rupee. Check which one is meant before you add or subtract.
- Using approximations when options are close. 10% against 4% inflation gives 5.77% exactly and 6% by the shortcut. If the options are 5.8 and 6, compute exactly.
- Forgetting that a good return in foreign currency can vanish when converted. CUET 2024: an investment yielding 10% in rupees while the rupee falls 10% (₹80 to ₹88) yields exactly 0% in dollars, since 1.10 ÷ 1.10 − 1 = 0.
- Confusing covered with uncovered. Covered uses the forward rate and is enforced by arbitrage. Uncovered uses an expectation and is a theory about behaviour.
- Treating absolute PPP as a law of nature. GATE 2024 gives a list of false statements: the “arbitrage pricing theory” (not the law of one price) is blamed for equal prices, and interest rates on similar assets are not always equal, since parity relates them to the exchange rate.
JAM 2024 · moderate (numerical)
An Indian company borrowed $300 at the start of the year at ₹70 per dollar and repaid it in dollars with 12% annual interest. With zero inflation in both countries, the real cost of borrowing is zero if the exchange rate at repayment is ₹______ per dollar.
- What was borrowed, in rupees300 × 70 = ₹21,000.
- What is repaid, in dollars300 × 1.12 = $336.
- Real cost zero means the same rupees go back336 × e = 21,000, so e = 62.5.
Answer
₹62.5 per dollar. That is a 12% appreciation of the rupee (70 ÷ 62.5 = 1.12), which exactly cancels the 12% interest.
Back to the puzzle
The opening question, solved
UGC NET Dec 2023 · hard
Rates are 3% in India and 1% in the US; spot ₹80, 3-month forward ₹81. Which are true of covered interest arbitrage flows?
- Compare covered returnsHome: 3% ÷ 4 = 0.75% a quarter. Abroad, covered: (1.0025) × 81 ÷ 80 − 1 = 1.50% a quarter. Abroad wins, so money leaves India.
- Trace the tradesArbitrageurs sell rupees and buy dollars on the spot market: the rupee depreciates spot (A true). They sell those dollars forward, which pushes the forward rate down, so the rupee appreciates forward (B false).
- Trace the interest ratesMoney leaving India raises Indian interest rates (C false). Money arriving in the US lowers American rates (D true).
Answer
(A) and (D) only.
Revision
The page on one screen
The lesson on one screen
- Arbitrage
- A riskless profit. The four parity conditions are what is left after it has been traded away.
- Absolute PPP
- S = P ÷ P*. The real exchange rate is 1. Fails for non-tradables; poor countries look cheap (Balassa-Samuelson).
- Relative PPP
- S₁ ÷ S₀ = (1 + π) ÷ (1 + π*), so %ΔS ≈ π − π*. Higher inflation at home means depreciation.
- PPP income
- Convert at the basket’s rate, not the market rate.
- Forward premium
- (F − S) ÷ S on the foreign currency. Annualise by multiplying by periods per year.
- Covered parity
- F ÷ S = (1 + i) ÷ (1 + i*). Premium ≈ i − i*. Enforced by arbitrage.
- Uncovered parity
- Expected %ΔS ≈ i − i*. The higher-rate currency is expected to depreciate. A rate rise first appreciates the currency, then it drifts down.
Practice
Check yourself
Numbers
Use the exact form unless the question asks for the shortcut.
Relative PPP
Expected inflation is 8% in India and 3% in the US. Using the shortcut, by how much is the rupee expected to depreciate against the dollar? (Percent.)
Absolute PPP
A basket costs ₹3,000 in India and $60 in the US. What exchange rate does absolute PPP imply, in rupees per dollar?
Forward premium
The spot rate is ₹80 per dollar and the three-month forward rate is ₹82. What is the forward premium on the dollar, annualised by multiplying the quarter’s figure by 4? (Percent.)
Covered parity
Spot ₹80 per dollar, one-year interest rates 7% in India and 3% in the US. What one-year forward rate satisfies covered parity? (₹ per $, two decimals.)
Uncovered return
India pays 6%, the US 4%. The spot rate is ₹80 and you expect ₹84 in a year. What is the expected rupee return on a one-year US deposit? (Percent, one decimal.)
Which condition?
1Inflation in India rises relative to inflation in Canada. According to relative PPP,
2Under uncovered interest parity, if the interest rate is higher in country X than in country Y,
3A forward contract removes
4Absolute PPP is equivalent to saying that
Past papers
Past papers, by level
Graded by each paper’s own difficulty. Questions with a key that conflicts with another paper’s (Hyderabad 2018 on the rate rise) or an ill-defined formula (UGC NET 2026) are left out; the reasons are listed at the top of the source file.
Foundation
Definitions, directions and one-step sums. Most CUET, UGC NET and Hyderabad questions sit here.
5 questionsPractise →
Standard
Classification traps, multi-step numericals and statement questions, up to the hardest JNU, JAM and GATE items.
18 questionsPractise →
Delhi School level
The Delhi School entrance asks this topic harder than any other exam. These questions are kept apart from the ordinary sets.
1 questionTry it →
All 23 past-paper questions on ppp and interest rate parity, covered and uncovered, easiest first: practise the full set →
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
Quick answers
Quick answers
What is the difference between absolute and relative PPP?
Absolute PPP says the exchange rate equals the ratio of price levels, S = P/P*, so a basket costs the same in both countries when measured in one currency. Relative PPP is the weaker version about changes: the exchange rate changes by the difference in inflation rates, so the home currency depreciates by about π − π*. Relative PPP can hold even when absolute PPP does not.
What is covered interest parity?
Covered interest parity says that, once the exchange-rate risk is removed with a forward contract, lending at home and lending abroad must earn the same return. That requires the forward premium on the foreign currency to equal the interest differential: F/S = (1 + i)/(1 + i*), approximately (F − S)/S = i − i*. If it fails, riskless arbitrage profits exist.
What is the difference between covered and uncovered interest parity?
Covered parity uses the forward rate fixed today, so there is no risk and the condition is enforced by arbitrage. Uncovered parity uses the spot rate you merely expect, so it involves risk and holds only if investors are indifferent to that risk. Uncovered parity says the currency with the higher interest rate is expected to depreciate by about the interest differential.
How do you calculate the forward premium?
The forward premium on the foreign currency is (F − S)/S, with both rates in home currency per unit of foreign currency. With a spot rate of 80 and a three-month forward rate of 81 it is 1/80 = 1.25% for the quarter, or 5% a year when annualised by multiplying by four. GATE 2024 gives a forward rate of 20 against a spot of 8, so the premium is 12/8 = 1.5, or 150%.
Why does PPP fail in practice?
Because many goods and most services are not traded, transport costs and trade barriers separate prices, firms price differently in different markets, and price indices contain different baskets. Poorer countries also tend to have lower price levels for services, which is the Balassa-Samuelson effect, so market exchange rates understate their purchasing power.
How do you compare incomes across countries using PPP?
Convert one income into the other currency at the PPP rate, the rate that makes a standard basket cost the same, instead of the market rate. If a burger costs ₹120 in India and £3 in the UK, the PPP rate is ₹40 per pound, so a £5,000 salary is worth ₹2,00,000 in Indian purchasing power, less than a ₹2,50,000 Indian salary even though the market rate of ₹100 per pound says the opposite.
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