A currency has two prices at once: the spot rate today, and the forward rate you lock in now for a year ahead. Interest rates tie the two together by pure arbitrage. Prices tie the rate to inflation instead, but only loosely and only in the long run. Knowing which relation is airtight and which is a tendency separates a clean exchange-rate answer from a hand-wave — and it brings second-years to an economics tutor in Paris.
1 · From the law of one price to purchasing power parity
Start with a single good. The law of one price says an identical, freely traded good costs the same everywhere once you convert currencies. Quote the rate as the market quotes EUR/USD: S dollars per euro. If a good costs P€ euros in the euro area and P$ dollars in the US, the law reads P$ = S × P€; any gap is an arbitrage.
Absolute PPP stacks that across a whole basket: S = P$ / P€, the rate as the ratio of price levels. A basket at $120 in the US and €100 in the euro area gives a PPP rate of 120 / 100 = $1.20 per euro.
Relative PPP is weaker but survives the data. It keeps only the rates of change: the rate moves with the inflation gap, %ΔS ≈ π$ − π€. The higher-inflation currency depreciates, its money losing value faster.
2 · The real exchange rate, and why PPP only anchors the long run
The real exchange rate strips out the money illusion: q = S × P€ / P$, the foreign basket priced in home baskets. Under absolute PPP q = 1. When q ≠ 1, PPP is violated. At a market rate of $1.26 against the $1.20 PPP rate, q = 1.26 × 100 / 120 = 1.05 — euro-area goods are 5% dearer in real terms, so the euro is overvalued.
Why does PPP fail outside the long run? Non-tradables. A haircut or a flat cannot be shipped to close a price gap, so such prices drift apart and drag the basket with them. Add transport costs and pricing-to-market, and even traded goods obey the law only loosely. Deviations shrink over years.
One deviation is systematic — Balassa–Samuelson. A rich economy is far more productive in tradables than non-tradables, which lifts wages economy-wide, raises non-tradable prices, and pushes up its whole price level. So richer countries are dearer, and PPP understates them predictably.
3 · Covered interest parity: pricing the forward
Now add interest rates and a forward contract — a deal struck today to swap currencies at a fixed rate F in one year. Covered interest parity is pure arbitrage, so it holds almost exactly. Keep the convention: S and F are dollars per euro.
Start with one euro and ask for dollars in a year. Two routes. Invest, then cover: deposit at the euro rate for (1 + i€) euros, then sell forward at F for F(1 + i€) dollars. Convert, then invest: sell spot for S dollars, deposit at the dollar rate for S(1 + i$) dollars. Both are riskless from the same euro, so they end equal:
F(1 + i€) = S(1 + i$), so F = S (1 + i$) / (1 + i€).
Read it off. If the dollar pays more (i$ > i€), then F > S: a forward premium on the euro, a discount on the dollar. The extra dollar interest is handed back through a weaker forward — nothing left on the table, as the rectangle in the figure shows.
4 · Uncovered interest parity and the carry trade
Covered parity uses a locked forward, so it is airtight. Uncovered interest parity swaps that forward for a guess — the expected future spot rate E[Sʹ] — and assumes investors hold either currency:
E[Sʹ] / S = (1 + i$) / (1 + i€).
So the higher-interest currency is expected to depreciate by roughly the interest gap — high yield offset by expected capital loss, no free lunch. The data disagree: higher-interest currencies often hold or even rise. That is the forward premium puzzle, and it powers the carry trade — borrow a low-interest currency, invest in a high-interest one, pocket a gap that should vanish. The catch is crash risk: carry unwinds violently when funding currencies snap back, so the average return looks like payment for risk, not arbitrage. (Whether a shared currency removes this margin is a separate question — optimal currency areas.)
Worked example — a covered arbitrage in euros and dollars
Step 1 — The fair forward. Spot S = $1.26 per euro, euro rate 5%, dollar rate 8% (one-year). Covered parity pins the forward: F* = 1.26 × 1.08 / 1.05 = $1.2960, a slight euro premium.
Step 2 — The mispricing. A dealer instead quotes F = $1.35, above fair value — the euro is too dear forward, so there is money to take.
Step 3 — The trade. Borrow $1,260,000 for a year at 8%; at maturity you owe 1,260,000 × 1.08 = $1,360,800.
Step 4 — Convert and invest. Change the loan to euros at spot: 1,260,000 / 1.26 = €1,000,000. Deposit at 5%: 1,000,000 × 1.05 = €1,050,000 in a year.
Step 5 — Cover forward. Lock their sale forward at 1.35: 1,050,000 × 1.35 = $1,417,500 in a year. Every leg is now riskless.
Step 6 — Resolve. Repay the loan, keep the rest: 1,417,500 − 1,360,800 = $56,700, riskless, on no capital of your own.
Step 7 — Interpretation. The profit exists only because F ≠ F*. At the fair forward $1.2960 the euros deliver exactly $1,360,800 — the loan — so profit is $0. Trading like this drags F back to parity, which is why covered parity holds.
Can you walk the four legs of a covered trade and show why the profit vanishes the moment the forward returns to parity? Deriving the forward from the no-arbitrage rectangle — rather than quoting the formula — is what separates a clean exchange-rate answer from a hand-wave. A one-on-one economics tutor drills covered against uncovered parity and the honest limits of PPP with you until they come out cleanly under exam pressure. Book a trial session.
Practice
Q1. Spot S = $1.04 per euro, one-year euro rate 4%, dollar rate 7%. (a) Find the no-arbitrage forward. (b) A bank instead quotes $1.10; borrowing $520,000, what riskless profit does the covered trade earn?
Q2. US inflation is 5% over the year, euro-area inflation 2%, spot $1.26 per euro. Using relative PPP, predict next year’s spot and say which currency depreciates.
Answers. Q1: (a) F = 1.04 × 1.07 / 1.04 = $1.0700, a euro forward premium. (b) Owe 520,000 × 1.07 = $556,400; convert to €500,000; grow to €520,000; sell forward at 1.10 for $572,000; profit = $15,600. Q2: %ΔS ≈ 5% − 2% = 3%, so Sʹ ≈ 1.26 × 1.03 = $1.2978 — the dollar depreciates, eroded faster by inflation.
Key takeaways
- Absolute PPP sets S = P$ / P€; relative PPP keeps only the inflation gap, %ΔS ≈ π$ − π€. The higher-inflation currency depreciates.
- The real exchange rate q = S P€ / P$ equals 1 under PPP. Non-tradables keep it off 1 — Balassa–Samuelson makes richer countries dearer systematically — so PPP is a slow long-run anchor, not a spot forecast.
- Covered interest parity is arbitrage: F = S(1 + i$) / (1 + i€). The higher-interest currency trades at a forward discount that cancels its yield edge exactly.
- Uncovered interest parity swaps the forward for an expectation and fails empirically — the forward premium puzzle — so the carry trade earns a risky, crash-prone return.
Why Paris students choose our economics tutoring
- Parity conditions built, not quoted: sessions derive the forward from the no-arbitrage rectangle, so you can reconstruct it under exam pressure instead of memorising a formula.
- The distinctions examiners reward, drilled: covered versus uncovered parity, absolute versus relative PPP, nominal versus real exchange rate — the pairs that decide a mark.
- One-on-one and matched to your course: a tutor works from your own notation and past papers, whether your module follows Krugman and Obstfeld, Feenstra and Taylor, or Burda and Wyplosz.
FAQ
Q: Is the forward rate the market’s forecast of the future spot rate?
A: Not under covered parity. The forward is fixed by today’s spot and the two interest rates — arbitrage, not prediction. It matches the expected future spot only if uncovered parity holds too, which the data rarely support.
Q: Why does the higher-interest currency trade at a forward discount?
A: Otherwise there is free money: you could borrow the low-yield currency, invest in the high one, hedge with a forward, and profit for certain. The discount is exactly the size that removes that profit.
Q: What is the difference between absolute and relative PPP?
A: Absolute PPP fixes the level of the rate as the price-level ratio, so a basket costs the same everywhere. Relative PPP ties only its change to the inflation gap, and holds far better in the data.
Q: If PPP is so often wrong, why learn it?
A: Because it is the long-run anchor: over years, currencies drift toward their inflation differentials and large misalignments correct. It is a poor forecast for next quarter and a good discipline for the next decade.
Q: What exactly is the carry trade?
A: Borrowing a low-interest currency to invest in a high-interest one, unhedged. Uncovered parity says the high-yielder should depreciate and erase the gain; it often does not, so carry earns a positive average return, with sharp sudden losses.
Book an economics tutor in Paris or online
Exchange rates reward the student who can derive the parity condition, not just name it — the rectangle, the forward, and the honest limits of PPP. One-on-one sessions build that fluency on your own past papers and notation. Tell us your course and exam date, and we will match you with the right tutor this week.
