Interest Rate Parity

The arbitrage that pins the forward rate

Suppose you have one dollar today and want one dollar in one year. You have two choices.

1. Stay home. Lend at the US one-year rate i_d. End with (1 + i_d) dollars.
2. Round-trip through Japan. Convert at the spot rate S (yen per dollar) to get S yen; lend in Japan at i_f, ending with S(1 + i_f) yen; sell the yen forward today at rate F, locking in $S(1 + i_f) / F$ dollars in one year.

Both strategies are riskless — strategy 2 only because the forward locks in the conversion now. If they didn't yield identically, an arbitrageur would borrow in the cheaper one and lend in the richer one until the gap closed. No-arbitrage demands:

(1 + i_d) = (S / F) · (1 + i_f)

which rearranges to the canonical form:

F / S = (1 + i_d) / (1 + i_f)

This is covered interest parity (CIP). The "covered" qualifier means the forward exchange rate has been contracted in advance — there is no FX risk in the round-trip.

A useful approximation when interest rates are small: F/S ≈ 1 + (i_d − i_f), so the forward premium (F − S)/S equals the interest-rate differential.

Worked example: USD vs JPY one-year

Take a snapshot:

• Spot S = 150 yen per dollar.
• One-year US rate i_d = 5.0% (Treasury or USD LIBOR-equivalent).
• One-year Japanese rate i_f = 0.5%.

Covered IRP says the one-year forward must satisfy F/S = (1.050) / (1.005) ≈ 1.04478. With S = 150, that gives F ≈ 156.72 yen per dollar.

The forward yen is weaker than the spot yen — a forward discount on the yen, equivalent to a forward premium on the dollar. Why? Because lending in yen pays less; to make the round-trip break even, the dollar you get back at the forward must be cheaper to buy in yen, i.e., one dollar must cost more yen forward than it does spot.

If a market-maker quoted F = 154 yen per dollar instead, the trade would be: borrow $1 at 5%, convert to 150 yen, lend in Japan at 0.5% to get 150.75 yen in a year, convert back at F = 154 to get $0.9789. Owe $1.05. Loss. Reverse direction: borrow 150 yen at 0.5%, convert to $1, lend at 5% to get $1.05, convert back at F = 154 to get 161.7 yen, owe 150.75 yen. Profit of 10.95 yen per dollar of notional, riskless. Arbitrageurs flood in until F drifts up to ~156.72.

Covered vs uncovered IRP

The forward F is a contractually locked-in price. The expected future spot E[S_{t+1}] is just a guess about where the market will be in a year. Equating them gives uncovered interest parity (UIP):

E[S_{t+1}] / S_t = (1 + i_d) / (1 + i_f)

UIP is a stronger claim. It says the expected appreciation of the foreign currency just offsets its lower interest rate, so investors are indifferent between currencies at the prevailing rates. That requires investors to be risk-neutral about exchange-rate risk and to have rational expectations.

	Covered IRP (CIP)	Uncovered IRP (UIP)
Equation	F/S = (1+i_d)/(1+i_f)	E[S']/S = (1+i_d)/(1+i_f)
Variable on LHS	Forward rate F (contractual)	Expected future spot E[S']
Risk in the trade	None — forward locks the rate	FX risk (can't lock in expectations)
Why it should hold	Pure arbitrage	Risk-neutrality + rational expectations
Empirical fit, pre-2008	Held to within bid-ask	Routinely violated; carry trade earns positive returns
Empirical fit, post-2008	Persistent basis 20-50 bp	Still violated, but carry returns lower and more volatile
Famous diagnostic	Cross-currency basis swap	Forward premium puzzle / Fama regression

The forward premium puzzle

Eugene Fama (1984) ran a simple regression: future spot change Δs_{t+1} on the forward premium (F_t − S_t)/S_t. UIP predicts the slope coefficient is +1. Across major currency pairs, Fama found slopes that were typically negative — usually around −2 to −4. High-yield currencies, far from depreciating to wipe out their carry, tended to appreciate on top of paying higher interest. The carry trade was profitable in expectation.

The literature offers three families of explanations:

• Risk premia. High-yield currencies pay a premium for crash risk. They earn returns in normal times but crash hard in stress (the AUD/JPY plunge in 2008). On a probability-weighted basis with risk aversion, the puzzle softens.
• Peso problems. Investors price in rare large devaluations that haven't happened in the sample. The data underweights the tail outcomes.
• Limits to arbitrage. Even risk-neutral arbitrageurs face leverage limits and value-at-risk constraints, especially around regulatory dates.

Real-world institutions and instruments

• FX swaps. The instrument that enforces CIP. Two parties exchange currencies at the spot rate today and reverse at the forward rate at maturity. Daily turnover roughly $4 trillion (BIS Triennial 2022) — the largest single FX instrument.
• Cross-currency basis swap. A longer-dated FX swap that exchanges interest payments in two currencies. The "basis" is the spread quoted on top of one leg's reference rate that makes the swap fair. A non-zero basis is the direct measure of CIP failure. The EUR-USD basis hit −80 basis points at the 2011 European bank funding crunch and again −50 bp in March 2020.
• Carry-trade vehicles. Retail products like FXCM "yield" portfolios in the 2000s; institutional carry baskets sold by major banks; ETFs such as DBV. Long-run pre-fee returns 5-7% with Sharpe ratios near 0.6 — but with severe drawdowns in 2008 and 2020.
• NDFs (non-deliverable forwards). For currencies with capital controls (CNY, INR, KRW), a forward settles in dollars based on the official fixing. The NDF rate prices the offshore market's IRP-implied expectation; gaps from onshore forwards are a direct measure of capital-control friction.
• Onshore vs offshore yuan. CNY (onshore) and CNH (offshore) trade with persistent gaps because cross-border arbitrage is restricted. Both anchor to PBOC fixings but with very different IRP-implied yields.
• Basel III leverage ratio. The supplementary leverage ratio (SLR) made cross-currency arbitrage on bank balance sheets capital-expensive. Quarter-end and year-end CIP deviations widen as banks window-dress balance sheets — a regulatory artefact, not a market signal.

Variants and refinements

• Continuous-compounded form. F/S = exp((r_d − r_f) · T), where r_d, r_f are continuously compounded yields and T is the maturity. Algebraically cleaner; standard in derivatives pricing.
• Real interest rate parity. Replaces nominal rates with real (inflation-adjusted) rates and predicts the real exchange rate to follow real interest differentials. Holds even more loosely than nominal UIP.
• International Fisher Effect. Combines UIP with PPP: nominal-interest differentials equal expected inflation differentials equal expected currency depreciation. Mostly a textbook construct; empirically weak.
• Cross-currency basis as a bank-funding signal. When the basis widens against the dollar, it indicates dollar funding stress in non-US banks. Central-bank swap lines (the Fed-ECB-BoJ network) are deployed precisely to compress this basis.
• Capital-control corrections. In economies with restrictions, IRP holds within local markets but not across the onshore/offshore boundary. The onshore-offshore spread is the price of the control.
• Sticky-price extensions. Models with menu costs and infrequent price-resetting can generate persistent UIP failures consistent with the data — a research thread led by Engel, Alvarez-Atkeson-Kehoe, and Itskhoki-Mukhin.

Common pitfalls

• Confusing CIP with UIP. CIP uses the forward rate F (locked in); UIP uses the expected future spot E[S']. CIP holds tightly by arbitrage; UIP routinely fails. Mixing them is the most common conceptual error in the topic.
• Treating the forward rate as a forecast. The forward rate is whatever no-arbitrage requires given S, i_d, i_f. It is the median forecast only under risk-neutrality, which is empirically false. Bloomberg's forward-implied "expected spot" is therefore not a real expectation.
• Confusing direction of the rate convention. Quote F and S consistently. If S = 150 yen per dollar, then i_f is the yen rate and i_d is the dollar rate; flipping either flips F. Half the textbook errors are sign errors here.
• Ignoring transaction costs. Bid-ask spreads on spot, forward, and money-market quotes can fully account for small CIP deviations of 5-10 bp.
• Using LIBOR-style rates with secured forwards. Pre-2008, banks borrowed unsecured at LIBOR and the IRP held. Post-2008, many counterparties post collateral; the relevant rate becomes the secured rate (OIS, SOFR), and using LIBOR creates a fictitious "CIP gap."
• Believing the carry trade is free money. Carry returns Sharpe ratios are decent in calm periods but the strategy carries severe negative skew. Most years it earns 5-8%; in a crisis it can lose 25% in weeks.