Interest Rate Parity (IRP): Concepts, Formulas, and Arbitrage

Interest Rate Parity (IRP) — definition

An arbitrage condition that must hold when international financial markets are in equilibrium; it's a manifestation of the Law of One Price applied to international money market instruments, and it links interest rates in two different countries.

Setup: two ways to invest $1 for one year

(1) Invest in a U.S. dollar security at the dollar interest rate i$. (2) Invest in a British pound security at the pound rate i£, and hedge the exchange risk by selling the maturity value of the pound investment forward.

Route 1 — maturity value of investing $1 at home

$1(1 + i$).

Route 2 — the three steps to invest $1 abroad (hedged)

(1) Exchange $1 for £(1/S) at the spot rate S. (2) Invest the pound amount at i£, maturing to £(1/S)(1+i£). (3) Simultaneously sell that maturity value forward at rate F, locking in a predetermined dollar amount.

Route 2 — the "effective" dollar interest rate formula

(F/S)(1 + i£) − 1 — this is what the hedged foreign investment actually yields once redenominated back into dollars.

Why Route 2 is fully hedged / riskless

Because the investor delivers exactly the pound amount owed on the forward contract, the net pound position at maturity is zero — the investor is assured of receiving a predetermined DOLLAR amount, just like Route 1.

IRP Formula — Equation 6.1

(1 + i$) = (F/S)(1 + i£), or equivalently F = S[(1+i$)/(1+i£)] — since Routes 1 and 2 require the same investment, same risk (zero), and same horizon, arbitrage equilibrium forces their dollar payoffs to be equal.

IRP in "foreign-currency-per-dollar" terms

If S and F are quoted as units of foreign currency per dollar (instead of dollar price of foreign currency), IRP becomes (1+i$) = (S/F)(1+i£).

Historical note on IRP

The relationship has been known among currency traders since the late 19th century, but it only became widely known to the public through the writings of John Maynard Keynes and other economists in the 1920s.

Arbitrage portfolio — definition

A portfolio built to have (1) no net investment and (2) no risk; market equilibrium then requires that it generate no net cash flow at maturity — this is an alternative way to derive IRP.

Arbitrage portfolio — the three legs used to derive IRP

(1) Borrow $S (just enough to buy £1 at the spot rate). (2) Lend that £1 at the pound interest rate. (3) Sell the maturity value of the pound investment forward.

IRP Formula — Equation 6.2 (arbitrage-portfolio form)

(1+i£)F − (1+i$)S = 0 — the first term is the dollar proceeds from the pound loan at maturity; the second is the dollar amount needed to repay the original loan. Rearranging gives the same result as Equation 6.1.

Why the arbitrage portfolio's cash flow at t=0 is zero

Borrowing $S to fund buying/lending £1 is exactly self-financing — it costs nothing out of pocket to hold the portfolio.

Why the arbitrage portfolio's maturity cash flow must be zero in equilibrium

None of S, F, i$, or i£ is uncertain, so the maturity payoff is known with certainty; if it weren't zero, someone could earn a riskless, costless profit — which can't persist in equilibrium.

IRP Formula — Equation 6.3 (general foreign-rate version)

(1+i$) = (F/S)(1+i*), or F = S[(1+i$)/(1+i*)] — identical to Equation 6.1, just written with a generic foreign interest rate i* instead of i£.

IRP Approximation — Equation 6.4

(i$ − i*) ≅ (F − S)/S — the interest rate differential approximately equals the forward premium or discount; this simplified form is often quoted in practice.

Forward discount ⇒ what it implies about interest rates

If the dollar is at a forward discount (F > S, meaning the dollar is expected to depreciate against the foreign currency), the dollar interest rate must be HIGHER than the foreign rate — otherwise nobody would hold dollar securities.

Forward premium ⇒ what it implies about interest rates

If the dollar is at a forward premium (F < S), the dollar interest rate must be LOWER than the foreign rate.

What Equation 6.1 implies about F vs. S whenever rates differ

The forward rate will deviate from the spot rate any time the two countries' interest rates are not equal to each other.

Covered Interest Arbitrage (CIA) — definition

Borrowing at one interest rate and simultaneously lending at another, with the exchange risk fully covered by a forward hedge — this is the mechanism that enforces IRP whenever it's violated.

Decision rule when IRP does NOT hold

If (1+i$) > (F/S)(1+i£): better off investing in dollar securities. If (1+i$) < (F/S)(1+i£): better off investing in pound securities (hedged). When borrowing, choose whichever currency has the lower effective cost.

Example 6.1 — the given data

i$ = 5%/yr, i£ = 8%/yr, spot S = $1.80/£, one-year forward F = $1.78/£. The arbitrager can borrow up to $1,000,000 or its spot equivalent, £555,556.

Example 6.1 — Step 1: checking whether IRP holds

Compute [F/S](1+i£) = (1.78/1.80)(1.08) = 1.068 and compare to (1+i$) = 1.05. Since 1.068 ≠ 1.05 — specifically (1+i$) < [F/S](1+i£) — IRP is violated and a profitable arbitrage exists.

Example 6.1 — which direction to arbitrage, and why

Because the dollar side is relatively "cheap" and the hedged pound side pays relatively more, the arbitrager should borrow in the United States (low effective cost) and invest in the U.K. (high effective return).

Example 6.1 — the four CIA transactions

(1) Borrow $1,000,000 (owe $1,050,000 in one year). (2) Convert to £555,556 at the spot rate. (3) Invest £555,556 at 8% → matures to £600,000. (4) Sell £600,000 forward at $1.78/£, locking in $1,068,000.

Example 6.1 — the arbitrage profit

$1,068,000 (forward-contract proceeds) − $1,050,000 (loan repayment) = $18,000 — realized with zero net investment and zero risk.

Example 6.1 — the profit shortcut formula

Arbitrage profit = (effective interest rate differential) × (amount borrowed) = (1.068 − 1.05) × $1,000,000 = $18,000.

Exhibit 6.1 — Covered Interest Arbitrage cash flow table

Shows the transaction is fully self-financing: net cash flow at the START is $0 (borrowing exactly funds the pound purchase/investment), and net cash flow AT MATURITY is +$18,000 — proof the profit is riskless.

How mass CIA activity restores IRP (Example 6.1's adjustment)

As everyone rushes to do the same trade: (1) the U.S. interest rate rises, (2) the U.K. interest rate falls, (3) the pound appreciates in the spot market, (4) the pound depreciates in the forward market — narrowing the gap until IRP holds again.

Exhibit 6.2 — the Interest Rate Parity Diagram

Plots the forward premium/discount (F−S)/S on the vertical axis against the interest rate differential (i$−i£) on the horizontal axis; the IRP line runs through the origin at a 45° slope. Any point off the line signals a CIA opportunity.

Exhibit 6.2 — reading point A (from Example 6.1)

At point A: interest rate differential = i$ − i£ = 5% − 8% = −3%; forward premium = (F−S)/S = (1.78−1.80)/1.80 = −1.11%. The point sits well off the IRP line, and a dotted arrow traces the path back toward it as CIA unwinds the mispricing.

Example 6.2 — the given data

3-month interest rates: 8.0%/yr in the U.S., 5.0%/yr in Germany. Spot rate €0.8000/$, 3-month forward rate €0.7994/$ (quoted in European terms). The arbitrager can borrow $1,000,000 or €800,000.

Example 6.2 — why you must convert the quotes first

Equation 6.1 requires the DOLLAR price of the foreign currency (American terms) and PERIOD-matched (not annualized) interest rates — so the €/$ quotes must be inverted, and the annual rates divided by 4 for a 3-month horizon.

Example 6.2 — the converted inputs

i$ = 8.0/4 = 2.0%; i€ = 5.0/4 = 1.25%; S = 1/0.8000 = $1.2500/€; F = 1/0.7994 = $1.2509/€.

Example 6.2 — Step 1: checking whether IRP holds

[F/S](1+i€) = (1.2509/1.2500)(1.0125) ≈ 1.0132, versus (1+i$) = 1.0200. Since 1.0132 < 1.02, the euro side is relatively "cheap" to borrow — so the arbitrage should involve borrowing in Germany and lending in the U.S.

Example 6.2 — the four CIA transactions

(1) Borrow €800,000 (owe €810,000 in 3 months at 1.25%). (2) Convert €800,000 to $1,000,000 at spot. (3) Invest $1,000,000 in the U.S. at 2.0% → matures to $1,020,000. (4) Buy €810,000 forward (needed to repay the euro loan) for $1,013,229.

Example 6.2 — the arbitrage profit

$1,020,000 (U.S. investment proceeds) − $1,013,229 (cost of buying back the euros owed) = $6,771.

How mass CIA activity restores IRP (Example 6.2's adjustment)

As everyone does the same trade: the German interest rate rises, the U.S. interest rate falls, the dollar appreciates spot, and the dollar depreciates forward — IRP is restored partly by a bigger forward premium and partly by a smaller interest differential (illustrated as point B in Exhibit 6.2).

IRP and the Spot Rate — Equation 6.6

S = [(1+i£)/(1+i$)] × F — given the forward rate, the spot rate is pinned down by relative interest rates.

What Equation 6.6 implies about a rate hike

All else equal, a HIGHER U.S. interest rate leads to a stronger (higher foreign-exchange value) dollar today, because higher rates attract capital inflows, boosting dollar demand; a lower U.S. rate weakens the dollar.

Forward Rate as Market Forecast — Equation 6.7

F = E(St+1 | It) — if FX markets are efficient, today's forward rate is the market's best conditional forecast of the future spot rate, given all currently available information It (money supplies, interest rates, trade balances, etc.).

Combining Eq. 6.6 and 6.7 — Equation 6.8

S = [(1+i£)/(1+i$)] × E(St+1 | It) — the current spot rate depends on relative interest rates AND the market's expectation of the future spot rate.

Two big takeaways from Equation 6.8

(1) Expectations are SELF-FULFILLING: if people expect the rate to rise, it rises now. (2) Exchange rates are driven by continuous NEWS updates, so they behave in a dynamic, volatile way in the short run, since news is by definition unpredictable.

Uncovered Interest Rate Parity — Equation 6.9

(i$ − i£) ≈ E(e), where E(e) = [E(St+1) − St]/St — obtained by replacing the forward rate F with the expected future spot rate E(St+1) in the approximate IRP formula; the interest rate differential approximates the expected % change in the spot rate.

Uncovered IRP worked example (textbook)

If the U.S. rate is 5%/yr and the U.K. rate is 8%/yr, uncovered IRP implies the pound is expected to DEPRECIATE against the dollar by about 3%/yr (5% − 8% = −3%). This same relationship is also called the international Fisher effect.

Why it's called "uncovered" IRP

Because, unlike regular (covered) IRP, there is no forward contract hedging the exchange-rate exposure — it relies purely on unhedged expectations, so (unlike covered IRP) it is NOT enforced by riskless arbitrage and often fails to hold.

Currency Carry Trade — definition

Borrowing in a currency with a LOW interest rate and investing in a currency with a HIGH interest rate, WITHOUT hedging the exchange risk — essentially a bet against uncovered interest rate parity.

Popular funding currencies for carry trade (textbook examples)

The Japanese yen (rates near zero since the mid-1990s), the Swiss franc, and — more recently — the U.S. dollar, during the Federal Reserve's low-rate policy after the Great Recession.

Popular investment currencies for carry trade (textbook examples)

The Australian dollar, the New Zealand dollar, and currencies of some developing economies, all offering relatively high interest rates.

Carry Trade Profitability Condition (yen-funded, A$-invested example)

Profitable as long as i_A$ − i_¥ (the interest rate spread) is GREATER than e_A$,¥ (the rate of appreciation of the yen against the Australian dollar) during the carry period.

Why carry trade can be self-reinforcing in the short run

If enough investors sell yen to buy Australian dollars at once, the yen can depreciate further — the OPPOSITE of what uncovered IRP predicts — which can make the trade even more profitable, at least temporarily.

Exhibit 6.3 — the AU$/¥ carry trade chart (2000-2018)

Plots the six-month interest-rate spread (i_A$ − i_¥) against the six-month rate of change in the exchange rate (e_A$,¥). The carry trade was mostly profitable during 2000-2007 and 2009-2014 (the yen often depreciated), but often unprofitable at other times due to sharp, sudden yen appreciations.

Carry trade going wrong — the textbook's cautionary example

The yen appreciated very sharply in the second half of 2008, acting as a "safe-haven" currency during the global financial crisis — generating a significant loss for anyone running the A$/¥ carry trade at that time.

Accominotti et al. (2019) — carry trade research finding

Using data back to 1919, they found that large carry trade returns occur exclusively in currencies operating under FLOATING exchange rate regimes.

DB G10 Currency Harvest Fund (DBV)

An ETF introduced in 2006 that let individual (not just institutional) investors participate directly in currency carry trade strategies.

Household "accidental" carry trade — the general pattern

Households that take out mortgages denominated in a foreign, low-interest currency (to save on interest) are effectively acting as carry traders — and bear the same currency risk if that funding currency later appreciates.

Household carry trade — Austria example

Austrian households began borrowing in low-rate foreign currencies in the late 1980s; by 2007, almost ONE-THIRD of Austrian household borrowing was in foreign currencies, mostly Swiss francs.

Household carry trade — Hungary example

When the Swiss franc appreciated sharply against the Hungarian forint in 2009, thousands of Hungarian families couldn't make mortgage payments; in 2011 the Hungarian government passed legislation letting households convert those loans back into forint at favorable, predetermined rates.

Household carry trade — Poland example

Polish households with Swiss-franc mortgages saw payments spike when the franc appreciated against the zloty; the resulting wave of borrower lawsuits led Poland's Supreme Court to begin hearings on how to treat foreign-currency loan cases in late 2021.

Two main reasons IRP deviates from holding exactly

(1) Transaction costs (bid-ask spreads in both interest rates and FX rates). (2) Capital controls imposed by governments.

Akram, Rime, and Sarno (2008) — deviation study finding

Using high-frequency data on USD rates against the euro, pound, and yen over 7 months in 2004, they found NO profitable arbitrage on AVERAGE, but did document numerous short-lived covered-IRP deviations that were large and long enough for traders to profitably exploit.

Lothian and Wu (2011) — very-long-run IRP study

Using ultra-long data spanning two centuries on the French franc/pound sterling and the U.S. dollar/sterling, they concluded uncovered IRP holds over the very long haul, but can deviate from holding for long stretches of time.

Transaction costs and the bid-ask spread mechanism

The arbitrager borrows at the higher "ask" interest rate and lends at the lower "bid" rate; similarly, currencies are bought at the ask FX price and sold at the bid FX price — these spreads eat into any apparent arbitrage profit.

IRP with bid-ask spreads — Equations 6.10 & 6.11

Forward FX dealers must quote so that Fᵃ ≥ Sᵇ[(1+i$ᵇ)/(1+i£ᵃ)] and Fᵇ ≤ Sᵃ[(1+i$ᵃ)/(1+i£ᵇ)] — otherwise a riskless, profitable arbitrage would exist even after accounting for transaction costs.

Exhibit 6.4 — Interest Rate Parity with Transaction Costs

The IRP line is surrounded by a shaded "no-arbitrage band" (bounded by dashed lines). Point D (inside the band) does NOT represent a profitable opportunity once transaction costs are considered; point C (outside the band) DOES represent one.

Capital controls — how they cause IRP deviations

Governments restrict inbound/outbound capital flows (via taxes or outright bans), often to manage balance-of-payments or exchange-rate targets; this impairs the arbitrage process, letting IRP deviations persist rather than being quickly traded away.

Capital controls — the Japan case study setup

Japan imposed capital controls on and off until December 1980, when it liberalized international capital flows; Otani and Tiwari (1981) studied the effect on IRP deviations from 1978-1981.

Deviations from Interest Rate Parity — Equation 6.12 (DIRP)

DIRP = [(1+i¥)S / (1+i$)F] − 1, using 3-month Gensaki bond rates (¥), 3-month Eurodollar deposit rates ($), and Tokyo spot/forward yen-dollar rates — an empirical yardstick for how far the market deviates from strict IRP.

Exhibit 6.5 — Deviations from IRP: Japan, 1978-1981

If IRP held strictly, DIRP would hover randomly around zero. Instead, deviations were largest in 1978, shrank in 1979 (as controls eased), rose again in 1980 (as new controls were added), and settled near zero in early 1981, right after Japan's new liberalization law.

Interpreting Japan's 1978 & 1980 IRP deviations

These deviations do NOT represent unexploited profit opportunities — they reflect real, binding barriers to cross-border arbitrage (capital controls), not market inefficiency.

Du, Tepper, and Verdelhan (2018) — modern covered-IRP violations

Found that covered IRP has been violated among 10 major currencies since the 2008 financial crisis, driven by post-crisis regulatory constraints on financial intermediaries and persistent imbalances in cross-currency funding/investment demand.

Bank for International Settlements (BIS) — explanation for modern deviations

Attributes post-2008 deviations from covered IRP to newly imposed constraints on arbitrage activity and the evolving demand for FX hedges — especially the growing demand for dollar hedges