Module 11

Interest Rate Parity, CIRP, UIP, and Currency Carry Trade

This module focuses on forward-rate pricing, covered and uncovered interest parity, covered arbitrage logic, and how interest-rate gaps relate to currency carry trades.

Theme
ParityNo-arbitrageFX

Interest rate parity basics

Core idea

Interest Rate Parity (IRP) links the spot exchange rate, the forward exchange rate, and the interest-rate difference between two currencies. In a covered setting, investors should not be able to lock in an easy extra return by borrowing in one currency, converting into another, investing there, and using a forward contract to remove exchange-rate risk.

F = S × (1 + iq × T) / (1 + ib × T)
In a quote like EUR/USD, EUR is the base / foreign currency and USD is the quoted / home currency. So the quoted-currency rate goes in the numerator and the base-currency rate goes in the denominator.
Open the JUFinance IRP calculator ↗

Why it matters

  • It explains how forward exchange rates are priced.
  • It shows whether a quoted forward is consistent with current spot and rates.
  • It helps identify covered arbitrage opportunities.
  • It clarifies why a forward premium is not automatically a free lunch.
Covered parity is about locked-in returns. If the forward is mispriced relative to spot and rates, traders can borrow, lend, convert, and hedge until the mispricing closes.
Equation mapSimple IRPEUR/USD

Simple IRP equation

Keep one quote convention all the way through. For EUR/USD, EUR is the base currency and USD is the quoted currency. So EUR/USD = 1.0800 means 1 euro costs $1.08.

F = S × (1 + iq × T) / (1 + ib × T)
Use this same equation for all three cases:
1 year: T = 1   |   90 days: T = 90/360   |   180 days: T = 180/360
Fforward rate
Sspot rate today
iqquoted-currency interest rate
USD in EUR/USD
ibbase-currency interest rate
EUR in EUR/USD
Ttime in years
1, 90/360, or 180/360
Open the JUFinance IRP calculator ↗
VideoInterest Rate Parity

Interest Rate Parity video

Watch this video for a clear walkthrough of interest rate parity and how spot rates, forward rates, and interest-rate differences fit together.

VideoUncovered IRP

Uncovered Interest Rate Parity (UIP) video

This one is different from the covered version. UIP is unhedged: it compares expected exchange-rate movement with the interest-rate gap, rather than using a forward contract to lock in a covered return.

UIPFYINot main focus

Uncovered Interest Parity (UIP) — just for reference

This is not the main focus of this module. The main focus is still covered interest rate parity and covered arbitrage. But it is useful to see the uncovered version because it connects the interest-rate gap to the expected future spot rate.

CIRP uses the forward rate and is a no-arbitrage condition.
UIP uses the expected future spot rate and is more of an expectation idea, not a locked-in arbitrage condition.

UIP equation

E(St+T) = S × (1 + iq × T) / (1 + ib × T)

Same quote rule as above: for EUR/USD, USD is the quoted currency, so iq is the U.S. interest rate, and EUR is the base currency, so ib is the euro interest rate.

Example

Suppose EUR/USD = 1.0800, U.S. interest rate = 5%, euro interest rate = 3%, and T = 1 year.

E(St+1) = 1.0800 × (1 + 0.05) / (1 + 0.03) = 1.1010
Interpretation: under UIP, the expected future spot rate is about 1.1010 USD per euro.
CalculatorIRP

IRP calculator and covered-arbitrage check

Spot / forward calculator

Enter values and click a button.
This simple calculator uses the same quote convention all the way through: F = S × (1 + iq × T) / (1 + ib × T).
Use it to solve a missing spot rate, a missing forward rate, or to check whether the market forward is above or below the parity level.
ICEIRPHomework-style practice

In-class exercises — IRP drill + homework-style practice

Part A. Quick IRP drill

Exercise 1

Given: i$ = 8%, iSF = 4%, S = 0.68 $/SF. Find F90.

The quote is $/SF, so the Swiss franc is the base currency and the dollar is the quoted currency.

T = 90/360 = 1/4
F90 = 0.68 × (1 + 0.08/4) / (1 + 0.04/4) = 0.6867 $/SF

Exercise 2

Given: i$ = 8%, iyen = 4%, S = 0.0094 $/yen. Find F180.

The quote is $/yen, so yen is the base currency and the dollar is the quoted currency.

T = 180/360 = 1/2
F180 = 0.0094 × (1 + 0.08/2) / (1 + 0.04/2) = 0.0095843 ≈ 0.0096 $/yen

Exercise 3

Given: i$ = 4%, i£ = 2%, S = $1.5/£, F = $2/£. Does IRP hold? How can you arbitrage? What is the equilibrium forward rate?

F* = 1.5 × (1.04 / 1.02) = 1.5294 $/£
The market forward is $2/£, which is too high.

  1. Borrow dollars
  2. Convert dollars to pounds at the spot rate
  3. Invest pounds in the U.K.
  4. Sell pounds forward at the overpriced forward rate
Equilibrium forward ≈ $1.5294/£

Exercise 4

Given: i$ = 2%, i£ = 4%, S = $1.5/£, F = $1.1/£. Does IRP hold? How can you arbitrage? What is the equilibrium forward rate?

F* = 1.5 × (1.02 / 1.04) = 1.4712 $/£
The market forward is $1.1/£, which is too low.

  1. Borrow pounds
  2. Convert pounds to dollars at the spot rate
  3. Invest dollars in the U.S.
  4. Buy pounds forward cheaply to repay the pound loan
Equilibrium forward ≈ $1.4712/£

Part B. Homework-style ICE for homework 1–4

These are similar to homework 1–4, but with different numbers and more detailed steps so students can practice before doing the homework.

Homework-style ICE 1 — solve the spot rate

Given: The one-year U.S. interest rate is 6.0%, the one-year euro interest rate is 4.0%, and the one-year forward rate is $1.24/€. What is the spot rate?

  1. Use the IRP equation: F = S × (1 + i$) / (1 + i€).
  2. Rearrange to solve for spot: S = F × (1 + i€) / (1 + i$).
  3. Substitute the numbers.
S = 1.24 × 1.04 / 1.06 = 1.2166 $/€
Interpretation: one euro should cost about $1.2166 today if the quoted forward is consistent with IRP.
Given forward and rates F = 1.24, i$ = 6%, i€ = 4% Solve for today’s spot S = F × (1 + i€) / (1 + i$) S = 1.2166 $/€ Covered euro path Buy € spot → invest at 4% Sell € forward at 1.24 Triangle closes under IRP With S = 1.2166, the covered euro return matches the U.S. return rearrange IRP test covered euro strategy equal covered payoffs = no arbitrage
Capital-flow read: this is a no-arbitrage triangle. Once you solve the correct spot rate, borrowing dollars and staying in dollars gives the same covered payoff as converting into euros, investing in euros, and selling euros forward.

Homework-style ICE 2 — find the spot rate that eliminates arbitrage

Given: You can borrow either $1,100,000 or €1,000,000 for one year. The one-year interest rates are i$ = 3% and i€ = 5%. The one-year forward rate is $1.18/€. What spot rate would eliminate arbitrage?

  1. Again use IRP: F = S × (1 + i$) / (1 + i€).
  2. Rearrange to get the no-arbitrage spot rate.
S = 1.18 × 1.05 / 1.03 = 1.2029 $/€
If the actual spot is not about $1.2029/€, then covered interest arbitrage would exist.
No-arbitrage spot S = 1.2029 $/€ Dollar borrowing route Borrow $1.1m → repay $1.133m Compare against covered euro route Euro covered route Borrow €1.0m → repay €1.05m Hedge with forward F = 1.18 Why arbitrage disappears here At S = 1.2029, the covered euro borrowing cost and dollar borrowing cost line up compare dollar path compare euro + forward path if actual S differs, capital flows toward the cheaper covered route
Capital-flow read: if the actual spot is below 1.2029, euros are too cheap today relative to the forward, so the euro covered route becomes too attractive. If the actual spot is above 1.2029, the dollar route becomes relatively attractive. The no-arbitrage spot is the value that closes the triangle.

Homework-style ICE 3 — find arbitrage profit on a €10,000 contract

Given: S0 = $1.45/€, F360 = $1.42/€, U.S. interest rate = 2%, euro interest rate = 3%, and the contract size is €10,000. What profit can be locked in at maturity?

  1. First compute the fair forward from IRP.
F* = 1.45 × 1.02 / 1.03 = 1.4359 $/€
Since the actual forward is 1.42, the market forward is too low.

  1. Borrow euros.
  2. Convert euros into dollars at the spot rate.
  3. Invest dollars in the U.S.
  4. Buy euros forward at the cheap forward price to repay the euro loan.
Profit per euro = S × (1 + i$) − F × (1 + i€)
Profit per euro = 1.45 × 1.02 − 1.42 × 1.03 = 0.0164 dollars
Total profit = 0.0164 × 10,000 = $164
So the locked-in arbitrage profit is about $164, ignoring transaction costs.
Forward is too low Market F = 1.42 < fair F* = 1.4359 Borrow and convert euros Borrow €10,000 today Spot convert → $14,500 Invest and hedge repayment U.S. investment → $14,790 Buy €10,300 forward for $14,626 Locked-in arbitrage profit $14,790 − $14,626 = about $164 borrow € because forward is cheap spot convert → invest dollars buy € forward to repay the loan
Capital-flow read: the cheap forward lets you lock in a low future dollar cost of buying back euros. That is why the profitable triangle is borrow € → spot into $ → invest in $ → buy € forward. The positive gap at the end is the arbitrage profit.

Homework-style ICE 4 — solve for the U.S. interest rate

Given: Italy’s annual interest rate is 2%. The spot rate is $1.10/€ and the one-year forward rate is $1.15/€. Use IRP to find the U.S. annual interest rate.

  1. Start with F = S × (1 + i$) / (1 + i€).
  2. Rearrange to solve for the U.S. rate.
1 + i$ = (F / S) × (1 + i€)
1 + i$ = (1.15 / 1.10) × 1.02 = 1.0664
i$ = 0.0664 = 6.64%
So the implied U.S. annual interest rate is about 6.64%.
Given market inputs S = 1.10, F = 1.15, i€ = 2% Covered euro deposit path Start with $1 → buy € spot Invest at 2% → sell € forward at 1.15 U.S. deposit must match Domestic dollar path = $1 × (1 + i$) Solve the i$ that makes both paths equal Implied U.S. annual rate i$ = 6.64% closes the parity triangle build the covered euro payoff match it with the U.S. payoff if i$ were different, funds would shift toward the better covered return
Capital-flow read: this triangle solves for the missing domestic rate. Once the U.S. rate is 6.64%, a U.S. dollar deposit and a covered euro deposit produce the same future dollar payoff, so there is no covered arbitrage.
Quick shortcut: if the actual forward is too high, you want to end up selling the base currency forward. If the actual forward is too low, you want to buy the base currency forward.
HomeworkChapter 7Due with final

Homework Chapter 7 (due with final)

Questions only are shown here.

1) Solve the spot rate given the forward rate

Suppose the one-year interest rate is 5.0% in the U.S. and 3.5% in Germany, and the one-year forward exchange rate is $1.3/€. What must the spot exchange rate be?

2) What spot rate eliminates arbitrage?

You can borrow either $1,000,000 or €800,000 for one year. The one-year interest rate is i$ = 2% and i€ = 6%. The one-year forward rate is $1.20 = €1.00. What must the spot rate be to eliminate arbitrage opportunities?

3) Futures / forward contract of €10,000

Given: S0 = $1.45/€, F360 = $1.48/€, U.S. interest rate = 4%, euro interest rate = 3%, and one contract has value €10,000. What profit can you make at maturity?

4) Use IRP to solve for the U.S. interest rate

Interest rate in Italy is 3%. Spot rate is $1.20/€ and one-year forward rate is $1.18/€. Use IRP to calculate the U.S. annual interest rate.

5) Carry trade and JPY/USD

JPY interest rate is near 0%, USD interest rate is around 5.5%, the exchange rate is stable, and the yen has been slowly weakening. Would a carry trade look attractive? What could happen if Japan suddenly raises rates?

This homework is due with the final. Use the simple IRP equation above and the calculator on this page for practice.