Covered Interest Parity Arbitrage Calculations

Comparing Returns After Hedging the Exchange Rate

An investor choosing between domestic and foreign interest-bearing assets faces an exchange-rate risk: the future domestic value of a foreign investment is uncertain if the currency must be exchanged later at the then-prevailing spot rate. A forward contract can set that future exchange rate in advance. Once the currency exposure is covered, the two investments can be compared in the same currency.

Covered interest parity (CIP) states the no-arbitrage relationship between the spot exchange rate, forward exchange rate, and interest rates in two currencies. This tutorial uses that relationship to calculate covered returns and, if market quotes depart from parity, to identify the theoretical arbitrage trades.

Definition: Let $S$ be the spot exchange rate quoted as units of domestic currency per one unit of foreign currency, and let $F$ be the forward exchange rate using the same quote convention for the same maturity. Let $i_d$ and $i_f$ be the domestic and foreign interest rates earned over that maturity. The covered domestic-currency return from investing one domestic currency unit through the foreign asset and hedging its proceeds is $$ (1+i_f)\frac{F}{S}. $$ The domestic investment returns $1+i_d$. Under covered interest parity, with frictionless markets, these returns are equal: $$ 1+i_d=(1+i_f)\frac{F}{S}, \qquad F=S\frac{1+i_d}{1+i_f}. $$

The rates in these equations must describe the same investment period and use compatible compounding conventions. They may be annual effective rates for a one-year investment, or period returns directly stated for a shorter maturity. Do not use an annual rate as though it were a six-month return unless the problem specifies how to convert it.

The exchange-rate quote matters. Here, $S=\$1.20/\text{€}$ means one euro costs $1.20. Multiplying euros by $F$, quoted in dollars per euro, converts the forward-settled euro amount into dollars. Reversing the quote without also changing the formula is a common source of errors.

Why the Parity Relationship Holds

Start with one unit of domestic currency and compare two ways to invest it for the same maturity. The domestic investment produces $1+i_d$ domestic currency units. Alternatively, converting at the spot rate gives $1/S$ foreign currency units; investing those produces $(1+i_f)/S$ foreign units. Selling that amount forward at $F$ produces $(1+i_f)F/S$ domestic units.

If these two certain, currency-hedged payoffs differ in a frictionless market, an investor can borrow through the lower-payoff route and place the proceeds through the higher-payoff route. The forward contract fixes the currency conversion needed to repay the borrowing. Competition for this trade pushes the prices and rates toward equality. Setting the payoffs equal and solving for $F$ gives the parity formula above. This is a no-arbitrage argument; it does not say that an unhedged foreign investment has the same return.

Compare like with like.
Confirm that the spot and forward quotes use the same currency order and that both interest rates and the forward contract have the same maturity.

Calculate the covered return.
For one domestic currency unit, calculate $(1+i_f)F/S$, then compare it with $1+i_d$.

Choose the trade direction.
If the covered foreign return is higher, borrow domestic currency and invest abroad. If it is lower, borrow foreign currency and invest domestically.

Hedge the repayment or proceeds.
Use a forward contract for the amount of foreign currency due at maturity, including interest on foreign borrowing when applicable.

Worked Arbitrage Calculations

Worked Example: The Covered Foreign Return Is Higher

A hypothetical investor can borrow or invest dollars for one year at an effective annual rate of 4%, or borrow or invest euros for one year at 2%. The spot rate is $S=\$1.20/\text{€}$, and the one-year forward rate is $F=\$1.23/\text{€}$. Assume the quoted rates are available for the stated transactions, with no transaction costs, taxes, credit risk, or restrictions. Determine whether a covered parity gap exists and calculate the trade profit from borrowing $1,000,000.

Method and return comparison. For each dollar invested through the euro route, first convert $1 into $1/1.20=0.833333$ euros. After one year, the euro investment is worth $0.833333(1.02)=0.850000$ euros. The forward contract converts this to $0.850000(1.23)=\$1.0455$. The dollar investment instead returns $1.04$ dollars. The covered euro route therefore pays more: $1.0455>1.04$.

Trade. Borrow $1,000,000 for one year and owe $1{,}000{,}000(1.04)=\$1{,}040{,}000$ at maturity. Convert the borrowed dollars at spot:

\frac{\$1{,}000{,}000}{\$1.20/\text{€}}=\text{€}833{,}333.33.

Invest the euros at 2%:

\text{€}833{,}333.33(1.02)=\text{€}850{,}000.

At the same time, agree to sell €850,000 forward at $1.23 per euro. The maturity proceeds are $850{,}000(1.23)=\$1{,}045{,}500$. After repaying the dollar borrowing, the theoretical profit is:

\$1{,}045{,}500-\$1{,}040{,}000=\$5{,}500.

Check. The excess return per dollar borrowed is $1.0455-1.04=\$0.0055$; on $1,000,000 this is $5,500. The calculation avoids the mistake of comparing the euro interest rate directly with the dollar interest rate without accounting for the spot and forward exchange rates.

Worked Example: The Covered Foreign Return Is Lower

In a hypothetical one-year market, the dollar interest rate is 5%, the pound interest rate is 2%, the spot rate is $S=\$1.25/\text{£}$, and the forward rate is $F=\$1.28/\text{£}$. Assume frictionless transactions and rates available for borrowing as well as investing. Identify the trade direction and find the profit from borrowing £1,000,000.

Method and return comparison. One dollar converted at spot buys $1/1.25=0.8$ pounds. The pounds grow to $0.8(1.02)=0.816$ pounds, and selling them forward produces $0.816(1.28)=\$1.04448$. A dollar invested domestically grows to $1.05$ dollars. Since $1.04448<1.05$, the covered foreign route has the lower return. The trade borrows pounds and invests the converted proceeds in dollars.

Borrow £1,000,000 and convert it at spot to $1{,}000{,}000(1.25)=\$1{,}250{,}000$. Invest those dollars at 5% to receive $1{,}250{,}000(1.05)=\$1{,}312{,}500$. The pound borrowing must be repaid with interest, so the amount due is $1{,}000{,}000(1.02)=\text{£}1{,}020{,}000$. Buy that amount of pounds forward at $1.28 per pound. The forward purchase costs $1{,}020{,}000(1.28)=\$1{,}305{,}600$.

After purchasing the pounds needed to repay the loan, the remaining dollars are:

\$1{,}312{,}500-\$1{,}305{,}600=\$6{,}900.

Check. The return gap per dollar initially invested is $1.05-1.04448=\$0.00552$. The initial dollar investment in this trade is $1,250,000, so $1{,}250{,}000(0.00552)=\$6{,}900$. A common error is to forward-buy only the original £1,000,000; the forward contract must cover £1,020,000, the amount owed after interest.

Worked Example: A Six-Month Parity Gap

A hypothetical investor compares six-month investments. The six-month dollar return is 2.4%, the six-month foreign-currency return is 1.2%, the spot rate is $S=\$0.80/\text{FC}$, and the six-month forward rate is $F=\$0.81/\text{FC}$. Assume the investor can borrow or invest $2,000,000 at the stated dollar rate, with no transaction costs or other frictions. Calculate the covered return, determine the trade, and find the profit.

Method and return comparison. These are six-month returns, so use them directly; do not convert them to annual rates. One dollar buys $1/0.80=1.25$ units of foreign currency. After six months, this becomes $1.25(1.012)=1.265$ foreign currency units. Selling that amount forward yields $1.265(0.81)=\$1.02465$ per dollar. The domestic investment returns $1.024$ dollars. Because $1.02465>1.024$, borrow dollars, invest in the foreign asset, and sell the investment proceeds forward.

Borrow $2,000,000 and repay $2{,}000{,}000(1.024)=\$2{,}048{,}000$ after six months. At the spot rate, the borrowed dollars purchase $2{,}000{,}000/0.80=2{,}500{,}000$ units of foreign currency. Investing at 1.2% produces $2{,}500{,}000(1.012)=2{,}530{,}000$ units. Sell those units forward at $0.81 each, for dollar proceeds of $2{,}530{,}000(0.81)=\$2{,}049{,}300$.

Profit and check. The net profit is $\$2{,}049{,}300-\$2{,}048{,}000=\$1{,}300$. The difference in covered returns is $1.02465-1.024=0.00065$ dollars per dollar; multiplying by $2,000,000 also gives $1,300. The result is a six-month profit, not an annualized profit.

Parity as a Benchmark, Not a Guaranteed Trading Profit

A market forward rate above or below the parity-implied rate can indicate a covered return difference, but the simple calculation assumes that the quoted rates are actually available to the investor. In practice, borrowing and lending rates can differ, and foreign exchange transactions have bid–ask spreads. Fees, taxes, credit limits, collateral requirements, and settlement risk can also reduce or eliminate a calculated profit.

For a practical comparison, use the investor’s borrowing rate on the currency being borrowed, the investable rate on the currency being invested, and exchange rates that match the direction of each transaction. The calculations in this tutorial deliberately assume frictionless markets; they describe theoretical arbitrage under those assumptions, not a claim that a particular real-world transaction is risk-free or profitable.

Common exam trap: Do not forward-contract only the original amount borrowed in a currency if that borrowing accrues interest. Borrowing $B$ foreign units at rate $i_f$ requires repayment of $B(1+i_f)$ foreign units. Also, keep the quote $S$ or $F$ in domestic currency per foreign unit throughout the calculation. A reversed quote requires a correspondingly reversed conversion.

Check Your Understanding

For each problem, use the stated quote convention and compare returns over the specified investment period.

A hypothetical one-year market has $S=\$0.72/\text{FC}$, $F=\$0.75/\text{FC}$, a domestic interest rate of 4%, and a foreign interest rate of 1%. Calculate the foreign investment’s covered domestic return and identify which route has the higher return. (Answer: The covered foreign return is $(1.01)(0.75/0.72)=1.0521$, or 5.2083%; it exceeds the 4% domestic return.)
For one year, $S=\$1.40/\text{FC}$, the domestic rate is 2%, and the foreign rate is 5%. Find the parity forward rate and state whether a forward rate of $1.36 per FC creates a covered parity gap under the frictionless assumptions. (Answer: $F=1.40(1.02/1.05)=\$1.36/\text{FC}$; the quote is at parity, so there is no covered return gap.)
A hypothetical one-year market has $S=\$0.90/\text{FC}$, $F=\$0.92/\text{FC}$, a domestic rate of 3%, and a foreign rate of 2%. An investor borrows $500,000. Calculate the theoretical arbitrage profit and name the trades. (Answer: The covered foreign return is $1.02(0.92/0.90)=1.0426667$, above 1.03; borrow dollars, buy and invest FC, and sell the maturity proceeds forward. The profit is $500{,}000(1.0426667-1.03)=\$6{,}333.33$.)
A hypothetical one-year market has $S=\$0.50/\text{FC}$, $F=\$0.49/\text{FC}$, a domestic rate of 4%, and a foreign rate of 6%. An investor borrows 100,000 FC and invests the spot-converted proceeds in dollars. What amount of FC must be bought forward, and what is the theoretical profit? (Answer: Repayment is $100{,}000(1.06)=106{,}000$ FC. The domestic investment grows from $50,000 to $52,000; buying 106,000 FC forward costs $106{,}000(0.49)=\$51{,}940$, leaving a $60 profit.)
A six-month investment has $S=\$1.10/\text{FC}$, $F=\$1.12/\text{FC}$, a six-month domestic return of 2%, and a six-month foreign return of 1%. For a $120,000 domestic borrowing, calculate the theoretical profit from the covered foreign investment. (Answer: The covered return is $1.01(1.12/1.10)=1.0283636$, compared with 1.02 domestically. Borrow dollars, invest in FC, and sell the proceeds forward; the profit is $120{,}000(1.0283636-1.02)=\$1{,}003.64$.)

What You'll Learn