Covered Interest Parity For Currency Forwards

Why Currency Forward Rates Depend on Two Interest Rates

A currency forward rate is linked to more than the current exchange rate. If investors can borrow or lend in either currency, they can arrange today to obtain a specified amount of foreign currency at a future date without entering a forward contract: they can invest in that currency now. A forward price inconsistent with this alternative creates an arbitrage opportunity in an idealized market.

The relationship is called covered interest parity. “Covered” means the exchange rate for the future currency exchange is fixed in advance, so the investor is not exposed to an uncertain future spot rate. The derivation below is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $S_0$ be the spot exchange rate, quoted as domestic currency per one unit of foreign currency. Let $F_0(T)$ be the forward exchange rate for delivery at time $T$, using the same quote. Let $i_d$ and $i_f$ be the domestic and foreign annual effective interest rates for that term. Under the frictionless no-arbitrage model, $$ F_0(T)=S_0\frac{(1+i_d)^T}{(1+i_f)^T}. $$ The quote direction determines which rate is domestic and which is foreign.

For example, a quote of $1.30$ domestic dollars per foreign dollar means $S_0=1.30$. The currency named in the numerator of the quote is domestic; the currency being priced in the denominator is foreign. If the quote is reversed, the roles of the currencies and the numerical forward rate change.

Deriving Covered Interest Parity

Consider obtaining one unit of foreign currency at time $T$. There are two ways to arrange this, measured in domestic currency. One is to agree today to buy one unit of foreign currency forward at rate $F_0(T)$. The other is to build the foreign currency through spot exchange and investment. No-arbitrage requires that these alternatives have the same domestic-currency cost at time $T$.

Find the foreign investment needed.
Invest $1/(1+i_f)^T$ units of foreign currency today. At time $T$, this grows to exactly one unit of foreign currency.

Convert its initial cost to domestic currency.
At spot, the foreign investment costs $S_0/(1+i_f)^T$ domestic currency units.

Finance that cost domestically.
Borrow $S_0/(1+i_f)^T$ domestic currency units today. The amount to repay at time $T$ is $S_0(1+i_d)^T/(1+i_f)^T$.

Compare the future domestic-currency costs.
The forward obtains one foreign unit for $F_0(T)$ domestic units. The spot-and-invest strategy obtains the same foreign unit and requires repayment of $S_0(1+i_d)^T/(1+i_f)^T$. No-arbitrage equates these amounts.

F_0(T)=S_0\frac{(1+i_d)^T}{(1+i_f)^T}

The argument depends on replicating the same future delivery of foreign currency and comparing the domestic amounts required to obtain it. If the forward rate were above the replication cost, one could obtain foreign currency through spot investment and sell it forward at the higher rate. If the forward rate were below the replication cost, one could borrow and sell foreign currency at spot, invest the proceeds domestically, and buy the foreign currency forward to repay the foreign borrowing.

Worked Example: Calculating a Two-Year Forward Quote

The spot rate is $1.20$ U.S. dollars per euro. The annual effective rate is $5\%$ in the United States and $2\%$ in the euro area. Find the two-year parity forward rate in U.S. dollars per euro. State the method and identify the quote-direction mistake to avoid.

Method. The quote is U.S. dollars per euro, so U.S. dollars are domestic and euros are foreign. Thus $i_d=0.05$, $i_f=0.02$, and $T=2$. The spot-and-investment replication gives $$ F_0(2)=1.20\frac{(1.05)^2}{(1.02)^2}. $$

Calculate the accumulation factors: $$ (1.05)^2=1.1025,\qquad (1.02)^2=1.0404. $$ Then $$ F_0(2)=\frac{1.20(1.1025)}{1.0404} =\frac{1.323}{1.0404} =1.271626\ldots $$ The parity forward rate is $1.2716$ U.S. dollars per euro, rounded to four decimal places.

Check. The U.S. dollar rate exceeds the euro rate, so a euro bought forward costs more U.S. dollars than it costs at spot. The computed forward rate is above $1.20$, consistent with that direction. A common mistake is to put the foreign rate in the numerator despite the quote being domestic currency per foreign currency; that would reverse the relative-interest adjustment.

Using the Quote Consistently

A currency quote and its reciprocal describe the same exchange rate in opposite units. If $S_0$ is domestic currency per foreign currency, the reciprocal spot quote is $1/S_0$, foreign currency per domestic currency. The reciprocal forward quote is $1/F_0(T)$. Consequently, if the rates and term are unchanged,

\frac{1}{F_0(T)} =\frac{1}{S_0}\frac{(1+i_f)^T}{(1+i_d)^T}.

The reciprocal formula is not a different economic result: it is the original relationship expressed in the opposite quote. The interest rates must be assigned to the currencies after the quote direction is identified. In particular, “domestic” does not mean the currency of the country where a trader happens to be located; it means the numerator currency under the chosen quote convention.

Worked Example: Finding an Implied Foreign Interest Rate

A market quotes $103$ units of currency A per unit of currency B at spot and $105$ units of A per B for one-year forward delivery. The annual effective rate in currency A is $5\%$. Assuming parity holds, find the implied annual effective rate in currency B. Explain why the rate labels matter.

Method. The quote is A per B, so A is domestic and B is foreign. For $T=1$, rearrange covered interest parity to isolate the foreign accumulation factor: $$ 1+i_f=\frac{S_0(1+i_d)}{F_0(1)}. $$

Substitute the stated values: $$ 1+i_f=\frac{103(1.05)}{105} =\frac{108.15}{105} =1.03. $$ Therefore, $$ i_f=1.03-1=0.03. $$ The implied annual effective interest rate in currency B is $3\%$.

Check. Substituting $i_d=5\%$, $i_f=3\%$, and $S_0=103$ into the parity formula gives $103(1.05)/(1.03)=105$, the stated forward quote. The common mistake is to assign the domestic rate to currency B merely because B is the currency being acquired; the quote makes A domestic.

Compounding Conventions and the Model

The displayed formula uses annual effective rates and a term measured in years. For example, under this convention, a deposit of one unit grows to $(1+i)^T$ units over $T$ years. The interest rates must cover the same term, and each rate must be expressed using a compatible compounding convention before they are compared.

If both rates are nominal annual rates convertible $m$ times per year, denoted $i_d^{(m)}$ and $i_f^{(m)}$, the corresponding accumulation factors over $T$ years are $(1+i_d^{(m)}/m)^{mT}$ and $(1+i_f^{(m)}/m)^{mT}$. Covered interest parity becomes

F_0(T)=S_0 \frac{\left(1+i_d^{(m)}/m\right)^{mT}} {\left(1+i_f^{(m)}/m\right)^{mT}}.

With constant forces of interest $\delta_d$ and $\delta_f$, the accumulation factors are $e^{\delta_dT}$ and $e^{\delta_fT}$, giving

F_0(T)=S_0e^{(\delta_d-\delta_f)T}.

These are equivalent ways to express the same replication principle, not additional adjustments to the relationship. Do not mix a nominal rate in one currency with an annual effective rate in the other without first converting them to comparable accumulation factors.

Testing a Forward Quote for Arbitrage

The parity rate is the no-arbitrage benchmark in a frictionless market. Suppose a quoted forward rate $K$, in domestic currency per unit of foreign currency, exceeds the parity rate $F_0(T)$. A covered strategy can borrow domestic currency, buy and invest foreign currency at spot, then sell the foreign currency forward. The forward sale proceeds exceed the domestic loan repayment. If $K<F_0(T)$, reverse the strategy: borrow foreign currency, sell it at spot, invest the domestic proceeds, and buy foreign currency forward to repay the foreign loan.

Worked Example: Calculating a Covered Arbitrage Profit

The spot rate is $1.25$ Canadian dollars per U.S. dollar. The one-year annual effective rates are $4\%$ in Canada and $2\%$ in the United States. A dealer quotes a one-year forward rate of $1.28$ Canadian dollars per U.S. dollar. Determine whether an arbitrage exists and calculate the profit from a covered strategy involving $1{,}000{,}000$ U.S. dollars at delivery. Assume borrowing and lending are available at the stated rates, with no transaction costs or other frictions.

Method. Canadian dollars are domestic and U.S. dollars are foreign. The parity rate is $$ F_0(1)=1.25\frac{1.04}{1.02} =\frac{1.30}{1.02} =1.274509804\ldots $$ Since the dealer quote $1.28$ exceeds parity, the forward is overpriced. Borrow Canadian dollars, convert to U.S. dollars, invest in the United States, and sell the resulting U.S. dollars forward.

To have $1{,}000{,}000$ U.S. dollars at the end of the year, invest $$ \frac{1{,}000{,}000}{1.02} =980{,}392.1569 $$ U.S. dollars today. Their spot cost is $$ 980{,}392.1569(1.25) =1{,}225{,}490.1961 $$ Canadian dollars. Borrow this amount in Canada. At the end of the year, the Canadian-dollar loan repayment is $$ 1{,}225{,}490.1961(1.04) =1{,}274{,}509.8039. $$ The U.S. investment is now worth $1{,}000{,}000$ U.S. dollars. Deliver that amount under the short forward contract and receive $$ 1{,}000{,}000(1.28)=1{,}280{,}000 $$ Canadian dollars. The net profit is $$ 1{,}280{,}000-1{,}274{,}509.8039 =5{,}490.1961. $$ The covered arbitrage profit is $\$5{,}490.20$ Canadian dollars, rounded to the nearest cent.

Check. The forward proceeds exceed the loan repayment by exactly the difference between the quoted and parity rates, multiplied by the foreign-currency delivery amount: $(1.28-1.274509804)(1{,}000{,}000)=5{,}490.196$. The strategy has no unhedged exchange-rate exposure because the foreign investment’s maturity value is committed for delivery under the forward.

Assumptions and Common Exam Traps

The parity equation is an idealized no-arbitrage result. Its direct use assumes that the spot exchange and forward contract are available at the stated rates; borrowing and lending can occur in either currency at the stated interest rates; the rates and forward delivery date match; and there are no transaction costs, taxes, capital controls, credit constraints, or other market frictions. When these assumptions fail, a single exact parity quote need not be executable. Actual borrowing and lending rates may differ, for example, creating a range of feasible forward rates rather than one frictionless benchmark.

Reversing the quote. For a quote in domestic currency per foreign currency, use the domestic accumulation factor in the numerator. Reversing the quote also reverses the rate ratio.
Using the wrong interest rates. The domestic and foreign labels follow the quote, not the investor’s location or which currency is being purchased.
Mixing term units. If $T$ is in years, use rates and accumulation factors applicable to $T$ years. Do not use a two-year rate as if it were an annual rate.
Comparing a quoted rate with the wrong benchmark. Both the market quote and parity rate must use the same currency units and delivery date.
Ignoring execution assumptions in an arbitrage calculation. A theoretical price gap is not necessarily a realizable arbitrage if borrowing, lending, conversion, or forward trading is restricted or costly.

Key takeaway: First write the exchange-rate quote in words. Identify domestic and foreign currencies from that quote, match the accumulation terms, and then apply $F_0(T)=S_0(1+i_d)^T/(1+i_f)^T$. This sequence prevents the most common sign, unit, and rate-label errors.

Check Your Understanding

For each problem, state the quote direction before calculating. Assume the stated markets are frictionless unless the problem says otherwise.

The spot rate is $1.30$ U.S. dollars per Canadian dollar. The annual effective rates are $4\%$ in the United States and $2\%$ in Canada. Find the two-year parity forward rate in U.S. dollars per Canadian dollar, rounded to four decimal places. (Answer: $1.3515$ U.S. dollars per Canadian dollar)
The spot rate is $0.75$ euros per pound sterling. The one-year annual effective rates are $3\%$ in the euro area and $5\%$ in the United Kingdom. Find the one-year parity forward rate, rounded to four decimal places. (Answer: $0.7357$ euros per pound sterling)
The spot rate is $120$ Mexican pesos per U.S. dollar, and the one-year forward rate is $126$ pesos per dollar. If the annual effective rate in Mexico is $8\%$, find the implied annual effective U.S. dollar rate, rounded to four decimal places as a percentage. (Answer: $2.8571\%$)
The spot rate is $2.00$ New Zealand dollars per Australian dollar. The one-year annual effective rates are $6\%$ in New Zealand and $2\%$ in Australia. A dealer quotes $2.10$ New Zealand dollars per Australian dollar forward. For a $100{,}000$-Australian-dollar delivery, find the covered arbitrage profit, rounded to the nearest cent. (Answer: $NZ\$2{,}156.86$)
The spot rate is $0.80$ U.S. dollars per euro. The two-year annual effective rates are $4\%$ in the United States and $1\%$ in the euro area. Find the parity forward rate in U.S. dollars per euro and its reciprocal quote in euros per U.S. dollar, both rounded to four decimal places. (Answer: $0.8482$ U.S. dollars per euro; $1.1789$ euros per U.S. dollar)

What You'll Learn