Deriving Covered Interest Rate Parity

Why Compare Domestic and Hedged Foreign Investments?

An investor deciding where to place funds across borders faces two sources of return: the interest earned on the investment and the currency conversion needed to bring the proceeds home. A forward exchange contract can fix the future conversion rate in advance. Covered interest rate parity describes the relationship among the spot exchange rate, the forward exchange rate, and the two countries’ interest rates when the investments are hedged this way.

The key comparison is not between an uncertain foreign-currency return and a domestic return. It is between two strategies whose final values are both known in domestic currency: invest at home, or convert to foreign currency, invest abroad, and lock in the conversion back with a forward contract. Under the assumptions below, the two strategies must have the same return.

Definition: Let $S_0$ be the spot exchange rate, quoted as domestic-currency units per one foreign-currency unit. Let $F_{0,T}$ be the forward exchange rate with the same quotation for delivery at time $T$. Let $R_d(T)$ and $R_f(T)$ be the accumulation factors over that same period for risk-free domestic and foreign investments, respectively. Covered interest rate parity is the no-arbitrage condition $$ F_{0,T}=S_0\frac{R_d(T)}{R_f(T)}. $$

For example, if the domestic currency is the dollar and the foreign currency is the euro, a spot quote of $S_0=1.20$ means one euro costs 1.20 dollars. The forward quote $F_{0,T}$ uses the same units: dollars per euro. Keeping the quotation fixed is essential; reversing the quote changes the equation.

Deriving the Parity Equation

Assume an investor starts with one unit of domestic currency. There are two ways to obtain a known domestic-currency amount at time $T$. The domestic investment grows directly at the domestic accumulation factor. Alternatively, the investor converts the initial amount at the spot rate, invests the resulting foreign currency, and agrees today to exchange the future foreign proceeds back into domestic currency at the forward rate.

Invest at home.
One domestic-currency unit grows to $R_d(T)$ domestic-currency units at time $T$.

Convert and invest abroad.
At the spot rate, one domestic unit buys $1/S_0$ foreign units. After foreign investment, this becomes $R_f(T)/S_0$ foreign units.

Hedge the currency conversion.
The forward contract exchanges each foreign unit for $F_{0,T}$ domestic units. The foreign strategy therefore ends with $F_{0,T}R_f(T)/S_0$ domestic units.

Equate the covered returns.
With no arbitrage, the two known domestic-currency terminal amounts must match. Solving that equality gives covered interest rate parity.

R_d(T)=\frac{R_f(T)}{S_0}F_{0,T} \qquad\Longrightarrow\qquad F_{0,T}=S_0\frac{R_d(T)}{R_f(T)}.

The reasoning is a no-arbitrage argument: if the known terminal amounts differed, an investor could borrow through the lower-return strategy and invest through the higher-return strategy, with the forward contract fixing the exchange conversion. The resulting difference would be a sure gain before costs. Consequently, parity is an equality in the frictionless model, not a claim that either investment is risk-free under every real-world circumstance.

Model assumptions: The domestic and foreign investments are risk-free over the matching horizon; the relevant borrowing and lending rates are the same; the investor can trade at the stated spot and forward rates; the forward contract matches the investment maturity and amount; and there are no transaction costs, taxes, capital controls, default risk, or other barriers. If these assumptions do not hold, an observed market relationship can depart from exact parity.

Using Interest Rates for the Same Horizon

The accumulation factors must describe the same period as the forward contract. If $i_d$ and $i_f$ are effective annual rates and the maturity is one year, then $R_d(1)=1+i_d$ and $R_f(1)=1+i_f$. If the maturity is $n$ years and the annual effective rates remain fixed, the factors are $(1+i_d)^n$ and $(1+i_f)^n$. For other compounding conventions, construct each accumulation factor using that convention and the actual number of periods.

F_{0,n}=S_0\frac{(1+i_d)^n}{(1+i_f)^n} \quad\text{for fixed annual effective rates and an \(n\)-year maturity.}

This form makes clear why comparing an annual rate directly with a forward rate for a different maturity is invalid. First calculate each investment’s accumulation over the forward contract’s full term; then take the ratio of those accumulation factors.

Worked Example: One-Year Parity with a Domestic-Per-Foreign Quote

Suppose the spot rate is $1.2500$ domestic currency units per foreign unit. The domestic risk-free effective annual rate is $4\%$, and the foreign risk-free effective annual rate is $2\%$. Both investments compound annually, and a forward contract has a one-year maturity. Find the parity forward rate and verify it by comparing terminal values from one domestic-currency unit.

Method. Use the one-year accumulation factors $R_d(1)=1.04$ and $R_f(1)=1.02$. The forward rate in this example must remain in domestic currency per foreign currency.

F_{0,1}=1.2500\frac{1.04}{1.02} =\frac{1.3000}{1.02} =1.2745098\ldots

Rounded only at the final step, the parity forward rate is 1.2745 domestic currency units per foreign unit to four decimal places.

Domestic strategy. One domestic unit invested for one year grows to $1(1.04)=1.04$ domestic units.

Hedged foreign strategy. Converting one domestic unit at the spot rate produces $1/1.2500=0.8000$ foreign units. Investing at $2\%$ gives $0.8000(1.02)=0.8160$ foreign units at year-end. Selling those units forward at the unrounded parity rate gives $0.8160(1.2745098\ldots)=1.0400$ domestic units.

The check succeeds: both strategies produce $1.04$ domestic units. Common mistake avoided: multiplying the forward rate by the domestic investment return would not describe the foreign strategy; the foreign investment first grows by its own accumulation factor, and only then is converted at the forward rate.

Worked Example: A Two-Year Forward Contract

A hypothetical investor observes a spot quote of $1.1000$ domestic currency units per foreign unit. The domestic and foreign effective annual risk-free rates are $6\%$ and $2\%$, respectively. Assume these rates apply in each of the next two years, with annual compounding, and a forward contract matures in two years. Calculate the parity forward rate and check the two-year domestic terminal value from one domestic unit.

Method. Accumulate both investments for two years before taking their ratio. The domestic factor is $(1.06)^2=1.1236$; the foreign factor is $(1.02)^2=1.0404$.

F_{0,2} =1.1000\frac{1.1236}{1.0404} =\frac{1.23596}{1.0404} =1.1879661669\ldots

The two-year parity forward rate is 1.1880 domestic currency units per foreign unit, rounded to four decimal places.

The domestic strategy grows one domestic unit to $1.1236$ domestic units. For the foreign strategy, one domestic unit converts into $1/1.1000=0.9090909\ldots$ foreign units. After two years, that amount is $0.9090909\ldots(1.0404)=0.94581818\ldots$ foreign units. Selling it forward at the unrounded parity rate gives $0.94581818\ldots(1.1879661669\ldots)=1.1236$ domestic units.

The terminal values agree. Common mistake avoided: using the one-year rate ratio only once would ignore the second year of compounding and produce the wrong maturity-specific forward rate.

Worked Example: Six Months with Quarterly Compounding

Suppose the spot rate is $1.2000$ domestic currency units per foreign unit. The domestic nominal annual rate convertible quarterly is $8\%$, and the foreign nominal annual rate convertible quarterly is $4\%$. A forward contract matures in six months, or two quarters. Find the parity forward rate and compare the terminal values from an initial $1{,}000$ domestic-currency units.

Method. A nominal rate convertible quarterly gives a quarterly rate equal to the nominal annual rate divided by four. Thus the domestic quarterly rate is $0.08/4=0.02$, and the foreign quarterly rate is $0.04/4=0.01$. Over two quarters, the accumulation factors are $(1.02)^2=1.0404$ and $(1.01)^2=1.0201$.

F_{0,0.5} =1.2000\frac{1.0404}{1.0201} =\frac{1.24848}{1.0201} =1.2238800\ldots

The six-month parity forward rate is 1.2239 domestic currency units per foreign unit, rounded to four decimal places.

The domestic investment grows to $1{,}000(1.0404)=1{,}040.40$ domestic units. Converting $1{,}000$ at the spot rate gives $1{,}000/1.2000=833.3333\ldots$ foreign units. The foreign investment grows to $833.3333\ldots(1.0201)=850.0833\ldots$ foreign units. Converting those proceeds at the unrounded forward rate gives $850.0833\ldots(1.2238800\ldots)=1{,}040.40$ domestic units.

The check agrees with the domestic result. Common mistake avoided: applying a full year of interest to a six-month contract would use four quarters rather than the two quarters that actually elapse.

Quotation Direction and Common Traps

With the specified domestic-per-foreign quotation, the domestic accumulation factor is in the numerator and the foreign factor is in the denominator. This direction follows directly from the conversion steps: one domestic unit buys $1/S_0$ foreign units, and the final foreign amount is multiplied by the forward quote $F_{0,T}$ to return to domestic currency. A reciprocal exchange-rate quotation requires rewriting the relationship consistently; the same displayed formula cannot be used unchanged with a reversed quote.

A useful diagnostic is to write the units at each stage. The spot quote $S_0$ has units of domestic currency per foreign currency, so $1/S_0$ converts domestic units to foreign units. The forward quote $F_{0,T}$ converts foreign proceeds back to domestic units. The factors $R_d(T)$ and $R_f(T)$ are dimensionless growth factors, not exchange rates.

Reversing the rate ratio: with a domestic-per-foreign quote, using $S_0R_f(T)/R_d(T)$ reverses the parity relationship.
Mixing quotations: the spot and forward rates in the formula must use the same currency order.
Mixing horizons: compare accumulation factors over the forward contract’s maturity, not just the quoted annual rates.
Mixing rate conventions: effective annual rates and nominal rates convertible at a stated frequency must be converted into matching-period accumulation factors before use.

Key takeaway: For a forward rate quoted in domestic currency per foreign currency, covered interest rate parity is $F_{0,T}=S_0R_d(T)/R_f(T)$. The equation follows by equating the known domestic terminal values of a domestic investment and a foreign investment hedged with a matching forward contract.

Check Your Understanding

For each problem, use the stated domestic-per-foreign quotation and match the investment horizon to the forward maturity. Round exchange rates to four decimal places where requested.

The spot rate is $0.8000$ domestic units per foreign unit. One-year effective annual risk-free rates are $3\%$ domestically and $5\%$ abroad. Find the one-year parity forward rate. (Answer: $0.7848$ domestic units per foreign unit)
The spot rate is $1.3500$ domestic units per foreign unit. Effective annual risk-free rates are $4\%$ domestically and $1\%$ abroad for each of the next two years. With annual compounding, find the two-year parity forward rate. (Answer: $1.4314$ domestic units per foreign unit)
The spot rate is $1.1000$ domestic units per foreign unit. Nominal annual risk-free rates convertible semiannually are $6\%$ domestically and $2\%$ abroad. A forward contract matures in six months. Find the parity forward rate. (Answer: $1.1218$ domestic units per foreign unit)
An investor has $5{,}000$ domestic units. The spot rate is $1.2500$ domestic units per foreign unit; one-year effective annual risk-free rates are $4\%$ domestically and $2\%$ abroad. Find the one-year parity forward rate and the common terminal value under the domestic and hedged foreign strategies. (Answer: Forward rate $1.2745$ domestic units per foreign unit; common terminal value $5{,}200.00$ domestic units)

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