Uncovered Interest Rate Parity

Why Interest Rates and Expected Exchange Rates Are Linked

An investor comparing deposits in two currencies faces two sources of return: the interest earned and any change in the exchange rate when the investment is converted back. Uncovered interest rate parity (UIP) connects these sources by stating that, under specified assumptions, expected returns measured in one currency are equal. The relationship is useful for interpreting how an interest-rate difference corresponds to an expected exchange-rate movement.

The word uncovered matters. The investor does not lock in the future exchange rate with a forward contract. The foreign investment therefore has an uncertain domestic-currency payoff. UIP is an expected-return condition, not a guaranteed return or an arbitrage rule. This distinguishes it from covered interest parity, where a forward exchange rate fixes the conversion rate.

Definition: Let $S_t$ be the spot exchange rate at time $t$, quoted as units of domestic currency per unit of foreign currency. Let $i_d$ and $i_f$ be the domestic and foreign effective interest rates for the same investment period. Under uncovered interest rate parity, the expected domestic-currency returns on the two investments are equal: $$ 1+i_d=(1+i_f)\frac{E_t[S_{t+1}]}{S_t}. $$ Here $E_t[S_{t+1}]$ is the expected future spot rate based on information available at time $t$. This condition assumes comparable investments, aligned interest-rate periods, and investors who require no additional expected return for bearing the foreign-exchange risk.

With this quote, a rise in $S_t$ means that one unit of foreign currency costs more domestic currency: the foreign currency has appreciated against the domestic currency. A fall means it has depreciated. The quote convention must be stated before interpreting the sign of an expected exchange-rate change.

Deriving the Implied Expected Spot Rate

The formula follows by comparing the two strategies for investing one unit of domestic currency over one period. The domestic investment produces a known payoff. The foreign investment must be converted into foreign currency first and then converted back at the uncertain future spot rate.

Invest domestically.
One unit of domestic currency earns interest $i_d$, producing $1+i_d$ units of domestic currency at the end of the period.

Invest abroad without a forward contract.
At the initial quote $S_t$, one domestic unit buys $1/S_t$ foreign units. After foreign interest, the balance is $(1+i_f)/S_t$. Converting at the future spot rate gives a domestic payoff of $(1+i_f)S_{t+1}/S_t$.

Compare expected returns.
Taking the expectation of the foreign payoff gives $(1+i_f)E_t[S_{t+1}]/S_t$, since the initial exchange rate and interest rate are known at time $t$. Equating that expected payoff to $1+i_d$ yields UIP.

E_t[S_{t+1}]=S_t\frac{1+i_d}{1+i_f}

This rearranged form calculates the expected future spot rate. It is exact for the stated one-period gross-return model; it does not require replacing the interest rates with their difference. In particular, if the interest rates are close to zero, the approximate expected percentage change in the quote is

\frac{E_t[S_{t+1}]-S_t}{S_t} =\frac{1+i_d}{1+i_f}-1 \approx i_d-i_f.

The approximation drops the product and higher-order effects in the ratio. The exact percentage change is $(i_d-i_f)/(1+i_f)$, so the interest-rate difference is not generally the exact percentage change. A log-change approximation is also common: for small rates and changes, the change in the logarithm of the quote is approximately $i_d-i_f$. These are approximations, whereas the gross-return ratio is the direct UIP calculation.

Applying UIP to One-Period Rates

For a one-year calculation, both interest rates must be effective annual rates, and the expected future spot rate must refer to one year ahead. The following examples use hypothetical market conditions. Each states the currency quote explicitly so that the direction of the expected movement is unambiguous.

Worked Example: A Higher Domestic Interest Rate

Suppose the spot rate is $1.20$ domestic currency units per euro. The one-year domestic interest rate is $5\%$, and the one-year euro interest rate is $2\%$. Under UIP, calculate the expected spot rate in one year and the implied expected percentage change in the quote.

Method. The quote is domestic currency per euro, so the domestic rate is $i_d=0.05$, the foreign rate is $i_f=0.02$, and $S_t=1.20$. Apply the gross-return ratio:

E_t[S_{t+1}] =1.20\frac{1+0.05}{1+0.02} =1.20\frac{1.05}{1.02} =1.20(1.0294118) =1.2352941.

To find the expected percentage change, compare the expected rate with the current rate:

\frac{1.2352941-1.20}{1.20} =\frac{0.0352941}{1.20} =0.0294118 =2.94118\%.

To four decimal places, the expected future spot rate is 1.2353 domestic currency units per euro, and the quote is expected to rise by 2.9412%. With this quote, that means the euro is expected to appreciate. The return check is $1.02(1.2352941/1.20)=1.05$, matching the domestic gross return. A common mistake is to call the expected change exactly $5\%-2\%=3\%$; that is only an approximation, not the exact change from the UIP ratio.

Worked Example: A Lower Domestic Interest Rate and a Different Quote

A hypothetical spot quote is $0.80$ euros per U.S. dollar. The one-year euro interest rate is $3\%$, and the one-year U.S. interest rate is $6\%$. Use UIP to find the expected spot quote and explain which currency is expected to appreciate under this quote.

Method. The euro is the domestic currency because the quote is euros per dollar. Thus $i_d=0.03$, $i_f=0.06$, and $S_t=0.80$. The expected quote is

E_t[S_{t+1}] =0.80\frac{1.03}{1.06} =0.80(0.9716981) =0.7773585.

The percentage change in the quoted rate is

\frac{0.7773585-0.80}{0.80} =\frac{-0.0226415}{0.80} =-0.0283019 =-2.83019\%.

To four decimal places, the expected spot rate is 0.7774 euros per dollar. The quote is expected to fall, so one U.S. dollar is expected to buy fewer euros; equivalently, the dollar is expected to depreciate against the euro. The return check is $1.06(0.7773585/0.80)=1.03$. The trap is to assume that the currency named first in a sentence must be domestic. Here the domestic currency is identified by the units in the quote, not by the order in which the currencies are mentioned.

Extending the Calculation Across Multiple Years

For a multi-year calculation, the interest-rate accumulation must cover the same horizon as the expected future spot rate. If the effective annual rates are constant for $n$ years, and the UIP condition is applied to the cumulative investment horizon, the corresponding relationship is

E_t[S_{t+n}] =S_t\frac{(1+i_d)^n}{(1+i_f)^n}.

This expression compounds each rate for $n$ years. It assumes the quoted rates are annual effective rates that remain applicable over the horizon. If rates vary by year, use the product of the annual gross returns for each currency instead. Do not compare a two-year expected spot rate with a one-year interest-rate factor.

Worked Example: A Two-Year Expected Spot Rate

Suppose $S_t=150$ units of domestic currency per unit of foreign currency. The domestic effective annual rate is $4\%$, the foreign effective annual rate is $1\%$, and both are assumed constant for two years. Calculate the expected spot rate two years from now under the cumulative UIP relationship.

Method. The two-year domestic gross accumulation is $(1.04)^2=1.0816$. The two-year foreign gross accumulation is $(1.01)^2=1.0201$. Therefore:

E_t[S_{t+2}] =150\frac{1.0816}{1.0201} =150(1.0602882) =159.04323.

To four decimal places, the expected two-year spot rate is 159.0432 domestic currency units per foreign currency unit. The expected quote change is $(159.04323-150)/150=0.0602882$, or $6.0288\%$. Check the accumulated returns: $1.0201(159.04323/150)=1.0816$, which matches the domestic accumulation. The common error is to use $2(4\%)$ and $2(1\%)$ as though simple interest applied. The stated rates are effective annual rates, so they compound.

What UIP Does—and Does Not—Claim

UIP is a condition about expected returns, not realized returns. The future spot rate is uncertain, so the actual foreign investment payoff can be above or below the domestic payoff even when the expected payoffs are equal. Equality of expected returns does not mean that either investment is risk-free.

That uncertainty also explains why UIP is not the same as a no-arbitrage condition. With a forward contract, an investor can lock in the conversion rate and compare known returns, as in covered interest parity. Without that contract, the foreign-currency payoff depends on the future spot rate. UIP therefore needs assumptions about how investors assess that uncertainty; it does not follow from a guaranteed arbitrage profit being unavailable. A treatment of the extra expected return investors may require for exchange-rate risk is a separate issue.

For economic interpretation, the sign of the interest differential gives a directional prediction conditional on the quote. If the domestic rate exceeds the foreign rate, the ratio $(1+i_d)/(1+i_f)$ exceeds one, so UIP predicts an increase in domestic currency per foreign currency. If the domestic rate is lower, the ratio is below one, so the quote is predicted to fall. In either case, state the quote before naming which currency is expected to appreciate.

Key takeaway: Under UIP, expected domestic-currency returns are equalized by an expected exchange-rate movement. With $S_t$ quoted as domestic currency per foreign unit, calculate the expected future quote using $S_t(1+i_d)/(1+i_f)$. Treat the result as an expectation under the model, not as a certain forecast or an arbitrage guarantee.

Common Exam Traps

Reversing the quote. For domestic currency per foreign unit, $S$ rising means the foreign currency appreciates. Reversing the quote reverses the interpretation.
Reversing the interest rates. The numerator is the domestic gross return and the denominator is the foreign gross return for this quote. Deriving the relationship from the two investment payoffs prevents a memorized formula from being applied backward.
Using a difference as an exact result. The exact expected percentage change is $(1+i_d)/(1+i_f)-1$. The interest-rate difference is an approximation when rates are small.
Mismatching horizons. Use rates and expected exchange rates for the same period. Compound effective annual rates when the horizon is more than one year.
Calling UIP arbitrage. The future spot rate is not locked in. UIP equates expected returns under assumptions; it does not promise equal realized returns.

Check Your Understanding

For each problem, identify the domestic currency from the quote before calculating. Unless stated otherwise, treat the rates as effective annual rates and assume UIP holds.

A one-year spot quote is $2.50$ Canadian dollars per pound. The Canadian rate is $4\%$, and the U.K. rate is $1\%$. Find the expected one-year spot rate and the expected percentage change in the quote. (Answer: $2.5743$ Canadian dollars per pound; $+2.9703\%$.)
The spot rate is $90$ yen per New Zealand dollar. The Japanese rate is $0.5\%$, and the New Zealand rate is $4.5\%$. Find the expected one-year quote and state the direction of the expected movement. (Answer: $86.5550$ yen per New Zealand dollar; the quote falls, so the New Zealand dollar is expected to depreciate against the yen.)
A spot quote is $0.72$ U.S. dollars per Swiss franc. The U.S. rate is $5\%$, and the Swiss rate is $2\%$. Calculate the expected one-year quote and its exact percentage change. (Answer: $0.7412$ U.S. dollars per Swiss franc; $+2.9412\%$.)
The spot rate is $1.30$ Australian dollars per U.S. dollar. The annual rates are $3\%$ in Australia and $5\%$ in the United States, constant for two years. Calculate the expected two-year spot rate using compounded returns. (Answer: $1.2509$ Australian dollars per U.S. dollar.)
The spot rate is $8.00$ units of domestic currency per foreign unit. The expected one-year spot rate is $8.24$, and the foreign rate is $2\%$. Under one-year UIP, find the domestic rate. (Answer: $5.06\%$.)

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