Interest-Rate Differentials and Exchange Rates

Why Interest-Rate Differences Matter for Exchange Rates

An investor comparing deposits in two countries cannot compare interest rates alone. A foreign deposit pays interest in its own currency, so the investor also cares about how much that currency will be worth when the funds are converted back. A higher foreign interest rate may be offset by an expected fall in the foreign currency’s value, and uncertainty about the exchange rate adds risk to the comparison.

The key is to keep the exchange-rate quotation consistent. In this tutorial, $S_t$ is the spot exchange rate at time $t$, measured as units of domestic currency per one unit of foreign currency. Thus, if $S_t$ rises, one unit of foreign currency costs more domestic currency: the foreign currency has appreciated relative to the domestic currency. The rates compared below are effective rates over the same investment horizon, and the investment principal is assumed to be free of default risk unless stated otherwise.

Definition: Let $i_d$ be the domestic interest rate, $i_f$ the foreign interest rate, and $S_0$ the current spot exchange rate in domestic currency per unit of foreign currency. If one unit of domestic currency is invested abroad, its foreign-currency value is $1/S_0$. At the end of the period, the foreign investment’s domestic-currency value depends on the future spot rate $S_1$.

Calculating the Return on a Foreign Investment

Start with one unit of domestic currency. At the current exchange rate, it buys $1/S_0$ units of foreign currency. After earning the foreign rate $i_f$, the investment is worth $(1+i_f)/S_0$ foreign currency units. Converting back at the future spot rate $S_1$ gives a domestic-currency amount of $(1+i_f)S_1/S_0$. If $S_1$ is uncertain, this expression is a random return; replacing $S_1$ by its expected value calculates an expected return.

\text{Domestic gross return on foreign investment} = (1+i_f)\frac{S_1}{S_0}

E[\text{foreign investment return}] = (1+i_f)\frac{E[S_1]}{S_0}-1

The domestic deposit’s gross return is $1+i_d$, so its net return is $i_d$. These formulas assume the same investment horizon and that conversion can occur at the stated spot rates without transaction costs. In practice, fees, taxes, restrictions on moving funds, credit risk, and differences in liquidity may also affect the comparison.

Worked Example: Inferring an Expected Exchange-Rate Change

A hypothetical investor can place funds in a domestic deposit paying 5% for one year or a foreign deposit paying 3%. The current spot rate is 1.20 domestic currency units per unit of foreign currency. Assume investors are risk-neutral, the deposits have the same maturity and negligible default risk, and there are no transaction costs or capital restrictions. Under uncovered interest parity, what expected one-year spot rate makes the two investments equally attractive?

Method. Uncovered interest parity equates the domestic gross return with the expected domestic gross return on the foreign deposit. Solve for the expected future spot rate:

1+i_d=(1+i_f)\frac{E[S_1]}{S_0} \qquad\Longrightarrow\qquad E[S_1]=S_0\frac{1+i_d}{1+i_f}

Substitute $S_0=1.20$, $i_d=0.05$, and $i_f=0.03$:

E[S_1] = 1.20\frac{1.05}{1.03} = \frac{1.26}{1.03} = 1.2233009709\ldots

The expected percentage change in the exchange rate is

\frac{E[S_1]-S_0}{S_0} = \frac{1.2233009709\ldots-1.20}{1.20} = 0.0194174757\ldots

The expected exchange rate is 1.22330097 domestic currency units per foreign unit, shown to eight decimal places, and the expected increase is 1.94%, rounded to two decimal places. With this quotation, the increase means that the foreign currency is expected to appreciate and the domestic currency to depreciate. The expected foreign investment gross return is $1.03(1.2233009709\ldots/1.20)=1.05$, matching the domestic gross return of 1.05. The calculation avoids the mistake of interpreting a rising $S$ as foreign-currency depreciation.

Uncovered Interest Parity and Its Interpretation

Uncovered interest parity (UIP) is a benchmark relationship between interest rates and the expected future spot rate. It says that, under the assumptions used in the previous example, investors require equal expected returns on comparable domestic and foreign deposits. “Uncovered” means the future exchange rate is not fixed by a forward contract: investors remain exposed to the eventual spot rate.

Uncovered interest parity: If comparable investments have no material default or liquidity differences, conversion is frictionless, capital can move freely, and investors are risk-neutral, then equal expected returns imply $$ 1+i_d=(1+i_f)\frac{E[S_1]}{S_0}, \qquad \frac{E[S_1]}{S_0}=\frac{1+i_d}{1+i_f}. $$ This is a model benchmark, not a guarantee that actual exchange rates will move as predicted.

The reason for the relationship is the return calculation itself. A domestic deposit turns one unit of domestic currency into $1+i_d$. The foreign investment first converts the unit into $1/S_0$ foreign currency, earns $1+i_f$, and is expected to convert back into $(1+i_f)E[S_1]/S_0$ domestic currency. Equal expected returns require setting these two gross amounts equal. If the expected rates do not satisfy the equality, the model predicts a difference in expected returns; whether investors act on it depends on risk and frictions.

UIP also gives a directional interpretation. If the domestic interest rate exceeds the foreign rate, then $E[S_1]/S_0$ exceeds 1 under UIP. With the quotation used here, the domestic currency is expected to depreciate against the foreign currency. This expected depreciation offsets the domestic rate advantage when an investor compares returns from the two currencies. The exact relationship uses gross rates; the familiar approximation that the expected percentage change in $S$ is about $i_d-i_f$ is most useful when rates and exchange-rate changes are small.

Expected Returns Are Not Guaranteed Returns

An expected future spot rate is not a promise about the rate that will prevail. Two possible future exchange rates can have the same expected value but produce very different outcomes for an investor. In addition, investors may dislike exchange-rate risk and may require compensation for bearing it. The risk-neutral equality in UIP therefore need not hold exactly in observed data.

A useful distinction is between the expected excess return on a foreign investment and a risk premium. The expected excess return is a calculated difference between two expected returns. A risk premium is compensation investors require for bearing risk; its size and sign depend on how the investment covaries with risks that matter to investors. The calculated excess return is not automatically equal to a known or constant risk premium.

Worked Example: Comparing Expected Returns and Currency Risk

A hypothetical investor compares a domestic one-year deposit paying 3% with a foreign deposit paying 4%. The current exchange rate is 2.00 domestic currency units per foreign unit. At the end of the year, the exchange rate will be 1.80 with probability 0.50 or 2.20 with probability 0.50. Assume the foreign deposit has no default risk and there are no transaction costs. Calculate both expected returns and the standard deviation of the foreign investment’s net return.

Method. Calculate the foreign investment’s domestic gross return in each exchange-rate outcome, then use the probabilities to find its mean and standard deviation.

If $S_1=1.80$, the gross return is $1.04(1.80/2.00)=0.936$, so the net return is $0.936-1=-0.064$, or $-6.4\%$. If $S_1=2.20$, the gross return is $1.04(2.20/2.00)=1.144$, so the net return is $1.144-1=0.144$, or $14.4\%$.

The expected net return is

$$ E[R_f] = 0.50(-0.064)+0.50(0.144) = -0.032+0.072 = 0.040 $$

Thus, the foreign investment’s expected return is 4.0%, compared with 3.0% on the domestic deposit. Its expected excess return is $4.0\%-3.0\%=1.0$ percentage point.

The two outcomes differ from the mean return of 0.040 by $-0.104$ and $0.104$. Therefore,

\operatorname{Var}(R_f) = 0.50(-0.104)^2+0.50(0.104)^2 = 0.010816

The standard deviation is $\sqrt{0.010816}=0.104$, or 10.4 percentage points. The expected return is higher, but the foreign investment can lose 6.4% in the first outcome. The calculation avoids the mistake of treating a positive expected excess return as a guaranteed gain or as sufficient information to determine which investment a risk-averse person should choose.

Covered Interest Parity: A Different Comparison

If an investor uses a forward contract to lock in the exchange rate for the future conversion, the comparison is covered, not uncovered. Let $F_0$ denote the forward exchange rate for delivery at the investment horizon, also quoted in domestic currency per unit of foreign currency. A forward contract sets the conversion rate in advance, so the foreign deposit’s domestic gross return is $F_0(1+i_f)/S_0$, rather than a return depending on the uncertain future spot rate.

In a frictionless market with comparable default-free deposits, no-arbitrage requires the covered returns to match:

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

This follows by comparing two ways of investing one unit of domestic currency: leave it in the domestic deposit, or convert it at $S_0$, invest abroad, and contract to convert the foreign proceeds back at $F_0$. If the two certain domestic amounts differed, an investor could borrow in the lower-return route and invest in the higher-return route, subject to the model’s assumptions. Covered parity concerns a contractual forward rate; UIP concerns an expected future spot rate. The two rates should not be substituted for each other.

Worked Example: Checking a Covered Return

A hypothetical domestic deposit pays 2% over one year, and a foreign deposit pays 5%. The current exchange rate is 1.50 domestic currency units per foreign unit. A one-year forward rate is quoted at 1.47 domestic currency units per foreign unit. Assume there are no transaction costs, credit differences, or restrictions on borrowing and lending. Compare the two covered gross returns and identify whether the quoted rates satisfy covered interest parity.

Method. Calculate the domestic deposit’s gross return and the hedged foreign investment’s gross return using the contractual forward rate.

The domestic gross return is $1+0.02=1.02$. Starting with one domestic unit, the investor obtains $1/1.50$ foreign units. After the foreign deposit pays 5%, the amount is $1.05/1.50=0.70$ foreign units. Converting that amount at the forward rate gives

$$ 0.70(1.47)=1.029 $$

The covered foreign gross return is 1.029, or a net return of 2.9%. This exceeds the domestic gross return of 1.02 by $1.029-1.02=0.009$, which is 0.9 percentage point of the initial investment. Thus, the quoted rates do not satisfy covered interest parity under the stated assumptions. The calculation avoids using an expected future spot rate: the forward rate is already specified by contract. In actual markets, transaction costs, bid-ask spreads, credit exposure, and other frictions can prevent an apparent return gap from being a risk-free arbitrage opportunity.

Common Mistakes and Practical Limits

Reversing the exchange-rate quotation. Here, $S$ is domestic currency per foreign unit. A rise in $S$ means the foreign currency appreciates. With the inverse quotation, the directional interpretation reverses.
Comparing rates without currency conversion. A higher $i_f$ alone does not show that a foreign investment has a higher domestic-currency return. The exchange-rate factor $S_1/S_0$ also matters.
Confusing expected and realized changes. UIP relates rates to an expected exchange rate under assumptions. The actual exchange rate can differ from that expectation.
Mixing covered and uncovered comparisons. Use $F_0$ for a contractual forward conversion and $E[S_1]$ for an expected, unhedged future spot rate.
Assuming risk does not matter. Currency risk, default risk, and liquidity differences can affect the return investors require. A model prediction does not by itself establish what a risk-averse investor will choose.

Interest-rate differentials can influence cross-border demand for currencies because investment decisions involve both the yield and the expected conversion value. But the relation is conditional: its exact interpretation depends on the exchange-rate quotation, the horizon, whether currency exposure is hedged, and the assumptions about risk and market frictions. This framework is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each question, use domestic currency per unit of foreign currency as the exchange-rate quotation. Assume rates refer to the stated horizon.

Domestic and foreign one-year rates are 6% and 2%, respectively. The current exchange rate is 0.80 domestic units per foreign unit. Under UIP, calculate the expected future spot rate and its percentage change from today. Give the expected rate to eight decimal places and the percentage change to two decimal places. (Answer: $E[S_1]=0.83137255$; expected increase $=3.92\%$.)
The domestic rate is 2%, the foreign rate is 5%, and the current exchange rate is 10.00 domestic units per foreign unit. Under UIP, find the expected future exchange rate and state which currency is expected to depreciate. (Answer: $E[S_1]=9.71428571$; the foreign currency is expected to depreciate by 2.86% against the domestic currency.)
The domestic rate is 3%, the foreign rate is 7%, and $S_0=4.00$. If the expected future exchange rate is 4.10, calculate the foreign investment’s expected domestic-currency return and its expected excess return over the domestic deposit. (Answer: foreign expected return $=9.675\%$; expected excess return $=6.675$ percentage points.)
A foreign deposit pays 2%, $S_0=5.00$, and $S_1$ is 4.50 with probability 0.40 or 5.50 with probability 0.60. The domestic deposit pays 3.5%. Calculate the foreign investment’s expected return and expected excess return. (Answer: expected foreign return $=4.04\%$; expected excess return $=0.54$ percentage point.)
The domestic rate is 4%, the foreign rate is 1%, and $S_0=1.25$. Under risk-neutral UIP, calculate the expected future exchange rate and its percentage change. Give the rate to eight decimal places and the percentage change to two decimal places. (Answer: $E[S_1]=1.28712871$; expected increase $=2.97\%$.)

What You'll Learn