Risk Premia, Expectations, and Carry Trades

Why Currency Risk Changes the Return Comparison

An investment in another country earns more or less than its stated interest rate once its proceeds are converted back into the investor’s currency. The exchange-rate outcome is uncertain, so an investor cannot generally compare two interest rates alone and conclude which investment will have the higher realized return.

Uncovered interest rate parity (UIP) gives a benchmark for the expected exchange-rate movement that would equalize expected returns. But an unhedged foreign investment exposes the investor to currency risk. If investors require compensation for bearing that risk, expected returns need not be equal: UIP can fail even when no risk-free arbitrage is available. A carry trade seeks to benefit from an interest-rate difference while accepting this exchange-rate risk.

The discussion uses a one-period model, with no transaction costs, taxes, capital controls, or default risk. There are no mortality-contingent cash flows, so no mortality assumption applies. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Let $S_t$ be the spot exchange rate at time $t$, quoted as units of domestic currency per unit of foreign currency. Define $R=S_1/S_0$, the gross change in this exchange-rate quote over one period. If $R<1$, one unit of foreign currency buys fewer units of domestic currency at the end of the period: the foreign currency has depreciated against the domestic currency. Let $i$ be the domestic interest rate and $i^*$ the foreign interest rate, both effective rates for the same period. The foreign investment’s domestic-currency gross return is $(1+i^*)R$.

For example, a domestic investor who converts one unit of domestic currency into foreign currency at $S_0$, invests at rate $i^*$, and converts the proceeds back at $S_1$ ends with $(1+i^*)S_1/S_0=(1+i^*)R$ units of domestic currency. The comparable domestic investment ends with $1+i$. Because $R$ is uncertain, the foreign investment’s expected gross return is $(1+i^*)E[R]$.

UIP and the Risk Premium

In this notation, UIP is the condition that the expected domestic-currency gross return on the foreign investment equals the domestic gross return. It implies a particular expected exchange-rate ratio. This is a statement about expected returns, not a risk-free arbitrage condition: the foreign investment’s final domestic-currency value is unknown when the investment is made.

Definition: In the one-period model, the arithmetic currency risk premium is the expected foreign investment’s gross return minus the domestic gross return: $$ \pi=(1+i^*)E[R]-(1+i). $$ UIP holds when $\pi=0$, or equivalently when $E[R]=(1+i)/(1+i^*)$. A positive $\pi$ means the foreign investment has a higher expected domestic-currency return; a negative $\pi$ means it has a lower expected return. The premium is a return difference, written as a decimal or converted to percentage points.

The direction of the exchange-rate ratio matters. With domestic currency per unit of foreign currency, UIP requires $E[R]$ to be less than one when $i^*>i$: the foreign currency is expected to depreciate enough to offset its higher interest rate. If the exchange rate is quoted in the inverse direction, the corresponding formula must also be inverted.

Write each investment’s gross return in the same currency.
The domestic investment returns $1+i$. The foreign investment returns $(1+i^*)R$ in domestic currency.

Take expectations for the unhedged investment.
Since $i^*$ is fixed and $R$ is uncertain, $E[(1+i^*)R]=(1+i^*)E[R]$.

Compare expected returns.
Subtract the domestic gross return. The difference is $\pi=(1+i^*)E[R]-(1+i)$; setting it to zero yields the UIP condition.

The key reason equality may fail is that expected return is not the same as a risk-free return. The exchange-rate outcome can matter most in states when an investor especially values wealth. An investor may therefore require a higher expected return to hold a foreign-currency exposure, or may accept a lower expected return if that exposure provides valuable protection in adverse states. The premium depends on how the currency risk is valued, not simply on the amount of exchange-rate volatility. No-arbitrage reasoning alone does not force two risky, unhedged investments to have equal expected returns.

This also distinguishes UIP from covered interest parity. Covered interest parity compares returns after the exchange rate has been fixed in advance with a forward contract. UIP instead uses an uncertain future spot exchange rate. A forward rate is therefore not automatically equal to the expected future spot rate: a risk premium can separate them.

Worked Example: Calculating a Positive Risk Premium

A domestic investor compares a domestic deposit paying 4% for one year with a foreign deposit paying 7%. The exchange-rate ratio $R$ is 0.97 with probability 0.5 and 0.99 with probability 0.5. Calculate the expected risk premium and compare it with UIP. Assume no transaction costs or taxes and no mortality contingency.

Method. First calculate $E[R]$ from the stated discrete distribution, then compare the expected foreign return with the domestic return. The interest rates are $i=0.04$ and $i^*=0.07$.

The expected exchange-rate ratio is $E[R]=0.5(0.97)+0.5(0.99)=0.485+0.495=0.98$. Thus the expected foreign gross return is $1.07(0.98)=1.0486$, or a net expected return of 4.86%. The domestic gross return is $1.04$, or 4%.

The risk premium is $\pi=1.0486-1.04=0.0086$, equal to 0.86 percentage points. For a hypothetical initial investment of $100,000, expected foreign proceeds are $$100{,}000(1.0486)=$104{,}860$; domestic proceeds are $$100{,}000(1.04)=$104{,}000$. The expected difference is $860.

Under UIP, the expected ratio would instead be $1.04/1.07=0.9719626$ (to seven decimal places). The assumed expectation, 0.98, is higher, so the foreign investment has the positive expected-return premium calculated above. Check: $1.07(0.98)-1.04=0.0086$, consistent with the $860 difference on $100,000.

Common mistake avoided: Subtracting the interest rates gives $7\%-4\%=3\%$, but that is not the currency-adjusted expected premium. The foreign currency is expected to depreciate, and accounting for that exchange-rate movement reduces the premium to 0.86 percentage points.

Carry Trades: Expected Return and Risk

A carry trade typically borrows in a lower-interest-rate currency and invests in a higher-interest-rate currency. In the notation here, a domestic investor borrows domestic currency at rate $i$, converts it to foreign currency, and invests at $i^*>i$. At the end of the period, the investor converts the foreign proceeds back to domestic currency and repays the domestic borrowing.

For one unit of domestic currency borrowed, the investment produces $(1+i^*)R$ units of domestic currency at the end of the period, while repayment is $1+i$. The trade’s net profit per unit borrowed is therefore $(1+i^*)R-(1+i)$. Its expected net profit per unit borrowed is $\pi$, and its return on the borrowed amount is that net profit. A higher foreign interest rate does not guarantee a profit: a sufficiently large depreciation of the foreign currency can outweigh the interest-rate advantage.

The distribution of the exchange-rate ratio is useful for assessing both expected profit and risk. With a fixed foreign interest rate, the variance of the foreign investment’s gross domestic-currency return is $(1+i^*)^2\operatorname{Var}(R)$. Subtracting the fixed domestic repayment changes the mean but not the variance.

Result: For a carry trade with fixed one-period rates, the net return per unit of domestic currency borrowed is $C=(1+i^*)R-(1+i)$. Thus $E[C]=\pi$, and $\operatorname{Var}(C)=(1+i^*)^2\operatorname{Var}(R)$. A positive expected carry is not a guarantee of a positive realized profit.

Worked Example: Expected Profit and Exchange-Rate Risk

An investor hypothetically borrows $50,000 in domestic currency for one year at 3%, converts it at $S_0=2.00$ domestic currency units per foreign unit, and invests the proceeds at 8% abroad. The one-year ratio $R=S_1/S_0$ is 0.92 with probability 0.25 and 1.02 with probability 0.75. Calculate the expected profit after repaying the borrowing and the standard deviation of the trade’s return on the borrowed amount. Assume no fees, taxes, or mortality contingency.

Method. Compute the exchange-rate expectation and variance, then calculate the domestic-currency payoff in each state. Here $i=0.03$, $i^*=0.08$, and the initial conversion gives $$50{,}000/2.00=25{,}000$ foreign currency units.

After earning 8%, the foreign investment is $25{,}000(1.08)=27{,}000$ foreign currency units. The domestic loan repayment is $$50{,}000(1.03)=$51{,}500$.

In the $R=0.92$ state, $S_1=2.00(0.92)=1.84$. The investment converts to $27{,}000(1.84)=$49{,}680$, so profit after repayment is $$49{,}680-$51{,}500=-$1{,}820$. In the $R=1.02$ state, $S_1=2.00(1.02)=2.04$. Proceeds are $27{,}000(2.04)=$55{,}080$, so profit is $$55{,}080-$51{,}500=$3{,}580$.

The expected exchange-rate ratio is $E[R]=0.25(0.92)+0.75(1.02)=0.23+0.765=0.995$. Expected investment proceeds are $27{,}000[2.00(0.995)]=$53{,}730$. Subtracting the $51,500 repayment gives expected profit of $2,230. Relative to $50,000 borrowed, that is an expected return of $2{,}230/50{,}000=0.0446$, or 4.46%.

For the risk calculation, $\operatorname{Var}(R)=0.25(0.92-0.995)^2+0.75(1.02-0.995)^2=0.25(0.005625)+0.75(0.000625)=0.001875$. Therefore, the standard deviation of the trade’s return is $1.08\sqrt{0.001875}=1.08(0.0433013)=0.0467654$, or 4.68% to two decimal places. The state returns provide a check: $-1{,}820/50{,}000=-3.64\%$ and $3{,}580/50{,}000=7.16\%$; their probability-weighted mean is $0.25(-3.64\%)+0.75(7.16\%)=4.46\%$.

Common mistake avoided: The interest-rate spread is $8\%-3\%=5\%$, but it is not the expected carry-trade return. The expected return is 4.46% because the foreign currency’s expected exchange-rate movement offsets part of the spread. The standard deviation also shows why the expected result is not a guaranteed outcome.

Reading the Premium and Avoiding Common Traps

The risk premium is a signed difference in returns, not a percentage of the exchange-rate ratio. For instance, a value of $\pi=-0.0197$ means a negative difference of 1.97 percentage points between expected foreign and domestic returns. It does not mean $-1.97$ when the formula is expressed in decimal return units.

When using a risk premium to infer an expected exchange-rate ratio, rearrange its definition:

E[R]=\frac{1+i+\pi}{1+i^*}.

This relationship makes the role of the premium explicit. When $\pi=0$, it reduces to the UIP expected ratio. When $\pi\ne0$, the expected ratio differs from the UIP benchmark. The result is conditional on the stated quote and one-period return convention; do not carry it over unchanged to an inverse quote or a different time interval.

Do not treat the interest-rate spread as the carry return. The exchange-rate ratio multiplies the foreign investment’s gross return.
Do not reverse the exchange-rate quote. Here $R<1$ means foreign-currency depreciation. An inverse quote moves in the opposite direction.
Do not confuse an expected outcome with a guaranteed outcome. A positive expected premium can coexist with losses in some exchange-rate states.
Do not infer risk-free arbitrage from unequal expected returns. An unhedged foreign investment is exposed to uncertain currency movements; covered interest parity concerns a different, hedged comparison.

Key takeaway: Calculate a foreign investment’s return in the investor’s currency before comparing it with a domestic return. UIP is the zero-risk-premium benchmark, while a carry trade accepts exchange-rate risk in pursuit of an expected return difference.

Check Your Understanding

For each problem, use the domestic-currency-per-foreign-unit quote and the one-period return convention in this tutorial. Assume no transaction costs, taxes, or mortality contingency.

A domestic deposit pays 5% and a foreign deposit pays 9%. The exchange-rate ratio is 0.95 with probability 0.25 and 1.01 with probability 0.75. Find $E[R]$ and the risk premium. (Answer: $E[R]=0.995$; $\pi=0.03455$, or 3.455 percentage points.)
The domestic rate is 3% and the foreign rate is 8%. The exchange-rate ratio is 0.94 with probability 0.25 and 0.98 with probability 0.75. Calculate the foreign investment’s expected gross return and its risk premium. (Answer: expected gross return $=1.0476$; $\pi=0.0176$, or 1.76 percentage points.)
An investor borrows $20,000 domestically at 4% and invests abroad at 10%. The exchange-rate ratio is 0.90 with probability 0.20 and 1.05 with probability 0.80. Find expected proceeds, the repayment, and expected profit. (Answer: expected proceeds $=$22{,}440$; repayment $=$20{,}800$; expected profit $=$1{,}640$.)
A foreign investment pays 5%, while the domestic borrowing rate is 2%. The exchange-rate ratio is 0.94 or 1.02, each with probability 0.50. Calculate the standard deviation of the carry trade’s return on the borrowed amount. (Answer: $0.042$, or 4.20%.)
The domestic rate is 1.5%, the foreign rate is 4.5%, and the risk premium is $0.012$. Find the implied expected exchange-rate ratio and state the implied expected percentage change in the quote. (Answer: $E[R]=1.027/1.045=0.9828$ to four decimals; the quote is expected to fall by approximately 1.7225%.)

What You'll Learn