Uncovered interest rate parity: Definition & Examples

Uncovered interest rate parity (UIP) is the hypothesis that comparable domestic and foreign currency investments have equal expected returns when both returns are measured in the domestic currency and the foreign investment is left unhedged. It is a benchmark for separating an interest-rate advantage from expected currency movement. The future conversion rate remains uncertain, so equal expected returns can coexist with substantial realised gains or losses.

“Uncovered” means the investor has not fixed the conversion back with a forward. Unlike covered interest rate parity, UIP imposes a restriction on expectations and compensation for currency risk. It is not a cash-flow identity enforced by arbitrage. Peter Isard’s IMF working paper develops this distinction and the difficulties of testing it.

Measure both investments in the same currency

Take USD as domestic and EUR as foreign. Let

S_t

be the spot rate in USD per EUR, and let

S_T

be the uncertain rate on the matching maturity date. This is the market pair EUR/USD: a higher

S_T

means the euro buys more dollars.

Let $G_d$ and $G_f$ be the known growth factors on comparable default-free USD and EUR investments between $t$ and $T$ . Ignore transaction costs, taxes and transfer restrictions. Investing one dollar domestically delivers $G_d$ dollars. Converting it into euros, investing and converting back at the future spot delivers a gross dollar return

R_{f\to d}=\frac{G_fS_T}{S_t}.

Write $\mathcal{F}_t$ for the information available today and $\mathbb{P}$ for physical, or real-world, probabilities. The zero-currency-risk-premium UIP restriction in this dollar-return comparison is

\mathbb{E}^{\mathbb{P}}[R_{f\to d}\mid\mathcal{F}_t]=G_d \quad\Longleftrightarrow\quad \mathbb{E}^{\mathbb{P}}[S_T\mid\mathcal{F}_t] =S_t\frac{G_d}{G_f}.

This is a conditional expectation, not a statement that the terminal spot must equal that number. Nor does it equate physical probabilities with a risk-neutral measure used for derivative pricing. Assuming zero expected currency excess return here is an economic restriction; risk-neutral valuation is a pricing construction.

A synthetic dollar–euro comparison

Assume one year, simple annual USD interest of 3%, simple annual EUR interest of 1%, and

S_t=1.20

USD per EUR. Both year fractions are one; the rates are fictional, not market observations. UIP gives

\mathbb{E}^{\mathbb{P}}[S_T\mid\mathcal{F}_t] =1.20\frac{1.03}{1.01} =\frac{618}{505} \approx1.223762376\ \text{USD per EUR}.

Starting with USD 100, the dollar deposit delivers USD 103. The euro route buys EUR

250/3

today and grows to EUR

505/6

at maturity. Its terminal dollar wealth is therefore

V_T=\frac{505}{6}S_T.

At the UIP conditional mean spot, its expected dollar wealth is exactly USD 103. Its actual dollar wealth depends on the realised rate:

Illustrative future spot, USD per EUR	EUR deposit converted to USD	USD deposit
1.10	92.58	103.00
1.20	101.00	103.00
1.35	113.63	103.00

Amounts are rounded to cents, half away from zero. These are conditional scenarios, with no probabilities assigned to the rows.

A synthetic USD 100 investment earns USD 103 in the USD deposit. The EUR deposit’s dollar value rises with future spot and crosses USD 103 at the UIP mean spot of 1.223762376 USD per EUR. The crossing is an expected-return benchmark, not a guaranteed future exchange rate.

Original synthetic one-year comparison. The solid line is EUR deposit wealth converted at each possible future spot; the dashed line is the USD deposit. The vertical guide locates the UIP conditional mean, without specifying a probability distribution.

The expected proportional increase in this USD-per-EUR quote is

\frac{\mathbb{E}^{\mathbb{P}}[S_T\mid\mathcal{F}_t]-S_t}{S_t} =\frac{0.03-0.01}{1.01} =\frac{2}{101} \approx1.980198\%.

The familiar “exchange-rate change equals the interest differential” uses a small-rate approximation here: the exact change is not the 2 percentage point rate gap. In these quote units, the lower-yielding euro is expected to appreciate against the higher-yielding dollar.

For the quotation mechanics underneath this comparison, the FX primer explains base/quote units, bid–ask conversions and the covered-parity formula. Its full lesson is available with a free signed-in account through the public preview entry.

Where carry returns and risk premia enter

A currency carry trade borrows the lower-yielding currency and invests in the higher-yielding one without covering the exchange rate. In this example, borrow EUR

250/3

, convert it into USD 100 and invest the dollars. At maturity the dollar asset is worth USD 103, while the EUR debt is

505/6

. Buying those euros at

S_T

leaves the USD-measured net payoff

\Pi_T=103-\frac{505}{6}S_T.

It is USD 2.00 if spot stays at 1.20, but −10.63 if spot reaches 1.35. Under the stated UIP restriction its conditional expected USD payoff is zero. This is a financed trade’s payoff, not a percentage return on zero initial equity, and a positive interest differential does not guarantee a profit.

A risk premium allows expected returns to differ. Define

\rho_t

specifically as the foreign investment’s expected gross dollar return minus the domestic growth factor:

\rho_t=\mathbb{E}^{\mathbb{P}}[R_{f\to d}\mid\mathcal{F}_t]-G_d.

Then the same cash-flow algebra gives

\mathbb{E}^{\mathbb{P}}[S_T\mid\mathcal{F}_t] =S_t\frac{G_d+\rho_t}{G_f}.

The sign follows this definition; an author defining a premium on the opposite currency position may use the opposite sign. Norges Bank Investment Management’s carry-trade research discusses historical positive average carry returns alongside prolonged losses and possible risk compensation. Historical averages are not a promise about the next trade.

Two equalities to keep separate

Under frictionless CIP with the same funding factors, today’s locked forward satisfies

F_{t,T}=S_tG_d/G_f

. If the additional UIP restriction holds, that forward also equals the physical conditional expected spot in this comparison. CIP by itself gives no such forecast equality. A locked forward and an uncertain future spot play different roles even when their benchmark numbers coincide.

Quote inversion needs equal care. Changing the quote to EUR per USD introduces $1/S_T$ inside any expected return that uses it. In general,

\mathbb{E}^{\mathbb{P}}[1/S_T\mid\mathcal{F}_t] \ne \frac{1}{\mathbb{E}^{\mathbb{P}}[S_T\mid\mathcal{F}_t]}.

For positive rates and finite expectations, convexity gives a weakly greater left side, strictly greater when there is genuine conditional uncertainty. Re-derive returns in the chosen currency instead of inverting an expected rate. Finally, a single realised miss does not refute an expectation restriction: empirical tests need repeated observations and assumptions about how expectations are formed. Isard’s survey explains why those tests involve more than comparing one forecast with one outcome.

Sources

Peter Isard, “Uncovered Interest Parity,” IMF Working Paper 06/96 (2006), §§II–IV, printed pp. 3–7: expected-return formulation, reciprocal-rate caveat and empirical interpretation. The paper uses the opposite quote direction; this article derives its dollar-return convention explicitly.
Norges Bank Investment Management, “Currency Carry Trade” (2014): carry mechanics, historical return evidence and risk explanations.
John C. Hull, Options, Futures, and Other Derivatives, 9th Global ed., Pearson, 2018, §5.10, pp. 139–141: currency quote units and covered funding replication. The numerical example and realised-outcome figure here are original synthetic calculations with simple interest.

Measure both investments in the same currency

A synthetic dollar–euro comparison

Where carry returns and risk premia enter

Two equalities to keep separate

Sources

Related terms

Conditional expectation

Covered interest rate parity

Currency carry trade

Risk premium