Home > Exchange rates, Macro > The Uncovered Interest Parity Condition

## The Uncovered Interest Parity Condition

There is no point choosing to hold foreign money over domestic money – if you aren’t going shopping abroad there is nothing you can do with foreign money. But you may be interested in holding foreign interest-paying assets.

Consider the option of holding US and UK bonds. If the rate on UK bonds this year is $i_t$ then if you buy a UK bond, in a year’s time it will be worth $1+i_t$. If instead you spend the same amount of money that you would use to buy a UK bond, on US bonds, you would be able to get $E_t$ US bonds. If the rate on US bonds was $i*$ then you would get $E_t(1+i*)$ in a year’s time. But you would have to convert this back into pounds, so you would have to divide it by the exchange rate in a year’s time. So your overall return would be $\frac{E_t(1+i*)}{E_{t+1}}$.

Arbitrage will dictate that if both US and UK bonds are selling on the markets, that the expected return for both must be the same according to the uncovered interest parity condition:

$(1+i_t)=(1+i*_t)\frac{E_t}{E^e_{t+1}}$.

Say the spot (current) exchange rate was £1=$1.62. The markets expect that in a year’s time, the exchange rate will be £1=$1.64. US bonds pay a rate of 3.2%. The UIP condition will imply that $(1+i_t)=(1.032)\frac{1.62}{1.64}=1.01941$ so UK bonds would pay a rate of 1.94%.

Say you had £100. You could buy £100 of UK bonds and after a year they would be worth £100 x 1.01941 = £101.94. Alternatively, £100 could buy you £100 x 1.62 = $162 of US bonds now. In a year’s time they would be worth$162 x 1.032 = $167.184. When you converted that back into pounds (assuming the exchange rate was the same as had been expected) you would have 167.184/1.64 = £101.94. If the expected exchange rate next year was £1=$1.64 and UK bonds paid an interest rate higher than 1.94% then nobody would hold US bonds, they may as well hold UK bonds instead. If UK bonds paid an interest rate lower than 1.94% then nobody would hold UK bonds, they may as well hold US bonds. The fact that both are selling on the markets implies that arbitrage has equalised their expected return – although individual buyers may choose UK or US bonds because they expect that the exchange rate will be higher or lower than the general market expectation.

The UIP relation plays a central role in the real world workings of currency fluctuations. It says that the nominal exchange rate will rise if the domestic interest rate rises, or if the future expected exchange rate rises. It will fall if the foreign interest rate rises.

When rates are small we can make an approximation to the UIP condition:

$i_t \approx i*_t - \frac{E^e_{t+1}}{E_t}$

ie the domestic interest rate = foreign interest rate minus expected appreciation of the domestic currency

There are a few assumptions contained in the UIP – investors are assumed to always
want to hold the bonds with the highest expected return, and take no account of the relative risk. There are also assumed to be no transaction costs. And we are taking the expected future exchange rate as exogenous.