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## The Uncovered Interest Parity Condition

July 21, 2011 Leave a comment

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.

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Categories: Exchange rates, Macro

## Real exchange rates

July 21, 2011 Leave a comment

Some goods (eg cars, computers) are called tradable goods because they can be traded between countries, and consumers in one country have the choice of buying domestically produced goods or importing them from foreign firms. Other goods (eg tourism, haircuts) can’t be traded and so are non-tradable goods.

Usually smaller countries have higher percentages of their GDP that are made up of imports/exports, they must specialise in what they are good at and rely on imports for other products.

For consumers, the choice between buying domestic or foreign goods depends on the relative prices, any tariffs which distort the relative prices, and the real exchange rate (purchasing power in terms of goods of one currency in another).

The real exchange rate is the nominal exchange rate multiplied by the ratio of price levels:

Real exchange rate: $\epsilon = \frac{EP}{P*}$.

This notation is saying the real exchange rate is equal to the nominal exchange rate multiplied by the domestic price level, divided by the foreign price level.

The nominal exchange rate is simply the rate at which you trade one currency for another, eg £1:€1.13 is a nominal exchange rate saying you get 1.13 Euros to the Pound Sterling.

If the real exchange rate is high, foreign goods are relatively cheap and domestic goods are relatively expensive. If the real exchange rate is low, foreign goods are relatively expensive and domestic goods are relatively cheap.

Think in terms of one good, eg imagine the same type of car costs £8000 or €9000. The nominal exchange rate is £1=€1.14. So the real exchange rate is $\epsilon = \frac{1.14(8000)}{9000}=1.013$. In other words, we can exchange 1 British car for 1.01333 European cars. Now imagine a year later, the nominal exchange rate is £1=€1.26. The car sells in Britain for £8300 and in Europe for €9600. Now the real exchange rate is $\epsilon = \frac{1.26(8300)}{9600}=1.0894$. which means that now we can exchange 1 British car for 1.0894 European cars. The good in question (the car) now exchanges on better terms for British customers.

We can extend this principle to thinking of a generic basket of goods. Imagine a general basket of goods in the UK being given an ‘index’ of 100. If the same basket of goods in the US was 98 then we could conclude the basket was slightly cheaper in the US. Assume the nominal exchange rate is £1=$1.62. So the real exchange rate is $\epsilon = \frac{1.62(100)}{98}=1.653$. Because we are using index numbers, on its own this figure is arbitrary and uninformative. All it tells us is that in terms of the basket of goods we are comparing, £1 would buy$1.65306 worth of goods. Now imagine inflation in the UK is 4% and in the US is 2%, but a year later, the nominal exchange rate stays the same, £1=$1.62. Now the real exchange rate is $\epsilon = \frac{1.62(104)}{99.96}=1.685$. Now £1 would buy$1.685 worth of goods in the US, which sounds like it is a better deal for UK consumers, yet inflation has been higher here than in the US? This is because we kept the nominal exchange rate the same. We can consider what would happen if we assumed the real exchange rate stayed the same and the nominal exchange rate adjusted to reflect the differing levels of inflation in each country.

You can rearrange the real exchange rate formula to give $E = \frac{\epsilon P*}{P}$ so if we assume the real exchange rate in our example stayed the same, then $E = \frac{1.653(99.96)}{104}=1.588$. So as a result of the higher inflation in the UK compared to the US, the pound has fallen against the dollar. You now don’t get as many dollars for each pound as you did before, assuming the purchasing power in terms of goods has stayed the same in the two countries.

In practice there are likely to be shifts in both the nominal and real exchange rate.

If the real exchange rate appreciates or depreciates against another currency over time then that tells us that goods are less or more expensive in one country against another. The real exchange rate is basically a relative price and so it affects the demand for goods. If the real exchange rate is low then domestic goods are relatively cheaper so imports are likely to be lower and exports are likely to be higher. If the real exchange rate is high then foreign goods are relatively cheaper so imports are likely to be higher. In the example before of the cars, the rise in real exchange rate for British consumers means they will probably import more cars from Europe in the second year than they did in the first.

Categories: Exchange rates, Macro