Chapter31: International Corporate Finance
31.4 Interest Rate Parity, Unbiased Forward Rates, and the International Fisher Effect
The next issue we need to address is the relationship between spot exchange rates, forward exchange rates, and interest rates. To get started, we need some additional notation:
As before, we shall use S_{0} to stand for the spot exchange rate. You can take the home currency nominal riskfree rate, R_{HC}, to be the home country Tbill rate.
Covered Interest Arbitrage
From Fig. 31.1 we observe the following information about the British pound and the US dollar in the market:
where R_{FC} is the nominal riskfree rate in the United States. The period is one year, so F_{1} is the 360day forward rate.
Do you see an arbitrage opportunity here? Suppose you have £10,000 to invest, and you want a riskless investment. One option you have is to invest the £10,000 in a riskless UK investment such as a 360day Tbill. If you do this, then in one period your £1 will be worth
Alternatively, you can invest in the US riskfree investment. To do this, you need to convert your £10,000 to US dollars and simultaneously execute a forward trade to convert dollars back to pounds in one year. The necessary steps would be as follows:
Convert your £10,000 to £10,000 × S_{0} = $14,417.
At the same time, enter into a forward agreement to convert US dollars back to pounds in one year. Because the forward rate is $1.4415, you will get £1 for every $1.4415 that you have in one year.
Invest your $14,417 in the United States at R_{FC}. In one year, you will have
(K) Convert your $14,456 back to pounds at the agreedupon rate of $1.4415 = £1. You end up with
(K)
Notice that the value in one year resulting from this strategy can be written as
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The return on this investment is apparently 0.28 per cent. This is lower than the 2.13 per cent we get from investing in the United Kingdom. Because both investments are riskfree, there is an arbitrage opportunity.
To exploit the difference in interest rates, you need to borrow, say, $10 million at the lower US rate and invest it at the higher British rate. What is the roundtrip profit from doing this? To find out, we can work through the steps outlined previously:
Convert the $10 million at $1.4417/£ to get £6,936,256.
Agree to exchange dollars for pounds in one year at $1.4415 to the pound.
Invest the £6,936,256 for one year at R_{UK} = 2.13 per cent. You end up with £7,083,998.
Convert the £7,083,998 back to dollars to fulfil the forward contract. You receive £7,083,998 × $1.4415/£ = $10,211,583.
Repay the loan with interest. You owe $10 million plus 0.27 per cent interest, for a total of $10,027,000. You have $10,211,583, so your roundtrip profit is a riskfree $184,583.
The activity that we have illustrated here goes by the name of covered interest arbitrage. The term covered refers to the fact that we are covered in the event of a change in the exchange rate, because we lock in the forward exchange rate today.
Interest Rate Parity
If we assume that significant covered interest arbitrage opportunities do not exist, then there must be some relationship between spot exchange rates, forward exchange rates, and relative interest rates. To see what this relationship is, note that in general strategy 1 from the preceding discussion, investing in a riskless home currency investment, gives us 1 + R_{HC} for every unit of home currency we invest. Strategy 2, investing in a foreign riskfree investment, gives us S_{0} × (1 + R_{FC})/F_{1} for every unit of home currency we invest. Because these have to be equal to prevent arbitrage, it must be the case that:
Rearranging this a bit gets us the famous interest rate parity (IRP) condition:
There is a very useful approximation for IRP that illustrates clearly what is going on, and is not difficult to remember. If we define the percentage forward premium or discount as (F_{1} − S_{0})/S_{0}, then IRP says that this percentage premium or discount is approximately equal to the difference in interest rates:
Loosely, what IRP says is that any difference in interest rates between two countries for some period is just offset by the change in the relative value of the currencies, thereby eliminating any arbitrage possibilities. Notice that we could also write
In general, if we have t periods instead of just one, the IRP approximation is written like this:
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Forward Rates and Future Spot Rates
In addition to PPP and IRP, there is one more basic relationship we need to discuss. What is the connection between the forward rate and the expected future spot rate? The unbiased forward rates (UFR) condition says that the forward rate, F_{1}, is equal to the expected future spot rate, E(S_{1}):
With t periods, UFR would be written as
Loosely, the UFR condition says that, on average, the forward exchange rate is equal to the future spot exchange rate.
If we ignore risk, then the UFR condition should hold. Suppose the forward rate for the South African rand is consistently lower than the future spot rate by, say, 10 rand. This means that anyone who wanted to convert euros to rand in the future would consistently get more rand by not agreeing to a forward exchange. The forward rate would have to rise to get anyone interested in a forward exchange.
Similarly, if the forward rate were consistently higher than the future spot rate, then anyone who wanted to convert rand to euros would get more euros per rand by not agreeing to a forward trade. The forward exchange rate would have to fall to attract such traders.
For these reasons, the forward and actual future spot rates should be equal to each other, on average. What the future spot rate will actually be is uncertain, of course. The UFR condition may not hold if traders are willing to pay a premium to avoid this uncertainty. If the condition does hold, then the oneyear forward rate that we see today should be an unbiased predictor of what the exchange rate will actually be in one year.
Putting it All Together
We have developed three relationships – PPP, IRP, and UFR – that describe the interactions between key financial variables such as interest rates, exchange rates, and inflation rates. We now explore the implications of these relationships as a group.
Uncovered Interest Parity To start, it is useful to collect our international financial market relationships in one place:
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We begin by combining UFR and IRP. Because we know that F_{1} = E(S_{1}) from the UFR condition, we can substitute E(S_{1}) for F_{1} in IRP. The result is
This important relationship is called uncovered interest parity (UIP), and it will play a key role in our international capital budgeting discussion that follows. With t periods, UIP becomes
The International Fisher Effect Next we compare PPP and UIP. Both of them have E(S_{1}) on the left side, so their right sides must be equal. We thus have
This tells us that the difference in returns between the home country and a foreign country is just equal to the difference in inflation rates. Rearranging this slightly gives us the international Fisher effect (IFE):
The IFE says that real rates are equal across countries.^{2} The conclusion that real returns are equal across countries is really basic economics. If real returns were higher in, say, Britain than in the Eurozone, money would flow out of Eurozone financial markets and into British markets. Asset prices in Britain would rise and their returns would fall. At the same time, asset prices in Europe would fall and their returns would rise. This process acts to equalize real returns.
Having said all this, we need to note a couple of things. First, we haven’t explicitly dealt with risk in our discussion. We might reach a different conclusion about real returns once we do, particularly if people in different countries have different tastes and attitudes towards risk. Second, there are many barriers to the movement of money and capital around the world. Real returns might be different in two different countries for long periods if money can’t move freely between them.
Despite these problems, we expect that capital markets will become increasingly internationalized. As this occurs, any differences in real rates will probably diminish. The laws of economics have little respect for national boundaries.
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