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Chapter18: International aspects of financial management

18.4 Exchange rates and interest rates

   The next issue we need to address is the relationship between spot exchange rates, forward exchange rates and nominal interest rates. To get started, we need some additional notation:

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For exchange rates, see
<www.ozforex.
com.au
>
.
p. 550

As before, we will use S0 to stand for the spot exchange rate. You can take the Australian nominal risk-free rate as the short-dated government debt.

COVERED INTEREST ARBITRAGE

Suppose we observe the following information about Australian and Hong Kong currency in the market:

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where RHK is the nominal risk-free rate in Hong Kong. The period is one year, so F1 is the 365-day forward rate.

   Do you see an arbitrage opportunity here? There is one. Suppose you have $1 to invest, and you want a riskless investment. One option you have is to invest the $1 in a riskless Australian investment such as a one-year government bond. We will call this Strategy 1. If you do this, then, in one period, your $1 will be worth:

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   Alternatively, you can invest in the Hong Kong risk-free investment. To do this, you need to convert your $1 to HKD and simultaneously execute a forward trade to convert HKD back to dollars in one year. We will call this Strategy 2. The necessary steps would be as follows:

1.

 

Convert your $1 to $1 × S0 = HKD6.00.

2.

 

At the same time, enter into a forward agreement to convert HKD back to dollars in one year. Since the forward rate is HKD5.625, you get $1 for every HKD5.625 that you have in one year.

3.

 

Invest your HKD6.00 in Hong Kong at RHK. In one year you will have:

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4.

 

Convert your HKD6.30 back to dollars at the agreed-upon rate of HKD5.625 = $1. You end up with:

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   Notice that the value in one year from this strategy can be written as:

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The return on this investment is apparently 12%. This is higher than the 10% we get from investing in Australia. Since both investments are risk-free, there is an arbitrage opportunity.

   To exploit the difference in interest rates, you need to borrow, say, $5 million at the Australian rate and invest it at the Hong Kong rate and enter into a forward-exchange-rate agreement. To find out what the round-trip profit is from investing $5 million in this strategy, we can work through the steps above:

1.

 

Convert the $5 million at HKD6.00 = $1 to get HKD30 million.

2.

 

Agree to exchange Hong Kong dollars for Australian dollars in one year at HKD5.625 to the dollar.

p. 551

3.

 

Invest the HKD30 million for one year at RHK = 5%. You end up with HKD31.5 million.

4.

 

Convert the HKD31.5 million back to dollars to fulfil the forward contract. You receive HKD31.5 million/5.625 = $5 600 000.

5.

 

Repay the loan with interest. You owe $5 million plus 10% interest, for a total of $5.5 million. You have $5 600 000, so your round-trip profit is a risk-free $100 000.

   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 since 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 above—investing in a riskless Australian investment—gives us (1 + RAU) for every dollar we invest. Strategy 2, investing in a foreign risk-free investment, gives us S0 × (1 + RFC)/F1 for every dollar we invest. Since these have to be equal to prevent arbitrage, it must be the case that:

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   Rearranging this a bit gets us the famous interest rate parity (IRP) The condition stating that the interest rate differential between two countries is equal to the percentage difference between the forward exchange rate and the spot exchange rate. condition:

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   There is a very useful approximation for IRP that illustrates very clearly what is going on and is not difficult to remember. If we define the percentage forward premium or discount as (F1S0)/S0, then IRP says that this percentage premium or discount is approximately equal to the difference in interest rates:

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Very 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:

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In general, if we have t periods instead of just one, the IRP approximation will be written as:

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EXAMPLE 18.5
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PARITY CHECK
Suppose the exchange rate for Japanese yen, S0, is currently ¥120 = $1. If the interest rate in Australia is RAU = 10% and the interest rate in Japan is RJ = 5%, then what must the one-year forward rate be to prevent covered interest arbitrage?

   From IRP we have:

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Notice that the yen will sell at a premium relative to the dollar (why?).

   

p. 552

   

CONCEPT QUESTIONS

18.4a  

What is interest rate parity?

18.4b  

Do you expect that interest rate parity will hold more closely than purchasing-power parity? Why?

     

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