Interest parity condition

In international finance, the interest parity condition is the basic identity that describes the equilibrium in interest rates and exchange rates in economic models. It can be stated in words as: The foreign exchange market is in equilibrium when deposits of all currencies off the same expected rate of return.

The IPC assumes that financial assets are perfectly mobile and similarly risky. The basic interest parity condition is:

<math> i_$ = i_D + \frac {e_$^e - e_$} {e_$} (1 + i_D) <math>

where: <math> i_$ <math> is the interest rate in dollars, <math> i_D <math> is the interest rate in Currency 2, <math> e_$^e <math> is the expected exchange rate of dollars to Currency 2, and <math> e_$ <math> is the current exchange rate of dollars to Currency 2.


Another way to express the interest rate parity is,

<math>\mathbf { }(1 + i_$) = (F/S)(1 + i_c) <math>

where:

i$ , is the interest rate in the US
ic , is the interest rate in the foreign country
F , forward exchange rate between $ and foreign currency c, i.e. $/c
S , spot exchange rate between $ and foreign currency c, i.e. $/c

From the equation it is clear that when the parity condition does not hold, there is a possibility of an arbitrage opportunity.

For example let us assume that the LHS < RHS

i.e. <math> (1 + i_$) < (F/S)(1 + i_c) <math>

This would imply that one dollar invested in the US < one dollar converted into a foreign currency and invested abroad. Such an imbalance would give rise to an arbitrage opportunity, where in one could borrow at the lower effective interest rate in US, convert to the foreign currency and invest abroad. (see covered interest arbitrage below).

A more approximate version is sometimes given, although it is less correct for countries with high exchange rates.

<math> i_$ = i_D + \frac {e_$^e - e_$} {e_$} <math>

The chief implication of the IPC is that if a country's interest rates are relatively low compared to other countries, then that country's currency will tend to appreciate. Conversely, if the country's interest rates are relatively high, then the country's currency will tend to depreciate. If these conditions are not met, there exist arbitrage opportunities.

An example for the uncovered interest parity condition: Consider an initial situation, where interest rates in the US and a foreign country (e.g. Japan) are equal. Except for exchange rate risk, investing in the US (home) or Japan would yield the same return. If the dollar depreciates vis-a-vis the yen, an Investment in Japan would become more profitable than an US-Investment: For the same amount of dollars, more yen can be purchased. That is, the possible investment in Japan is higher than in the initial situation. In order to persuade an Investor to invest in the US nonetheless, the dollar interest rate would have to be higher than the yen interest rate by an amout equal to the devaluation (a 20% depreciation of the dollar implies a 20% rise in the dollar interest rate).

Covered Interest Parity

Why is there a close connection between forward and spot rates in the real world?

It follows directly from the interest parity condition that a forward exchange rate equals the spot exchange rate expected to prevail on the forward contract's value date.

Let's assume you wanted to pay for something in Yen in a months time. There are two ways to do this. (a) You could avoid exchange rate risk by buying some Yen now and selling your Yen forward for 30 days (for example in a Japanese 30 day fixed deposit). This is called covering because you now have covered yourself and have no exchange rate risk. (b) You could also invest the money in dollars and change it for Yen in a month. According to the IPC you would get the same number of Yen as with (a) (but you would still have some exchange rate risk.)

Without going into too much detail, these two methods are similar to what an exchange trader would do to get the 30 day forward rate (method a) and the expected spot rate in 30 days (method b). This links the two rates.

(As an aside, it is only the expected rate in 30 days that method (b) relies on. This may be different to the actual rate on that day, of course).

Covered Interest Arbitrage Example

The following is a rudimentary example to understand CIA (Covered Interest Rate Arbitrage)

Consider the Interest Rate Parity (IRP) equation,

<math>\mathbf { }(1 + i_$) = (F/S)(1 + i_c) <math>

Assume,

the 12-month interest rate in US is 5%, per annum
the 12-month interest rate in UK is 8%, per annum
the current Spot Exchange is 1.5 $/£
the current Forward Exchange is 1.5 $/£

From the given conditions it is clear that UK has a higher interest rate than the US. Thus the basic idea of Covered interest arbitrage is to borrow in the country with lower interest rate and invest in the country with higher interest rate. All else being equal this would help you make money riskless. Thus,

  • Per the LHS of the interest rate parity equation above, a dollar invested in the US at the end of the 12-month period will be,
$1 * (1 + 5%) = $1.05
  • Per the RHS of the interest rate parity equation above, a dollar invested in the UK (after conversion into £ and back into $ at the end of 12-months) at the end of the 12-month period will be,
$1 * (1.5%/1.5%)(1 + 8%) = $1.08

Thus, one could carry out a Covered Interest Rate (CIA) arbitrage as follows,

  1. Borrow $1 from the US bank at 5% interest rate.
  2. Convert $ into £ at current spot rate of 1.5$/£ giving 0.67£
  3. Invest the 0.67£ in the UK for the 12 month period
  4. Purchase a foward contract on the 1.5$/£ (i.e. cover your position against exchange rate fluctuations)

At the end of 12-months

  1. 0.67£ becomes 0.67£(1 + 8%) = 0.72£
  2. Convert the 0.72£ back to $ at 1.5$/£, giving $1.08
  3. Pay off the initially borrowed amount of $1 to the US bank with 5% interest, i.e $1.05
  • Making an arbitrage profit of $1.08 - $1.05 = $0.03 or 3 cents per dollar.

Obviously, any such arbitrage opportunities in the market will close out almost immediately.

In the above example, any one or combination of the following may occur to re-establish the equilibrium of the IRP to close out the arbitrage opportunity,

  • US interest rates will go up
  • Forward exchange rates will go down
  • Spot exchange rates will go up
  • UK interest rates will go down
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