# Interest Rate Parity

Interest rate parity is a theory proposing a relationship between the interest rates of two given currencies and the spot and forward exchange rates between the currencies. It can be used to predict the movement of exchange rates between two currencies when the risk-free interest rates of the two currencies are known.

Interest rate parity theory assumes that differences in interest rates between two currencies induce readjustment of exchange rate. However, exchange rates are determined by several other factors and not just the interest rate differences, therefore interest rate parity theory cannot predict or explain all movements in exchange rates. But it does serve as a useful guide nonetheless.

Interest rate parity theory is based on assumption that no arbitrage opportunities exist in foreign exchange markets meaning that investors will be indifferent between varying rate of returns on deposits in different currencies because any excess return on deposits in a given currency will be offset by devaluation of that currency and any reduced return on deposits in another currency will be offset by appreciation of that currency.

Covered interest rate parity exists when forward contract rates of currencies can be used to prove that no arbitrage opportunities exist. If forward exchange quotes are not available the interst rate parity exists but it is called uncovered interst rate parity.

## Formula

Covered interest rate parity may be presented mathematically as follows:

$$\frac{Forward\ Rate}{Spot\ Rate}=\left(\frac{1+i_{quot}}{1+i_{base}}\right)^n$$

Where iquot is the interest free rate of return on deposits of quote currency, ibase is that rate for base currency and n are the number of years until the date of foraward rate.

If exchange rate is quoted as USD/EUR i.e. Euros per 1 US Dollar, the USD is the base currency and EUR is the quote currency.

Interest rate parity can be used to estimate forward rates between two currencies by rearranging the above equation to:

$$Forward\ Rate=Spot\ Rate\times\left(\frac{1+i_{quot}}{1+i_{base}}\right)^n$$

## Example

Suppose mid-market USD/CAD spot exchange rate is 1.2500 CAD and one year forward rate is 1.2380 CAD. Also the risk-free interest rate is 4% for USD and 3% for CAD. Check whether interest rate parity exist between USD and CAD?

Solution:

$$Ratio\ of\ Forward\ to\ Spot = \frac{1.2380}{1.2500} = 0.9904$$

$$Ratio\ of\ Returns = \left(\frac{1+3\%}{1+4\%}\right)^1 \approx 0.9904$$

Since the two values are approximately equal, therefore interest rate parity exists.