 # Interest Rate Parity By Yuriy Smirnov Ph.D.

## Definition

The interest rate parity theorem implies that there is a strong relationship between the spot exchange rate and the forward exchange rate based on the interest rate differential between two countries. As a result, investors in both countries are indifferent as to where to invest their money. In other words, appreciation of one currency will be offset by a lower interest rate, and a higher interest rate will be offset by a devaluation of other currency.

## Assumptions

The theory of interest rate parity assumes the following assumptions are met:

1. Mobility of capital. There are no restrictions on capital flows between two countries, which means that any investor is allowed to transfer all its money from one country to another and vice versa at any time.
2. Assets are perfectly interchangeable. It is assumed that an investor from one country will be able to find a similar asset of the same risk and liquidity in another country.
3. There is no arbitrage. There are no arbitrage opportunities in the interest rate differential between two countries.
4. Foreign exchange market is in equilibrium. As a rule, the foreign exchange market is in equilibrium, but during financial crises, the opportunity of arbitrage may arise.

## Formula

### Uncovered interest rate parity

If there is no contract related to the forward exchange rate, the interest rate parity is called uncovered. The equation describing it is as follows:

 (1 + rA) = St+k (1 + rB) St

where rA is the interest rate in Country A, rB is the interest rate in Country B, St+k is the expected spot exchange rate at the time (t+k), and St is the current spot exchange rate at the time t.

### Covered interest rate parity

If there is a related forward contract, i.e., the forward exchange rate is known in advance, the interest rate arbitrage is called covered. In such cases, the equation above should be transformed as follows:

 (1 + rA) = Ft (1 + rB) St

where Ft is forward exchange rate at the time t.

It should be noted that a forward contract, which is used to lock the forward exchange rate, eliminates the arbitrage opportunity. Moreover, the interest rate differential determines the size of the forward premium or discount on a foreign exchange rate.

## Example

An investor from the United States has an opportunity to invest \$1,000,000 USD at 4.25% in the domestic market or invest in the European Union market at 3.60%. What should be the 6-month forward exchange rate between the Euro and the U.S. dollar if the current spot exchange rate is EUR/USD 1.3275?

### Solution

We should use the equation of covered interest rate parity to compute the forward exchange rate. The formula to find it is as follows:

 Ft = St(1 + rUSD) (1 + rEUR)

Let’s put all the data available in the formula above, but we should first adjust the annual interest rate to a semiannual basis. Thus, the semiannual interest rate in the U.S. is 2.125% (4.25%÷2), and it is 1.80% in the EU.

 Ft = 1.3275 × (1 + 0.02125) = 1.3317 1 + 0.0180

Let’s use the example above to illustrate how interest rate parity works. An investor has two options:

1. Invest \$1,000,000 USD in the local market for 6 months at an annual interest rate of 4.25%.
2. Convert USD for EUR and invest in the EU market for 6 months at an annual interest rate of 3.60%.

### Option I

Investing \$1,000,000 USD at 4.25% will yield \$1,021,250 USD in the 6-month period.

\$1,000,000 × (1 + 0.0425 ÷ 2) = \$1,021,250 USD

### Option II

Alternatively, an investor may convert \$1,000,000 USD at the current spot exchange rate of 1.3275 and get 753,295.67 EUR. This amount will yield 766,855 EUR in the 6-month period.

753,295.67 × (1 + 0.036 ÷ 2) = 766,855 EUR

If there is no arbitrage opportunity and interest rate parity exists, \$1,021,250 USD must be equal to 766,855 EUR. Thus, the 6-month forward exchange rate will be 1.3317.

 6-month forward exchange rate = \$1,021,250 = 1.3317 766,855

## Criticism

The theory of interest rate parity is strongly criticized because the assumptions it is based on do not exist in real markets, which very often face a situation when an increase in demand for a currency with a high interest rate results in its appreciation against other currencies with lower interest rates. This contradicts the key assumptions of the theory stating that an arbitrage opportunity does not exist.

It should be noted that covered interest rate parity can be met quite accurately for the short term between advanced economies, but some discrepancies always exist because of transaction costs.