According to the interest rate parity theory, the currency margin is dependent mainly on the prevailing interest rate (for investment for the given time period) in the two currencies. The forward rate can be calculated by the following formula:

**F/S = (1+Rh)/ (1+Rf)**

Where, F and S are future and spot currency rate. Rh and Rf are simple interest rate in the home and foreign currency respectively.

Alternatively, if we consider continuously compounded interest rate then forward rate can be calculated by using the following formula:

**F = S*e (rh- rf)*t**

Where**, rh and rf** are the continuously compounded interest rate for the home currency and foreign currency respectively,** T** is the time to maturity and **e** = 2.71828 (exponential).

If the following relationship between the futures rate and the spot rate does not hold, then there will be an arbitrage opportunity in the market. This will force the futures rate to change so that the relationship holds true.

Let us assume that risk free interest rate for one year deposit in India is 7% and in USA it is 3%. You as smart trader/ investor will raise money from USA and deploy it in India and try to capture the arbitrage of 4%. You could continue to do so and make this transaction as a non ending money making machine. Life is not that simple! And such arbitrages do not exist for very long.

We will carry out the above transaction through an example to explain the concept of interest rate parity and derivation of future prices which ensure that arbitrage does not exist.

Assumptions:

- Spot exchange rate of USD/INR is 50 (S)
- One year future rate for USD/INR is F
- Risk free interest rate for one year in USA is 3% (RUSD)
- Risk free interest rate for one year in India is 7% (RINR)
- Money can be transferred easily from one country into another without any restriction of amount, without any taxes etc)

You decide to borrow one USD from USA for one year, bring it to India, convert it in INR and deposit for one year in India. After one year, you return the money back to USA.

On start of this transaction, you borrow 1 USD in US at the rate of 3% and agree to return 1.03 USD after one year (including interest of 3 cents). This 1 USD is converted into INR at the prevailing spot rate of 50. You deposit the resulting INR 50 for one year at interest rate of 7%. At the end of one year, you receive INR 3.5 (7% of 50) as interest on your deposit and also get back your principal of INR 50 i.e., you receive a total of INR 53.5. You need to use these proceeds to repay the loan taken in USA.

Two important things to think before we proceed:

- The loan taken in USA was in USD and currently you have INR. Therefore you need to convert INR into USD
- What exchange rate do you use to convert INR into USD?

At the beginning of the transaction, you would lock the conversion rate of INR into USD using one year future price of USDINR. To ensure that the transaction does not result into any risk free profit, the money which you receive in India after one year should be equal to the loan amount that you have to pay in USA.

We will convert the above argument into a formula:** **

**S(1+RINR)= F(1+RUSD) **

Or,** **

**F/ S = (1+RINR) / (1+RUSD)**

Another way to illustrate the concept is to think that the INR 53.5 received after one year in India should be equal to USD 1.03 when converted using one year future exchange rate.

Therefore, F/ 50 = (1+.07) / (1+.03) F= 51.9417

Approximately, F is equal to the interest rate difference between two currencies i.e.

**F = S + (RINR- RUSD)*S **

This concept of difference between future exchange rate and spot exchange rate being approximately equal to the difference in domestic and foreign interest rate is called the “Interest rate parity”. Alternative way to explain, interest rate parity says that the spot price and futures price of a currency pair incorporates any interest rate differentials between the two currencies assuming there are no transaction costs or taxes.

A more accurate formula for calculating, the arbitrage – free forward price is as follows.

**F = S*(1 + RQC * Period) / (1 + RBC * Period) **

Where

- F = forward price
- S = spot price
- RBC = interest rate on base currency
- RQC = interest rate on quoting currency
- Period = forward period in years

For a quick estimate of forward premium, following formula mentioned above for USD/INR currency pair could be used. The formula is generalized for other currency pair and is given below:

**F = S + (S * (RQC – RBC) * Period)**

In above example, if USD interest rate were to go up and INR interest rate were to remain at 7%, the one year future price of USD/INR would decline as the interest rate difference between the two currencies has narrowed and vice versa.

Traders use expectation on change in interest rate to initiate long/ short positions in currency futures. Everything else remaining the same, if USD interest rate is expected to go up (say from 2.5% to 3.0%) and INR interest rate are expected to remain constant say at 7%; a trader would initiate a short position in USD/INR futures market.

**Illustration: **Suppose 6 month interest rate in India is 5% (or 10% per annum) and in USA are 1% (2% per annum). The current USD/INR spot rate is 50. What is the likely 6 month USD/INR futures price?** **

As explained above, as per interest rate parity, future rate is equal to the interest rate differential between two currency pairs. Therefore approximately 6 month future rate would be:

Spot + 6 month interest difference = 50 + 4% of 50 = 50 + 2 = 52

The exact rate could be calculated using the formula mentioned above and the answer comes to 51.98 = 50 * (1+0.1/12 * 6) / (1+0.02/12 * 6)

**Interpretation:** Future price of USD/INR depends upon the interest rate of each country. If USD price is appreciated then borrower has to pay more USD in return but if USD is depreciated then borrower has to pay less USD dollar.

**Interpretation of Concept of premium and discount:** Therefore one year future price of USDINR pair is 51.94 when spot price is 50. It means that INR is at discount to USD and USD is at premium to INR. Intuitively to understand why INR is called at discount to USD, think that to buy same 1 USD you had to pay INR 50 and you have to pay 51.94 after one year i.e., you have to pay more INR to buy same 1 USD. And therefore future value of INR is at discount to USD. Therefore in any currency pair, future value of a currency with high interest rate is at a discount (in relation to spot price) to the currency with low interest rate.