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# Relationship Among Forward, Interest and Spot Rates

The interest rate difference between the two countries affects the spot and forward rates. Using a single period analogy, an investor who has funds to invest in treasury securities, has two alternatives:

• invest at the domestic risk-free rate ($$i_d$$); or
• invest at the foreign risk-free rate ($$i_f$$).

If the investor takes the former option, the fund held at the end of the period would be ($$1 + i_d$$). Alternatively, the investor could convert the domestic currency to be invested in a foreign currency using the spot rate $$S_{f/d}$$. It is important to note that ($$f/d$$) is the currency quoting convention that expresses the number of foreign units per single domestic unit.

At the end of the investment period, $$S_{f/d}(1 + i_f)$$ units of foreign currency would be held by the investor. Then, the funds would have to be converted back into the domestic currency using the initial forward rate. Note that the two investment alternatives are risk-free because they are invested in risk-free assets.

Since these investment alternatives are equal by considering the risk characteristics, the returns must also be equal. As such, we have the following relationship:

$$1+i_d =S_{f/d}(1 + i_f)(\frac {1}{F_{f/d}})$$

Note that $$\frac{1}{F_{f/d}}$$ is the number of units of domestic currency for each unit of foreign sold forward.

The relationship above can be rearranged to get the formula for a forward rate as:

$$F_{f/d}=S_{f/d} (\frac{1 + i_f}{1+i_d})$$

This formula shows the relationship between the spot rate, the forward rate, and the interest rate in foreign and domestic countries.

### Example: Relationship Among Forward, Interest, and Spot Rates

Given that the spot exchange $$S_{f/d}$$ is 1.502, the domestic risk-free rate for 12-month is 4%, and the 12-month foreign risk-free rate is 6.2%, the forward rate $$F_{f/d}$$ is:

$$F_{f/d}=S_{f/d} (\frac{1 + i_f}{1+i_d})$$

$$F_{f/d}=1.502 (\frac{1+0.062}{1+0.04})=1.5338$$

## International Fisher Effect in Spot vs. Forward Rates

According to the Fisher effect, interest rate differences between two countries reflect the difference in the inflation rates of these two countries. High-interest rate countries experience higher inflation rates, and so the same uninvested dollar today is worth much less in the future. Therefore, there is the need to have a higher interest rate to compensate for the loss of purchasing power.

## Interest Rate Parity (IRP) in Spot vs. Forward

The interest rate parity is a theory which states that the difference between the interest rates of two countries is the same as the difference between the spot exchange rate and the forward exchange rate. This theory plays a major role in foreign exchange markets since it connects the dots between the interest rates, the spot exchange rates, and the foreign exchange rates.

## Purchasing Power Parity (PPP) in Spot Rates vs. Forward Rates

According to the purchasing power parity principle, a country’s currency fluctuates as the inflation rate of another country fluctuates. Therefore, the depreciation rate in a currency is roughly equal to the excess inflation rate in the domestic country over another country’s inflation rate.

### Question

Which of the following theories states that the interest rate differences between two countries reflect the difference in the inflation rates of these two countries?

1. The interest rate parity (IRP).
2. The international Fisher effect.
3. The purchasing power parity (PPP).

Solution

The international Fisher effect suggests that the estimated appreciation or depreciation of two countries’ currencies is proportional to the difference in their nominal interest rates.

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