# Derivation of the Interest Rate Parity (IRP)

Suppose that you consider investing in the home or foreign country for one period. It means that you have some amount of money now (present value or PV) and, given an interest rate, you want to make some amount of money in the future (future value or FV). The basic relation between PV and FV for one period is

Because you know how much money you have (PV) and what the interest rate (R) is now, the unknown is how much money you will make in the future (FV). You rewrite the preceding formula to have the unknown variable in the left-hand side and get:

FV=_{H}PV(1 +R)_{H}

Here, *R** _{H}* and (1 +

*R*

*) are the nominal interest rate and the interest factor (1 +*

_{H}*R*

*) in the home country (*

_{H}*H*), respectively. For simplicity, assume a $1 investment so that you can simplify your (dollar) earnings to the following:

FV= (1 +_{H}R)_{H}

Similarly, your (euro) earnings in the foreign country by investing €1 in Eurozone are shown here:

FV= (1 +_{F}R)_{F}

Here, *R** _{F}* and (1 +

*R*

*) imply the foreign country’s (*

_{F}*F*) nominal interest rate and interest factor (in this case, Eurozone’s), respectively.

You can’t directly compare *R** _{H}* and

*R*

*because the home and foreign country’s interest rates are denominated in different currencies. Therefore, you need a conversion mechanism.*

_{F }You can convert your earnings in euro into dollars by multiplying the interest factor in foreign currency with the percent change in exchange rate. But in order to calculate the percent change in the exchange rate, you need to know the current exchange rate and the expected exchange rate.

While the current exchange rate is observable, there is no explicit series called expected exchange rate. Therefore, you need a measure for the expected exchange rate. The exchange rate on a forward contract (namely, the forward rate) would be a good proxy for the expected exchange rate.

Therefore, express the nominal version of the MBOP’s parity condition as follows:

In this equation, F and S are the forward rate and spot rate, respectively. You can further write the forward rate (F) in a way that shows the relationship between F and S:

F=_{t}S(1 + ρ)_{t}

This equation states that the difference between the forward rate and the spot rate is related to a factor ρ (rho). The variable ρ can be interpreted as the percentage difference between the forward rate and the spot rate. Inserting the previous definition of the forward rate

and eliminating the spot rate in the bracket of the equation, you have:

(1 +

R) = (1 +_{H}R) x (1 + ρ)_{F}

This equation is a different way of expressing interest rate parity. It implies that investors are indifferent between home and foreign securities denominated in home and foreign currencies if the nominal return in the home country equals the nominal return in a foreign country, including the change in the exchange rate.

Look at this equation also from the viewpoint of which variables are known and which variable should be calculated. In the equation, you observe the home and foreign nominal interest rates and want to know what ρ* *is. Therefore, you divide both sides by (1 + *R** _{F}*) and find

Conceptually, ρ implies the percent change in the exchange rate. Because the previous derivation was based on the change between the forward rate and the spot rate, you refer to ρ as a forward premium or forward discount.

The terms *forward premium* and *forward discount* refer to the other currency. You can explain this by considering the sign of ρ. Clearly, ρ can be positive or negative. If the home nominal interest rate (*R** _{H}*) is larger than the foreign nominal interest rate (

*R*

*), the ratio of the home and foreign interest factor [(1 +*

_{F}*R*

*) / (1 +*

_{H}*R*

*)] becomes larger than 1, which makes ρ positive.*

_{F}Because higher nominal interest rate in a country is consistent with higher inflation rates, a positive ρ is *forward premium* on the foreign currency.

If the home nominal interest rate (*R** _{H}*) is lower than the foreign nominal interest rate (

*R*

*), the ratio of the home and foreign interest factor [(1 +*

_{F}*R*

*) / (1 +*

_{H}*R*

*)] becomes less than 1, which makes ρ negative. Because lower nominal interest rates in a country is consistent with lower inflation rates, a negative ρ is*

_{F}*forward discount*on the foreign currency.