# 5.3: Forex Equilibrium with the Rate of Return Diagram

- Page ID
- 20237

learning objective

- Use the rate of return plots to represent the interest rate parity equilibrium in the foreign exchange market.

An alternative graphical approach is sometimes used to depict the equilibrium exchange rate in the foreign exchange (Forex) market. The graph is called the rate of return diagram since it depicts rates of return for assets in two separate countries as functions of the exchange rate. The equilibrium condition depicted in the diagram represents the interest rate parity condition. In effect, the diagram identifies the equilibrium exchange rate that must prevail to satisfy the interest rate parity condition.

Recall the rate of return formulas for deposits in two separate countries. Consider an investor, holding U.S. dollars, comparing the purchase of a one-year certificate of deposit (CD) at a U.S. bank with a one-year CD issued by a British bank. The rate of return on the U.S. deposit works out simply to be the U.S. interest rate shown below:

*\[R_{0}R_{$} = i_{$}\]*.

The rate of return on the British asset, however, is a more
complicated formula that depends on the British interest rate
(*\(i_{£}\)*), the spot exchange rate
(*\(E_{$/£}\)*), and the expected exchange rate
(*\(E_{$/£}^e\)*). In its simplest form it is written as
follows:

In Figure 5.5 , we plot both *\(R_{0}R\)* equations
with respect to the exchange rate (*\(E_{$/£}\)*).
Since *\(R_{0}R_{$}\)* is not a function (i.e.,
not dependent) on the exchange rate, it is drawn as a vertical line
at the level of the U.S. interest rate (*\(i_{£}\)*). This
simply means that as the exchange rate rises or falls,
the *\(R_{0}R_{$}\)* always remains immutably
fixed at the U.S. interest rate.

Figure \(\PageIndex{1}\): Rate of Return Diagram

The *\(R_{0}R_{$}\)*, however, is a function of the
exchange rate. Indeed, the relationship is negative
since *\(E_{$/£}\)* is in the denominator of the
equation. This means that as
*\(E_{$/£}\)* rises, *\(R_{0}R_{$}\)* falls,
and vice versa.

The intuition behind this negative relationship is obtained by
looking at the alternative (equivalent) formula for
*\(R_{0}R_{£}\)*:

Recall that the exchange rate ratio represents the expected
percentage change in the value of the pound. Suppose, as an
example, that this term were positive. That would mean the investor
believes the pound will appreciate during the term of the
investment. Furthermore, since it is an expected appreciation of
the pound, it will add to the total rate of return on the British
investment. Next, suppose the spot exchange rate
(*\(E_{$/£}\)*) rises today. Assuming ceteris paribus, as we
always do in these exercises, the expected exchange rate remains
fixed. That will mean the numerator of the exchange rate expression
will fall in value, as will the value of the entire expression. The
interpretation of this change is that the investor’s expected
appreciation of the pound falls, which in turn lowers the overall
rate of return. Hence, we get the negative relationship between the
*\($/£\)* exchange rate
and *\(R_{0}R_{£}\)*.

The intersection of the two RoR curves in the diagram identifies
the unique exchange rate *\(E_{$/£}'\)* that
equalizes rates of return between the two countries. This exchange
rate is in equilibrium because any deviations away from interest
rate parity (IRP) will motivate changes in investor behavior and
force the exchange back to the level necessary to achieve IRP. The
equilibrium adjustment story is next.

key takeaways

- The rates of return are plotted with respect to the exchange rate. The domestic rate of return does not depend on the exchange rate and hence is drawn as a vertical line. The foreign rate of return is negatively related to the value of the foreign currency.
- The intersection of the rates of return identifies the exchange rate that satisfies the interest rate parity condition.

exercise

**Jeopardy Questions**. As in the popular television game show, you are given an answer to a question and you must respond with the question. For example, if the answer is “a tax on imports,” then the correct question is “What is a tariff?”

- Of
*positive*,*negative*, or*zero*, the relationship between the U.S. interest rate and the rate of return on U.S. assets. - Of
*positive*,*negative*, or*zero*, the relationship between the exchange rate (*E*_{$/£}) and the rate of return on U.S. assets. - Of
*positive*,*negative*, or*zero*, the relationship between the exchange rate (*E*_{$/£}) and the rate of return on British assets. - The name of the endogenous variable whose value is determined at the intersection of two rate of return curves.