## 1 Exchange Rates and Interest Rates in the Short Run: UIP and FX Market Equilibrium

A review of UIP and its importance for the short-run exchange rate1. Risky Arbitrage
UIP requires the expected dollar rate of return on dollar deposits to equal the expected dollar rate return on euro deposits:i\$ = i + (Ee\$/€ - E\$/€)⁄E\$/€. UIP is the fundamental equation of the asset approach. Given interest rates and the expected future spot rate we can deduce the current spot rate.
a. Short-Term Interest Rates
Traders observe i\$ and i. The section will develop a theory of how they are determined.
b. Exchange Rate Expectations
The monetary model is a theory of the long-run behavior of the exchange, so it provides a forecast of the future spot rate.

2. Equilibrium in the FX Market: An Example
Depict equilibrium using the FX market diagram, which depicts dollar returns from the two investments as functions of the current spot rate. The dollar return to dollar deposits (DR) is independent of the spot rate. The dollar return to euro deposits (FR) is decreasing in the spot rate: If the dollar depreciates, the euro investment becomes more expensive, so the dollar return decreases. Equilibrium occurs at the spot rate where DR and FR intersect.

3. Adjustment to Forex Market Equilibrium
Arbitrage makes UIP hold immediately, so that FX is always in equilibrium.

4. Changes in Domestic and Foreign Returns and FX Market Equilibrium
How the spot rate responds to exogenous changes in interest rates and expectations.
a. A Change in the Domestic Interest Rate
If i\$ increases, DR shifts up. Traders buy dollars, so the dollar appreciates.
b. A Change in the Foreign Interest Rate
If i falls, FR shifts down. Traders buy dollars and the dollar again appreciates.
c. A Change in the Expected Future Exchange Rate
If Ee(\$/€) falls, FR shifts down. Because traders expect the dollar to appreciate in the future, they buy it now, causing the dollar to appreciate now.

5. Summary
The spot rate is determined by interest rates and the expected future spot rate.

To begin our study of exchange rates in the short run, let’s recap the crucial equilibrium condition for the forex market. In our earlier presentation, we considered a U.S. investor with two alternative investment strategies: a one-year investment in a U.S. dollar account with an interest rate i\$, or a one-year investment in a euro account with an interest rate i. Here are the essentials.

## Risky Arbitrage

They should be familiar with UIP by now, so just note that it suggests the beginning of a theory of exchange rate determination: the exchange rate should be affected by interest rates and expectations of future spot rate.

For the case of risky arbitrage, the forex market is in equilibrium when there is no expected difference in the rates of return on each type of currency investment in the two countries or locations. As before, we assume Home is the United States (U.S.); and Foreign is Europe (EUR, meaning the Eurozone). As we saw in the approximate uncovered interest parity condition—Equation (2-3), and repeated in this section as Equation (4-1)—this outcome requires that the dollar rate of return on the home investment (the dollar deposit) equal the expected dollar rate of return on the foreign investment (the euro deposit),

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where each interest rate is an annual rate, E\$/€ is today’s exchange rate (the spot rate), and is the expected future exchange rate that will prevail one year ahead.

The uncovered interest parity (UIP) equation is the fundamental equation of the asset approach to exchange rates, and from now on, we use it in the form of Equation (4-1). As we have seen, by rearranging this equation, we can solve it for the spot exchange rate, if we know all of the other variables. Thus, the asset approach employs the UIP equation to determine today’s spot exchange rate, as illustrated in Figure 4-1. Note that the theory is useful only if we know the future expected exchange rate and the short-term interest rates. Where does that knowledge come from?

As in Chapter 3, this kind of flow chart is very useful in clarifying causal linkages.

Short-Term Interest Rates The first assumption is that we know today’s interest rates on deposit accounts in each country, the dollar interest rate i\$, and the euro account interest rate i. Market participants can observe these short-term interest rates. But how are these interest rates determined, and, in particular, what is their relation to economic policy? In the next section, we explore that question to develop a fuller understanding of how exchange rates are determined.

Exchange Rate Expectations The second assumption is that we know the forecast of the future level of the exchange rate . The asset approach itself does not provide the answer, so we must look elsewhere. Where to look? In the long-run monetary approach to the exchange rate presented in the previous chapter. We can now see how the asset approach and monetary approach fit together.

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## Equilibrium in the FX Market: An Example

To explore the concepts we’ve just studied, let’s work through a numerical example to show how equilibrium in the forex market is determined.

Suppose that the current European interest rate i is 3%, and the current U.S. interest rate i\$ is 5%. Suppose also that we have made a forecast (using the long-run monetary model of exchange rates) that the expected future exchange rate (in one year’s time) is 1.224 dollars per euro.

Now examine Table 3-1 to see how, for various values of the spot exchange rate E\$/€, we can calculate the domestic rate of return and expected foreign rate of return in U.S. dollars. (Remember that 5% = 0.05, 3% = 0.03, etc.) As you work through the table, remember that the foreign returns have two components: one due to the European interest rate i and the other due to the expected rate of depreciation of the dollar, as in Equation (4-1).

Figure 4-2 presents an FX market diagram, a graphical representation of these returns in the forex market. We plot the expected domestic and foreign returns (on the vertical axis) against today’s spot exchange rate (on the horizontal axis). The domestic dollar return (DR) (which is Home’s nominal interest rate) is fixed at 5% = 0.05 and is independent of the spot exchange rate.

Instead of working through this numerical example, it might be quicker just to draw FR as a function of E. The negative slope will then be obvious, but make sure students understand the intuition (below) for the slope.

According to Equation (4-1), the foreign expected dollar return (FR) depends on the spot exchange rate and it varies as shown in Table 3-1. For example, we can infer from the table that a spot exchange rate of 1.224 implies a foreign return of 3% = 0.03. Hence, the point (1.224, 0.03) is on the FR line in Figure 4-2. This is the special case in which there is no expected depreciation (spot and expected future exchange rates equal 1.224), so the foreign return equals the foreign country’s interest rate, 3%.

Also note that FR shouldn't be a straight line, but should be convex.

More generally, we see that the foreign return falls as the spot exchange rate E\$/€ increases, all else equal. Why? This is clear mathematically from the right side of Equation (4-1). The intuition is as follows. If the dollar depreciates today, E\$/€ rises; a euro investment is then a more expensive (and, thus, less attractive) proposition, all else equal. That is, \$1 moved into a European account is worth fewer euros today; this, in turn, leaves fewer euro proceeds in a year’s time after euro interest has accrued. If expectations are fixed so that the future euro–dollar exchange rate is known and unchanged, then those fewer future euros will be worth fewer future dollars. Hence, the foreign return (in dollars) goes down as E\$/€ rises, all else equal, and the FR curve slopes downward.

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What is the equilibrium level of the spot exchange rate? According to Table 3-1, the equilibrium exchange rate is 1.20 \$/€. Only at that exchange rate are domestic returns and foreign returns equalized. To illustrate the solution graphically, domestic and foreign returns are plotted in Figure 4-2. The FX market is in equilibrium, and foreign and domestic returns are equal, at point 1 where the FR and DR curves intersect.

## Adjustment to Forex Market Equilibrium

This is just revisiting the arbitrage argument for UIP; emphasize that this happens pretty much instantly.

Our forex market equilibrium condition and its graphical representation should now be clear. But how is this equilibrium reached? It turns out that arbitrage automatically pushes the level of the exchange rate toward its equilibrium value.

To see this, suppose that the market is initially out of equilibrium, with the spot E\$/€ exchange rate at a level too low, so that the foreign return—the right-hand side of Equation (4-1)—exceeds the domestic return (the left-hand side).

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At point 2 in Figure 4-2, foreign returns are well above domestic returns. With the spot exchange rate of 1.16 \$/€ and (from Table 3-1) an expected future exchange rate as high as 1.224 \$/€, the euro is expected to appreciate by 5.5% = 0.055 [=(1.224/1.16) − 1]. In addition, euros earn at an interest rate of 3%, for a whopping foreign expected return of 5.5% + 3% = 8.5% = 0.085, which far exceeds the domestic return of 5%. At point 2, in other words, the euro offers too high a return; equivalently, it is too cheap. Traders want to sell dollars and buy euros. These market pressures bid up the price of a euro: the dollar starts to depreciate against the euro, causing E\$/€ to rise, which moves foreign and domestic returns into equality and forces the exchange rate back toward equilibrium at point 1.

The same idea applies to a situation in which the spot exchange rate E\$/€ is initially too high. At point 3 in Figure 4-2, foreign and domestic returns are not equal: the exchange rate is 1.24 \$/€. Given where the exchange rate is expected to be in a year’s time (1.224 \$/€), paying a high price of 1.24 \$/€ today means the euro is expected to depreciate by about 1.3% = 0.013 [=1.224/1.24 − 1]. If euro deposits pay 3%, and euros are expected to depreciate 1.3%, this makes for a net foreign return of just 1.7%, well below the domestic return of 5% = 0.05. In other words, at point 3, the euro offers too low a return; equivalently, it is too expensive today. Traders will want to sell euros.

Only at point 1 is the euro trading at a price at which the foreign return equals the domestic return. At point 1, the euro is neither too cheap nor too expensive—its price is just right for uncovered interest parity to hold, for arbitrage to cease, and for the forex market to be in equilibrium.

## Changes in Domestic and Foreign Returns and FX Market Equilibrium

Remind them that UIP implies the spot rate should depend upon three exogenous variables, the two interest rates and the expected future spot rate. Explain that this FX diagram now allows us to predict how E changes endogenously when these exogenous things change.

When economic conditions change, the two curves depicting domestic and foreign returns shift. In the case of the domestic return curve, the movements are easy to understand because the curve is a horizontal line that intersects the vertical axis at the domestic interest rate. If the domestic interest changes, the curve shifts up or down. Shifts in the foreign returns curve are a bit more complicated because there are two parts to the foreign return: the foreign interest rate plus any expected change in the exchange rate.

To gain greater familiarity with the model, let’s see how the FX market example shown in Figure 4-2 responds to three separate shocks:

• A higher domestic interest rate, i\$ = 7%
• A lower foreign interest rate, i = 1%
• A lower expected future exchange rate, = 1.20 \$/€

These three cases are shown in Figure 4-3, panels (a), (b), and (c). In all three cases, the shocks make dollar deposits more attractive than euro deposits, but for different reasons. Regardless of the reason, however, the shocks we examine all lead to dollar appreciations.

Go through each of these cases geometrically and explain the causal mechanism: For example, i increases -> demand US deposits increases (and demand for euro deposits decreases) -> demand for dollars increases -> E decreases. Students find these little flow charts very helpful.

A Change in the Domestic Interest Rate In Figure 4-3, panel (a), when i\$ rises to 7%, the domestic return is increased by 2% so the domestic return curve DR shifts up by 2% = 0.02 from DR1 to DR2. The foreign return is unaffected. Now, at the initial equilibrium spot exchange rate of 1.20 \$/€, the domestic return (point 4) is higher than the foreign return. Traders sell euros and buy dollars; the dollar appreciates to 1.177 \$/€ at the new equilibrium, point 5. The foreign return and domestic return are equal once again, and UIP holds once more.

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Another example of a flow chart: falls (and demand for euro deposits increases) -> demand for \$ falls -> E increases

A Change in the Foreign Interest Rate In Figure 4-3, panel (b), when i falls to 1%, euro deposits now pay a lower interest rate (1% versus 3%). The foreign return curve FR shifts down by 2% = 0.02 from FR1 to FR2. The domestic return is unaffected. Now, at the old equilibrium rate of 1.20 \$/€, the foreign return (point 6) is lower than the domestic return. Traders sell euros and buy dollars; the dollar appreciates to 1.177 \$/€ at the new equilibrium, point 7, and UIP holds once more.

A Change in the Expected Future Exchange Rate In Figure 4-3, panel (c), a decrease in the expected future exchange rate lowers the foreign return because a future euro is expected to be worth fewer dollars in the future. The foreign return curve FR shifts down from FR1 to FR2. The domestic return is unaffected. At the old equilibrium rate of 1.20 \$/€, the foreign return (point 6) is lower than the domestic return. Again, traders sell euros and buy dollars, causing the dollar to appreciate and the spot exchange rate to fall to 1.177 \$/€. The new equilibrium is point 7.

## Summary

The FX market diagram, with its representation of domestic returns and foreign returns, is central to our analysis in this chapter and later in the book. Be sure that you understand that domestic returns depend only on the home interest rate i\$ but that foreign returns depend on both the foreign interest rate i and the expected future exchange rate . Remember: any change that raises (lowers) the foreign return relative to the domestic return makes euro deposits more (less) attractive to investors, so that traders will buy (sell) euro deposits. The traders’ actions push the forex market toward a new equilibrium at which the dollar will have depreciated (appreciated) against the euro.

To check your understanding, you might wish to rework the three examples and the figures for the opposite cases of a decrease in i\$, an increase in i, and an increase in ; constructing the equivalent of Table 3-1 for each case may also prove helpful.