Exchange Rate Parity Relationships

TL;DR

Exchange rate parity relationships explain how exchange rates, interest rates, and inflation rates are theoretically linked across countries. These theories predict how exchange rates should adjust over time to eliminate arbitrage opportunities. While these relationships don't always hold perfectly in the real world, they provide a valuable framework for understanding currency movements.

1. The Mental Model

Think of these relationships as different ways for a "law of one price" to apply to financial assets and goods across borders. They describe how markets should adjust to prevent someone from making risk-free profits by exploiting price differences.

2. The Core Material

Exchange rate parity relationships include:
* Purchasing Power Parity (PPP): Connects inflation and exchange rates.
* Interest Rate Parity (IRP): Connects interest rates, spot rates, and forward rates.
* Fisher Effect: Connects real interest rates, nominal interest rates, and inflation.
* International Fisher Effect (IFE): Connects nominal interest rates and expected exchange rate changes.

Let's break them down.

Purchasing Power Parity (PPP)

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PPP suggests that identical goods should cost the same in different countries when expressed in a common currency.
* Absolute PPP: The exchange rate between two currencies should equal the ratio of the countries' price levels.
$S_0 = \frac{P_{domestic}}{P_{foreign}}$
where $S_0$ is the spot exchange rate (domestic currency per unit of foreign currency), $P_{domestic}$ is the domestic price level, and $P_{foreign}$ is the foreign price level.
* Relative PPP: The difference in inflation rates between two countries should be equal to the change in their exchange rate.
$\% \Delta S \approx \text{Inflation}_{domestic} - \text{Inflation}_{foreign}$
This means if domestic inflation is higher than foreign inflation, the domestic currency should depreciate.

Interest Rate Parity (IRP)

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IRP states that the difference between interest rates in two countries should be equal to the difference between the forward exchange rate and the spot exchange rate. This prevents arbitrage.
* Covered Interest Rate Parity (CIRP): This is an arbitrage-free condition where using a forward contract to lock in an exchange rate ensures a risk-free return.
$F_T = S_0 \times \frac{(1 + i_{domestic})}{(1 + i_{foreign})}$
or, approximately: $F_T - S_0 / S_0 \approx i_{domestic} - i_{foreign}$
where $F_T$ is the forward rate (domestic per foreign), $S_0$ is the spot rate, and $i$ are the interest rates.
* Uncovered Interest Rate Parity (UIRP): This involves expected future spot rates instead of actual forward rates and assumes investors are risk-neutral.
$E(S_1) = S_0 \times \frac{(1 + i_{domestic})}{(1 + i_{foreign})}$
or, approximately: $E(\% \Delta S) \approx i_{domestic} - i_{foreign}$
where $E(S_1)$ is the expected future spot rate.

Fisher Effect

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The Fisher Effect describes the relationship between nominal interest rates, real interest rates, and expected inflation within a single country.
$1 + i = (1 + r)(1 + \pi_e)$
or, approximately: $i \approx r + \pi_e$
where $i$ is the nominal interest rate, $r$ is the real interest rate, and $\pi_e$ is the expected inflation rate.

International Fisher Effect (IFE)

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The IFE extends the Fisher Effect to an international context, combining it with UIRP. It suggests that the difference in nominal interest rates between two countries should be equal to the expected percentage change in the exchange rate. This assumes real interest rates are equal across countries.
$E(\% \Delta S) \approx i_{domestic} - i_{foreign}$
This is the same approximation as UIRP, implying that a country with a higher nominal interest rate is expected to see its currency depreciate.

Here's how these concepts link together:

graph TD
    A["Inflation Rate Differential"] -->|Relative PPP| B["Expected Spot Rate Change"];
    C["Nominal Interest Rate Differential"] -->|International Fisher Effect| B;
    C -->|Uncovered IRP| B;
    C -->|Covered IRP| D["Forward Premium/Discount"];
    A -->|Fisher Effect| E["Nominal Interest Rate (Country A)"];
    A -->|Fisher Effect| F["Nominal Interest Rate (Country B)"];
    E & F --> C;

3. Worked Example

Let's use Covered Interest Rate Parity (CIRP).

Suppose the spot exchange rate for USD/EUR is 1.1000 (meaning 1.1000 USD per 1 EUR).
The 1-year USD interest rate is 3.0%.
The 1-year EUR interest rate is 1.0%.

What should the 1-year forward exchange rate (USD/EUR) be according to CIRP?

Using the formula:
$F_T = S_0 \times \frac{(1 + i_{domestic})}{(1 + i_{foreign})}$

In our case, USD is the domestic currency (numerator interest rate) and EUR is the foreign currency (denominator interest rate).
$F_{1-year} = 1.1000 \times \frac{(1 + 0.03)}{(1 + 0.01)}$
$F_{1-year} = 1.1000 \times \frac{1.03}{1.01}$
$F_{1-year} = 1.1000 \times 1.01980198$
$F_{1-year} \approx 1.1218$ USD/EUR

So, the 1-year forward rate should be approximately 1.1218 USD per EUR. If the actual forward rate in the market deviates significantly from this, an arbitrage opportunity might exist.

4. Key Takeaways

• These parity relationships are theoretical models describing how markets should behave without arbitrage.
• Purchasing Power Parity (PPP) links inflation differences to exchange rate changes.
• Interest Rate Parity (IRP), especially Covered IRP, is a no-arbitrage condition for forward rates.
• The Fisher Effect relates nominal rates, real rates, and expected inflation within a country.
• The International Fisher Effect (IFE) suggests that interest rate differentials predict expected exchange rate changes.
• While useful, these relationships don't always hold perfectly in the real world due to transaction costs, capital controls, and other market imperfections.

Common Mistakes to Avoid:
- Confusing spot rates with forward rates or expected future spot rates.
- Mixing up which currency is "domestic" and "foreign" when applying the formulas, especially for interest rates.
- Assuming these relationships always hold perfectly in practice; they are theoretical guides.
- Forgetting that the Fisher Effect and IFE involve expected inflation/spot rates, making them harder to test empirically.

5. Now Try It

You're given the following information:
Spot exchange rate: JPY/USD = 110.00 (110 JPY per 1 USD)
1-year Japanese interest rate: 0.10%
1-year US interest rate: 2.50%

What is the 1-year forward exchange rate for JPY/USD according to Covered Interest Rate Parity? What would you do if the market offered a 1-year forward rate of 113.00 JPY/USD?

Success looks like calculating the theoretical forward rate and briefly explaining whether an arbitrage opportunity exists and in which direction.