# International Parity Conditions

International parity conditions refer to the economic theories that link exchange rates, price levels (inflation), and interest rates. These theories describe the interrelationships that help determine long-run fluctuations in exchange rates, interest rates, and inflation.

### Assumptions of International Parity Conditions

• There are no transaction costs.
• Market participants have access to perfect information.
• Market prices can easily be adjusted.
• Transactions occur in a risk-neutral environment.

## 1. Covered Interest Rate Parity

This no-arbitrage condition states that an investment in a foreign market that is entirely hedged against exchange rate risk should give the same return as a similar investment in a domestic market. Mathematically, it is represented as:

$$F_{f/d}=S_{f/d}\left(\frac{1+i_f\left[\frac{\text{Actual}}{360}\right]}{1+i_d\left[\frac{\text{Actual}}{360}\right]}\right)$$

Where:

$$i_d$$ = The interest rate in the base currency (domestic country).

$$i_f$$ = The interest rate in the foreign currency or the quoted currency.

$$S_{f/d}$$ = The current spot exchange rate.

$$F_{f/d}$$ = The forward foreign exchange rate.

Under the covered interest rate parity, the interest rate differential between any two currencies in the cash money markets should equal the differential between the forward and spot exchange rates. In other words, any forward premium or discount exactly offsets differences in interest rates. As a result, an investor would earn the same return investing in either currency.

For this condition to hold, it is assumed that:

• There is no transaction cost.
• The domestic and foreign market instruments should be identical in terms of liquidity, time to maturity, and default risk.

#### Example: Covered Interest Rate Parity

The U.S. dollar interest rate is 10%, and the GBP interest rate is 8%. The spot USD/GBP exchange rate stands at $1.40 (per GBP), and the 1-year forward rate is$1.48. Determine whether a profitable arbitrage opportunity exists.

Solution

According to the CIP equation, the one-year forward rate should be $1.43, i.e., $$1.40(1.1 / 1.08) = 1.4259$$. As such, there is an arbitrage opportunity, and here’s how it could be exploited: At the onset (time 0): Step 1: Borrow$1,000 at 10%.
Step 2: Use the borrowings to purchase 1,000 / 1.40 = 714.29 GBPs in the spot market.
Step 3: Invest the pounds at 6%.
Step 4: Enter a forward contract to sell the expected proceeds at the end of one year (i.e., 714.29
× 1.08 = 771.43 pounds), at $1.48 each. After one year: Step 1: Sell the 771.43 pounds under the forward contract at$1.48 to get $1,141.72. Step 2: Repay the$1,000 loan plus 10% interest, which requires $1,100. Step 3: Keep the difference of$41.72 as an arbitrage profit.

## 2. The Uncovered Interest Rate Parity

This condition postulates that the expected yield from a risky foreign investment must be equal to that of an equivalent domestic currency investment. While using the (f/d) notation with the domestic (d) currency as the base currency, assume that an investor has a choice of venturing either into a one-year domestic market investment or a risky (unhedged) foreign market investment over a similar horizon.

The uncovered parity condition compels the investor to weigh between the certain return from domestic investment and the expected return from the risky foreign investment (in terms of foreign currency). The foreign investment return in domestic currency will be given by:

$$\left(1+i_f\right)\left(1-\%\Delta S_{f/d}\right)-1$$

Where:

$$i_f$$  = Foreign rate of interest.

$$\%\Delta S_{f/d}$$  = Percentage change in the spot rate.

In case we were to interpret the above equation, we’d say that the investor’s return on foreign investment is a function of both the foreign interest rate and the change in the spot rate. Note that a depreciation in the foreign currency reduces the investor’s return.

The percentage change in $$S_{f/d}$$ enters with a minus sign. This is because a decline in the value of the foreign currency occasions an increase in $$S_{f/d}$$, consequently reducing the all-in return from the domestic currency perspective of the investor.

The all-in return given above can also be approximated as:

$$\approx i_f-\%\Delta S_{f/d}$$

Also, the uncovered interest rate parity implies that the expected change in the spot exchange rate over the investment period should be equal to the difference between the foreign and domestic interest rates. This is mathematically represented as:

$$\%\Delta S_{f/d}^e=i_f-i_d$$

Where $$\Delta S^e$$ is the future change in the spot exchange rate.

Therefore, the assumption brought forward by the uncovered interest rate is that when a country has higher interest rates, its currency will depreciate. Currency depreciation offsets the higher yield and brings the return of the investment to the level of the other country’s return.

#### Example: Uncovered Interest Parity

The spot exchange rate quote for the Kenyan shilling (KES) versus the U.S. dollar is 110.125 (KES/USD = 110.125, where USD is the base currency). The one-year nominal rate in the U.S. is 8%, while the one-year nominal rate in Kenya is 12%. Using uncovered interest rate parity, the expected percentage change in the exchange rate over the coming year is closest to:

#### Solution

The shilling interest rate is higher than the dollar interest rate. Thanks to this, uncovered interest rate parity predicts that the value of the shilling will fall. In a year, it will take more shillings to buy one dollar as a result of the higher interest rate in Kenya.

The dollar is, therefore, expected to appreciate by approximately 4% (= 12% − 8%) relative to the shilling.

This means that the exchange rate will change from 110.125 to 114.53.

## 3. Forward Rate Parity

This condition states that the forward exchange rate will be an unbiased projector of the future spot exchange rate if the covered interest rate parity and the uncovered interest rate parity hold. Note that the covered interest rate parity is given by:

$$F_{f/d}=S_{f/d}\left(\frac{1+i_f\left[\frac{\text{Actual}}{360}\right]}{1+i_d\left[\frac{\text{Actual}}{360}\right]}\right)$$

This can be rearranged to give the formula for the forward premium/discount:

$$F_{f/d}-S_{f/d}=S_{f/d}\left(\frac{\left[\frac{Actual}{360}\right]}{1+i_d\left[\frac{Actual}{360}\right]}\right)(i_f-i_d)\ldots\ldots\ldots (1)$$

For easy understanding, let’s use an investment horizon of one year as an example. In this instance, equation 1 transforms to:

$$F_{f/d}-S_{{f}/{d}}=S_{{f}/{d}}\left(\frac {i_f-i_d} {1+i_d}\right)(i_f-i_d)\ldots\ldots\ldots (2)$$

Since $$1+i_d\approx1$$, equation (2) becomes:

$$F_{f/d}-S_{f/d}\approx S_{f/d}(i_f-i_d)$$

Rearranging this gives:

$$\frac{F_{{f}/{d}}-S_{{f}/{d}}}{S_{f/d}}\approx(i_f-i_d)\ldots\ldots\ldots (3)$$

Remember that if the uncovered interest rate parity holds, then:

$$\%\Delta S_{f/d}^e=(i_f-i_d)\ldots\ldots\ldots (4)$$

Combining equations (3) and (4) gives:

$$\frac{F_{{f}/{d}}-S_{{f}/{d}}}{S_{f/d}}=(i_f-i_d)=\% \Delta S_{f/d}^e$$

The forward premium or discount is expressed as a percentage of the current spot exchange rate. When you analyze the resulting equation, you will easily see that it should be analogous to an appreciation or depreciation of the domestic currency if the uncovered interest rate parity remains intact. If both the covered interest rate parity and uncovered interest parity hold, then the forward exchange rate will be an unbiased estimator of the future spot exchange rate. That is:

$$F_{{f}/{d}}=S_{f/d}^e$$

#### Example: Forward Rate Parity

Let us revisit the example above.

$$\begin{array}{c|c|c|c} \textbf{Currency} & \textbf{Libor} & \textbf{Currency} & \textbf{Spot Rate} \\ & \textbf{(annualized)} & \textbf{Combinations} & \\ \hline USD & 0.30\% & USD/EUR & 1.6975 \\ \hline EUR & 5.00\% & JPY/EUR & 0.0085 \\ \hline JPY & 0.30\% & JPY/USD & 82.25 \end{array}$$

If the uncovered interest rate parity holds, the approximate forward rate of JPY/EUR currency one year from now is closest to:

#### Solution

Since we have assumed that the uncovered interest rate parity holds, the forward rate parity holds. That is, a one-year spot rate should be equal to the one-year forward rate. That is,

$$F_{f/d}=S_{f/d}\left(\frac{1+i_f}{1+i_d}\right)=0.0085\left(\frac{1.003}{1.05}\right)=0.0081$$

## 4. The Purchasing Power Parity (PPP)

This condition reflects the link between the exchange rates and the difference in countries’ inflation rates. However, there are different versions of this condition, whose foundation is based on the law of one price. They include:

• Absolute PPP.
• Relative PPP.
• Ex Ante PPP.

The law of one price states that the price of a foreign good $$x$$ denoted as $$P_f^x$$ must be equal to the price of a similar good in a domestic country, $$P_d^x$$, using the spot rate $$S_{f/d}$$. Here, we have used the (f/d) notation for simplicity. Put mathematically,

$$P_f^x=S_{f/d}\times P_d^x$$

For instance, a product in Canada costs CAD 100. The nominal exchange rate for USD/CAD is 0.76. The same product will, therefore, cost 0.76 × 100 = USD 76 in the U.S.

Now, let’s discuss each version.

### Absolute PPP

This version of the PPP amplifies the law of one price to include a broader range of goods and services and not just good $$x$$. The act of one price equation transforms into:

$$P_f=S_{f/d}\times P_d$$

Where:

$$P_f$$ = Price level of the foreign country.

$$P_d$$ = Price level of the domestic country.

$$S_{f/d}$$ = Nominal exchange rate.

Making $$S_{f/d}$$ the subject of the formula, we get:

$$S_{f/d}=\frac{P_f}{P_d}$$

Therefore, the absolute PPP states that the ratio of the national price level of the two countries determines the equilibrium in the exchange rates. However, if the transaction cost is coupled mainly with the non-tradeable nature of some goods, this condition might not hold.

### Relative PPP

This version assumes that the transaction costs and other trading difficulties are constant. This justifies the fact that the exchange rate and national price changes are interrelated. That is, according to this version, the deviation of the exchange rate is entirely determined by the difference between the inflation rates (of foreign and domestic countries). Mathematically,

$$\%\Delta S_{{f}{d}}\approx \pi_f-\pi_d$$

Where:

$$\%\Delta S_{{f}/{d}}$$ = Change in the spot exchange rate.

$$\pi_f$$ = Foreign inflation rate.

$$\pi_d$$ = Domestic inflation rate.

### Ex Ante PPP

This version states that the expected (future) differences between the national inflation rates (domestic and foreign countries) determine the changes in the spot exchange rate. Mathematically, it is represented as:

$$\%\Delta S_{{f}/{d}}^e\approx\pi_f^e-\pi_d^e$$

That is, countries with long-term high inflation rates should see their currencies depreciate. On the other hand, those with relatively low inflation rates should see their currencies appreciate over time.

## 5. The International Fisher Effect

The fisher effect postulates that the nominal interest rate of a country can be divided into a real interest rate and an expected inflation rate. That is:

$$i=r+\pi^e$$

Where:

i = Nominal interest rate.

r = Real interest rate.

$$\pi^e$$ = Expected inflation rate.

Now, if we write this for both domestic (d) and foreign (f) countries, we have:

\begin{align*} i_d &=r_d+\pi_d^e\ldots\ldots\ldots(1) \\ i_f & =r_f+\pi_f^e\ldots\ldots\ldots(2) \end{align*}

Subtracting (1) from (2):

$$i_f-i_d=\left(r_f-r_d\right)+(\pi_f^e-\pi_d^e)$$

Note that this is still consistent with the Fisher effect. If we make the difference between real interest rates the subject of the formula, we get:

$$\left(r_f-r_d\right)=\left(i_f-i_d\right)-\left(\pi_f^e-\pi_d^e\right)\ldots\ldots\ldots(3)$$

But from the uncovered interest rate parity, we have:

$$\%\Delta S_{f/d}^e=i_f-i_d$$

And the ex-ante PPP formula is:

$$\%\Delta S_{{f}/{d}}^e\approx\pi_f^e-\pi_d^e$$

From these equations, it is easy to see that:

$$i_f-i_d=\pi_f^e-\pi_d^e$$

Substituting this in equation (3), we get:

$$\left(r_f-r_d\right)=0$$

So, provided that the uncovered interest parity and ex-ante hold, the real interest rates of the different markets will always converge at the same value. That is:

$$r_f=r_d$$

This is sometimes simply described as the interest rate parity.

It follows immediately that the difference between the foreign and the domestic inflation rates determines the domestic and foreign nominal interests. Put simply:

$$i_f-i_d=\pi_f^e-\pi_d^e$$

This is termed the international Fisher effect.

## Question

Which one of the following is the best reason for the unreliability of the absolute version of purchasing power parity (PPP)?

1. Real interest rates tend to converge across different countries.
2. Inflation rates are different for each country.
3. Various goods and services are consumed in markets with trade barriers.

Solution

The correct answer is C.

According to the absolute version of purchasing power parity (PPP), all goods and services are tradable, and the same proportion of products and services in a country is used to determine the price level.

Reading 8: Currency Exchange Rates: Understanding Equilibrium Value

LOS 8 (e) Explain international parity conditions (covered and uncovered interest parity, forward rate parity, purchasing power parity, and the international Fisher effect).

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