# International Parities

International parities are concerned with the relationships between the values of two or more currencies and the respective economic conditions in these countries, and the way in which these relationships respond to the changing economic conditions in these countries. International parities are important since they establish relative currency values and their evolution in terms of economic circumstances, and cross border arbitrage may be possible when they are violated.

There are three international parity conditions:

• Purchasing power parity PPP: which is concerned with the exchange rate of two currencies and the prices in the two countries.
• Covered Interest rate parity CIRP: which identifies the relationships between the spot exchange rate, the forward exchange rate, and the interest rates in the two countries.
• Uncovered Interest rate parity UIRP or the International Fisher effect IFE: which establishes the relationship between changes in the exchange rate and the interest rate differential in the two countries, exploiting the relationship between the interest rate differentials and the inflation differentials.

In international finance, PPP means that the same goods or basket of goods should sell at the same price in different countries when measured in a common currency, in absence of transactions costs.

There are two versions of PPP. One is absolute PPP and the other is relative PPP. The former studies the exchange rate for the two currencies in terms of the absolute prices for the same basket of goods in the two countries, and the latter examines how the exchange rate changes over time in response to changes in the price levels in the two countries.

## Absolute PPP

Absolute PPP is the application of the law of one good, one price in international finance:

$P_d = S \times P_f$

where $P_d$ is the price for a good or basket of goods in the domestic or home country, $P_f$ is the price for the same good or basket of goods in the foreign country, and $S$ is the exchange rate expressed as the units of the home currency per foreign currency unit. This equation indicates that the exchange rate is the ratio of the prices for the same goods in the two countries.

In the case that the law of one good one price, or absolute PPP, is violated, and consequently, there might be arbitrage opportunities for cross border trade activities, given the absence of transaction costs. Absolute PPP is a sufficient condition for no arbitrage in international trade and finance, but it is not a necessary condition. Upholding of absolute PPP guarantees the elimination of arbitrage, while a violation of absolute PPP may or may not give rise to arbitrage opportunities, depending on the level of associated costs in the international trading process.

Having studied absolute PPP, we can progress to introduce the concept of the real exchange rate. The real exchange rate is the exchange rate adjusted by the price levels in the two countries, defined as follows:

$Q = S\times \frac{P_f}{P_d}$

where $Q$ is real exchange rate. Real exchange rates are one if absolute PPP holds. Domestic currencies are over valued if $Q < 1$ and are under valued if $Q >1$. Foreign currencies are under valued if $Q < 1$ and are over valued if $Q >1$.

## Relative PPP

Relative PPP examines the relationship between changes in exchange rates and changes in the aggregate price levels in the two countries involved. Taking log differences of the absolute PPP equation yields:

$\Delta s_t \approx \Delta p_{d,t} - \Delta p_{f,t} = \pi_{d,t} - \pi_{f,t}$

where $\Delta s_t = \ln(S_t) − \ln(S_{t −1} )$ is the (percentage) change in exchange rates in period $t-1$ to $t$; $\pi_t = \Delta p_t = \ln(P_{t}) − \ln(P_{t-1})$ is the (percentage) change in the the price levels, or the inflation rate in the same period. Price indices, usually consumer price indices (CPI), are used for the measurement of inflation.

Relative PPP establishes an evolution path for exchange rate changes, which is a consequence of relative price developments in the two countries. Suppose absolute PPP holds at time $t-1$ and relative PPP holds in the period $t-1$ to $t$, then absolute PPP holds at time $t$ also. The size of the change in the exchange rate reflects the relative purchasing power gain of one of the two currencies, or the relative purchasing power loss of the other currency, to the right extent when relative PPP holds.

Relative PPP indicates that:

1. the domestic currency will depreciate if inflation in the domestic country is higher than that in the foreign country, and
2. the domestic currency will depreciate to the extent equal to the inflation differential between the two countries.

For example, if inflation in the US was $2\%$ and inflation in Japan was $0\%$ in 2000, the yen should have appreciated by 2% vis-à-vis the dollar in the same period, according to relative PPP.

# Covered Interest rate parity

Interest rate parities are concerned with expected exchange rate changes and the interest rate differential between the two involved countries or currency zones dur- ing a certain period. The idea is that the expected exchange rate change can be covered by entering a forward contract for the foreign exchange transaction at a future time. Subsequently, it is called covered interest rate parity (CIRP), which identifies the relationships between the spot exchange rate, the forward exchange rate, and the interest rates in the two countries in a time period.

CIRP states that the forward premium must be equal to the two countries’ interest rate differential, otherwise there exist exploitable profitable arbitrage opportunities. The following relationship (parity) between the spot exchange rate $S_0$, forward exchange rate $F_{0,1}$, and the interest rates in the two countries must hold to eliminate any arbitrage opportunities:

$\frac{F_{0,1}}{S_0} = \frac{1+r_d}{1+r_f}$

With mathematical arrangement, we can infer forward premium as follows:

$\frac{F_{0,1} - S_0}{S_0} = \frac{r_d -r_f}{1+r_f} \approx r_d - r_f$

# Uncovered Interest rate parity

Without a forward contract on future foreign exchange transactions, the expected change in exchange rates is uncovered, and then an appropriate relationship between expected exchange rate changes and the interest rate differential between the two countries during a certain period is justified and established by uncovered interest rate parity.

While CIRP holds, there are no profitable opportunities if:

$F_{0,1} = E_0(S_1)$

UIRP implies that the forward exchange rate is an unbiased predictor of the future spot exchange rate. From CIRP formula and forward premium, we obtain:

$\frac{E_0(S_1) - S_0}{S_0} \approx r_d - r_f$

The above equation states that the expected (percentage) change in the spot exchange rate, exactly speaking, is equal to the two countries’ interest differential adjusted by a factor of $(1 + r_f )$.

# International Fisher effect

Fisher effect is associated with the relationship between the real interest rate, the nominal interest rate, and inflation, in a domestic economic setting:

$r \approx a + E(\pi)$

where $a$ is the real interest rate, $r$ is the nominal interest rate, and $E(\pi)$ is the expected inflation rate in a certain period $(t-1, t]$.

Applying the Fisher effect to two concerned countries leads to the international Fisher effect.

$E(\Delta s_t) = r_d - r_f$

given $a_d = a_f$.

IFE suggests that the expected change in exchange rates be equal to the interest rate differential between the two countries, which is a statement of UIRP derived under different circumstances.

According the IFE, expected inflation is a channel to cause nominal interest rates to change, the differential in the expected inflation of the two countries is a channel to result in the interest rate differential between the two countries, leading to expected relative changes in purchasing power and subsequent adjustment in exchange rate expectations.

In contrast, UIRP relies on the forward exchange rate as an unbiased predictor of the future spot exchange rate and the validity of CIRP.