 # Future Exchange Rates and Parity Conditions

## Uncovered Interest Rate Parity

It states that the change in spot rate over the investment period should be averagely equal to the difference between the interest rates in two different countries. Put another way, the expected appreciation or depreciation should approximately offset the difference in interest rates.

While using the (f/d) notation (domestic (d) currency as the base currency), assume that an investor has a choice of venturing into a one-year domestic market investment and a risky (unhedged) foreign market investment. 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:

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

This also can be represented as:

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

Also, the uncovered interest rate parity implies that the anticipated change in the spot rate over the investment period should show 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 rate.

This change can be added to (or subtracted from) the current spot rate to get the new spot rate.

### The Purchasing Power Parity (PPP)

This condition reflects the link between the exchange rates and the difference in countries’ inflation rates. Note, however, that there are different versions of this condition. Each of these versions has its foundation anchored upon the law of one price. The versions include:

1. Absolute PPP.
2. Relative PPP.
3. 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 $$S_{f/d}$$ (We have used the (f/d) notation for simplicity). Put mathematically,

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

#### Absolute PPP

This version of PPP amplifies the law of price to include a broader range of goods and services and not just good x. To attain this, the law of one price equation transforms into:

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

where:

$$P_f$$-the price level of the foreign country.

$$P_d$$-the price level of the domestic country.

$$S_{f/d}$$-nominal exchange rate.

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

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

Therefore, the absolute PPP states that the equilibrium in the exchange rates is determined by the ratio of the national price level of the two countries in question. It is imperative to note that if the transaction cost is coupled mainly with the non-tradable 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 difference between the inflation rates of the foreign and domestic countries entirely determines the deviation of the exchange rate. 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:

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

In conclusion, if all international parity conditions were to hold at all times, then the spot exchange rate expected change would be equal to:

1. The nominal yield spread between countries.
2. The forward discount or premium.
3. The difference between expected national inflation rates.

## Question

Below is data derived from an interbank market:

$$\begin{array}{c|c|c|c} \textbf{Currency} & \textbf{Libor} & \textbf{Currency} & \textbf{Spot Rate} \\ & \textbf{(annualized)} & \textbf{Combinations} & \\ \hline USD & 0.70\% & USD/EUR & 1.8975 \\ \hline EUR & 7.00\% & JPY/EUR & 0.0075 \\ \hline JPY & 0.50\% & JPY/USD & 82.25 \end{array}$$

Assuming that the Uncovered Interest Parity holds, the expected size of fluctuation in the JPY/USD pair over one year is closest to:

1. -0.2%.
2. 0.2%.
3. 0%.

#### Solution

According to the uncovered interest rate parity condition, the expected change in a spot exchange rate is equivalent to the difference between the interest rates corresponding to each currency (Libors). That is,

$$\%\Delta S_{f/d}^e=i_f-i_d=\left(0.5-0.7\right)\%=-0.2\%$$

So JPY/USD exchange rate reduced by 0.002.

Reading 8: Currency Exchange Rates: Understanding Equilibrium Value

LOS 8 (g) Evaluate the use of the current spot rate, purchasing power parity, and uncovered interest parity to forecast future spot rates of exchange.

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