Lecture 3, revised

Parity Conditions

• Parity Conditions provide an intuitive explanation of the movement of prices and interest rates in different markets in relation to exchange rates.

• Parity conditions rely on ARBITRAGE to hold.

• The derivation of these conditions requires the assumption of Perfect Capital Markets (PCM).

  • no transaction costs
  • no taxes
  • complete certainty

Purchasing Power Parity

• PPP is based on the notion of arbitrage across goods markets and the basic building block of PPP is given by the Law of One Price (LOP)
• LOP states that the price of an identical good should be the same in all markets (assuming no transactions costs).

The law of one price

• States that a while product’s price may be stated in different currency terms, but the price of the product in a common currency should remain the same.
  • Comparison of prices would only require conversion from one currency to the other:

$$P_{\$} = S_{\$/\pounds}\cdot P_{\pounds}$$

Conversely, the exchange rate could be deduced from the relative local product prices:

$$S^{PPP}_{\$/\pounds} = \frac{P_{\$}}{P_{\pounds}}$$

Absolute Purchasing Power Parity (PPP)

• A less extreme form of the Law of One Price is the ABSOLUTE PPP which says that the price of a basket of goods would be the same in each market.
• The PPP exchange rate between the two countries would then be:

$$S^{A/B}_{t} = \frac{PI_{A, t}}{PI_{B,t}}$$

• $PI_{A, t}$ and $PI_{B, t}$ are the price indices of the two countries (i.e. CPI) at time $t$

Violations

• Violations of Absolute PPP occur in the short run, but it tends to hold in the long run (several years).

Relative PPP

• Relative PPP claims that exchange rate movements should exactly offset any inflation differential between two countries

Exchange rate differential = inflation rate differential

$$\therefore \frac{S^{A/B}_{t+1}-S^{A/B}_{t}}{S^{A/B}_{t}} = \frac{\pi_{A}-\pi_{B}}{1+\pi_{B}}$$

• $\pi_{A}$, $\pi_{B}$ are the inflation rates of countries A and B, respectively

Relative PPP:

$$\frac{S^{A/B}_{t+1}}{S^{A/B}_{t}} = \frac{1+\pi_{A}}{1+\pi_{B}}$$

Approximate inflation rate differential

$$\pi_{A}-\pi_{B}$$

Applications of relative PPP

1. Forecasting future spot exchange rates.
2. Calculating appreciation in “real” exchange rates. This will provide a measure of how expensive a country’s goods have become (relative to another country’s).
   • More expensive relative to another country creates challenges - more competition, etc.

Forecasting Future Spot rates

• Suppose the ¥/$ spot exchange rate and expected inflation for Japan and Australia are:

$$S_{\yen/\$, t_{0}}=\yen87.86 ;$$

$$\pi_{AUS} = 1.9\%$$

$$\pi_{Japan} = 1\%$$

• What is the expected ¥/$ exchange rate if relative PPP holds?

$$S_{\yen/\$, t_{1}} = S_{\yen/\$, t_{0}} \cdot (\frac{1 + \pi_{\yen}}{1+\pi_{\$}}) = (87.86)\cdot (\frac{1.01}{1.019}) = 87.08 \;\yen/\$$$

The Real Exchange Rate

• The real exchange rate measures deviations from PPP.
• Changes in the spot exchange rate that do not reflect differences in inflation rates between the two currencies in question.

Real Exchange Rate

$$E = \frac{S^{Actual}_{t+1}}{S^{PPP}_{t+1}}$$

$$E = \text{1 + \% over-under valuation of denominator currency}$$

• When E = 1, the denominator currency is valued correctly. The competitiveness of this country is unaltered.
• When E < 1, the denominator currency is undervalued. Products from the other country seem expensive relative to the base year. That is, the competitiveness of the denominator country improves.
• When E > 1, the denominator currency is overvalued. Products from the other country seem cheap relative to the base year. That is, the competitiveness of the denominator country deteriorates.

Interest Rate Parity

• Interest rate parity (IRP) is an arbitrage condition that provides the linkage between the foreign exchange markets and the international money markets.

$$\frac{F^{A/B}_{t, t+1}}{S^{A/B}_{t}} = \frac{1+i_{A}}{1+i_{B}}$$

percentage forward premium = interest rate differential

$$\therefore \frac{F^{A/B}_{t, t+1} - S^{A/B}_{t}}{S^{A/B}_{t}} = \frac{i_{A} - i_{B}}{1+i_{B}}$$

• The currency trading at a forward premium is the one from the country with the lower interest rate.
• The currency trading at a forward discount is the one from the country with the higher interest rate.

Why Parity Holds?

• This must hold by arbitrage. Otherwise, riskless profits could be made. This is known as covered interest arbitrage (CIA) and occurs whenever IRP does NOT hold.

• Covered interest arbitrage (CIA) should continue until interest rate parity is re-established, because the arbitrageurs are able to earn risk-free profits by repeating the cycle.

• But their actions nudge the foreign exchange and money markets back toward equilibrium:

The Fisher Effect

• The Fisher effect (also called Fisher-closed) postulated by Irving Fisher states:

$$(1+i) = (1+r) \cdot (1+\pi)$$

$$i = r + \pi + r\pi$$

$\pi$ - inflation

$r$ - real interest rate

$i$ - nominal interest rate

This relation is often presented as a linear approximation stating that the nominal interest rate (i) is equal to a real interest rate (r) plus expected inflation ($\pi$)

$$i \approx r + \pi$$

The International Fisher Effect

• The International Fisher Effect (also called Fisher-open or Uncovered Interest rate parity condition) states that the spot exchange rate should change to adjust for differences in interest rates between two countries:

$$\frac{S^{A/B}_{t, t+1}}{S^{A/B}_{t}} = \frac{1+i_{A}}{1+i_{B}}$$