Lecture 3 - Parity conditions

Parity conditions

Purchasing Power Parity

The law of one price

S: exchange rate $/pound

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

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Absolute PPP

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The Big Mac Index

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How well does it work?

Relative PPP

π\pi : the inflation rate of country A, B

Relative PPP:

St+1A/BStA/B=1+πA1+πB\frac{S^{A/B}_{t+1}}{S^{A/B}_{t}} = \frac{1+\pi_{A}}{1+\pi_{B}}

Exchange rate differential = inflation rate differential

∴St+1A/B−StA/BStA/B=πA−πB1+πB\therefore \frac{S^{A/B}_{t+1}-S^{A/B}_{t}}{S^{A/B}_{t}} = \frac{\pi_{A}-\pi_{B}}{1+\pi_{B}}

Approximate inflation rate differential

πA−πB\pi_{A}-\pi_{B}

Relative PPP

Given inflation rates of 1.9% and 3% in Australia and the UK respectively, what is the prediction of PPP with regards to $A/GBP exchange rate?

St−St−1St−1=πA−πB1+πB=0.019−0.031+0.03=−0.010678=−1.07%\frac{S_{t}-S_{t-1}}{S_{t-1}}=\frac{\pi_{A} - \pi_{B}}{1+\pi_{B}} = \frac{0.019-0.03}{1+0.03} = -0.010678 = -1.07\%

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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

S¥/$,t0=¥87.86;S_{\yen/\$, t_{0}}=\yen87.86 ;

πAUS=1.9%\pi_{AUS} = 1.9\%

πJapan=1%\pi_{Japan} = 1\%

S¥/$,t1=S¥/$,t0⋅(1+π¥1+π$)=(87.86)⋅(1.011.019)=87.08  ¥/$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

Real Exchange Rate

E=St+1ActualSt+1PPPE = \frac{S^{Actual}_{t+1}}{S^{PPP}_{t+1}}

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

Real Exchange Rate

Interest Rate Parity

Ft,t+1A/BStA/B=1+iA1+iB\frac{F^{A/B}_{t, t+1}}{S^{A/B}_{t}} = \frac{1+i_{A}}{1+i_{B}}

AND

Ft,t+1A/BStA/B−1=1+iA1+iB−1\frac{F^{A/B}_{t, t+1}}{S^{A/B}_{t}} - 1 = \frac{1 + i_{A}}{1 + i_{B}} - 1

percentage forward premium = interest rate differential

∴Ft,t+1A/B−StA/BStA/B=iA−iB1+iB\therefore \frac{F^{A/B}_{t, t+1} - S^{A/B}_{t}}{S^{A/B}_{t}} = \frac{i_{A} - i_{B}}{1+i_{B}}

Example

$A(1+i$A4)\$A(1+ \frac{i_{\$ A}}{4})

1S$/CHF(1+iCHF4)⋅F3($/CHF)\frac{1}{S_{\$/CHF}}(1+ \frac{i_{CHF}}{4}) \cdot F_{3(\$ / CHF)}

$A(1+i$A4)=1S$/CHF(1+iCHF4)⋅F3($/CHF)\$A(1+ \frac{i_{\$ A}}{4}) = \frac{1}{S_{\$/CHF}}(1+ \frac{i_{CHF}}{4}) \cdot F_{3(\$ / CHF)}

Example

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Why Parity Holds?

Example 2

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Covered Interest Arbitrage

  1. Purchase of Pounds in the spot market and sale of pounds in the forward market narrow the premium on forward pounds.
  2. The demand for pound-denominated securities causes the pound interest rates to fall, while the higher level of borrowing in Australia causes dollar interest rates to rise.

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The Fisher Effect

(1+i)=(1+r)⋅(1+π)(1+i) = (1+r) \cdot (1+\pi)

i=r+π+rπi = r + \pi + r\pi

π\pi - inflation

rr - real interest rate

ii - 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 (π\pi)

i≈r+πi \approx r + \pi

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The International Fisher Effect

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Uncovered Interest Rate Parity

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Forward Exchange Expectations

Ft,t+1=Et(St+1)F_{t, t+1} = E_{t}(S_{t+1})

All parities example

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