The Interest Parity Theory

Transcription

1 Kapitel 13 The Interest Parity Theory Proposition: Two Assets with the same risk should have the same rate of return 13.1 The Covered Interest Parity (CIP) German: Gedeckte Zinsparität Mechanism: Interest rate arbitrage. Economic agents arbitrage between two assets until the rate of return is equalized. Investment Home Investment Foreign (1 + i) 1 (1 + i ) F t F t Spot Exchange Rate Forward Exchange Rate Amount of domestic currency per unit of foreign currency 1 + i > F t (1 + i ) Invest at home Capital imports (Balance of Payments) 1 + i < F t (1 + i ) Invest in USA Capital Exports (Balance of Payments) 1

2 Money would cease to ow when equality is achieved (1 + i) = (1 + i ) F t (13.1) Approximation (Small i ): i i = F t (13.2) The relationship (13.2) between the spot exchange rate, the forward exchange rate F t and the interest rates in two countries must hold to eliminate any arbitrage opportunity. In other words: The interest parity theorem says that the interest rate dierential i i must be equal to the forward premium Ft St > 0 (or forward discount Ft St < 0). If the interest rate is in favor of the domestic currency, this must be compensated by a forward premium, making it more attractive to buy foreign currency spot and sell it forward. German: Ft St heisst Swapsatz If CIP does not hold, it implies that there are riskless arbitrage possibilities. Taking Logarithms: f t s t = i t i t (13.3) f t = ln (F t ) ; s t = ln ( ) 2

3 1. Along 45 line, interest parity holds; no arbitrage possibility 2. Left of 45 line, Point A ρ t < i i Example ρ t = 1%; i i = 2% The gain of a foreign exchange transaction (forward premium) is smaller than the opportunity costs of going abroad (i i = 2%) - remain at home 3. Right of 45 line, Point B i i < ρ t The foward premium is greater than i i ; it is protable to exploit the arbitrage opportunity by investing in foreign country Point B - go foreign 4. Point C, i t i > ρ t No arbitrage possible remain at home Regression Equation McCallum, International Monetary Economics (1996) p.188; Log's (s t, f t ) 3

4 f t s t = α + β (i t i t ) + ɛ t OLS (13.4) Hypothesis: α = 0 ; β = 1 BRD: α = ; β = ; R 2 = Very good t! CIP is conrmed; 13.2 The uncovered interest rate parity (UIP) The UIP introduces a hypothesis how the foreign exchange rate is determined. If the forward market is ecient it must reect the prevailing expectations about the future spot rate. F t = E t +1 (13.5) (13.5) (13.1): in Log's: (1 + i t ) = (1 + i t ) E t+1 (13.6) i t i t = E t s t+1 s t (13.7) Interpretation 1. If the domestic currency is expected to depreciate in future E t +1 >, the domestic interest rate must exceed the foreign interest rate. The interest rate dierential needs to compensate economic agents for an expected capital loss from holding domestic assets. 2. The UIP is a forward looking model. If agents have formed their future expected exchange rate [for example E t (S 2 )] the interest parity condition will uniquely determine the exchange rate (spot). Economic agents forecasts of the future exchange rate will determine today's exchange rate 4

5 3. The economy is continously disturbed by new shocks. Agents will continously re-evaluate the eects for the future value of the currency. This concept explains the short term variability of the exchange rates. 4. From (13.7) follows that the equilibrium path of the exchange rate remains indeterminate. Given the interest rate dierential (i t i t ), there is an innite number of paths linking the present spot rate with the future rate. Spot E t etc The equilibrium of nancial markets is compatible with an innite number of dierent exchange rates. 5. Expectations about the future exchange rates are self-fulllling. 5

6 No empirical evidence for the UIP Is there empirical evidence for the UIP to hold? NO. See McCallum op. cit. p estimates the following relation ship (OLS) E t s t+1 s t = α + β (f t s t ) (13.8) Of course there are no observations of E t s t+1. Assume rational expectations, one has: RE : s t+1 = E t s t+1 + ɛ t+1 s t+1 s t = α + β (f t s t ) + ɛ t+1 (13.9) Hypothesis: The forward premium (f t s t ) is an unbiased predictor of the expected change of the exchange rate. This hypothesis does NOT hold, not even approximately (which is a serious problem). s t+1 s t = }{{} (f t s t ) (13.10) (1.70) Example Germany R 2 = 0.04 Equation (13.9) implies the hypothesis: α = 0.0 β = 1.00 Conclusion: The UIP holds if there is no risk or foreign and domestic investors have the same risk (this is generally not the case). Assumption: Returns on domestic assets are known with certainty; foreign returns are unknown; since the future spot exchange rate is unknown, a risk premium is needed; i.e. some excess-return on foreign investment to compensate investors for the higher risk. Introducing a risk premium π, we obtain: (1 + π) = (1 + i ) E t s t+1 (13.11) (1 + i) s t 6

7 (1 + i) = (1 + i ) E t +1 (13.12) 1 + π Rewrite: i t i t = E t s t+1 s t π t (13.13) (13.13) makes it clear that the interest dierential may arise because of the expected depreciation (appreciation) but also because of the existence of a risk premium. i t = i t + µ t (13.14) µ t = expected rate of change of the exchange rate µ > 0 expected depreciation UIP time continuous and π = 0 i t = i t + µ t π t (13.15) π t = i t + µ t i t (13.16) π t (risk-premium) is the excess - return on foreign assets. Unfortunately, the risk-premium is extremely volatile. We lack a theory which is able to explain the determinants of π t The international Fisher Equation The following relationship is the Fisher-Equation: 1 + i = (1 + r) (1 + E(π)) (13.17) i = nominal interest rate E(π) = expected ination r = real interest rate 7

8 Approximation 1 + i = 1 + r + E(π) + re(π) (13.18) i r + E(π) (13.19) Fisher: The nominal interest rate reects inationary expectations given the real interest rate Two Countries i = r + E(π) (13.20) i = r + E(π ) (13.21) The PPP - version of exchange rate expectations: E t ( s t+1 ) = E(π t ) E(π t ) (13.22) E t s t+1 = E t s t+1 s t E t ( s t+1 ) = (i t r t ) (i t r t ) (13.23) E t ( s t+1 ) = i t i t (13.24) Assume the real interest rates (rates of return on capital) are equalized across countries: r t = r t The international Fisher Equation suggests that the expected change in the exchange rate to be equal to the interest rate dierential between two countries, which is the same statement as UIP derived under dierent circumstances. 8