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1、Covered and Uncovered Interest Rate Parity,WONG Ka Fu26th January 2000,Comparing Local and Foreign Prices,Prices within a countryPrices across countriesP(in home currency)P*(in foreign currency)1 HD=x FD=1/e FD,i.e.,e HD=1 FDP vs.eP*,t,t+1,Time,For example,t=January,t+1=February,Buy Asset:pay Pt,Ret
2、urn on home asset,Sell asset:get Pt+1,May receive dividend Dt+1between time t and time t+1,Return=Pt+1+Dt+1-Pt,Rate of Return=(Pt+1+Dt+1-Pt)/Pt,Return on a home asset,Pt+1-Ptdividends or any interest payments to the asset holder Dt+1 Pt+1-Pt+Dt+1,Rate of Return on a home asset,Return/cost of asset a
3、t time of purchase/yearHome asset in home currency(Pt+1+Dt+1-Pt)/Pt=(Pt+1+Dt+1)/Pt-1,t,t+1,Time,For example,t=January,t+1=February,Buy Asset:pay etPt*,Return on foreign asset,Sell asset:get et+1Pt+1*,May receive dividend Dt+1*between time t and time t+1,Return=et+1(Pt+1*+Dt+1*)/-etPt*,Rate of Return
4、=et+1(Pt+1*+Dt+1*)/-etPt*/etPt*,Return on a foreign asset,A foreign investor invests in a foreign asset Pt+1*-Pt*+Dt+1*=Pt+1*+Dt+1*-Pt*A home investor invests in a foreign assetet+1(Pt+1*+Dt+1*)-etPt*,Rate of Return on a foreign asset,Foreign asset in home currency et+1(Pt+1*+Dt+1*)-etPt*/(etPt*)=et
5、+1(Pt+1*+Dt+1*)/(etPt*)-1=(et+1/et)(Pt+1*+Dt+1*)/Pt*-1,Expectations,Lottery 10.5 probability to win 10000.5 probability to win 0Expect to win0.5 1000+0.5 0=500,Expectations,Lottery 20.2 probability to win 10000.3 probability to win 5000.5 probability to win 0Expect to win0.2 1000+0.3 500+0.5 0=350,E
6、xpectations,Lottery 3Pi=f(yi)probability to win yiExpect to winEt(y)=i Pi yi=i f(yi)yi,Expectations,Lottery 4f(y)probability to win yExpect to winEt(y)=E(y|information available at time t)=y f(y)y dy,Replacing assets with deposits greatly simplifies the algebra:,Some unknown quantities become known:
7、Pt+1=1 Pt=1 Dt+1=Rt=home interest ratePt+1*=1 Pt*=1 Dt+1*=Rt*=foreign interest rateThe only unknown at time t is et+1,Expected return and expected rate of return,Expected return on a home asset:Et(Pt+1+Dt+1-Pt)=Et(Pt+1+Dt+1)-PtExpected rate of return on a home asset:Et(Pt+1+Dt+1)/Pt-1=Et(Pt+1+Dt+1)/
8、Pt-1,Rate of return of home deposit,Et(Pt+1+Dt+1)/Pt-1 Pt+1=1 Pt=1 Dt+1=Rt=home interest rateEt(1+Rt)/1-1=Rt,Expected return and expected rate of return,Expected return on a foreign asset:Et et+1(Pt+1*+Dt+1*)-etPt*=Et et+1(Pt+1*+Dt+1*)-etPt*Expected rate of return on a foreign assetEt(et+1/et)(Pt+1*
9、+Dt+1*)/Pt*-1=Et(et+1/et)(Pt+1*+Dt+1*)/Pt*-1,Rate of return on foreign deposit,Et(et+1/et)(Pt+1*+Dt+1*)/Pt*-1 Pt+1*=1 Pt*=1 Dt+1*=Rt*=foreign interest rateEt(et+1/et)(1+Rt*)/1-1=Et(et+1)/et(1+Rt*)-1,Rate of return on foreign deposit,Et(et+1)/et(1+Rt*)-1=(1+Rt*)Et(et+1)-et+et/et-1=Et(et+1)-et/et+1+Rt
10、*Et(et+1)-et/et+Rt*-1 Et(et+1)-et/et+Rt*,RHS=Et(et+1)-et/et+Rt*,Suppose Et(et+1)and Rt*fixed,larger et implies smaller RHSSuppose et and Rt*fixed,larger Et(et+1)implieslarger RHS,RHS=Et(et+1)-et/et+Rt*,Suppose Et(et+1)and et fixed,larger Rt*implieslarger RHS,Uncovered Interest Parity,Suppose we care
11、 only about expected return(say,we are risk neutral)Deposit in home currency if and only if the rate of return on the deposit in home currency is not less than the deposit in foreign currencyRt Et(et+1)-et/et+Rt*Equilibrium if Rt=Et(et+1)-et/et+Rt*,Uncovered Interest Parity floating exchange rate re
12、gime,If Rt Et(et+1)-et/et+Rt*both home and foreign investors will deposit in home currency impliessupply foreign currency and demand home currencyinitially,e=y HD=1 FDnow,e=z HD=1 FD,z yhence larger RHS,i.e.,towards equality,Uncovered Interest Parity floating exchange rate regime,Also because both h
13、ome and foreign investors will deposit in home currency I.e.,larger supply of home deposit and smaller supply of foreign deposithence,home interest rate Rt decreases,i.e.towards equalityRt*increases,i.e.towards equality,Uncovered Interest Parity floating exchange rate regime,In general,both interest
14、 rates and exchange rate will adjust to restore the equality.Can the CBs fix the interest rates at some desired level?Yes.If so,the exchange rate alone will do the job.,Uncovered Interest Parity floating exchange rate regime,Rt=Et(et+1)-et/et+Rt*Hence,the Uncovered Interest Parity can be used to det
15、ermine the exchange rate.And,given any three of the variables we can compute the remaining one.E.g.,given Et(et+1),et,and Rt*,we can compute Rt=Et(et+1)-et/et+Rt*.,Recall rate of return on foreign deposit decreases with et,et,RHS,RHS=Et(et+1)-et/et+Rt*,Recall rate of return on foreign deposit increa
16、ses with Rt*,et,RHS,Recall rate of return on foreign deposit increases with Et(et+1*),et,RHS,Determination of et given Rt,Et(et+1)and Rt*,et,RHS,Return on home deposit,Effect of an increase in the home deposit interest rate,et,RHS,Return on home deposit,Effect of an increase in the foreign deposit i
17、nterest rate,et,RHS,Return on home deposit,Fixed exchange rate regime,To maintain the exchange rate at a fixed level,CB has to commit to buying whatever the market supply of home currency and selling whatever the market demand for home currency,Uncovered Interest Parity fixed exchange rate regime,If
18、 Rt Et(et+1)-et/et+Rt*=Rt*both home and foreign investors will deposit in home currency and supply foreign currency and demand home currencyCB is committed to a fixed exchange rate and hence has to sell home currency and buy foreign currency official foreign reserves increase,Uncovered Interest Pari
19、ty fixed exchange rate regime,Both home and foreign investors will deposit in home currency I.e.,larger supply of home deposit and smaller supply of foreign deposithence,home interest rate Rt decreases,i.e.towards equalityRt*increases,i.e.towards equalityuntil Rt=Rt*,Effect of an increase in the for
20、eign deposit interest rate,et,RHS,Return on home deposit has to increase,Uncovered Interest Parity fixed exchange rate regime,Note that only interest rate will adjust to restore the equalityCan the CBs fix the interest rates at some desired level?No.Not without restrictions on capital flow.Under pur
21、e fixed exchange rate regime a CB does not have monetary policy.,Uncovered Interest Parity floating exchange rate regime,Recall that in Rt=Et(et+1)-et/et+Rt*investors faces exchange rate risk when invested in foreign deposits.Investors are generally not risk-neutral.In general,Rt Et(et+1)-et/et+Rt*R
22、t=Et(et+1)-et/et+Rt*+risk premiuminvestors spend huge amount of money trying to forecast et+1,Forecasting et+1,Uncovered interest parity:Rt=Et(et+1)-et/et+Rt*impliesa forecast of et+1 is Et(et+1)=et(Rt-Rt*)+1 Nave forecast(random walk):et+1=et+ut+1Et(et+1)=et,Forecasting et+1,Use linear time series
23、models such as ARIMA ECO3131:Applied Forecasting MethodsUse non-linear time series modelsHowell Tong:Non-linear Time Series a Dynamical System ApproachJournal of Forecasting,Uncovered Interest Parity fixed exchange rate regime,In general,Rt Rt*Why?Because Et(et+1)-et 0Because investors may expect th
24、e(fixed)exchange rate to changeand hence risk premium 0,Covered Interest Parity,Is there a way to avoid the exchange risk?Yes!By using a forward contract to offset your position on foreign exchange.Hence,may replace Et(et+1)with the time t one-period ahead forward rate ft,1Rt=ft,1-et/et+Rt*,Forecast
25、ing et+1,Forward rate suggest:ft=Et(et+1)impliesa forecast of et+1 is Et(et+1)=ft,Empirical evidence of the interest parity,Covered interest parity:generally supported by dataUncovered interest parity:generally not supported by data,Empirical evidence of the interest parity,Figures from Moosa and Bh
26、atti(1997):Uncovered interest parity:Domestic returns and foreign returnsFigure 1.9(a),Figure 1.9(b)Figure 1.10(a),Figure 1.10(b),Figure 1.11(a),Figure 1.11(b),Figure 1.12(a),Figure 1.12(b)Covered interest parity:Actual forward rate and CIP forward rate.Figure 1.5(a),Figure 1.5(b)Figure 1.6(a),Figur
27、e 1.6(b),Figure 1.7(a),Figure 1.7(b),Figure 1.8(a),Figure 1.8(b),Uncovered interest parity is,rate of return on home deposit=rate of return on foreign depositRt=Et(et+1)/et(1+Rt*)-11+Rt=Et(et+1)/et(1+Rt*)et(1+Rt)/(1+Rt*)=Et(et+1)ln Et(et+1)=ln et+ln(1+Rt)-ln(1+Rt*)Et(ln et+1)ln et+(Rt-Rt*)ln et+1 ln
28、 et+(Rt-Rt*)+tt=ln et+1-Et(ln et+1)ln et+1-ln et=+(Rt-Rt*)+t,Empirical test of uncovered interest parity,ln et+1-ln et=+(Rt-Rt*)+tTest=0 and=1.Ordinary Least Squares regression 0,Test of unbiased hypothesis,Et(et+1)=ft Et ln et+1 ln ftln et+1 ln ft+tt=ln et+1-Et(ln et+1)ln et+1-ln et=ln ft-ln et+tln
29、 et+1-ln et=+(ln ft-ln et)+tTest=0 and=1.Reject hypothesis.,Why did the regression test fail so badly?,Poor approximation?Risk premium?Expectation not rational?Monetary policy working on the short-term interest rate?Omitted variables?Buy and sell exchange and interest rates are different?Any arbitra
30、ge opportunities?,Dealing with assets returns directly is complicated:,Expected return on a home asset:Et(Pt+1+Dt+1)-PtExpected return on a foreign asset:Et et+1(Pt+1*+Dt+1*)-etPt*=Covt et+1(Pt+1*+Dt+1*)+Et et+1 Et Pt+1*+Dt+1*-etPt*using the formula Cov(x,y)=E(xy)-E(x)E(y).Thus,to forecast et+1(Et e
31、t+1),we would need to know Et Pt+1*+Dt+1*,Et Pt+1*+Dt+1*and the Covt et+1(Pt+1*+Dt+1*),Dealing with assets returns directly is complicated:,Et(Pt+1+Dt+1)-Pt may depends on the economic growth,fiscal and monetary policy(interest rate)of home country.Et Pt+1*+Dt+1*may depends on the economic growth,fi
32、scal and monetary policy(interest rate)of foreign country.Covt et+1(Pt+1*+Dt+1*)is the covariance between et+1 and(Pt+1*+Dt+1*).If they move in the same direction,Covt et+1(Pt+1*+Dt+1*)is positive.If they move in opposite directions,Covt et+1(Pt+1*+Dt+1*)is negative.,Dealing with assets returns dire
33、ctly is complicated:,There are too many unknowns.Solving these quantities would probably require us to build another economic model(s).It is difficult for us to get a clear intuition and forecast of future exchange rate.,Want to know more about it?,Krugman and ObstfledChapter 13 for the theorypp.675
34、-679 for empirical evidence Chapter 11 of LeviSearch“interest parity”in EconLit 165 items returnedFor examples.,Want to know more about it?,Moosa,Imad A.;Bhatti,Razzaque H.(1997):International Parity Conditions:Theory,Econometric Testing and Empirical Evidence,Macmillan Press.Marston,Richard C.(1997):“Tests of Three Parity Conditions:Distinguishing Risk Premia and Systemic Forecast Errors,”Journal of International Money and Finance,16(2),pp.285-303.,
