第10讲Covered and Uncovered Interest Rate Parity(国际金融(香港大学,WONG Ka Fu)
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19
RHS = [Et (et+1 ) - et ] / et + Rt*
Suppose Et (et+1 ) and et fixed, larger Rt* implies larger RHS
20
Uncovered Interest Parity
Suppose we care 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 currency Rt [Et (et+1 ) - et ] / et + Rt* Equilibrium if Rt = [Et (et+1 ) - et ] / et + Rt*
8
Expectations
Lottery 1 0.5 probability to win 1000 0.5 probability to win 0 Expect to win 0.5 × 1000 + 0.5 × 0 = 500
9
Expectations
Lottery 2 0.2 probability to win 1000 0.3 probability to win 500 0.5 probability to win 0 Expect to win 0.2 × 1000 + 0.3 × 500 + 0.5 × 0 = 350
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 asset Et { [(et+1 / et ) (Pt+1* + Dt+1* ) / Pt* ] - 1 } = Et [(et+1 / et ) (Pt+1* + Dt+1* ) ] / Pt* - 1
18
RHS = [Et (et+1 ) - et ] / et + Rt*
Suppose Et (et+1 ) and Rt* fixed, larger et implies smaller RHS
Suppose et and Rt* fixed, larger Et (et+1 ) implies larger RHS
16
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 rate Et [(et+1 / et ) (1 + Rt* ) ] / 1 - 1 = [Et (et+1 ) / et ] (1 + Rt* ) - 1
22
Uncovered Interest Parity floating exchange rate regime
Also because both home and foreign investors will deposit in home currency I.e., larger supply of home deposit and smaller supply of foreign deposit hence, home interest rate Rt decreases, i.e. towards equality Rt* increases, i.e. towards equality
Covered and Uncovered Interest Rate Parity
WONG Ka Fu 26th January 2000
1
Comparing Local and Foreign Prices
Prices within a country
Prices across countries P (in home currency) P* (in foreign currency)
17
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* [Et (et+1 ) - et ] / et + Rt * -1 [Et (et+1 ) - et ] / et + Rt*
t+1
Return = et+1(Pt+1* +Dt+1* )/ - etPt* Rate of Return = [et+1(Pt+1* +Dt+1* )/ - etPt* ]/ etPt*
6
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 asset et+1 (Pt+1* + Dt+1* ) - etPt*
4
Rate of Return on a home asset
Return / cost of asset at time of purchase / year
Home asset in home currency ( Pt+1 + Dt+1 - Pt )/ Pt = [ ( Pt+1 + Dt+1 ) / Pt ] - 1
5
Time
Return on foreign asset
Buy Asset: pay etPt* May receive dividend Dt+1* between time t and time t+1 Sell asset: get et+1Pt+1*
t
For example, t=January, t+1=February
7
Rate of Return on a foreign源自文库asset
Foreign asset in home currency [ et+1 ( Pt+1* + Dt+1* ) - etPt* ] / ( etPt* ) = [ et+1 ( Pt+1* + Dt+1* ) / ( etPt* ) ] - 1 = (et+1 / et ) [ ( Pt+1* + Dt+1* ) / Pt* ] - 1
t
For example, t=January, t+1=February
t+1
Return = Pt+1+Dt+1 -Pt Rate of Return = (Pt+1+Dt+1 - Pt)/Pt
3
Return on a home asset
Pt+1 - Pt dividends or any interest payments to the asset holder Dt+1 Pt+1 - Pt + Dt+1
Expect to win Et(y) = E(y| information available at time t)
= y f(y) y dy
12
Replacing assets with deposits greatly simplifies the algebra:
Some unknown quantities become known: Pt+1 = 1 Pt = 1 Dt+1 = Rt= home interest rate Pt+1*= 1 Pt* = 1 Dt+1*= Rt*= foreign interest rate The only unknown at time t is et+1
14
Rate of return of home deposit
Et (Pt+1 + Dt+1) / Pt - 1 Pt+1 = 1 Pt = 1 Dt+1 = Rt= home interest rate Et (1 + Rt) / 1 - 1 = Rt
15
Expected return and expected rate of return
10
Expectations
Lottery 3
Pi = f(yi) probability to win yi
Expect to win
Et(y) = i Pi yi = i f(yi) yi
11
Expectations
Lottery 4
f(y) probability to win y
23
Uncovered Interest Parity floating exchange rate regime
In general, both interest 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.
1 HD = x FD = 1/e FD, i.e., e HD = 1 FD P vs. eP*
2
Time
Return on home asset
Buy Asset: pay Pt May receive dividend Dt+1 between time t and time t+1 Sell asset: get Pt+1
13
Expected return and expected rate of return
Expected return on a home asset: Et (Pt+1 + Dt+1 - Pt ) = Et (Pt+1 + Dt+1) - Pt
Expected rate of return on a home asset: Et [(Pt+1 + Dt+1) / Pt - 1 ] = Et (Pt+1 + Dt+1) / Pt - 1
21
Uncovered Interest Parity floating exchange rate regime
If Rt > [Et (et+1 ) - et ] / et + Rt* both home and foreign investors will deposit in home currency implies supply foreign currency and demand home currency initially, e = y HD = 1 FD now, e = z HD = 1 FD , z < y hence larger RHS, i.e., towards equality
24
Uncovered Interest Parity floating exchange rate regime
Rt = [Et (et+1 ) - et ] / et + Rt* Hence, the Uncovered Interest Parity can be used to determine 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* .
RHS = [Et (et+1 ) - et ] / et + Rt*
Suppose Et (et+1 ) and et fixed, larger Rt* implies larger RHS
20
Uncovered Interest Parity
Suppose we care 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 currency Rt [Et (et+1 ) - et ] / et + Rt* Equilibrium if Rt = [Et (et+1 ) - et ] / et + Rt*
8
Expectations
Lottery 1 0.5 probability to win 1000 0.5 probability to win 0 Expect to win 0.5 × 1000 + 0.5 × 0 = 500
9
Expectations
Lottery 2 0.2 probability to win 1000 0.3 probability to win 500 0.5 probability to win 0 Expect to win 0.2 × 1000 + 0.3 × 500 + 0.5 × 0 = 350
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 asset Et { [(et+1 / et ) (Pt+1* + Dt+1* ) / Pt* ] - 1 } = Et [(et+1 / et ) (Pt+1* + Dt+1* ) ] / Pt* - 1
18
RHS = [Et (et+1 ) - et ] / et + Rt*
Suppose Et (et+1 ) and Rt* fixed, larger et implies smaller RHS
Suppose et and Rt* fixed, larger Et (et+1 ) implies larger RHS
16
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 rate Et [(et+1 / et ) (1 + Rt* ) ] / 1 - 1 = [Et (et+1 ) / et ] (1 + Rt* ) - 1
22
Uncovered Interest Parity floating exchange rate regime
Also because both home and foreign investors will deposit in home currency I.e., larger supply of home deposit and smaller supply of foreign deposit hence, home interest rate Rt decreases, i.e. towards equality Rt* increases, i.e. towards equality
Covered and Uncovered Interest Rate Parity
WONG Ka Fu 26th January 2000
1
Comparing Local and Foreign Prices
Prices within a country
Prices across countries P (in home currency) P* (in foreign currency)
17
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* [Et (et+1 ) - et ] / et + Rt * -1 [Et (et+1 ) - et ] / et + Rt*
t+1
Return = et+1(Pt+1* +Dt+1* )/ - etPt* Rate of Return = [et+1(Pt+1* +Dt+1* )/ - etPt* ]/ etPt*
6
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 asset et+1 (Pt+1* + Dt+1* ) - etPt*
4
Rate of Return on a home asset
Return / cost of asset at time of purchase / year
Home asset in home currency ( Pt+1 + Dt+1 - Pt )/ Pt = [ ( Pt+1 + Dt+1 ) / Pt ] - 1
5
Time
Return on foreign asset
Buy Asset: pay etPt* May receive dividend Dt+1* between time t and time t+1 Sell asset: get et+1Pt+1*
t
For example, t=January, t+1=February
7
Rate of Return on a foreign源自文库asset
Foreign asset in home currency [ et+1 ( Pt+1* + Dt+1* ) - etPt* ] / ( etPt* ) = [ et+1 ( Pt+1* + Dt+1* ) / ( etPt* ) ] - 1 = (et+1 / et ) [ ( Pt+1* + Dt+1* ) / Pt* ] - 1
t
For example, t=January, t+1=February
t+1
Return = Pt+1+Dt+1 -Pt Rate of Return = (Pt+1+Dt+1 - Pt)/Pt
3
Return on a home asset
Pt+1 - Pt dividends or any interest payments to the asset holder Dt+1 Pt+1 - Pt + Dt+1
Expect to win Et(y) = E(y| information available at time t)
= y f(y) y dy
12
Replacing assets with deposits greatly simplifies the algebra:
Some unknown quantities become known: Pt+1 = 1 Pt = 1 Dt+1 = Rt= home interest rate Pt+1*= 1 Pt* = 1 Dt+1*= Rt*= foreign interest rate The only unknown at time t is et+1
14
Rate of return of home deposit
Et (Pt+1 + Dt+1) / Pt - 1 Pt+1 = 1 Pt = 1 Dt+1 = Rt= home interest rate Et (1 + Rt) / 1 - 1 = Rt
15
Expected return and expected rate of return
10
Expectations
Lottery 3
Pi = f(yi) probability to win yi
Expect to win
Et(y) = i Pi yi = i f(yi) yi
11
Expectations
Lottery 4
f(y) probability to win y
23
Uncovered Interest Parity floating exchange rate regime
In general, both interest 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.
1 HD = x FD = 1/e FD, i.e., e HD = 1 FD P vs. eP*
2
Time
Return on home asset
Buy Asset: pay Pt May receive dividend Dt+1 between time t and time t+1 Sell asset: get Pt+1
13
Expected return and expected rate of return
Expected return on a home asset: Et (Pt+1 + Dt+1 - Pt ) = Et (Pt+1 + Dt+1) - Pt
Expected rate of return on a home asset: Et [(Pt+1 + Dt+1) / Pt - 1 ] = Et (Pt+1 + Dt+1) / Pt - 1
21
Uncovered Interest Parity floating exchange rate regime
If Rt > [Et (et+1 ) - et ] / et + Rt* both home and foreign investors will deposit in home currency implies supply foreign currency and demand home currency initially, e = y HD = 1 FD now, e = z HD = 1 FD , z < y hence larger RHS, i.e., towards equality
24
Uncovered Interest Parity floating exchange rate regime
Rt = [Et (et+1 ) - et ] / et + Rt* Hence, the Uncovered Interest Parity can be used to determine 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* .