Less Reliable International Parity Conditions

Transcription

1 The International Parity Conditions The Law of One Price Interest Rate Parity Less Reliable International Parity Conditions The Real Exchange Rate 1

2 The International Parity Conditions Though this be madness, yet there is method in it. William Shakespeare 2

3 Notation Upper Case Symbols = Prices lower case symbols = changes in a price P d t p d i d ʀ d S t s t F t = price of an asset in currency d at time t = inflation rate in currency d (% change in CPI) = nominal interest rate in currency d = real interest rate in currency d = spot exchange rate at time t between d and f = change in the spot rate during period t = forward exchange rate between d and f E[ ] = expectations operator (e.g., E[S t $/ ]) Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 3

4 The law of one price Equivalent assets sell for the same price (also called purchasing power parity, or PPP) Seldom holds for non-traded assets Can t compare assets that vary in quality May not hold precisely when there are market frictions Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 4

5 Purchasing power (dis)parity: The Big Mac Index As of October 14, 2010 Relative P $ = P f / S f/$ P f S f/$ P $ to the $ USA ($) % Brazil (BRL) % Britain ( ) % Canada (C$) % Euro-zone ( ) % China % Japan ( ) % S. Korea (Won) % Switzerland (SFr) % Sources: The Economist ( The Big Mac Index, Oct. 14, 2010) and Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 5

6 An example of PPP: The price of gold Suppose P = 940/oz in London P $ = $1504/oz in New York The law of one price requires: P t = P t$ S t /$ S t /$ = P t /P t$ = ( 940/oz) / ($1504/oz) = /$ or S t $/ = 1 / S t /$ = 1 / ( /$) = $1.6000/ If this relation does not hold, then there may be an opportunity to lock in a riskless arbitrage profit. Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 6

7 An example with transactions costs London gold dealer FX dealer $1.599/ bid $1.601/ ask New York gold dealer $1522/oz Ask $1510/oz Bid Sell high to New York 940/oz Ask Buy low from London 930/oz Bid Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 7

8 Arbitrage profit in gold Buy 1000 oz at London s 940/oz offer price +(1000 oz) - 940,000 Sell 1000 oz at NY s $1510/oz bid price +$1,510,000 Arbitrage profit + 3,161 -(1000 oz) Cover your position at the $1.601/ ask price for pounds + 943,161 = ($1,510,000)/($1.601/ ) -$1,510,000 Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 8

9 Cross exchange rate equilibrium S d/e S e/f S f/d = 1 If S d/e S e/f S f/d < 1, then either S d/e, S e/f or S f/d must rise For each spot rate, buy the currency in the denominator with the currency in the numerator If S d/e S e/f S f/d > 1, then either S d/e, S e/f or S f/d must fall For each spot rate, sell the currency in the denominator for the currency in the numerator Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 9

10 Cross exchange rates and triangular arbitrage Suppose S Rbl/$ = Rbl 5.000/$ S $/Rbl = $0.2000/Rbl S $/ = $ / S = 100.0/$ S /Rbl = 20.20/Rbl S Rbl/ Rbl / S Rbl/$ S $/ S /Rbl = (Rbl 5/$)($.01/ )( 20.20/Rbl) = 1.01 > 1 Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 10

11 Cross exchange rates and triangular arbitrage S Rbl/$ S $/ S /Rbl = 1.01 > 1 Currencies in the denominators are too high relative to the numerators, so sell dollars and buy rubles sell yen and buy dollars sell rubles and buy yen Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 11

12 An example of triangular arbitrage S Rbl/$ S $/ S /Rbl = 1.01 > 1 Sell $1 million and buy Rbl 5 million Sell 100 million yen and buy $1 million Sell Rbl million and buy 100 million Profit of 50,000 rubles = $10,000 at Rbls5.000/$ or 1% of the initial amount Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 12

13 International parity conditions that span both currencies and time Interest rate parity Less reliable linkages F t / S 0 = [(1+i d )/(1+i f )] t = E[S t ] / S 0 = [(1+E[p d ])/(1+E[p f ])] t where S 0 E[S t ] F t i p = today s spot exchange rate = expected future spot rate = forward rate for time t exchange = a nominal interest rate = an expected inflation rate Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 13

14 Interest rate parity F t /S 0 = [(1+i d )/(1+i f )] t Forward premiums and discounts are entirely determined by interest rate differentials. This is a parity condition that you can trust. Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 14

15 Interest rate parity Invest in dollars V 0 $ Convert to euros at S 0 $/ FV t$ = PV 0$ (1+i $ ) t Moving $s today into s tomorrow Convert to euros at F t $/ Time 0 Invest in euros FV t = PV 0 (1+i ) t V t Time t Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 15

16 Interest rate parity: Which way do you go? If F t /S 0 > [(1+i d )/(1+i f )] t then F t must fall S 0 must rise i d must rise i f must fall so... Sell f at F t Buy f at S 0 Borrow at i d Lend at i f Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 16

17 Interest rate parity: Which way do you go? If F t /S 0 < [(1+i d )/(1+i f )] t then F t must rise S 0 must fall i d must fall i f must rise so... Buy f at F t Sell f at S 0 Lend at i d Borrow at i f Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 17

18 Interest rate parity is enforced through covered interest arbitrage An Example: Given: i $ = 7% S 0 $/ = $1.20/ i = 3% F 1 $/ = $1.25/ F 1 $/ / S 0 $/ > (1+i $ ) / (1+i ) > The fx and Eurocurrency markets are not in equilibrium. Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 18

19 Covered interest arbitrage 1. Borrow $1,000,000 at i $ = 7% 2. Convert $s to s at S 0 $/ = $1.20/ 3. Invest s at i = 3% +$1,000, ,333 -$1,000, Convert s to $s - 833,333 at F $/ 1 = $1.25/ 5. Take your profit: $1,072,920-$1,070,000 = $2,920 -$1,070, ,333 +$1,072, ,333 Less reliable parity conditions Notation and definition Parity in commodities markets Cross exchange rate equilibrium Interest rate parity & covered interest arbitrage 19

20 Relative purchasing power parity (RPPP) Let P t = a consumer price index level at time t Then inflation p t = (P t - P t-1 ) / P t-1 E[S t ] / S 0 = (E[P td ] / E[P tf ]) / (P 0d /P 0f ) = (E[P td ]/P 0d ) / (E[P tf ]/P 0f ) = (1+E[p d ]) t / (1+E[p f ]) t where p d and p f are geometric mean inflation rates. Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 20

21 Relative purchasing power parity (RPPP) E[S t ] / S 0 = (1+E[p d ]) t / (1+E[p f ]) t Speculators will force this relation to hold on average The expected change in a spot exchange rate should reflect the difference in inflation between the two currencies. This relation only holds over the long run. Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 21

22 RPPP over monthly intervals 15% S 1 /S % 5% 0% (1+p )/(1+p $ ) - 1-5% -10% -15% -2% -1% 0% 1% 2% Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 22

23 RPPP over a 5-year horizon ( ) S 1 f/$ /S 0 f/$ - 1 (1+p f )/(1+p $ ) - 1 Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 23

24 Relative purchasing power parity (RPPP) Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 24

25 Forward rates as predictors of future spot rates E[S t ] / S 0 = F t / S 0 Speculators will force this relation to hold on average For daily exchange rate changes, the best estimate of tomorrow's spot rate is the current spot rate As the sampling interval is lengthened, the performance of forward rates as predictors of future spot rates improves Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 25

26 Yen-per-$ forward parity over monthly intervals 15% S 1 /S % 5% 0% F 1 /S % -10% -15% -1% 0% 1% Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 26

27 International Fisher relation (Fisher Open hypothesis) The Fisher equation provides a starting point (1+i) = (1+ʀ)(1+p) i = nominal interest rate ʀ = real interest rate p = inflation rate or i ʀ + p for small ʀ and p Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 27

28 International Fisher relation (Fisher Open hypothesis) [(1+i d )/(1+i f )] t = [(1+p d )/(1+p f )] t where (1+i) = (1+ʀ)(1+p) from the Fisher relation If real rates of interest are equal across currencies (ʀ d = ʀ f ), then [(1+i d )/(1+i f )] t = [(1+ʀ d )(1+p d )] t / [(1+ʀ f )(1+p f )] t = [(1+p d )/(1+p f )] t Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 28

29 International Fisher relation (Fisher Open hypothesis) [(1+i d )/(1+i f )] t = [(1+p d )/(1+p f )] t Speculators will force this relation to hold on average If real rates of interest are equal across countries ( d = f ), then interest rate differentials merely reflect inflation differentials This relation is unlikely to hold at any point in time, but should hold in the long run Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 29

30 Summary: Int l parity conditions International Fisher relation Interest rates [(1+i d )/(1+i f )] t Interest rate parity F t / S 0 Forward-spot differential Inflation rates [(1+p d )/(1+p f )] t Relative PPP E[S t ] / S 0 Expected change in the spot rate Forward rates as predictors of future spot rates Less reliable parity conditions Relative purchasing power parity Forward parity The international Fisher relation Summary 30

31 adjusts the nominal exchange rate for differential inflation since an arbitrarily defined base period Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 31

32 Change in the nominal exchange rate Example S 0 = 100/$ S 1 = 110/$ E[p ] = 0% E[p $ ] = 10% s 1 = (S 1 S 0 )/S 0 = 0.10, or a 10 percent nominal change Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 32

33 The expected nominal exchange rate But RPPP implies E[S 1 ] = S 0 (1+ p )/(1+ p $ ) = 90.91/$ What is the change in the nominal exchange rate relative to the expectation of 90.91/$? Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 33

34 Actual versus expected change 130//$ S t 120//$ 110//$ 100//$ 90//$ Actual S 1 = 110/$ E[S 1 ] = 90.91/$ time Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 34

35 Change in the real exchange rate In real (or purchasing power) terms, the dollar has appreciated by ( 110/$) / ( 90.91/$) - 1 = or 21 percent more than expected Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 35

36 Change in the real exchange rate (1+x t ) = (S t / S t-1 ) [(1+p tf )/(1+p td )] where x t = percentage change in the real exchange rate S t = the nominal spot rate at time t p c t = inflation in currency c during period t Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 36

37 Change in the real exchange rate Example S 0 = 100/$ S 1 = 110/$ E[p ] = 0% and E[p $ ] = 10% x t = [( 110/$)/( 100/$)][1.10/1.00] -1 = 0.21, or a 21 percent increase in real purchasing power Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 37

38 Behavior of real exchange rates Deviations from purchasing power parity - can be substantial in the short run - and can last for several years Both the level and variance of the real exchange rate are autoregressive Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 38

39 Behavior of selected currencies against the USD in 2010 Mean level = 0% for each series Less reliable parity conditions Change in the real exchange rate The behavior of real exchange rates 39