Chapter 5 The International Parity Conditions 5 1

Chapter 5 The International Parity Conditions 5. 1 The Law of One Price 5. 2 Exchange Rate Equilibrium 5. 3 Interest Rate Parity 5. 4 Less Reliable International Parity Conditions 5. 5 The Real Exchange Rate 5. 6 Exchange Rate Forecasting 5. 7 Summary Appendix 5 -A Continuous Time Finance Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -1

Prices appear as upper case symbols P td = price of an asset at time t in currency d Std/f = spot exchange rate at time t in currency d Ftd/f = forward exchange rate between currencies d and f E[…] = expectation operator (e. g. E[St€/$]) Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -2

Rates of change Changes in a price appear as lower case symbols rtd = an asset’s return in currency d during period t ptd = inflation in currency d in period t td = real interest rate in currency d in period t std/f = change in the spot rate during period t Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -3

The law of one price Equivalent assets sell for the same price (also called purchasing power parity, or PPP) Ø Ø Ø Seldom holds for nontraded assets Can’t compare assets that vary in quality May not hold precisely when there are market frictions Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -4

An example: The world price of gold Suppose P€ P£ = £ 250/oz in London = € 400/oz in Berlin The law of one price requires: Pt£ = Pt€ St£/€ £ 250/oz = (€ 400/oz) (£ 0. 6250/€) or 1/(£ 0. 6250/€) = € 1. 6000/£ Ø If this relation does not hold, then there is an opportunity to lock in a riskless arbitrage profit. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -5

An example with transactions costs Gold dealer A Gold dealer B € 401. 40/oz Offer FX dealer € 401. 00/oz Bid Sell high to B € 1. 599/£ bid € 1. 601/£ ask Buy low from A £ 250. 25/oz Offer £ 250. 00/oz Bid Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -6

Arbitrage profit Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -7

Cross exchange rate equilibrium Sd/e Se/f Sf/d = 1 If Sd/e. Se/f. Sf/d < 1, then either Sd/e, Se/f or Sf/d must rise For each spot rate, buy the currency in the denominator with the currency in the numerator If Sd/e. Se/f. Sf/d > 1, then either Sd/e, Se/f or Sf/d must fall For each spot rate, sell the currency in the denominator for the currency in the numerator Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -8

A cross exchange rate table £ UK pound Canadian $ Euro Japanese yen Swiss Franc US Dollar C$ € ¥ SFr $ 1. 000 2. 487 1. 518 191. 6 2. 221 1. 609 0. 402 1. 000 0. 612 77. 24 0. 893 0. 647 0. 659 1. 634 1. 000 126. 1 1. 460 1. 057 0. 0052 0. 0130 0. 0079 1. 0000 0. 0116 0. 0084 0. 451 1. 120 0. 685 86. 48 1. 000 0. 724 0. 622 1. 546 0. 947 119. 4 1. 381 1. 000 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -9

Cross exchange rates and triangular arbitrage Suppose SRbl/$ = Rbl 5. 000/$ Û S$/Rbl= $0. 2000/Rbl S$/¥ Û S¥/$ = ¥ 100. 0/$ = $0. 01000/¥ S¥/Rbl = ¥ 20. 20/Rbl Û SRbl/¥ » Rbl 0. 04950/¥ SRbl/$ S$/¥ S¥/Rbl = (Rbl 5/$)($. 01/¥)(¥ 20. 20/Rbl) = 1. 01 > 1 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -10

Cross exchange rates and triangular arbitrage SRbl/$ 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 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -11

An example of triangular arbitrage SRbl/$ S$/¥ S¥/Rbl = 1. 01 > 1 Sell $1 million and buy Rbl 5 million Sell ¥ 100 million yen and buy $1 million Sell Rbl 4. 950 million and buy ¥ 100 million Profit of 50, 000 rubles = $10, 000 at Rbls 5. 000/$ or 1% of the initial amount Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -12

International parity conditions that span both currencies and time Interest rate parity linkages Ftd/f / S 0 d/f= [(1+id)/(1+if)]t Less reliable = E[Std/f] / S 0 d/f = [(1+pd)/(1+pf)]t where S 0 d/f = today’s spot exchange rate E[Std/f] = expected future spot rate Ftd/f = forward rate for time t exchange i = a country’s nominal interest rate p = a country’s inflation rate Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -13

Interest rate parity Ftd/f/S 0 d/f = [(1+id)/(1+if)]t Ø Forward premiums and discounts are entirely determined by interest rate differentials. Ø This is a parity condition that you can trust. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -14

Interest rate parity: Which way do you go? If Ftd/f/S 0 d/f > [(1+id)/(1+if)]t then Ftd/f must fall S 0 d/f must rise id must rise if must fall so. . . Sell f at Ftd/f Buy f at S 0 d/f Borrow at id Lend at if Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -15

Interest rate parity: Which way do you go? If Ftd/f/S 0 d/f < [(1+id)/(1+if)]t then Ftd/f must rise S 0 d/f must fall id must fall if must rise so. . . Buy f at Ftd/f Sell f at S 0 d/f Lend at id Borrow at if Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -16

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£) 1. 041667 > 1. 038835 The fx and Eurocurrency markets are not in equilibrium. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -17

Covered interest arbitrage 1. Borrow $1, 000 +$1, 000 at i$ = 7% 2. Convert $s to £s at S 0$/£ = $1. 20/£ 3. Invest £s at i£ = 3% 4. Convert £s to $s at F 1$/£ = $1. 25/£ -$1, 070, 000 +£ 833, 333 -$1, 000 +£ 858, 333 -£ 833, 333 +$1, 072, 920 -£ 858, 333 5. Take your profit: $1, 072, 920 -$1, 070, 000 = $2, 920 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -18

Forward rates as predictors of future spot rates Ftd/f = E[Std/f] or Ftd/f / S 0 d/f = E[Std/f] / S 0 d/f Forward rates are unbiased estimates of future spot rates. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -19

Forward rates as predictors of future spot rates E[Std/f ] / S 0 d/f = Ftd/f / S 0 d/f Speculators will force this relation to hold on average Ø For daily exchange rate changes, the best estimate of tomorrow's spot rate is the current spot rate Ø As the sampling interval is lengthened, the performance of forward rates as predictors of future spot rates improves Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -20

Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -21

Relative purchasing power parity (RPPP) Let Pt = a consumer price index level at time t Then inflation pt = (Pt - Pt-1) / Pt-1 E[Std/f] / S 0 d/f = = = (E[Ptd] / E[Ptf]) / (P 0 d /P 0 f) (E[Ptd]/P 0 d) / (E[Ptf]/P 0 f) (1+E[pd])t / (1+E[pf])t where pd and pf are geometric mean inflation rates. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -22

Relative purchasing power parity (RPPP) E[Std/f] / S 0 d/f = (1+E[pd])t / (1+E[pf])t Speculators will force this relation to hold on average Ø The expected change in a spot exchange rate should reflect the difference in inflation between the two currencies. Ø This relation only holds over the long run. Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -23

Relative purchasing power parity (RPPP) Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -24

International Fisher relation (Fisher Open hypothesis) [(1+id)/(1+if)]t = [(1+pd)/(1+pf)]t Recall the Fisher relation: (1+i) = (1+ )(1+p) If real rates of interest are equal across currencies, then [(1+id)/(1+if)]t [(1+ f)(1+pf)]t = [(1+ d)(1+pd)]t / = [(1+pd)/(1+pf)]t Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -25

International Fisher relation (Fisher Open hypothesis) [(1+id)/(1+if)]t = [(1+pd)/(1+pf)]t Speculators will force this relation to hold on average Ø If real rates of interest are equal across countries ( d = f ), then interest rate differentials merely reflect inflation differentials Ø This relation is unlikely to hold at any point in time, but should hold in the long run Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -26

International Fisher relation Difference in realized quarterly inflation Difference in 3 -month interest rates Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -27

Summary: Int’l parity conditions International Fisher relation Interest rates [(1+id)/(1+if)]t Interest rate parity Ftd/f / S 0 d/f Forward-spot differential Inflation rates [(1+pd)/(1+pf)]t Relative PPP E[Std/f] / S 0 d/f Expected change in the spot rate Forward rates as predictors of future spot rates Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -28

Purchasing power (dis)parity The Big Mac Index Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -29

The real exchange rate Ø The real exchange rate adjusts the nominal exchange rate for differential inflation since an arbitrarily defined base period Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -30

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 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -31

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/$? Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -32

Actual versus expected change ¥ 130//$ St¥/$ ¥ 120//$ ¥ 110//$ Actual S 1¥/$ = ¥ 110/$ ¥ 100//$ ¥ 90//$ E[S 1¥/$] = ¥ 90. 91/$ time Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -33

Change in the real exchange rate ØIn real (or purchasing power) terms, the dollar has appreciated by (¥ 110/$) / (¥ 90. 91/$) - 1 = +0. 21 or 21 percent more than expected Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -34

Change in the real exchange rate (1+xtd/f) = (Std/f / St-1 d/f) [(1+ptf)/(1+ptd)] where xtd/f Std/f ptc = percentage change in the real exchange rate = the nominal spot rate at time t = inflation in currency c during period t Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -35

Change in the real exchange rate Example S 0¥/$ = ¥ 100/$ S 1¥/$ = ¥ 110/$ E[p¥] = 0% and E[p$] = 10% xt¥/$ = [(¥ 110/$)/(¥ 100/$)][1. 10/1. 00] -1 = 0. 21, or a 21 percent increase in real purchasing power Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -36

Behavior of real exchange rates Ø Deviations parity from purchasing power - can be substantial in the short run - and can last for several years Ø Both the level and variance of the real exchange rate are autoregressive Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -37

Real value of the dollar (1970 -1998) Mean level = 100 for each series Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -38

Appendix 5 -A Continuous time finance Ø Most theoretical and empirical research in finance is conducted in continuously compounded returns Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -39

Holding period returns are asymmetric r 1 = +100% 200 r 2 = -50% 100 (1+r. TOTAL) = (1+r 1)(1+r 2) = (1+1)(1 -½) = (2)(½) = 1 r. TOTAL = 0% Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -40

Continuous compounding Let r = holding period (e. g. annual) return r = continuously compounded return r = ln (1+r) = ln (er ) Û (1 + r) = er where ln(. ) is the natural logarithm with base e » 2. 718 Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -41

Continuous returns are symmetric +69. 3% 200 -69. 3% 100 r. TOTAL = Ln[(1+r 1)(1+r 2)] = r 1+r 2 = +0. 693 - 0. 693 = 0. 000 r. TOTAL = 0% Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -42

Properties of natural logarithms (for x > 0) eln(x) = ln(ex) =x ln(AB) = ln(A) + ln(B) ln(At) = t * ln(A) ln(A/B) = ln(AB-1) = ln(A) - ln(B) Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -43

Continuously compounded returns are additive rather than multiplicative ln[ (1+r 1) (1+r 2). . . (1+r. T) ] = r 1 + r 2 +. . . + r. T Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -44

The international parity conditions in continuous time Over a single period ln(F 1 d/f / S 0 d/f ) = id– if = E[pd ] – E[pf ] = E[sd/f ] where s d/f, p d, p f, i d, and i f are continuously compounded Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -45

The international parity conditions in continuous time Over t periods ln(Ftd/f / S 0 d/f ) = t (i d – i f ) = t (E[pd ] – E[pf ]) = t E[sd/f ] where s d/f, p d, p f, i d, and i f are continuously compounded Kirt C. Butler, Multinational Finance, South-Western College Publishing, 3 e 5 -46