Chapter 4 The International Parity Conditions and their Consequences Learning objectives  The Law of One Price – Exchange rate equilibrium  International parity conditions – These relate exchange rates to cross-currency interest rate and inflation differentials – Interest rate parity is the most important relation  The real exchange rate – Measuring inflation-adjusted currency values

yet there is method in it. 4.1 The Law of One Price Though this be madness, yet there is method in it. William Shakespeare

4.1 The Law of One Price Notation Upper Case Symbols = Prices lower case symbols = changes in a price Vtd = value of an asset in currency d at time t rtd = return (% change) in currency d during period t id = nominal rate of interest in currency d pd = inflation rate in currency d ʀd = real (inflation-adjusted) interest rate in currency d Std/f = spot exchange rate between currencies d and f at time t std/f = percentage change in the spot rate during period t Ftd/f = forward rate between d and f for exchange at time t ftd/f = percentage change in the forward rate during period t E[…] = expectations operator (e.g., E[Std/f])

4.1 The Law of One Price The law of one price (also called purchasing power parity, or PPP) is the principle that equivalent assets sell for the same price: Seldom holds for nontraded assets. Can’t compare assets that vary in quality. May not be precise when there are market frictions. Market participants promote the law of one price through speculation and through riskless arbitrage.

4.1 The Law of One Price Arbitrage Riskless arbitrage is a profitable position obtained with no net investment and no risk. The no-arbitrage condition refers to an absence of arbitrage opportunities, so that the law of one price holds within the bounds of transaction costs.

4.1 The Law of One Price A purchasing power parity (PPP) example Suppose V£ = £940/oz in London V$ = $1504/oz in New York The law of one price requires that… St£/$ = Vt£ / Vt$ = (£940/oz) / ($1504/oz) = £0.6250/$ Alternatively, St$/£ = 1 / St£/$ = 1 / (£0.6250/$) = $1.6000/£ If this relation does not hold within the bounds of transactions costs, then there may be an opportunity to lock in a riskless arbitrage profit.

London gold dealer New York gold dealer 4.1 The Law of One Price A purchasing power parity (PPP) example London gold dealer New York gold dealer Equilibrium prices in New York should be set such that V$ = V£ S$/£ = (£940/oz)($1.6/oz) = $1504/oz S$/£ $1.6000/£ £940/oz $1504/oz

London gold dealer New York gold dealer 4.1 The Law of One Price A PPP example with transactions costs London gold dealer New York gold dealer $1522/oz Ask FX dealer $1.599/£ bid $1.601/£ ask $1510/oz Bid Sell high to New York £940/oz Ask Buy low from London £930/oz Bid

4.1 The Law of One Price Arbitrage profit in gold Arbitrage profit Buy 1000 oz at London’s £940/oz offer price +(1000 oz) -£940,000 Arbitrage profit +£3,161 Sell 1000 oz at NY’s $1510/oz bid price +$1,510,000 -(1000 oz) Cover your £ position at the $1.601/£ ask price for pounds +£943,161 = ($1,510,000)/($1.601/£) -$1,510,000

4.2 Exchange Rate Equilibrium A cross exchange rate table Sd/f Sf/d = 1  Sd/f = 1 / Sf/d USD EUR JPY GBP CHF CAD AUD CNY USD U.S.A. 1 1.115 0.008 1.517 1.029 0.754 0.692 0.157 EUR Euro 0.897 1 0.008 1.361 0.923 0.676 0.620 0.141 JPY Japan 119.0 132.7 1 183.2 122.4 89.70 82.28 18.72 GBP U.K. 0.659 0.735 0.006 1 0.678 0.497 0.456 0.104 CHF Swiss 0.972 1.084 0.008 1.475 1 0.733 0.672 0.153 CAD Canada 1.327 1.479 0.011 2.013 1.365 1 0.917 0.209 AUD Australia 1.446 1.613 0.012 2.195 1.488 1.090 1 0.228 CNY China 6.356 7.088 0.053 9.645 6.538 4.791 4.395 1

4.2 Exchange Rate Equilibrium Cross exchange rate equilibrium Sd/e Se/f Sf/d = 1 If Sd/eSe/fSf/d < 1, then either Sd/e, Se/f or Sf/d must rise. Þ For each spot rate, buy the currency in the denominator with the currency in the numerator. If Sd/eSe/fSf/d > 1, then either Sd/e, Se/f or Sf/d must fall. Þ For each spot rate, sell the currency in the denominator for the currency in the numerator.

4.2 Exchange Rate Equilibrium Cross exchange rates and triangular arbitrage An example with U.S. dollars ($), Japanese yen (¥), and Egyptian pounds (E£): SE£/$ = E£5.000/$ Û S$/E£ = $0.2000/E£ S$/¥ = $0.01000/¥ Û S¥/$ = ¥100.0/$ S¥/E£ = ¥20.20/E£ Û SE£/¥ » E£0.04950/¥ SE£/$ S$/¥ S¥/E£ = (E£5/$)($0.01/¥)(¥20.20/E£) = 1.01 > 1

4.2 Exchange Rate Equilibrium Cross exchange rates & triangular arbitrage SE£/$ S$/¥ S¥/E£ = 1.01 > 1 Þ Currencies in the denominators are too high relative to the numerators, so… Sell dollars and buy Egyptian pounds. Sell yen and buy dollars. Sell Egyptian pounds and buy yen.

4.2 Exchange Rate Equilibrium An example of triangular arbitrage Arbitrage profit = +E£50,000 or about 1% of the initial amount Sell $1m and buy E£5m at E£5/$ +E£5m -$1m Sell ¥100m and buy $1m at $0.01/¥ +$1m -¥100m Sell E£4.95m and buy ¥100m at ¥20.20/E£ +¥100m -E£4.95m

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage International parity conditions that span both currencies and time Less reliable linkages = E[Std/f] / S0d/f = [(1+E[pd])/(1+E[pf])]t Interest rate parity Ftd/f / S0d/f = [(1+id)/(1+if)]t where S0d/f = today’s spot exchange rate E[Std/f] = expected future spot rate Ftd/f = forward rate for time t exchange i = a nominal interest rate p = an expected inflation rate

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Interest rate parity Ftd/f / S0d/f = [ (1+id) / (1+if) ]t Forward premiums and discounts are entirely determined by interest rate differentials. This is a parity condition that you can trust.

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Suppose yield curves are flat S0$/€ = $1.20/€ i$ = 7% and i€ = 3% What are forward rates F1$/€, F2$/€, and F3$/€? Ft$/€ = S0$/€ [(1+i$)/(1+i€)]t F1$/€ = ($1.20/€) [(1.07)/(1.03)]1 ≈ $1.2466/€ F2$/€ = ($1.20/€) [(1.07)/(1.03)]2 ≈ $1.2950/€ F3$/€ = ($1.20/€) [(1.07)/(1.03)]3 ≈ $1.3453/€

Moving dollars today into euros tomorrow 4.3 Interest Rate Parity and Covered Interest Arbitrage Interest rate parity Invest in dollars FVt$ = PV0$ (1+i$)t V0$ Dollars Moving dollars today into euros tomorrow Convert to euros at S0$/€ Convert to euros at Ft$/€ Euros Vt€ Invest in euros FVt€ = PV0€ (1+i€)t Time 0 Time t

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Interest rate parity: Which way do you go? If Ftd/f/S0d/f > [(1+id)/(1+if)]t then so... Ftd/f must fall Sell f at Ftd/f S0d/f must rise Buy f at S0d/f id must rise Borrow at id if must fall Lend at if

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Interest rate parity: Which way do you go? If Ftd/f/S0d/f < [(1+id)/(1+if)]t then so... Ftd/f must rise Buy f at Ftd/f S0d/f must fall Sell f at S0d/f id must fall Lend at id if must rise Borrow at if

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Interest rate parity is enforced through “covered interest arbitrage” An Example: Given: i$ = 7% S0$/£ = $1.20/£ i£ = 3% F1$/£ = $1.25/£ F1$/£ / S0$/£ > (1+i$) / (1+i£) 1.041667 > 1.038835 The currency and Eurocurrency markets are not in equilibrium.

and Covered Interest Arbitrage 4.3 Interest Rate Parity and Covered Interest Arbitrage Covered interest arbitrage 1. Borrow $1,000,000 at i$ = 7% 2. Convert $s to £s at S0$/£ = $1.20/£ 3. Invest £s at i£ = 3% 4. Convert £s to $s at F1$/£ = $1.25/£ 5. Arbitrage profit = $2,920 +$1,000,000 -$1,070,000 +£833,333 -$1,000,000 +£858,333 -£833,333 +$1,072,920 -£858,333

4.4 Less Reliable Parity Conditions 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] / S0d/f = (E[Ptd] / E[Ptf]) / (P0d / P0f) = (E[Ptd] / P0d) / (E[Ptf] / P0f) = (1+E[pd])t / (1+E[pf])t where pd and pf are geometric mean inflation rates.

4.4 Less Reliable Parity Conditions Relative purchasing power parity (RPPP) E[Std/f] / S0d/f = (1+E[pd])t / (1+E[pf])t Speculators will force this relation to hold on average. The expected change in a spot rate should reflect the difference in inflation between the two currencies. This relation only holds over the long run.

4.4 Less Reliable Parity Conditions RPPP over monthly intervals S1¥/$/S0¥/$ - 1 (1+p¥)/(1+p$) - 1

4.4 Less Reliable Parity Conditions

4.4 Less Reliable Parity Conditions

4.4 Less Reliable Parity Conditions

4.4 Less Reliable Parity Conditions Forward rates as predictors of spot rates E[Std/f ] / S0d/f = Ftd/f / S0d/f 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.

4.4 Less Reliable Parity Conditions Yen-per-$ forward parity over monthly intervals S1¥/$/S0¥/$ - 1 F1¥/$/S0¥/$ - 1

4.4 Less Reliable Parity Conditions 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

4.4 Less Reliable Parity Conditions International Fisher relation (Fisher Open hypothesis) [(1+id)/(1+if)]t = [(1+pd)/(1+pf)]t where (1+i) = (1+ʀ)(1+p) from the Fisher relation If real rates of interest are equal across currencies (ʀd = ʀf), then… [(1+id)/(1+if)]t = [(1+ʀd)(1+pd)]t / [(1+ʀf)(1+pf)]t = [(1+pd)/(1+pf)]t

4.4 Less Reliable Parity Conditions 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. If real rates of interest are equal across countries, then interest rate differentials merely reflect inflation differentials. This relation is unlikely to hold at a point in time, but should hold in the long run.

4.4 Less Reliable Parity Conditions Summary: The international parity conditions International Fisher relation Interest rates [(1+id)/(1+if)]t Inflation rates [(1+pd)/(1+pf)]t Interest rate parity Relative PPP Ftd/f / S0d/f Forward-spot differential E[Std/f] / S0d/f Expected change in the spot rate Forward rates as predictors of future spot rates

4.4 Less Reliable Parity Conditions The parity conditions are useful even if they are poor short-run predictors of the FX rate. Interest rate differentials reflect the difference in the opportunity cost of capital between two currencies. Forward prices similarly reflect the relative opportunity cost of capital, and have predictive ability over long horizons. Inflation reflects change in a currency’s purchasing power and inflation differentials are a key element of the real exchange rate.

4.5 The Real Exchange Rate The real exchange rate adjusts the nominal exchange rate for differential inflation since an arbitrarily defined base period. Example S0¥/$ = ¥100/$ S1¥/$ = ¥110/$ p¥ = 0% p$ = 10% s1¥/$ = (S1¥/$–S0¥/$)/S0¥/$ = 0.10, or a 10 percent nominal change

4.5 The Real Exchange Rate The expected nominal exchange rate But RPPP implies E[S1¥/$] = S0¥/$ (1+ p¥)/(1+ p$) = ¥90.91/$ What is the change in the nominal exchange rate relative to the expectation of ¥90.91/$?

4.5 The Real Exchange Rate Actual versus expected change St¥/$ ¥130/$ ¥120/$ ¥110/$ Actual S1¥/$ = ¥110/$ ¥100/$ E[S1¥/$] = ¥90.91/$ ¥90/$ time

4.5 The Real Exchange Rate Change in the real exchange rate In real (or purchasing power) terms, the dollar has appreciated by (¥110/$) / (¥90.91/$) - 1 = +0.21 or 21 percent more than expected.

4.5 The Real Exchange Rate Change in the real exchange rate (1+xtd/f) = (Std/f / St-1d/f) / [(1+ptd)/(1+ptf)] where xtd/f = percentage change in the real exchange rate Std/f = the nominal spot rate at time t ptd = inflation in currency d during period t ptf = inflation in currency f during period t

4.5 The Real Exchange Rate Change in the real exchange rate Example S0¥/$ = ¥100/$  S1¥/$ = ¥110/$ p¥ = 0% and p$ = 10% (1+xt¥/$) = [(¥110/$)/(¥100/$)] / [1.00/1.10] = 1.21 = a 21 percent increase in relative purchasing power

4.5 The Real Exchange Rate Consequences of real exchange rate changes A real appreciation of the domestic currency increases the purchasing power of domestic residents: Domestic companies and consumers benefit from lower prices Pd on foreign goods; Pd = Pf Sd/f. This helps importers and hurts exporters. A real depreciation of the domestic currency decreases the purchasing power of domestic residents: Domestic companies and consumers benefit from higher prices Pd on foreign goods; Pd = Pf Sd/f. This hurts importers and helps exporters.

4.5 The Real Exchange Rate Behavior of real exchange rates Deviations from PPP… can be substantial in the short run and can last for several years Both the level and the variance of the real exchange rate are autoregressive.

4.5 The Real Exchange Rate Real exchange rates (Xf/d) 2010 base year Source: bis.org/statistics/eer/ The Bank for International Settlements uses 2010 as the base year in their “effective exchange rates”

4.5 The Real Exchange Rate Real exchange rates (Xf/d) rescaled average = 100 Source: bis.org/statistics/eer/

Appendix 4-A Continuous Compounding Most theoretical and empirical research in finance is conducted in continuously compounded returns.

Appendix 4-A Continuous Compounding Holding period returns are asymmetric r1 = +100% r2 = –50% 200 100 100 (1+rTOTAL) = (1+r1)(1+r2) = (1+1)(1–½) = (2)(½) = 1  rTOTAL = 0%

Appendix 4-A Continuous Compounding Let r = holding period (e.g. annual) return r = continuously compounded return r = ln (1+r) Û (1 + r) = er where ln(.) is the natural logarithm with base e » 2.718

Appendix 4-A Continuous Compounding Properties of natural logarithms (for x > 0) eln(x) = ln(ex) = x ln(AB) = ln(A) + ln(B) for positive values A and B ln(At) = (t) ln(A) ln(A/B) = ln(AB-1) = ln(A) - ln(B)

Appendix 4-A Continuous Compounding Continuous returns are symmetric r1 = ln(1+1) = +69.3% r2 = ln(1-½) = -69.3% 200 100 100 rTOTAL = ln [ (1+r1)(1+r2) ] = ln(1+r1) + ln(1+r2) = r1 + r2 = +0.693 -0.693 = 0.000  rTOTAL = 0%

Appendix 4-A Continuous Compounding Continuously compounded returns are additive ln [ (1+r1) (1+r2) ... (1+rT) ] = ln(1+r1) + ln(1+r2) ... + ln(1+rT) = r1 + r2 +... + rT

Appendix 4-A Continuous Compounding The international parity conditions in continuous time Over a single period ln(F1d/f / S0d/f ) = i d – i f = E[p d ] – E[p f ] = E[s d/f ] where s d/f, p d, p f, i d, and i f are continuously compounded returns

Appendix 4-A Continuous Compounding The international parity conditions in continuous time Over t periods ln(Ftd/f / S0d/f ) = t (i d – i f) = t (E[p d ] – E[p f ]) = t (E[s d/f ]) where s d/f, p d, p f, i d, and i f are continuously compounded returns