International Fixed Income Topic IIB International Parity Relations

International Fixed Income Topic IIB: International Parity Relations

Outline • • • Introduction Forward Contracts Interest Rate Parity Purchasing Power Parity International Fisher Relation

I. Introduction • International parity conditions provide the relation between interest rates, exchange rates and inflation between two countries. – They are the cornerstone of international finance, and the basis behind understanding international fixed income. – Four main results: • Covered and uncovered interest rate parity • International Fisher relation • Purchasing power parity

International Parity Conditions International Fisher relation Interest rates [(1+rd)/(1+rf)]t Interest rate parity Inflation rates [(1+pd)/(1+pf)]t Purchasing power parity Ftd/f / S 0 d/f E[Std/f] / S 0 d/f Forward-spot Expected change The exp’ differential hypothesis in the spot rate Forward rates as predictors of future spot rates r-int. rate p-inflation S-exch. rate F-forw. rate

II. Forward Contracts Ft, T - the current forward rate for T periods hence exchange Example: the spot rate is the 1 year forward rate is S$/FF = $0. 20/FF F$/FF = $0. 25/FF We can sell forward FF 40, 00 in 1 yr, and guarantee the receipt of $10, 000

Forward Premiums/Discounts Percentage forward premium/discount = (F 1 d/f - S 0 d/f ) / S 0 d/f • Forward premium: nominal value in the forward exchange market is higher than in the spot exchange market • Forward discount: nominal value in the forward exchange market is lower than in the spot exchange market

Forward Premium/Discount: Example Suppose S$/FF = $0. 20/FF and F$/FF = $0. 25/FF Franc forward premium = ($. 25/FF-$. 20/FF)/($. 20/FF) = +25% so the franc is selling at a 25% forward premium. Alternatively, S* = (1/S) = SFF/$ = FF 5. 00/$ F* =(1/F) = FFF/$ = FF 4. 00/$ Dollar forward premium = (FF 4/$-FF 5/$)/(FF 5/$) = -20% so the dollar is selling at a 20% forward discount.

III. Pricing Forwards: Interest Rate Parity Ftd/f/S 0 d/f = [(1+rd)/(1+rf)]t Forward premiums and discounts are entirely determined by interest rate differentials. • This is a parity condition that you can trust. • It holds by ARBITRAGE. . . that is, if it didn’t, you could make an infinite profit

covered interest arbitrage =>Interest Rate Parity An Example: Given: r$ = 7% r£ = 3% S 0$/£ = $1. 20/£ F 1$/£ = $1. 25/£ F 1$/£ / S 0$/£ = 1. 041667 1. 038835 The forward premium > > (1+r$) / (1+r£) = The “synthetic” forward premium FX and Eurocurrency markets are not in equilibrium WHAT TO DO? ? ?

Interest Rate Parity: Arbitrage Opportunities If Ftd/f/S 0 d/f > [(1+rd)/(1+rf)]t then Ftd/f must fall, S 0 d/f must rise, rd must rise, rf must fall so Sell f at Ftd/f, Buy f at S 0 d/f, Borrow at rd , Lend at rf If Ftd/f/S 0 d/f < [(1+rd)/(1+rf)]t then Ftd/f must rise, S 0 d/f must fall, rd must fall, rf must rise so Buy f at Ftd/f, Sell f at S 0 d/f, Lend at rd, Borrow at rf

Covered Interest Arbitrage Example (cont’d) 1. Borrow $1, 000 at 7% +$1, 000 2. Convert $s to £s at S 0$/£ = $1. 20/£ 3. Invest £s at 3% -$1, 070, 000 +£ 833, 333 -$1, 000 -£ 833, 333 4. Convert £s to $s at F 1$/£ = $1. 25/£ 5. Take your profit $1, 072, 920 - $1, 070, 000 = $2, 920 +£ 858, 333 +$1, 072, 920 -£ 858, 333

Conclusion • Covered Interest Rate Parity must hold by no arbitrage. • There is a corresponding relation, uncovered interest rate parity, sometimes known as the expectations hypothesis for exchange rates: What about this relation?

Forwards as Predictors of Future Spot (or Uncovered Parity) …just like the expectations hypothesis in the context of interest rates, we have here: The “expectation hypothesis” for XRs Forward equals expected spot: Ft, 1 d/f = Et[St+1 d/f] or Forward premium equals expected appr/depr: Ft, 1 d/f / Std/f = Et[St+1 d/f] / Std/f

“Uncovered Interest Parity” Strategy 1: invest in US$ E[$ROR]=r d Strategy 2: invest in FX E[$ROR]=(1/St) * r f * E[S t+1] E[ROR]’s must be equal: r d =(1/St) * r f * E[S t+1] => E[St+1]= Ft, 1

Uncovered Interest Parity • This condition does not hold by no arbitrage; like the expectations hypothesis, it is a theory for how interest rates and exchange rates evolve. – It, for example, assumes no risk premium. – Independent of the risk premium, it also assumes rational expectations.

IV. The Law of One Price (purchasing power parity, or PPP) “Equivalent assets sell for the same price” Ptd = price of an asset in domestic currency Ptf = price of the same asset in foreign currency Ptd /Ptf = Std/f Û Ptd = Ptf Std/f • seldom holds for nontraded assets • cannot be used to compare assets that vary in quality • may not hold when there are market frictions

An Example of the Law of One Price: The World Price of Gold Suppose P$ = $500/oz in New York PDM = DM 800/oz in Berlin The law of one price requires: Pt$ /Pt. DM = St$/DM ($500/oz)/(DM 800/oz) = $. 6250/DM +If this relation does not hold, then there is an opportunity to lock in a riskless arbitrage profit

How Well Does the LOP Hold? • For the LOP to hold, we need a number of conditions What are they? • Example: “Big Mac. Currencies” Country Price Implied PPP USA $2. 25 1 Japan Y 380 169 Ger DM 4. 30 1. 91 Actual XR 1 135 1. 67 %+/0 -20 -13

http: //www. economist. com/editorial/freeforall/focus_bi gmac_tframeset. html • P(BMindonesia)/P(BMusa) =rupiah 3900/$ • Actual rate: rupiah 14, 100/$ ==> • * We can buy 4 MBs in Indonesia for the price of one in the US • * The rupiah is 72% undervalued

Big Macs in Japan • From a PPP point in 1990, a combination of weak $ and strong Yen resulted in 100% overvaluation in 1995, followed by a sharp reversal

Big Macs in Emerging Markets • The effect of devaluation in the far east

The Rally of the GBP

Extending the LOP to PPP • The “Price Level” is: P = Swi. Pi a sum of prices of various consumption goods • A natural extension of the LOP is hence Pt$ = St$/DM Pt. DM the US price level should be the same as in Ger, adjusting for the XR • How sound is theory?

The Cost of Living Location cost of living index Tokyo, Japan 162. 36 Geneva, Switzerland 122. 95 Frankfurt, Germany 108. 82 Hong Kong 107. 54 Paris, France 101. 82 New York, United States. . . . 100 London, United Kingdom 92. 91 Los Angeles, United States 83. 42 Montreal, Canada 80. 01 Mexico City, Mexico 77. 22 Prague, Czech Republic 65. 93 From “Working Out Where to Live, ” Richard Donkin, London Financial Times, October 20, 1995, page 27.

Relative Purchasing Power Parity (RPPP) Let Pt = consumer price index level at time t. Then inflation pt = (Pt - Pt-1)/Pt-1. Suppose PPP were to hold well, then P 0 = S 0 P 0* Pt = S t Pt * The expected change in the spot exchange rate should reflect the difference in inflation between two currencies: 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

Relative PPP E[Std/f] / S 0 d/f = (1+E[pd])t / (1+E[pf])t Example: p. US=5%, p. UK=10% ==> (1+p. US)/(1+p. UK)=-5% ==> E[St. USD/GBP] / S 0 USD/GBP =. 95 What does it mean?

RPPP and RPPP Deviations • Consider again: p. US=5%, p. UK=10%, St=$1. 800/L • According to RPPP: E[St+1]=1. 80(1. 05/1. 10)=$1. 718/L • Suppose that in fact St+1=$1. 75/L ==>The $ has appreciated, but________

The Importance of RPPP Deviations • Consider again: p. US=5%, p. UK=10%, St=$1. 800/L • According to RPPP: E[St+1]=1. 80(1. 05/1. 10)=$1. 718/L • Suppose that in fact St+1=$1. 718/L ==> RPPP HOLDS!!!

RPPP Deviations: Real Vs. Nominal Rate Changes • RPPP deviations have great economic importance affecting economic competitiveness, inflationary pressures, etc. . • REAL EXCHANGE RATES measure RPPP deviations In the first example, the US$ appreciated in NOMINAL TERMS (from 1. 80 to 1. 75) BUT depreciated in REAL TERMS In the second example, the US$ appreciated in NOMINAL TERMS (from 1. 80 to 1. 718) BUT did not change in REAL TERMS

Deviations from RPPP states that: DSt+1 d/f = (1+pd) / (1+pf) • Note that this is an exchange rate determination model (the first one we see) • What is the model’s empirical record? – Var(DSt+1 d/f) >> Var[ (1+pd) / (1+pf) ] This phenomenon is often described as “excess volatility”, that is, XRs change by more than is warranted by fundamentals. “excess” is normally interpreted as evidence of irrationality, with the exception of the “overshooting XRs model”. – DSt+1 d/f is highly unpredictable, (1+pd) / (1+pf) is highly predictable

Evidence on RPPP Deviations

The empirical Record of RPPP in the long run

Change in the Nominal Exchange Rate EXAMPLE RPPP Þ St¥/$ ¥ 130/$ ¥ 120/$ ¥ 110/$ ¥ 100/$ ¥ 90/$ ¥ 80/$ S 0¥/$ = ¥ 100/$, E[p¥] = 0%, E[p$] = 10% E[S 1¥/$] = S 0¥/$ (1+ p¥)/(1+ p$) = ¥ 90. 91/$ Actual S 1¥/$ = ¥ 110/$ E[S 1¥/$] = ¥ 90. 91/$ t=0 t=1 In real (purchasing power) terms, the dollar has appreciated by (¥ 110/$)/(¥ 90. 91/$)-1 = 21% more than expected.

The Real Exchange Rate • The real exchange rate (RXR) adjusts the nominal exchange rate for differential inflation since an arbitrarily defined base period: 1+xtd/f = (Std/f / St-1 d/f) [(1+ptf)/(1+ptd)] where Std/f = the nominal spot rate at time t p tc = inflation in currency c during period t • What should the RXR be under RPPP?

Percentage Change in the RXR Xtd/f = level of the real exchange rate xtd/f +1 = (Std/f / St-1 d/f) [(1+ptf)/(1+ptd)] = [(¥ 110/$)/(¥ 100/$)][(1. 10)/(1. 00)] = 1. 21, or a 21% increase X 1¥/$ = X 0¥/$ (1+ x 1¥/$) = 1. 21 = 121% Xt¥/$ Base: X 0¥/$ = 1. 00 = 100% + The real value of the dollar has appreciated by 21%.

Another Example • Let’s go back to the previous example – p. US=5%, p. UK=10%, St=$1. 800/L – According to RPPP: E[St+1]=1. 80(1. 05/1. 10)=$1. 718/L – Suppose that in fact: St+1=$1. 75/L • 1+xtd/f = (Std/f / St-1 d/f) [(1+ptf)/(1+ptd)] = (1. 75/1. 80)(1. 10/1. 05)=“R$” 1. 018/ “RL” “the $ has depreciated in real terms” • . . . and in the second example = (1. 718/1. 80)(1. 10/1. 05)=“R$” 1. 00/ “RL”

Real Value of the Dollar (1970 -1995) 180 German mark 160 140 120 British pound 100 80 60 Japanese yen 40 20 0 1975 1980 1985 1990 Mean level = 100 for each series 1995

PPP: an Exchange Rate Determination Theory • Causality is Goods market ==> Currency Market The adjustment mechanism: real$ is weak and domestic prices are low => export will rise & import will fall => a trade surplus will generate a real and nominal $ appreciation • We completely ignore – the role of financial transaction – the forward-looking nature of exchange rates – the stickiness of prices

V. International Fisher Relation The Fisher relation: (1+r) = (1+RR)(1+p) nominal returns are inflation times real return (denote RR) If real rates are equal across countries (RRd=RRf), then interest rate differentials reflect inflation differentials: (1+rd)/(1+rf) = [(1+RRd)(1+pd)] / [(1+RR)(1+pf)] = (1+pd)/(1+pf) +This relation may not hold at any particular point in time, but does hold in the long run. WHY?

International Fisher Relation Difference in realized quarterly inflation (p£-p$) Difference in 3 -month Eurocurrency interest rates (i£-i$)

Summary: International Parity Conditions International Fisher relation Interest rates [(1+rd)/(1+rf)]t Interest rate parity Inflation rates [(1+pd)/(1+pf)]t Purchasing power parity Ftd/f / S 0 d/f E[Std/f] / S 0 d/f Forward-spot Expected change The exp’ differential hypothesis in the spot rate Forward rates as predictors of future spot rates r-int. rate p-inflation S-exch. rate F-forw. rate