International Parity Relationships

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International Parity Relationships

• Chapter 6 (134-149)

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Lecture Objectives

• Examine several key international parity
• relationships, such as
• interest rate parity
• purchasing power parity

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Interest Rate Parity

• Interest Rate Parity Defined
• Covered Interest Arbitrage
• Interest Rate Parity Exchange Rate
  Determination
• Reasons for Deviations from Interest Rate Parity

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Interest Rate Parity Defined

• IRP is an arbitrage condition.
• If IRP did not hold, then it would be possible
  for an astute trader to make unlimited amounts of
  money exploiting the arbitrage opportunity.
• Since we dont typically observe persistent
  arbitrage conditions, we can safely assume that
  IRP holds.

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Interest Rate Parity Defined

• Suppose you have 100,000 to invest for one year.
• You can either
• invest in the U.S. at i. Future value
  100,000(1 i)
• or
• trade your dollars for yen at the spot rate,
  invest in Japan at i and hedge your exchange
  rate risk by selling the future value of the
  Japanese investment forward. The future value
  100,000(F/S)(1 i)
• Since both of these investments have the same
  risk, they must have the same future
  valueotherwise an arbitrage would exist,
  therefore (F/S)(1 i) (1 i)

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Interest Rate Parity
100,000
100,000(1 i)

1. Trade 100,000 for at S

100,000(F/S)(1 i)

1. One year later, trade for at F

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Interest Rate Parity Defined

• Formally,
• (F/S)(1 i) (1 i)
• or if you prefer,

IRP is sometimes approximated as
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Interest Rate Parity Carefully Defined

• Depending upon how you quote the exchange rate (
  per or per ) we have

or
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IRP and Covered Interest Arbitrage

• If IRP failed to hold, an arbitrage would exist.
  Its easiest to see this in the form of an
  example.
• Consider the following set of foreign and
  domestic interest rates and spot and forward
  exchange rates.

Spot exchange rate S(/) 1.6000/
360-day forward rate F360(/) 1.5922/
U.S. discount rate i 2.0
British discount rate i 2.5
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IRP and Covered Interest Arbitrage

• A trader with 1,000 to invest could invest in
  the U.S., in one year his investment will be
  worth 1,020 1,000?(1 i) 1,000?(1.02)
• Alternatively, this trader could exchange 1,000
  for 625 at the prevailing spot rate, (note that
  625 1,0001.6/) invest 625 at i 2.5
  for one year to achieve 640.625. Translate
  640.625 back into dollars at F360(/)
  1.5922/, the 640.625 will be exactly 1,020.

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Interest Rate Parity
1,000
1,020

1. Trade 1000 for 625

1,020

1. One year later, trade 640.625 for at F360(/)
   1.5922/

1. Invest 625 at 2.5 i

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Interest Rate Parity Exchange Rate
Determination

• According to IRP only one 360-day forward rate,
• F360(/), can exist. It must be the case that
• F360(/) 1.5922/
• Why?
• If F360(/) ? 1.5922/, an astute trader could
  make money with one of the following strategies

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Arbitrage Strategy I

• If F360(/) gt 1.5922/
• i. Borrow 1,000 at t 0 at i 2.0.
• ii. Exchange 1,000 for 625 at the prevailing
  spot rate, (note that 625 1,0001.60/)
  invest 625 at 2.5 (i) for one year to achieve
  640.625.
• iii. Translate 640.625 back into dollars, if
• F360(/) gt 1.5922/ , 640.625 will be more
  than enough to repay your dollar obligation of
  1,020.

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Arbitrage Strategy II

• If F360(/) lt 1.5922/
• i. Borrow 625 at t 0 at i2.50 .
• ii. Exchange 625 for 1,000 at the prevailing
  spot rate, invest 1,000 at 2.0 for one year to
  achieve 1,020.
• iii. Translate 1,020 back into pounds, if
• F360(/) lt 1.5922/ , 1,020 will be more
  than enough to repay your obligation of
  640.625.

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Observations

• IRP provides a linkage between interest rates
  differential and forward premium
• Interest rates are more stable the XRs. Thus, in
  IRP the F and S change to accommodate the demand
  and supply for currencies
• If F360(/FC) gt S (1 ius)/ (1 iFC) borrow in
  and lend in FC
• If F360(/FC) lt S (1 ius)/ (1 iFC) lend in
  and borrow in FC
• If (1 ius) gt (F/ S )(1 iFC) lend in and
  borrow in FC
• If (1 ius) lt (F/ S )(1 iFC) lend in FC and
  borrow in

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Uncovered -IRP
Covered IRP arbitrage condition
Un-Covered IRP
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IRP and Hedging Currency Risk

• You are a U.S. importer of British woolens and
  have just ordered next years inventory. Payment
  of 100M is due in one year.

Spot exchange rate S(/) 1.60/
360-day forward rate F360(/) 1.5922/
U.S. discount rate i 2.00
British discount rate i 2.50
IRP implies that there are two ways that you fix
the cash outflow to a certain U.S. dollar
amount a) Put yourself in a position that
delivers 100M in one yeara long forward
contract on the pound. You will pay
(100M)(1.5922/) 159.22M b) Form a forward
market hedge as shown below.
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IRP and a Forward Market Hedge
To form a forward market hedge Borrow 156.098m
in the U.S. at 2 p.a. (in one year you will owe
159.22m). Translate 156.098m into pounds at the
spot rate S(/) 1.60/ to receive
97.561m. Invest 97.561m in the UK at i 2.5
for one year. In one year your investment will
have grown to 100 millionexactly enough to pay
your supplier.
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Forward Market Hedge
Where do the numbers come from? We owe our
supplier 100 million in one yearso we know that
we need to have an investment with a future value
of 100 million. Since i 2.5 we need to
invest 97.561 million at the start of the year.
How many dollars will it take to acquire 97.561
million at the start of the year if S(/)
1.60/?
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Reasons for Deviations from IRP

• Transactions Costs
• The interest rate available to an arbitrageur for
  borrowing, ib, may exceed the rate he can lend
  at, il.
• There may be bid-ask spreads to overcome, Fb/Sa lt
  F/S
• Ex If (1 i) lt (F/ S )(1 i) exists, the
  arbitrage strategy involves lending in and
  borrowing in .
• With transaction costs, it is possible that
• (Fb/Sa)(1 i l) ? (1 i b) ? 0
• Capital Controls
• Governments sometimes restrict import and export
  of money through taxes or outright bans.

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Purchasing Power Parity

• Purchasing Power Parity (PPP)
• A concept regarding the equilibrium value of a
  currency based upon its purchasing power in
  different countries
• General idea is that prices in different
  countries once converted to a common currency
  should be equal

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Purchasing Power Parity

• Three concepts of PPP are used
• The Law of One Price for individual goods
• Absolute PPP for general price levels
• Relative PPP for changes in the general price
  level (inflation rates)

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The law of one price

• 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
• may not hold when there are market frictions
• cant be used to compare assets that vary in
  quality

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An example of the law of one priceThe world
price of gold

• Suppose P 500/oz in New York
• P 550/oz in Berlin
• The law of one price requires
• Pt /Pt st/
• Þ (500/oz)/(550/oz) .91/
• If this relation does not hold, then there is an
  opportunity to lock in a riskless arbitrage
  profit.

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The Big Mac index (The Economist, 6 Jan 2010)

• According to Burgernomics, the Chinese yuan is
  still undervalued!
• Based on the theory of PPP exchange rates should
  equalize the price of a basket of goods in
  different countries.
• The exchange rate that leaves a Big Mac costing
  the same in dollars everywhere is its fair-value
  benchmark.
• Most overvalued currency against the dollar is
  the Norwegian kroner, which is 96 above its PPP
  rate. In Oslo you can expect to pay around 7 for
  a Big Mac.
• Most undervalued currency against the dollar is
  the Chinese yuan, which is undervalued by 49.

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Absolute Purchasing Power Parity

• Absolute PPP
• Identical baskets of consumer goods in two
  countries will have exactly the same price (in a
  given currency)
• Generally economists consider the basket of goods
  that comprise the CPI
• The exchange rate is the ratio of the price
  levels
• S(/ ) P/ P

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Absolute Purchasing Power Parity

• Example
• Let CPIUS PUS and CPIUK PUK
• PUS S(/) x PUK S(/)
  PUS/PUK
• The same number of dollars that buys the basket
  of goods in the U.S. also buys the basket of
  goods in Britain
• Both currencies have the same purchasing power in
  terms of a basket of goods

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Relative Purchasing Power Parity

• Relative PPP states that the rate of change in
  the exchange rate is equal to the differences in
  the rates of inflation

If U.S. inflation is 5 and U.K. inflation is 8,
the pound should depreciate by about 3.
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Important Points

• If U.S. inflation gt U.K. inflation, the should
  appreciate against the , i.e. the should
  weaken
• If U.K. inflation gt U.S. inflation, the should
  depreciate against the , i.e. the should
  strengthen
• If U.K. inflation U.S. inflation, there should
  be no change in the exchange rate.

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PPP Deviations and the Real Exchange Rate

• The real exchange rate is

If q lt 1 competitiveness of domestic country
improves with currency depreciations. If q gt 1
competitiveness of domestic country deteriorates
with currency depreciations.
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Evidence on PPP

• PPP probably doesnt hold precisely in the real
  world for a variety of reasons.
• Haircuts cost 10 times as much in the developed
  world as in the developing world.
• Film, on the other hand, is a highly standardized
  commodity that is actively traded across borders.
• Shipping costs as well as tariffs and quotas can
  lead to deviations from PPP.

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Problems with PPP

• Transaction costs and restrictions on trade
• high transportation costs, nontradables, import
  tariffs
• Departures from free competition
• product differentiation and market segmentation
  can produce large price differences
• International differences in the measurement of
  price levels
• different preferences and consumption baskets
  across countries

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Learning outcomes

• Explain how IRP (covered) is derived
• Calculate a forward rate under the assumptions
  of IRP
• Determine whether an arbitrageur can profit from
  given interest rates, and spot and forward
  rates. If yes, explain in detail the arbitrage
  strategy and calculate the profits.
• Define uncovered IRP. Explain the differences
  between covered and uncovered IRP
• Discuss the reasons for deviations from IRP
• Define and discuss the law of one price, and the
  absolute and relative PPP
• Define and discuss the real exchange rate
• Discuss the implications of the deviations from
  PPP for countries competitive position
• Discuss the evidence on and problems with PPP
• Discuss the three approached to forecast
  exchange rates