© 2018 Cambridge University Press 6-1
Michala Moravcová
Department of Finance, Masaryk University
International Finance
INTERNATIONAL PARITY CONDITIONS
International Finance
© 2018 Cambridge University Press 6-2
• The interest rate differential between two currencies approximately equals the
percentage spread between the currencies’ forward and spot rate.
• The intuition behind interest rate parity
• Future value of one unit of currency depends on interest rate for that currency
• Interest rate parity
• Covered interest rate parity refers to a theoretical condition in which the relationship between
interest rates and the spot and forward currency values of two countries are in equilibrium.
• The covered interest rate parity condition says that the relationship between interest rates and
spot and forward currency values of two countries are in equilibrium.
• Relationship between forward/spot rates and the interest rate differential between two countries
• 𝐹(ℎ/𝑓)/𝑆(ℎ/𝑓) = (1 + 𝑖!)/(1 + 𝑖")
• Why there must be interest rate parity
• If not, arbitrage possibilities would exist (borrowing any government controls)
1. The Theory of Covered Interest Rate Parity
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© 2018 Cambridge University Press 6-3
• $10M to invest, iUS = 8%; iUK = 12%; S = $1.60/£; F1-year = $1.53/£
• Buy 1year US bond or 1year UK bond? (US investor wants to have the highest US
return)
• Steps:
• Convert using spot rate: $10𝑀 ÷ ($1.60/£) = £6.25𝑀
• Invest at foreign interest rate: £6.25𝑀×1.12 = £7𝑀
• Convert back at forward rate: £7𝑀×($1.53/£) = $10.71𝑀
• Compare to what you could have earned by just investing in your home nation:
• $10𝑀×(1 + 0.08) = $10.8𝑀
• Investing at home (U.S.) is more profitable for Kevin.
• But what if he could borrow or lend? Is the answer still the same?
1. The Theory of Covered Interest Rate Parity
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© 2018 Cambridge University Press 6-4
• $10M to invest, iUS = 8%; iUK = 12%; S = $1.43/£; F1-year = $1.53/£
• Steps:
• Borrow pounds: £1𝑀×1.12 = £1.12𝑀 (what Kevin owes at end of investment term)
• Convert pounds to dollars: £1.12𝑀×($1.43/£) = $1.6𝑀
• Invest at U.S. interest rate: £1.6𝑀×1.08 = $1.728𝑀
• Convert back at forward rate: $1.728𝑀/($1.53/£) = £1,129,411.76
• Profit: £1,129,411.76 - £1,120,000 = £9,411.76
• Kevin would make £9,411.76 (Step 4 – Step 1) profit for every £1M that is borrowed!
• Think of doing the transaction the other way.
• This arbitrage activity would quickly eliminate the profit opportunity by rising pound IR
and the sale of pounds for dollar would lower dollar–pound spot exchange rate (pound
depreciates).
1. The Theory of Covered Interest Rate Parity
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© 2018 Cambridge University Press 6-5
• Deriving interest rate parity
• When the forward rate is priced correctly, an investor is indifferent between investing at home or
abroad
• General expression for interest rate parity
• [1 + 𝑖 ∗] = [1/𝑆] × [1 + 𝑖]×𝐹
• i *  the foreign currency interest rate appropriate for one period, i the domestic currency interest rate
appropriate for one period, S the spot exchange rate (domestic currency per foreign currency), F the
one-period forward exchange rate (domestic currency per foreign currency)
• Interest rate parity and forward premiums and discounts
• (1 + 𝑖 ∗)/(1 + 𝑖) = 𝐹/𝑆
• Subtracting 1 from each side and simplifying we obtain
• (i ∗ −𝑖)/(1 + 𝑖) = (𝐹 − 𝑆)/𝑆
• If this equation is (+), the forward is selling at a premium
• If it is (-), the forward is selling at a discount
• With continuously compounded interest rates
• (𝑖 − 𝑖 ∗) = ln(𝐹) − ln(𝑆)
1. The Theory of Covered Interest Rate Parity
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© 2018 Cambridge University Press 6-6
• Suppose the 5-year interest rate on a dollar-denominated pure discount bond is
4.5% p.a., whereas in France, the euro interest rate is 7.5% p.a. on a similar pure
discount bond denominated in euros. If the current spot rate is 0.9259
(EUR/USD), what is the value of the forward exchange rate that prevents covered
interest arbitrage?
• F (t,EUR/USD) = S(t,EUR/USD) x
(,-.!)
(,-.")
, 𝑖/ = home/base currency
• F (t,5,EUR/USD) = S(t, EUR/USD) x
(,-.(0,2,345)#
(,-.(0,2,467)#
• F (t,5,EUR/USD) = 0.9259 x
(,,892)#
(,,8:2)# = 0.868
Problem solving
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© 2018 Cambridge University Press 6-7
USD 3 months (90 days) Term Deposit Interest Rate : 3.5%
JPY 3 months (90 days) Term Deposit Interest Rate : 0.1%
Spot Rate USD/JPY = 120.10
• F-S = S x (
.#$% ;.467
,-.&'(
)
• F-S = 120.1 X {(0.001 - 0.035) X 90/360} / (1 + 0.035 X 90/360) = -1.02
• F = 120.1 - 1.02 = 119.08
Problem solving
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© 2018 Cambridge University Press 6-8
• If the 30-day yen interest rate is 3% p.a., and the 30-day euro interest rate is 5% p.a., is
there a forward premium or discount on the euro in terms of the yen? What is the
magnitude of the forward premium or discount?
• We know that the high interest rate currency must sell at a forward discount when priced
in the low interest rate currency to prevent a covered interest arbitrage. Therefore the
euro is at a discount in the forward market.
, -!"#
$%& ./ -!"#
$%&
/ -!"#
$%&
=
0!"# .0123
(450123)
The de-annualized interest rates are 0.0025 = (3/100) x (30/360) for the yen and 0.004167
= (5/100) x (30/360) for the euro. The right-hand side of the above expression is therefore
– 0.00166. The annualized value is -0.00166 x (100) x (360/30) = -1.99%. We therefore say
that the euro sells at an annualized discount of 1.99%
Problem solving
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© 2018 Cambridge University Press 6-9
• Spot exchange rate: ¥146.03/EUR
• 1-year forward exchange rate: ¥141.9021/EUR
• EUR interest rate: 3.5200% p.a.
• JPY interest rate: 0.5938% p.a.
• % change of FX rate:
<;6
6
=
,9,.=8>, ;,9?.8@
,9?.8@
= 2.82%
• Interest rate differential (Euro - Yen) is: 3.52% - 0.5938% = 2.9262%, which is
approximately equal to the forward premium. (1-iJPY )/(1+iEUR)
• Premium compensates for the lower interest rate that yen investments offer.
1. The Theory of Covered Interest Rate Parity
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© 2018 Cambridge University Press 6-10
Diagram of Covered
Interest Arbitrage
For example:
If you are at the node representing
pounds today and you move 1
pound to the future, the future
pound revenue is (1 + i (£)).
If you place yourself at the dollar
node in the future, and you move 1
dollar to the present, you receive 1
/ (1 + i($)) dollars in the current
period.
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© 2018 Cambridge University Press 6-11
Interest Rates in the
External Currency
Market
• The external currency market is a bank
market for deposits and loans that are
denominated in currencies that are not the
currency of the country in which the bank is
operating.
• The first of these deposits and loans were
called eurodollars because they were dollar
denominated deposits at European banks.
• To determine the appropriate interest rate
for a 3-month basis, we must “de-annualize”
the quoted interest rates.
• Bid – deposit, Ask- loan
• Quated as % p.a
• 𝐷𝑒 − 𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑠𝑒𝑑 𝑟𝑎𝑡𝑒 =
𝐴𝑛𝑛. 𝑟𝑎𝑡𝑒×
1
100
×
𝑛𝑢𝑚. 𝑜𝑓 𝑑𝑎𝑦𝑠
360
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© 2018 Cambridge University Press 6-12
• Cross currency market
• External currency market influences rates elsewhere
• Loans to investors/corporations are based on these interbank rates
• Most important interbank reference rate is LIBOR. LIBOR is the average interbank interest
rate at which a selection of banks on the London money market are prepared to lend to one
another. Since the beginning of 2022, LIBOR comes in max 5 maturities (from overnight to 12
months) and in 3 different currencies. The official LIBOR interest rates are announced once
per working day at around 11:45 a.m.
• For actual Libor rates see: https://www.wsj.com/market-data/bonds
2. Covered Interest Rate Parity in Practice
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Covered
Interest Rate Parity
with Bid-Ask Rates
(transaction costs)
Transaction costs
• Reduced regulatory burden
and strong competition result
in very small spreads
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© 2018 Cambridge University Press 6-14
An Example with
Transaction Costs
• Convert $10M to yen:
• $10𝑀×¥82.67/$ = ¥826.7𝑀
• Invest for 3 months
• 0.46×(1/100)×(90/360) = 0.00115
• ¥826.7𝑀×1.00115 = ¥827,650,705
• Sell forward (enter into forward contract)
• (¥827,650,705)/(¥82.6495/$) = $10,013,983
• Compare to what we would make in US
• $10,013,983 − $10𝑀×1.002275
• 0.91x(1/100)x90/360) = 0.002275
• Lose money this way
• No arbitrage, but borrowing yen also results in losses
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© 2018 Cambridge University Press 6-15
• Does covered interest rate parity hold?
• Prior to 2007, documented violations of interest rate parity were very rare
• Akram, Rime, and Sarno (2008) – multiple short-lived deviations that persist for only a few minutes
• Frequency, size and duration of apparent arbitrage opportunities do increase with market
volatility
• 2007-2009 financial crisis
2. Covered Interest Rate Parity in Practice
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© 2018 Cambridge University Press 6-16
• Too good to be true?
• Default risks
• Risk that one of the counterparties may fail to honor its contract
• Exchange controls
• Limitations
• Taxes (in interests)
• Political risk
• A crisis in a country could cause its government to restrict any exchange of the local currency for
other currencies
• Investors may also perceive a higher default risk on foreign investments
• Liquidity
3. Why Deviations from Interest Rate Parity May
Seem to Exist
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© 2018 Cambridge University Press 6-17
Covered Interest Parity
Deviations During the
Financial Crisis
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© 2018 Cambridge University Press 6-18
• Zachy’s: Importing wine for €4M, payable in 90 days
• S: $1.10/€; F(t+90): $1.08/€; i($, t+90): 6.00% p.a; i(€, t+90): 13.519% p.a.
• Choice #1: Enter into a forward contract
• Cost in 90 days: €4𝑀× $1.08/€ = $4.32𝑀
• Choice #2: Money Market hedge
• You have euro liability, you must acquire euro asset
• Invest X amount now that becomes what you owe in 90 days
• 𝑋 = €4𝑀/[1 + (
'(.*'+
',,
)(
+,
(-,
)] = €3,869,229.71 (PV of future euro payment €4𝑀)
• 𝑋 𝑎𝑡 𝑠𝑝𝑜𝑡 𝑟𝑎𝑡𝑒 = $3,869,229.71×$1.10/€ = $4,256,152.68 (I need to buy €3,869,229.71 at
spot market)
• PV of forward hedge (1st option)
• $4.32𝑀/ 1 +
).++
,++
-+
.)+
= $4,256,157.64
• Forward contract is more expensive by $4.96
4. Hedging Transaction Risk in the Money Market
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• Shetlant: Receive ¥500M in 30 days, you are British manufacturer
• S: ¥179.5/£; F(t+30): ¥180/£; i(£, t+30): 2.70% p.a; i(¥, t+30): 6.01% p.a.
• Choice #1: Sell yen forward
• Earn: ¥500𝑀/ ¥180/£ = £2,777,778
• Choice #2: Money Market hedge
• Borrow PV of ¥500M, and sell at spot
• 𝑃𝑉 = ¥500𝑀/[1 + (
).+,
,++
)(
.+
.)+
)] = ¥497,508,313
• £ 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 = ¥497,508,313/(¥179.5/£) = £2,771,634
• FV of forward hedge
• £2,771,634× 1 +
/.0+
,++
.+
.)+
= £2,777,785
• Forward contract is more expensive by £6,151
4. Hedging Transaction Risk in the Money Market
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• The term structure of interest rates
• Description of different spot interest rates for various maturities into the future
• Rates derived by:
• For shorter maturities, spot interest rates are directly observable because they are widely quoted
by banks.
• For longer maturities, the spot interest rates derived from the market prices of coupon-paying
bonds.
• Recent empirical evidence suggests that covered interest rate parity does not hold perfectly at
longer horizons.
5. The Term Structure of Forward Premiums and
Discounts
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Yield Curves for Four
Currencies
• The slope of the yield curve gives an
idea of future interest rate changes
and economic activity
• Three main shapes of yield curve
shapes: normal (upward sloping
curve), inverted (downward sloping
curve - point to economic recession),
and flat.
• Watch video
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• A review of bond pricing
• Pure discount bonds (simplest bonds) promise a single payment of, say, $1,000 at the
maturity of the bond.
• Price of a 10-year pure discount bond with a face value of $1,000 is $463.19
• What is the spot interest rate for the 10-year maturity expressed in percentage per annum?
• $463.19×[1 + 𝑖(10)],+ = $1,000
• 𝑖 = 8%
• The spot interest rate as the market interest rate that equates the price of a pure discount
bond to the present value of the face value of the bond.
5. The Term Structure of Forward Premiums and
Discounts
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• Yield to maturity
• The discount rate that equates the present value of the n coupon payments plus the final
principal payment to the current market price
• A 2-year bond with face value of $1,000, an annual coupon of $60, and a market price of $980
• If the 1-year spot rate is 5.5%, the 2-year spot rate is found by solving:
• $980 = ($60/1.055) + ($1060/1 + 𝑖(2)/)
• 𝑖(2) = 7.1574%
5. The Term Structure of Forward Premiums and
Discounts
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• Long-term forward rates and premiums
• Let 𝑖(2, ¥) and 𝑖(2, $) denote the spot interest rates for yen and dollar investments with 2-year
maturities
• If no arbitrage opportunities exist, then the rate of yen per dollar for the 2-year maturity must be:
• 𝐹(2) = 𝑆×[1 + 𝑖(2, ¥)]$/[1 + 𝑖(2, $)]$
• Spot: ¥110/$; i(2,$) = 5% p.a.; i(2, ¥) = 4% p.a.; ¥10M to invest
• Investing in Japan: (you invest  10,000,000 JPY in a 2-year yen pure discount bond. At the end of 2 years,
your investment will grow to:
• ¥10𝑀×(1.04)$
= ¥10.816𝑀
• $90,909.09 at current spot the dollar cost of ¥10𝑀
• Investing at home:
• $90,909.09×(1.05)$
= $100,227.27
• You would be indifferent between the two investments if the forward sale of ¥10.816𝑀 for dollars
provides you with the same dollar return as investing directly in dollars ¥10.816𝑀/F(2) = $100,227.27
F(2) = ¥107.9147/$
5. The Term Structure of Forward Premiums and
Discounts
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• Relative Purchasing Power Parity
• Exchange rates adjust in response to differences in inflation across countries
• Logic is that inflation lowers the purchasing power of money, so a change in the nominal
exchange rate to compensate for different levels of inflation should occur
• Purchasing power is the power of money expressed by the number of goods or services that
one unit can buy, and which can be reduced by inflation. RPPP suggests that countries with
higher rates of inflation will have a devalued currency.
• The Big Mac Index
6. Relative Purchasing Power Parity
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Relative PPP
• Relative PPP, which states that the
percentage change in the exchange
rate is equal to the difference in the
percentage changes in average
prices—that is, the inflation rate.
(Formally: (Et-Et-1)/Et-1= π(CH)t-π(USA)t,
where π(x)t is the inflation rate in
country x at time t.)
• Source and more reading about RPPP
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• The real exchange rate
• Adjusted for inflation:
• 𝑅𝑆 𝑡, $/€ =
.(0,$/€)×7(0,€)
7(0,$)
• Real appreciations and real depreciations
• Three basic movements:
• An increase in the nominal exchange rate ($/£), holding $ prices and £ prices constant
• An increase in the £ prices of goods holding the $ prices of goods constant
• An increase in the $ prices of goods holding the £ prices of goods constant
• Trade-weighted real exchange rates
• Useful when looking at how forex changes will affect trade balance
7. The Real Exchange Rate
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• The International Parity Conditions
• CIRP – Covered Interest Rate Parity
• Links forward rates, spot rates, and interest rate differentials
• UIRP or Unbiasedness – Uncovered Interest Rate Parity
• Sometimes called International Fisher Effect / Relationship
• Links expected exchange rate changes and interest rate differentials
• PPP
• Links inflation rates and rates of changes in forex rates
8. Parity Conditions and Exchange Rate Forecasts
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© 2018 Cambridge University Press 6-29
• The Fisher Hypothesis long-run link between inflation and interest rates the nominal interest rate
is the sum of the expected real interest rate and the expected rate of inflation.
• Real rates of return – measures how much your purchasing power has increased over time
• The ex post real interest rate
• The ex ante real interest rate
• Expected real interest rate
• Expected rate of inflation
• Fisher hypothesis – decomposition of nominal int. rates (real interest rates + inflation)
8. Parity Conditions and Exchange Rate Forecasts
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© 2018 Cambridge University Press 6-30
• Suppose the nominal interest rate in Mexico is 10%, and the expected rate of
inflation in Mexico is 7%.
• What is the expected real rate of return in Mexico?
• Simplified: re = 10% - 7% = 3%
• By investing pesos at a nominal interest rate of 10% when the expected rate of inflation is 7%,
the investor expects to earn a 3% real rate of return. The investor expects to have 3% more
purchasing power over goods and services at the end of the year for every peso invested.
• The real interest rate is important because it influences investment decisions.
Problem solving:
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Average Long-Term
Government Bond Yields
and Inflation Rates
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An Example of International
Parity Conditions: covered
interest rate parity (CIRP),
uncovered interest rate parity
(UIRP), and
purchasing power parity
(PPP), the Fisher hypothesis
The United Kingdom and
Switzerland
If the parity conditions all hold simultaneously, real interest
rates are equal across countries.
If uncovered interest rate
parity and PPP hold, the nominal interest rate differential
between the United Kingdom and
Switzerland reflects only an expected inflation differential.
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© 2018 Cambridge University Press 6-33
• International parity conditions
• CIRP, UIRP (or Unbiasedness) and PPP
• If they hold, real interest rates are the same everywhere
• Empirical studies suggest that beyond CIRP, none hold in either long-run or short-run (CIRP
may hold in the short run)
• If International Fisher Relationship holds, the interest rates below would be equal – closer in longrun,
but why not equal?
• Significant deviations in PPP values
• Returns in different currencies can have different risk premiums
• Political risk and threat of capital control
8. Parity Conditions and Exchange Rate Forecasts
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© 2018 Cambridge University Press 6-34
• If interest rate parity is satisfied, there are no opportunities for covered interest
arbitrage. What does this imply about the relationship between spot and forward
exchange rates when the foreign currency money market investment offers a
higher return than the domestic money market investment?
• Describe the sequence of transactions required to do a covered interest arbitrage
out of Japanese yen and into U.S. dollars.
• Suppose you are the French representative of a company selling soap in Canada.
Describe your foreign exchange risk and how you might hedge it with a money
market hedge.
• What do economists mean by the external currency market?
• What is the term structure of interest rates?
Questions:
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