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```					International Finance                       Dr. Angela Ng
FINA 342                                          HKUST

Class Notes 6

INTERNATIONAL PARITY CONDITION II:
COVERED AND UNCOVERED INTEREST
RATE PARITY

I. COVERED INTEREST RATE PARITY

EXAMPLE

 Mr. David Sylvian, an arbitrageur for the Hong Kong &
Shanghai Banking Corporation in Hong Kong, arrives at
work on Tuesday morning to be faced with the following
quotations on his Reuters’ screen:
Spot rate: ¥125.00/\$
One year forward rate: ¥130.00/\$
Euroyen one year interest rate: 10%
Eurodollar one year interest rate: 5%

What should Mr. Sylvian do?

 Solution:

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DERIVATION OF A GENERAL EXPRESSION FOR
COVERED INTEREST RATE PARITY

 Notations:

it = U.S. interest rate

it* = “comparable” foreign interest rate

St = spot exchange rate (\$/FC)

Ft+1 = forward exchange rate (\$/FC)

 An investor has two options:

1. Invest in U.S.  \$ terminal wealth = 1+ it

2. Invest abroad  \$ terminal wealth = (1/St)(1+it*)Ft+1

To avoid arbitrage opportunities, we need equality of dollar
returns so that one statement of interest rate parity, with
exchanges rates expressed in “American terms”, is
Ft 1 (1  i* )
(1  i t )              t
St

Dividing both sides of the above expression by (1+it*) gives
Ft 1 1  i t

St     1  i*
t

Now subtracting 1 from both sides gives
Ft 1  St i t  i*
        t
St      1  it*

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 Alternatively, the covered interest rate parity can be written
as:
Ft  1  i 
t

        
S0  1  i* 
where i and i* are the geometric mean US and foreign
interest rates over the t periods.

 Forward premiums and discounts are entirely determined by
interest rate differentials.

 Suppose StFC/\$ is the spot rate in European terms and Ft+1FC/\$
is the forward rate in European terms. How would the interest
rate parity formula look like?

 Exercise: Spot (FF/\$) = 7.4825
90-day Forward = 7.5650
90-day Eurodollar interest rate = 8.2% p.a.
90-day Eurofranc interest rate = 12.7% p.a.
Does the covered interest rate parity hold?

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 If Ftd/f/S0d/f > [(1+id)/(1+if)]t, then Ftd/f must fall, S0d/f must rise,
id must rise, and/or if must fall.
 Sell FC at Ftd/f. Buy FC at S0d/f. Borrow at id. Lend at if.

If Ftd/f/S0d/f < [(1+id)/(1+if)]t, then Ftd/f must rise, S0d/f must fall,
id must fall, and/or if must rise.
 Buy FC at Ftd/f. Sell FC at S0d/f. Lend at id. Borrow at if.

Example: i\$ = 7%             i£ = 3%
S0\$/£ = \$1.20/£     Ft\$/£ = \$1.25/£
How could one make a riskless arbitrage profit?

 Covered interest arbitrage is the profit-seeking activity that
forces interest rate parity to hold.

 Interest rate parity line (see Exhibits 6.1 & 6.2)

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RELAXING THE BASIC ASSUMPTIONS

 Transaction costs

If you are holding a dollar asset, foreign investment requires
four transaction costs:
1. Brokerage fees on sale of dollar security  t
3. Buy foreign security – brokerage fee  t*

How do you construct the neutral band within which covered
interest arbitrage transactions will not occur, when
transaction costs are present?

Illustration: Exhibit 6.3

 Taxes

Consider the case where income is taxed at a rate of y and
capital gains at k. On an after-tax basis, the parity condition
is modified as
Ft 1  St              it  it*
(1   k )           (1   y )
St                  1  it*
or
Ft 1  St it  it* 1   y

St      1  it* 1   k

Illustration: Exhibit 6.4

 Uncertainty

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 Notations: Spot rate (\$/¥)          St b – St a
Forward rate (\$/¥)       Ft,Tb – Ft,Ta
Interest on \$            iTb – iTa
Interest on ¥            iTb* – iTa*

 To see whether there is a riskless profit, there are two
approaches.

(1) Borrow \$
 Borrow \$ at the rate of iTa.
 Get \$ 1/(1+ iTa)
 Convert this \$ amount into ¥
 Have ¥ 1/[(1+ iTa)Sta]
 Invest this ¥ amount
 Obtain ¥ (1+ iTb*)/[(1+ iTa)Sta]
 Sell all of the ¥ amount in the forward market
 End up with \$ [(1+ iTb*) Ft,Tb]/[(1+ iTa)Sta]

So, if this final amount is great than 1, then there is a
profit opportunity. To eliminate arbitrage, we require
1  ia
F S 
b
t,T
a
t
T
*
1  iT
b

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(2) Borrow ¥
 Borrow ¥ at the rate of iTa*.
 Get ¥ 1/(1+ iTa*)
 Convert this amount into \$
 Have \$ Stb/(1+ iTa*)
 Invest this \$ amount in the US
 Obtain \$ Stb(1+ iTb)/(1+ iTa*)
 Sell this amount of \$ forward
 End up with ¥ Stb(1+ iTb)/ Ft,Ta(1+ iTa*)

To eliminate arbitrage, we require
1  iT
b
F S 
a
t,T
b
t            *
1  ia
T

IMPORTANCE OF IRP

 One-way arbitrage:

One-way arbitrage is simply picking the lowest-priced
alternative when buying or the highest-priced alternative
when selling.

Example: Consider an investor who holds US\$ cash and
wants to make a DM payment in 3 months. The investor is
given the following quotations:
Spot \$/DM: 0.6793 – 0.6803
3-month Forward \$/DM: 0.6800 – 0.6830
3-month Euro DM: 4.00 – 4.125% p.a.
3-month Euro \$: 6.0625 – 6.1875% p.a.

Can the investor make a one-way arbitrage profit?

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 The search for one-way arbitrage opportunities will limit
the possibility of round-trip, covered interest arbitrage
profits.

 Cost of hedging for firms:

Consider a manager wishes to own the Japanese yen in 6
months from now. The following two hedging strategies are
possible:
1. Bank forward contract
2. Money market hedge

Firms with lower credit rating and firms with lower
borrowing capacity prefer to hedge using a bank forward.

 Country risk:

IRP provides a market determined measure of political risk
differences between countries. To evaluate risky cash flows
from foreign countries, one needs a risk-adjusted discount
rate. IRP provides a market determined measure of this
discount rate.

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II. INTERNATIONAL FISHER EFFECT

 Fisher Relation: (1  i)  (1  r)(1  p)

Nominal returns are part real return and part inflation.

 Uncovered Interest Rate Parity or International Fisher Effect
states that returns in different countries’ money markets
should be equalized “in an expected sense.”

DERIVATION OF A GENERAL                             EXPRESSION           FOR
INTERNATIONAL FISHER EFFECT

 Notations:

it = U.S. interest rate

it* = “comparable” foreign interest rate

St = spot exchange rate, in American terms (\$/FC)

 An investor has two options:

1. Invest in U.S.  \$ terminal wealth = 1+ it

2. Invest abroad and covert at future spot  \$ terminal
wealth = St+1(1+ it*)/St

International Fisher Effect (in American terms) states:
         S                  E[St 1 ]
1  i t  E (1  i* ) t 1   (1  i* )
t                  t
          St                  St

E[St 1 ] 1  i t              E[St 1 ]  St i t  i*
                                or                             t
St       1  i*
t                   St         1  it*

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 What would be the formulation for the International Fisher
Effect in European terms?

 Foreign investment proposed in the IFE contains exchange
rate risk.

 When the U.S. interest rates are higher than foreign interest
rates, the dollar is expected to depreciate, and vice versa.

 International Fisher and PPP combined: real interest rate
parity

Based on PPP, the expected percentage change in the
exchange rate is
E[St 1 ]  St E[Ptus1 ]  Ptus E[Ptf1 ]  Ptf
    
us
        f
 p us1  p ft 1
t
St             Pt               Pt
where pt+1us and pt+1f are the expected inflation rates.

Combining with the International Fisher Effect gives
p us  p f  i  i*
Thus, the nominal interest differential is equal to the expected
inflation differential.

 Alternatively, if we assume that real interest rates are equal
across countries, then the Fisher relation implies
1  i 1  p us

1  i* 1  p f
Measured over t periods, the relation becomes
t
 1  i  1  p 
t        us
                 
 1  i*   1  p f 
        

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 The IFE does not hold at any particular point in time, but
does hold in the long run.

THE FORWARD RATE UNBIASED CONDITION

 Recall that the International Fisher Effect is given by
1  it
E[S t 1 ]           St
1  i*
t

This is equivalent to
Ft  E[St ]
Thus, the forward rate is an “unbiased predictor” of the future
spot rate.

 If speculators think the spot rate will close above the forward
rate, they can buy the forward contract and settle in the spot
market at a positive expected profits. This type of speculative
activity forces the forward price to rise and ensures that
forward exchange rates are close to expected future spot
rates.

 Consider a 90-day forward exchange contract to purchase the
British pounds. Think of an investor as intentionally
speculating, that is, as wanting to bear the risk. If you
contract to buy pound forward at \$1.50/£ and you do not hold
any pounds, then you are taking on exchange risk. You will
profit if the future spot exchange rate is greater than \$1.50/£,
because you could sell the pounds for more than it takes to
buy them. Your profit at time t+90 from a contract entered at
time t is St+90 - Ft,90. (See Exhibit 6.5)

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 If the forward rate is an unbiased predictor of the future spot
exchange rate, the forecast error made by the forward rate has
a mean of zero.

This has two important practical implications:

1. When you make a sequence of purchases in the forward
market over a long time period, profits average out to zero.

2. At any time t, we can describe the possible future spot
rates in any 90 days by a probability distribution. This
gives us the probability of profits and losses.

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III. INTERNATIONAL PARITY CONDITIONS: WRAP-
UP

 Summary

 PPP

 Absolute version: link exchange rate and price ratio

 Relative version: link inflation rates and exchange rate
changes

 Interest Rate Parity

interest differential

 Uncovered Interest Rate Parity or International Fisher
Effect: link expected exchange rate changes and interest
rates

 The PPP, CIRP and IFE relations are sometimes referred to as
the International Parity Conditions. (See Exhibit 6.6)

If they all hold simultaneously, they imply equal real interest
rates across countries.

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Exhibit 6.1
The Interest Rate Parity Line
Equilibrium and Disequilibrium Points

Note: Points A and B are on the interest rate parity line. Points off the parity line generate economic incentives for capital
to flow out of investments in one currency and into another currency. The arrows indicate the marginal impact of
arbitrage transactions on prices of F, S is, and iforeign.

Exhibit 6.2
Covered Interest Rate Parity in European Terms

i*  i t
t                                             45° line
1 it

Region of Arbitrage

Ft  S t
St

Region of Arbitrage

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Exhibit 6.3
The Interest Rate Parity Line
Transaction Costs and the Neutral Band

Note: Points A, A”, and B are inside the neutral band and considered equilibrium points with no arbitrage profit
possibility. For points A’, B’ and B”, the profit opportunity is greater than the cost of executing arbitrage transactions.

Exhibit 6.4
Interest Rate Parity Lines
Pretax (PT) and After-Tax (AT) Lines

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Exhibit 6.5
Speculative Long Position of a £ Forward Contract

Profit per unit of £

0.10

1.40
Future Spot
Rate (\$/£)
1.50   1.60

-0.10

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Exhibit 6.6
International Parity Conditions

Forecast Change in
Spot Exchange Rate

(E)                                            (A)

Foreign Currency                  (C)                  in Inflation Rates

(D)                                            (B)

Difference of Nominal
Interest Rates

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