International Finance
FINA 5331
Lecture 7:
Forward contracts
Interest rate parity
Read: Chapter 5 (125-129)
Chapter 6 (125-129)
Aaron Smallwood Ph.D.
Review: Terminology
• Long vs short:
– A trader is long in a currency when they are owed that currency
in the future. A trader is short in a currency when they owe that
currency in the future.
• Hedging
– The act of reversing a natural short or long position, such that no
net position is taken.
• Speculation
– The act of intentionally creating a short or long position in an
attempt to profit on currency movements. This exposes the
trader to risk.
• Arbitrage
– The act of simultaneously buying and selling an asset, such that
no net position remains.
Forward contracts
• Forward contracts: Allow traders to buy/sell
foreign currency for delivery in the future, for a
price that is known TODAY.
• We considered both outright forward contracts
and non-deliverable forward contracts (NDFs).
• Example: Suppose the 3 month forward
contract for Swiss Franc: $1.1280. 3 months
from today, we could buy or sell SF at this price.
Forward contracts
• Let’s review aspects of deliverable
forward contracts:
• 1. A forward contract is a derivative asset, based off of a spot
contract. The forward market is known as an “over the counter”
(OTC) market, as compared to futures markets, where trade
typically occurs on a centralized exchange.
• 2. The terms of a forward contract are negotiated between a
bank/dealer and their client. At least in theory, except where capital
controls may be present, a forward contract can be written for any
currency.
• 3. Based on 2, a forward contract can be written for delivery at any
point in the future.
• 4. Again, based on 2, a forward contract can be written in any “lot
size.”
Forward rates
• When the forward rate exceeds the spot rate,
the foreign currency (currency in the
denominator) is trading at a “forward
premium.”
– Example: S($/SF)= $1.1271…F($/SF)3m=$1.1280,
SF is selling at premium.
• …When the forward rate is less than the spot,
the foreign currency is said to be selling at a
“forward discount.”
– Example: S($/£)=$1.6725… …F($/£)3m=1.6714, the
pound is selling at a discount.
Forward rates
• Like spot rates, banks will buy forward foreign
currency at one price (the bid price) and sell it
forward at another price (the ask price).
• When the spot quotation is provided, the dealer
will simply quote basis points for the associated
forward rates.
– For the basis points, the trader will understand that if
the first quote exceeds the second, the basis points
are subtracted from the spot quotations.
– If the first number in the quote is less than the
second, the basis points are added.
• A basis point is defined as 0.0001.
Example
• Suppose we call our dealer and ask for forward
quotes relative to the spot rate for Canadian $.
Our dealer supplies the following information:
• Spot: $1.0480-86
–
–
–
–
–
–
1 month forward “12-10”
3 month forward “14-8”
6 month forward “16-2”
One month forward rates: $1.0468-$1.0476
Three month forward rates: $1.0466 - $1.0478
Six month forward rates: $1.0464 - $1.0484.
Forward premia/discount
• In some sense, the premium or discount
provides information related to how the value
of a currency changes over time. We can
express these values in percentage terms.
We will typically apply annualization. The
calculation:
Ft  St 360
%Premia/Dis count 

,
St
days
• Where “days” is the number of days until the
contract matures.
Example
• Suppose the three month forward rate on
yen today is $0.009779. The current spot
rate $0.009774. Suppose a contract written
for yen matures 92 days from now. The yen
is trading at a premium:
Ft - St 360
´
,
St
days
0.009779 - 0.009774 360
=
´
= 0.00200
0.009774
92
%Premia/Disc. =
• The yen is selling at 0.200% premium.
Premia/Discount Cont.
• If F>S, in some sense, this might signal
that we expect foreign currency to
appreciate in value.
• If F<S, this might indicate that we expect
the foreign currency to depreciate in value.
Interest Rates and Exchange Rates
• One of the most important relationships in
international finance is the relationship between
interest rates and exchange rate.
• The setup:
• Suppose a trader has the ability to borrow or
lend in both the domestic market and a foreign
market.
– Denote the domestic annualized interest rate as it and
denote the foreign annualized interest rate as it*.
– Denote the spot and forward domestic currency price
of the foreign as St and Ft. Suppose the forward
contract matures in M days.
Interest Rate Adjustment
• The forward contract matures in M days.
• Interest rates are quoted in annualized
terms. We need to adjust interest rates to
allow a comparison:
M ~*
~
* M
it  it
, it  it
360
360
Borrowing in the domestic currency; lending
in the foreign
• If I borrow one unit of the domestic currency,
in M days, I will repay:
~
1  it
• To lend in the foreign currency, I must
convert domestic currency into foreign
currency. For each unit of domestic currency
I have, I receive, 1/St units of the foreign
currency.
Lending
• Now I lend the proceeds in the foreign country…I
have 1/St units of the foreign currency…I will
receive:
1
1  ~i  S
*
t
t
• Problem…these proceeds are in foreign currency units…I
want the proceeds in domestic currency. I could have
acquired a forward contract, to sell (forward) foreign currency
proceeds in M periods. The result:
1  ~i  SF
*
t
t
t
The result:
• Suppose


~
~* Ft
1  it  1  it
St
• Then, to profit, I could borrow in the domestic
currency, convert the proceeds into foreign
currency, lend in the foreign market, and convert
proceeds back into domestic currency using a
forward contract.
• What if,


~
~* Ft
(1  i ) t  1  it
St

~ S
~*
 (1  i ) t t  1  it
Ft

I can still profit…Start by borrowing in the foreign
currency.
Implications
• The no arbitrage condition implies:


~
~* Ft
1  it  1  it
St
• The equation, known as the no arbitrage condition, has important
implications.
• To illustrate suppose the equation didn’t hold.
• Example, suppose:
• it: 6.00% (annualized interest rate in the US for an asset maturing
in one month).
• it*: 5.25% (annualized interest rate in Germany for a similar asset
maturing in 1 month).
• St: $1.36537 (dollar price of the euro on the spot market).
• Ft: $1.30 (assume asset matures in 30 days time).
An arbitrage opportunity exists:
• First, interest rates are adjusted:
• We have: ~i  0.06 30  0.005
t
360
30
~*
it  0.0525
 0.004375
360
• As thus:
~
1  it  1.005
1.30
~* F
(1  it ) t  (1.004375)
 .956288
St
1.36537
• PROFIT TIME!
How do we profit
• Start by borrowing in the foreign country. Let’s
do it big! Let’s borrow €10,000,000.
– We will have to repay:
– €10,000,000*1.004375= €10,043,750
• Note, as a result of our actions, demand for loanable funds in Germany
increases. Foreign interest rates increase.
• Convert euros and lend in the US.
– €10,000,000*$1.36537 = $13,653,700.
– Lend at .5% yielding:
– 13,653,700*(1.005) = $13,721,968.50.
• Note, two things happen here. On the spot market, supply of
euros increases, driving down St.
• Supply of loanable funds increases in the US, driving down it.
Last step…
• Finally, you use the pre-existing forward
contract to sell the dollar proceeds for
euros. The result:
$13,721,968.50/1.30 = 10,555,360.38.
Profit: €10,555,360.38 - €10,043,750 =
€511,610.38.
Note, in the final step, you sell forward dollars.
You are buying forward euros. This likely
causes, Ft to rise.
No arbitrage opportunities?
• NOT ONCE YOU HAVE LEFT THE MARKET!
• Recall, our arbitrage opportunity existed because:


~
~* F
(1  it )  1  it t
St

~ S
~*
 (1  it ) t  1  it
Ft

• However, as a result of your actions:
–
–
–
–
1.
2.
3.
4.
Foreign interest rates rise.
The spot rate falls.
Domestic interest rates fall.
The forward rate rises.
No arbitrage
• Thus, we can expect smart traders will eliminate
profitable arbitrage opportunities quickly when they
exist. Thus, as a rule:


~
~* Ft
(1  it )  1  it
St
• Implications: Suppose domestic interest rates fall as a
result of, say, monetary policy.
• To ensure equilibrium:
– 1. Foreign interest rates could also fall…
– 2. and/or The forward rate could fall.
– 3. and/or…The spot rate COULD rise. An increase in the spot rate
implies a DOMESTIC CURRENCY DEPRECIATION.
Covered Interest Rate Parity
• The no arbitrage condition is frequently rearranged in a more convenient way:




~ ~*
it  it
Ft  S t
~* 
St
1  it
or
~ ~* Ft  S t
it  it 
St
Deviations from CIRP?
• Transactions Costs: Suppose we thought we could
profit from borrowing in the US and lending in Japan.
– The interest rate available to an arbitrageur for borrowing,
ib,may exceed the rate she can lend at, il.
– There may be bid-ask spreads to overcome, Fb/Sa < F/S
– Thus
(Fb/Sa)(1 + i¥l)  (1 + i$ b)  0
• Capital Controls
– Governments sometimes restrict import and export of
money through taxes or outright bans.
• Taxation differences on capital gains.
FRUH and UIP
• Ft = E(St+1) if investors are risk neutral.
• Since investors are assumed to be rational,
E(St+1) = St+1 + εt+1 where εt+1 is a random
(unforecastable) forecast error.
• Then Ft = St+1 + εt+1 and the forward rate is
an unbiased predictor of the future spot rate.
• From CIP Ft – St
(it – i*t)
=
St
(1 + i*t)
FRUH and UIP
• Then it must be the case that
E(St+1) – St
St
*)
(i
–
i
t
= t
(1 + i*t)
This is the Uncovered Interest Parity
condition.
E(st+1) – st=it-it*
It will only hold if investors are risk neutral or
equivalently they do not care about the currency
denomination of the assets they hold
FRUH and UIP
• If uncovered interest parity (UIP) holds
then FRUH is true and investors are risk
neutral.
• Risk neutrality implies that investors
have no currency preference in which
their investments are denominated.
• Assets with identical risk characteristics
but denominated in different currencies
will be viewed as perfect substitutes.
Purchasing Power Parity
• Purchasing Power Parity and Exchange
Rate Determination
• PPP Deviations and the Real Exchange
Rate
Purchasing Power Parity
in a Perfect Capital Market
• Purchasing power parity (PPP) is built on
the notion of arbitrage across goods
markets and the Law of One Price.
• The Law of One Price is the principle that
in a PCM setting, the same goods will sell
for the same price in two markets, taking
into account the exchange rate.
PUS,wheat = PChina,wheat ´ S$/RMB
Purchasing Power Parity
• Let PUS and PCHINA represent the weighted
average price level for goods in the U.S.
and Chinese market baskets respectively.
• Absolute PPP predicts that these two price
measures will be equal after adjusting for
the exchange rate: PUS = S$/RMB  PCHINA
• Absolute PPP requires that the
consumption baskets are identical across
the two countries.
Purchasing Power Parity and Exchange Rate
Determination
• The exchange rate between two currencies should equal the
ratio of the countries’ price levels:
P$
S($/RMB) = P
CHINA
For example, an ounce of gold on March 12 cost $1370.70 in the
U.S. and RMB8,413.36 in China. Then the price of one dollar in
terms of RMB should be:
PRMB
S(RMB/$) =
=
P$
RMB8,413.36
$1,370.70
= ¥6.138
Relative Purchasing Power Parity
• Note, that %Δp = π, the rate of inflation
• Relative PPP states that the rate of
change in the exchange rate is equal to
the differences in the rates of inflation:
%Δs =
($ – ¥)
(1 + ¥ )
≈  $ – ¥
If U.S. inflation is 5% and Chinese inflation is 8%, the yuan
should ordinarily depreciate by 2.78% or approximately
3%.
Ex-Ante PPP
• Ex-Ante PPP says that relative PPP will
hold in an expected value sense, i.e.
E (% st 1 )  E ( t 1 )  E (
*
t 1
)
Where E is the expectations operator signifying
that E(·) is an expected value.
PPP Deviations and the Real Exchange Rate
The real exchange rate is
St Pt *
RERt =
Pt
If absolute PPP holds then
Pt
St = * ® RERt = 1
Pt
Absolute PPP implies that price indices in all countries are
computed with the same weights and the same basket of
goods. Since this is never true, relative PPP is the more
appropriate form. Under relative PPP, RERt will be constant
but differ from unity.
Purchasing Power Parity and Overvalued or
Undervalued Currencies
Example
Base period nominal exchange rate = $1.50/£
Prices of U.S. goods had risen by 8%
Prices of U.K. goods had risen by 4%
PPP spot rate = $1.50/£  1.08/1.04 = $1.5577/£
•
A nominal exchange rate of $1.5577/£ would reestablish PPP
in comparison to the base period.
•
Nominal exchange rates greater than $1.5577/£ represent £
“overvaluation” ($ undervaluation), while rates less than
$1.5577/£ represent $ “overvaluation” (£ undervaluation).
Purchasing Power Parity and Overvalued or
Undervalued Currencies
Nominal exchange rates greater than the PPP
implied exchange rate represent foreign
currency overvaluation, while nominal
exchange rates less than the PPP implied
exchange rate represent domestic
overvaluation (or foreign undervaluation).