SUGGESTED READINGS
3.4 FOREIGN EXCHANGE RATES AND QUOTATIONS
APPLICATION The notation used in Multinational Finance
UPPERCASE SYMBOLS ARE USED FOR PRICES Lowercase symbols are used for changes in a price
■ Pdt =price of an asset in currency d at time t
■ pd=inflation rate (i.e., change in the consumer price index) in currency d
■ id =nominal interest rate in currency d
■ d
=real (or inflation-adjusted) interest rate in currency d
■ Sdt/f =spot exchange rate between currencies d and f at time t
■ sd/ft =change in the spot rate between currencies d and f during period t
■ Fd/ft =forward exchange rate between currencies d and f for exchange at time t
■ ftd/f=change in the forward rate between currencies d and f during period t Note: Time subscripts are dropped when it is unambiguous to do so.
Allocational Efficiency Because of its operational and informational efficiency, the interbank market in major currencies is the most allocationally efficient market in the world. Markets for less liquid currencies are less efficient in their allocation of capital. Fixed exchange rate systems also are less efficient, because governments intentionally disrupt the flow of capital in the pursuit of their policy objectives.
when you should be dividing. This seems simple enough now, but as our discussion of FX instruments and positions becomes more complex, it will become imperative to include the currency units wherever they appear in an equation. This is such an important point that it has its own rule.
Rule #1
Keep track of your currency units.
A related problem in currency trading is in keeping track of which currency is being bought and which is being sold. References to currency values invariably have the value of a single currency in mind. The statement, ‘‘Thedollar fellagainst the yen,’’ refers to the dollar. Conversely, the statement, ‘‘The yen roseagainst the dollar,’’ refers to the yen. The currency that is being referred to is called thecurrency of reference,or thereferent currency.
Buying or selling currency is like buying or selling any other asset. It is easiest to think of the currency in thedenominatoras the asset being traded. Currency values are then just like the price of any other asset. For example, you could substitute
‘‘unit’’ for dollar and think of the euro price of the dollar as 0.80/unit (or, in this case, 0.80/$). You might just as well be buying bottles of wine.
Rule #2
Think of buying or selling the currency in the denominator of an exchange rate.
Figure 3.6 provides an example. Suppose you buy 1 million euros at a price of
¥115.4/ and then sell 1 million euros at a price of ¥115.7/ . Remember, you are buying and selling euros — the currency in the denominator. The net result is that you spend (¥115.4/ )( 1,000,000)=¥115,400,000 to buy 1 million euros and then sell them for (¥115.7/ )( 1,000,000)=¥115,700,000, for a profit of ¥300,000.
The bottom panel of Figure 3.6 illustrates what can go wrong. Suppose euro-per-yen rates are quoted as 1/(¥115.7/ ) ≈ 0.008643/¥ and 1/(¥115.4/ )
≈ 0.008666/¥. If you buy 1 million (in the numerator!) at the ‘‘low’’ price of 0.008643/¥, your cost is in fact ( 1,000,000)(¥115.7/ ) = ¥115,700,000.
If you then sell at the ‘‘high’’ price of 0.008666/¥, your payoff in yen is ( 1,000,000)(¥115.4/ ) = ¥115,400,000. This results in a net loss of ¥300,000.
The simplest way to avoid this pitfall is to follow Rule #2 and think of the denominator as the currency of reference.
Foreign exchange quotations can be easy to understand if you follow these two rules. Make sure that you conscientiously apply them as you practice the end-of-chapter problems. You’ll see that following these rules will help you avoid
Exchange rates ⇔
⇔
An example following Rule #2
“Buy €1 at a price of ¥115.400/€ and sell it for ¥115.700/€” Buy €1 at ¥115.400/€ ⇔ Sell ¥s at €0.008666/¥
Sell €1 at ¥115.700/€ ⇔ Buy ¥s at €0.008643/¥
An example of what can go wrong
“Buy ¥1 at a price of ¥115.400/€ and sell it for ¥115.700/€”
Buy ¥1 (sell euros) at ¥115.400/€ ⇔ Sell €s (buy yen) at €0.008666/¥
Sell ¥1 (buy euros) at ¥115.700/€ ⇔ Buy €s (sell yen) at €0.008643/¥
S¥/€ = ¥115.400/€ S¥/€ = ¥115.700/€
S€/¥ = €0.008666/¥
S€/¥ = €0.008643/¥
⇒ ¥0.3/€ profit ⇒€0.000023/¥ profit
⇒ ¥0.3/€LOSS! ⇒€0.000023/¥ LOSS!
FIGURE 3.6 Buying Low and Selling High.
Outright quote Outright quote Mid-rates quoted* (European terms) (American terms) in the financial press
Bid Offer Bid Offer SFr/$ $/SFr
Spot rate 1.7120 1.7130 0.5838 0.5841 1.7125 0.5839
1-month forward 1.7169 1.7179 0.5821 0.5824 1.7174 0.5823
3-month forward 1.7256 1.7267 0.5791 0.5795 1.7261 0.5793
6-month forward 1.7367 1.7379 0.5754 0.5758 1.7373 0.5756
*Mid-rates are averages of bid and ask rates.
FIGURE 3.7 Swiss Franc per Dollar Exchange Rate Quotations.
many careless mistakes as the problems become more complex in the chapters that follow.
Foreign Exchange Quotation Conventions
In practice, FX quotations follow a variety of conventions. Because the referent currency is not always in the denominator, some of these conventions can be difficult to interpret. The two most common conventions distinguish either between the U.S. dollar and another currency, or between the domestic and a foreign currency.
These two conventions are described in this section.
European and American Quotes for the U.S. Dollar Interbank quotations that include the U.S. dollar conventionally are given inEuropean terms,which state the foreign currency price of one U.S. dollar, such as a bid price of SFr1.7120/$ for the Swiss franc in Figure 3.7.2
European terms state the foreign currency price of one U.S. dollar.
The U.S. dollar is the most frequently traded currency, and this convention is used for all interbank dollar quotes except those involving the British pound or the
currencies of a few former colonies of the British Commonwealth. The SFr1.7120/$
quote could be called ‘‘Swiss terms.’’ It is convenient to the Swiss in that it treats the foreign currency (the U.S. dollar) just like any other asset. The ‘‘buy low and sell high’’ rule works for a resident of Switzerland that is buying or selling dollars in the denominator of the quote.
When this bank is buying dollars, it is simultaneously selling francs. Conse- quently, the dollar bid price must equal the Swiss franc ask price. Following Rule
#2, we could treat the Swiss franc as the currency of reference and place it in the denominator.
S$/SFr=1/SSFr/$=1/(SFr1.7120/$)≈$0.5841/SFr
Conversely,American termsstate the dollar price of a unit of foreign currency.
This is convenient to a U.S. resident because the foreign currency (the Swiss franc) is in the denominator.
European and American quotes are not possible for transactions that do not include the U.S. dollar. For these transactions, we need an alternative quotation convention, such as one based on domestic versus foreign (rather than U.S. versus non-U.S.) currencies.
Direct and Indirect Quotes for Foreign Currency The most straightforward way to quote bid and offer prices from a domestic perspective is withdirect quotes,stating the price of a unit of foreign currency in domestic currency terms.
Direct quotes state the domestic currency price of one unit of foreign currency.
This is a natural way to quote prices for a domestic resident, because the foreign currency is in the denominator. For a U.S. resident, a direct quote for the Swiss franc might be
$0.5838/SFr Bid and $0.5841/SFr Ask
This bank is willing to buy francs (and sell dollars) at $0.5838/SFr or sell francs (and buy dollars) at $0.5841/SFr. The bank’s bid-ask spread is $0.0003/SFr.
Nevertheless, the convention in many countries is to useindirect quotes,which state the price of a unit of domestic currency in foreign currency terms, such as SFr1.7120/$ for a U.S. resident. For example, an indirect Swiss franc quote to a U.S.
resident might be
SFr1.7120/$Bid and SFr1.7130/$ Ask
In this example, the bank is willing to buy dollars in the denominator (and sell francs in the numerator) at the SFr1.7120/$ price. It is also willing to sell dollars (and buy francs) at the SFr1.7130/$ price. The bank’s bid-ask spread is SFr.0010/$.
What If a Quote Doesn’t Follow Rule #2? In this example, the bank could quote SFr1.7130/$ Bid and SFr1.7120/$ Ask
In this case, the bid is higher than the ask. Does this mean that the bank is willing to lose money on every purchase and sale? Not at all. By quoting a higher bid price than ask price, the bank is indicating that it is willing to buy francs (in the numerator!) at SFr1.7130/$ or sell francs at the SFr1.7120/$ rate. This is, of course, equivalent to buying dollars at SFr1.7120/$ and selling dollars at SFr1.7130/$. The rule for determining the currency that is being quoted is as follows:
■ When the bid is lower than the offer, the bank is buying and selling the currency in thedenominatorof the quote.
■ When the bid is higher than the offer, the bank is buying and selling the currency in thenumeratorof the quote.
Note that this indirect quote to a U.S. resident is equivalent to 1/(SFr1.7120/$)
≈$0.5841/SFr and 1/(SFr1.7130/$)≈$0.5838/SFr. Swiss banks quoting these bid and offer prices to a Swiss resident with an indirect quote might quote
$0.5838/SFr Bid and $0.5841/SFr Ask
This bank is willing to buy Swiss francs (and sell dollars) at 58.38 cents per franc or sell Swiss francs (and buy dollars) at 58.41 cents per franc. Alternatively, the bank might quote
$0.5841/SFr Bid and $0.5838/SFr Ask
which means that the bank is willing to buy dollars (in the numerator) at the bid price and sell dollars (in the numerator) at the ask price. These quotes are equivalent.
Each of these examples makes sense if (and only if) you follow Rule #2 and think of the denominator as the currency of reference.
The Special Case of the British Pound Exchange rates for the British pound sterling (and of countries associated with the British empire, such as Australia) often are quoted as the foreign currency price per pound, such as $1.4960/£. The reason for this is historical. Prior to 1971, one British pound was worth 20 shillings and each shilling was worth 12 pence. The convention of keeping the pound in the denominator was convenient at that time because fractions of a pound were not easily translated into shillings and pence.
Forward Premiums and Discounts
Forward premiums and discounts reflect a currency’s forward price relative to its spot price. Again, it is easiest to keep the currency of reference in the denominator of the FX quote.
A forward premium is when a forward price is higher than the spot price.
■ A currency is trading at a forward premium when the value of that currency in the forward market ishigherthan in the spot market.
■ A currency is trading at aforward discountwhen the value of that currency in the forward market islowerthan in the spot market.
Forward premiums and discounts can be expressed as a basis point spread. If the Swiss franc spot rate is $0.58390/SFr and the 6-month forward rate is $0.57560/SFr, then the franc is selling at a 6-month forward discount of $0.00830/SFr, or 83 basis points (in this case, a basis point is 1/100th of one Swiss cent). Common usage is to speak of the ‘‘forward premium’’ even when the forward rate is at a discount to the spot rate. This saves having to say ‘‘forward premium or discount’’ each time.
Forward premiums also are quoted as a per-period percentage deviation from the spot rate.
Foreign currency premium (periodic)=(Fd/ft −Sd0/f)/(Sd0/f) (3.1) In the example with S$0/SFr=$0.58390/SFr and F$1/SFr=$0.57560/SFr where one period equals six months, the 6-month forward premium is calculated as
(Fdt/f−Sd/f0 )/(Sd/f0 )=($0.57560/SFr−$0.58390/SFr)/($0.58390/SFr)
= −0.014215
or−1.4215 percent per six months. Note this formula works only for the currency in the denominator.
This 6-month forward premium can be stated as an annual forward premium in several ways. The formula used in the United States and Canada is
Foreign currency premium (annualized)=(n)[(Fd/ft −Sd/f0 )]/(Sd/f0 ) (3.2) where n is the number of compounding periods per year. Multiplying by n translates the forward premium into an annualized rate with n-period compounding. For example, a 6-month forward premium is annualized by multiplying the 6-month forward premium by n=2 semiannual periods per year. Similarly, a 1-month forward premium is annualized by multiplying the 1-month forward premium by
n=12. In the example with S$/SFr0 =$0.58390/SFr and F$1/SFr=$0.57560/SFr, the forward premium is calculated as
(n)[(Fd/ft −Sd0/f)]/(Sd0/f)=(2)[($0.57560/SFr−$0.58390/SFr)]/($0.58390/SFr)
=(−0.014215/period)(2 periods)
= −0.028430
or−2.8430 percent on an annualized basis with semiannual compounding.
In much of the rest of the world, forward premiums are calculated as an effective annual rate, also called aneffective annual percentage rate(APR), according to
Foreign currency premium (APR)=(Fd/ft /Sd0/f)n−1 (3.3) Under this convention, the annual forward premium is
(Fd/ft /Sd/f0 )n−1=(($0.57560/SFr)/($0.58390/SFr))2−1
=(0.985785)2−1
= −0.028227
or an effective annual rate of−2.8227 percent. This is the same as−2.8430 percent with semiannual compounding, simply stated under an alternative compounding convention.
Percentage Changes in Foreign Exchange Rates
In a floating exchange rate system, an increase in a currency value is called an appreciationand a decrease is a depreciation. Changes in currency values in fixed exchange rate systems are called revaluations or devaluations. Calculation of a percentage change in a FX rate is similar to that of a forward premium. The value of the currency in the denominator of an exchange rate quote changes according to the formula
Percentage change in a foreign currency value=(Sd/f1 −Sd/f0 )/Sd/f0 (3.4) Suppose the dollar-per-franc rate changes from S$/SFr0 =$0.5839/SFr to S$/SFr1 =
$0.5725/SFr over a 6-month period. The percentage change in the Swiss franc in the denominator of the quote is
[($0.5725/SFr−$0.5839/SFr)]/($0.5839/SFr)≈ −0.0195
The Swiss franc in the denominator depreciated 1.95 percent over the 6-month period.
If the franc falls, the dollar must rise. Rule #2 says that to find the dollar appreciation, we first should place the dollar in the denominator. The beginning spot
rate is 1/($0.5839/SFr)≈SFr1.7126/$ and the ending rate is 1/($0.5725/SFr)≈ SFr1.7467/$. The percentage rise in the dollar (in the denominator) is then
[(SFr1.7467/$−SFr1.7126/$)]/(SFr1.7126/$)≈ +0.0199 That is, the dollar appreciated 1.99 percent over the 6-month period.
Percentage changes in direct and indirect FX rates are related, as an appreciation in one currency must be offset by a depreciation in the other. Applying the equality Sdt/f=1/Sf/dt and simplifying the result yields
Sd/f1 /Sd/f0 =1/(Sf/d1 /Sf/d0 )
Alternatively, we can let (Sd/f1 /Sd/f0 )=(1+sd/f), where sd/f is the percentage change in the d-per-f spot rate during the period. This can then be rewritten as
(1+sd/f)=1/(1+sf/d) (3.5) For a+1.99 percent change in the dollar that is offset by a−1.95 percent change in the Swiss franc, the algebra looks like this.
(1+s$/SFr)=(1−0.0195)≈1/(1+0.0199)=1/(1+sSFr/$)
Note that an appreciation in one currency is offset by a depreciation of smaller magnitude in the other currency. This asymmetry is an unfortunate but essential part of the algebra of holding period returns.3
A Reminder: Always Follow Rule #2
The intuition ‘‘buy low and sell high’’ works only for the currency in the denominator of a foreign exchange quote. Thus, there is a simple remedy for keeping things straight — just follow Rule #2. If the currency that you would like to reference is in the numerator, simply move it to the denominator according to Sd/f=1/Sf/d. Following this convention will help you avoid needless confusion. (Actually, this rule is entirely self-serving. If you conscientiously follow Rule #2, your teachers — me included — will be spending less time on the phone answering your questions!)