116 CHAPTER 3. SPOT MARKETS FOR FOREIGN CURRENCY
3.6 TekNotes
Technical Note 3.1 What’s wrong with theFC/HCconvention, in a text-
book?
In the text just below Example 3.2 we claimed that using thefc/hcconvention would mean all the
familiar formulas from Finance would have to be abandoned. Here’s this message in math. Letr∗
denote the risk-free interest rate earned onfc, andS ̃ 1 the (random) future value, inhcof one unit
offc. If you buy one unit offc, you’ll have 1 +r∗of them next period, worthS ̃ 1 (1 +r∗) inhc.
Standard finance theory then says that the current price,S 0 , should be the future value discounted
at a rate E( ̃rS) that takes into account this risk ofS ̃ 1 :
S 0 =E(
S ̃ 1 )(1 +r∗)
1 + E( ̃rS). (3.19)This looks quite normal and well behaved. Now look at what would happen if we had used the
inverse rate,X:=S−^1 , and if we wanted a theory about howX 0 is set. First substituteX=S−^1
into the equation and then solve forX 0 :
1 /X 0 =E(1/
X ̃ 1 )(1 +r∗)
1 + E( ̃rS) ⇒X^0 =1 + E( ̃rS)
E(1/X ̃ 1 )(1 +r∗)
.All connection with finance is gone. The discount rate is on top (?!), and the expectation is below,
and the expectation is about the inverse ofX. Clearly, this makes no sense in a finance textbook.