APPENDIX 6A: WHAT ARE LOGARITHMS, AND WHY ARE THEY USED IN FINANCIAL RESEARCH?
APPENDIX 6A: WHAT ARE LOGARITHMS, AND WHY ARE THEY USED IN FINANCIAL RESEARCH?
Although this is not a course in mathematics, there are certain techniques that are so prevalent in modern financial research that not to use them would be a disservice to the student. Logarithms are a prime example. The most important reason for the use of logarithms is that they show the “true” percentage distance. In addition, they facilitate calculations in financial relationships
What are logarithms? Logarithms are a way of transforming numbers to simplify mathematical anal-
ysis of a problem. One way to view a logarithm is as the power to which some base must be raised to give a certain number. For example, we all know
that the square of 10 is 100, or 10 2 5 100. Therefore, if 10 is our base, we know that 10 must be raised to the second power to equal 100. We could then say that the logarithm of 100 to the base 10 is 2. This is written as
log 10 100 5 2
What then is the log 10 of 1000? Of course, log 10 1000 5 3, because
10 3 10 3 10 5 10 3 5 1000: In general, any number greater than 1 could serve as the base by which
we could write all positive numbers. Picking any arbitrary number designated as a, where a is greater than 1, we could write any positive number b as
log a b5c
where c is the power to which a must be raised to equal b. Rather than pick any arbitrary number for our base a, there is a particular number that arises naturally in economic phenomena. This number is approximately 2.71828, and it is called e. The value of e arises in
the continuous
compounding
of
interest. Specifically,
Exchange Rates, Interest Rates, and Interest Parity 127
the base of the natural logarithms. Financial researchers utilize logarithms to the base e. Rather than write the log of some number b as log e b 5 c, it is
common to express log e as ln, so that we write lnb 5 c
In all uses of logarithms in this text, we assume log b is actually ln b or the natural logarithm; it is for convenience that we drop the e subscript
and simply write log b rather than log e b.
Why use logarithms in financial research? If the lesson so far has seemed rather esoteric and unrelated to your inter-
ests, here is the payoff.
A useful feature of logarithms is that the change in the logarithm of some variable is commonly used to measure the percentage change in the variable (the measure is precise for compound changes and approximate for simple rates of change). If we want to calculate the percentage change in the yen/$ exchange rate (E) between today (period t) and yesterday
(period t21), we could calculate (E t 2E t 21 )/E t21 . Alternatively, we could calculate ln E t 2 ln E t21 . For example, let us assume that the E t21 5 80 for the yen/$ in the past value and E t 5 125 for the yen/$ in the current period. What is the percentage change in the value of the yen? If we use the formula (E t 2E t21 )/E t 21 then the percentage change is 56.25%, but if we use the ln E t 2 ln E t21 5 44.63%. The two values are near each other, but not quite the same. Now assume that the exchange rate was quoted in inverse form instead as $/yen. The rates become E t21 5 0.0125 for the yen/$ and E t 5 0.008. Note that these are identical to the rates quoted above. What is the percentage change
in the value of the yen? If we use the formula (E t 2E t21 )/E t21 then the per- centage change is 236%, but if we use the ln E t 2 ln E t21 5 244.63%. So when we use natural logarithms the percentage distance becomes identical no matter how currencies are quoted.
128 International Money and Finance
Natural logarithms also have some convenient mathematical proper- ties. Three extremely helpful properties of logarithms that are used fre- quently in international finance are:
1. The log of a product of two numbers is equal to the sum of the logs of the individual numbers:
ln ðMN Þ 5 ln M 1 ln N
2. The log of a quotient is equal to the difference of the logs of the indi- vidual numbers:
ln ðM =N Þ 5 ln M 2 ln N
3. The log of some number M raised to the N power is equal to N times the log of M:
ln ðM N Þ 5 Nðln MÞ
Since many relationships in financial research are products or ratios, by taking the logs of these relationships, we are able to analyze simple, linear, additive relationships rather than more complex phenomena involving products and quotients.
This appendix serves as a brief introduction or review of logarithms. Rather than provide more illustrations of the specific use of logarithms in international finance, at this point it is preferable to study the examples that arise in the context of the problems, as analyzed in subsequent chap- ters. More general examples of the use of logarithms may be found in Wainwright and Chiang (2004).