The natural logarithms used in finance
The Sunday Nation
Nairobi,
04 November 2012
Githuku Mungai (no relation to me) is an accounts professional and he says that every now and then he has to use a mathematical function know as the natural logarithm. His question is therefore straightforward: I would like Mr. Mungai Kihanya to tackle the natural logarithm in his column, for it is applicable in finance, and leaves us scratching our heads.
Before explaining
what a natural logarithm is, we must first understand the meaning of
logarithm in the first place.
I tackled that question in June last year, but as a quick reminder, the
logarithm of X to base Y is the power to which Y can be raised in order
to get X.
For example, we know
that 2 x 2 x 2 x 2 x 2 x 2 = 64; that is, 2 raised to the power of 6 is
64. In this case, X = 64 and Y = 2; therefore, the logarithm of 32 to
base 2 is 6.
But we also know that
4 x 4 x 4 = 64; in other words, 4 raised to the power of 3 is 64. Thus
now X = 64 but Y = 3; therefore, the logarithm of 64 to base 4 is 3.
Notice that the
logarithm of a number changes when the base is changed. In most cases,
people count in base 10; thus when asking for the logarithm of a number
X, what they want to know is: to what power can we raise 10 in order to
get X? Now 10 raised to power 1.8062 is 64; thus the logarithm of 64 to
base 10 is 1.8062.
In the case of
natural logarithms, the base is the special (but irrational) number
2.7182818284590452353602874713527
Mathematicians call it the
exponential constant, or simply the exponential. Since it is irrational
(meaning that its exact value CANNOT be determiner!), it is usually
represented by the letter e.
The origin of this
special number can be understood by considering a example from finance.
Suppose you invest Sh100,000 in a ten-year fixed deposit account where
the bank promises to pay back Sh200,000 on maturity. The interest earned
in that period is 100 per cent.
Seeing that you are
getting 100 per cent in ten years, you may ask the bank to pay 10 per
cent per year; but, instead of giving you the interest in cash, you may
ask them to reinvest it in the same account at the same rate.
In that case, your
balance would be Sh110,000 after the first year; then Sh121,000 after
the second and so on up to Sh259,374 at the end of the tenth year. If
you look at the workings carefully, you will notice that we are
multiplying 1.10 by itself ten times and then by Sh100,000; that is 1.10
raised to power 10 (hint, hint; can you see a logarithm somewhere in
there?)
What if you wanted
this calculation to be done monthly? The monthly interest is 0.8333 per
cent and there are 120 months in ten years. Thus raising 1.00833 to
power 120 we get 2.70704; therefore, your account will have Sh270,704
after ten years. This is better than Sh259,374
and much better than
Sh200,000.
If the interest is
calculated daily, the daily rate would be 0.0274 per cent and raising
1.000274 to power 3,650 gives 2.71791. Therefore your balance after ten
years will be Sh271,791 you gain only Sh1,087!
If we do the same
hourly we get Sh271,827; if every minute the result is Sh271,828.16 and
so on. Now compare this last result with the irrational number e = 2.
7182818284590452353602874713527
; the first seven digits are identical!
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