March 08, 2018 Thursday
Bedtime Story
Jacob Bernoulli Compounding Problem
Even though Napier and more rightly
Oughtred used the number e in the logarithmic tables, it is to Jacob Bernoulli
that the credit for the discovery of the constant e is given when in 1683 he
was pondering upon a very banal and practical problem of compound interest.
The problem that he was considering went
something like this in his mind:
‘The question is, if some lender were to
invest a sum of money at interest, let it accumulate, so that at every moment
it were to receive its proportional part of its annual interest; how much would
he be owed at the end of the year?’
Bernoulli constructed a power series and
then went on to write this:
“if a = b, then the lender will be owed
more than 21/2a and less than 3a or in other words if a = b, the geometric
series reduces to the series for a x e, so 2.5 < e < 3.”
This might not have made much sense to you
so let me phrase the problem that Jacob Bernoulli was considering in more
modern way.
So let us suppose you are an investor with
100 units of fiat currency.
Now you wish to invest these 100 units of
your currency into a ponzi scheme that promises 100% interest at the end of
every year.
So how much will your investment become at
the end of one year with this rate of interest?
The answer is simple.
It will double to 200 (unless of course the
ponzi schemer vanished off with your and others money).
Now the question that Jacob further
proposed to himself was what would be the value of my investment at the end of
one year if the interest is computed and credited more frequently than a year
say every 6 months, or every 4 four months or even better, quarterly.
You must note that the annual interest
remains the same, i.e. 100%, only that it will be computed more frequently.
So if it is computed every 6 months, then
the interest rate would become 50% for each 6 months.
This essentially means that the 100 units
of currency will become 150 at the end of six months and then 50% of the
interest will be computed on this amount of 150 for the next six months.
50% of 150 turns out to be 75.
So then when 100 units is computed at 6
monthly interval for annual interest rate of 100%, the final amount at the end
of one year would be 150 + 75 which is 225.
Another more general way of compounding a
principal amount is to multiply the amount x with (1 + 1/n)n where n
is the interval at which the compounding is done.
We will see with the examples given below
how this formula works.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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