Hi Kalid
thanks so much for your response. What I was trying to say is elaborated below:
Let us say we have a sum in the bank invested at 100% growth per year compounded every 6 months. If we dive straight in with our formulas involving e we would say:
We have 100% growth over 1 year rate x time = 1
Therefore growth factor = e^1 = 2.71828182846
However using our compound growth formula:
Growth factor = (1 +1/2)^2 = 2.25
A fair size inaccuracy has crept in:
So we always have to be mindful of how many times the we have compounded over the period in question.
If there was continuous growth, of course, there would be no problem.
Now let us consider rates different from 100%; for example, a growth rate of 1% compounded 100 times over a year.
We can jump in with our e formula and say
that the growth factor will be:
e^(1/100) = 1.01005016708
However, if we use our compound growth formula we get a growth factor of:
(1+.01/100)^100 = 1.01004966209
Why the discrepancy?
Well let us recall how we derived the e formula for growth rates that are different from 100%
We said that a continuous compound could be approximated to a number of discrete compounds.
Applying this to the problem above: we will approximate 100 discrete compounds to 1 discrete compound.
In other words we are stating that:
1+.01/100)1^100 is close to
(1 + .01/1)^1
We then rearrange this to give:
(1 + 1/100)^(100x(1/100))
But (1+1/100)^100 is nearly e
so the answer must be e^(1/100)
right so let us examine the areas where we have allowed inaccuracies to creep in.
first we said:
1+.01/100)1^100 is close to
(1 + .01/1)^1
In fact it is a little bit larger
Then we said:
(1+1/100)100 is close to
e,
In fact it is a little smaller
So we’ve exaggerated in opposite directions allowing us to get an answer but it is an approximate one.
If we had continuous growth instead of our 100 discrete compounds over the time period in question, all of these inaccuracies would disappear.
We can see this if we look at the same problem again but with continuous growth over the year.
Let us approximate an infinite number of discrete compounds to 1000 compounds
The compound growth formula gives:
1+.01/infinity)^infinity
This is close to
(1 + .01/1000)^1000 (closer than the last approximation because the series converges.)
This can be rewritten as:
(1 + 1/10000)^(100000x(1/100))
making our e formula even more accurate because (1+1/10000)^10000 is closer to e than (1+1/100)^100
You can use your imagination to determine that as we approach continuous growth the e formulas become more and more accurate.
However very few systems in the real world exhibit continuous growth so caution must be exercised.
I am an amateur mathematician so this could be a load of rubbish please chip in with your two cents. Thanks for such a brilliant web site Kalid.