I am not a math professional but I know the formula for continuous compounding and also (finally) just studied its derivation using the limit when n --> infinity; how r, n, t all play together in the formula when n/r = m and all of that which accounts for situations when r (growth rate) is not equal to 100%. I understand it but don't understand the intuition behind the substitution n=mr. How in the world one would intuitively understand and get an idea like that -i.e. by substituting n=mr, we can pull out r from inside the limit to outside?
What I fail to understand is the intuitive part behind the merging of the rate and time. Using a simple example as a case in point. Assuming a 100% continuous compounding rate, a 1 dollar grows to e at the end of year 1. At the end of year 2, it grows to e.e = e^2. This is fine so far.
Now I say that I cannot wait two years, but I want the same e^2 amount at the end of year 1 instead of year 2 end. This implies that the rate of growth has to be naturally higher since there is less time. The formula says because (e^2)^1 = (e^1)^2 the rate and time can be interchanged. So a 200% rate is required.
I understand the formula, but the intuition is lacking. If the amount at the end of two years is e^2, the amount after half the time (i.e year 1 end) is square root of (e^2) which is e. I understand this part. While 200% fits the formula, I don't intuitively understand how the 200% growth rate (continuous compounding, that is) is required during year 1 to make the amount e^2 at the end of year 1.