## Annual Exponential Growth

I was Googling the “law of 72” which is used in compound interest to determine how many times your initial investment is required to be compounded at a given interest rate to double itself. In Investopedia website they defined it as “a simple way to determine how long an investment will take to double given a fixed annual rate of interest”.

When I read the words “annual rate”, I laughed out loud. What if the rate was for a given “cycle” which less than a year, perhaps just over a month? Lets say there are 10 of these cycles in a year, and lets assume the rate per cycle is 7%, using the “law of 72”, it would be 72/7 or 10.28 cycles or in other words after 11 cycles, your initial investment would double.

Let us try a hypothetical scenario, where the minimum investment was \$1,000, and you could increase it in increments of \$100, so it would be \$1,000, \$1,100, \$1,200, etc. For this illustration we can use 7% as the fixed rate per cycle. We shall use start with \$2,000 for this example. After one cycle, you would get 7% or \$140. Since the investment steps are in the nearest \$100, you could increase the investment in the next cycle by \$100, and save the \$40 until it grows to \$100 or more. Now let’s tabulate it over 11 cycles.

Cycle 1 Profit \$140 Next investment amount: \$2,100 Remainder : \$40
Cycle 2 Profit \$147 Next investment amount: \$2,200 Remainder : \$87
Cycle 3 Profit \$154 Next investment amount: \$2,400 Remainder : \$41**
** \$87 + \$54 = 141 (so, we can add another \$100 to invest)
Cycle 4 Profit \$168 Next investment amount: \$2,600 Remainder : \$9**
Cycle 5 Profit \$182 Next investment amount: \$2,700 Remainder : \$91
Cycle 6 Profit \$189 Next investment amount: \$2,900 Remainder : \$80**
Cycle 7 Profit \$203 Next investment amount: \$3,100 Remainder : \$83
Cycle 8 Profit \$217 Next investment amount: \$3,400 Remainder : \$0**
Cycle 9 Profit \$238 Next investment amount: \$3,600 Remainder : \$38
Cycle 10 Profit \$252 Next investment amount: \$3,800 Remainder : \$90
Cycle 11 Profit \$266 Next investment amount: \$4,100 Remainder : \$56**

As you can see after 11 cycles, the total amount is at \$4,156 just a little over double the initial amount. If you could fit 10 cycles in a year, you can guess that a cycle is 5 weeks, as there are 52 weeks in a year. So, 11 cycles is a year and 3 weeks. In just that time, your initial investment has doubled, and in that same amount time a year later, it will double again. \$2K becomes \$4K, becomes \$8K, \$16K, \$32K, \$64K. This is what we call annual exponential growth.

Somehow this seems too good to be true, and yet truth is stranger than fiction.