Imagine there is a pond. One day, someone introduces a fast-growing lily to the pond. This lily grows such that its population doubles daily. On the first day there is 1 lily and on the second day there are 2 lilies, and so on. If it takes 37 days for the lilies to completely cover the pond, how much of it was covered on the 35th day? Take a few minutes to work it out own your own.
If the population doubles daily and the pond was 100% covered on the 37th day, that means it only covered 50% of the pond on the 36th day, because 2 times 50% is 100%. By the same logic, the pond was covered half as much on day 35 vs day 36. Half of 50% is 25%, so the pond was 25% covered on the 35th day.
If we make a table of growth, we get:
Day Percent of Coverage
37 100%
36 50%
35 25%
34 12.5%
33 6.25%
32 3.125%
31 1.5625%
...
1 less than 1%, let us assume we measure it to be 0.001%
(the area of the pond is 100,000 times bigger than the area of 1 lily)
As you can see, the bulk of the growth (87.5%) in just the last 3 days. If we made a graph of the growth with time on the x-axis, we get something like this:
Here, exp() is short for the exponential function. This is a graph showing exponential growth with rate of 31% per day with an initial size of 0.001. It can also be written as e^(rt) where e is Euler's number, equal to about 2.718, r is the rate of growth, and t is the time elapsed.
What we have here is a graph of exponential growth for something with an initial size of 0.001 (how much of the pond 1 lily covered) that increases at a rate of about 31% per day.
The basic idea of exponential growth is that something doubles in size at regular intervals. In this case, the number of lilies doubles about every three days.
Another way to think about this is with the Rule of 72. It states that the time needed for something that grows exponentially to double is the same as 72 divided by the growth rate. If something is growing exponentially at 6% per year, it will take 12 years to double in size because 72/6 = 12. This method can also be used to find growth rates. If we want to know the growth rate required for something to double in 8 years, we simply evaluate 72/8 to get 9%.
There's a lot of talk of inflation these days. Because inflation is exponential just like compound interest, even a small rate of inflation can have a big effect over a relatively short period of time.
With an annual inflation rate of just 4%, it would only take 18 years for the price of everything to double (72/4 = 18). The US Federal Reserve bank has an inflation target of 2% per year, but the actual rate is higher. If we use an online inflation calculator, we see that:
So something that cost $1 in 1995 costs $2 today. That means there was a 100% price increase over 28 years ( from 1995 to 2023). 100/28 = 3.57 about, so the real rate of inflation during that period was 3.57% per year.
To put it another way, if you owe $1,000 on a credit card which charges 4% interest per year, in 18 years, you'll owe $2,000. Most credit cards these days charge around 24% interest, which means if you owe $1,000 on such a card, in 3 years, you'll owe $2,000.
“Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it.”
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