The exponential growth formula is used to calculate the future value [P(t)] of an amount given initial value [P_{0}] given some rate of growth [r] over some period of time [t].
This formula is absolutely core to understanding compound interest. It can be used for anything that is growing exponentially! Try entering any numbers in to see their value:
You can change which variable you solve for any time and the solved for variable will update as you make changes.
So what do these variable means?
- P(t) = the value at some future time, t
- P_{0} = initial amount at time t = 0
- r = the growth rate as a percentage (1% = 0.01)
- t = time – the number of periods (intervals). This could be months or years – just depends on when the rate compounds.
This form is solving for P(t), or the future value. You can also shift this formula around and solve for any other variable! If you have any 3 values, you can solve for the remaining one.
The hard part (for me at least) was solving the equation for each variable. What if you know the current value, future value, and rate, but you want to solve for time? Or what if you want to solve for rate?
Let’s check out how to use this same formula solved for each variable!
Solving for Future Value – P(t)
So, let’s look at an example. Assume you have $1,000 [P_{0}] and want to know what the value of that money will be in 10 years [t], assuming a 7% growth rate [r].
This works great for our initial formula! We’re already solving for future value [P(t)], so we can use this one:
You can substitute in the values from the above paragraph and see the result:
So after 10 years of 7% a year growth, the initial $1,000 would almost double to $1,967.15. In other words, the doubling time is just about a decade.
This falls in line with the Rule of 72, which states if you divide 72 by the rate, it’ll take that many years to double your money. 72/7 = 10.28 years. If you’re looking to make quick double calculations, the Rule of 72 is an easier way than the exponential growth formula.
Solving For Current Value – P_{0}
If you know what the future value is, but want to solve for the current value, you can rebalance this equation to solve for P_{0}, current value:
Where P_{0} is the current value, P(t) is the future value, r is the rate, and t is time. This is useful if you want to see what initial investment is needed today to reach a future state with a known time and growth rate.
Solving For Rate – r
When I make predictions I tend to be somewhat pessimistic – or more likely realistic. When calculating out the growth of my portfolio, I use a rate of 7%, which isn’t too far from what others have calculated the real rate of return on the stock market to be.
The formula looks kind of crazy, but it’s the same one as above just reformatted to solve for r. You can solve for r using this calculator:
Curious about how to earn 7% a year in the stock market? I have a free course that shows exactly how to get started doing it using safe, diversified index funds that track the larger market. This strategy is the same one Warren Buffet and John Bogle (creator of Vanguard and the Index fund) recommend.
Solving For Time – t
One thing I used a lot when going down my path to financial independence was solving for time. For instance, if I know the current value, how much I need to save, and can assume a 7% rate of return, how much time will it take to get there?
I built a much more elaborate visualization of this in The Interactive Guide to Early Retirement and Financial Independence which leans heavily on this formula. The formula for calculating t gets tricky, so hold on.
This one looks a bit crazy. It’s because in order to move that exponent down to where we can work with it, we need to log both sides of the equation. The result comes out looking like this. Time is also called duration, interval or period – it’s the total number of time intervals it takes.
If you’re curious to learn more about how the work goes to solve for this, I’d recommend reading this paper on compound interest that has a few more examples on how to use formula.
Exponential Growth vs Compound Interest
These two terms are the same, but their definitions are different. Both involve non-linear growth that’s greater than constant growth. The growth curves look the same – up and to the right with a growing rate of change.
Exponential growth implies any type of growth where the calculation of the “next step” in growth is based on the current value. This can apply to money, logarithmic functions, algebra, calculus, acceleration number of bacteria in a sample, how fast a virus could spread, human population growth, Moores’s law for growth in transistors on a computer – you get the idea.
It’s a formula for seeing the growth over time.
Compound interest, on the other hand, is a specific application of the exponential growth formula to calculate future value. In the above calculators I’ve added dollar signs, percent signs and “years” to the fields to make them more easily understandable – but that does make all of these calculators compound interest calculators. You can use them for any of the above exponential growth calculations if you just ignore the units.
Have you ever needed to use the exponential growth formula manually to calculate out something? What did you use it for? Let me know!