I don’t usually post on here, but as a math professor, I feel especially compelled to chime in on this one.

The discrepancy that Chris is pointing out is due to the subtle but important difference between having an “*r% Compound Annual Growth Rate*” (CAGR) vs. an investment growing at “*r%, compounded continuously*”.

The Rule of 72 assumes that interest is being compounded *continuously*. And it’s also an approximation. To find the **exact** amount of time it takes to double, you actually need to divide by 100*ln(2)=69.314718… (that’s “natural logarithm”, i.e., log base e, where **e=2.7182818…**)

So why is it the “Rule of 72”, and not the “Rule of 69”? Because many numbers divide evenly into 72 (e.g., 2, 3, 4, 6, 8, 9, 12), so it’s easy to compute in your head. This is also why some people use the “Rule of 70” instead, especially for 7% or 10% rates.

If you’re curious, let’s look at an example to understand *why* this difference occurs. Suppose you start at time t=0 with P(0)=$100, and I make the ambiguous statement that your “growth rate is 100%”. How much do you have after 1 year? What this means could depend on your payment schedule:

Annually: 1 payment at 100% interest = $100 * (2.0)^1 = $200

Semi-annually: 2 payments at 50% interest = $100 * 1.5^2 = $225

Quarterly: 4 payments at 25% interest = $100 * 1.25^4 = $244.14

Monthly: 12 payments at 8.333% interest = $100 * 1.0833^12 = $261.30

Daily: 365 payments at 0.274% interest = $100 * 1.00273^365 = $270

Minutely: 525600 payments at 0.00019% interest = $100 * 1.0000019^525600 = $271.83

Continuously: Take the “limit” as the number of payments, n–>infinity, (*do some nifty Calculus here,…*), and you get $100*e^(1) = $271.83818185…

The first of these above is CAGR, which is what most of us use when computing our investment returns. If we know our profits, it’s really easy to compute our average rate on a napkin. In contrast, the last one is what “compounded continuously” actually means, and it’s what your bank will give you if they promise, e.g., a fixed 3% interest rate. And it’s not so easy on a napkin. For those familiar with calculus, it’s basically the difference between “average rate of change” vs. “instantaneous rate of change”. And as Chris pointed out, this gets much more pronounced as the rates increase.

To see just how much more, let’s use the same example of P(0)=100 dollars. After one year:

at “3% growth”, you’ll have

P(1) = 100 * 1.03 = 103, if your CAGR is 3%

P(1)= 100*e^(.03) = 103.05, if your rate is 3%, compounded continuously

at “8% growth”, you’ll have

P(1) = 100 * 1.08 = 108, if your CAGR is 8%

P(1)= 100*e^(.08) = 108.33, if your rate is 8%, compounded continuously

at “20% growth”, you’ll have

P(1) = 100 * 1.20 = 120, if your CAGR is 20%

P(1)= 100*e^(.2) = 122.14, if your rate is 20%, compounded continuously

at “72% growth”, you’ll have

P(1) = 100 * 1.72 = 172, if your CAGR is 72%

P(1)= 100*e^(.72) = 205.44, if your rate is 72%, compounded continuously

So in summary, the Rule of 72 (actually, of 69.315) isn’t busted, but as Chris observed, it’s not as accurate when one tries to use it for CAGR for large growth rates.

Sorry to continue this discussion in a somewhat off-topic direction for this board, but I just wanted to add some clarity and resolution to the initial observation made in this thread, and I hope some of you got something useful or insightful out of it.