The Rue of 72 Busted

The Rule of 72 Busted

The rule of 72 provides a quick way to determine the number of years it takes your money to double given a certain rate of return (compounded). For example, a rate of 10% will double your money in about 7 years. This can be quickly calculated by dividing 10 into 72 giving you 7.2 years. The following table shows how this works and it also shows that this rule is an approximation. The actual results differ slightly from result predicted by the rule.

``````
Rate	Divided by 72	Actual return after predicted period
3%	24		103.3%
4%	18		102.6%
6%	12		101.2%
8%	 9		 99.9%
10%	 7.2		 98.6%
12%	 6		 97.4%
15%	 4.8		 95.6%
20%	 3.6		 92.8%

``````

As you will have noticed, the rule is just about spot on for an 8% return and is still pretty close for returns of 3-15%. For returns above 8%, the rule underestimates the years required for a double, and for returns below 8% the rule overestimates the years required for a double. This phenomenon is going to cause the most distortion of the rule for large rates of compounded rates of return.

Perhaps the rule was developed when expected long term rates of return were in the 6-12% range. Well, Saul’s Investing Discussions has made the rule of 72 obsolete! Using myself as an example, my YTD return hit 72% sometime in May or June suggesting that I should have doubled my money (72/72=1). But I was only up 72%. To get an accurate time period it takes to double your money in Saul’s world, you will need 100% return in a year. 50%, 60%, 70% 80%, or even 90% just won’t do it. The rule of 72 is no longer useful for us!

Chris

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I bookmark free sites like this for a quick CAGR % calculation.
https://www.investopedia.com/calculator/cagr.aspx

On CNBC, I noticed a link touting the best investments since man landed on the moon, in the spirit of celebrating the 50th moon landing anniversary.
https://www.cnbc.com/2019/07/19/mcdonalds-is-among-5-stocks-…

Understanding CAGR really helps to determine what is a good return or not.
Saul was one of the first I can think of that said the concept of “baggers” was kind of pointless, and after thinking thru the math, I found I agreed and understood.

If you buy a stock at \$10, and it goes to \$20, you have a double…whoo!
But if the stock goes up \$10 every year until it reaches \$100, you get lower and lower CAGR returns.

So the CNBC top example given was McDonalds with an astounding 82,000% appreciation…wowza!
However, after you do the math, over 50 years that comes out to about 14% CAGR. Still great by S&P/market standards, but not exactly shocking compared to what most reading this board have done in the past 2-3 years. (which is probably unsustainable, btw)

Getting 70%+ CAGR every year is pretty unlikely, I think most would reasonably agree and understand.
Saul created a sensation when he announced his 2-decade or so CAGR was over 20%, according to knowledgebase, I believe.

I am still trying to model conservative goals, but definitely keep moving the goal posts the past two years, which is a great problem to have. But I think it is still beneficial to have longer-term goals that are somewhere between the market indices and what Saul was able to do (pre-SaaS stocks) and then everything over that just becomes gravy.

If my CAGR link is accurate, I show it takes about 41% CAGR to double your money in 2 years or about 26% CAGR to double your money in 3 years.

Given how much our stocks have expanded their multiples, and how others like Bear have echoed sentiments I share that eventually stocks should track their growth rates (whether rev, FCF, or profit, depending on what stage of growth the company is in). So I am trying to find stocks that I can model a decreasing P/S multiple over time that still nets me around 25%+ CAGR.

An example might be ESTC, currently at a 27 P/S and 7.5b mkt cap, based on 270m+ TTM.
I believe they will finish their current fiscal year around \$425m, which at a 25 P/S would equal a \$10.6b mkt cap, or a 41% stock price appreciation from today’s prices.

This is simplistic, and doesn’t take into account things like dilution, but gives me an easy way to gauge how well I think the stock can do as an investment the next 12 months.

When stocks have multiples so high that you need to go out 2 years to justify the CAGR, I get more antsy as a lot can happen over those extra 12 months in the technology world. But you just have to balance that risk/reward as suits your tolerance level.

Happy to have my math pointed out as incorrect, which wouldn’t be shocking.

Dreamer

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I put my starting amount in my calculator and then multiply it times 1 plus % a year that I want to know about. Every time I hit the multiply button counts for a year. Very simple and roughly approximates my ability to tell someone “25% a year will double your money in 3 years, 33% a year will double your money in 2 years”. That is how I explain compound interest in money slob terms.

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And to go backwards we take roots. For example, if I want to know what CAGR I need to double my money in 4 years, take the 4th root of 2. (Answer is 1.189, or 18.9% CAGR). Or if I want to build my portfolio by 6X in 13 years (which I do), I need 14.8% CAGR, which is the 13th root of 6.

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If I had 70+% returns for 2 years from today (after already being up 66% YTD), I would in fact… retire at the ripe old age of 45 (in 2 years).

I’m neither greedy nor interested in dying upon my rat race wheel.

Costa Rica November to May. Portland, Maine May through October.

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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.

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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

For my 2.5 cents on the topic, for folks needing ‘refresher’ help, Khan Academy has several videos on the topic. I have referred both youngsters and oldsters to the site on many topics.

Excellent resource!

Compound interest introduction (video) | Khan Academy

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I’m neither greedy nor interested in dying upon my rat race wheel.

Costa Rica November to May. Portland, Maine May through October.

Leaving for Costa Rica and Brazil and maybe Columbia August 1st.

I was able to retire 1.5 years earlier than planned thanks to first Motley Fool Stock Advisor and now Saul et al.

Dinero Haragán*

*Spanish for Money Slob

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Pretty quickly off topic, yes?

I’ll offer that this board (and other sources) have taught me it’s OK to sell. It’s OK to sell losers, to take profits. It’s also OK to buy more of your winners.

That’s been a really hard lesson for me to learn. I’m grateful to Saul and all on his board for helping me learn. I promise to pay it forward the best I’m able.

Kip

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This thread is about arithmatic, not about analyzing our individual stocks. It’s really off-topic. We’ve played with it for a dozen or so posts but lets STOP THE THREAD NOW. Thanks
Saul

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Perhaps the rule was developed when expected long term rates of return were in the 6-12% range. Well, Saul’s Investing Discussions has made the rule of 72 obsolete! Using myself as an example, my YTD return hit 72% sometime in May or June suggesting that I should have doubled my money (72/72=1). But I was only up 72%. To get an accurate time period it takes to double your money in Saul’s world, you will need 100% return in a year. 50%, 60%, 70% 80%, or even 90% just won’t do it. The rule of 72 is no longer useful for us!

Wow, starting with a modest \$20,000.00 in a Roth account and doubling it every year for 20 years, you’d then have an after-tax net worth of over \$20.9 billion.
Just do the math!
*bill

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