I saw this chart on LinkedIn and the call of mental math immediately lured me onto the rocks.
Since 1972, the SP500 is up 250x.
So what’s the CAGR (compound annual growth rate)?
One can open the calculator on their phone and type 250^(1/52) - 1
I cannot. I am impelled to estimate the answer. By what? I don’t know, probably the same ghost that makes people play games like Wordle.
I figured I’d share how I did this because practicing mental math is fun.
Recipe:
Immediately recognize that 250x return is about 8 doublings (2⁸ = 256)
If we estimate a 10% log return per year, Rule of 72 predicts in 50 years you double 7 times since at 10% you double every 7 years.
Log returns are handy because they are proportional to time. If you doubled 8/7 or 14% more cumulatively, then your return per year must have been 14% greater than 10% or 11.4% compounded.
This reasoning got us quite close to 250^(1/52) - 1 = 11.2%
Shortcut for estimating a 50-year compounded return: 10% (1 + doublings/7)
Compounding feels unnatural. If we compound at 11.2% instead of 10%, over 50 years we get an additional doubling!
At 10% CAGR wealth grows by 2^7 or ~250x
At 11.5% CAGR wealth grows by 2^8 or ~500x
At 13% CAGR, wealth grows by 2^9 or ~1000x
Rule of 72
Formally, the rule of 72 says that if you earn 10% per year, it will take 7.2 years for your money to double. A derivation of rule of 72 can be found here.
If you already know how it’s derived then you can guess that I prefer rule of 69.
[Let’s the Beavis chuckle pass]
If you double wealth, then your log return is ln(2) or .69
.69 / 7 years ~ 10% annual log return
The rule of 69 is very close to rule of 72 but is derived from continuous compounding instead of annual.
I prefer log returns because they are directly proportional to time.
The expression ex is a total quantity of growth. It’s actually assumed to be
e 1 * x where the 1 represents 100% continuously compounded growth and X represents a unit of time. The natural log or ln(ex) then solves for how much time (ie x) did it take to arrive at the total quantity of growth assuming 100% continuous compounding.
A key insight is that we don’t need to assume a 100% rate and x to be time. We can simply think of x as the product of “rate multiplied by time”. This allows us to substitute any rate for the assumed rate of 100% to find the time. Once again we turn to BetterExplained:
If you review that section of the post a few times and make up a few examples for yourself, you’ll never get confused about e or ln again. You might even start thinking about all numbers in terms of their logs.
For any number X:
log (base-10) ~ “how many orders of magnitude to get to X?”
log (base-2) ~ “how many doublings to get to X?”
ln (aka log base e) ~ “compounding continuously at 100% how long will it take to get X”
In this last example, it takes one unit of time to get to 2.718 compounding continuously at 100%. If our unit of time was a year, then 100% return compounded continuously would turn $1 into $2.718
In the earlier examples, if you compounded continuously at 69% you’d double your money in 1-year. At 6.9% continuous compounding, it takes 10 years. At .69% continuous compounding it takes 100 years.
As long as you keep this in terms of doublings, ie log₂(wealth), then you can compute the compounded growth rate in your head.
Testing it remembering that the SP500 250x return was about 8 doublings in 50 years:
70% * (8/50)
70% * 16% = 11.2%
If you can count doublings then you can easily dazzle your friends with how fast you estimate a growth rate for any number of years.
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When I was introduced to the math of Pascal and the elementary probability, I saw immediately how important this math was. My math teacher had no idea that he’d come to a part of the math that was very important in the regular world to everybody, but I saw it immediately and I just utterly mastered it. And I used it. I’m still using it. I used it routinely all my life quite intensely.
And when I got to study in the Harvard Business School, in the early days at the Harvard Business School, they were proudest of something called decision tree theory. And they taught it at the Harvard Business School, a lot of pomp and ceremony and many examples, all these graduate students.
Decision tree theory, it’s a Harvard Business School — in those early days, what they were teaching you was that Pascalian probability math works in real life. Here’s the Harvard Business School needing to do remedial high school math to a bunch of graduate students, and they weren’t wrong. They were right in those days to teach decision tree theory because other people hadn’t mastered probability math the way it should be mastered.
My teacher in high school, if you don’t pay attention to anything else, this stuff you ought to master. And he should explain how carny operators and casinos take advantage of ordinary people. It should have been taught, and it wasn’t taught right in high school, and it wasn’t taught right in college and it wasn’t taught right. Finally, the Harvard Business School got so they taught high school math to graduate students. And you can say how could that be correct? But it’s because the earlier education was so ineffective.
This is the best speech that Charlie Munger ever gave. He explains how an investor can beat the stock market. The 20-page transcript is a treasure trove and a must-read for all. Today, we're sharing the key points and the full speech…
The following excerpt reinforces the spirit of the compounding math above (emphasis mine):
Another very simple effect I very seldom see discussed either by investment managers or anybody else is the effect of taxes. If you're going to buy something which compounds for 30 years at 15% per annum and you pay one 35% tax at the very end, the way that works out is that after taxes, you keep 13.3% per annum.
In contrast, if you bought the same investment, but had to pay taxes every year of 35% out of the 15% that you earned, then your return would be 15% minus 35% of 15% or only 9.75% per year compounded. So the difference there is over 3.5%. And what 3.5% does to the numbers over long holding periods like 30 years is truly eye-opening. If you sit back for long, long stretches in great companies, you can get a huge edge from nothing but the way that income taxes work.
Even with a 10% per annum investment, paying a 35% tax at the end gives you 8.3% after taxes as an annual compounded result after 30 years. In contrast, if you pay the 35% each year instead of at the end, your annual result goes down to 6.5%. So you add nearly 2% of after-tax return per annum if you only achieve an average return by historical standards from common stock investments in companies with tiny dividend payout ratios.[Kris: Stop investing specifically for dividend cash flow…if the company’s earnings are growing, its price will increase and you can sell shares to create your own dividend on your own schedule. Plus companies that don’t pay dividends but whose earnings are growing are presumably re-investing internally at higher returns on capital then what you are going to do with the taxed cash. This is not a blanket truth, but the blanket love of dividends is worse, especially if companies cater to investors who crave them.]
Munger does warn:
But in terms of business mistakes that I've seen over a long lifetime, I would say that trying to minimize taxes too much is one of the great standard causes of really dumb mistakes. I see terrible mistakes from people being overly motivated by tax considerations.
Money Angle For Masochists
Volatility risk premiums or VRPs are measures of implied volatility relative to realized volatility. Implied volatility typically trades a bit rich. Also, most dogs don’t bite. It’s a broad statement. I am presenting it without a prescription because that’s not the point of this note.
Instead, I just want to share a recent example of how a very fat VRP ended up compressing back to typical levels. As a ratio, an extended VRP can normalize by any mix of the numerator (implied vol) falling and/or the denominator (realized vol) rising.
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Curse you for planting this idea in my head. I was at a tour of Drexel with my son today and their student run investment fund turned 250k in 2007 into $5M today, a 20x return. So not I only did I have to do this in my head (and make my son follow along) but I had to do it a different way, because well you know. I got just over 4 doublings in 17yrs or a double every 4.25 years so I just divided 72 by 4.25 to get ~17%. Close enough to the actual 23% for monkey math but not as accurate as your method.
Curse you for planting this idea in my head. I was at a tour of Drexel with my son today and their student run investment fund turned 250k in 2007 into $5M today, a 20x return. So not I only did I have to do this in my head (and make my son follow along) but I had to do it a different way, because well you know. I got just over 4 doublings in 17yrs or a double every 4.25 years so I just divided 72 by 4.25 to get ~17%. Close enough to the actual 23% for monkey math but not as accurate as your method.