# The 2nd Law of Large Numbers

February 27, 2023

Statisticians developed the Law of Large Numbers. It states that as a sample size grows, its mean (average) approaches the mean of the whole population. Here it is in English. If you flip a coin, you have 50% odds of heads and 50% odds of tails. Flip a coin twice and anything can happen. All heads is as likely as heads then tails. Flip a coin 10 times and the probability of all heads is 0.977%. Probability of all heads on 100 flips is zero. Well, not technically zero, but it requires 30 zeros after the decimal to notate and I don’t feel like typing all those zeros. Bottomline: if your sample size is small (two flips), anything can happen. If your sample size is large (100 flips), your results will pretty closely approximate reality. Right around 50 heads and 50 tails.

I’m proposing a 2nd Law of Large Numbers. There are multiple Laws of Thermodynamics, I see no reason there shouldn’t be another Law of Large numbers. Here it is:

The 2nd Law of Large Numbers: when dealing with large numbers, our brains explode.

We struggle differentiating between a million, a billion, and a trillion. Our language makes them sound the same. Just one or two different letters. We know there is a difference of three zeros and one comma when those numbers are notated. But all three of those numbers translates roughly the same way in our head: “a lot.” We don’t really comprehend the difference. An example…

One million seconds is roughly 12 days.

One billion seconds is 32 years.

One trillion seconds is 32,000 years. For context, all recorded human history spans about 5,000 years.

I was in a conversation the other day about the valuation of NBA teams. Someone wondered how much Mark Cuban made on his purchase of the Dallas Mavericks. Here’s how the conversation went:

“NBA team valuations have gone nuts since Ballmer bought the Clippers for a few billion.”

“Cuban bought the Mavs awhile ago – I bet he’s made a lot of money”

“He paid \$285 million in 2000. Now they’re valued at \$3.3 billion.”[i]

“WOW! What a killer investment!”

It is very difficult to conceptualize the difference between \$285M and \$3.3B. Our brains translate both of those numbers roughly as “a lot.” 285M seconds works out to about nine years. 3.3B seconds is about 106 years. A massive difference. Cuban is worth a lot more now than when he made the purchase 23 years ago. We know \$3.3B is more than \$285M. But did you realize it was that much more? Probably not. The 2nd Law of Large Numbers. When the numbers are large our brains explode.

The 2nd Law is applicable to more than magnitude. It also holds true regarding rates of return when large numbers are involved. We’re very bad at conceptualizing rates of return with large numbers. Back to Cuban’s Mavs purchase. \$285M to \$3.3B over 23 years – that’s a killer investment, right?

It works out to an annualized return of…

11.2%.

For some context, the S&P 500 has historically returned an average of 10%/yr.[ii]

Yes, he beat the S&P. Barely. Admittedly I imagine owning a sports team is about the most fun you can have with an investment. Certainly more fun than owning a handful of ETFs. If I had a few hundred million laying around in 2000 I would have thought about buying a basketball team too. My point is that when the numbers are large, we lose any sense of context. If I told you I invested \$285 in the stock market in 2000 and it grew to \$3,300 by 2023, you may say something like “well at least you probably beat inflation.” But if I tell you I invested \$285,000,000 in 2000 and it grew to \$3,300,000,000 by 2023 you will probably say something like “WOOOWWWW THAT’S AMAZING.” The same 11.2% annual return. Slap a few zeros on the end of the numbers and we all freak out.

When the numbers are large, we underestimate the magnitude of the numbers and overestimate rates of return. The 2nd Law of Large Numbers. Our brains explode.

I see this in discussions about real estate often as well. You’ve probably heard some variation of this story:

“My parents bought their home for \$100,000 in 1980. Now it’s worth \$519,000! Can’t beat real estate!”

5x’ing your money is great. But over a 22-year time period that works out to a 4% annualized return (appropriate, given housing has historically returned ~4%/yr.).[iii] Four percent seems more like just ok? That is of course before the expenses of owning a home. Net out taxes, insurance, and maintenance and you’ll find that 4% annual return ground down quickly.

But we all have the same gut response to that story. Growing \$100K to \$500K – WOW, LET’S HEAR IT FOR REAL ESTATE! Growing \$100 to \$500 over 22 years doesn’t seem like a big deal. Add three zeros to the end of the numbers and it sounds like the investment of a lifetime. If I tell you I have an illiquid asset with high carrying costs and a 4% annualized return, you will pity me for my poor investing acumen. If I tell you I bought a house for \$100K in the 80s and solid it for half a million, you will admire my investing brilliance. The 2nd Law of Large Numbers.  When the numbers are large our brains explode.

“Our brains explode” is admittedly vague. A more formal definition for the 2nd Law of Large Numbers is we misattribute an asset’s growth to its rate of return when the driving source of the growth is its original magnitude (Cuban’s purchase price) or time (home bought in 1980).

The 2nd Law is one of the reasons building wealth is challenging. Wealth building is more a function of time and magnitude than rate of return. When you start from nothing, the financial (and psychological) rewards are small. If you let \$10,000 compound at 10%/yr. for 10 years, it will grow to \$25,937. ~2.5x your initial investment. That’s fine. Let it compound for 50 years and it grows to \$1,173,908. 117x your investment. That’s better. When your nest egg is \$10,000 and your portfolio returns 10% in a year, you make \$1,000. That’s fine. When it’s grown to \$1,000,000 and your portfolio returns 10% in a year, you make \$100,000. That’s more like it. The same return, but now it’s real money. You’re not earning an impressive return; you’re earning a decent return on a large sum of money.

It's not about the rate of return. It’s about magnitude and time. The 2nd Law of Large Numbers.

Remember the Mavericks are worth \$3.3B. If that asset value grows by another 11.2% in 2023, Cuban’s net worth will grow by \$369,600,000. The rich get richer not because they’ve unlocked some tax secret (maybe sometimes) or they have access to some unique investment opportunity (occasionally the case) but because of this simple equation:

Massive sum of money x average return = massive growth

\$3,300,000,000 x 11.2% = \$369,600,000

The most important factor in the billions of dollars the Mavs have added to Cuban’s net worth is the fact that he bought them for a lot of money! Not that NBA teams are an asset with a uniquely high return. You’re probably still a bit skeptical reading that (c’mon, it’s the Mavs! Luka!). But that’s the 2nd Law of Large Numbers:

When dealing with large numbers, our brains explode.

Sean Cawley, CFP®

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[iii] YCharts, Fundamental Chart, Case Shiller Home Index: National