The Linear Thinking Pitfall
By: Teng Rong
Tags: Investing, Thinking, Mathematics, Psychology, Compound Interest

The Linear Thinking Pitfall

Reality rarely draws straight lines, but your mind doesn’t know it (yet)

“Compound interest is the eighth wonder of the world. He who understands it, earns it. He who doesn’t, pays it.” — Albert Einstein (Allegedly)

There is a Twitter screenshot going around that says if you saved $10,000 every day since the height of Ancient Egypt, you wouldn’t have as much money as the top five billionaires today. This is true. The Giza Pyramids were built in the 4th dynasty, around 4600 years ago. If you saved $10,000 every day for 4600 years, you would have $16,790,000,000, or around 17 billion dollars. At 16.8 billion, you would rank just under Ray Dalio at number 85 on the billionaire rankings. (1)

The point of the screenshot was to point out how ridiculous it was for individuals to amass so much wealth. Implicit in this screenshot is the idea that nobody can make and save $10,000 every day for over four thousand years, and thus the billionaire wealth is illegitimate. However, if we examine the presuppositions of the screenshot we see that it conveniently falls victim to the linearity bias. Billionaires don’t “make and save” money. Their money compounds. They get rich through the power of exponentiation.

I am not here to defend billionaires (for I am neither one nor do I know any). However, I would like to offer an alternative example: suppose that you have $10,000 right now, and I offered you one of two choices:

Use the 10k to purchase an investment that returns 10k every day for the next 50 years, OR Use the 10k to purchase an investment that compounds at 30% for the next 50 years

Which option do you pick? We are naturally inclined to pick option #1: 10k a day! That means you make $3.6 million each year, or $182 million over the 50 year timespan. Sounds good, right?

Indeed, though option #1 is a pretty slick investment in the abstract, it becomes nothing when you compare it to option #2. If you are compounding yearly at 30%, you will end up with $5 billion dollars by the end of the 50 years. How does this work, you ask?

The first year, you end up with $13,000. The second year, $16,900. The third, $21,970. It sounds pretty lame compared to option #1, which, by the end of the third year has accumulated over $10 million. However, option #2 always grows at 30% every year while the other is constant. By year 18, option #2 will have finally made you a millionaire — and by year 50, one of the richest people in the world. What does this look like?

Option 1 is the sum of 10,000 for 356 days for x number of years. Option 2 is 10,000*(1.3)^x where x is the number of years

Now, lets zoom in to the first 40 years, and see where the two choices cross over.

We see that had you picked option #2, you would have lived a mediocre life until you got extremely wealthy, seemingly overnight. Does this happen in real life? Indeed it does. Just take a look at the wealth curve of legendary investor Warren Buffett, for example:

We see that Warren Buffett’s wealth follows the exact same trend as option #2. Over time, the power of exponentiation prevails and Mr. Buffett sees almost overnight wealth. What we don’t see here is that the linear relationship is hidden behind the exponential curve.

With the first option, what is linear is the rate of accumulation. With the second option, what is linear is the time it takes to double an amount of money. With a linear rate of accumulation, the time it takes to double becomes longer and longer as the base amount grows. With a linear time of doubling, the amount of accumulation grows at a higher and higher rate. With the first option, your money doubles once a day to begin with, then once a week, then once a year, then, once every ten years, and so on. With the second option, your money is doubled once every 3 years or so, guaranteed.

Perhaps then it would be better to rephrase the two options:

Use the 10k to purchase an investment that takes longer and longer to double over time Use the 10k to purchase an investment that doubles over a constant time period

This is illustrated through a log graph of both options:

Suddenly, option 2 becomes linear while option 1 plateaus.

Through reframing of the question, we are suddenly able to shed the bias of “wow! 10k a day”, and reach the conclusion that option 1 is an inferior investment because it does not take advantage of exponentiation.

Non-linear thinking in the real world

We are accustomed to seeing the world in linear terms. Intuitively, we think linearly: if I buy a bag of popcorn for $3, then I assume twenty bags are $60. If it takes me 20 days to get 100 Instagram followers, then I assume it’ll take 180 days to get to 1000 followers.

This is sometimes true, but far more often, it is a form of linear bias. In the real world, the current value of a variable often depends on its prior value. For example, the more Instagram followers you have, the easier it is to get followers. It is far easier to get from 18k to 19k than it is to get from zero to the first thousand. This could be the result of many factors. Maybe people are more likely to follow large accounts. Maybe the Instagram algorithm allows your content to reach more people, the more followers you have. Whatever the cause may be, the result is that followers at time x and followers at time y are not independent of one another.

In other words, we have a non-linear relationship. What is likely true in the Instagram follower example is that the doubling period is constant. if it took you 10 days to get from 50 to 100, it will probably take you 10 days to go from 100 to 200. Eventually it will slow down, of course, but in the beginning the trend will hold.

This means that for the majority of activities, an amount of effort in the beginning will produce orders of magnitude less results than the same amount in the end. Metaphorically, its not so much that climbing the mountain gets easier after each hike, but rather your steps literally become larger each time.

This realization has major real world implications. For example, whenever you are deciding to spend money you are making a trade-off between present enjoyment and future security. What is the trade off when you spend $50 on a new pair of shoes you don’t need? Your linear brain would think that $50 of enjoyment now is about $50 of enjoyment later, so your might as well spend it now. But this is actually not true! If you put $50 in the stock market, in an index fund will return on average 7% per year. This means you will see a 32 fold increase in your money in “later” money (more specifically, 50 years later). This means that if instead of buying a pair of shoes, you invest the $50, you will likely have $1,600 worth of enjoyment when you are older.

However, you will not see this money for a very long time. Indeed, in the first year of saving, you end up with $54 — and you might think to yourself, “darn, four bucks. I should have bought the shoes.”

Like how first time investors often think that their returns are measly, and how you might see a $4 trade-off on buying shoes versus saving as not worth it, in an exponential relationship the initial effort (or time) often seems insignificant and not worth it. However, it is worth knowing that the majority of the returns will be found at the end of the road. Exponential relationships are an exercise of great patience.

Practicing non-linear thinking

“Rule №1: Never lose money. Rule №2: Don’t forget rule №1” — Warren Buffett (rules for investing)

Besides the obvious fact that you are losing money, why is this the number one rule? A linear mind might think, “if I can make a million dollars, surely I can make it back?”

Yet, those who think this way are the same people who have never understood this enlightened quote by Warren Buffett. Warren thinks non-linearly. He realizes that the effort it takes to make x amount of money is proportional to the amount of money you have in the beginning. It took him 30 years to get from a million to a billion, and if he goes back to a million, it will take him another 30 years.

Indeed, to lose money is to squander the effort it took to produce returns in the beginning. Because only through growing the first 10 followers can you get to the next 100, and only through growing the first 1,000 can you get to the 100k you have always desired. If effort is rewarded non-linearly, as it is the case for investing, then the most important thing suddenly becomes protecting the fruits of your time and effort at all costs. Then, it must be that Warren Buffett realizes that losing money is the equivalent to losing time.

Perhaps we can reframe the quote: rule no. 1, never squander the time you have already spent; rule no. 2, never forget rule 1.

We can see that the reason behind Warren’s rule for investing is the exact same reason behind Robert Greene’s 48 Laws of Power. In particular, Law 5 says:

“Protect Your Reputation at All Costs.”

In this example we can see that reputation is a non-linear function, similar to followers or money. It is supremely difficult to gain a good reputation with one person. It is somewhat easier with 10. And, it is extremely easy to spread your good reputation if you have a million believers. Then, to squander this non-linear resource is to waste the time and effort it took to develop the first x number of pieces of good will.

Practice thinking non-linearly will greatly reward your future self, whether it is with reputation, or followers, health, or money. It may even make you better at making every day decisions. Consider this final example from the brilliant HBR article on linear thinking:

If you have two cars, one at 10 mpg and one at 20 mpg, and have only enough budget to change one car to become more efficient, which car do you change to maximize your savings?

Change the 10 mpg to a 20 mpg Change the 20 mpg to a 50 mpg

It turns out that despite the greater absolute degree of efficiency change in the second option, the first option saves far more money. Over 10,000 miles, you save 500 gallons in the first option and only 200 gallons in the second option.

With non-linear thinking, you can be wealthy, healthy, reputable, well-followed, and environmentally conscious. Give it a try!

All my articles are dedicated to the public domain under the terms of the Creative Commons Zero licence. Please translate, copy, excerpt, share, disseminate and otherwise spread it far and wide. You don’t need to ask me, you don’t need to tell me. Just do it!