The Butterfly Effect is a poetic name to say that in some situations, small changes in initial conditions can lead to major variations of the final outcome.
It was first observed in the 1960s by an American mathematician and meteorologist named Edward Lorenz, in a quest to forecast weather using computer simulation.
As he replayed the same simulation, Lorentz decided to enter numbers with a lesser precision, certainly to save calculation time. Computers were very slow back then. Each decimal would use up significant computing power. Scientists and programmers had a strong incentive to find ways to speed up calculations.
Legend has it that Lorentz truncated a number such as 0.506127 to 0.506. That would have been like a carpenter shaving 1/40 of an inch off an 8 feet beam (0.6 mm off 2.44 meters). In most practical real-life applications, such correction has no incidence whatsoever, engineers rarely use measurements with a precision beyond 3 significant figures.
Everything is different when repeated in loop
There are instances, when calculations are repeated over and over and each calculation is starting from its previous result (this is called an iteration) then a small change in initial conditions can lead to major variations in final outcome. In a weather model, everything is recalculated by the hour so 24 times per day of forecast.
The chart below shows a sample calculation made with the same two numbers. We ran it with 2 different degrees of precision. The calculation is repeated 20 times, each time using the result of the previous round. The calculation is very simple: we take a number, calculate its square, add the constant. The result of this calculation is used to do the same thing again (take the result, calculate its square, add the constant) 30 times. The original number is 1, and the constant is respectively -1.618034 and -1.618 (the same number, rounded to 6 and 3 decimals)
In the beginning, both lines follow the same path and only the red is visible. At some point, the green line starts to show up and follow its own path.
As you can see using 3 or 6 decimals has an impact after 25 iterations, the error accumulates and make the results diverge. This with only two numbers, and two basic operators (square and addition). Imagine the consequences if it were a few dozens of formulas and data entries like weather models have.
Butterfly effect in finance ...
Compare two mortgages: same amount, same duration, slightly different rates:
$400,000 borrowed for 30 years at respectively 6% and 5.5%. What that zero point five percent can do? Over the life of the loan, one loan would cost $45,800 more.
... and in real life
That is what the butterfly effect is all about. A tiny difference that in itself seems to have zero or little impact, can have dramatic consequences after several iterations of a repetitive process. This phenomenon is observable in many domains in real life, like how bad a document can become after successive copies or how being consistently late a few minutes makes someone fired one day.
Same is true with good habits and how having higher standards - demanding a little bit more of oneself on a regular basis- can lead to a higher life overall.