How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg is a fascinating exploration of how mathematical principles apply to everyday life, from personal decisions to societal issues. Ellenberg, a mathematician, aims to show readers how mathematical thinking can help solve problems, avoid common fallacies, and understand the world in a deeper, clearer way. He emphasizes that mathematics isn't just a series of abstract formulas but a tool for making more rational decisions.
How Not to Be Wrong – The Power of Mathematical Thinking, shares a powerful story that we’ll look at from two perspectives.
From your perspective
Imagine receiving a newsletter from a stockbroker in Baltimore. The newsletter predicts that a certain stock is going to rise, and a week later, it does. The following week, the newsletter predicts that the price of another stock is going to fall, and sure enough, it does.
This pattern continues for ten weeks, with each prediction coming true. Eventually, you receive a solicitation from the stockbroker to invest money with him. Although there is a hefty commission, it seems like a good deal since the broker has been so accurate with his predictions.
The math may match your intuition. If we assume that each stock pick had a 50/50 chance of being correct, the odds of the stockbroker being so consistently successful are extremely low – in fact, it would be a 1 in 1,024 chance of hitting that kind of string of success.
This must be a demonstration of pure skill, or is it?
From The Stockbroker’s Perspective
From the perspective of the stockbroker, the story looks different. In the first week, the newsletter was sent to 10,240 people, half of whom received a prediction that the stock would rise (including you) and the other half received a prediction of a fall.
In the next week, the newsletter was only sent to 5,120 people, which included those who received a correct prediction of a rise. Those who received the opposite prediction were not contacted again.
This pattern continued, with the number of recipients decreasing each week for those who received correct predictions in a row. After ten weeks, only a “lucky” few, including you, received ten straight winning picks.
Of course, it is advisable to steer clear of this stockbroker. His only apparent skill seems to be manipulating people. But I suppose you can appreciate his patience and mathematical insight. While situations like this may not exist in the real world to the same extreme, companies can still employ similar tactics to deceive investors.
Ellenberg goes on to describe a common practice in the financial industry. A company can be preparing to launch a mutual fund to the public. They might maintain the fund in-house for some time before making it publicly available. This practice is called incubation.
What happens behind the scenes is that the company incubates many funds at once. This gives the company the chance to experiment with different strategies and asset allocations.
Many funds will not survive this incubation period. That’s not unexpected or undesirable. It’s having the one or few funds that have outsized returns that matters. Those can be packaged and marketed to the public. The trouble is that funds that survive the incubation process may be no more likely to outperform once public.
This all comes down to the concept of survivorship bias.
A Brief Look at Survivorship Bias
Survivorship bias is a logical error that occurs when one focuses only on the people or things that "survived" a process while ignoring those that did not, leading to skewed or overly optimistic conclusions. This bias can cause incorrect conclusions by analyzing only the successes and ignoring the failures that might offer crucial insights.
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Common Areas Where Survivorship Bias Occurs:
Business and startups: Focusing on the stories of successful companies without considering the many failures that never became well-known.
Investing: Looking at funds or stocks that have performed well over time, while ignoring those that failed or were removed from consideration.
History and war: Glorifying surviving figures or strategies without acknowledging those that perished or failed under similar conditions.
Self-help and motivation: Highlighting successful individuals who overcame odds while disregarding those who tried and failed under similar circumstances.
Survivorship bias can distort understanding and decision-making, emphasizing success over the often overlooked lessons from failure.
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