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# Investing Lessons From the Birthday Paradox

I was recently with a group of friends, and we were astonished to learn that two pairs of us had shared birthdays – out of 10 people! We were all blown away by this and assumed that the likelihood of even one pair of us having a matching birthday would be close to zero. Naturally, we turned to Google rather than our calculators, and found that a matching birthday is not nearly as rare as you would imagine. With just 10 people, there is over an 11% chance that two people will share a birthday! Perhaps even more surprisingly, the odds that two people will share a birthday are better than a coin flip within a group of 23 people and essentially guaranteed with a random group of 57! This “phenomenon” is known as the Birthday Paradox because it seems counterintuitive. But it started to be less surprising after reading about it.

Rather than thinking about the odds of getting a specific date (out of 365) to match, we need to consider the pairs of individuals present – because each person needs to be paired up against every other person in the group to see if they share a birthday. As an example, with two people there is only one pair (and very low odds of a birthday match). But with 10 people, there are 45 pairs! The number of pairs grow so quickly that with 57 individuals there are 1,596 pairs – hence the almost certain probability of a match.

Our brains have an extremely difficult time understanding this, because probabilistic and exponential calculations are not how we usually think about the world, birthdays, or money.

But perhaps they should be – even if we cannot wrap our minds around it fully! Thomas Edison’s quote, “The strongest force in the universe is compound interest,” is surprising in the same way that the Birthday Paradox catches our mind off guard. We’ve all heard the question: Would you rather have \$1,000,000 today or a single penny that doubles in value each day for 30 days? And the answer…the power of compound interest is exhibited by the \$5,368,709.12 you would have after a month if you opted for the penny.

Getting our minds to consider the world like this can be difficult, but it is important when it comes to investing.

To me, the lessons from the Birthday Paradox can be applied to market pricing efficiency. In liquid, transparent markets, pairs consisting of buyers and sellers trade constantly based on current information and their best future expectations. If only a few trades occurred for a stock, the odds of finding the “right price” may be low. But for markets like the S&P 500, there have been between 4-10 billion trades each day over the first three months of the year according to S&P Global Market Intelligence. Everyone is trying to assess the value of those 500 stocks, every day. And while there are certainly instances where individual stocks do not seem to trade “normally,” the market as a whole is very efficient in finding the matching price to value a company. New information and even expectations of future developments (like the odds of the next inflation reading being higher or lower than expected) are constantly assimilated into the price. This makes it very difficult to consistently outsmart the market through stock picking or market timing.

If you can’t beat ‘em, join ‘em! Despite the difficulty our brains have in understanding probabilities and exponential functions, we can still use them as we participate in the stock market. Taking Edison’s advice, saving earlier and often results in greater wealth down the road as we use compound growth to our advantage. And considering market efficiency, usually the best way to invest for the long-term is through more passive investment approaches and investing excess cash right away – because while no one knows what the future holds, our collective best guess is already “priced in.” The fruits of disciplined investing are not seen every week, month, quarter, or year, but over time will serve you well.