Broker Check

Can 'Math' Be Wrong?

August 17, 2022

The proposition below has been floating around Twitter the last couple days and the replies are fascinating (probably my own fault for going into the comment section of any tweet, but nonetheless here we are).

Anyone with a basic calculator can run the numbers.  The red button has an expected value of $1m while the green button's expected value is $25m.  Anybody who has watched a bit of late night poker knows that you always take the higher expected value play in a cash game, every time.  Easy decision, right?

While mathematically correct, if this hypothetical ever appeared in reality, I believe it's Red all day for most. 

The reason?  One word - Variance. While something can be reliable in the long run, the outcomes over shorter samples can vary drastically.  The NBA's 2018 Houston Rockets were a great example of this concept.  The Rockets realized that they shot something less than 50% on 2-point shots that weren't layups (let's say 45%) and 35% (again, approximate) on 3-point attempts.  This gave them an expected value of 0.9 points on long 2-pointers and 1.05 points on 3-pointers.  As you might imagine, they leaned into only taking very short layups and 3-pointers.  They cruised to the #1 seed in the Western Conference Playoffs over their 82 game regular season.  Unfortunately, in Game 7 of the Western Conference Finals against the Golden State Warriors they missed a statistically improbable 27 consecutive 3-point attempts and lost a game in which they led by 15 at halftime.

In a full season, the math worked quite nicely for the Rockets.  In shorter runs, the results can vary.  This phenomenon impacts our risk management decisions during financial planning. For example, the stock market goes up roughly 4 out of 5 years, a good reason to be continually invested in the market to accomplish long term wealth-building goals.  But say you're buying a house in 12 months, in our rough example there is only a 20% chance the market is lower a year from now.  In that 20%, however, you potentially can't afford to purchase anymore or you have to settle for 'less' house then you wanted - both likely huge disappointments.  The other 80%, you either have a bit more money left over after your downpayment or you get to buy slightly more house.  I'd argue the life impact of that decision (to stay invested or keep the downpayment money in cash) skews significantly to the downside.  80% you win a little, 20% you lose a lot. 

Proper risk management can oftentimes seem overly conservative even when the math seems to be 'in your favor.'  For most clients, pursuing most objectives, falling meaningfully short of a goal is far more impactful than finishing meaningfully ahead and we help our clients manage as such.