The Most Misunderstood Force in the Universe

I once believed Einstein called compound interest the most misunderstood concept in the world.  He didn’t.  Google claims he actually said, “compound interest is the most powerful force in the universe”.¹ Well, I’m going to build on Einstein’s supposed quote and state that: 

“Random compound interest is the most misunderstood force in the universe.”

The Geometric Average

The geometric average defines random compound interest.  The geometric average is a higher order average than the more commonly used arithmetic average.  The arithmetic average uses addition to combine a group of numbers, and then divides by the quantity of those numbers.

(1+2+3+4+5)/5 = 3

The geometric average works at one order higher, combining a group of numbers through multiplication and then rooting those numbers by their quantity.

(1 x 2 x 3 x 4 x 5)^1/5 = 2.6

Compound interest works through repeated multiplication, therefore, the geometric, not the arithmetic, average applies. Most investors don’t seem to understand this.

Study of Geometric Concepts

In Math Games I gave examples of how the concept can confuse people.  We’re going to build on that foundation here to make sure you, the reader fully grasps this concept. 

Concept #1.

Imagine a bet that goes up by 10% and then falls by 10%.  How much are you left with?  One percent less than you started with:

110% X 90% = 99%

Our mind likes addition.  It feels odd to go forward and backwards the same amount, yet end up further behind.  But that’s exactly what happens when you compound varying numbers.

Concept #2

If you lose 50% of your wealth, it takes a doubling of your wealth to get back to whole.

50% X 200% = 100%

Once again this feels odd.  If feels like this trade off should work to your advantage, but it doesn’t.

Concept #3

Winning 25% twice outperforms wining 50% once followed by breaking even (0%).

125% X 125% = 156%

150% X 100% = 150%

Here we see stable returns work out better in the long run than variable returns.  In “Math Games” we dealt with games that had the potential for loss.  This example illustrates that the math doesn’t care if there are losses or gains.   It’s the existence of variation in the compounding returns that matters.

S&P 500’s Geometric Average.

The S&P 500 averaged 5.13% return per year (excluding dividends) over the last 20 years.  Yet nobody received that return.  Investors actually received the geometric return of 3.63% per year, commonly called the compound annual growth rate (CAGR).  The concepts illustrated above explain this discrepancy.  The days with losses cause more harm than the equivalent days with gains provide a benefit (concept #1 and #2).  Smoother returns generate better overall results than volatile ones (concept #3).  The investment industry calls this phenomenon volatility drag.

The equation for equating the arithmetic average, geometric average and volatility for a normal distributions is:

Arithmetic Average – Volatility² / 2 = Geometric Average

We can observe two key factors from this equation:

1.  The geometric average is always less than the arithmetic average
2. The “volatility drag” grows exponentially when volatility increases  

Therefore, an investment strategy should strive to contain volatility as much as possible, and especially to stay far away from extreme levels of volatility.  Volatilities very existence reduces the expected performance of every single portfolio.

In the case of the S&P 500 over the last 20 years you find:

Arithmetic Average – Volatility² / 2 = Geometric Average

5.13%      -–    17.08%²  /  2   =        3.67%

Which mirrors the actual geometric average of 3.63%.  You can’t escape the geometric average.

Repetition

If you play a game once, you will expect returns in line with the arithmetic average.  The expectation in “Math Games” of a single play is 5% profit.  However with every repetition, the expected return degrades towards the geometric average.  Below is a chart showing the expected compounded return of a game, depending on the number of games played.  I used Game #3 described in Math Games as the repeated game.  As a reminder the rules of the game are:

Bet $100.  Flip a coin, heads you win 50% of the bet (receive $150 back, for a gain of $50), tails you lose 40% of your money (get $60 back, for a loss of $40).  The game repeats with the entire payout of the previous round until I decide to stop the game.

With each repetition the expected compounded return degrades slowly towards the geometric average.  The fewer repetitions, the better the expected result.  If the game was more complicated, like say stock market returns, the transition from arithmetic to geometric return takes much longer than a handful of plays as in this example.  But given enough plays, it will always degrade toward the geometric average.  This is very key.  With enough repetition, every game will degrade to its geometric average.

You only get one chance to play.

Even though you can clearly see above evidence of convergence to the geometric average in the market,  many people have a really hard time grasping the geometric return’s superiority.  An enormous amount of financial research ignores the geometric average.  The financial community has a hard time looking past the arithmetic average of these games since arithmetic average actually never changes with repetition.  If you could play many games simultaneously, the geometric average wouldn’t matter.  But there are only so many simultaneous games available.  There is a distinct mathematical difference between playing multiple games simultaneously vs. playing one game repeatedly. ²   Consequently, you are never going to receive the arithmetic return.

To understand this, lets re-visit the table of 20 repetitions from Math Games.

There are only 20 repetitions.  But 20 repetitions is 2²⁰ = 1,048,576 different unique outcomes.  Play 1,000,000 games simultaneously, and you can expect to achieve the arithmetic average. Of course that’s impossible.  There are limits to the number of simultaneous games.  The arithmetic average is a pipe dream.  A lottery ticket.  You should never pay attention to it. The geometric average determines your fate and is all that matters in investment strategy.

Yet, the investment community is infatuated with arithmetic returns often ignoring the expected geometric returns entirely. Investing research commonly touts an amazing strategy with a glorious back test that ultimately fails to live up to the hype in the real world.  Check the “research”. Its likely the back test was reporting arithmetic returns.  Real life returns are geometric , hence the under performance and my statement:

“Random compound interest is the most misunderstood force in the universe.”

1-Plenty of people believe this quote is an urban legend. But it does sound good.

2-This is best explained through Nassim Taleb’s Russian roulette example.  If six people play Russian roulette at the same time, one will likely die.  You personally will have an 83% chance of living.  If you play all 6 games yourself back to back, I think you can figure out the likely outcome.  The arithmetic average deals with the first version of roulette.  The geometric average, and all things in life, deal with the second.