The Road Not Taken

You have a journey to make.  You need to head west 15 miles as the crow flies over a mountain ridge, and then go 15 miles down the valley on the other side. So you head out for a drive.

The start of the trip entails climbing up and over the ridge, and is full of back and forth switchbacks.  You’re driving 30 mph complying with the speed limit as you climb up and over. The road is curvy and it’s fun to drive.  But it’s clear as you climb the mountain you aren’t actually going very far.  Much of the driving is wasted on the back and forth motion in the swtichbacks. 

Same is true as you descend down the other side of the ridge.  Now you can see the destination in the distance, but even though the minutes go by it’s not really growing closer at the rate your speed would lend you to believe.

Finally, you make it to the valley.  The road isn’t perfectly straight here. There are still a few curves, but they are much softer. You could safely go faster, but for some reason the speed limit is still 30 mph.  Even so, you’re gaining on the destination now. Each time you crest over small hills, you can see the goal approaching faster and faster in the distance. 

The Straighter Path is Faster

Everyone has taken a similar drive before.

Driving through the mountains may be fun and scenic, but is frustrating if you’re actually trying to get somewhere.  The contrast  between the back and forth travel through the slopes and the much more direct path in the flatlands is stark.  The switchbacks are also mentally taxing if you’re not up for it.  It’s hard constantly driving around corners, with limited visibility on narrow roadways. 

If time was a priority and you had to choose, everyone would prefer the straighter path.

The Math of the Switchbacks.

The problem with the mountain roads is much of the distance traveled by the car is wasted going in the wrong direction.  You could model your path through the mountains as a ziz-zag. 

I’m going to assume most people know the Pythagorean theorem explaining the relationship between three sides of a triangle.  Here the car is traveling on the “C” leg of triangle, and the “B” leg is the width of the switchback.  The “A” leg of the triangle is how much closer you actually move to the final destination. 

Looking at a map, you see that the road has you drive 2 tenths of a mile before switching back (“C”), and that each run takes you 1.73 tenths of a mile sideways (“B”).  Using the Pythagorean theorem you see that during this drive, you’re only gaining on the destination by 1 tenth of a mile (“A”):

2² – 1.73² = 1²

The speed in the direction you care about, “A”, is half the speed you are actually traveling. Even though the car is going 30 mph, you’re only gaining on the final destination at 15 miles per hour. Therefore it takes a full hour to get through the mountains. 

Math of the Valley

But in the valley, the curves are much smaller.  Down here the corners are still every 2 tenths of a mile (“C” leg) , but now, the “B” leg is only 1 tenth of a mile wide.  The “A” leg is much longer at 1.73 tenth miles.  This means the car now travels 26 mph towards its destination, traversing the valley in 34 minutes. 

2² – 1² = 1.73²

I Thought This Was an Investing Blog?

Maybe you’ve recognized the relationship already?

I write about the geometric return’s superiority in investing, and how volatility eats away at this return. The car’s velocity is constantly changing directions going through the switchbacks. Its direction is volatile. Therefore it’s not approaching the true objective as fast as one would hope.

The volatility of the car’s heading means that the realized traveling velocity does not match the speed of the car itself. “The heading volatility” is eating away at the actual progress of the journey.

Now let’s go one step deeper.

The equation I’ve used for determining the geometric average is an approximation, but one that is nearly perfect when the arithmetic return is small (near 1).

Geometric Average = Arithmetic Average – Volatility² / 2

A more accurate equation¹ is this:

Geometric Average = √(Arithmetic Average² – Volatility² )

Which when rearranged is equivalent to this:

Geometric Average² + Volatility² = Arithmetic Average²

Wait a minute.  That’s the Pythagorean theorem!

Yes it is.  The true equation for the geometric return is really just the Pythagorean theorem applied to growth rates. 

The arithmetic return equates to the car’s actual speed in whatever direction it is heading.  The volatility relates to the car’s speed perpendicular to the end goal.  And the geometric return is the velocity in the direction of the true destination.

I’ve said before the geometric return is all that matters.  This is why.  Some of the arithmetic return is wasted going back and forth. 

The geometric return is therefore the “speed” you truly experience toward your destination of increased wealth. 

Unlike Roads, You Pick Your Investing Path

The real difference in this analogy: when driving, you do not get to design the road. But you do get to choose your own path through the stock market. You can design the road.

You don’t have to take a curvy road and experience all the ups and downs of volatile investments. Sometime the terrain gets a little bumpy, like it did in February and March, but you can still find gentle curves through those mountains. There are straighter roads available.³

Sure you can try and drive 80 mph on a windy road. But if you can take a straighter road at 60 mph to get to the same place, in the same amount of time, with less stress, wouldn’t you?⁴

Eighty mph through the curves would be fun from time to time.  But it’s dangerous if repeated. There might be a crash along the way.  It’s also going to be mentally taxing and stressful.  However straighter roads are there.  Everyone may not know about them, or understand why they provide a better journey, but they are there waiting to be taken.

The Road Less Traveled

When it comes to investing,
Lots of people take the curvy road.
Lots of people are told to take the curvy road.
Many have the curvy road taken for them.

Two roads diverged in a wood, and I—

I took the one less traveled by,

And that has made all the difference.

From “The Road Not Taken” by Robert Frost

Which investing road will you take?

1-Three things: Some could say the correct equation is (1+ arithmetic return)². This depends on the view of the the arithmetic return as 10% or 1.1. From a pure math view it is probably more correct to say its (1+10%)², especially if you are thinking about continuous compounding using e^{t * 10%} . But I prefer to think of it as 1.1 x 1.1 x 1.1, etc., which lends itself to what I wrote.

Second, the foundation of this equation comes from the following: Imagine there was a coin flip proving a 20% return when you win, and no return when you lose. The arithmetic return is 1.1 and the standard deviation is 0.1. A win returns the arithmetic return + the standard deviation. The loss is the arithmetic return – the standard deviation. A full cycle of these two outcomes multiplies them together:

Third, this equation is useless for highly skewed distributions. Thankfully most investments are fairly symmetrical in their returns.

3-There are plenty other strategies than my own that work by reducing volatility to increase the geometric return. 60/40 operates simply under the same principle. So does the permanent portfolio. And therer are a handful of fancy, more complicated, strategies are designed around the same principle.

4-You can drive faster more safely on straight roads. If speed is really your desire, pick the less volatile path and use leverage.