Factor of Safety

Surprisingly, designing a bridge or building has similarities to constructing an investment portfolio. We’re going to discuss some engineering concepts for a bit. Bear with me, because I will connect each and every one back to the investing world.

Imagine you have to design a structural beam. To design this beam the engineer must understand the forces on the structure, select a material, and use that information to determine the shape and size of the beam.

Materials have parameters of strength, which are often shown in a “stress-strain curve“. Each material has it’s own properties and it’s own specific curve. Some have unique little quirks to them. Think of the description below as a curve for a generic metal.

Stress Strain Curve

The x-axis shows the strain, or simply the amount the material is stretching or deflecting. It’s the percent elongation in the size of the beam. The y-axis is the stress, or the force on the beam over it’s area. A sort of pressure if you will.

The chart shows the relationship between how much the material moves or deflects, and the force on the material. More deflection requires more force. This ratio — stress/strain — through the elastic zone is called Young’s modulus.

Elastic Zone-Yield Strength

The first part of this curve is called the elastic zone. Here, the curve rises fairly straightly, sometimes with a slight curve. It is “elastic” because if the force is removed, the material will return to it’s original shape. Think of pulling on a spring.

This condition exists until you reach the yield strength. The yield strength represents the highest stress you can put on a material before it deflects permanently.

Strain Hardening-Ultimate Strength

At this point, the curve bends over. The part stretches (deflects) further. If you take the force off the part here it won’t return to it’s original shape. It has been hardened into a new, often stronger condition. Think of forging metal with a hammer, or cold rolling steel.

The process continues until reaching the ultimate strength (UTS). The ultimate strength is peak strength of the material.

Necking-Fracture

After the ultimate strength, the curve turns down. Here the material begins necking. Instead of stretching uniformly, it greatly deforms one area of the material.

The material necks further and further until failure occurs and the part fractures. This area is bad news for an engineer, and should be avoided.

Potential Errors Everywhere

How do engineers use this information to design and size beams? Each material has a clearly known ultimate strength. So just match the ultimate strength with the stress on the beam? If only it was that simple.

The problem is the chart is idealized. Engineers and scientists have tested most materials thoroughly, and the chart itself is going to be nearly 100% correct. However, it has to be applied to real life where there are errors everywhere. For starters:

• What are the dimensions of the beam? Do we know them for certain? How precisely? Was the part made to those dimensions? To what tolerance?
• Is the material formed perfectly? Is it pure? Are there any voids?
• How sure are you about the stresses and forces that will be placed on the beam? Maybe they are known today, but will they change over time? If you’re building a bridge, will cars get heavier? How much heavier?

Potential errors everywhere. If you just take what you believe to be true, and size the beam for the ultimate strength, those errors could easily push the actual part into the necking/fracture zone.

Fracture.

That’s not acceptable.

Factor of Safety

So engineers can’t size anything for the ultimate strength (or the yield strength if the part must stay elastic). Ok then, just estimate the errors, and size with those errors in mind?

But how do you estimate the errors confidently? For some errors maybe this is possible. Others not as much. And what of errors you don’t even know about?

Therefore to deal with the the potential errors in the system, engineers design with a “Factor of Safety“. Instead of designing a part around the ultimate strength, they design it around a fraction of that strength.

Factor of Safety = Ultimate Stress / Designed Stress

The specific factors vary depending on the material, application, and the consequences of failure.

Buildings commonly use a factor of safety of 2.0 for each structural member. The value for buildings is relatively low because the loads are well understood and most structures are redundant. Pressure vessels use 3.5 to 4.0, automobiles use 3.0, and aircraft and spacecraft use 1.2 to 3.0 depending on the application and materials. Ductile, metallic materials tend to use the lower value while brittle materials use the higher values.

https://en.wikipedia.org/wiki/Factor_of_safety#Margin_of_safety

So as an example when using a factor of safety of 2, instead of sizing the structural components for the ultimate strength, engineers will size the beam for half (one divided by two) the ultimate strength.

The factor of safety moves the target point for sizing the structure conservatively along the curve “to the left” of the ultimate strength, ensuring the part does not fracture.

Portfolio Stress Strain Curves

You may already see the parallels here to investing. The stress strain curve maps almost perfectly to the geometric frontier. Let’s work through the comparison.

Engineered Stress-Geometric Return

The stress correlates to the geometric return.

Strain-Standard Deviation

I quite like this comparison. Many times the volatility feels like a strain placed on a “portfolio”.

Young’s Modulus-Sharpe Ratio

The Young’s modulus tells you the relationship between stress and strain, similar to the strength of a spring. The Sharpe ratio is relationship between return and volatility. It is often used to indicate the “strength” of a portfolio.

Yield Strength-Tangency Point

The yield strength is the point where the stress-strain curve starts to bend over. The material actually starts to change here. The same thing happens with the geometric frontier at the tangency point. The curve bends over and the portfolio evolves from employing cash to adjusting the ratio of the risky assets.

Ultimate Strength-Maximum Geometric Return

The ultimate strength is the maximum strength the material can provide. The maximum geometric return is the maximum return the assets can provide.

Fracture Point-Negative Returns

Push too far down the stress strain curve and the part will literally break. Push to far down the geometric frontier, and your portfolio will break too, potentially producing negative returns.

True Stress-Arithmetic Return

I really like this one.

The stress stain curves above are for engineered stress and strain. What is that?

Simply, it’s a type of stress and strain that have been converted from the true stress and strain to help engineers design functioning structures. The engineering stress-strain changes the true stress and strain of the material into one that actually represents what happens to the engineered part.²

This curve sits above the engineered stress strain curve.

The true stress strain curve is not necessarily straight for all materials, but it’s much straighter than the engineered stress strain curve. Importantly it never rolls over. There is no peak.

Similarly, the arithmetic frontier (think Markowitz) sits above the geometric frontier. And it too never rolls over. There is no peak.

Because it never rolls over, the ultimate strength does not exist on true stress strain curves. And on the arithmetic frontier, the geometric maximum return point doesn’t exist either.

The geometric return changes the arithmetic return into a return that represents the true path of the portfolio, showing a peak.

Factor of Safety-Partial Kelly

Most importantly, engineers design with a factor of safety. They don’t approach the ultimate strength with their designs because the inputs used to size those structures are full of errors.

When sizing a portfolio, similar errors exist. These errors have the same potential to cause a portfolio failure. Therefore portfolio construction should employ a factor of safety as well.³

What is the investment portfolio’s factor of safety? When looking at it from a geometric point of view, I would say the portfolio factor of safety equates to “partial Kelly”.

A “full Kelly” portfolio sizes the portfolio at the top of the curve. Most people don’t invest here. Some invest left of the peak. It’s very common to hear people recommending investing at “half Kelly”. Half Kelly is analogous to using a factor of safety of two.⁴ But partial Kelly could be any fraction, 3/4th, 1/3rd, 9/10th, or whatever you want.

Partial Kelly acts just as a factor of safety, ensuring the portfolio doesn’t end up right of the peak into a failure zone, or insane investment zone.

Now I want to bring up a big distinction here about partially Kelly investing. Most people reason using a partially Kelly portfolio because the ride is too volatile, too bumpy at full Kelly.⁵ They can’t handle the up and down swings.

I’m not using the same logic to justify a partial Kelly portfolio. I’m saying the first reason to employ a partially Kelly portfolio comes from errors in portfolio construction, not volatility reduction. It’s a different philosophy.

Use a Factor of Safety

I’d like for there to be some really deep meaning at the heart of the comparison between the engineered stress-strain curves and the geometric frontier. Some kind of “grand theory of everything” type of wisdom. I’m not sure there is.

But still the similarities are everywhere. The curves look nearly identical. Just as stress-strain curves are unique to each material, the geometric frontier is unique to each market, always changing. And when engineers build systems with stress-strain curves they:

• Recognize the danger in designing past the peak of the curve.
• Acknowledge the errors in their inputs.
• Realize these errors are really hard to estimate.
• Design the system with a factor of safety to prevent crossing into the failure zone.⁶

Every engineering concept here applies to portfolio construction. Therefore, you to should consider applying a factor of safety to your own portfolio.

I know I’ve left you hanging a bit on how to apply partially Kelly to the geometric frontier, but this got long and just means you will read the next posts when they come out.

1-This is the infamous Tacoma Narrows Bridge. It’s failure was much more complicated than the concepts discussed in this post, but it is a great image.

2-The reasoning for the difference is best understood in the necking zone. Engineering stress and strain considers the size of the part as it begins. Therefore, when the part necks down, the stress (the pressure) looks like it goes down, and the curve bends over. Within the part however the stress actually continues to increase because the area (stress = force / area) is shrinking as it necks. But this is meaningless in application as once the part begins to neck, it no longer has the required size needed to resist the full force (material stress x necked area = less force).

3-Now it’s true that the consequences of failure her are far different. People can die from engineering mistakes. First order consequences of poor portfolio sizing are simply losing money and maybe bankruptcy. Not necessarily the same.

4-Half Kelly is probably a bit more conservative than I would envision for dealing with portfolio errors. But really the proper factor of safety depends on the type of portfolio just as it depends on the type of engineering application. For example, a portfolio of penny stocks should maybe be traded at quarter Kelly due to the high likelihood of large errors.

5-I find this reasoning silly. The portfolio isn’t volatile because it’s invested at full Kelly. It’s volatile because that specific portfolio at full Kelly has high maximum geometric return and a high volatility. This is not a function of Kelly betting. It’s a function of the volatility of the portfolio. I’ve shown that a portfolio without leverage built from common investment choices to maximize geometric return is not volatile, and does not produce crazy swings. On the contrary actually. It is calmer than most other portfolio strategies because it avoids high volatility due to it’s destructive nature.

6-Another benefit, it can make calculation errors less costly.