The Earliest Advocate for the Geometric Average

As a side effect of writing publicly, when I find others work quite similar to my own I often wonder if they were influenced by one of my posts. Normally I assume it’s probably a coincidence.  But sometimes the topic is so unusual–and an approach not often found in investing– that the coincidence seems less likely. 

I got this feeling when reading Mark Spitznagel’s new book Safe Haven.

Safe Haven

Mark Spitznagel is a famous hedge fund manager specializing in tail risk black swan type funds.  He built his fund, Universa, working with Nassim Taleb, and has produced quite spectacular results since its founding in the late 00’s.  This year he published his second book discussing his investment philosophy.

Many parts of the book had déjà vu characteristic for me and my blog.  There are discussions on random chance (he went with dice instead of coins), failures of traditional portfolio construction, and talk of expected returns. He is a huge advocate for the importance of the geometric average. He even threw in a reference to “The Road Not Taken”, although in a different framing than my own. But what struck me the most was he dedicated an entire chapter to dissecting Daniel Bernoulli’s paper on expected returns and the measurement of risk.

Tying a 300 year old paper, which most attribute to economics (not finance), to investment strategies is not something you see every day. The paper is famous for introducing utility theory, not as a treatise on how to invest. Most finance articles that quote Bernoulli do so to talk about personalized investor risk tolerance and risk aversion. Spitznagel came with very different approach, ignoring the traditional view of the paper and instead focusing on the geometric return.

Last December I put together a long twitter thread on Bernoulli as a sort of side project.  It’s one of my most read twitter threads. So when I saw his book spend 28 pages on the same esoteric topic, detailing Daniel Bernoulli’s hidden manifesto on the importance of the geometric return (not personal “utility”), and how it applies to modern investing, I took note.

Mark has a slightly different flavor to his analysis of the paper than I do, but at their essence the take-a-ways are the same: Bernoulli was writing about the superiority of the geometric return.  I actually constructed the twitter thread into a blog post, but for some reason, never finished editing it.  But if Mr. Spitznagel found the topic important enough for his book, then I might as well finish it and put it on Breaking the Market. 

It probably is a coincidence that we ended up writing about the same thing in a similar way. When you start looking at the world through the geometric return, and Mark certainly views it that way, you perceive things differently. But who knows, maybe he’s a reader of the blog just like you.

Bernoulli Didn’t Care About Utility, He Cared About the Expected Value

I recently re-read Bernoulli’s 1738 paper which is the foundational paper of Expected Utility Theory.

It’s amazing.

Interestingly it’s wildly different than Expected Utility Theory–the foundation of modern economics that this paper supposedly started– so much so that it’s hard to believe this was its beginning. Let’s see if you agree.

Who was Daniel Bernoulli?

Daniel Bernoulli was one of the greatest mathematicians, physicists and thinkers who ever lived. I first learned about him in my college fluid mechanics classes. He wrote Hydrodynamica, essentially inventing the field of fluid mechanics. As an engineer I routinely use his equations to design and evaluate piping systems. The concepts from this book were expanded into evaluating the aerodynamics of airplanes, as well as forming the basis for the kinetic theory of gases.

He accomplished an enormous amount with his life. On the side, he dabbled in probability theory.¹ This side hobby is what lead him to write “Exposition of a New Theory on the Measurement of Risk” in 1738. Future economists latched on to this paper in creating modern day economic utility theory.

What is the Expected Value?

However, the paper isn’t about utility. It’s about expected value. Up until this point in history, people used the arithmetic average as the expected future outcome for a risky proposition. In this way the arithmetic average became known as the “expected value”.

Bernoulli thought this was wrong. But he needed a way to explain to the world why the arithmetic average was not the expected value.

Therefore, Bernoulli developed the concept of utility to get the reader to abandon the traditional view of expected value (arithmetic average). He then used utility as an intermediate step to derive the proper equation for managing risk. In doing so, he created a new method to measure risk that doesn’t use utility at all.

Part 1: The Arithmetic Average isn’t Expected

Bernoulli starts the paper confirming that traditional risk evaluation comes from expected values, calculated from the arithmetic average.  Notice the rule here in italics is about expected values. 

Then he points out that this average ignores anything about the specific financial circumstances of the participant. Simply put, someone with little wealth will value a gamble differently than someone with great wealth.

Therefore the orginal rule for determining expected values must not be correct.  An item is valued by its usefulness (utility) therefore the expected value must also reflect the usefulness it yields. 

The paper was originally in Latin, so while I’m not sure of the translation, I love the word yields here, as it implies return on something.

I also love that Bernoulli noticed a disconnect between using the arithmetic average as the expected value and how people act in the real world.  Instead of calling people flawed because they clearly don’t follow it’s logic, he chose to call the theory flawed instead. 

As stated above, the arithmetic average was entrenched as the foundation of decision making at the time. So Bernoulli chose to use the concept of utility (which he called “emolumentum” in Latin)² to divorce people of this belief that the arithmetic is “expected”. What matters isn’t price, but growth.

Part 2: Growth is Utility

After diving into some extreme examples of how the arithmetic average clearly doesn’t work, he returns from this side discussion, stating we should focus on what’s typical for most people. He says that most people find usefulness from money in inverse proportion to their current wealth.  As in: usefulness = new money / wealth. 

After running off on another tangent about income, he returns to the previous point to double down on it.

New money / wealth is simply growth.  Bernoulli is saying we derive usefulness from growth, not absolute price.³

This is his foundational point. Nearly everybody will derive the usefulness of a risky proposition from the growth it provides them.

Part 3: Building the New Expected Value Model

So now that the hypothesis is clear–usefulness comes from growth, not absolute gains–he begins to derive a formula for measuring growth.  He starts out with a generic “utility curve”.  It’s clearly a logarithm, but it’s not stated or derived yet.  He’s just dropping logical constraints now.

The x axis is real world wealth.  The Y axis is utility.  B is the current wealth. 

Bernoulli finds the “utility” of each outcome and averages them. The PO line is this average, from the utility of 4 possible outcomes GC, DH, EL, and FM.  This is the point where EUT often stops. 

But Bernoulli aims to then translate P back to real world wealth at O.  He went into utility space, did some math, and then left utility, returning to the real world.

Next he calculates the “expected gain” by subtracting this value O from the original value B.⁴

He’s saying to go into the utility function (which he hasn’t made clear is a logarithm yet), average the results, then leave utility to calculate a new expected value. Bernoulli doesn’t care about “utility”, he’s only using it to calculate the correct expected value in our world!

Part 4: Deriving the Formula

Bernoulli then maps the current world’s beliefs of expectation risk measurement into his chart method.  Remember, he hasn’t yet said the chart should be a logarithm. It could be anything.  So he shows that for the expected value to equal the arithmetic average, the curve isn’t a curve but a straight line. He still works back to real world values (BP).

Notice he ends with “which is in conformity with the usually accepted rule”. He’s re-iterating this is how people currently believe gains should be evaluated (many still do). But as he diligently explained above, this view of the world is wrong.

The paper next returns back the key hypothesis:

usefulness of gains is inversely proportional to current wealth

and derives what this curve looks like.  This is where he shows his hypothesis leads to a logarithmic utility curve. 

He then takes this logarithm curve and goes back to his original chart method of determining the expected value, combines them, and simplifies, producing his final equation and ultimate solution.

This equation “AP = ….” is the equation for the geometric average. 

Part 5: The Geometric Average is the Expected Value

It’s eye opening that Bernoulli’s final equation doesn’t include utility at all. It’s been simplified out. Furthermore, the equation is meant to find the value of the risky proposition. It’s supposed to convey the risk’s worth in actual real dollars, not an imaginary utility value.

It’s also eye opening that his ultimate rule compares the geometric return of the potential outcomes to the original wealth. He has replaced the arithmetic return with the geometric return. 

The geometric average indicates the value of the risky proposition.

Bernoulli only used utility to convince people they were valuing risk incorrectly, and then to derive the final solution that does value risk correctly. 

But utility isn’t part of his final solution, it was just a tool to get there.

The solution Bernoulli found is the power of the geometric average.

Part 6: Utilizing the Geometric Average

The paper then tackles multiple examples, showing ways of employing the geometric average to value risk.  First, he works through an example of a “fair bet”. Risk 50 ducats to gain 50 ducats has an expected loss of 13 ducats.

Start with 100 ducats.    Sqrt(50 x 150) = 86.6 ducats.      86.6-100 = -13.4 ducats

And now he takes a jab at those looking for arithmetically fair games. I love that he calls those using the arithmetic return to evaluate fair bets irrational. I wonder if economists realize one of the foundational documents of their profession says they are the irrational ones for valuing propositions with the arithmetic average, not their subjects that they classify as risk averse. 

After the irrationality jab, he solves for the properties of a true, geometric “fair game”.  It takes a bit of rearranging, but this formula — x = alpha*a /(alpha+a) — matches a version of the Kelly Criterion discovered 200 years later by John Kelly.

Part 7: Investment Insurance

Then he tackles a much more complicated problem of shipping insurance.  His solution is entirely about comparing geometric averages. Not utility.  He calls the averages the expectation.  This is a “size of the bet calculation”, once again similar to Kelly.

(side note, Mark Sptiznagle goes into this section very deeply, which is not surprising as his preferred method of investing is to control risk by purchasing out of the money puts, which act like insurance contracts on your portfolio.)

There are people today who think insurance is irrational because it’s expected value is negative. They come up with all sorts of behavioral reasons why people buy insurance, when Bernoulli laid the math out clearly in the 1700’s.

Part 8: Diversification

Next he explores the usefulness of diversification, calculating the benefit of diversification with the geometric average, and calling the calculation his expectation.

Bernoulli 300 years ago may have been the first person to prove, and quantify, the benefits of diversification.

(Side note: Spitznagel thinks pirates would still get both ships, which is why he prefers insurance).

Part 9: The St. Petersburg Paradox

Finally to the St. Petersburg Paradox. Funny how the key puzzle this paper is known for comes last, and could be dropped entirely with no effect to the overall message. Bernoulli of course solves the paradox with the geometric average, and interestingly points out that the value of something you own is different than something you purchase. 

Solving the Endowment Effect

Bernoulli’s math says the amount the buyer ought to pay for something is lower than amount it is worth to them if they already own it. This a very interesting observation that goes against all traditional thinking of rational behavior in of modern economics. Economics thinks that you should value something you already own the same as if you didn’t already own it.

But people actually do value things they own more than something they don’t. Studies show people resist selling items they already own, even though they wouldn’t buy the same item for the same price. Economists have called this behavior the endowment effect, and have labeled it as an irrational bias.

Yet here we are 300 years earlier and Bernoulli uses the math of compound growth and the geometric average to show why people should behave this way. The endowment effect isn’t a bias, it’s logical, correct, and rational.⁵

Bernoulli Was Way Ahead of His Time

This paper is just spectacular. Bernoulli was way ahead of his time, and he may even still be ahead of our time. I think Mark Spitznagel would agree, but you should read Safe Haven to see for yourself.⁶

Bernoulli’s core tenat is:

The geometric average, not the arithmetic average, is the expected value and decisions involving risk should be judged from this value.

So why do economists use utility, when Bernoulli condensed it out of his equation and didn’t use it to solve problems? Why do they measure risk aversion from the arithmetic average?  Why does everyone continue to use the arithmetic average as the expected value? 

Do you think about your own investment portfolio in terms of arithmetic return or geometric return? Are you valuing risk from the geometric return? If not, maybe you should listen to one of the greatest minds ever, and measure risk through the power of the geometric average.

The full is paper here Bernoulli.pdf (ucsb.edu)

Footnotes

1-I do mean on the side as he wrote “I would elaborate it into a complete theory, as has been done with the traditional analysis, were it not that, despite it’s usefulness and originality, previous obligations do not permit me to undertake this task.”

2-Modern translation of emolumentum into English seems to often be “gain” or “advantage”, not utility.

3-To give you an analogy from today’s world, imagine Bernoulli surrounded by people quoting Dow moves in points (arithmetic return).  This drove him nuts and his response was, “nobody gives a shit about points, you should quote it in percent gain.”

4-Some have said Bernoulli made a mistake in this section. I find this interesting as, one, Daniel Bernoulli was one of the smartest people to ever walk this earth, and two, they are applying modern thoughts on Economic Utility Theory from the 1900’s to a document that pre-dates their field by two hundred years. What Bernoulli is doing here is determining the value of something he already owns, not something he want’s to purchase. So in short, Bernoulli didn’t make a mistake. More on this later.

5-Let’s work through this in an example: Say you have $200, and you already own the outcome of a proposition that will either pay you $100 or $200. What would you sell for? Well ($300 * $400)^0.5 – $200= $146.4. This is what the gamble is worth to you and it’s what you should accept from someone who wants to pay you for the outcome.

Now how much would you pay to own this gamble? This is how most economists look at the problem. This equation is (($300-X)*($400-X))^0.5=$200. X = $143.8.

$146.6 is more than $143.8, hence you value the risky proposition you own more than you would pay for a risky proposition you don’t already possess. No bias. No irrationality. Just the math of the geometric return.

6-OK, fine a couple of screenshots of the book here.