I just re-read Bernoulli’s 1738 paper “Exposition of a New Theory on the Measurement of Risk” which is the foundational paper of Expected Utility Theory.

It’s Amazing

It’s so wildly different than EUT that its hard to believe this was its beginning.

Let’s see if you agree.
The paper isn't about utility. It’s about expected value.

Bernoulli used the utility concept to get the reader to abandon the traditional view of expected value(arithmetic average), and then used it to derive the equation for valuing risk.

The final equation doesn’t use utility
He starts out the paper identifying that tradition evaluation of risk come from expected values, which are calculated with the arithmetic average.

Notice the rule here in italics is about expected values.
Then points out that this average ignores anything about the specific financial circumstances of the participant. It’s not a risk aversion point, but that a poor person values money differently than a rich person.
Therefore the 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.

This was originally Latin, so not sure of the translation, but I love the word yields.
I also love that Bernoulli noticed a disconnect between using the arithmetic average as the expected value and how the world really works.

Instead of calling people flawed for not following it, he chose to call the theory flawed instead.
The idea of the arithmetic average being expected and the foundation of rational decisions was pretty well entrenched at the time.

So he has to use the concept of utility here to break people of this belief that the arithmetic was expected.
Essentially explaining that people should have different usefulness for the money.

He runs through various examples to convince people that the arithmetic average can’t possibly describe the true risk, outlining some extreme cases to emphasize the flaw in the arithmetic average.
After he does that though, he returns to normal and say, most of these examples were unusual. We should focus instead on what’s typical, which he says is that people find usefulness from money in proportion to their wealth.

As in usefulness = money / wealth.
Then he runs off on a tangent about income, and then returns to the previous point to double down on it, repeating it nearly identically.

Money/wealth is simply growth.

So he’s saying we derive usefulness from growth, not absolute price.

This is his foundational point.
To give you an analogy from today’s world, Imagine Bernoulli surrounded by a everyone quoting Dow moves in points.

This drove him nuts and his response was, “nobody cares about points, you should quote it in percent gain.”
So now that the hypothesis is clear--usefulness comes from growth, not absolute gains--he begins to derive the formula. He starts out with a generic “utility curve”. It’s clearly a logarithm, but thats 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 averages this mythical utility. The PO line is this average, from the utility of 4 possible outcomes GC, DH, EL, and FM. This is where EUT starts comparing things
But Bernoulli aims to then to 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.
He says 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 calculate the correct expected value in our world!
Bernoulli then maps the current world’s beliefs of expectation into his chart method. Remember, he hasn’t yet said it’s a logarithm.

So he shows that in the arithmetic view of expected values, the curve isn’t a curve but a straight line. He still works back to real values(BP).
The paper next returns back the key hypnosis:

Usefulness of gains is inversely proportional to current wealth

and derives the equation for this curve.

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 expected value from utility, combines them, and simplifies.

Producing his final equation and ultimate solution for a new expected value.

It is a geometric average.
Its eye opening that Bernoulli’s final equation doesn’t include utility at all. Its 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 wrong, and then as necessary to derive the final solution.

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

The solution is the geometric average.
The paper then tackles examples and shows some interesting finding from employing the geometric average to value risk.

First, he shows that “fair bet” of risk 50 to gain 50 has an expected loss of 13.

Start with 100. Sqrt(50 x 150) = 86.6. 100-86.6 =13.4
I love that he then calls those using the arithmetic return to evaluate fair bets irrational. I wonder if most economists realize one of the foundation documents of their professions says they are the irrational ones, not their subjects that they classify as risk averse.
After the irrationality jab, he solves for the properties of a “fair game” for his new view of the expected gain and loss. It takes a bit or rearranging, but this formula matches the Kelly Criterion discovered 200 years later.
Then he tackles a much more complicated problem of shipping insurance.

His solution is entirely about comparing geometric averages. No utility.

He calls the averages the expectation. This is a “size of the bet calculation”, once again similar to Kelly.
Next he explores the usefulness of diversification, calculating the benefit of diversification with the geometric average, and calling the calculation his expectation.

300 years go Bernoulli proved, and quantified, the diversification benefit.
Finally to the St. Petersburg Paradox. Funny how the key puzzle is last, and could be dropped entirely with no effect.

He of course solves the paradox with the geometric average, and interestingly points out the value of something you own is different than something to purchase
This paper is just spectacular, and so much of it is ignored. Its essential point 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?
Will it be 3 more centuries until everyone realizes to expect the geometric average?

Full is paper here.
Bernoulli.pdf (ucsb.edu)

Read it, and then read economist description of what they think he said. See if you agree Bernoulli prescribed a different solution

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More from @breakingthemark

14 Dec
If everyone is society optimized for arithmetic return, or linear utility, then society would grow wonderfully at first. Society's geometric return would be high. Some people would win big, some would lose big, and the average would be good because many are involved.
Through time though, many people would get unlucky by losing a few times in row and would fall out of contributing because they don’t have much capital/resources/access any longer to help. So now the number of contributors to society’s growth is smaller.
If people keep basing decisions on linear utility, with fewer and fewer winners each round and more and more losers, a funny thing starts to happen. Probability says society stops growing as more people fall out of the game, leaving fewer people capable of creating growth.
Read 7 tweets
4 Jul
There are two side to the Kelly Criterion which I think often get equated as the same when they really are not.

Traditional Kelly betting is about limiting your exposure to a risky bet. The bet in question is usually a "bet" in that when you lose, you lose everything you expose.
So you scale back and don't risk everything. Most casino games fit this description as do some financial instruments like options.

The optimal leverage here is less than 1. You want to hold cash on the side to buffer the future losses.
But standard investment assets, don't work this way.

I showed here, that individual stocks are effectively full Kelly bets.

Just buying one stock is the appropriate "size", as they have an optimal leverage of 1.

Read 7 tweets
1 Jun
Lots of tail hedge articles these days. I feel many miss the point. They keep studying returns as if they add with each other through time. They don’t. The math of lose 3% in 9 calm years, make 25% in the one volatile year = -5% return is meaningless.
The average through time is meaningless because investment returns don’t ADD. They COMPOUND.

A tail hedge that reduce the average return of a portfolio (as a tail hedge often does) but reduces the portfolio variance by more than twice as much, leads to higher geometric returns.
Now is this really complicated and difficult to get right with options based tail risk hedges? Yes. There are so many ways to implement it and returns are skewed and convex. And if you don’t understand why tail hedges are useful, you could easily butcher the implementation.
Read 8 tweets
21 May
I’ve really enjoyed the Asness-Taleb feud. Some of the best parts are the comments by the people supporting their “guy”. I’m drawn to the similarity of their views:

-Both sides think they are the counterpuncher. Both sides think they were attacked first.
-Both sides think the other’s intellectual prowess is overrated.

-Both sides think their investment strategy is superior.

-Both sides think the other often acts like a bully.

-Both sides think the other is acting unhinged and triggered in their response.
-Both sides think the other often gets very angry and blocks people.

-Both sides think their “guy” is making clear obvious points.

-Both sides think the followers of the other are brainwashed, but are slowly coming around to the truth.
Read 4 tweets
20 Apr
A year ago today I started reading @ole_b_peters and @alex_adamou 's ergodicity economic lecture notes.

They were so good I finished it by the end of the next day.

There's lots of math, but as I've said before, this stuff is going to change the world.

My blog is about trying to create the best investment strategy, and isn’t EE centric. But the concepts I use are very similar to those EE discusses. I’m an engineer, so I’m focused more on application than pure theory.

Many posts come from similar concepts as EE does. This post on stochastic efficient is the most similar, as it’s my proof that EE’s work on the subject is correct.

Read 14 tweets
11 Dec 19
Let’s tweak the game a tiny bit: Flip a coin. Heads, Double money. Tails, lose 51%

Arithmetic return=24.5%

Geometric return= -1%

100 coin flips through time. Now the values trend downward toward 0.

Arithmetic mean, the dark blue straight line, leaves the picture early on.
Here is the average wealth of the 100 coin flips. It climbs, and climbs quite aggressively due to the positive arithmetic return. But the wealth ends up peaking, and then falling back down to where it started. The portfolio can’t escape the negative geometric return.
Three more random trials. These coins are a great examples of news traveling slowly between the coins. Early on some coins don’t realize others are landing heads. Once they do, they shoot up. Then later the coins hesitate to realize they are supposed to start losing money.
Read 7 tweets

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