A Sexist Gives A Math Lesson
Since leaving my trading career nearly 18 months ago I’ve allowed myself a lot of space to explore. I’ve been wary of narrowing my focus prematurely. However, the last few months have started revealing opportunities that warrant deeper dives. I’ll discuss them when it makes sense to. In the meantime, I felt compelled to write about the meta-framework that has helped me filter opportunities.
It’s a long post (it might need to be split into 2 posts for your sake) that is also personal. But I used the personal stuff as a way to move from the concrete to the abstract1. I saw how to generalize my thinking so you can map it to your own needs. It took a long time for me to see the framework because recent experiences revealed it, instead of me just conjuring it (which I hope means it’s more durable). For those that are in the mental place to receive the post (slight condolences for being in that place — there’s a good chance you’re agitated), I hope it helps.
But, there’s one problem. The dog ate my homework.
The post is only 85% done and still unedited. I’m traveling, then camping most of the next week so inshallah I get it to you by next weekend.
Instead, I give you this.
How The Need For Coherence Drives Us Mad (5 min read)
There’s no point in describing this post. Be careful.
Let’s use this section to learn a math concept.
We begin with a question:
You drive to the store and back. The store is 50 miles away. You drive 50 mph to the store and 100 mph coming back. What’s your average speed in MPH for the trip?
[Space to think about the problem]
[If you think the answer is 75 there are 2 problems worth pointing out. One of them is you have the wrong answer.]
[The other is that 75 is the obvious gut response, but since I’m asking this question, you should know that’s not the answer. If it’s not the answer that should clue you in to think harder about the question.]
[You’re trying harder, right?]
[Ok, let’s get on with this]
The answer is 66.67 MPH
If you drive 50 MPH to a store 50 miles away, then it took 60 minutes to go one way.
If you drive 100 MPH on the way back you will return home in half the time or 30 minutes.
You drove 100 miles in 1.5 hours or 66.67 MPH
Congratulations, you are on the way to learning about another type of average or mean.
You likely already know about 2 of the other so-called Pythagorean means.
Simple average. Used when trying to find a measure of central tendency in a set of values that are added together.
The geometric mean or geometric average is a measure of central tendency for a set of values that are multiplied together. One of the most common examples is compounding. Returns and growth rates are just fractions multiplied together. So if you have 10% growth then 25% growth you compute:
1 x 1.10 x 1.25 = 1.375
If you computed the arithmetic mean of the growth rates you’d get 17.5% (the average of 10% and 25%).
The geometric mean however answers the question “what is the average growth rate I would need to multiply each period by to arrive at the final return of 1.375?”
In this case, there are 2 periods.
To solve we do the inverse of the multiplication by taking the root of the number of periods or 1.375^1/2 - 1 = 17.26%
We can check that 17.26% is in fact the CAGR or compound average growth rate:
1 x 1.1726 * 1.1726 = 1.375
Have a cigar.
The question about speed at the beginning of the post actually calls for using a 3rd type of mean:
The harmonic mean
The harmonic mean is computed by taking the average of the reciprocals of the values, then taking the reciprocal of that number to return to the original units.
That’s wordy. Better to demonstrate the 2 steps:
“Take the average of the reciprocals”
Instead of averaging MPH, let’s average hours per mile then convert back to MPH at the end:
50 MPH = “it takes 1/50 of an hour to go a mile” = 1/50 HPM
100 MPH = “it takes 1/100 of an hour to go a mile” = 1/100 HPM
The average of 1/50 HPM and 1/100 HPM = 1.5/100 HPM
“Take the reciprocal of that number to return to the original units”
Flip 1.5/100 HPM to 100/1.5 MPH. Voila, 66.67 MPH
Ok, right now you are thinking “Wtf, why is there a mean that deals with reciprocals in the first place?”
If you think about it, all means are computed with numbers that are fractions. You just assume the denominator of the numbers you are averaging is 1. That is fine when each number’s contribution to the final weight is equal, but that’s not the case with an MPH problem. You are spending 2x as much time as the lower speed as the higher speed! This pulls the average speed over the whole trip towards the lower speed. So you get a true average speed of 66.67, not the 75 that your gut gave you.
I want to pause here because you are probably a bit annoyed about this discovery. Don’t be. You have already won half the battle by realizing there is this other type of mean with the weird name “harmonic”.
The other half of the battle is knowing when to apply it. This is trickier. It relies on whether you care about the numerator or denominator of any number. And since every number has a numerator or denominator it feels like you might always want to ask if you should be using the harmonic mean.
I’ll give you a hint that will cover most practical cases. If you are presented with a whole number that is a multiple, but the thing you actually care about is a yield or rate then you should use the harmonic mean. That means you convert to the yield or rate first, find the arithmetic average which is muscle memory for you already, and then convert back to the original units.
When you compute the average speed for an entire trip you actually want to average hours per mile (a rate) rather than the rate expressed as a multiple (mph) before converting back to mph. Again, this is because your periods of time at each speed are not equal.
You can’t average P/E ratios when trying to get the average P/E for an entire portfolio. Why? Because the contribution of high P/E stocks to the average of the entire portfolio P/E is lower than for lower P/E stocks. If you average P/Es, you will systematically overestimate the portfolio’s total P/E! You need to do the math in earnings yield space (ie E/P). @econompic wrote a great post about this and it’s why I went down the harmonic mean rabbit hole in the first place:
The Case for the Harmonic Mean P/E Calculation (3 min read)
Consider this example of when MPG is misleading and you actually want to think of GPM. From Percents Are Tricky:
Which saves more fuel?
1. Swapping a 25 mpg car for one that gets 60 mpg
2. Swapping a 10 mpg car for one that gets 20 mpg
You know it’s a trap, so the answer must be #2. Here’s why:
If you travel 1,000 miles:
1. A 25mpg car uses 40 gallons. The 60 mpg vehicle uses 16.7 gallons.
2. A 10 mpg car uses 100 gallons. The 20 mpg vehicle uses 50 gallons
Even though you improved the MPG efficiency of car #1 by more than 100%, we save much more fuel by replacing less efficient cars. Go for the low-hanging fruit. The illusion suggests we should switch ratings from MPG to GPM or to avoid decimals Gallons Per 1,000 Miles.
The Tom Brady “deflategate” controversy also created statistical illusions based on what rate they used. You want to spot anomalies by looking at fumbles per play not plays per fumble.
Why Those Statistics About The Patriots' Fumbles Are Mostly Junk (14 min read)
The most important takeaway is that whenever you are trying to average a rate, yield, or multiple consider
a) taking the average of the numbers you are presented with
b) doing the same computation with their reciprocals then flipping it back to the original units. That’s all it takes to compute both the arithmetic mean and the harmonic mean.
If you draw the same conclusions about the variable you care about, you’re in the clear.
Just knowing about harmonic means will put you on guard against making poor inferences from data.
For a more comprehensive but still accessible discussion of harmonic means see:
This post is so good, that I’m not sure if I should have just linked to it and not bothered writing my own. You tell me if I was additive.
A few years ago Harvard conducted those “implicit bias” tests and used the results as evidence that even allegedly unbiased people show evidence of unconscious prejudice. I’m not climbing into that cement mixer. But I’ll share 2 quick stories of me catching myself being biased on gender.
Last year, I spoke to a local coding school that said they would hold an in-person class for my son if I could get 5 kids to join the class so it would make sense for them to staff a teacher. I email blasted some parents I knew to ask if their sons would be interested.
After sending the email, I was kinda shaken that I was looking for “sons”. I told Yinh about it and how bad I felt. In her opinion, I wasn’t biased, she just looked at the list of parents I sent it to and thought it made sense but also since we have 2 boys our parents’ group is skewed (the running joke is if we meet girl parents we really like, it’s like “nice hanging out see you in 18 years”). Yinh didn’t think my actions were evidence of bias but I was suspicious of myself and the verdict is irrelevant. That feeling that I acted in a way that does not accord with my beliefs will help me be better next time. Mistake + growth = all we can ask for.
There’s a project I’m working on in a half-assed way (what else is new?). I was thinking about what it might need if I got more serious. Part of that exercise prompted me to think of who I’d want to be on the board if it became a thing. I wrote a list of names.
Then I realized it was all men.
Was this off-the-top-of-my-head list the best list? I opened my CRM.
[Aside: I keep a database in Notion of everyone I meet, including where they live, what they are working on, what type of help they may need, ie investors or collaborators etc. I recommend doing this. Just makes you a better connector. I have also been keeping a list of every restaurant I have ever to for the past 10 years by location. That way if anyone asks me for a rec, I can look at my phone instead of suddenly going blank. I’m pretty sure I was born to be a librarian or serial killer.]
When I scanned the CRM, I found multiple women who would hands-down be better choices than the men on the list. Not only that, one of them actually played a part in the brainstorming of the project. WTF Kris.
I suck. I don’t wanna suck, and I still suck.
So is implicit bias a thing? It is for me. So I put a checkbox field in my CRM table:
Sounds heavy-handed right? Well, you are free to tell me how I can help myself otherwise, but this is the only way I’ve ever known how to change. Make the thing I want to improve more explicit. What gets measured gets managed.
The subtlety of bias makes it hard to address. What cultural nudges have I silently absorbed that undermine the lessons from the strongest bonds in my direct experience? Consider these following personal facts that have always been top of mind for me :
I grew up with a single mother for much of my childhood.
I have 1 sibling. A sister who is a brilliant, kind high-achiever.
My closest older-than-me relatives are all blood aunts. I have profound respect for their beautiful hearts and can-do-immigrant-tenacity. I have uncles too of course, but only a small percentage of them relative to the women hold my love or respect. (I’m not saying this because I want anyone to generalize about the nature of men vs women, but just to describe how my observations of women have been unusually favorable — on the conscious level).
And then I’m married to a woman who inspires me. My wife is an absolute boss but more importantly, has a trail of people who have been touched by her generosity. But she’d never accept recognition for any of it no matter how much they’d want to shout her. However driven she is in professional or measurable ways, her highest priority is her family and friends.
In other words, for the entirety of my existence, I’ve been surrounded by lionesses.
And still, I fail to give women equal consideration without an extra effort. So either the stories we, both men and a special hell-circle of women, tell about why females getting a lesser deal is justified are ego-protecting rationalizations…or I’m alone in being a sexist?
You all have women in your lives that you love. You all have women in your lives you have tremendous respect for or might consider role-models. But that acknowledgment is not a free pass to think you are untouched by bias. Try to catch yourself. Spend some attention on it. And then ask yourself, do you really think unequal outcomes are fully explained by the smart-sounding justifications?
Did fewer girls get interested in coding because I forgot to add them to the carpool? Did the ones who get to hacker camp get discouraged because they were surrounded by farty boys like mine when they arrived? Did the moms not bring their daughters to camp because that idea was less familiar to them since any of the bias that exists today is a fraction of what they faced growing up?
I do believe that over time our awareness of these ideas enlightens us. It just happens over generations. But we shouldn’t take it for granted because it’s not natural to suppress bias. A strained society does not have the energy to suppress bias or prioritize equality. Fear peddlers are opportunists who see people’s pain and sell them scapegoats not solutions. Because solutions to the complexities that weaken a society are hard to see. Our differences are not. They are the first culprits when our insecurities turn to fear.
The word “woke” is stretched to its excess by our mind’s habit of substituting the extremes for the typical. If you label both basic civil rights activism and a fringe effort to have furries considered a protected class as “woke” the word is going to break under tension. So when we push for progress (is that word broken too?), remember there are always people who will pretend there’s none that needs to be made (or that we’ve made so much progress that we need to go backwards).
Maybe I’d listen if they stepped up and told me how bias shows up in their own behavior. If they say it doesn’t, I’ll feel bad for being the only one. But I’ll also know those people aren’t credible.
I get that negative screens work both ways. I’ll let you know how many people unsub. I’ve said my piece. These writers made that easier to do. The cost we incur is still nothing compared to the cost the discouraged bear:
The Uphill Battle Women Still Face in High Finance (12 min read)
by Benn Eifert
Markets, discrimination, and "lowering the bar" (12 min read)
by Dan Luu
I’ve already wrecked myself today so I’ll comment on this one. Dan uses the term “teenage libertarian”. What a fitting word for the hospital-grade dose of Randian ideology that characterizes so much of Wall Street. I would have described myself as a libertarian at one point but posts like this demonstrate that “market efficiency” is a dangerous expression when used without reference to what conditions must be present for its existence and the degree to which such conditions are unfilled. Libertarianism is as idealized as communism. In their god-given instantiations, both sound beautiful. In reality, one leads to totalitarianism and remains mostly irrelevant as a majority ideal in the US, while the other is insidious because it hasn't been refuted as the self-serving ideology of those who have been served more than their share (especially if you believe in equality-equity tradeoffs as I do).
This thread by Benn explains further.Never mind the fact that even in undergrad economics you learn that there is an equality/efficiency tradeoff. Sure Ken might be a better allocator, but this is hardly an ethical mandate to give him all the capital. Plus everything else Benn says here
Benn Eifert (🧙🏿♂️, 🥷) @bennpeifertIf there's one point from the bullshit-in-investing complex I'd emphasize above all others, it is that the classical Chicago economics view that creating markets in absolutely everything leads to an efficient allocation of investment capital is sheer lunacy.
[For the option traders in the room, did anyone else notice the Easter egg in that Dan Luu post? There’s a roll call of thank you’s at the end and one of the people is a Barone-Adesi!]
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