Moontower #64

Friends,

Comedians Unfiltered

• From the NYTimes:

  Dave Chappelle released a lacerating new special, “8:46” — the length of time that a police officer held his knee on George Floyd’s neck as Floyd pleaded for his life — that has become among the first live shows in the Covid era to reckon with the protests gripping the nation…

  …The show was taped in Ohio on June 6, and Chappelle’s performance isn’t much of a comedy set, because, as he notes, there aren’t really any jokes.

  I’m a Chappelle fan so I’m biased, but this was my top link this week. Watch it for the emotion, the personal coincidences, his take on the role of celebrity right now, his anti-terrorism/revenge angle, and the Kobe/LeBron views. (Link)

• The next link is from Trevor Noah.

  18 minutes of candid talk into a webcam. I’m always impressed by his thinking. (Link)

  Some of the interesting angles:

  1. While rioting violates our social contracts, to many, Gladwell’s 3 requirements for legitimacy have been violated long rioting was the issue.

  2. His discussion of leadership by example. He explains how police set the price of life by example and how some in the “hood” internalize that price when dealing with each other.

  I find his thread on leadership quite poignant. The importance of leadership by example. It’s somehow both a cliche and an underrated point. Parents, police, politicians. Everyone takes their cues from leaders. We are built to learn socially and by imitation. Recent actions of powerful people just feel disheartening.

Record scratch…

Something totally different. Just some math fun because I noticed a random observation this week while thinking through 2 unrelated problems.

Combinations

If you recall the formula for a combination where n is total items and r is how many you are choosing from the set:

So if you have 12 musical notes how many 3 note chords can you create?

12! / (3! * 9!) or 220.

If you have spent time playing games the combination formula is as ingrained as the Pythagorean theorem (We are doing some backyard remodeling, I literally needed to compute a hypotenuse this weekend so Pythagoras was top of mind).

Anyway, here’s the random thing I noticed.

Suppose the question was how many 2 note chords can we create out of 12 notes? Well, a correlation matrix answers the same question when r = 2. A correlation matrix is a bunch of pairs. Pairs are just combinations of 2.

Look at the matrix. The grey boxes are just double-counting the green ones. A,E is the same chord as E,A. And the diagonal axis is not a pair so they don’t count either. So if you add up the green boxes you get 66 boxes.

But if we used the combination formula you also get 66 because 12! / (2! x 10!) = 66

Very neat. This makes sense. Let’s break it down.

Generalizing

If you need to compute a combination of 12 notes choosing 2 you can just visualize the matrix.

1. Start with 12 x 12 boxes = 144
2. Subtract 12 diagonal boxes to arrive at 132.
3. Divide by 2 because of double counting. 66.

   Generalizing that process. (N²-N)/2 or N(N-1)/2

To prove that it’s equivalent you can notice that:

C(N,2) = N!/ ((N-2)! x 2)

Substitute N!/(N-2!) for N(N-1). That is kosher since 12×11 is the same as 12!/10!

So N!/ ((N-2)! x 2) == (N²-N)/2

Permutations

With permutations, we say that “order matters”. A,D and D,A are distinct. They must be counted as 2 pairs, not just 1. So if you go back to the correlation matrix you start with the N² terms and subtract the N diagonals. You’re done! No need to divide the duplicates by 2 since A,D and D,A both need to be counted as distinct pairs.

To check you can look at the permutation formula which is the same as the combination formula but you don’t need to divide by r. You don’t need to divide out the number of ways the chosen items can be re-arranged as you do with the combination formula which doesn’t care about order.

So the permutation formula says there are 12×11 permutations or 132. Same as the matrix method which says there are 144 pairs minus 12 diagonal boxes.

When it comes to music chords permutations might be the more relevant count if you consider chord inversions.

One last trick

Remember when you were summing the green boxes…the columns were made of 11 boxes, 10 boxes, 9 boxes and so on…

Here’s the trick to summing the numbers 1 through N or in this case 1 thru 11.

N(N+1)/2

So 11×12/2 = 66

Let’s try another. Sum the numbers 1 through 100.

100×101/2 = 5050

Why does this work?

Pair the ends off.

100 + 0
99 + 1
98 + 2
97 + 3
96+4
…continue until 51+49

What do you end up with:

• 50 pairs summing to 100
• The middle “50” left over.

  50×100+50 = 5050.

That maps to (N/2) x N+1 or “the middle number occurs N + 1 times”. That 1 term is the “middle 50”.

The first time I encountered that question was playing a game that required us to sum the ranks of a suit of cards. So Jacks are 11, Queens 12, Kings are 13 and Aces are 1s. So 13 total ranks.

What’s the sum of 1 to 13?

Perhaps another time I will explain how we used to mock trade options on the sum of ranks of cards held in people’s hands as cards were “flopped” into collective view. (Susq alum are having flashbacks to “after-work mock” now).

The Money Angle

The response to my post Lessons From The .50 Delta Option was flattering. Readers thought it broke down delta in ways they hadn’t seen before. This encouraged me to turn some of my personal scribbles into digestible posts on a few other investment topics.

This week I published 2 posts that simplify concepts that are most commonly associated with options. They were similarly well-received for being approachable. I’m aiming for lucidity over completeness in all my finance education posts so you won’t find anything beyond middle-school math. If you get something out of them I will have felt I did something good.

Why Option Traders Focus On Vega (Link)

Most investors are directional players. They are betting on whether stocks will go up or down. Even when they use options they tend to use them directionally. But there is a much smaller segment of investors who “trade volatility”. The exposure they are most concerned about is not how long or short the market they are. In fact, they usually try to be market or “delta neutral”. They are more concerned with vega. 

What does it mean to “trade volatility”?
What is vega?
What can the vega of a position tell us about risk?

Where Does Convexity Come From? (Link)
“Unless you are a bond or derivatives trader, the term “convexity” usually just makes you want to say “beat it, nerd”.  As if to frustrate us further it has many aliases: “curvature”, “non-linearity”, “gamma”.  It turns out, for such a fancy word “convexity” is quite approachable.

In this post, we will learn:

• what it looks like
• common places to find it
• why you should care

But first, let’s see what convexity isn’t…”

In case you didn’t hear, Hertz got approval to sell $1B of common stock to the public which is bidding up its shares. Hertz however is in bankruptcy with its debt trading for less than $.50 on the dollar. Issuing stock in such a situation is unprecedented. These are unprecedented times I guess. The fact that stock traders are bidding for shares which are subordinate to debt that isn’t even going to be made whole feels awfully kangarooey.

• Matt Levine’s write-up is funny and educational. (Link)
• Alex Danco describes Hertz as the first example of new type of bubble — a view that the future doesn’t matter. This type of bubble is a new branch “on the financial tree of life”. He also expects this type of bubble to only happen once. (Link)

Danco’s description of the bubble dynamic in Hertz is a sign of just how meta-weird the world can feel sometimes. The dynamics of markets have become their selling point. Not their role. Automobiles were a transformative breakthrough as a transportation function. But anything that covers distance that quickly is even more fun to race and crash. To thrill-seekers, the convenience of faster transport isn’t even a secondary attribute. It doesn’t even figure into their framework of “what is a car for?”. If you find enough of those people, you have found a source of demand for the same product for different reasons. Hertz as roulette not equity. No veil of “investment” pretense.

So the question that remains — has anyone ever bought chips to play in a bankrupt casino? That’s what those new shares will be. Just don’t be the last one to cash out at the cage.

Last Call

• I liked this whole thread. Alexey Guzey starts, “Something I wish I knew as a teenager…” (Link)
• This post stood out to me in the competitive genre of blogs that catalog so-called mental models. It just made me see them a tad better or in a more sticky way. Scott Young is pretty good at that. Here’s his brief look at 10 of his favorites. (Link with my highlights)
• Another fun list. Maggie Appleton’s Neologisms.

  These are freshly coined words or phrases inching their way into common usage, but not fully mainstream yet. Neologisms emerge when we are unable to express or conveniently refer to an emerging collective idea or experience. (Link)

• A Moontower reader with a background in law enforcement wrote a highly educational response to my quip about what defect rate we can tolerate in police activity. The demands of the person’s current profession require anonymity so I published it on my site with permission. (Link)

From my actual life 

• Those wooden circles are randomly placed on the board (each player gets the same coordinates on the grid). The dice allow 62,208 combinations of wood placement. Every puzzle is solvable.
• Then ready, set, go — the first person who can fill the board with the Tetris pieces wins.