
I should probably note here that November 20 was my last day as a Mozilla employee. In theory, that shouldn't really change much, given the opensource nature of Mozilla. In practice, of course, it does. I did successfully set up a nonstaff account and migrate things to that, so I still retain some level of access. I intend to continue contributing; however, my contributions will likely be restricted to things that don't require paging in huge chunks of code, or require large chunks of time. In other words, mostly cleanuptype stuff, or smaller bugfixes/enhancements.
I still believe that the Mozilla mission to make the Internet healthier is important, but over the course of the past year, I've come to realize that there are other problems facing society are perhaps more important and fundamental. That, combined with more companies opening up remote positions, provided me with an opportunity that I decided to take. In January, I'll be starting work on the Cash App Platform team at Square, and hopefully will be able to help them move the needle on economic empowerment.
Working at Mozilla was in many ways a dream come true. It was truly an honour to work alongside so many worldclass engineers, on so many different problems. I'm going to miss it, for sure, but I am also excited to see what the future holds.
A final note: if you follow this blog via Planet Mozilla, keep in mind that only posts I tag as "mozilla" show up there, and those posts will be much fewer in number going forward (not that they were particularly numerous before...). I have an RSS feed (yes, RSS is still a thing) if you care to follow that.
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Something that has bugged me for a while is how brokerages never seem to provide an annualized rate of return for stocks/portfolios. They tell you things like the book value, market value, and closed profit/loss, but computing a rate of return from that totally discounts the time factor and incremental/partial investments, and gives you a rate of return that is not meaningfully comparable across different stocks or portfolios.
I've written my own financial tracking software (this should come as a surprise to noone who knows me) and I figured I should build this in. It turned out to be a somewhat interesting problem to come up with something reasonable, so I thought it would be worth describing.
First we have to figure out what kind of rate of return can be meaningfully compared across different investment portfolios. Consider an easy scenario:
t=0: Buy $100 of stock A
t=1 year: Sell stock A for $110
What should be the rate of return here? The obvious answer is 10% because we sold for 10% more than we bought. But compare against this:
t=0: Buy $100 of stock B
t=1 year: Sell half of stock B for $55
t=2 years: Sell remaining stock B for $60
We can think of this as two substocks, B1 and B2:
t=0: Buy $50 of B1, $50 of B2
t=1 year: Sell B1 for $55
t=2 years: Sell B2 for $60
If we want to be consistent with our calculation for stock A, then B1 got a rate of return of 10%, and B2 got 20% but over 2 years. You could treat that as equivalent to 10% per year, but... if B1 was worth $55 at t=1 year, then so was B2. Which means B2 went up from $55 to $60 over the second year, which is less than 10% per year. So that seems like a contradiction.
With a bit of thought it seemed to me that what we really want here is a "continuous compounding" rate of return rather than a "yearly compounding" (or "no compounding") rate of return. The continuous compounding formula is:
P_{t} = P_{0} * e^{r * t}, where P_{0} is the initial investment, r is the rate of return and t is the time.
So let's look at B1 and B2 again:
Formula for B1: 55 = 50 * e^{r * 1} which produces r as 9.53%.
Formula for B2: 60 = 50 * e^{r * 2} which produces r as 9.12%.
This makes sense since we know B did worse in the second year than it did in the first year, so B2's rate over 2 years should be lower than B1's rate over the first year. (Note that with this formula we also get 9.53% for stock A above, which seems consistent.)
What about combining B1 and B2 back into a single rate of return for B? They're over different time periods so just taking the arithmetic mean doesn't seem right. I realized instead that we can think of the investments as putting money into and taking money out of an imaginary savings account with continuous compounding. That's similar to the stock market (or other investment vehicles) with the difference that the stock market fluctuates a lot and the instantaneous value at any given time is pretty meaningless, so we want to avoid using that anywhere in our calculations. We want to get away with just using the cash flows in and out of the investment.
So to compute the return for B, we could do something like this:
(((100 * e^{r * 1})  55) * e^{r * 1})  60 = 0
This says the $100 grew at continuouslycompounded rate r for one year, at which point we removed $55 and let the rest grow for another year at the same r, and then removed $60 and ended up with $0 left. And here solving for r gives us 9.25% which seems like a reasonable number given our values for B1 and B2 above.
This solution can be extended for all sorts of complex scenarios spanning different time periods and with many cash flows in and out. I don't know if there's a closedform solution to this but I ended up writing some code that did a binary search to converge on r.
Another interesting factor to consider is related transactions that don't actually affect the "stored value" in the investment. This includes things like dividend payouts (excluding reinvested dividends) or transaction commissions taken by the broker. I wasn't quite sure how to fit these in, but eventually decided that they should just be treated as noncompounding. So, for example, if we have this scenario:
t=0: Buy $100 of stock C
t=1 year: Receive $10 dividend from C
t=2 years: Sell all of C for $130
We'd use this formula:
(100 * e^{r * 2})  130 = 10
The left side is what we'd normally put in for the buy/sell transactions, but the right side is the net result of all the related transactions (in this case, a $10 dividend payout). In this case, it gives us a 16.82% rate of return, versus a 13.12% return without the dividend. So again, seems reasonable since the net value at the end is $40, versus $30 without the dividend.
Accounting for dividends this way makes it so that the time at which we receive the dividend doesn't make a difference to the overall rate of return  we could receive the dividend right at the beginning, or even after we sell the stock, and our rate of return will be the same. I considered the argument that dividends that arrive sooner are better, because we have access to the money earlier. Upon further reflection, I think that's only true if we actually invest that money in something. If that something is part of the portfolio we're evaluating, that dividend is effectively a reinvested dividend and shouldn't get counted as a dividend at all. And if that something is outside the portfolio we're evaluating, then it should get counted towards that other portfolio. I'm not totally sold on this bit yet but even with this caveat the overall approach seems to work well enough.
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In a a previous post I described a puzzle. A couple of people I've talked to since have mentioned that they thought about it but couldn't figure out the answer, so here it is. (If you don't want spoilers, stop reading now!)
The "magic square" chosen by the Devil is one of 64 possibilities. Or, in information theory terms, it's 6 bits of information (since each bit encodes one of two possibilities, and 2^6 is 64). So we need to somehow convey 6 bits of information to our friend, yet do so by flipping at most one token on the board.
The way to do this is to define 6 "parity sets" such that each parity set gives you 1 bit of information, and overlap them such that with a single token flip you can control the bit produced by each parity set. A parity set is simply an area of the board where you count up the number of "up" tokens. The parity (even or odd) of that number produces the bit of information.
So for example, consider a parity set that is the top half of the board (the first four rows). If there are an odd number of "up" tokens in that half of the board, the bit produced by that parity set is a 1. If there are an even number, the bit produced is a 0. By flipping any token in the top half of the board, you can change the bit produced from 1 to a 0 or viceversa. And now consider a second parity set that is the left half of the board (the first four columns). Likewise that parity set produces a 1 or a 0. Importantly, if you flip a token in the topleft quarter of the board, you will change the bits produced by both parity sets. If you flip a token in the topright quarter of the board, you will change the bit of only the first parity set and not the second. Flipping a bit in the bottomleft quarter will change the bit of only the second parity set and not the first.
We can extend this concept to create the following six parity sets:
 rows 1,2,3,4
 rows 1,2,5,6
 rows 1,3,5,7
 columns 1,2,3,4
 columns 1,2,5,6
 columns 1,3,5,7
Flipping the token in row 1, column 1 will change the parity of all six sets, while (for example) changing the token in row 5, column 6 will change the parity of sets 2, 3, and 5.
So the complete solution is like so: with your friend beforehand, you decide on the 6 parity sets (the above is one possibility) and their interpretation. One interpretation is that you take a 1 for an odd number of "up" tokens in the set, or a 0 for an even number, and glue together those six bits into a 6bit number (e.g. 011001). That number then encodes the position of the "magic square", as it can represent 64 different values. Then, when you are in the room with the Devil, and he selects the "magic square", you work backwards to figure out the 6bit number you want to encode, and flip the appropriate token so that the six parity sets produce the bits you need. Tada!
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Book #28 of 2019 is Theories of Childhood, by Carol Mooney. At long last, a book that succintly describes some of the different theories of early child development, without a lot of prescriptive advice. It let me build some mental models of how children develop so that I have a foundation for evaluating other howto articles and such. It's a short book and doesn't have too much detail but is a good starting point, and I can probably now find other books that build on the different theories in more detail.
The only question I have is why the author chose these particular 5 theories and if there any other ones that might be good to know as well.
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Book #27 of 2019 is Your Baby and Child by Penelope Leach. I was recommended the book on the grounds that it explained child development processes, which is something I am very interested in. Sadly this turned out not to be the case, but is just another opinionated/prescriptive parenting book from which you have to tease out the development process yourself.
That being said, it's a pretty comprehensive book and covers a lot of ground (I just skimmed some of it, specially the later sections). I did like how it splits the material into different stages (newborn/settled baby/toddler/child/etc.) rather than use explicit age ranges, because those age ranges vary a lot in practice. But the book is dated, and some of the material is no longer "best practice" or has been rejected by the latest scientific research. And that material is mixed in with everything else, so it's hard to take anything the book says at face value.
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Book #26 of 2019 is Joy in the Morning, last of my Wodehouse binge. Certainly very similar to the previous books, in that the same scenarios appear over and over but are glued together in different sequences. While it was fun I'm glad I'm done with this series for now.
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Books #22, #23, #24, and #25 of 2019 are Something Fresh, Heavy Weather, The Inimitable Jeeves, and The Code of the Woosters, all by P. G. Wodehouse. The last of the four I found the best, but they all were pretty good. Amusing as they are, there does appear to be some amount of repetition of themes, more than I would expect.
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Book #21 of 2019 is The Information by James Gleick. It's a comprehensive but easytoread book on information. It starts with the transition from oral to written history, and goes all the way to quantum information theory concepts, spending the most amount of time on Claude Shannon's work on developing information theory. I found it quite good, although it took me a while to get through as I had to stop periodically and absorb stuff. There was a bunch of stuff in there that made for interesting thoughtfodder. Wouldn't recommend it to a general public though; good for somebody with a general interest in information theory.
As a tangent, here's a (variant of a) puzzle I was forwarded not too long ago on WhatsApp. Usually I dislike those things, but this puzzle intrigued me as it seemed impossible to solve at first and took me a few days to figure out.
You, your friend, and the Devil play a game. You and the Devil are in the room with a 8x8 chess board with 64 tokens on it, one on each square. Meanwhile, your friend is outside of the room. The token can either be on an up position or a down position, and the difference in position is distinguishable to the eye. The Devil randomizes the tokens on the board (so it's a random mix of up and down) and chooses one of the 64 squares and calls it the magic square. Next, you may choose one token on a square and flip its position. Then, you leave the room, and your friend comes in and must guess what the magic square was by looking at the state of the board. You and your friend may agree on some strategy beforehand, but there are no "side channels" for leaking information other than the tokens on the chessboard.
Bonus points if you can explain the solution without using concepts from information theory (I couldn't).
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Book #20 of 2019 is Thank you, Jeeves by P. G. Wodehouse. Hiliarious! The writing style kind of reminded me of Douglas Adams or Terry Pratchett, but the content is somewhat different. I very much enjoyed it though, and parts had me LOL'ing.
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Book #19 of 2019 is Investing: The Last Liberal Art. I saw this randomly while browsing in a library and it sounded interesting so I picked it up. It was a bit of a rollercoaster, because:
(1) What I expected based on the jacket was that it would give a quick overview of the main ideas from different disciplines in a way that would encourage me to learn more about them.
(2) After reading the first chapter, I was very disappointed, because it seemed like it was really "pick a concept from a discipline and shoehorn it into some theory/explanation of how the stock market works". To be specific, the first chapter chose the concept of "equilibrium" from physics. Which just really rubbed me the wrong way, because it seemed like he was taking ideas totally out of context and misapplying them.
(3) After reading the rest of the chapters, I understand a bit more what the author was trying to do. I still don't think he did a particularly good job, but at least the book pointed me to some interesting ideas that I hadn't thought about before, and can guide me to other interesting books.
Still not a book I would recommend overall, but I'm glad I didn't quit after the first chapter since the later ones redeemed the book a bit.
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