For the past 3 years, the Berkeley Chess School has done a weekly lesson in our backyard. About 15 kids attend. They are mostly 2nd and 5th graders because we know the families through our kids.
I walk home from school with all the kids on chess day. I usually throw out a math question or riddle and by now the kids just ask for them on our strolls. Lately, the questions deal with rates, percentages/fractions, exponents/roots, or some basic number play stuff (“How do you know if a number is divisible by 9?”)
- Start with $100. If you earn 10% then lose 10%, how much money do you have?
- What’s larger 3⁴ or 4³ and similar questions?
- If you travel 5 miles in 12 minutes, how fast are you going?[I haven’t dropped this one on them: “If I drive around a one-mile track and average 30 mph for the first lap, how fast do I have to drive the second lap to average 60 mph for both laps combined?” (Solution)]
- If you have 5 kids on your team and only 4 kids start how many possible starting lineups are there?
- If you lose 25%, how much do you have to make to get back to even?[I like that one because it’s a useful elasticity idea. If X is the loss percentage, you need to earn back X/(1-X).
If you are selling lemonade and you cut your price by 20%, you need to sell 25% more cups to be revenue-neutral.]
I started giving these questions because, well it’s just play. Riddles are inherently satisfying. But I’m also conscious of imparting some durable concepts. It’s not quite as deliberate as hiding the dog’s pill in the peanut butter but there’s some overlap. Honestly, the main motivation is keeping the kids who aren’t into sports engaged. There’s about 40 minutes between the kids getting to my house and the start of chess. The 2nd graders usually play soccer and the 5th graders hoop it up. A few kids aren’t into either but they gravitate to the riddles (my boys just wanna play sports and my 5th grader, Zak, with sass tells his friends “This is the stuff I deal with all the time”).
Lately, I’ve been doing more exponent stuff because I know they aren’t doing that yet in 5th grade but it’s reachable for them. Zak is taking the online Pre-Algebra I course on Art of Problem Solving which is comprised of 7 chapters. He just wrapped chapter 2 which is all about exponents so that’s been top-of-mind for coming up with the questions.
The challenge question to end the chapter was to solve for the 2 possible values of X (see below). But keep in mind, they haven’t learned how to compute square roots or any other kind of roots. You can do this without any involved computations and without roots. You can find the solution at the end of the post.
Find the 2 possible values of x:
Anyway, I didn’t give the kids that question but by this past week, they understood the basics of computing a simple exponent or taking a square root. So I took a stab at base 10 logarithms. I just explained it as the power you need to raise 10 to get to the target number.
“So if our target is 100, what do you have to raise 10 to? How about a target of 1,000?”
They had no trouble with this. So I explained how both the Richter and decibel scales were log scales that compressed a wide range into a smaller ruler. A 6 on the Richter is not twice the energy of a 3 but 1000x more energy. Every integer increase in the scale is just a higher order of magnitude.
The most pleasant thing happened. The kids that gravitated to this stuff were stoked. As it it settled in their brains they were all Keanu Reeves “Whoa, that’s so cool”.
I texted one of the parents:
Putting aside the pure joy of watching a kid unlock, exponents and logs are fundamentally important operations like adding and subtracting. Our first formal introduction to them outside a math class is usually science (exponential growth/decay) but more prosaic to this audience is the topic of investing, specifically the idea of compound growth. It’s an idea you’d love to see people internalize as young as possible.
Typically when someone (and I’ve done this too) writes about compounding they reference Einstein’s 8th Wonder of the World quote or talk about how our minds think linearly and find exponential growth unintuitive. [This was a common conversation at the start of the COVID pandemic with VCs patting themselves on the back for lateral thinking about how coefficients of virality applied to…the domain where exponential growth is usually people’s first contact with the topic. Like twisting a eulogy into a chance to talk about yourself. I’m not even mad, it’s the whole wheel of cheese].
With that in mind, I’ll leave you with an excerpt from Grant Sanderson, the mathematician behind the 3Blue1Brown YouTube channel. This is from his appearance on
excellent Lunar Society Podcast:
Have you come across those studies where anthropologists interview tribes of people that are removed enough from normal society that they don’t have the level of numeracy that you or I do? But there’s some notion of counting. You have one coconut or nine coconuts like you have a sense of that. But if you ask what number is halfway between one and nine, those groups will answer three whereas you or I or people in our world would probably answer five and because we think on this very linear scale.
It’s interesting that evidently the natural way to think about things is logarithmically, which kind of makes sense. The social dynamics of as you go from solitude to a group of 10 people to a group of 100 people have roughly equal steps in increasing complexity more so than if you go from 1 to 51 to 102 and I wonder if it’s it’s the case that by adding numeracy in some senses we’ve also like lost some numeracy or lost some intuition in others, where now if you ask middle school teachers what’s a difficult topic to teacher for students to understand they’re like logarithms. But that should be deep in our bones right so somehow it got unlearned
What a cheeky observation. Gives a second entendre to the expression “natural log”.
I’m using this space to invite you to play PitBulls (formerly StockSlam) online this coming Friday evening. It’s totally free but space is limited.
Reserve your spot (choose Friday November 3rd if you want to play with me)
- Click here to run through a tutorial. <— the app improves daily; you can now practice on your own against bots!
- Host your own game for free
This is my testimonial for the game:
One of the most fortuitous decisions I made in my career was accepting a job from SIG out of college. Back in 2000 going to Silicon Valley or I-Banking was all the rage. While trading was a coveted job, the idea of going to an exchange floor to sling options was not a mainstream career choice. And SIG was not a well-known company outside this narrow world.
The job offer I got from them was the lowest paying, but the interview process stood apart from the banks and other firms I talked to. It was clear that working for SIG meant a serious education in decision-making commensurate with the objective — to take responsibility for risking the partners’ own money after as little as 9 months of training.
I was placed at the American Stock Exchange where I would learn from senior traders including their head of education in NY, Mike Steiner, simply known as Steiner.
Steiner was a natural teacher, able to communicate complex ideas with simplicity and frankly, joy. It was no surprise when I discovered 20 years later he retired to become a physics teacher in public school.
In 2022, we reconnected and he showed me the prototype for what would become Pitbulls. Pitbulls is a game distilling the essence and mechanics of the mock trading program we used in training. Pitbulls is a fast-paced game requiring players to think quantitatively while building intuition and understanding investor psychology. Steiner focused on making it fun — it’s a game first. But when I saw it I was immediately struck by its potential to bring investing principles to life!
Skills that will immediately develop from the very first game:
- market making in an open outcry market
- tracking multiple quotes from competitors
- managing a rudimentary portfolio
- reacting to new information
Deeper concepts embedded in Pitbulls:
- arbitrage pricing, inter-dependent pricing
- expected value and probability
- the concept of edge as the foundation of a business
- how you can be profitable without relying on prediction
- making trading decisions under uncertainty
- risk and diversification
- the role, wisdom, and conditions of a healthy market
- an introduction to derivatives
- a bridge from trading to investing principles
Steiner has been hosting in-person playshops for years and I helped organize larger events in NYC, SF, and Chicago to overwhelmingly positive feedback. Unsurprisingly, the most universal suggestion was “give us an online version”. People wanted to play on their own and host their own sessions.
Starting now, you can play online!
I’ve been writing about financial education topics for years. These posts go into how Pitbulls and its underpinnings can improve your thinking in powerful ways.
- Finance As A Laboratory For Decision-Making [StockSlam Preview]
- If You Wait For All The Info You’ll Be Too Late
- Mock Trading
- The Benefits Of Betting Culture
- Lessons From Susquehanna
- Thinking Like A Trader
- Calibrated Confidence
Money Angle For Masochists
Quant legend Peter Muller wrote a candid, somewhat irreverent post 20 years ago that holds timeless wisdom.
🔗Proprietary trading: truth and fiction (Link)
- But the most important risk is the possibility of our models not working correctly. To minimize that risk, we set loss targets for strategies — if we lose more money than the pre-specified target then the strategy is re-evaluated and shut down for a while (perhaps forever). This is not that different from the old school of proprietary trader management: ‘Go ahead and trade, don’t do anything too risky, and if you lose more than $x we’re going to shut you down. ’Our strategies are evaluated by looking at reward/risk measures. For symmetric, market-neutral strategies without significant tail events, the Sharpe Ratio (SR) is probably the best ex-ante measure. SR is defined as the portfolio annual excess return divided by the annualized standard deviation of that return. Our benchmark is cash, hence measuring excess returns is appropriate for our portfolio. For long-only managers, the Information Ratio—which measures excess returns relative to a benchmark—is more appropriate. When we evaluate past performance, we also look at peak-to-trough drawdowns (a measure of the maximum drop between consecutive maximum and minimum values of return over the life of the strategy) as an additional risk variable. This can help pick up serial correlation in portfolio returns that the Sharpe Ratio doesn’t capture. Also of interest is the fraction of expected gross profits consumed by expected transaction costs. The higher this number, the more money we expect to lose if our model stops working. At least some of our edge comes from opportunities that are created in the market by institutional managers who trade too much. Their trading is usually based on either an exaggerated view of how well they can predict investment returns or a misunderstanding of how trading costs increase with size. The strategies of institutional managers can still be perfectly rational despite providing us with opportunities through over-trading, simply because of the huge agency issues in portfolio management.
- In Grinold and Kahn’s book on Active Portfolio Management, the authors describe the ‘FundamentalLaw of Active Management’: a strategy’s Sharpe Ratio is proportional to the number of independent bets taken by the strategy multiplied by the correlation of those bets with their outcome. To get a higher SR, you need to increase the number of your bets or increase the strength of your forecasts. In my opinion, it is far better to refine an individual strategy by increasing both the number of bets within the strategy and the strength of the forecasts made in the strategy, than to attempt to put together lots of weaker strategies. Depth is more important than breadth for investment strategies…I would much rather have a single strategy with an expected Sharpe Ratio of 2 than a strategy that has an expected Sharpe Ratio of 2.5 formed by putting together five supposedly uncorrelated strategies each with an expected Sharpe Ratio of 1. In the latter case, you’re faced with the risk that the strategies are more correlated than you realize (think Long Term Capital). There is also the increased effort of ascertaining whether each individual strategy really has a Sharpe Ratio of 1.
- An important choice for many proprietary traders is whether to focus on shorter or longer-horizon strategies. Typically, shorter horizon strategies get their edge from providing temporal liquidity to a marketplace or predicting short-term trends that arise from efficient trading. Longer-term models focus on asset pricing inefficiencies. How does the implementation of these strategies compare? Shorter-horizon investment strategies are desirable because they tend to create higher Sharpe ratios. If your average holding period is a day or a month, you have the opportunity to place many more bets than if you hold positions for three months to a year or longer. On the flip side, shorter horizon strategies tend to have capacity issues (it’s easy to make a small amount of money with them, but harder to make a lot of money). Shorter horizon strategies also require serious investments in trading infrastructure, since quick and inexpensive execution is much more important than for longer horizon strategies. Risk management for shorter-horizon strategies tends to occur through position trading rather than portfolio construction. Assets are not held for long periods of time and portfolio characteristics change quickly. The biggest risk for shorter horizon strategies model risk, or the risk that the trading strategy deployed has stopped working. Since even the best trading strategies experience periodic drawdowns, the hardest challenge for the short-term model-based trader is to figure out whether his model is going through a regular drawdown or has stopped working altogether. Longer-horizon model-driven investment strategies have different issues. Since assets are held for longer periods of time, execution costs (although still important) are not the primary focus. Statistical inference becomes more difficult and the danger of overfitting or mining data becomes larger. Risk management for longer-term strategies happens in portfolio construction: since rebalancing occurs less frequently, more care needs to be taken to ensure the portfolio is not exposed to unintended sources of risk. Because they tend to have lower Sharpe ratios, longer horizon strategies have a different kind of capacity issue—the capacity for pain. However, there is one advantage: because trading occurs less frequently it’s possible to lead a much better lifestyle than if you’re running shorter horizon strategies!
2 ways I came up with to solve the AoPS question
x could also be -243 since it’s being raised to an even power.