My grandma is $24.05 bid, stop embarrassing yourself

Last week’s Getting Paid To Flip Million Dollar Coins wondered how much the right to flip a coin for $50mm would trade for. I argued why it would go for something close to $25mm, certainly more than $24mm.

Based on the messages I received, the reaction was a barbell:

  1. “Duh”

  2. Some version of “you’re reckless or stupid”

I very much stand by my argument and take no offense to the reactions. I’ll share one of the critical responses and my reply. (emphasis mine)


Hi Kris,

I received your latest post. 

I analyse and value companies daily to determine whether we would or should invest in them. I noticed a couple of errors in your latest post and as you offer help with business and investment analysis for a fee, I thought I should bring the errors to your notice, as I offer advisory services myself.

“The red button is worth $25mm so our risk-neutral friend Spock would not pay more than $24mm…” – The red button is not worth $25mm. The expected value of possible outcomes resulting from if that button is pushed is $25mm. Also, there is a crucial difference between a weighted mean of possible theoretical outcomes that is probabilistic and what someone would pay and not the logical inference you make.

“$24mm to someone worth $100b is the same as $24 is to someone with $100k.” It’s not the same because of quantitative and qualitative materiality – with wide-ranging financial implications – and because of potential multiplicative effects of the $24mm/$24 difference, beyond the first order. Using your earlier equivalence of expected value and “worth”, the expected value of a $24mm decision is highly likely very different from that of a $24 decision.

My reply:

The proposition is actuarially worth $25mm. I agree that doesn’t equate to market value of the proposition but by competition for arbitrage, it would trade very close to that. 

The entire business of index and futures arbitrage looks like this. I mean if that was a real proposition in the marketplace and you didn’t buy it for $24mm that would be grounds for getting fired. Traders will bet huge size for way less edge and in fact big sports gamblers will too. But the real-life caveat is it’s rare that a prop is so actuarially obvious. Seeing an opportunity with that much edge would have you checking assumptions before pulling a trigger. 

And to your point its tradeable value can differ from actuarial value based on the competitive landscape (how many entities can afford the risk and get a look at the trade), the capital of those entities, and how many ways they have to lay off the risk. 

[Inserting an observation: Markets are not democracies. Because 99.9% of people wouldn’t pay $24mm is irrelevant. If a single trader is willing to absorb the risk he or she will bid one penny more than the best lowball bid. But the moment there are 2 capable entities the bid ratchets much higher assuming they don’t collude. We’ll talk about competition more later but bidding behavior is not a linear function of quantity of bidders.]

In fact, you can imagine a situation where the person offering the proposition is a valuable customer and a bank or trading firms does the trade for actuarial value or even for negative edge as a loss leading trade to get more business. I have not only watched that play out in the options market, I myself have traded at fair value with counterparties to make sure the brokers keep giving me looks. 

I did want to sanity check myself so I administered a poll. 6,500 people responded and it set off a cascade of discussions that I won’t re-hash here.

Link to tweet

Link to tweet

link to tweet

And finally this is the thread with the trading lesson.

Concluding remarks

I think the answers to the poll and discussions are a revealing litmus test for seeing how people think risk is priced in competitive markets. It appears there are people walking around thinking the market is way dumber than it is. Which explains why way too many think they can beat it.

It reminds me of when a trader would bid something like $24 while the rest of the pit was $24.10 bid. The broker: “Oh you think? My grandma is $24.05 bid, stop embarrassing yourself.” Your voice is immediately discredited when you’re that clueless about value.

(My grandma, probably)

If you think this coin prop trades for less than $24mm you are underestimating the competition and confessing overconfidence in what you think constitutes a good trade. A practical question to improve your tuning is to ask yourself, “Am I folding when I should raise? Am I raising when I should fold?”

Here’s a benchmark to consider when evaluating that coin flip trade:

If you had access to buying such a proposition many times a day every day you’d be very rich in just a short number of years. If you’d pass on this trade, this means you think you are doing better trades. In which case, the proof of your assertion demands being super rich from trading. (Or your boss, since again this is aimed at the ideal case where you have an adequate bankroll).

And finally to address a common rebuttal — “I’d pay more if I could do the trade many times”. Let’s interpret this generously. The rebuttal understands that whether you do the trade once or many times doesn’t change the expectancy, just the risk. But it is still a repeat game at the meta level even if not at the object level. Even if a firm were never see this coin again their biz is to put a price on risk. This is just another in a long chain of trades of decisions and decisions are bets. This particular trade is as easy as it gets.

[Again assuming the coin is fair, no credit risk — I’m not trying to make this about “gotchas”, just the platonic ideal of the math. As a pedagogical test, it’s useful to consider the theoretical asymptote because there’s no caveats to hide behind. See Can Your Manager Solve Betting Games With Known Solutions?You start with the platonic idea and work backwards through the practical realities. If you can’t solve the solved case (or hire someone who can) what should we conclude about the rest of your reasoning?]

I wish everyone on this list could have come to the Pitbull/StockSlam sessions. Within a few rounds, some people can make markets very tight (minimum increment wide) but what’s more interesting is how a teacher or experienced player could quickly spot who understands pricing and risk and who’s not getting it. But the thing that makes the game valid is anyone who is a market-maker, despite never having played this particular game before, is immediately good at it.

The principles of sound trading are universal. The devilish thing is applying them is hard because it’s not easy to get the inputs. But the problem is not symmetrical. Not understanding the principles or failing to apply them is definitely the route to failure over enough reps.

More On Bet Sizing

To put a bow on our discussion of bet sizing from last week I will just emphasize a few overarching ideas from the Moontower curation 🏇🏽Kelly Criterion Resources.

Via Nick Yoder’s amazing post:

Two keys are needed to unlock success in professional gambling, trading and investing:

  1. Profitable opportunities

  2. Sizing investments/bets (correctly)

A trader with a mediocre strategy and a great risk model will become fairly successful. A trader with a great strategy and a mediocre risk model will become bankrupt.

The Kelly Criterion only defines the ‘Optimal’ bet to maximize return. It does not use caution or assign value to risk. It is limit not a goal.

This is why I make such a big deal about managers who might understand their markets but don’t understand gambling and money management.

A trader with a mediocre strategy and a great risk model will become fairly successful. A trader with a great strategy and a mediocre risk model will become bankrupt.

This is a candidate for the most profound idea in investing. “How much” matters more than “what”. Most professional investors who lay an egg fail at bet sizing moreso than security selection. If you want to become a better investor you’d be better off learning about gambling than finance. On average, it would be easier to teach an advantage gambler to make money in markets than an MBA.

Here’s an exercise a group of Chicago traders I know give to trainees. Saddle them with a random position and see how they manage it. The subtext is — trading and risk management is a general toolset. This is exactly why I am very suspicious of specialists who know a lot about a certain domain but are inexperienced in general risk-taking acumen. This came up a lot in crypto years ago where FOMO investors were dazzled by super-smart whiz kids who grok the tech but had no experience in actually managing an investment book.

No thanks. Don’t care how smart you are.

That’s not the vector that drives the outcome. I wanna see lots of reps. Not “I HODL’d and won and you should trust that I’ll continuously figure out the right thing to HODL.” Falling for that shtick as a fiduciary is borderline malpractice.

Lessons from a naive sizing recipe

And here’s Victor Haghani (yes that Victor Haghani for the finance nerds) with a short case study with the best attributes: simple and deeply insightful.

Bitcoin is Nothing Either Good or Bad, but Sizing Makes It So (4 min read)

Takeaways of note:

  1. We can’t say that an investment is good or bad without considering how we will manage its sizing over time: sizing is as important as evaluating an investment’s expected risk and return.

  2. While there are an infinite set of investment strategies involving a given asset, we can learn a lot from focusing on the simplest strategy: Constant Proportion investing.

  3. Among Constant Proportion investment strategies, there will be a range of investment sizes that will be profitable, with sizing above and below that rapidly becoming increasingly unprofitable. And the range of profitable sizes is strongly related to the quality of the investment.

  4. For a given investment, the realistic strategies which turn a profit are typically quite a small subset of the infinite number of total strategies to choose from.

This post ties right back into the idea that outcomes have more to do with sizing than anything else. If it sounds heretical or so unconventional maybe investing education under-prioritizes the practical stuff that is much higher in the hierarchy of what “impacts” your performance.

How much time have you spent thinking about DCFs vs sizing?

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