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Giuseppe Paleologo’s book, Advanced Portfolio Management, makes a fun point about portfolio stop rules: a trader running a portfolio will have a cutoff at which half of their capital is taken away, and another cutoff where the rest of it is yanked, and they are (probably) fired or (possibly) given a second chance later on. And these rules tend to be set pretty tightly—older numbers floating around were 5% for losing half your capital and 10% for losing all of it, but Bluecrest did 3% and 6%.

His book notes that these rules are frustrating, since they force managers to buy high and sell low, or sell high and buy low. But it’s also frustrating because those cutoffs exist whether or not they’re written down—someone who doesn't have a firm rule like this doesn't know when they'll get fired, but knows that, if they lose money for a while, it's a live possibility. Per Samuel Johnson, “Depend upon it, sir, when a man knows he is to be hanged in a fortnight, it concentrates his mind wonderfully.” But phrasing is important: knowing exactly what leads to a hanging is good for focus; knowing that it's a live possibility and that the rules are obscure leads to a distracting level of dread.

But this raises an interesting meta question: rigorous portfolio construction means setting up bets in proportion to bankroll: if you have high odds of a small win, betting big is the only way to get meaningful returns, but for a given expected value for an outcome, the lower the probability the smaller the share of bankroll that should be invested in it.

For example, if you're using the Kelly Criterion to size bets, and you're betting "full Kelly," (i.e. not adjusting the size of your bet down to account for the intrinsic uncertainty of your odds estimates), then given a 55% chance of a 2:1 payoff, you should bet 10% of your bankroll each time. If you compare that to a bet with a higher expected value, but much lower odds of winning—say, a 1% chance of a 1000:1 payoff—your optimal bet size is lower, at 0.9% of bankroll. The Kelly Criterion formalizes the intuition that it's hard to recover from big losses; a 50% drop must be offset by a 100% gain to get back to even. What Kelly betting solves for is the optimal bet size for rapidly compounding your money.

In practice, those fractional-Kelly approaches are very common because you just can't know the odds or payoffs for a given betting category—even in games of chance where the rules are stated in advance and the odds can be calculated. That’s because any gap between the theoretical odds and the applied ones—anything from misreading your hand one time to accidentally dropping a poker chip on the floor shifts a full-Kelly bettor to over-betting. In investing, naturally, the scope of uncertainty is much higher. Not only can you be wrong about your calculation of the odds, but the existence of other people doing the same analysis (and making the same bets) can change the payoff function.¹

But, per Paleologo, the size of the bankroll is also an unknown! For a trader managing $500m at Bluecrest, one option is to view the $500m as the bankroll, which makes the optimal size of a trade with 55% odds and a 2:1 payoff $50m. But that trader also gets fired if they lose 6% of their money, so in practice perhaps their real bankroll is $30m, and the real optimal trade size is just $5m.

But that, too, is too punitive, and in two directions. First, the expected value of the trade is $1.95m, or 39 basis points of return. And that's a trade where we're assuming a pretty favorable setup! If each trade lasts a year, the trader in question needs to find two great ideas every three trading days to be fully deployed; for a shorter timeline, the effort is even more onerous. The other, bigger problem is that hitting a drawdown limit is not the end of the world, or the end of a career. Sure, it can be career-ending to get fired, of course, but the most common outcome afterward tends to be either getting a similar job at a different fund or getting a slightly more junior one somewhere else.

So the effective bankroll for the trader is larger than the $30m they can lose before getting fired, because there will be another $30m out there, or another $10m, or at least another six- to seven-figure job without as much stress involved.

This is easiest to look at when asking a different question: suppose someone has gotten their first job, saved their first thousand dollars, and wants to know what they should allocate it to. What's their bankroll? The simplest answer is that it's $1,000, and they should allocate some of it to cash, some to low-risk bonds, and some to equity. But really, their bankroll is the net present value of their future earnings less necessary expenses, which is a number a few orders of magnitude higher than $1,000 (and, as academic research suggests, they should be levering up as much as possible). They won't run with that kind of leverage forever, but if the net present value of your future savings is $100,000, and your Robinhood account has $1k, using 3x leverage just means putting 2.97% of your wealth into stocks instead of 0.99%. This is a fairly small shift, and is still well below optimal.

Of course, with that kind of leverage, you run the risk of a wipeout.² But for a healthy young person with a job that can cover their minimal bills with a little bit extra, $0 in a bank account or brokerage does not mean a $0 bankroll for risk-taking purposes. It doesn't mean they should gamble, of course, but it does mean that they're limited in how much financial risk they can take relative to the intangible but very real economic asset of their future income.³

That story should illustrate yet another bankroll complexity. Is someone who just lost all of their liquid savings in a state of mind that will lead them to make cool, rational decisions? Probably not. It’s easy to be shell-shocked by a big loss (this is one reason for stop-loss rules: the last thing you want to do is keep on trading through it when you’re losing money and don’t know why!). Meanwhile, the opposite bias also appears: The Missing Billionaires, a very Kelly-driven book, has a subchapter about how one of the authors disastrously doubled-down on his position in Long-Term Capital Management; it wasn’t about misunderstanding the risks, but about refusing to calculate the risk-adjusted payoff of a concentrated bet. The psychological bankroll is hard to measure—no one is good at accurately imagining how they’d behave during a 50% drawdown on their effective bankroll, but the level of drawdown that makes someone bad at careful risk assessment is itself a measure of the real bankroll they’re operating with.

Zooming back out to the original question: it is very useful to frame opportunities in a Kelly-like model, where you allocate based on both the expected value and the relative uncertainty of the bet you're making. But when you do that, it's very important to know the level at which you're making these optimized bets. It can be convenient to anchor your bankroll number to a specific dollar value in your account, but it's never going to be accurate—a young person should be overbetting, while an older person with more fixed obligations should, relative to the optimum, be underbetting. Given the intrinsic uncertainty of the shape of the true probability distribution you’re betting it, it feels like the hard part of the Kelly Criterion is estimating the edge and odds, with the bankroll as the easy constant. But it turns out that the size of the bankroll may be the most uncertain variable of all.

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Read More in The Diff

We’ve discussed portfolio construction in The Diff many times, both at the level of investors and of companies. (It’s a complicated setup: a limited partner in a multi-strategy fund that invests in such companies is managing a portfolio of investments that includes an investment in a portfolio of trading strategies, each of which is a portfolio of companies that includes a portfolio of subsidiaries.)

• “Pod People” ($) is a breakdown of what multi-strategy hedge funds do, which turns out to be heavily dependent on optimal bet sizing.

• For more on that, see this review of Advanced Portfolio Management ($).

• At the company level, when you buy equity in a good company you’re really buying an option on that company’s future reinvestment ($).

• One driver of optimal portfolio construction is how much it costs to make a given bet ($).

1. Specifically, crowded trades made by levered investors can mean-revert very dramatically. This has happened many times in statistical arbitrage in both equities and fixed income, where investors need extremely levered portfolios to get meaningful returns. As a trade or strategy gets more popular, the left tail of the distribution fattens, but this is only apparent after the fact.

2. In fact, someone who read that paper when it came out and ran the strategy would have lost all their savings in the space of a few months, since the paper was published in June 2008. That would definitely have been annoying. On the other hand, they also would have been heavily levered during the great 2009-2020 bull market, and as their wealth rose and their leverage dropped, they would have done quite well.

3. In fact, you can think of the average person's financial lifecycle as a gradual process of converting that intangible asset into something more quantifiable and measurable—a set of measurable assets that implicitly consists of claims on the output of younger workers, which can be used to purchase outputs produced by the same.

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