Money Math

after this post you will be sizing bets in your head

One of the most important concepts in risk-taking is bet sizing. Which is unfortunate because people are quite bad at it, while the effort to be way above average is quite low.

A jarring and famous demonstration of this is the Haghani-Dewey Coin Flipping study, which showed how even college grads with business, economic, and technical backgrounds incinerated their capital or massively underperformed the expected profits presented to them by a game they knew was rigged in their favor.

You can read my synopsis in Bet Sizing Is Not Intuitive.

For a binary wager (ie win or lose), if you know the payoffs and the probability of winning, both of which were known to the participants, the solution is to use the Kelly Criterion.

The tragedy is that it is incredibly simple to compute and applies to many conventional gambles and decisions (the examples in the quiz will span various life situations!).

If something is both easy and widely relevant, it should be common knowledge. So let’s fix that today. I’ll show you how easy it is to use, and you’ll forever be able to do it in your head.

First, a succinct definition:

Kelly is the bet size, as a fraction of bankroll, that maximizes the long-run compounded growth rate of your wealth. It’s a mathematical solution to bet size that doesn’t seek to maximize expected profit per trial, but the size that optimally balances compounding rate and survival.

If you want to go deep on this, see Moontowerquant’s Kelly Criterion Resources, but today’s focus is on getting straight to usability.

We will use this formulation of Kelly because it’s general:

f* = p − q/b

where:

p = probability of winning

q = 1−p or probability of losing

b = the odds you’re getting → what you win divided by what you risk

The easiest way to learn it is just jump right in with a few worked examples:

Fair coin wager (even odds style bet)

p =50%

q= 50%

b =1 (ie even money, for a $1 bet you either lose a $1 or make a $1 profit)

f* = 50% – 50% / 1 = 0 → bet nothing, zero edge

Coin biased in your favor (even odds style bet)

p =60%

q= 40%

b =1 (ie even money, for a $1 bet you either lose a $1 or make a $1 profit)

f* = 60% – 40% / 1 = .20 → bet 20% of your bankroll

Roll a 6 on a die (underdog bet where you get odds)

p =1/6

q= 5/6

b =8 (for a $1 bet, you either lose a $1 or make an $8 profit)

f* = 1/6 – (5/6) / 8

f* =8/48 – 5/48 = 3/48 → bet 6.25% of your bankroll

If f* is 0 or negative, you have no edge, so not betting is prescribed

Sports moneyline (betting as a favorite where you lay odds)

A −200 favorite. You risk $2 to win $1, and the line implies 2/3, but you think it’s closer to 3 in 4.

p = 75%
q = 25%
b = 0.5 (getting 50% return on the amount you risk)

f* = 75% − 25% / 0.5 = 75% − 50% = 25% → bet 25% of your bankroll

Wait a minute, these are large bets?!!

If these bet sizes seem surprisingly large for the given advantages, then your senses are well-tuned. For most people, “full” Kelly is too big!

Kelly maximizes long-run growth on the assumption your probability is correct. Well, it probably isn’t because the world is messy. We can inject some humility by using a fraction of Kelly:

“Half Kelly” gives up about a quarter of the growth rate and roughly halves the drawdowns. If you invert that, you see that the Kelly scaling law means as you bet bigger, you get diminishing returns per unit of risk. Extrapolating that logic, betting more than “full Kelly” is incinerating compounded wealth even if the individual bet has positive EV.
“Quarter Kelly” or less is far more common in practice.

Please don’t let the equation scare you, it’s intuitive and easy to remember

Look at the equation again:

f* = p − q/b

It’s just “how often you win” minus “how often you lose.” It’s just that the second term incorporates the payoff. The loss term gets divided by b, which represents the return you collect when you’re right.

When b = 1, you’re getting even money. A 100% return. Dividing by 1 leaves q alone, and the equation collapses to pure hit rate: p − q. That’s the coin case where you bet $1 to make $1.
When b > 1, you’re getting long odds. The division shrinks the loss term. This is why the die works. You lose 5 out of 6 rolls. Straight subtraction says you’re down 66 cents on the dollar and should never play, but you’re paid 8-to-1, so that 5/6 becomes 5/48, and suddenly the 1/6 win percentage is the bigger number. Long odds forgive a bad hit rate.
When b < 1, you’re laying odds. Now division stretches the loss term. The moneyline: you only lose a quarter of the time, but at −200 each loss costs you two units to earn back one, so that 25% loss percentage behaves like 50%. Being right three times out of four barely clears the bar. Lay enough odds and even a very good record is a losing proposition.

The graphic shows how you can think of the odds (the denominator) as shrinking or inflating q, as you collapse your thinking to a comparison of p vs q.

A word on b

b trips people up because “odds” is loaded gambler jargon. A wider interpretation of b is that it’s a percent return.

It’s what you make divided by what you risk. Even money is b = 1: risk a dollar, make a dollar. That’s a 100% return on the amount at stake. 3-to-1 is b = 3, a 300% return. Laying −200 is b = 0.5 because if you risk two to make one, it’s a 50% return.

[Return is a profit, while multiples don’t subtract your initial risk. It’s the difference between “I 2x’d my money” vs “I made 100%” or “I 10x’d my money” vs “I made 900%”. The percent return is the multiple minus one because we subtract our initial risk.]

The reason it’s a return and not just “the odds” is that Kelly assumes a loss wipes out the whole stake. The denominator is always the same number: everything you put up. b is comparable across a coin, a die, and a moneyline because it’s the return on risk, always measured against a total loss.

b = 1 is a natural reference point. At 100% return, a win exactly cancels a loss, so you need to win more than half the time. The breakeven hit rate changes with b.

Set f* = 0 and you get p = 1/(1+b).

Read the table as a menu of the hit rates you’re allowed to have. At b = 24 you can be wrong 24 times out of 25 and still be flat. At b = 0.25 you can be right four out of five and still be flat

Applying to real life: when is Kelly the right tool?

Kelly needs a few inputs: a bankroll, a payoff you know, and a probability estimate.

Which of these is a Kelly problem?

A prediction market contract trading at 30¢. You think it’s worth 45¢.
You’re all-in-or-fold on the river with a read that you’re good 40% of the time, getting 3-to-1 from the pot.
How much of your 401(k) to put in equities.
Writing checks as an angel investor across 30 startups.
Buying a weekly call on a biotech ahead of an FDA decision date.
Whether to take the new job.
Your buddy offers you 5-to-1 that it rains in Oakland tomorrow. The forecast says 30%.
Buying homeowners insurance. The premium is clearly more than the expected loss — that’s how the insurer stays in business.
Your neighbor doesn’t carry homeowners coverage. She banks the premium instead.
Your auto policy offers a $500 deductible or a $2,500 deductible, for a $340/yr discount on the premium.
You’ve got vested startup options. Exercising costs $40k out of pocket in strike, and you think there’s maybe a 15% chance the company gets somewhere that makes them worth $1M.
A merger arb spread. Target’s at $46, deal price is $50, and it trades back to $38 if the deal breaks. You think it closes 90% of the time.
Your agency spends 20 hours of unbilled time on a speculative pitch. You win about a quarter of them, and a win is worth 80 billable hours.

Solutions to Kelly Problems

Yes. Cleanest case there is. Binary, known payoff, and the price provides b directly. Risk 30¢ to make 70¢, so b = 2.33. f* = 45% − 55%/2.33 = 21%.
Yes. This is the canonical one. p = 40%, b = 3, f* = 40% − 60%/3 = 20% of your stack. The wrinkle is that in poker your stack isn’t really your bankroll. There’s a whole literature on pros using Kelly for bankroll management across sessions rather than for a single river decision.
No. Not this version of it. Stock returns aren’t generally binary so there’s no p, q, or b. There’s a continuous analog called Merton’s Share, which is similarly rooted in reward vs variance. What Gamblers Can Teach the Buy-and-Hold Crowd can get you started.
Sort of. The structure is right: repeated, roughly binary, long odds. The problem is that p is a guess and b is a bigger guess, and Kelly is violently sensitive to overestimating your edge. Garbage in, garbage out.
Approximately. If you treat it as approve/reject it’s binary enough to size with. If the expiry aligns with the date such that you are betting strictly on the terminal intrinsic value, the option piece will inherit the binary modeling you imposed on the stock.
No. The variables are too opaque.
Yes. p = 30%, b = 5, f* = 30% − 70%/5 = 16%. Note, you’ll lose this bet more than twice as often as you win it, so your most likely scenario is losing 16%. You can shrink the Kelly fraction if this makes you uncomfortable.
Wrong side of the equation. Run f* on this and you get a negative number, because you’re buying a negative-EV bet.
Yes. It’s the same policy, so notice that the bet only exists on the insurer’s side of it! Every year your neighbor doesn’t buy, she collects a premium and writes a tail. Rebuild cost $500k, premium $3,000, call it a 1-in-500 chance of a total loss.
p = 99.8%
q = 0.2%
b = 3,000 / 500,000 = 0.006 (risk $500k to win $3,000)

f* = 99.8% − 0.2%/0.006 = 99.8% − 33.3% = 66.5%

Positive, as expected since insurers price premiums well above fair value. In this case, her bet size is the $500k house. If the house is most of her net worth, she’s at 100% on a bet capped at 66%. Rather than overbet, she should buy the policy. If she’s worth $5M, she’s betting 10% when she’s allowed 66%, which puts her near quarter Kelly (~14%), and she could skip the insurance. If she’s worth $1mm, it’s a 50% bet, which is more than half Kelly. I’d say take the insurance but I’m a wimp. There are other considerations (would she have the liquidity to rebuild the home or maybe taking the insurance with a high deductible is a better fit), but just doing this exercise gives you a sense of how risky or conservative your choices are relative to the bet share that maximizes long-term wealth.
Yes. Raising the deductible is like you writing a $2,000 policy and collecting $340 a year for it. You’re the insurer again, so work out what you need to believe. Take the high deductible and save $340. Have a claim, and you’re out $2,000 more, but you already banked the $340, so the loss is $1,660.
b = 340 / 1,660 = 0.205

f* = p − q/0.205 = p − 4.88q

Set that to zero, and you get p = 4.88q, which, with p + q = 1, means q = 17%. Your breakeven is a claim every 5.9 years. Anything less frequent and you’re the one with the edge.

Let’s say real-world collision frequency is more like 6%. So p = 94%:

f* = 94% − 6%/0.205 = 94% − 29.3% = 64.7%

Which says risk at most ~65% of your bankroll. The risk here is $1,660. That clears as long as you have about $2,600 in liquid savings, which is to say the sizing check is trivially satisfied for almost everyone so you should generally opt for the higher deductible. For quarter Kelly, we’d need savings of $1,660/(.647 * .25) = $10,262.
Approximately. It’s not truly binary, but if you frame it in a way where you are comfortable with the no consolation prize of a medium outcome, you can see it as paying $40k for a 15% shot at $1M. b = 24, so f* = 15% − 85%/24 = 11.5% of your liquid net worth. This is quite sensitive to your estimate of p of course.
Yes, a classic example of binary-type risk in markets. You risk $8 to make $4, so b = 0.5 — you’re laying odds, same as the moneyline. f* = 90% − 10%/0.5 = 70%. That number is only as good as the 90%. Revise p to 75% and f* is 25%.
Yes, in a subtle way! Your bankroll is capacity, not cash. b = 80/20 = 4, so f* = 25% − 75%/4 = 6.25%. Twenty hours has to be 6% of what you’re working with, which means you can’t run this pitch out of a 100-hour month. If you’re the manager, you can put it in dollar terms by converting to wages.

Finally, I strongly recommend William Poundstone’s book Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street

Description:

In 1956, two Bell Labs scientists discovered the scientific formula for getting rich. One was mathematician Claude Shannon, neurotic father of our digital age, whose genius is ranked with Einstein’s. The other was John L. Kelly Jr., a Texas-born, gun-toting physicist. Together they applied the science of information theory—the basis of computers and the Internet—to the problem of making as much money as possible, as fast as possible.

Shannon and MIT mathematician Edward O. Thorp took the “Kelly formula” to Las Vegas. It worked. They realized that there was even more money to be made in the stock market. Thorp used the Kelly system with his phenomenally successful hedge fund, Princeton-Newport Partners. Shannon became a successful investor, too, topping even Warren Buffett’s rate of return. Fortune’s Formula traces how the Kelly formula sparked controversy even as it made fortunes at racetracks, casinos, and trading desks. It reveals the dark side of this alluring scheme, which is founded on exploiting an insider’s edge.

Shannon believed it was possible for a smart investor to beat the market—and William Poundstone’s Fortune’s Formula will convince you that he was right.

And this is from my notes, Insights From Fortune’s Formula:

a gripping narrative full of 20th century trivia that ties together the birth of information theory, some of the greatest scientific minds of the 1900s, the rise of quantitative finance, and the role of organized crime. These topics come alive in a fresh, memorable way when discovered through the lens of its colorful characters.

It chronicles the history of the efficient market hypothesis (MIT, U Chicago, Paul Samuelson). You can organize its conclusion around this excerpt:

There is much truth in the efficient market hypothesis. The controversy has always been over just how far the claim can be pressed. Asking whether markets are efficient is like asking whether the world is round. The best way to answer depends on the expectations and sophistication of the questioner. If someone is asking whether the world is round or flat, as fifteenth-century Europeans might have asked, then “round” is a better answer. If someone knows that and is asking whether the earth is a geometrically perfect sphere, the answer is no.

editor