Bankroll Management – Part I
Bankroll management is arguably the most important concept to understand to maximize your chances of success (or rather, minimize your chances of failure).
Did you know that with a $1,000 bankroll and a model with a 55.0% winning percentage, if you bet $100 per game at -110 lines, you would go broke ~14.0% of the time after 100 bets? After 1,000 bets the chances of you going broke are a more staggering ~31.0%.
Why does this happen? Despite a positive expected value, you’re betting too much. And this gives you a high risk of ruin.
The Kelly Criterion states that 5.5% of your bankroll is the ideal wager size to maximize the median return of your portfolio. So, what if we bet $55 instead, which represents 5.5% of our bankroll. What’s our risk of ruin then?
After 100 bets? ~2.0% After 1,000 bets? ~13.0%.
Better, but still significant risk of ruin.
Some might be surprised to see any risk of ruin at a 5.5% bankroll allocation. One of the assumptions, however, that the Kelly Criterion relies on is that bet sizes are a percentage allocation of your portfolio and not a fixed amount. Among sports bettors, a fixed bet amount is frequently referred to as a bet “unit”.
Bet Units vs Bet Allocation
Record: 72-53 +13.7 units
Patriots -7.5 2 units
Sports bettors love to measure their performance or display their picks as a function of “units”. Most people use it and because of its widespread adoption, it’s easy to communicate between parties. Since it’s become the de facto unit of measurement for sports bettors, it is widely accepted that the best way to practice bankroll management is to 1) determine your wager size and 2) never deviate from that bet size.
Let me explain the risks behind that strategy and why Cleat Street doesn’t recommend it.
Flat Betting $55: Expected Value of 1,000 Bets
We all know how to calculate the expected value, or EV, of a single bet. All you need is three inputs:
1) Payoff of a win (Pw): $50
2) Payoff of a loss (PL): -$55
3) Probability of winning (p): 55.0%
So - if we want to determine the EV of 1,000 bets, can we just multiply $2.75 x 1,000 and get an EV of $2,750?
If you had unlimited funds, then yes. While there is variance around our expected win percentage, our ending bankroll would be normally distributed with a median of $3,750 ($1,000 starting bankroll + $2,750 EV). Without the constraint of going broke, the distribution of the ending bankroll looks as follows:
However, most of us don’t have unlimited funds. We are constrained by our bankroll, so we must account for the possibility that we lose our entire bankroll at some point between Bet #1 and Bet #1,000. As a result, we might not get the chance to finish making all of the bets.
Monte Carlo Simulation – Flat Betting
To determine the likelihood and impact of going broke at some point between Bet #1 and Bet #1,000, we can use a Monte Carlo simulation. We simulated the 1,000 bet opportunities 10,000 times resulting in the following risk-return profile:
Risk of Ruin: ~13.0%
Expected Return: ~4.8%
Median Return: ~ $2,645
Expected Portfolio ROI: ~265%
Without the benefit of an unlimited bankroll, the risk of ruin decreases our EV by nearly 5%, decreasing from $2,750 to ~$2,645. Starting with a bankroll of $1,000, our median ending bankroll is ~$3,645 but has a distribution as displayed below:
Bet Allocation of 5.5%: Expected Value of 1,000 Bets
When you bet a percentage of your bankroll, the expected value calculation changes a bit. Your payoff outcomes are now framed as a percentage:
1) Payoff of a win (Pw): 5.0%
2) Payoff of a loss (PL): -5.5%
3) Probability of winning (p): 55.0%
To determine the EV of 1,000 bets, however, we cannot just multiply 0.275% x 1,000 and get an EV of 275%. This is because each bet compounds on one another when you are betting a percentage of your bankroll.
Ok – so instead we determine the expected value by saying that you expect to win 550 bets (55% x 1,000) and lose 450 bets (45% x 1,000) and calculate by compounding the returns as follows:
The above computation reflects the median of the distribution of outcomes as well as the most likely outcome. Yes, the most likely outcome is that you win exactly 550 games, which would generate returns of $2,967. However, this scenario happens only 2.54% of the time.  The rest of the time, you either win more than 550 games or less than 550 games.
 Binomial probability inputs: Prob (Success): 55%, Num. Trials 1,000, Num. Successes, 550.
We get the following risk-return profile:
Risk of Ruin: 0.0%
Expected Return: 5.0%
Median Return: $2,967
Expected Portfolio ROI: ~297%
“So you’re telling me, I have no chance of losing my entire bankroll, and I can increase my EV? That sounds too good to be true.”
You’re right – the above metrics are true, but they don’t tell the whole story. Although the risk of ruin is zero, there are many scenarios where you could still walk away a loser. To properly assess, we need to take a closer look at the distribution of outcomes.
The returns generated by using a bet allocation bankroll management strategy follow a lognormal distribution. A lognormal distribution is frequently used to describe the price of financial assets and effectively states that 1) the lowest that your bankroll can go is zero, and 2) your returns have a long-tail to the right.
Visually, the distribution of the ending bankroll after 1,000 bets looks odd when plotted on a linear scale:
When plotted on a logarithmic scale, however the distribution appears normal (hence the name “lognormal”):
As you can see in the distribution above, there are scenarios where you still walk away a loser after 1,000 bets. In fact, betting 5.5% of your bankroll in this scenario will lead you to losing money approximately 20 percent of the time. To properly assess the risk-return profile, we’ll have to take a deeper look at the full distribution of outcomes in Part II.
What we’ll find is that although the Kelly Criterion is a betting strategy that maximizes median wealth in the long run, there are still considerable risks that may not make it ideal for most bettors. In Part II, we explore Kelly Criterion in further depth and show how you can use the same principles to tailor a bankroll management strategy that better fits your risk appetite.