What A Trip To The Casino Can Teach You About Investing And Risk
Learn the ins and outs of how gambling works from a quantitative investor, and use it to your advantage in investing for the long term.
For better or worse, there is a certain thrill involved with gambling that makes otherwise rational individuals make irrational bets. When the Mega Millions jackpot hit $1.6 billion in October 2018, over 370 million tickets were sold—that’s more than one ticket for every person in the United States.
Of course, the vast majority of those who bought the tickets knew their chances were slim of winning the jackpot in the end. They probably even realized that, on average, they’d lose money by buying a ticket. What you should know is that gambling is never an effective use of money, and gambling addiction can be a major problem. In this article, we’ll review some ways to think about light gambling, but if you gamble often, we encourage you to talk with an appropriate professional.
Even with small-time gambling, it’s important not to wager any more money than you can afford to lose. That said, if you set a reasonable, fixed limit of what you will gamble ahead of time (and stick to it!), an occasional night of gambling in Vegas or an entry into your office Fantasy Football pool can be fun.
Of course, it’s even more fun if you win. If you do gamble, here are two things you should keep in mind to help maximize your chances of returning, or exceeding, your stake: know your odds, and size your bets accordingly.
Part 1. Knowing Your Gambling Odds
One of the important skills to have is knowing exactly what your odds are, and how you can improve them.
In a skill-based gambling game, like poker, you can increase your chance of winning with practice, but even games that are completely dependent on luck can have different odds based on strategy. As an extreme example, let’s consider the Mega Millions lottery I mentioned earlier. In this game, players choose five numbers between one and 70, and a sixth number between one and 25. The lottery then draws numbers at random, and players win successively more money based on the amount of correctly matched numbers. Players can win the jackpot if they match all six numbers correctly, and if the jackpot isn’t won at a drawing, the money rolls over to the next drawing.
The first thing to note is how miniscule the chance of winning the jackpot is. The total number of possible combinations for the lottery is 302,575,350, meaning you have an eight-times higher chance of flipping two nickels and having them land on their side than to win the jackpot.
One strategy that may seem enticing at first is to closely monitor the jackpot and only buy when the value gets very high. Since every ticket costs $2, you could technically buy every combination of numbers for a bit more than $600m, seemingly guaranteeing a profit if the jackpot exceeds that number. However, an important caveat is that the jackpot is split between all winners, and a split jackpot would ruin what seemed like a sure win. Nonetheless, the prospect of a large jackpot is enticing. Using LottoReport.com’s data from the 97 Mega Millions drawings between December 2017 and November 2018, we found a pretty clear relationship between jackpot size and number of tickets sold:
Relationship between Mega Millions Jackpot Size and Tickets Sold
If we assume that each person randomly chooses the numbers on their ticket, we can pretty easily calculate the odds of a split jackpot (using the same set of data):
Mathematical Chance of Winning as Jackpot Increases in Value
At a jackpot size of just over $1 billion, it becomes more likely than not that someone will win the grand prize. Somewhat more counterintuitively, the probability that someone else will win the jackpot given that you also won the jackpot is about the same, i.e. around 50% if the jackpot is over $1 billion. This means that there is probably some “sweet spot” in which the jackpot is small enough that not as many people decide to buy tickets, but still large enough to maximize your possible returns.
Part 2. Sizing Your Bets
The second skill you should have is to know how to size your bets. Even with great odds, most people place odds that are too conservative, hampering their potential winnings, or too aggressive, increasing their risk of losing money. In one study from 2016, participants were given $25 and were allowed to bet on a biased coin that had a 60% chance of landing on heads. Even though the participants were given even odds and the probability of winning was high—i.e. the game was stacked in their favor—the majority of the participants performed suboptimally, with 30% of them going bust within 30 minutes. Their mistake was unsound bet sizing.
You can view bet sizing as a spectrum of risk-seeking behavior, where we have the most conservative option at one end—don’t bet at all, no matter how good the odds are—and the most aggressive on the other—bet the farm (or better yet, mortgage the farm and bet that money too!) on any gamble, no matter how long the odds are. Mathematically, both of these behaviors are well below optimal if you want to maximize your betting winnings, and the optimal strategy exists somewhere between the two.
The optimal bet-sizing strategy is a matter of math.
John Kelly found the optimal solution in 1956 for how to size your bets if you want to increase your wealth optimally in the long-run. The so-called “Kelly Criterion” can be described in a short formula. Then, we can test out the formula with simulations of what individuals’ gambling circumstances might actually be like. What we’ll find at the end is exactly what casinos already know well—that typically the best gambling strategy is to only gamble a small amount of your wealth, while diversifying your bets.
Using the Kelly Criterion
The mathematical formula for the Kelly Criterion is:
f = (pb – q) ÷ b
where f is the fraction of your bankroll that you should bet, p is your chance of winning, q is your chance of losing (i.e., 1 – p), and b are your net odds, i.e. the amount of money you would win by betting $1 and winning. For example, in the biased coin study described earlier, the participants should have bet (60% x $1 – 40%) / $1, which equals 20% of their bankroll.
The participant who follows this strategy could expect their wealth to grow by 2% per bet, on average.
Simulating Possible Outcomes
We can test this out by running a Monte-Carlo simulation, in which we simulate a strategy thousands of times to estimate the efficiency of a strategy. Assuming that each player can play 300 games in the 30 minutes allotted to them, and that the players’ “wealth” is capped at $250, how well do the players do if they follow the optimal bet sizing strategy? Let’s simulate the results for one player first:
This player had a very volatile and bumpy start, but by following the Kelly Criterion they managed to maximize their wealth after a bit less than 100 coin tosses.
Let’s see what happens when we simulate the wealth of 10,000 people:
When we look at the summary results above, things look very rosy. Over 90% of players end up maxing out their wealth, and no-one went broke when they followed the Kelly Criterion. However, what is missing from this chart (compared to the previous figure) is the amount of volatility in the results. If you look back at the single individual, you can see just how volatile their wins and losses were. At any point, a bad run of three tails in a row would cut your wealth in half. This sort of drawdown is a rare event in stock markets and can make even the steadiest investor question their strategy, but we would expect a 50% drawdown to happen over 19 times if we played the coin-toss game 300 times as we posited earlier.
Casinos Play Optimally and Pool Money to Help Reduce Likelihood of Losses
While playing optimally—if emotion wasn’t part of the game—might help lead to better results, it can’t solve for the volatility in wins and losses. And yet, that’s what most players want: to win and keep on winning.
Maybe you could pool your money with many other players and share your collective winnings in order to reduce your likelihood of losing… Congratulations, you just invented the casino!
Positive expected returns and small bet sizing are exactly what allows a casino to reduce their risk and maximize returns, and why you see casinos impose maximum limits on bet sizes, even if the odds are in their favor.
Outsmart average. Be more like the casino with your investments.
When thinking of gambling, everyone wants to be James Bond—calmly going all-in on what seems like an impossible hand, and then winning with a royal flush in the end. But for each James Bond, there are thousands of gamblers losing their money in games specifically designed to be stacked against them. What’s really cool is to be the casino. Strict regulations and the astronomical amounts of cash needed to create your own casino makes it a hard business to crack in to, but what you can do is implement some of the factors that help make casinos successful into your own investment moves:
1. Diversify your bets.
Casinos never bet all their cash on one horse. Similarly, you can improve your investment returns by diversifying your investments across different asset classes and geographic markets.
2. Play the odds.
A casino will never enter into a bet in which they expect to lose money. Most individuals would probably say they behave the same way, and yet millions of Americans keep significant portions of their cash in checking or savings accounts with yields far below the current inflation rate, almost guaranteeing that they’ll lose money in real terms. If you don’t want to invest, help maximize your cash by putting it to work in a high-yield cash account, like Betterment Everyday™ Cash Reserve.
3. Turn Your Losses into Wins
A small consolation if you lose at the gambling table is that your losses are tax-deductible, up to the extent of your other gambling winnings. You can do even better, however, by harvesting and deducting your investment losses, while still retaining the potential for investment upside.
Betterment Everyday Cash Reserve
Betterment Everyday Cash Reserve (“Cash Reserve”) is offered by Betterment LLC. Clients of Betterment LLC may participate in Cash Reserve through their brokerage account held at Betterment Securities. Neither Betterment LLC nor any of its affiliates is a bank. Through Cash Reserve, clients’ funds are deposited into one or more banks (“Program Banks“) where the funds earn a variable interest rate and are eligible for FDIC insurance. Cash Reserve provides Betterment clients with the opportunity to earn interest on cash intended to purchase securities through Betterment LLC and Betterment Securities. Cash Reserve should not be viewed as a long-term investment option.
Funds held in your brokerage accounts are not FDIC‐insured but are protected by SIPC. Funds in transit to or from Program Banks are generally not FDIC‐insured but are protected by SIPC, except when those funds are held in a sweep account following a deposit or prior to a withdrawal, at which time funds are eligible for FDIC insurance but are not protected by SIPC. See Betterment Client Agreements for further details. Funds deposited into Cash Reserve are eligible for up to $1,000,000.00 (or $2,000,000.00 for joint accounts) of FDIC insurance once the funds reach one or more Program Banks (up to $250,000 for each insurable capacity — e.g., individual or joint — at up to four Program Banks). Even if there are more than four Program Banks, clients will not necessarily have deposits allocated in a manner that will provide FDIC insurance above $1,000,000.00 (or $2,000,000.00 for joint accounts). The FDIC calculates the insurance limits based on all accounts held in the same insurable capacity at a bank, not just cash in Cash Reserve. If clients elect to exclude one or more Program Banks from receiving deposits the amount of FDIC insurance available through Cash Reserve may be lower. Clients are responsible for monitoring their total assets at each Program Bank, including existing deposits held at Program Banks outside of Cash Reserve, to ensure FDIC insurance limits are not exceeded, which could result in some funds being uninsured. For more information on FDIC insurance please visit www.FDIC.gov. Deposits held in Program Banks are not protected by SIPC. For more information see the full terms and conditions and Betterment LLC’s Form ADV Part II.
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