I follow a few blogs regularly. One is mebfaber.com. Meb uses simple, clear language to share data and insights. His recent podcast covered an interview with Ed Thorp, a pioneer in more ways than I can describe here. Listen to it if you have one hour to spare. If not, just read the summary.
The title of the podcast is also the title of this blog post. The consideration of “bet size” has been on my mind recently. Each year, I place a few bets that have small odds of producing large payouts. “Small odds” means that most of these bets don’t win. This is to be expected, but the process of identifying, placing, and following through on such bets is both mathematically complex and emotionally demanding. Ed Thorp is one of the world’s foremost experts on this kind of betting. And at age 84 he has a boatload of experience to share.
Ed says that if you bet too little, it won’t make an impact on your portfolio, which is obvious, but if you bet too much you’ll almost certainly be ruined. The second part of the statement – the part about losing a lot of money – is interesting for two reasons.
First, for a favorable bet with known odds, the Kelly criterion gives us the optimum bet size. The Kelly criterion is optimum in the sense that if the same bet is placed repeatedly, then the Kelly bet size will grow money faster than any other betting method. To illustrate the problem with large bet sizes, consider the following two bets.
- In the first bet (Bet 1), we have a 1% chance of increasing the betting money 200-fold, and a 99% chance of losing everything we’ve bet.
- In the second (Bet 2), we have a 50% chance of quadrupling our money, and a 50% chance of going bust.
Both bets have the same expected return. “On average”, both bets will double our money after a single bet, but stating it in this way is misleading. Here’s why. Kelly says that it is optimum to bet ~0.5% of our wealth if presented with the Bet 1, and ~33.3% of our wealth if presented with Bet 2. Intuitively, this is because the first bet doesn’t win very often, so we bet only small amounts and once in a blue moon we hit the jackpot. With the first bet, the maximum rate of compounding achievable in the long run is less than 0.2%, whereas the with the second bet, we can achieve 14.4%. These numbers can be calculated using some simple formulas that can be found here.
The conclusion from the above is that it is best to bet tiny amounts on profitable bets with rare large payouts (e.g. Bet 1), and one bet of this sort doesn’t contribute very much to long-term growth. This conclusion is perhaps counterintuitive, and in stark contrast to the excitement that the prospect of a 200-fold gain produces. A high-risk high-return bet resembles a lottery, and many people bet on lotteries even though they know that “on average” lotteries lose money. The less obvious point is that even Bet 1 and Bet 2, which have positive expected returns, will lose money if we bet too much.
This brings me to the point about “being ruined” in the title of this post. If, instead of betting the Kelly amount, we bet our favorite amount, say 10% of our wealth, on each bet. The first bet will lose money at a compounded rate of 7.4%. In fact, any bet size larger than 1.26% will lose money on the first bet. Betting less than the Kelly bet size is suboptimal, so it leaves some gains unutilized, but it doesn’t lose money.
There is another consideration here that is perhaps as important. In real life, the odds of profitable bets are seldom known. The odds of some games are known, but the universe is producing an endless stream of opportunities to profit whose odds can only be imperfectly estimated. Human nature messes with our ability to estimate these odds. Placing Kelly bets on mis-estimated odds can also lead to rapid loss of wealth. The fix is simple: be less greedy. By placing smaller bets than we otherwise would, we’ll have a better chance of coming out ahead.