In his book The Dhandho Investor, Mohnish Pabrai discusses the Kelly Formula, a strategy for determining the optimal fraction of a bankroll to bet on favorable odds. Using examples, Pabrai illustrates how to calculate the edge (expected value) and apply the formula to maximize returns while managing risk.
For instance, with an 80% chance to win $21, a 10% chance to win $7.50, and a 10% chance to lose, the Kelly Formula suggests betting 83% of a $10,000 bankroll ($8,300). Pabrai emphasizes the importance of balancing risk and reward, referencing works by William Poundstone and Michael Mauboussin to explain the formula’s practical application.
Here’s an excerpt from the book:
Let’s assume you were offered the following odds on a $1 bet:
– 80 percent chance of winning $21.00
– 10 percent chance of winning $7.50
– 10 percent chance of losing it all
Let’s further assume that you had $10,000 to your name and you were allowed to bet as much of that bankroll as you wanted. How much of that $10,000 would you be willing to put at stake to play this game once?
The answer is clearly not $10,000, as there is a solid 10 percent chance of being in the poorhouse. Betting $1 seems too conservative—it isn’t going to move the needle.
The good news is that exactly 50 years ago, a researcher at Bell Labs in New Jersey, Mr. John Larry Kelly Jr., pondered this question and published his findings. Kelly came up with what is now known as the Kelly Formula. Kelly calculated that the optimal fraction of your bankroll to bet on a favorable bet is:
Edge/odds = Fraction of your bankroll you should bet each time
There is a wonderful book written by William Poundstone entitled Fortune’s Formula that is well worth reading. Poundstone describes the Kelly Formula beautifully.
Michael Mauboussin of Legg Mason recently wrote a paper on the Kelly Formula where he used the following illustration: assume you’re offered a coin toss where heads means you get $2 and tails costs you $1. How much of your bankroll should you bet if you’re offered these odds?
According to the Kelly Formula, the edge is $0.50 [(0.5 × $2) + (0.5 × −$1)]. The odds are what you win, if you win, or $2. So the Kelly Formula suggests you bet 25 percent ($0.50/$2.00) each time.
For the first example, the edge is $17.45 [(0.8 × $21) + (0.1 × $7.50) + (0.1 × −$1)] and the odds are the maximum we can win, or $21. It’s $17.45/$21 or about 83 percent of your $10,000 bankroll, or $8,300.
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