Edward Thorp is one of the greatest investors in history as well as an extraordinarily interesting character. Judge for yourselves: Thorp was a student of mathematics who demonstrated how to apply mathematics in many areas of everyday life. He invented card counting and discovered the apparently impossible – how to beat the dealer in blackjack – and he practiced this very successfully. He played in casinos, oftentimes visiting them in disguise. After having been discovered and banned from casinos, he described it all in his first best-seller Beat the Dealer.
He then redirected his attention from cards to roulette and figured out how significantly to beat the casino also in roulette. This sounds incredible, because everyone knows that the winning odds in roulette are theoretically in the casino’s favour. Not necessarily so in practice, however. Thorp realised this and put his findings into practice using the first wearable computer.
Casino owners were livid, and Thorp began to fear for his life. He therefore left casinos to their fate and thought about which field would provide him with plentiful room for his further intellectual enjoyment. He chose the capital markets and stock exchange.
Thorp was a forerunner of quantitative investment methods, a pioneer of convertible arbitrage, and the actual first author of the option pricing method for which Merton and Scholes were later awarded a Nobel Prize. He described his investment practices in his next best-seller, Beat the Market. Above all else, he was not solely a theorist but also a practicing investor. His hedge fund Princeton/Newport Partners achieved an average annual return of 19.8% in the 19-year period between 1969 and 1988.
Thorp also was an early investor in Warren Buffett’s Berkshire Hathaway, and 17 years before the cover was blown off fraudster Bernie Madoff’s scheme Thorp already had demonstrated by way of fictitious results and trades that Madoff was a crook.
Whether it was about cards, roulette, or stocks, all of Thorp’s activities had one common denominator. He was always working in an environment where he endeavoured to find an optimal size for repeated investments (or bets) for a given set of investment opportunities with various expected returns and various probabilities for their occurrence. He often utilised the Kelly Criterion, named after its author, the physicist John Kelly:
f is the part of portfolio which should be placed into the investment,
b is the ratio of the return on investment, and
p is the probability of winning.
For example, if there is a 60% chance of winning (p = 0.60) and the investor makes 10% (b = 1.1), he or she should invest 24% of his or her portfolio in this manner. If the probability of winning increases to 70%, he or she should invest 42% of his or her portfolio, and if the expected return increases to 20% with a 60% probability of winning, he or she should invest 27% of the portfolio.
Generally speaking, the Kelly Criterion offers a good indication for how to calibrate individual investments. If we find an investment which we believe promises a solid expected return and above-average probability of its being achieved, then we must not hesitate and must invest a goodly amount.
When you tinker a little with the parameters in the formula, you will find that the size of the investment is influenced more by the probability of success than by the expected returns. Investors frequently make the mistake of being attracted by high potential returns while underappreciating their relatively low probability. We should avoid such investments and instead seek investments where the expected return may not be staggering but has a high probability of occurring. This is what the Kelly Criterion teaches us.