The key decision to be made for each successive wager concerns the fraction ‘f’ of one’s current bankroll to place at risk. William Poundstone (Fortune’s Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street) expresses Kelly’s (John Kelly and Claude Shannon, Bell Labs, 1956) criterion in the form of “f = edge/odds”, where “edge” refers to the expected gain from the current bet and “odds” refers to the multiple of the amount of the bet that the winning bettor stands to receive.
Starting with a $100 bankroll, the bettor is to toss a fair coin four times, getting back six times the amount of his bet each time he throws a head, and nothing each time he throws a tail. The expected gain on this highly favorable bet is 200% – players win $5 with every win and fall back $1 with every equally likely loss. The payoff odds are 5 to 1, because the successful bettor gets back five times the amount of the bet, in addition to the bet itself. Hence f= edge/odds = 2/5 = 40% in the present case; this Kelly Criterion instructs the bettor to wager 40% of his bankroll at each opportunity. By doing so, he stands 1 chance in 16 of ending up with a mere $12.96 in his bankroll, and an equal chance of possessing $8100. The intermediate possibilities are $64.80 and $1620, each with 4 chances in 16, and $324 with 6 chances in 16. The arithmetic and geometric means of these 16 possible bankrolls are $1049.76 and $324, respectively. The Kelly criterion for deciding on the fraction ‘f’ of a bankroll to bet is to choose the one that maximizes the geometric mean of these 16 possibilities. Poundstone illustrated the use of the formula with an example based on a pin-ball machine, based in turn on Pascal’s triangle.
Claude Shannon and Edward Thorp, Professors of Mathematics at MIT during the 60’s benefitted handsomely using Kelly Criterion in their own portfolios and personal income. Prof. Edward Thorp also created a successful hedge fund PNP (Princeton-Newport Partners) that beat the S&P 500 threefold between 1968-1988 by producing 20% annualized returns.
Kelly Criterion and Warren Buffett’s Investment
In November 1963 Mr. Buffett invested 40% of the Buffett Partnerships’ assets into a single business, American Express. American Express’ stock was cut in half due to “Salad Oil Crisis” just before Buffett’s large purchase. Mr. Buffett generated a three or four bagger return on this American Express investment over three years. The conservative odds of this bet are as follows:
Odds of a 200% or greater return in 3 years – 90%
Odds of a breakeven return in 3 years – 5%
Odds of a loss of up to 10% in 3 years – 4%
Odds of a total loss on the investment – 1 %
Based on these odds, the Kelly Criterion would suggest betting
f = edge/odds = (2*0.9+1*0.05-0.1*0.04-1*0.01) / 2 = 0.918
91.8% of the partnership’s assets of the funds. Mr. Buffett stayed well within the maximum suggested. Notwithstanding the sub-optimal betting the return on investment was at least 300% in 3 years!
Fortune Formula Kelly Criterion works!
Courtesy: Book Review by James Case & Dhandho Investor by Mohnish Pabrai