We are often told that diversification is the closest thing to a free lunch in investing. And yet some of the most successful investors have politely declined this free lunch. Instead, they have opted to, in Warren Buffett's words, "keep all [their] eggs in one basket, but watch that basket closely."

*Concentrated Investing: Strategies of the World's Greatest Concentrated Value Investors* by Allen C. Benello, Michael van Biema, and Tobias E. Carlisle (Wiley, 2016) re-visits "the subject of bet sizing [the Kelly Criterion] and portfolio concentration as a means to achieve superior long-term investment results." In eight chapters it analyzes the techniques of investors Lou Simpson; John Maynard Keynes; John Kelly, Claude Shannon, and Edward Thorp; Warren Buffett; Charlie Munger; Kristian Siem; Grinnell College; and Glenn Greenberg.

Holding a concentrated portfolio is not for the faint of heart and certainly not for the dabbler. It requires an extraordinary amount of skill, intense research, and preferably deep pockets since a concentrated portfolio can have large drawdowns. As a strategy, it goes in and out of fashion. "When times are good, portfolio concentration is popular because it magnifies [or can magnify] gains; when times are bad, it's often abandoned-after the fact-because it magnifies [or can magnify] volatility. Concentration has been out of favor since 2008, when investment managers began in earnest to avoid what they perceive as a risky business practice."

Central to most concentrated investing, and not just value investing, is the application of some version of the Kelly Criterion. Keynes was a "Kelly-type bettor" in that he bet big when he had an edge and wagered nothing when he didn't. Keynes, however, "shied away from attempting mathematical precision because 'our existing knowledge does not provide a sufficient basis for a calculated mathematical expectation.'"

Kelly, simplifying Shannon's rate of transmission theorem used in his information theory paper, offered an equation for the optimal bet size, one that maximized the exponential rate of growth of the gambler's capital. (By the way, he looked at the case of betting on baseball teams, not horse racing.) The formula was *f* = (bp - q)/b = (p(b+1) -1)/b,* where *f** is the fraction of the current bankroll to wager, *b* is the net odds received on the wager, *p* is the probability of winning, and *q* is the probability of losing. The equation can be further simplified to *edge/odds*. Kelly's formula assumes "the possibility of reinvestment of profits and the ability to vary the amount of money invested or bet."

The Kelly Criterion is a very aggressive position sizing strategy and can experience wrenching volatility in the short term. Some investors therefore halve its recommended position size, achieving "three-quarters of the compound return of Kelly with half the volatility." Or opt for some other fractional variation.

Value investors don't always use the Kelly Criterion, and those investors who use some variation of the Kelly Criterion aren't always value-oriented, Thorp being a case in point. But somewhere in the matrix of value investing and position sizing there is a sweet spot. As the authors conclude, "The lesson of this book condensed into a single sentence is: 'Bet seldom, and only when the odds are strongly in your favor, but when you do, bet big, hold for the long term, and control your downside risk.'"