Below are excerpts from an article written by Michael Mauboussin in 2006 on the importance of position sizing (Size Matters). For fans of the Kelly formula, this is a must-read. Mauboussin highlights a few very important flaws of the Kelly formula when applied to our imperfect, non-normally distributed world of investing. Sizing, Diversification

*“To suppose that safety-first consists in having a small gamble in a large number of different [companies] where I have no information to reach a good judgment, as compared with a substantial stake in a company where one’s information is adequate, strikes me as a travesty of investment policy. -- John Maynard Keynes, Letter to F.C. Scott, February 6, 1942”*

*“As an investor, maximizing wealth over time requires you to do two things: find situations where you have an analytical edge; and allocate the appropriate amount of capital when you do have an edge. While Wall Street dedicates a substantial percentage of time and effort trying to gain an edge, very few portfolio managers understand how to size their positions to maximize long-term wealth.”*

*“Position size is extremely important in determining equity portfolio returns. Two portfolio managers with the same list and number of stocks can generate meaningfully different results based on how they allocate the capital among the stocks. Great investors don’t stop with finding attractive investment opportunities; they know how to take maximum advantage of the opportunities. As Charlie Munger says, good investing combines patience and aggressive opportunism.”*

This is consistent with my belief that investors can differentiate himself/herself from the pack by going beyond security selection, and applying superior portfolio management tactics.

Sizing, Expected Return, Fat Tails, Compounding, Correlation

*“We can express the Kelly formula a number of ways. We’ll follow Poundstone’s exposition: Edge / Odds = F*

*Here, edge is the expected value of the financial proposition, odds reflect the market’s expectation for how much you win if you win, and F represents the percentage of your bankroll you should bet. Note that in an efficient market, there is no edge because the odds accurately represent the probabilities of success. Hence, bets based on the market’s information have zero expected value (this before the costs associated with betting) and an F of zero…if there is a probability of loss, even with a positive expected value economic proposition, betting too much reduces your expected wealth.”*

*"Though basic, this illustration draws out two crucial points for investors of all stripes:*
*• An intelligent investor needs an edge (a view different than that of the market); and*
*• An investor needs to properly allocate capital to maximize value when an investment idea does appear."*

*“In the stock market an investor faces many more outcomes than a gambler in a casino…Know the distribution. Long-term stock market investing differs from casino games, or even trading, because outcomes vary much more than a simple model suggests. Any practical money management system faces the challenge of correcting for more complicated real-world distributions. Substantial empirical evidence shows that stock price changes do not fall along a normal distribution. Actual distributions contain many more small change observations and many more large moves than the simple distribution predicts. These tails play a meaningful role in shaping total returns for assets, and can be a cause of substantial financial pain for investors who do not anticipate them.”*

*“…the central message for investors is that standard mean/variance analysis does not deal with the compounding of investments. If you seek to compound your wealth, then maximizing geometric returns should be front and center in your thinking…For a geometric mean maximization system to work, an investor has to participate in the markets over the long term. In addition, the portfolio manager must be able to systematically identify investment edges—points of view different than that of the market and with higher expected returns. Finally, since by definition not all market participants can have an edge, not all investors can use a Kelly system. In fact, most financial economists believe markets to be efficient. For them, a discussion of optimal betting strategy is moot because no one can systematically gain edges.”*

Notice in order for the Kelly Formula to work effectively, the devil (as usual) lies in the details. Get the odds wrong, or get the edge wrong, the sizing allocation will be wrong, which can reduce your expected wealth.

Another question that I’ve been pondered is how the Kelly formula/criterion accounts for correlation between bets. Unlike casino gambling, probability outcomes in investing are often not independent events.

Psychology, Volatility

*“The higher the percentage of your bankroll you bet (f from the Kelly formula) the larger your drawdowns.*

*Another important lesson from prospect theory—and a departure from standard utility theory—is individuals are loss averse. Specifically, people regret losses roughly two to two and a half times more than similar-sized gains. Naturally, the longer the holding period in the stock market the higher the probability of a positive return because stocks, in aggregate, have a positive expected value. Loss aversion can lead investors to suboptimal decisions, including the well-documented disposition effect.*

*Investors checking their portfolios frequently, especially volatile portfolios, are likely to suffer from myopic loss aversion. The key point is that a Kelly system, which requires a long-term perspective to be effective, is inherently very difficult for investors to deal with psychologically.”*

*“Applying the Kelly Criterion is hard psychologically. Assuming you do have an investment edge and a long-term horizon, applying the Kelly system is still hard because of loss aversion. Most investors face institutional and psychological constraints in applying a Kelly-type system.”*