I wrote the following at RealMoney on 4/10/2007

I’ve written two columns already about “spring cleaning” a portfolio, walking you first through the criteria I use when reshaping my holdings and then through the stocks and decisions those criteria pointed to. But there’s one aspect I didn’t cover: What’s the best way to size positions for a long-only equity portfolio?

In order to answer this question, I use the Kelly criterion (popularized in the excellent book Fortune’s Formula), which says that a position size should be equal to (edge/odds). Edge is the excess return that you expect to earn on average. Odds describes the likelihood of winning. A 50/50 bet makes odds equal to 1.

There is added complexity in applying this to stocks, because in gambling, each game is (mostly) uncorrelated with the last one. In investing, if you have a number of investments going at the same time, they are to some degree correlated with one another.

That’s particularly true for me, because I concentrate sectors. I believe that my portfolio acts more like a portfolio with half the number of positions because of the correlatedness of the various names in the portfolio. Thus, the 35 names in my portfolio behave more like 18 uncorrelated names.

The Kelly criterion applied to stock investing would recommend a fixed-weight portfolio. Optimally, you would rebalance daily to those fixed weights. But two factors interfere: First, there are costs to trading, and second, momentum tends to persist in the short run.

To me, those realities mean that you can have fixed weights, but that you set a band around those fixed weights for rebalancing. I use a 20% band, but the more I think about it, the band should be smaller, maybe 10%. My portfolio has gotten bigger over the past few years, and trading costs are a smaller-percentage cost factor. I’ll stick with 20% for now. It has served me well, but I will re-evaluate this.

I firmly believe that my eight rules tilt the odds in my favor. What are they?

1. Industries are under-analyzed relative to the market on the whole and relative to individual companies. Spend time trying to find good companies with strong balance sheets in industries with lousy pricing power, and cheap companies in good industries where the trends are not fully discounted.
2. Purchase equities that are cheap relative to other names in the industry. Depending on the industry, this can mean a low price-to-earnings ratio, low price-to-book value ratio, low price-to-sales ratio, low price-to-cash flow from operating activities, low price-to-free cash flow or low enterprise value-to-EBITDA.
3. Stick with higher-quality companies for a given industry.
4. Purchase companies appropriately sized to serve their market niches.
5. Analyze financial statements to avoid companies that misuse generally accepted accounting principles and overstate earnings.
6. Analyze the use of cash flow by management to avoid companies that invest or buy back their stock when it dilutes value, and purchase those that enhance value through intelligent buybacks and investment.
7. Rebalance the portfolio whenever a stock gets more than 20% away from its target weight. Run a largely equal-weighted portfolio because it is genuinely difficult to tell what idea is the best. Keep about 30-40 names for diversification purposes.
8. Make changes to the portfolio three or four times per year. Evaluate the replacement candidates as a group against the current portfolio. New additions must be better than the median idea currently in the portfolio. Companies leaving the portfolio must be below the median idea currently in the portfolio.

I need to calculate what I think my edge is. My procedure will be messy and somewhat ad hoc, but I would rather be approximately right than precisely wrong. How much is each rule worth?

The offensive rules are 1, 2, 7 and 8. Each of those generates roughly 2% of alpha annually. The defensive rules are 3, 4, 5 and 6. Each of those generates roughly 0.5% of alpha annually. Now, that would be an alpha of 10% annually for a portfolio that followed all those rules perfectly. My alpha over the last six years and seven months has been greater than that, but I will chalk up the overage to happenstance. Maybe more should go to happenstance, but I will discuss that below.

If my edge is 10% per year, what are my odds? I have long maintained that the idea I believe is best probably is as good as my 35th idea. Too many portfolio managers naively believe that they can identify their best ideas, when there are so many unknowns to an outside, passive, minority investor. My methods work on average, over the intermediate term. Most of my investments work if I give them two to three years to develop. Time is usually on my side with my methods. That would imply that my odds might be 50/50, or 1:1. That’s a coin flip, but one that, repeated often enough, leads to an extra 10% per year. Using the straight Kelly formula, that would mean I should have 10 positions; 10% divided by 1 is 10%, and for each position to be 10% of my portfolio, that would mean 10 positions. But I own 35 positions. What do I do?

As I mentioned above, because I concentrate industries, my 35 stocks behave more like 18. An equal-weight 18-stock portfolio has weights of 5.5%, and that’s still not 10%. Again, what do I do?

Good investing is based on humility, a willingness to accept that there are many things that you don’t know and that many successes may not be due to skill. Many who use the Kelly criterion decide to go “half Kelly” and cut position size in half because it also cuts volatility in half — but it also diminishes the expected returns by 25%. It’s a humble way to go.

Another way to think about it is that with quantitative investing, when you find a good strategy and have vetted it though back-testing, typically the alpha achieved when the strategy goes live is half of what the model would have predicted.? Though I have already given my edge a haircut, perhaps more of the success needs to be attributed to happenstance.

Whether due to happenstance or conservatism on my part, cutting the 5.5% in half leads to positions of 2.8% in size. How many equal-weighted positions is that? 36. That’s about what I have, and when conditions are bullish, I contract the number down to 30. When conditions are bearish, I hold up to 40 positions. Most of the time, I keep it at 35. That keeps me disciplined, which is a virtue that yields dividends in its own right.

That’s what I do. But what should you, my reader, do?

I encourage you to calculate your edge.

Look at your annualized returns over the past few years, and compare them with the index that you want to beat. Take the? excess (assuming there is one), and cut it in half. Some things went well for you that are not attributable to your skill.

I encourage you to calculate yours odds.

Look at your trading. Divide it into two categories, winners and losers. How often do you win (relative to the index)? When you win (vs. the index), how much do you win on average? When you lose (vs. the index), how much do you lose on average? (Make sure you use your annualized results for this exercise.)

If you consistently lose vs. the index, I’d buy the index instead of continuing to lose capital trying to do it myself. If you beat the index, then work out the calculation that I went through for my portfolio. Take into account your conservatism, but even if you are aggressive, I strongly discourage you from using portfolio weights higher than the full Kelly criterion; it’s too dangerous.

In general, I believe skilled investors with moderately sized portfolios are best served by diversification equal to what I use.? On the raw Kelly criterion, it’s equivalent to saying that one has an alpha between 2.5% and 3.3%. Hey, that’s hot stuff!? Most mutual fund managers would kill for those returns.

In summary, size your positions inversely to your expected alpha. To the extent that you are less certain of your skills, expand the number of your positions.