The Kelly Criterion

The Kelly Criterion is a formula for choosing how large a bet to make on each trade/investment/gamble. It works for the stock market, though it was originally developed for gambling. The formula is simple: bet the proportion of your investment as defined by the ratio of expected return divided by maximum return. Expected return is what you expect in the long run.

So, the formula is: P_invest = E(r) / M(r)
Proportion of portfolio to invest = P_invest
Expected return= E(r)
Maximum return = M(r)

Now, a couple of examples:

1. If you flip fair coin and win $1 if heads and lose $1 if tails, the expected return is $0 (.5 x $1 + .5 x -1). The maximum return is $1 (if heads). Therefore, the Kelly criterion suggests you bet no money ($0/$1). This makes sense, because you should not invest money where you expect to only break even.

2. You want to short Apple (AAPL) because you think there is an 80% chance the stock will go down in the next month. You think if that happens, the stock will go down 10%. You figure that there is a 20% chance that the stock will go up 5%. The expected return is 7% (.8 x 10% + .2 x -5%). The maximum gain is 10%. The Kelly formula suggests that you invest at most 70% (7/10) of your portfolio.

3. Same thing, shorting AAPL. You like the odds, so you increase your leverage by buying put options. You buy just out of the money options. Now, there is a 70% chance that your options expire worthless (-100% return) and a 30% chance that you make 300%. The expected return is +20% (.7 x -100 + .3 x 300). The maximum gain is 300%. The Kelly formula says that you should bet less than 1/15 (about 6.5%) of your portfolio (20/300).

One thing to consider is that the Kelly formula seeks only to maximize gains. If you wish to minimize portfolio variability as well, you should invest significantly less than the maximum allowed by the Kelly formula. Also, keep in mind that the formula is only as good as your guesses of probability.

I recommend a Legg Mason article on the Kelly Criterion, or this paper by Edward Thorp (who used it to great effect).

Visit Cisiova’s website for their advanced online Kelly Criterion calculator, which allows you to enter a large number of possible outcomes.

If you liked this post you may want to check out William Poundstone’s book Fortune’s Formula.

Disclosure: I own no Apple stock, long or short. Unfortunately, I did once lose money shorting AAPL. My disclosure policy never loses me money.

Cheap isn’t always cheap

I found American Realty Investors (ARL) by searching for cheap stocks–those trading below book value. It is down 15% since I found it, and it is now selling at half of tangible book value. I think that, if anything, the company’s book value understates the true value of its assets, because all its recent land and property sales have resulted in gains relative to the prices those properties were carried at on the books.

However, one man who is highly involved in ARL through various private companies that own a majority of ARL and manage it, Gene Phillips, is not exactly the kind of guy you would want to bring home to meet the parents. In fact, he and the outside company that manages ARL (Basic Capital Management) have previously been fined $850,000 by the SEC for failing to report certain stock holdings. Even that does not bother me much. But Gene Phillips has been involved in other problems, such as two high-profile bankruptcies of previous real estate companies: his private company, Phillips Development, went bankrupt in 1973 with $30 million in debt in what was then the largest bankruptcy in South Carolina. In the 1980s, Phillips helped to grow Southmark Corp. through prodigious deal-making. But by 1989 the house of cards collapsed and Southmark went bankrupt, ending with a book value of negative $759 million, a drop of over $1.5 billion from the company’s 1987 book value of $861 million. Through the entire 1980s there had been numerous instances of self-dealing and dealings with questionable characters, including loans to Herman K. Beebe Sr. or companies controlled by him; Beebe had months previously been convicted of defrauding the SBA (small business administration).

So, while American Realty Investors (ARL) may seem cheap, it is not a Goode value. Neither are the similar companies Transcontinental Realty Investors (TCI) or Income Opportunity Realty Investors (IOT). All three companies are advised by Basic Capital Management, trade below book value, and suffer from the same conflicts of interest and self-dealing that have pervaded other companies that Gene Phillips has touched.

While the last five years have been one of the largest real estate booms ever in the US, shareholders in ARL would have lost money over that time period. This article should serve as a reminder that it is a good idea to check out a company’s management (and controlling shareholders) before investing, lest your fate be similar to that of previous ARL shareholders.

Disclosure: I hold no shares of any of the companies mentioned. In fact, I own no REITs or public real estate companies, although I do directly own some real estate.