Information exists only when the sender is saying something that the recipient doesn't already know and can't predict. Because true information is unpredictable, it is essentially a series of random events like spins of a roulette wheel or rolls of dice.

The more improbable the message, the less "compressible" it is, and the more bandwidth it requires. This is Shannon's point: the essence of a message is its improbability.

Kelly concluded that a gambler should be interested in "compound return," much as an investor in stocks or bonds is. The gambler should measure success not in dollars but in percentage gain per race. The best strategy is one that offers the highest compound return consistent with no risk of going broke.

In the long run, "bet your beliefs" will earn you the maximum possible compound return--provided that your assessment of the odds is more accurate than the public's.

The Kelly formula says that you should wager this fraction of your bankroll on a favorable bet: edge/odds The edge is how much you expect to win, on the average, assuming you could make this wager over and over with the same probabilities. It is a fraction because the profit is always in proportion to how much you wager. Odds means the public or tote-board odds. It measures the profit if you win. The odds will be something like 8 to 1, meaning that a winning wager receives 8 times the amount wagered plus return of the wager itself. In the Kelly formula, odds is not necessarily a good measure of probability. Odds are set by market forces, by everyone else's beliefs about the chance of winning. These beliefs may be wrong. In fact, they have to be wrong for the Kelly gambler to have an edge. The odds do not factor in the Kelly gambler's inside tips.

*par·lay* v. [trans.] (parlay something into) turn an initial stake or winnings from a previous bet into (a greater amount) by gambling: it involved parlaying a small bankroll into big winnings. INFORMAL transform into (something greater or more valuable): a banker who parlayed a sizable inheritance into a financial empire; an excellent performance is quickly parlayed into lucrative contracts. n. a cumulative series of bets in which winnings accruing from each transaction are used as a stake for a further bet. late 19th cent.: from French paroli, from Italian, from paro 'like', from Latin par 'equal'.

The Kelly system manages money so that the bettor stays in the game long enough for the law of large numbers to work.

The short-seller is liable, potentially, for infinite losses.

A convertible bond is essentially a bond with a "bonus" stock option attached.

It turns out that Kelly's prescription can be restated as this simple rule: When faced with a choice of wagers or investments, choose the one with the highest geometric mean of outcomes. This rule, of broader application than the edge/odds Kelly formula for bet size, is the Kelly criterion.

Someone serious about making money should follow the (regular) Kelly gambler, who always maximizes the geometric mean.

In short, the Kelly criterion may risk money you need for gains you may find superfluous; it may sacrifice welcome gains for a degree of security you find unnecessary. It is not a good fit with people's feelings about the extremes of gain and loss. No buy-and-hold stock investor lives long enough to have a high degree of confidence that the Kelly system will pull ahead of all others. That is why the Kelly system has more relevance to an in-and-out trader than a typical small investor.

Kelly betting is a way of making all gambles and investments interchangeable. Given any gambling or investment opportunity, the Kelly wager converts it into a capital-growth-optimal gamble/investment. When the wager is too risky, the Kelly bettor stakes only a fraction of the bankroll in order to subdue the risk. When an investment or trade carries no possibility of a total loss, the Kelly bettor may use leverage to achieve the maximal return.

There are two ways to smooth the ride [in the stock market]. One is to stake a fixed fraction of the Kelly bet or position size. As before, you determine which opportunity or portfolio of opportunities maximizes the geometric mean. You then stake less than the full Kelly bet(s). A popular approach with gamblers is "half Kelly." You consistently wager half of the Kelly bet. This is an appealing trade-off because it cuts volatility drastically while decreasing the return by only a quarter.

Because it is better to be aggressive than insane, it is wise for even the most aggressive people to adopt a Kelly fraction of less than 1. In practical applications, there is always uncertainty about the true odds of the gambles we take. Human nature may further bias the estimation error in the direction desired.

Another method of taming the Kelly system is diversification. Blackjack players sometimes pool their bankrolls. Each takes a share of the group bankroll and plays it independently. At the end of the day they repool their winnings (or losses) and split them. By averaging out the players' luck, the team wins more consistently. Setbacks are fewer.

In a world where financial models can be so incredibly wrong, the extreme downside caution of Kelly betting is hardly out of place. For reasons mathematical, psychological, and sociological, it is a good idea to use a money management system that is relatively forgiving of estimation errors.

The core of John Kelly's philosophy of risk can be stated without math. It is that even unlikely events must come to pass eventually. Therefore, anyone who accepts small risks of losing everything will lose everything, sooner or later. The ultimate compound return rate is acutely sensitive to fat tails.

For true long-term investors, the Kelly criterion is the boundary between aggressive and insane risk-taking. Like most boundaries, it is an invisible line. You can be standing right on it, and you won't see a neat dotted line painted on the ground. Nothing dramatic happens when you cross the line. Yet the situation on the ground is treacherous because the risk-taker, though heading for doom, is liable to find things getting better before they get worse.

Claude Shannon was a buy-and-hold fundamental investor. He said that a smart investor should understand where he has an edge and invest only in those opportunities. In the early 1960s, Shannon had played around with technical analysis. He had rejected such systems: "I think that the technicians who work so much with price charts, with ‘head and shoulders formations' and ‘plunging necklines,' are working with what I would call a very noisy reproduction of the important data." Shannon emphasized "what we can extrapolate about the growth of earnings in the next few years from our evaluation of the company management and the future demand for the company's products…Stock prices will, in the long run, follow earnings growth." He therefore paid little attention to price momentum or volatility. "The key data is, in my view, not how much the stock price has changed in the last few days or months, but how the earnings have changed in the past few years." Shannon plotted company earnings on logarithmic graph paper and tried to draw a trend line into the future. Of course, he also tried to surmise what factors might cause the exponential trend to continue or sputter out.

Paul Wilmott wrote that "life, and everything in it, is based on arbitrage opportunities and their exploitation." This idiosyncratic view is interesting for its candor. The defenders of free markets are often at pains to insist that market prices are "fair" prices and no one "exploits" anyone. Wilmott proposes instead that many of the market's participants are always trying to take the maximum advantage of people who know less than they do. We are unlikely to get very far in understanding markets by pretending otherwise. The operative model is Kelly's gambler, or perhaps Dostoyevsky's The Gambler (who finds that "people, not only at roulette, but everywhere, do nothing but try to gain or squeeze something out of one another").