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CALCULATING ODDS

Calculating the odds is the first step in calculating the probabilities. The mathematical groundwork distinguishing odds from probability was done in the Renaissance by Girolamo Cardano. Writing in Against the Gods, Peter Bernstein maintains that Leonardo da Vinci was fascinated by the concepts, even though he would fail a Year 3 arithmetic test today. Leonardo spent many hours with Cardano until he finally understood the problem. If he understood the concepts then the following primer discussion should not present too much of a challenge to today’s calculator and spreadsheet-enabled trader.

Toss two dice, put money on the outcome and we enter the world of odds and probability. And it is not as far from this common wager to trading the financial markets as many people think. The probabilities of any outcome become different from the odds of any outcome in subtle ways.

Before we calculate the odds, we need first to determine the list of possible results. Each of the pair of dice has six faces giving a total of 11 possible results. Throwing a 0 or a 1 are not possible results. The lowest possible result is a 2 (one dot plus one dot). The highest possible result is a 12 (six dots plus six dots). This gives eleven possible results. All the possible results are shown in Figure A.1 column A – if only life, and trading, were so amenable to a similar process of neat tabulation.

Calculating the odds of any number between two and 12 appearing after a throw depends on the total number of possible outcomes, or the opportunity set. As our Renaissance scholar Cardano told Leonardo da Vinci, the odds are the ratio of favorable outcomes to unfavorable outcomes. If we want the number 7, a favorable result, then there are 30 other possible combinations in the opportunity set representing unfavorable results. In modern terms the odds of any turning up the number 7 are six (favorable results) to 30 (unfavorable results), or more commonly expressed as 1 to 5 (30 divided by 6).

Why not 6 to 36 (1 to 6)? We are saying that out of a total of six throws one of them will produce a 7 because there are six combinations out of a possible 36 combinations that give this result. The total of throws needed is six – one favorable outcome plus five unfavorable outcomes.

The full list of favorable outcomes is tabulated in column C, Figure A.1. The ratio of favorable to unfavorable outcomes, the odds, are in column D.

CREATING PROBABILITY

The probability of an event is the ratio of favorable outcomes to the total opportunity set. The difference is important. Calculating the odds compares favorable outcomes to unfavorable outcomes. Calculating probability compares favorable outcomes to all possible outcomes. Leonardo, like many high school students and ‘wanna-be’ traders, had difficulty in understanding the difference. I imagine Cardano dragged the dice out yet again to illustrate the difference.

The divergence between odds and probabilities appears in the ways the outcomes can be reached. There are 36 possible combinations, or ways, of reaching those 11 results. Column C, favorable outcomes, shows the number of combinations required for each result. Some results, such as number 12, only occur from one group of combinations – six dots and six dots. At the other extreme, number 7 is a result reached by six different combinations. They are 6 + 1, 5 + 2, 4 + 3, 3 + 4, 2 + 5 and 1 + 6. For those who do their mathematics simply, the total number of favorable outcomes is equal to the total opportunity set.

From these calculations we create a table of probability. It tells us there are six possible combinations that give a result of number 7. This is six out of the 36 combinations available. This ratio of favorable outcomes to the total opportunity set is written as a probability of 6/36. The probability of a 12 occurring is found in just one combination out of the 36 available and written as 1/36.

We make these calculations because we know the limits of the total opportunity set – lower limit 2 and upper limit 12 – required to produce the total opportunity set.

At a more complex level we can perform similar calculations for a pack of cards, or for the market. The professional card player knows these probabilities, recognizes them instantly and uses them to his advantage. The trader does the same by recognizing market conditions that tip the balance of probability in his favor.

In transferring the discussion of odds and probabilities from the dice to the market, we shift from ‘favorable outcomes’ to ‘events’. The father of Modern Portfolio Theory, Harry Markowitz, showed the mathematics required for this is much more powerful than a simple calculation of odds and probabilities. Although we know the effects of these events it is more difficult to attach a value to them. The extent of a rise is theoretically infinite, and a fall is limited only by zero. The total opportunity set is ill-defined so the calculation of probability that looks at the ratio of favorable outcomes to the total opportunity set is difficult, if not impossible, to complete.

This level of financial mathematics is beyond the scope of our discussion and readers who wish to follow this in more detail should refer to Against The Gods, or Hedge Funds by Richard Hills. Those equipped with the appropriate guidance equipment can make their way through A Treatise On Probability by John Maynard Keynes.

Extended discussions of risk and probability in wider contexts are found in Taking Risks: The Science of Uncertainty by Peter Sprent and Lady Luck: The Theory of Probability by Warren Weaver. The relationship between odds and probability form the basis of some linked trading and money management approaches. David Caplan in Trade Like a Bookie deals with these types of approaches in detail.

Difficult or not, the trader needs a rule of thumb to estimate the ratio of favorable outcomes to the total opportunity set – the probability. Chart tools and chart patterns point the way. What tips the balance of probability in favor of a particular outcome depends on market conditions and the section of the market you are trading. The way prices clump together with reduced trading ranges at particular levels suggests outcomes with a higher than normal probability.

Some trade these clumps for small profits using leverage from options and other derivatives. Equity traders look for trend and break-outs that deliver higher returns. How far does the trend go? This can be calculated using a number of techniques that put the balance of probability in our favor where one outcome is more likely than another.

We shift the balance of success when our trading strategy is based on moves from clump to clump, capturing a break-out as it happens and taking a profit as it slows. This rule of thumb serves the private trader well.