What expectancy describes

The average of a record that already exists 

Take a list of closed positions with the money result of each written beside it, gains as positive figures and losses as negative ones. Add the column and divide by the number of entries. What comes out is the average result per position across that list, and that figure is what the word expectancy names. There is nothing else in it. It is not a projection, it is not a rate, and it is not a property of a method. It is the mean of a column of numbers that have already happened.

The formula usually printed alongside the word looks more elaborate than that, and it is the same calculation rearranged. The proportion of recorded outcomes that closed as gains is multiplied by the average size of those gains. The proportion that closed as losses is multiplied by the average size of those losses. The second product is subtracted from the first. Splitting the column into two groups and weighting each group by how much of the column it occupies gives exactly the same number as adding the column and dividing by its length, which is worth verifying once, because a formula that has been checked stops looking like an oracle and starts looking like an average.

The rearranged form is the one that gets used, because it separates the two ways a record can differ from another record. One record can differ because outcomes closed as gains more often. Another can differ because the gains that did occur were larger. In the rearranged form those are two visible terms rather than a single mean with everything folded into it.

Two terms that pull against each other 

The two terms are not independent, and their relationship is the reason the figure gets quoted at all. A record in which a small proportion of outcomes closed as gains can produce exactly the same figure as a record in which a large proportion did, provided the average sizes differ in the opposite direction. That is the argument the term is conventionally deployed to make. The proportion of outcomes that closed as gains, on its own, says nothing about the average of the record, and the average size of a gain, on its own, says nothing either. Only the combination describes the record, and only the record it was computed on.

The same property cuts the other way, and that half is quoted far less often. Because one term compensates for the other, two records that look nothing alike compress to the same figure, and the compression cannot be undone. A single number cannot say whether it came from many small gains or from a few large ones separated by long stretches of losses, and those are very different sequences to hold a position through.

Why the figure needs a large sample 

Everything above describes a completed list, and as a description of a completed list the arithmetic is exact. The reason the term is treated carefully is that nobody computes it in order to describe a list. It is computed in the hope that the average of the entries so far says something about the entries still to come, and that hope is a statistical claim rather than an arithmetic one. The mean of a sample estimates the mean of whatever the sample was drawn from, with an error that shrinks as the sample grows, and it shrinks slowly.

The structure of the formula makes that worse rather than better, because every one of its inputs is itself estimated from the same short record. The proportion term moves in large steps when there are few entries, since one entry out of a handful shifts the proportion by a large fraction of itself. The two average sizes are means of even smaller sub lists: a record of a few dozen entries may contain only a dozen gains, so the average gain is a mean of a dozen numbers that one unusual entry can move a long way. The figure inherits the instability of all of its parts at once, and the instability compounds rather than cancelling.

The block shows the mechanism at its plainest. Reclassifying one entry moves the figure by three times the figure itself, changing its sign in one direction and multiplying it in the other. Nothing about anything follows from a quantity that fragile, and the fragility is not a flaw in the formula. It is what a mean of a very short list is.

The spread around the average 

Sample size is only half of the problem. Even where an underlying average genuinely exists and is known exactly, individual runs scatter around it, and the scatter is not small. A set of proportions describes how a very long run divides up. It places no constraint at all on how any particular short run divides up, and a short run is what a record is.

The distance between the extremes in that block is several times the total the inputs imply, from inputs that never changed. That distance is what variance means in practice. It is also why a figure computed on a short record is mostly a description of which arrangement happened to occur, rather than of the inputs that produced it, and why the same record extended by another stretch of entries can return a materially different figure without anything about the method having changed at all.

What the figure does not say 

Several things sit outside the arithmetic entirely, and each of them is routinely read into a figure that does not contain it.

• It looks backwards. A mean of results that have already been recorded describes those results. Nothing in the calculation carries forward, and the conditions the record was produced under, the instruments, what it cost to trade them and the person keeping the record, do not hold still while the record lengthens.
• It is silent on order. Two records with an identical figure can arrive in completely different sequences, and the sequence is what determines whether an account is still in a condition to reach the average. Position sizing and loss limits govern that. A mean does not.
• It absorbs the largest single entry by construction. A record containing one loss far larger than everything around it presents a figure that resembles the rest of the record and misdescribes the part of it that mattered most.
• It changes with cost. A figure computed on gross results is not the figure computed after the spread and the commission charged on every position, and the two are quoted interchangeably often enough that the difference is worth stating explicitly.

Where a position actually closed is also less determinate than a spreadsheet makes it look. A recorded loss can be larger than the level chosen in advance for closing the position, because a market that gaps or moves faster than an instruction can be filled does not stop at the level the instruction names, and losses on a leveraged contract are not limited to the amount deposited. The instructions that determine where a position closes, and what each of them does and does not fix, are set out in the order types reference.

Where practitioners disagree 

The first argument is about the unit. One convention records each entry as a money amount, which is what the account actually experienced. Another records it as a multiple of the amount risked on that position, which makes entries from different instruments and different sizes comparable to each other. The objection to the money version is that a column mixing large and small positions is dominated by the largest ones, so its mean describes the sizing rather than the method. The objection to the multiple version is that costs are charged in money and do not scale with the multiple, so a column of multiples quietly omits the one component that is certain. Neither convention converts into the other, and a figure quoted without saying which unit it used is not interpretable.

The second argument is about whether a single summary figure should be computed over a personal record at all. One tradition holds that the record is the only data a trader has, that reviewing it in aggregate is the discipline, and that refusing to compute anything leaves the review resting on memory, which is worse. Another holds that a personal record is short, is drawn from conditions that keep changing, and is assembled by the same person whose decisions it is meant to assess, so compressing it to one number produces more confidence than the sample can support. Both camps agree that the information lives in the individual entries. They disagree about whether compressing them helps or conceals.

The third argument is about the extremes. Some practitioners trim the largest entries before averaging, on the reasoning that an exceptional result is unrepresentative and distorts a mean computed over so few numbers. Others hold that the extremes are the distribution, that trimming them is the mechanism by which a record is made to look like something it is not, and that the trimming rule tends to be chosen after the extremes are known. That last objection is the one nobody has answered.

Why no figure appears here beside a method 

No expectancy figure is attached anywhere on this site to a strategy, a pattern, an indicator, an order type, an instrument or an account, and none appears in this lesson. That is a rule rather than an omission, and it has two separate reasons. The first is evidential. A figure describing how a method performs would need a verified source, and no such source exists, so publishing one would be publishing a number nobody can check. The second is that a figure attached to a method is a claim about that method's effectiveness, which is a claim this site does not make in any form, for any technique, at any size of sample.

The arithmetic above is complete without one. Every figure in the worked blocks is an assumption stated inside the block it appears in, chosen to make a property of the formula visible and attached to nothing. A figure computed from anybody's actual record describes that record, over the period it covers, in the conditions that prevailed, at the sample size it reached, and the sections above are an account of how little that is. The term is worth recognising, and worth knowing the limits of, which is the whole reason it appears here at all.