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Defining expectancy: KISS or Rocket Science? Which do you prefer and why?

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This question is more directed at craft and other f/a longer term players.

How do you determine your expectancy? Its all well and good for 100+round trip/day ticker hounds to plug it into an excel sheet and spit out a number... But what if you've only did 4 trades this month and your havent closed out any winners ?
 
The expectancy quoted in my signature is generated by my accounting software, i have no idea how it's calculated, guess that's keeping it pretty simple
 
So_Cynical said:
The expectancy quoted in my signature is generated by my accounting software, i have no idea how it's calculated, guess that's keeping it pretty simple
Click to expand...

Do you know whether unrealised pnl is included in that calculation?

Edit: scratch that question. The answer is already in the signature.
 
SQ and So C have just highlighted a further consideration with regard to expectancy. How is one's expectancy calculated in the event of increasing size of open exposure. Does one simply include a marked to market calculation for the open trades and then include that data in the expectancy calculations?
 
Greetings --

The thread starter seemed to be asking for a definition of mathematical expectation. The conversation has wandered a bit since the first few replies. I would like to make two comments.

One.
Expectation is the mean of the set of values being analyzed. Define the set, and define the formula to be used, and the expectation follows. In trading, a common definition of expectation is the arithmetic (or geometric) mean percentage return from a set of trades.

Two.
The balance of a trading account after a period of trading depends on much more than the expectation of the set of trades. There is some rocket science involved, along with some subjective analysis of the trader and his or her risk tolerance and personal utility function.

The expectation must be positive over the period of trading for the account balance to be higher at the end than at the beginning. There is no position sizing or money management scheme that will transform a system that has a negative expectation into a system that has a long-term winning profile. It is, however, easy to lose money even with a system that has a positive expectation through poor money management.

Account growth depends strongly on the distribution of individual trade results. Looking at the amount won when winning and the amount lost when losing is not adequate. The distribution matters, and the sequence matters. One of the premises of technical analysis is that the future resembles the past. We cannot expect the exact same trades to recur in the exact same order. But we do rely on the distribution of trades to be stationary. If it is not, then the results of the system developed over the in-sample period will not be continued in the out-of-sample and live trading period. So we must assume on a stationary distribution and work from there. Here is a link to a presentation I made a few months ago about stationarity:
https://www.youtube.com/watch?v=iBhrZKErJ6A

Given a distribution of trade-by-trade results, or equivalently of daily-marked-to-market results, final account equity after some period of time or number of trades is also a distribution. Pointedly, it is not a single number and it cannot be predicted with accuracy. There is a central value of the distribution of terminal equity -- a mean or median. And there is variation around that value. For a given distribution of individual trades, we can estimate the risk of drawdown of trading that system. Maximizing profit depends on the relationship between the risk of the system and the risk tolerance of the trader. Here is a link to a presentation I made a few months ago describing treatment of risk in trading:
https://www.youtube.com/watch?v=Vw7mseQ_Tmc

Speaking specifically to wins and losses. I have done considerable research into the characteristics of trading systems. One analysis is as follows:
1. Pick a set of trades to analyze. These can be any set -- real trades, out-of-sample trades, in-sample trades, hypothetical trades. Have enough trades to cover about four years. It will become evident very soon that more trades provide finer resolution than fewer trades. But a set of any size can be used.
A. Sort that set into ascending order by trade result. Largest losing trade at the left, largest winning trade at the right. Just the percentage won or lost on the basic unit of trading. No other position sizing. Call this set "A". It is the set that would normally be evaluated as the result of a backtest, for example.
B. Copy set "A" and save it as set "B". Remove the 5% of trades that are the biggest losers. Resave set "B".
C. Copy set "A" and save it as set "C". Remove the 5% of trades that are the biggest winners. Resave set "C".
D. Copy set "A" and save it as set "D". Add the 5% of trades from set "A" that are the biggest winners. Resave set "D".
E. Copy set "A" and save it as set "E". Add the 5% of trades from set "A" that are the biggest losers. Resave set "E".

You now have five sets of trades. Set "A" is the original. The others have either more or fewer winners or losers. Evaluate the risk, maximum safe position size, and profit potential of all five sets. One of the steps of the analysis is "risk normalization." When risk normalized, risk of drawdown over the period of forecast trading is equal for all alternatives. Given that, the set of trades that is preferred (by any rational trader / analyst) is the one that has the highest profit potential.

I have found, and you will find when you replicate my analysis, that the single most important thing you can do to maximize future profit is remove / avoid large losing trades. It is more important to avoid large losses than anything else -- even at the expense of missing profitable trades. The next important thing you can do is avoid a high percentage of losing trades. Since we cannot expect the order of trades in the past to be repeated in the future, there is a chance that several losing trades will occur in sequence. Without gains to allow the account equity to recover, these create a cumulative drawdown that will cause the trader's risk tolerance to be exceeded.

Each analyst needs to do this analysis him or herself for the system that will be traded. Then adjust the rules of the system so that the distribution of trades gives maximum profit potential for your personal risk tolerance.

Thanks for listening,
Howard
 
cynic said:
SQ and So C have just highlighted a further consideration with regard to expectancy. How is one's expectancy calculated in the event of increasing size of open exposure.
Click to expand...

You're making me think, over the years my open exposure is growing as my total portfolio value grows as a consequence of my position building and recycling capital policy, i have generally avoided locking in big profits to lower my tax bill.

Anyway - i think expectancy is a post trade completion measure, it's a figure derived from what has happened, open profit is something else - potential expectancy.
 

Anything under the number of occurrences that starts to reduce the impact of randomness and dissecting your profit into the expectancy components isn’t really telling you much. One of the things that make long term investing hard is how dam slow the information feedback loop is.

So often I see this circular argument of what an edge is (hence the question to quant) – The answer typically being a statistically proven expectancy. (besides the bastardry of statistics normally involved) the next proposition then is typically I have a statistically proven expectancy therefore I have an edge and around and around we go. That sort of circular reference can’t happen with such a slow feedback loop, so you better damn well know what makes you the profit before going long term.

Howards A-E experiment of position sizing is exactly what I was saying in the post below about win/loss ratio being important – It’s probably the missing piece of information I find the hardest to do without. In long term investing we address risk of ruin by diversification across positions held and there is a lot of debate over how concentrated you should be to maximise profit and minimise risk. Time to get a statistical valid sample size under your belt is so slow you’re probably dead before you have a valid answer.

I do things via a lot of individual parcels which ups the numbers of transactions a bit, so this year got the computer to spit out the numbers. Pretty meaningless really because the underlying decisions are still pretty small. For what it’s worth the numbers were posted here with a bit of following discussion.


https://www.aussiestockforums.com/forums/showthread.php?t=25070&page=2&p=874305&viewfull=1#post874305
 

Thanks Howard , you have expressed a lot of what I was trying to say . Everyone interested in system trading should have a look at Howards videos and read his book , absolutely full of brilliant information . I am sure it would even help every style traders .. repeating myself but this will help all traders regardless of style

" If you cant measure it , you probably can't manage it .... Things you can measure tend to improve "

Hopefully some will be more to willing listen to such a well regarded and peer reviewed expert such as Howard Bandy , We are very lucky to have him participating here at ASF
 
Thanks for taking the time to offer your input. I'm certain that I am not alone in being impressed by your thoroughness.

I find your comments about profit maximization and the challenges regarding estimation of future performance from stationary data sets interesting in that they appear to highlight some key challenges to the translation of theory into practice.

On the subject of sequential losses, would the incorporation of fixed fractional position sizing into one's system eliminate (at least in theory) the risks posed by distribution sequencing?
 

Yep, i think the main problem with discussions like this is people want a solution that fits all situations, and its just not going to happen. "Expectancy" and the discussion around it is specific to trading and very short term investing, and thats fine - those interested in this end of the spectrum should discuss its meaning and implementation in their strategies.

Its not really relevant to long term investing and trying to shoe horn it into a strategy is unlikely to create much more than noise.

I have read about it and learnt a little about how traders use it as part of their strategy, but I dont try to fit it in my box of tools. Still its an interesting discussion and its good to see a new face at the table!

I suspect that having made the rather obvious point about expectancy and its likely lack of relevance to long term investors, those of us of that ilk should probably let those to whom its important, continue the discussion while we observe from the sidelines!
 

Whilst I agree that the statistical approach to calculation of expectancy may be unsuitable for those trading larger timeframes, I believe expectancy as a concept does still hold relevance to business in general.

This discussion would be incomplete without input from those trading time frames that demand an alternative approach to measuring the health of one's business operation.
 
Greetings --

Cynic asked about incorporating fixed fractional position sizing in the model to help reduce the effect of drawdowns. My recommendation is to use the model solely to generate buy and sell signals. Then monitor performance during trading and apply position sizing in response to recent trade results. My view is that position sizing is a function of trading management, not of signal generation.

Continued good system performance relies on the continued good (stationary) relationship between the model and the data. I think of that relationship as synchronization. The system goes through periods where the model accurately identifies the patterns in the data you are looking for, resulting in good profits. And periods when the synchronization is inaccurate, resulting in poor profits. Overall, the system performance has a mean and a variance. The system can continue to be operating within its expected performance, but trades show losses due to random sequences of misinterpreted signals.

Eventually systems fail. That is almost always due to changes in the data. Those changes can be due to fundamental changes in the underlying market or company, or due to other traders also identifying the inefficiency your system is based on and reducing the profit potential for everyone.

We cannot tell when a drawdown is a normal excursion to the left side of the performance distribution or the start of a permanent breakdown. That is, at the first few losing trades we cannot tell whether there has been a change in stationarity or not. The best approach, in my opinion, is to treat every drawdown as a warning and reduce position size. If / when the system recovers, increase position size in response to the recovery. If not, take the system offline.

Assuming a system will recover from a drawdown is a statement of faith that I do not share. But many trading coaches do recommend "keeping the faith", "trading through the drawdown", "seeing an opportunity to double down and recover losses", etc. The gain resulting from trading through a drawdown that recovers is somewhat more than the gain of trading at a reduced position size during the drawdown. The loss resulting from trading through a drawdown that does not recover is serious loss of money. Consider the reward to risk ratio.

In the early days of statistical analysis, mid to late 1900s, frequentist statistics prevailed. Sir Ronald Fisher, of F-test fame, was the overwhelmingly strongest influence in statistical methods. He was a bit of a bully and repressed views other than his own. Beginning over a hundred years ago, and strongly gaining strength since Fisher's death, Bayesian statistical analysis has become more accepted. Almost all of us who studied statistics further in the past than a few years ago learned frequentist methods. There is a big difference between the two. A Bayesian walks past a yard on Tuesday and a dog rushes out to bite his ankles. On that same walk on Wednesday, the Bayesian is already alert for bad behavior. The frequentist waits until there have been 30 experiences, then decides with 95% probability that the dog is dangerous. In most of our decisions in life, we are Bayesians without thinking about the theory. We should apply Bayesian techniques to trading as well.

My point is that we can note trade-by-trade changes in trading behavior and make position size adjustments based on small samples of recent performance without waiting for 30 trades and near certainty. The position sizing decision is based on performance and on a trading management model. It is not a component of the signal generation model.

Thanks for listening,
Howard
 
Greetings All

There's one point that has always bugged me about these types of discussions, which is the use of the arithmetic mean trade result as expectancy (typically expressed as Avg Win * Win Rate - Avg Loss * Loss Rate). This value is only accurate in two situations: fixed position sizing (always buy $10,000 of stock at a time) or if every single trade return is identical. Assuming we're taking advantage of compounding and using profits for future trades, using either fixed fractional or dynamic position sizing, the geometric mean trade result is more accurate. That is, if we were to take our (final equity/initial equity) ^ (1/ #trades), this would be the actual "average" trade result.

I only bring this up because I've seen it suggested that the expected equity after N trades is
(1+arithmetic mean)^ (#trades). This is incorrect and can be verified by looking at any backtest.

For longer-term investors, you can use the daily/weekly/monthly returns of an account as a proxy for closed trades. Use these results to determine expectancy by more traditional methods. Also of note is the fact that systems that win frequently, even if average losses are larger than average wins, can utilize more aggressive position sizing safely because of the effects of compounding consecutive losses. Again, this can be proved out mathematically and Ralph Vince has done so in his books.

On a different topic, but one that Howard touched on, is how to determine whether a system is broken. I stole this approach from Kevin Davey's book, but I've found it to be effective. If you have a distribution of trades and use Monte Carlo analysis to predict future performance, you can generate confidence bands for your equity curve that create an envelope around where your equity curve could be at some point in time or after some number of trades if the system is behaving normally. If you plot your equity curve from live trading on top of these bands, you can determine with some degree of certainty (10%, 5%, etc.) whether the system is performing where you'd expect it to or if it is unusually bad.

Sorry for the rant
 
Greetings --

About arithmetic or geometric mean --
When a trading system issues signals and produces trades, or the equivalent marked-to-market daily returns, they are individual. When analyzed as a distribution, each trade contributes a percentage gained or lost (or point gain or loss for futures or forex). That distribution has moments -- mean, variance, kurtosis, skew. That mean is defined by the equations of moments to be the arithmetic mean. When a sequence of trades forms an equity curve, the analyst has some choices. If every trade is taken with all available funds (traded at full fraction), the equity at any point along the curve is the product of the relative gain of the trades. The final equity is the geometric mean raised to the power of the number of trades. Given a set of trades that are traded at full fraction, the final equity (terminal wealth relative -- TWR -- using Ralph Vince's terminology) is independent of the sequence. The drawdown, however, is Very sequence dependent. Making a decision of whether to use a system to trade or continue to develop it based on a single equity curve is simplistic and includes a selection bias that leads to over estimate of profit and under estimate of risk. That is the reason I recommend analysis of the distribution of trades to estimate the distribution of drawdowns at a given position size fraction, comparing the tail risk of the distribution of drawdowns to the individual trader's personal risk tolerance.

Assumptions made in Ralph Vince's materials --
Mr. Vince assumes:
1. Systems being analyzed are stationary. So do most (read 98% or more) of articles, books, and presentations. Systems are not stationary. If they were, drawdowns would be opportunities to increase position size anticipating the return to the mean through an increase in the winningness of the system. Whether a system is working or broken is one of the big questions of trading management. Translated to statistics, that big question is whether two distributions -- previous performance and current performance -- are the same or different.
2. The trader is comfortable with the high levels of drawdown that accompany high return. The position size indicated by Kelly, optimal-f, and several other position sizing techniques produces exceptionally high returns, but experiences drawdowns well in excess of what most traders can tolerate. As long as the trader has an indefinitely long time horizon, the system is stationary and guaranteed to recover, and the trader is able to take all trades at the recommended position size (similar to the table limits at a casino), drawdowns of 70% or higher are no problem. Traders with limited funds, who trade systems that are not stationary, or who might need to withdraw funds from their account within the next few years cannot afford position sizes that high.

About a system being broken --
I recommend a technique similar to that of Kevin Davey, but done on a trade-by-trade basis. Using the entire set of trades is correct as long as the system is stationary (the terminology of an earlier post is "behaving normally"). The danger we have is that the regime underlying the system changes and the recent trades are no longer coming from the same distribution. Frequentist statistical analysis waits for 30 (more or less) trades, then decides. This style of analysis is OK when looking at backtesting results where everything appears in one run rather than trade-by-trade over real time. Bayesian statistical analysis revises the working / broken response trade-by-trade. Several of my presentations illustrate the technique. My latest book has a detailed discussion, including computer code.

Thanks for listening,
Howard
 
It is best to apply KISS to long term trend following trading systems - this has served me well. The greater the applied time-frame, the less accuracy is required (ideal for a retail trader).

I believe most people over-complicate things, by trying to control/validate parameters in trading systems to accommodate their poor trading psychology (i.e. unable to stomach losses). The greater the potential risk (i.e. draw-downs), the greater the potential reward (i.e. annual returns) - this fundamental concept is overlooked far too often when searching for the "holy-grail".
 
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