How advanced mathematics and gaming theory can help you as a trader

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please look up the defintion of the expectancy (or mathematical expectation). You will find it in the trader's wiki here as well. It is not a ratio but an absolute value. The definition is

expectancy = winning probability * average winning trade - losing probability * average losing trade

This means that it is a $ amount. If you double your bet size this will neither change your winning nor your losing probability, but it will double the $ amount of your average winning trade and your average losing trade.

Doubling the bet size will therefore double your expectancy or mathematical expectation.

The following user says Thank You to Fat Tails for this post:

Average profit vs. average profit per dollar risked which one is more useful?

You could look at expectancy as a measure of the average profit, as you say and as it is stated on the wiki of this website.
You could, if it makes sense. Probably it does.
However, another approach is at least as rational as the above:
the measure of your system, so what your systems money making capability is.

I came across the term expectancy while reading Van Tharp. He used the formula you have mentioned in the first edition of Trade your way to financial freedom, but he corrected it later.
So the formula cited by me in my previous post is from Van Tharp, after correction.

It makes more sense to me as a measure of ROI of your system.

The following 3 users say Thank You to Arpad for this post:

thank you for your answer. As I have the 2nd edition of the book by van Tharp, I have checked his definitions for the expectancy. Indeed, he defines expectancy as the "mean R Multiple".

Mathematically, this is false. Mathematically the expectation is the arithmetic average of a random variable defined over a probability space. The expectation depends on two different factors

-> the winning probability
-> the size of the average winning trade versus the average losing trade

Now let us look at Van Tharp's definition. He divides the mathematical expectation by the factor, which is the amount risked, and declares that this is the expectancy. In doing so, he commits several errors:

(1) Let us look at his example with the marbles (pages 197 ff). He compares two cases with an "expectancy" of 0.2 R and 0.78 R. His results do not make sense, because he has selected R arbitrarily. In his first example all losing trades produce a loss of 1 R, in his second example the losers produce losses of 1, 2 and 3 R. This is totally inconsistent, as for the second case he should have used one of two possible approaches

(a) Define R as the maximum loss. In this case 3R becomes the real R and his "expectancy" should only be 0.26 R, and not 0.78 R

(b) Define R as the average loss (as you did in your definition). In this case 1.28 R becomes the real R and his "expectancy" should be 0.61 R

This was just to show that he does not properly use his own concept. Typically to evaluate any system you would rather use the maximum drawdown or loss within a given confidence limit rather than any average. So the second marble system may be still superior to the first, but Van Tharps results are false.

(2) If you look at a ratio - as van Tharp does - you introduce the concept of leverage. A reward-to-risk ratio assumes that a favorable ratio can be used to generate better returns than an approach with higher absolute returns, but a higher inherent risk. This is true if you use leverage. Once you decide to use leverage, the next question is how much leverage you use. The most common approach to this question is a fixed fractional position sizing, which is based on an optimal f, which in turn depends on the probability distribution.

Without going into further details, it can be shown that the performance of a system should be judged against the geometric mean of its returns. The geometric mean can be approximated by using the arithmetic mean and the standard deviation, which is a measure of dispersion. The concept of the R-Multiple does not correctly catch the dispersion of results, so basically the method can not produce any valid comparisons of two leveraged systems.

So at best Van Tharp introduces a set of simple ideas, such as the R-Multiple, which help beginning traders to overcome some of the psychological difficulties of trading, such as loss aversion. Loss aversion will let cut you profits short and let your losers run. Judging trades against R-Multiples is a highly effective medicine against loss aversion. The concepts presented by Van Tharp therefore have a high practical value as they address psychological difficulties faced by traders.

Mathematically his approach, the definitions and examples are mostly flawed.

The following 3 users say Thank You to Fat Tails for this post:

You can choose a time period for the number of trades. It’s arbitrary.
It’s obvious if you have short time frame system it will look much better compared to a long time frame system. In one respect this is OK, because you want to know how much money can your system make in a given time period. On the other hand, you will not get a real measure of your system if you want to compare systems across categories, i.e. you want to compare a day trading system to a swing trading system.

However, I think you could omit the second part of the above equation, and use only SQN = (Expectancy / Standard Deviation of R).

But if you only observe one kind of systems (e.g. only swing trading systems based on daily close price, or only 5 min. day trading systems), you could use the complete formula:
· SQN = (Expectancy / Standard Deviation of R) * square root of Number of Trades
· or you could use the shortened formula SQN = (Expectancy / Standard Deviation of R)and Expectunity, which is Expectancy * Opportunity.
In fact these two are nearly the same.

What do you think? What would be the best equation to measure a system?

The following 2 users say Thank You to Arpad for this post:

Van Tharp's definition of the System Quality Number uses his own definition of th expectancy, which is based on the R-multiples. As I have stated, I do not understand, what is R, it seems to be an arbitrary measure. So I am not familiar with his concept, and would first have to try to understand it.

If you look at any system - assuming that returns are reinvested and that position size is calculated on a fixed fractional basis - there are two parameters that should be used

(a) Return: geometrical mean of cumulated returns over the total holding period
(b) Risk: largest observed drawdown

The largest observed drawdown is not R, because it is the result of a string of (mostly) losing trades.

For the geometric mean I would use the following approximation: Geometric mean = SQRT (Arithmetic Mean x Arithmetic Mean - Variance). Divide this by the largest drawdown and you get a measure for the system quality. If you compare this to the formula of Van Tharp, it uses similar input variables!

SQN = E * SQRT(N) / (AL * SD)

RAGM = SQRT(E * E - SD * SD) / LDD

where

E = mathematical expectation
AL = average loss or R (as defined by Van Tharp)
SD = standard deviations of return
LDD = largest observed drawdown
N = number of trades

Criticism:

-> E/SD is not a good proxy for the geometric mean
-> AL is not a good proxy for the drawdown
-> SQRT(N) has been added to emphasize the sample size

The system quality number could be a rule of thumb method, rather than a method based on the risk adjusted geometric mean. But as I said before, I do not know how it was derived.

The following 3 users say Thank You to Fat Tails for this post:

well i think going to the casino is a waste of money unless your just going for pleasure,i spend my money at the movies or shopping so there really is no difference we are both doing something we enjoy,and @Fat Tails your a pretty smart guy...sharky

I believe that you are 100% correct except for the fact that every table has a maximum bet, preventing an indefinite execution of the plan. The casino also has the right to refuse any bet.