This is the continuation of previously presented notebooks. They are all in this thread and presented in chronological order. At times, you might need to refer to them to gain a better understand of where all this is going.

A trader wants to be in and out of a trade with a profit, and fast. The second thing he wants, is to do it many many times. As a matter of fact, a great number of times. Previous notebooks in this series showed: $n\cdot\bar x$, the average profit per trade was scaled to $n$, the total number of trades.

But even a trader should consider that it is not all there is.

Sure, only two variables is not much to work with, however, whatever is done trading will have to be explained within these constraints. It was observed that $\bar x$ tended to a limit as $n$ increase. It was never said what that limit was since it had a tendency to expand with time as was demonstrated in **Trade Decision Process**.

The expression: $\, n \cdot \bar x \,$ could be broken down in two parts: $ \, n \cdot \bar x \, = (n - \lambda) \cdot \bar x_{_+} + \, \lambda \cdot \bar x_{_-} \, $ with $\, \lambda\,$, the number of losing trades somewhere between 0 and $n$: $\,\, 0 \leq \,\lambda \, \leq n$, and where $\, \bar x_{_-} \, $ and $\,\, \bar x_{_+}\, $ represent respectively the average loss per losing trade and average profit per trade from winning trades.

Follows that $\,\bar x\,$ is between these extremes: $\,\bar x_{_-} \leq \,\bar x \, \leq \, \bar x_{_+}$. The limit as $\lambda \,$ tends to zero is $\bar x_{_+}$: $\displaystyle \lim_{\lambda\, \rightarrow \,0} \,\,(n - \lambda) \cdot \bar x_{_+} + \, \lambda \cdot \bar x_{_-} =\,\,n \cdot \bar x_{_+}$. And conversely, as $\lambda \,$ tends to $n$, the portfolio gets filled with losing trades: $\displaystyle \lim_{\lambda\, \rightarrow \,n}\,(n - \lambda) \cdot \bar x_{_+} + \, \lambda \cdot \bar x_{_-}\,=\,\,n \cdot \bar x_{_-}$.

A trader has this problem to solve: the limit of $\bar x$ can't be that big, he is not in the market long enough to have his $\bar x$ move that far away to the right of the distribution curve as illustrated in **Trade Decision Process**. As $n$ increases, this limit becomes more and more evident especially if the holding time is, on average, reduced to minutes. And in such cases, $n$ needs to be quite large. It is why there is some talk about equilibrium, not that you get to a fixed state, but that you need to make some compromises, adapt to the situation in order to take advantage of it.

You can only trade what is there. Someone on the other side of a trade needs to be there: either to propose you a trade or accept yours.

The trader has a service to render. It is: take the risk somebody else is not willing to take for a period of time. He serves as a price stability insurer: when someone might be interested in selling, he might be there to buy, and when someone wants to buy, he might also be there as a seller. He volunteers for the role simply by taking market positions. Give it the name you want, but he is putting money on the line, taking the risk. Making a gamble that his view of the risk involved is different than his counterparty. If he is right, he wins. If he is wrong, the surprise is: he might still win. He can buy stocks on his own set of criteria based on his evaluation of what is ahead.

Appraisal of risk is his business. He is not ready to take all risks, only the ones he considers as his better probability of winning scenarios. He can intervene when he wants for whatever reason he wants. He can be fast and take what is offered (market orders) or wait patiently for his price (limit orders).

What he is chasing is: $ \Delta_i p > 0$. The search for a probable profit: $\mathbf{E}[q_{i,j} \cdot \Delta_i p_{i,d}]= x_{i,j} > 0.$

In my trading strategies I use the notion of trading units. It is simple, it is a preset amount per trade: $u = p\cdot q$. The quantity to trade is determined by the prevailing price: $ \displaystyle {q = \tfrac {u}{p}}$. You can spread trade units over time or over one another to make bigger single bets. It has some uses. At least, I find them practical in my strategy designs.

A trade unit is not attached to any stock. It is a bet size, the dollar amount put on a trade, a part of a portfolio. You close a trade, you can open the next one on whichever stock you want. A trade unit can be viewed as an independent bet, and you are the one fixing its size. So, in a payoff matrix, what you would see is the outcome of a multitude of independent bets that have been carried out over the lifespan of the trading strategy.

What you want is that their sum be positive: $\,n \cdot \bar x > 0.$

Having expressed the total outcome of a stock trading strategy as: $n\cdot\bar x$, the more a short term trader should see the relevance of $n$, the number of trades. If you did a million trades a day for an average net profit of about $1.00 per trade. You would figure out that it is one million dollars a day in profit. It would become more than acceptable. You would be ready to put up the collocated low latency machines and trading software required to do the job. You know it won't cost a million a day to operate.

However, if you only do 10 trades or less a day, you better keep an eye on $\bar x$ since alone, it wou't be able to generate that million per day. You have to go for the feasible, and it is very much related to the capital at hand at all times. You can always invest less than what you have, which in itself will reduce your market exposure and thereby reduce portfolio volatility. There is a price to pay, it comes in two forms: opportunity costs and possible reduced performance in terms of long term portfolio CAGR.

We can study a portfolio's generated profits and losses separately. The average profit per trade $\bar x$ could also be expressed as: $\displaystyle \bar x = \,\,\bar x_{_+}\cdot (1-\frac {\lambda}{n}) + \lambda \cdot \bar x_{_-}$. Based on that equation, you want $\,\lambda\,$ to be only a small fraction of total trades. You also want: $ \left\vert \bar x_+ \right\vert > \left\vert \bar x_- \right\vert$.

You want $\,\bar x \,$, the average of it all to be positive. And therefore, there is a balance to be reached. But, nothing stops anyone from reaching for: $\lambda\, \rightarrow \,0$, since going the other way: $\lambda\, \rightarrow \,n$, will get you out of the game pretty fast with nothing to show for it.

In **Extracting Tradable Information** I use an elementary MACD system to show that by setting your own exit rule while the MACD was increasing could provide you with an expected positive outcome. In fact, the output would tend to market averages, would do a little less due to the lesser market exposure.

The question is: can I make an estimate of the outcome of that trading strategy? What is the math behind it. We can use all that has been elaborated in this series of articles. I see them more as chapters in a book. I write sequentially, never knowing where it will go, but still following a logical thread. Maybe it took all that preceded to make the following points.

**Extracting Tradable Information** was a MACD trading strategy using a moving average crossover entry with a profit target exit. More than just simple. I know it does not beat the market. What I am trying to show is the mechanics of the trading system, and the ability to make long term estimates of the trading strategy itself without the aid of predicting future prices. What some would consider practically an aberration.

A trading unit was defined as: $u = p\cdot q$. The quantity traded being: $ \displaystyle {q = \tfrac {u}{p}}$. A profit target could be defined as: $PT = \frac {\Delta p}{p}.\,\,$ And we have: $ \, n \cdot \bar x \, = (n - \lambda) \cdot \bar x_{_+} + \, \lambda \cdot \bar x_{_-} \, $ as expressed above.

In the MACD example the profit target was set at 10%, so we will use that again for illustrative purposes. The big question is, what it the expected value of $n\cdot \bar x$: $\,\mathbf{E}$$[\,n\cdot \bar x]$ ? Turns out that:

$$\,\mathbf{E}[\,n\cdot \bar x] \,= \,\mathbf{E}[ \, (n - \lambda) \cdot \bar x_{_+} + \, \lambda \cdot \bar x_{_-} \,] = u \cdot PT \cdot (n-1) + \,\mathbf{E}[u \cdot PT]$$And since $\,\mathbf{E}[u \cdot PT] = u \cdot PT $, we have: $\,\mathbf{E}[\,n\cdot \bar x] \,= u \cdot PT \cdot n\,$. And again, $n$ counts. But now, you are left with an entirely different problem than seen in usual trading strategies.

It appears that the outcome of the trading strategy depends on items under your control. Full control over: $u$, $PT$, and a partial control over $n$. Meaning that you could dictate the outcome of your trading strategy by fixing $u$, $PT$ and seeking as many trades $n$ as the trading strategy can deliver, meaning, you can detect to then take advantage of.

We have changed the problem. Knowing that: $\,\mathbf{E}[\,n\cdot \bar x] \,= u \cdot PT \cdot n\,$, we can now make long term estimates based on the method of play. The trading unit $\,u\,$ is the bet size, the amount put on a trade. $PT\,$ is a profit target expressed as a percent, and $\,n\,$ is the number of trades that can be extracted from a particular trading strategy. Say you set $u = 50k,\,\,PT = 0.10$ for a 10% profit target, all that is left to determine is $n$, the number of trades the trading strategy can do. And a backtest will give you an estimate of that.

If you make 100 trades under these conditions, you will get: $u \cdot PT \cdot n = 50,000 \cdot 0.10 \cdot 100\, = 500,000\, $ or 10 times the trading unit with a win ratio of 99%. The last trade might be with a partial win or at a loss.

You did not get a 99% win ratio because you were lucky, you simply designed the trading strategy to operate that way, to deliver winning trade after winning trade.

You can express your trading strategy based on your ability to extract from all the noise the line segments you want and stand ready to wait for their realization knowing in advance what will be the outcome. Not as hoping to be right, but knowing that the equation will stand.

A trading unit $u$ is independent of a stock, it is your bet size that you can increase or decrease at will. It does not say you have to trade in the same stock all the time either. You can select what you think are the best opportunities to realize your profit target $PT$. It was given in **Extracting Tradable Information** that the MACD scenario with profit target exit would tend to market averages. It still holds:

The interesting part is that you can now control where you are going. You can gradually increase either $u$, $PT$, $n$, in pairs, or all three at a time and you will simply exceed market averages: $\mathbf{E}[\,n\cdot \bar x]^\uparrow \, > \displaystyle \sum (H_{M}\,.^*\Delta P)\,$.

It transforms the trading problem into finding ways to increase expectations using these three variables with their respective enhancer functions to generate: $$\mathbf{E}[\,n\cdot \bar x] ^\uparrow \,= \, u \cdot(1+f_t(a)) \cdot PT \cdot(1+f_t(b)) \cdot n \cdot(1+f_t(c))$$

where $f_t(a) > 0$, $f_t(b) > 0$, and $f_t(c) > 0$. All stuff that is realizable. Evidently, if you set all three functions to zero: $f_t(a) = 0$, $f_t(b) = 0$, and $f_t(c) = 0$, you will be left with the market expectation, doing no better than market averages.

If you want to make more, you will have to do more. And it is in your hands, under your control.

© 2016, October 22nd. Guy R. Fleury