# The Prediction Dilemma¶

A lot of work is being done in trying to predict the direction of stock prices, or for that matter, any tradable security. The reason is simple: there is a lot of money involved and it can be linked to one's ability to forecast future prices, or at least take advantage of them.

However, predicting on small bets could be considered almost a trivial pursuit unless, there could be a great number of such bets with an upside bias. As the holding interval shrinks, so does the potential average price variation while at the same time its annualized equivalent CAGR increases. To compensate for the smaller price variations, one can respond with more trades.

That is the thing. In stock trading the payoff matrix could be reduced to: $n\cdot\bar x$ which says that the average net profit per trade $\bar x$ is scaled by the number of trades taken $n$. Therefore, one might have to predict $\bar x$, but could easily see that $n$ counts, pun intended.

In a prior notebook it was presented that the payoff matrix depended on $n$:

$$A(t) = A_0\, + \displaystyle{\sum_{i=1}^n }(H\,.^*\Delta_i{P}) = A_0 + n \cdot \bar x$$

and the larger $n$ could be, at whatever level of net profit per trade $\bar x$, it would be more beneficial than a smaller $n$. The relationship is simple.

$$\frac {{\sum_{i=1}^n }(H\,.^*\Delta_i{P})}{n}\,= \,\bar x \,> \,0$$

Clearly, looking for $\bar x$ > 0 is the mission of any portfolio manager. We do want a net $\bar x$ > 0 and show a net average profit per trade. But, often $n$ is neglected. In the sense that one accepts the number of trades generated by his trading strategy almost without question, when is should be evident that $n$ matters a lot. Every effort should be deployed to extract as many trades $n$ as demonstrated by a positive tradable edge. If something applies to one stock, it might also apply to others. It is in this generalization, in this search for added profitable trades where one needs to concentrate. Especially, if one goes for relatively short term trading and shorter holding times.

What are the techniques available that can increase $n$? First, a restriction will be: $A_0$, the initially available trading capital. If ever: $\,\,n\cdot \bar x < - A_0\,$, then game over, you lost. Evidently, if it happens, your capital is gone.

But this would be a simplistic view. Sure if you lose your initial trading capital you lost. But the problem is more pervasive. We should read: if ever $\,\,n\cdot \bar x \,< - A(t)\,$, then it is game over. Since then, at whatever level you might have increased your portfolio, it can all come crashing down. You would lose all accumulated profits: $\sum_{i=1}^n (H\,.^*\Delta_i{P})\,$ as big as they were, and even manage to lose $A_0$, your initial stake. Now, that is not good news. Losing it all after some 10+ years of work is not what you signed up to do. Nonetheless, a lot of advice we get is designed to get us there. We definitely should do our homework.

So yes, $\bar x$ needs to be positive: $\,n\cdot \bar x > 0$. But, you need more than that, $\,\bar x\,$ needs to be of such size as to outperform market averages or a Buy $\&$ Hold scenario: $\sum_{i=1}^n (H_{your}\,.^*\Delta_i{P})\, = \,n \cdot \bar x \,>\,\sum_{i=1}^j (H_M\,.^*\Delta_i{P})$.

It is on this premise that one needs to design his/her trading strategy. If somehow you design something that can not outperform averages due to structural defects or erroneous assumptions on your part, then you are simply shooting yourself in the foot. And, for not doing your homework, I would say: you deserved it.

Your objective, no matter what will be thrown at you (stock price wise) is to achieve: $n \cdot \bar x \,> \sum_{i=1}^j (H_M\,.^*\Delta_i{P})$. All you have at your disposal are these two variables with one not helping much since being a simple trade counter, but which can still have a major impact.

## The Rules You Want¶

It is not a rule, but you want to win, and this, no matter what is thrown at you. You accept that you won't win all the time, and that you will see some red in almost every trade you take. But, what you don't accept is lose the game. Since the market has for secular trend close to a 10% average CAGR (dividends included), it becomes de facto the minimum long term acceptable performance level. Anything lower should be considered as a waste of time and capital resources. Playing to underperform market averages should not be viewed as an objective.

A stock trading strategy needs to be monitored, an automated one, even more. Whatever you do trading, it will be time consuming. Even if it is a machine that will be doing the work, you will have the boring and boring job of monitoring it all the time it is active. Your job will be transformed into the machine watcher for years and years to come. So, better make it worthwhile... And it can be worthwhile only if you exceed market averages over the long term. Note that, with time, you will be able to delegate the watching job to someone else.

Failing 10 or 15 years down the road is not an option. Your trading programs should be able to withstand the test of time even before it can start trading. Otherwise, you might end up having nothing to show for all your efforts when all you had to do was make a better plan, design a better trading strategy from the start.

Another consideration, it might all be for you, but often, if you succeed at generating a worthwhile stock trading strategy that can last, it could also help others who could benefit from your work. But this does not change your primary goal: the need, for yourself, to significantly outperform market averages. You are the first person to convince that your trading strategy is worthwhile.

1. Define and select a tradable stock universe.
2. Set a generalized trading methodology.
3. Seek repeatable traits, especially generating $\Delta p$'s > 0.
4. Seek as large a number $n$ as possible.

Answer the question: Is the strategy worthwhile? If not, repair its weaknesses, add to its strengths, or throw it away. Don't throw away all you have done and start over, reuse what has proven beneficial and incrementally add new procedures that could improve on your design. Find out why, and know why your trading strategy succeeds or not. It is all part of the design process.

## What Will I Be Trading¶

I see trading as the search for $\Delta p$. With a profit or loss defined as: $q \cdot \Delta p = x$. I need to count all the trades, therefore, I can say: $q_i \cdot \Delta_i p = x_i$, where each trade is sequentially identified. I can express a profit or loss as an integral: $\displaystyle \int_{t=0}^T q \cdot dp\,\,$ giving each trade a start and end time. Having identified all executed trades, I could sum them up as in: $\displaystyle {\sum_{i=1}^n \int_{t=0}^T q_i \cdot d_ip = \sum_{i=1}^n x_i}$. Dividing by $n\,$ gives:

$$\displaystyle {\frac \sum_{i=1}^n \int_{t=0}^T q_i \cdot d_ip}{n} = \frac \sum_{i=1}^n x_i}{n} = \,\,\bar x$$

But this is all known stuff from over a century ago (see Bachelier 1900). There is another expression that could be used. Say we detrend a price series using a stochastic Itô process representation: $dp_i= \mu_i dt + \sigma_i dW_i$. Each stock would have its unique trend $\mu_i$ (up or down) to which would be added a random Wiener like process $W_i$ scaled by the stock's standard deviation $\sigma_i$.

There is only one problem with such an iteration, it is: $\,\sum _{i=1}^n\,\sigma_i dW_i \rightarrow 0$, as any stochastic i.i.d. Wiener process would show. There is also a conclusion to extract from that equation: there is nothing, no trading method that can extract anything from $\,\sigma_i dW_i \,$ except by chance. The more a price series will be random in nature, the less valuable information one will be able to extract. There is currently no one in the past millennia on this planet that has found a method to overcome the randomness of a simple game of heads or tails. You can end up a winner, but it will be by luck of the draw alone. Absolutely no skill, no alpha to be extracted.

All that is left is: $dp_i= \mu_i dt$, a linear time equation corresponding to the rate of return of each stock. Playing the game, that is what you can and will catch. The randomness section of the equation could be anything, but still tend to 0. Nonetheless, short term, the random component could have a major impact, even if it is by chance. However, due to its very nature, it remains hard to predict in any meaningful way.

So, one is bound to win just by catching the drift, and all that is required is exposure, purchasing $q_i$ shares at $p_i$ to then wait some time $\Delta t$. Exactly what a Buy $\&$ Hold investment strategy does. There is no need to look at daily, or minutely data, since most price variations could be characterized as some kind of randomness having on its own, a mean tending to zero: $\,W_{\mu _i} \rightarrow 0$.

## Probable Outcome¶

Having a random component to the equation did not change the desirable outcome. Whatever the trading strategy would be, it would still have to comply with: $\displaystyle {\sum_{i=1}^n \int_{t=0}^T q_i \cdot d_ip = \sum_{i=1}^n x_i}$.$\,\,$ From a return point of view where $\mu_i$ dominates the landscape, finding a portfolio's probable average rate of return would be: $\, {\displaystyle\sum_{i=1}^j \,\omega_{i,\,j}\cdot r_{i,\,j}} =\, \bar r$, the average portfolio return, provided that: $\,\displaystyle \sum_{i=1}^n \,\omega _{i} \,=\,1$. Now that you have $\bar r\,$, we can proceed to the outcome: $A(t) = A_0 \cdot (1+\bar r)^t\,$ which is equal to: $A_0 + n \cdot \bar x$. We can go full circle with:

$$\displaystyle \left ({\frac {A(t)}{A_0}}\right )^{\frac{1}{t}} - 1 =\, \bar r = \left ({\frac {A_0+n \cdot \bar x}{A_0}}\right )^{\frac{1}{t}} - 1$$

where again, $n$ and $\bar x$ are the dominant factors. My point is: whatever you do trading, better make $n$ as large as you can, otherwise be prepared to make $\bar x$ as large as you can. Somewhere in there, you have to find your own compromise. It won't be perfect, but that is not what you are seeking. What you are seeking is: $A(t) = A_0 + n \cdot \bar x \,>\,A_0 + \sum_{i=1}^j (H_M\,.^*\Delta_i{P})$.

It might be a lot easier to predict $n \cdot \bar x$ than trying to predict the future. A portfolio level simulation over a long term trading interval could provide a reasonable estimate: $\bf{E}$$[\,\hat n \cdot \hat {\bar x}\,] \rightarrow \,n \cdot \bar x \,$. Every simulation done provides those numbers. Every trading strategy has its own unique signature. And the future will certainly be different from the past, but, this does not mean we know absolutely nothing about it.

## Conclusion¶

The equation $\,A(t) = A_0 + n \cdot \bar x\,$ says: find an edge $\bar x$, make it as big as you can. Then trade it as often as you can. A rather simple recipe.

This makes it a bean counting operation. You won't be able to predict $\,n$ exactly but maybe make some rude approximations as to $\,\bar x$. If a napkin calculation like $\,n \cdot \bar x\,$ is a prediction, then let's call it that. But, I only see a guess, an estimate of things to come.

© 2016, October 11th. Guy R. Fleury