August 23, 2015

Any stock trading strategy should be basic common sense. A stock portfolio does not grow instantaneously. It takes years to build it up and nurture it. It is not enough to make a trade here in there without considering the size of the portfolio or the time span under which it will have to grow.

Making a 100% profit on a trade is good, but it simply might not be enough. If you risk 5% of your portfolio, it will grow by only 5%. If it took 2 weeks to make the 5%, great. If it took 2 years or more, positive, but not so great. An immediate question would be: what do you do after? Finding another 100% profitable trade! Well, those don't come by on a weekly basis... and even if some do, there might not be that many or even be predictable in some way.

If, for example, I play AXP, at whatever time frame, I will be faced with the same time series as everybody else. And with its 1 billion shares outstanding, I won't be the one moving the price either. This AXP time series can be expressed as p(t) = p(0) + ƩΔp (an initial starting price plus the sum of all price variations thereafter). My AXP interest, as in any other selectable stock, is in the positions I might, could, or can take. But overall, only the taken positions will impact the portfolio. The could have, should have, or would have don't generate profits. Only the have, did, done, and executed matter.

I could cut p(t) into thousands of pieces or in as many irregular time intervals as I want, each with its own entry p(in) and exit p(out) to define each trade. A profit or loss on a single trade is given by the quantity held over the time interval: q*Δp = q*(p(out) – p(in)). To handle thousands of trades, I can number them sequentially: q(i)*Δp(i) = q(i)*(p(out)(i) – p(in)(I)), which can show the profit or loss in the ith trade among many (i = 1, ..., n).

Profits and losses for all n trades would sum to: Ʃ(n) q(i)*Δp(i) = Ʃ(n) q(i)*(p(out)(i) – p(in)(i)). Technically, ending up with n price segments Δp(i) with inventory q(i). All price segments with no inventory (q=0), meaning no participation, just as a Δp(i) = 0, would have no trading value and, therefore, could not increase or decrease a portfolio's value.

To outperform a Buy & Hold, I need Ʃ(n) q(i)*Δp(i) > q(0)*(p(T) – p(0)), meaning that the sum of all price segments with inventory should be greater than the entire price series. If n is too small and/or the Δp(i)s are too small, then the sum of price variations might not be sufficient to outperform even a Buy & Hold. Making 50 trades with an average profit of \$1.00 per share will have the same impact as having 1 trade with a profit of \$50.00 over the same time span. However, making 100 \$1.00 trades would exceed the 1 trade \$50.00 profit.

Trading has constraints, one of which would be to at least take a sufficient number of price segments (trades) to exceed the price differential of the entire period of play. This means you should aim to at least beat the Buy & Hold over the long haul; otherwise, what's the use of trading?

Another constraint is slicing the time series: Δp(I), which translates to delta p what? What is the outcome of the ith future price interval? There is no way of knowing this, especially if I plan to do thousands of these trades in the future. The mathematical formula will account for the total generated profit or loss from the n trades: Ʃ(n) q(i)*Δp(i); this is irrespective of how I look at the past or the future. But it still won't help in predicting any of the n trades nor their price differential Δp(i) going forward. Neither will I be able to determine q(i) unless it is fixed, like using a constant or subject to a predefined function. Having a fluctuating stock inventory can have its benefits...

In payoff matrix notation, all the above translates to Σ(H.*ΔP), all of it resumed in one expression. Most people start by designing a trading strategy from whatever concept they might have, while I start with the background math and then figure out how I could take advantage of what I see. It gives a slightly different perspective to the portfolio management problem.

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