June 4, 2020

A stock trading strategy can often be simplified to its most basic components, and there are not that many of them. In fact, maybe just two. Those trading strategies cannot be considered that complicated either if, whatever their outcomes, they will end up as being the result of two numbers, namely, the number of trades executed over the life of the portfolio and the average net profit per trade. The continuous trading transforms the expected portfolio profit problem into a long-term, statistically driven, and dynamic inventory management problem under uncertainty.

Whatever the long-term objective, or the actual outcome of your trading strategy, you get the following payoff matrix equation:

F(t) = F_{0} + $X = F_{0} + Σ (**H** ∙ Δ**P**) = F_{0} + **E**[n] ∙ x_{avg} = F_{0} ∙ (1 + r_{m} + α)^{t}

where $X is the total generated profit or loss.

Was given, in a **previous article**, the formula to predict the expected number of trades over the investment period **E**[n], while now my interest will be on x_{avg}, the average net profit per trade over the strategy's long-term trading interval.

Evidently, to show an overall profit, no matter how many trades were executed or will be executed, you will need x_{avg} > 0 to win. Meaning that you will need, on average, something that can show a profit, something with a sustainable positive long-term edge. This does not change the fact that whatever your objective **E**[$X], it could be accomplished in many ways since **E**[$X] = **E**[n] ∙ x_{avg} is not a single solution equation. A huge number of trading scenarios will satisfy combinations of those two numbers.

It was shown (see related articles) that using a full exposure periodic portfolio rebalancing technique with a fixed fraction of equity, we could make estimates as to the number of trades that a trading strategy might do over the years. This estimate, based on the expected average turnover rate, could extend to long-term projections. With this, half of the problem seems to have been solved, namely, the estimate for the total number of trades that might be executed: **E**[n].

The task is more complicated in solving for the average net profit per trade: x_{avg}. It appears more of an unknown. However, there are, here too, estimates that can be made based on the structure and design of the trading strategy itself. Nothing extraordinary, but conclusions nonetheless due to the very nature of the stock market and the architecture of the program.

Without loss of generality, we could say that the expected U.S. long-term (20^{+} years) average rate of return for stocks is in the range of about 10% +/- 2 to 4% depending on which academic papers you are looking at. This is with reinvested dividends. We could average out the whole market to a single line on a chart which could also represent the expected average outcome of individual stocks.

A chart of this on a log scale would give a straight line, like in the following chart. In real life, this line is chaotic, to say the least. Nonetheless, it will still reach its endpoint over the long term.

(click to enlarge)

Any deviation from its long-term trend could be considered as market noise. On the chart, the red line was randomly generated to show one of those possible chaotic paths. It could have been any other path that reached close to the estimated outcome. The +/- α was set to +/- 4% to show the impact of management or program skills brought to the problem. It could be higher, which would accentuate the differences even more.

The expected long-term return **E**[r_{m}] on the above chart was set at 10%. The +/- alpha does not seem to be that significant, as illustrated on the log scale. It appears as a relatively small deviation from the expected outcome. However, that picture does change if you revert back to a normal scale as shown below.

(click to enlarge)

It can be observed that even a low alpha number can have a major impact over the long term, as portrayed in the above chart. Also, in the beginning (the first 5 to 15 years), even though the difference is there, it is much less significant than the last 5 years on that chart. Giving for a portfolio's first objective the need to last over the whole trading interval in order to reap the end rewards.

One of the things the above equation says is that you are, due to the continuous trading, dealing with averages. It is not you win one trade here and lose another there; it is the large number of trades that will be undertaken over the years that matters, especially in a weekly rebalancing scenario. From the illustrated strategy (see recent articles), using a 400-stock portfolio with equal weights also fixed part of how the trading strategy behaved. Trying to maintain this full-exposure portfolio and forcing it to maintain its equal weights did set its bet sizing function to 1/400 of ongoing equity.

It was expressed before that: x_{avg} = u(t) ∙ **E**[PT], where u(t) is this betting function and **E**[PT] is the average percent profit per trade on those bets. The average profit per trade cannot be that considerable for the simple reason that those profits are extracted from weekly price variations. The average weekly price change could be in the range of 0.12% to 0.27%, which would translate into an average compounding price change of 6% to 14% percent per year.

That is what the market has to offer over the long term. The average turnover rate will determine that only part of the portfolio will be traded at each rebalancing. Most of the trades will be the result of weight adjustments. And as the turnover rate indicates, it will not be every stock that will participate every week in the reweighing process.

The market will not follow your rebalancing schedule. It follows its own course, independent of what you do or think. You are not the one having an impact on the market.

Overall, the rebalancing operation made it a statistical problem due to the trading method used and the sheer number of trades. If you want more than market averages, you will have to do more.

**Related Recent Articles**:

**The Inner Workings Of A Stock Trading Program – Part III**

**The Inner Workings Of A Stock Trading Program – Part II**

**The Inner Workings Of A Stock Trading Program – Part I**

**Lessons From The Portfolio Rebalancing Gambit**

June 4, 2020, © Guy R. Fleury. All rights reserved.