June 25, 2020

The more you look at the stock market game, the more you realize you need to play for the long term even when you are making short-term trades. Also, the more you trade over the short term, the more those trades will be faced with random-like outcomes, and the more trades you will need to reach your long-term goals whatever they are. As if there was a contradiction in purpose and means to achieve those goals. Nonetheless, most often, it remains quantifiable. The presented equations will govern it all for some planned and preset strategies.

That you do 100 trades or 100,000+ over the next 20+ years or that you do a simulation over the past 20, you will be facing the same equation, but it will not have the same outcome, even if theoretically and with a very low probability, it could. The portfolio's underlying payoff matrix equation is:

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

where **E**[n] is the actual or expected number of trades, x_{avg} is the average net profit or loss per trade, **E**[r_{m}] is the expected average long-term market return, *α* is the strategy's added return contribution to the overall return and ex(t) is the annualized percent equivalent of expenses and fees incurred over the life of the portfolio. We could summarize the expected return as: **E**[g] = **E**[r_{m}] + *α* - ex(t).

If the expenses plus fees amount to 2-3% per year, then that is what is deducted from the above equation. The added alpha needs to be higher than the incurred expenses just to make it worthwhile. Otherwise, your excess return (*α*) is being eaten up by your trading expenses. If the alpha is zero, the expenses will tend to reduce your expected market return. And, it will be worse if your alpha is sub-zero. For instance, you get a sub-zero alpha when your strategy does not outperform market averages, and, over the long term, most professionals portfolio managers don't.

If your trading strategies can be explained using the above equation, then that equation becomes the most important formula in your arsenal of trading tools. In **The Portfolio Rebalancing Gambit III**, we did set the impact of the rebalancing procedure over the life of the portfolio. The estimated turnover rate **E**[tr] had a role to play in estimating the number of trades to be executed by the trading strategy. A single simulation of your trading strategy would give you this number.

The behavior of that trading strategy was predetermined from the onset by outside trading decisions. For instance, F_{0}, the initial capital, is not a program decision. Just as the number of years or the number of stocks that will be traded. Even the number of rebalancing per year is an outside decision. These numbers will greatly impact the strategy's outcome. And yet, they should be considered as simple management decisions. The what you want to do with what you have.

**Portfolio Equation**

Here is that part of the equation again:

F(t) = F_{0} + y ∙ rb ∙ j ∙ **E**[tr] ∙ u(t) ∙ E[PT]

where *y* is the number of years the strategy is applied, *rb* is the number of rebalancing per year, *j* is the number of stocks held in the portfolio, u(t) is the trade unit function, and **E**[PT] the estimated average profit per executed trade. The Portfolio Rebalancing Gambit series of articles dealt with the first part of the above equation: y ∙ rb ∙ j ∙ **E**[tr].

One thing that the short-term trader seems to not realize is that once a trade is completed, what does he/she do afterward? You are on a 20-year plus journey and your trade took only a few weeks, do you stop trading in fear you might lose the next trade? Evidently, one short-term trade is not enough. An automated trading program will need to do many many more.

When you want to automate your trade process you see the difficulties compounding rapidly. It becomes more and more difficult to find short-term trades that will help grow your portfolio at a reasonably higher rate of return than just the average long-term market return: **E**[r_{m}].

This is important since if you do not do better than the long-term market averages, you are working very hard to do less than these averages which is the same as throwing money away since you could have had those average market returns more than easily. It would be like working hard but still shooting yourself in the foot.

The payoff matrix equation says that whatever your trading strategy the equal signs will prevail, and therefore, whatever numbers are part of that equation, they will give the final result of your trading strategy. I do not know how to put it clearer, the equal sign in 2 + 2 = 4 will hold for quite some time to come. So will the above equal signs. Start analyzing your stock trading strategy in light of the above portfolio equation since there too the equal signs will hold.

I am often asked what is the recipe for the excess return in my trading simulations. The answer is: I use equations. And, I have provided them. The equation governing those simulations is the above-cited equation. The trading methods used try to enhance its variables. Improving any of those variables will enhance the final result since each variable in the equation is multiplied by the others. I also think there are millions and millions of possible solutions to that equation. Thereby, giving anyone with a good understanding of that equation the ability to design better trading methods, better suited to their own constraints and views of the market.

Of note, my recent simulations used prescheduled weekly rebalancing on 400 stocks. This determined how the strategy would behave over time. I usually work on the variables given in the above equation knowing quite well that they will give the ending portfolio value whatever the trading interval. And I also keep an eye on the portfolio's long-term perspective. If your trading strategy blows up over the long term, you simply had absolutely nothing to start with even if in the beginning, it generated some profits. Ending your long-term portfolio with zero in the trading account is not a good scenario. It should become evident that you will have to learn how to play the game if you want to be good at it.

The Portfolio Rebalancing Gambit series showed that we could make estimates as to the number of trades that might be executed over the life of a portfolio. It was the structure of the program itself that enabled calculating the estimated number of trades that might be executed over the years. All you needed was the estimated turnover rate since the rest was determined by the rebalancing procedures. However, that series of articles did not show where the profits were generated or provided an estimate.

**The Trading Unit**

Presently, the emphasis will be to estimate the other part of the equation: u(t) ∙ **E**[PT] which can help determine the profitability of the trading strategy over time. The function u(t) is the bet sizing function or the stock allocation function. Its initial value was easy to ascertain: u(t) = F(t=0) / j = F_{0} / j where *j* is the number of stocks in the portfolio. You put 100 stocks (*j* = 100) in your portfolio, the trading unit will represent 1% of equity. And if a stock goes bankrupt, you should lose about 1% of the portfolio's equity. So, not that much of a big deal. On a 400-stock portfolio, a bankrupt stock would represent only 0.25% of its portfolio equity. Again, not of dire consequence.

The trade unit function u(t) as defined is controlled by the number of stocks in the portfolio and by the ongoing liquidation value of the portfolio. From the portfolio's objective, and the trading methods used, we want F(t) to grow exponentially. That is the mission, and at full market exposure, the bet size will grow at the same rate as the portfolio. If the alpha is zero, the bet size might grow at the same rate as the expected long-term market return minus expenses as illustrated in the following equation.

F(t) = F_{0} + y ∙ rb ∙ j ∙ **E**[tr] ∙ u(t) ∙ **E**[PT] = F_{0} ∙ (1 + **E**[r_{m}] + α - ex(t))^{t}

It is by adding more skill to the game, by adding some alpha (α) that you can raise the bar by forcing a higher bet sizing function. The more your trading skills will raise the overall portfolio return, the more the bet sizing function will rise. However, making bigger and bigger bets does not make profits. It only sets how much you will put on the table depending on the value of F(t). And due to the fixed number of stocks *j*, each of those bets will be F(t) / j.

In a drawdown period, F(t) will decline, thereby, reducing the bet size proportionally. Whatever added skills you bring to the mix, as long as the generated alpha is greater than the added expenses α > |ex(t)|, it will raise overall returns above the expected market average **E**[r_{m}].

The added alpha is a requirement for outperformance. It should be noted that a negative alpha is detrimental to your portfolio. You would be trading and producing less than if you had bought a market surrogate, something like SPY, for instance. And producing less than market averages should not be considered an optimal scenario.

**Related Recent Articles**:

**Toward Playing A Smarter Stock Trading Game**

**Stock Trading Game - Gambling It Out**

**Trading Is Not That Simple, But**

**Winning The Stock Trading Game**

**Books**:

**Reengineering Your Stock Portfolio**

June 22, 2020, © Guy R. Fleury. All rights reserved.