March 16, 2020

We can represent stock trading systems with equations and not necessarily know that much about their future market returns except general expectations and/or educated guesses. However, with these equations, we can determine what is needed, over the long term, to trade and win.

I will go from not knowing much about future market returns to finding out what is required from simple portfolio equations. In my book: **Reengineering Your Stock Portfolio**, equations were arranged to explicitly give the bet sizing function needed over the life of a portfolio. While at the same time gaining some control over the process in order to generate more alpha. There is a reason for the outperformance. It had for reduced equation:

F(t) = F_{0} + Σ (**H** ∙ Δ**P**) = F_{0} + n ∙ x_{avg} = F_{0} ∙ (1 + g_{m} + α – ex)^{t}

where the alpha *α* is the excess return brought to the game due to the methods of play, and *ex* is the trading expenses incurred, and g_{m} is the average market return. Designing a better positive alpha can have a major impact over the long term, as illustrated in Reengineering Your Stock Portfolio. At the heart of those equations is the payoff matrix: Σ (**H** ∙ Δ**P**), where **H** is the inventory holding matrix, the trading strategy itself, and the most important part of the portfolio equation.

There is math in the stock trading game, whether you like it or not. Ignoring it is at your expense, not mine. Nonetheless, the math that is there can provide a solution as to what to do to perform better over the long term. But there too, it is a matter of choice. Do you want the added work or not? Because you most certainly will have to work harder.

You invest in or trade stocks, and you want the most out of it. Whatever you do can be expressed using a simple future value formula: F(t) = F_{0} ∙ (1 + g)^{t}. It says what the end result will be depending on the initial investment compounded over a number of years at a specified growth rate. It is useful in making past and/or future portfolio value estimates, and that's about it. The formula is good for any type of investment at whatever rate ±*g*. The same formula can be used to discount a future value: PV = F(t) / (1 + g)^{t}. The formula has been around for centuries, so nothing new there.

However, with stocks, to actually know the answer some years from now, you will have to wait to get there first. It could be quite different from your estimates since you do not have access to this ending future growth rate. Nonetheless, rearranging the equation gives: g = (F(t)/F_{0})^{(1/t)} – 1. With positive compounding at play, the more time you stay invested, the more this time can become valuable.

**Trading Is A Different Ballgame**

Trading is a ballgame all its own. It is still investing, but it acquires a more speculative tone due to the quasi-randomness of short-term price movements. You predict, make some assumptions, even hyperbolas at times, and give diverse knowledge attributes to explain those “expected” price gyrations. But whatever, even betting heads on the next flip of a biased coin (0.51) does not make it an assured win; it remains a probabilistic bet with only a slight positive expectancy.

Using past market data, it is relatively easy to do a software simulation of what could have been. If you are not satisfied with the simulation results, you can always make some changes to tweak the program and rerun it. But that is over past data.

In real life, there is no such reset button. There is no going back 10 or 20 years and starting over as if you had not lost anything. If you want to try again, you will need another 10-20^{+} years. Therefore, you better be right in the real-life approach you will be taking since you do not have that many 20+ years to throw away and start over.

Nor should you think that hitting the 65 retirement age is also the end of your portfolio. From 65 onward, you might have up to 40 more years where your source of revenue might just be that depleting portfolio. So, you better look at the problem from a long-term perspective. One thing you cannot do at any one time is blow up your trading account or the ones you manage for anyone else.

**A Trader's Formula**

There is a formula for a trader's future account value. It is the trading system's payoff matrix which can also be used for any type of long-term investor:

F(t) = F_{0} ∙ (1 + g)^{t} = F_{0} + Σ (**H** ∙ Δ**P**)

where **H** is the trading strategy, which could be partially controlled and, therefore, help in controlling part of the strategy's outcome. Whatever the trading program, the ongoing inventory matrix **H** is composed of a buy **B** and a sell **S** matrix: **H** = **B** – **S**. We could rewrite the fund equation as: F(t) = F_{0} + Σ (**S** ∙ **P**) - Σ (**B** ∙ **P**), where Σ (**B** ∙ **P**) represents all the stock purchases taken over the trading interval while Σ (**S** ∙ **P**) is the proceeds from all sales.

**The Payoff Matrix**

The payoff matrix can be any size we want. It can handle any number of stocks for as long as we want and trade as often as we want using whatever trading logic we want under the sole condition that we have the money to play the game (invest). For a trader, the payoff matrix makes it a statistically governed inventory management problem.

The outcome of the payoff matrix will be the total sum of profits and losses from all the trades taken over the entire trading interval *t*. It is not the growth rate *g* that determines this final outcome, it is **H**, the how you will play the game, *g* is just the equivalent compounding rate of return needed to get to the same end results.

Having added the payoff matrix as an equivalent to the future value formula, we have added the notion that it is entirely due to the trading methods used that we will determine the strategy's ultimate value. If Σ (**H** ∙ Δ**P**) = 0, there is no profit and also no loss giving *g* = 0.

Say you buy the whole market, what do you get? You would get the expected average market return over the period *t*: **E**[g_{m}]. You should not expect more, nor should you expect less. Your future return would tend to **E**[g_{m}]: F(t) = F_{0} + Σ (**H**_{m} ∙ Δ**P**) → F_{0} ∙ (1 + g_{m})^{t}. This should not be surprising, having bought everything, you would be the market.

**The Average Portfolio**

However, your own portfolio can only handle a very, very small fraction of what is out there, as if only taking a ridiculously small sample of what is available. And like in most sampling methods, the one you take will be part of the whole and most probably also lead to the expected average market return **E**[g_{m}], which translates right back to: F(t) → F_{0} ∙ (1 + g_{m})^{t}.

Evidently, you might not get the actual expected market average, but it is still something getting close to it. This says that by being fully invested and just reasonably participating in the game, you will get about the same as the other average portfolios out there. Not by choice, but simply as being one sample in the gazillions and gazillions of other samples which in aggregate will have for mean the expected average market return **E**[g_{m}].

It explains why there are so many indexed funds out there. Why most fund managers do not beat the averages. If you cannot beat the averages, then you might as well try to perform as close to the averages as possible.

**Trading Over The Long Term**

Trading can be more profitable than simply investing for the long term. Nonetheless, trading will still answer basic math. It is also covered by the portfolio payoff matrix above.

We could add another refinement to the equation:

F(t) = F_{0} ∙ (1 + g)^{t} = F_{0} + Σ (**H** ∙ Δ**P**) = F_{0} + n ∙ x_{avg}

where *n* is the number of executed trades, and x_{avg} is the average net profit per trade.

This is a lot more related to trading as it is only concerned with the actual number of trades taken and the average net profit per trade. If you do 10,000 trades, which will take time, and make on average a dollar, you wasted much time and effort for those $10,000 bucks.

If those two numbers are all that is needed for your bottom line, then every effort should be made to make them as large as possible within the limits of the game you intend to play and all within your own portfolio constraints.

**The Value of Time**

This does not shorten the time span needed for that compounded return, nor does it shorten the time needed for stock prices to rise. There is not much you can do about the price matrix **P**. It is there and is the same for everyone. If you look at AAPL, whether it be historical data or some future expectations, there is nothing you can do on your own to alter that price that much unless you are very big.

It is only when you take a position that you are at risk of losing or have a potential for profit. Entering or exiting a trade is a decision that either you or your computer program is going to take based on whatever criteria you consider worthwhile, even if, in fact, those same criteria might have no value or might not even be considered worthwhile by others.

The last equation is more revealing for a trader. It states that all 3 formulas give the same ending result. You will get your long-term positive compounding growth rate *g* on the condition that your payoff matrix is positive: Σ (**H** ∙ Δ**P**) > 0. It goes even further to state that your future outcome will depend entirely on the positive edge (x_{avg}) you designed in your trading strategy and how many times you will be able to take advantage of it. If F_{0} + n ∙ x_{avg} < F_{0} ∙ (1 + g_{m})^{t}, then you traded for nothing. It is even worse, you actually paid to make less money since your trading strategy produced less than having bought an index fund which required practically no work.

**Two Basic Numbers**

We have reduced an automated trading strategy to its two most basic components: the average net profit per trade and the number of trades your program was able to execute.

Anything else your program does has little value if it does not have an impact on those two numbers. Those two numbers do not say what were the trading methods used were and, therefore, could be about anything you want as long as they are consistent.

The number of trades is an ever-increasing positive monotonic function that acts as a simple counter. You can make it related to the market if you want or not. The objective is that within your portfolio constraints and methods of play, you can do as many trades as possible with, on average, a positive average net profit per trade.

From an unknown future growth rate *g*, we are able to extract, from the above equations, what to do. And because we intend to trade, the objectives become really simple. It does not say that they will be easy to implement, only that what is required can be expressed in a trading context. You do as many trades as possible (*n*), having an average positive edge expressed as the average net profit per trade (x_{avg}).

Any trading method will follow the above equations. It is not a question of opinion. There are equal signs all over the place. Disprove them, if you can!

March 16, 2020, © Guy R. Fleury. All rights reserved.