March 31, 2020

My previous article dealt with **The Making Of A Stock Trading Strategy**'s mathematical backdrop. Designing automated trading strategies with the objective of prospering over the long term. There are a multitude of ways of doing so. A trading portfolio, even with its short-term vision, needs to view its final outcome in light of a long-term compounded return. This is where a portfolio's average doubling time takes some importance.

Whatever the trading script, it will need to select some tradable candidates, analyze the available data, and make trading decisions within a portfolio's evolving constraints. However it is done, as stated in the above-cited article, it will have to follow the equal signs below:

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

All three formulations give the same answer, which is the ending value of a trading strategy.

What we should take from the set of equations is that the holding matrix **H** is the most critical element from which everything else is derived. You will get a positive return if and only if Ʃ (**H** ∙ Δ**P**) > 0, effectively saying that the strategy **H** needs to make a profit.

We could express the strategy's average compounded annual growth rate (its CAGR) over time as: g_{avg} = g_{m} + α – ex, where, to the market's average rate of return is added some positive alpha and subtracted trading expenses. Due to the equal signs, whatever the strategy **H**, there will be a corresponding average net return g_{avg}, that it be positive, negative, or zero.

Say your trading strategy has a long-term CAGR g_{avg }= 2.5%. Hopefully, with a low volatility measure, say something like 5.0% or less. With all the volatility we see in the market it could be a commendable outcome with its relatively smooth equity curve.

However, one should realize that such a low CAGR implies doubling one's portfolio every 28 years or so. Refer to the chart below on doubling times. In an environment where you have negative returns on long-term bonds, 2.5% does not look so bad, but is it really enough?

**Doubling Time**

(click to enlarge)

Based on the numbers, it would take 56.32 years to double twice, should the compounding hold on that long. That would be getting 4 times the initial investment: F(t) = 4 ∙ F_{0}. In reality, it is only 3 times in profits since the initial capital is included in the final result.

No matter how stable such a trading strategy could be, I find it extremely low-key. Especially when index funds could greatly exceed such a low return, analyze the numbers in the table below, where a set of CAGRs are computed over 5 to 50 years in 5-year increments. Time in a compounding environment is a major factor, and it becomes even more considerable as you increase the rate of return.

The above chart is like a reality check. It says how much you can get, CAGR-wise, and how long it can take to double one's capital based on the compounding rate of return. All of it will translate into: F(t) = F_{0} + n ∙ x_{avg}, the number of trades executed over the life of the portfolio and the strategy's average net profit per trade.

Whatever you want as the end value for your portfolio, it will have to be executable and produce those two numbers: n ∙ x_{avg} which can then be converted to an equivalent CAGR. The formula does not care that you do 1 or 500,000 trades or even more. What will matter is if you are making an average net profit of $5.00 or $1,000+ per trade.

**CAGR**s

Here is another view of the same thing. Using the $10 million initial stake as is often used in Quantopian simulations:

(click to enlarge)

The table can be scaled up or down by subtracting or adding zeros to the initial capital, which will result in moving the decimal point left or right in the table. It is just a matter of bet sizing: κ ∙ F(t) = κ ∙ F_{0} + Ʃ (κ ∙ **H** ∙ Δ**P**).

Each cell has the same general formula: F(t) = F_{0} + n ∙ x_{avg}. It is not only one combination of n ∙ x_{avg} that will provide an answer. There are gazillions of combinations for each cell. In all cases, to get there, it will be according to the above equations.

**Return Objectives**

From the above CAGR table, you could have elected to operate at a 2.50% CAGR over some 30 to 40 years. Or moved to an index fund in the vicinity of an average 8-10% return over the same period. Or, even better, followed Mr. Buffett with his 20% CAGR over the same time interval with the advantage of no management fees. Just look at the numbers and make your choice. Which strategy should or will you follow? Can you accept to pursue higher CAGRs, even with the incremental volatility? Can you do even better than the 20% CAGR?

Whatever your trading strategy does, and however it does it, the number of trades and the average net profit per trade are the crucial elements you need to concentrate on to not only control but also find ways to increase the outcome as much as you can within your own portfolio constraints.

In a way, you could pick any one of the cells in that table and then convert it into: F(t) - F_{0} = n ∙ x_{avg}, which would give you the needed numbers to achieve. Then, it is a matter of forcing your trading strategy to actualize those two numbers over the long term or find strategies that can do the job.

It is asking what combination of those two numbers will give you what you want. Can your program do it? And is it not only reasonable but also doable? Those are the questions. My recent articles have dealt with these considerations.

The above charts and equations illustrate that it is a compounding game and that the trading strategy **H** is at the center of it all. It is the what you do that counts. Fortunately, you can make it do what you want within your own views of the game and portfolio constraints.

**Related Articles**:

**The Making Of A Stock Trading Strategy**

**Dealing With Stock Portfolio Equations**

March 31, 2020, © Guy R. Fleury. All rights reserved.