The equation presented: A(t) = A(0) + n∙u∙PT, is derived from a portfolio's payoff matrix. In payoff matrix notation, the given equation is: A(t) = A(0) + Σ(H.∙ΔP). Where the expression: Σ(H.∙ΔP) is a simple element by element multiplication. It could also be viewed as the sum of a vector of n trades: Σ q∙Δp. It is the summation, the end result of all trades taken (profits & losses) over the duration of a portfolio, be it simulated or live.
The payoff matrix has for origin a simple integral: ∫ q(t)∙dp, and must be centuries old. It is in reading Schachermayer's 2000 course notes: Portfolio Optimization in Incomplete Financial Markets, that you will find in his equation (1.1) the above payoff matrix. That too is not new, it has been derived by other researchers before him.
When you decompose the payoff matrix equation above, which in passing I would like to state that it has an equation sign, you will find as a linear expression: Σ q∙Δp. It is only that as a payoff matrix, it represents a complete historical record of a portfolio's trading activity. All its trades, over the entire portfolio duration in a single block of data. Therefore, without even the notion of a doubt, I can categorically state: A(t) = A(0) + Σ(H.∙ΔP), just as those before me.
It would therefore be up to you to prove that the equal sign is wrong. Demonstrate that it is a not equal sign that should be there. On this, I look forward to your presentation.
The expression: A(t) = A(0) + n∙u∙PT, has the same meaning. It gives the exact same answer. And is reduced to just three known portfolio metrics.
Any time you do a portfolio simulation, you start with A(0), your initial capital. You end up with A(t), whatever the applied trading strategy produced. As for n∙u∙PT, it is derived from the payoff matrix itself. [A(t) – A(0)]/n = u∙PT, which will translate to the average profit per trade since the trading strategy will have taken n trades. That it be one trade or a million.
Your average profit per trade is simply derived from the percent profit made on your bet size u. I use fixed trading units which would make u a constant. And PT stands for the average percent profit on a trade unit. Σ(H.∙ΔP)/( n∙u) = PT.
All three of these numbers are given by any portfolio simulation you do. And this gives you an added tool to help you control your portfolio's outcome.
For whatever you want to do to improve on your trading strategy. You can count on only four numbers: A(0), n, u, and PT. There is nothing mysterious, or secret, or in need to buy a book to understand this. Although my book does explain all this in detail (307 pages).
You are the one to supply A(0), and my advice on that one is: make it as big as you can. You are playing a CAGR game and the outcome A(t) is proportional to: A(0)∙(1+r)^t. Make A(0) small, and I think you will be wasting your time and energy playing for peanuts. Note that A(0) is independent of what the market is. It is just an expression for the size of your initial trading account.
The trade unit is something you fix. It is not a mystery number either. Do you want to trade $100 dollars at a time or $10,000 or more. It is up to you, and it is also totally independent of the market, or the market outcome. It is just part of your method of play. The denomination of the chips you put on the table.
PT is your edge, your average percent profit per trade. If PT is less than zero, you lost the game. Since overall, you will lose: n∙u∙(- PT). You have to design a trading strategy with a positive edge: PT > 0. If you can't, then quit while you are still ahead.
We are left with n, the number of trades over the life of a portfolio. There is the mystery, if you want one. I have no means, just as anybody else, to know in advance how many trades a particular trading strategy might do over it lifetime. But, that is why “I” do simulations, to get an approximation for this number.
Most people are looking for how to make PT bigger (forecast it, increase their edge), while I am looking to make A(0), n, u, and PT larger, all at the same time. As long as they will think in terms of PT (trying to predict the price from period to period), they will not see what I am trying to do.
If my trading strategy does 100,000 trades over 20 years, I know the average profit per trade: Σ(H.∙ΔP)/n = u∙PT. And from this, I can deduce another approximation due to the large number of trades, an average number of trades per time unit: n/Δt. From which I can extrapolate an estimate of what a strategy might do going forward.
For instance, the initially posted trading strategy in this thread is scalable. If it is, then it has to respond to: 10∙A(t) = 10∙[A(0) + n∙u∙PT]. As simple as that. If it does not, then the strategy is not 100% scalable. A demonstration of this is easy, you simply do the tests:
The original trading strategy's scalability test:
http://alphapowertrading.com/quantopian/SPY_WVF_Orig_Init_Cap.png
As a conclusion, whatever trading strategy we want to design, if we want to improve upon it, we are left with very simple questions.
How can I make A(0) bigger? How can I increase the number of trades? How can I increase the trade unit? And, how can I increase my profit margin, my long term edge? That is it! That is what will make A(t) bigger, nothing else. One does not need a two week course to figure this out.
No mystery, no hocus pocus, no secret sauce, no the trick is. Just plain math.
What you see in the simulations I've provided is me answering those questions to my satisfaction.
Somebody else might prefer other routes. Like you stated before, you don't like leveraging. Then, don't use any. However, note that I went for 2x leverage on a 3x leveraged ETF portfolio which can easily explain part of the outperformance. This is equivalent to going 6x leverage. But then again, if it would not have worked, the strategy would have “crashed and burned”. Nonetheless, one could have stopped at anytime, closed the account and walk away with whatever the net liquidation value equity line was giving at the time.
This does not change the mission. You still have to maximize the equation: A(t) = A(0) + n∙u∙PT, that is the objective of any portfolio manager and strategy designer.
Personally, I opted to scale profits higher as in: A(t) = A(0) + k∙n∙u∙PT. All I need is k>0, and PT>0. I will let the trading profits finances the added trading which should generate further profits to finance further trades. It is that simple. Somebody else might prefer another solution.
What I say is: anybody, and I do mean anybody, can have their own unique solution to A(t) = A(0) + n∙u∙PT. And even from there, they can continue to improve on their strategy designs by pushing on those three portfolio metric levers as was demonstrated in the three successive tests I presented. What those test results showed is that it can be done. At least, over the testing interval, under those trading conditions, it did.
It really is only a matter of choice. After all, there are only there numbers of interest!