## Stock Trading Strategy Math II

January 20th, 2017

## Stock Trading Strategy Math II

In a previous article was put forward the notion of a trading strategy's signature. It was defined as the output of a long term automated stock trading strategy that traded a lot. The result of a program which executed what it was programmed to do over an extensive period of time.

If a stock trading strategy is designed to generate thousands upon tens of thousands of trades, it will asymptotically approach a kind of law of large numbers. Meaning that the numbers in n∙u∙PT will become more representative of the whole due to the sheer size of n.

These three numbers might start as metrics in your portfolio backtest, but due to the large number of trades taken, they also become the trading strategy's unique long term signature. You will be able to use the words: on average.

If your portfolio backtest was over the past 20 years, it should provide a pretty good approximation for the next 20.

It goes like this. As the number of trades increase, u∙PT tends asymptotically to a limit. This is easy to visualize. Take 100,000∙u∙PT. Then, add one more trade. You will not see PT move by much. In fact, the added trade's profit percent will tend to be close to PT, the average. Since 1/100,000 of the difference will be able to make it move. The same applies if you put in a million trades. The formula is: lim (n → large n) Σ(H.*ΔP) / n = u∙PT, the net average profit per trade.

A long term backtest is becoming a reasonable approximation of what your trading strategy will do in the future. We have an explicit equation to make this estimate: A(t) = A(0) + n∙u∙PT. This can be said even in the face of usual disclaimers like: "past results are not indicative of future performance" which in fact will hold if you keep the word performance as in return.

You will not know in which stock, at what time, or at what price trades will be taken going forward. But, you do know, right now, that your trading strategy will continue to execute trades as instructed and as programmed. It will produce, on average, about the same output per time unit going forward as it did in the past in its backtest.

This puts a lot of importance on the long term backtest. You want to know the approximation of the asymptotic limit of u∙PT. You want to know the trading strategy's distinctive signature. And, it will be given for a large n. Meaning for a trading strategy generating a lot of trades over an extensive period of time.

This will also give how much you had to put on the table to achieve that goal. That is: n∙u. Evaluating your performance becomes easy as well. It will be:

A(t) / A(0) = (A(0) + n∙u∙PT) / A(0)

It is also from here that you will see why most trading strategies slow down going forward, if not fail to maintain their performance level.

We already stated that due to a trading strategy's signature, and the law of large numbers we should see our trading strategies do about the same as they did in the past. This over extended periods of time. Time was a basic requirement for having a large number of trades.

Take for example a backtest done over the last 20 years producing: n∙u∙PT in total profits. It might not matter how large n∙u∙PT was. So, let's make it huge. Make it such an outstanding backtest that it could result in only one conclusion: this is the trading strategy to be used for the next 20 years. It is that good.

Then, as expected, over the next 20 years you get about the same results as the 20-year backtest. Great. Your trading strategy lived up to its expectations.

The total outcome is in: A(t) = A(0) + n∙u∙PT + n∙u∙PT. Over the added 20 years, you got about the same amount of profit as in the backtest.

The strategy was expected to do about the same number of trades, had a constant trade unit, and PT tended to a limit. Therefore, you do get about the same amount of profit in either of the 20-year periods. So, what your trading strategy did in the past did become indicative of what it would do in the future.

Doubling profits over a 20-year period is a 3.5% CAGR. Nothing more. Now, that is a bummer. Still, those are the numbers. All your trading strategy could do over the added 20 years would be to double its total profits, at most.

You could have done better during those 20 added years abandoning your trading strategy altogether. Not trade it at all, and buy index funds. This, no matter what your trading strategy's performance level would have shown over its backtest.

The nature of the market will probably not change during those added 20 years. Neither would your trading strategy, it would have been frozen in time. It would have kept a relatively stable u∙PT, due to the large number of trades. It would have generated: n∙u∙PT.

And, yet, you would have been better off to scrap your high performing trading strategy rather than having it go live, and underperform market averages.

I hope I am stressing the point enough.

You have to take measures to force your trading program to increase those three portfolio metrics to compensate for what is a built-in and normal performance degradation.

I have shown how this can be done in recent articles. Maybe most importantly, how this compensation can be done with ease.

If you do not do it, it will not happen.

If you do not compensate for these inherent structural gaming deficiencies, the market will not do it for you. Nor will your program.

Going forward, your trading strategy will simply fail over the long run. And this, no matter how good it was.

It will fail for doing what it did best.

It will fail, not by doing less, not by breaking down, not due to changes in regime. But rather because it will be doing the exact same things it did in its backtests.

Technically, it does not matter how good your trading strategy was, it will not be able to do more. Its upper limit is: A(t) = A(0) + 2∙n∙u∙PT over those 40 years. With the first 20 to show you could get: n∙u∙PT.

Your trading program is just a program. As it turns out, it would simply not be enough to have it do the same thing going forward as it did in its backtest.

One has to compensate. No alternative. No other choice. On the other hand, you might find that n∙u∙PT is enough.

This is part of the math of the game. You can ignore it. But... there is a price!

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