Jan. 24, 2019

**Reducing the Trading Interval**

As you reduce the average trade interval for the average trade, its potential average return is also reduced. But this can be compensated by the sheer number of trades that can be made with positive results. There are a lot more 1% moves on a daily basis than there are 10% moves. In the first case, you will find hundreds of them every trading day, while in the second, you could count them with both hands. Also, those 10+% moves appear more as outliers and are much more difficult to predict or anticipate. Whereas, a 1% move can be had, on a daily basis, on about a quarter of the listed stocks, meaning opportunities abound.

If there are so many opportunities, then why is there no one with extraordinary results? Already, 1% a day would translate to (1+0.01)^252 = 1,227.4% a year. Yet, you do not see any of that anywhere. Why is that so?

The most plausible reason would be related to the notion that most trade forecasters are not that good at it. And when you look at it closer, you might, in fact, find it is the most plausible excuse for non-performance. Actually, most forecasters have a hard time exceeding long-term market averages.

If you get the average or less, it is as if you were faced with randomness. All you can grab would be any kind of underlying long-term trend.

If you play a heads or tails game with a biased coin at 52:48, no matter how you would play, you should expect to average out at 52:48, too. The bias would be transferred to you even if your bets were determined by a fair coin. You would expect to get less if you insisted on playing tails all the time. Could you get more than 52 by playing heads only? Yes, but by luck of the draw. No skills required. But then, you might need to know that the coin was indeed biased to the upside.

Nonetheless, in a 52:48 game, you are still expected to get it wrong 48 times out of 100, 480 out of 1,000, or 24,000 times out of 50,000. Get the picture! In Leo's case (see tearsheet cited in **Part II**), the wins to losses came in at 48.98:51.02, indicating something close enough to random that you could not distinguish it from random. Could we assume that over the trading interval, the bias was more like 49:51? Probably. Could it be that the method of play resulted in 49:51, even if the game was 52:48? Possible. Taking another trading interval, the results could and would be different.

When the actual signal is buried deep in the surrounding noise, there are not that many tools that will help you extract that signal, determine its magnitude, and, in the case of stock prices, its direction. The distribution of price variation and, thereby, their percent change could be any type of distribution with a mean approaching zero but not necessarily zero. Something like an off-centered Paretian distribution with fat tails.

The problem, short-term, is that we do not know what tomorrow will bring. If your portfolio is composed of some 500 stocks that you follow on a daily basis, there is no way of knowing which of those stocks will go up or down and by how much. The most expected change is close to zero. Regardless, if you want to play the game, then you should plan and play to win.

Note: for a better view of the mathematical formulas, this part is also available in **HTLM** format.

**A Trading Strategy in Numbers**

The end result of any trading strategy is: Ʃ x_{i}. It states: add up all the profits and losses from all the trades taken over the entire trading interval. There are n numbers to add up, that's it. The average profit or loss per trade is this sum divided by n: Ʃ x_{i} / n = x_bar. It does not say how you got there, but it does say how much you made trading.

There are basic restrictions like at no time should you lose it all: Ʃ x_{i} < - F(t) where F(t) is your ongoing trading capital. The equity function would be: F(t) = F_0 + Ʃ x_{i}. From it, we could express a drawdown restriction such as at all times have: F_0 + Ʃ x_{i} > 0.90 ∙ F(t). Thereby allowing at most a 10% drawdown at any one time. However, doing so starts to limit what you can and cannot do. Added the fact that it might be difficult to do and bring with it other restrictions, which would tend to curtail performance even more.

There should be no surprise if you increase the number of trades or increase the average profit per trade to see the outcome get larger. With an increased number of trades, you have: Ʃ_(n+k) x_{i} = (n + k) ∙ x_bar > Ʃ x_{i} for x_bar > 0. Increasing the average profit per trade would result in: (1 + φ) ∙ Ʃ_(n+k) x_{i} > (n + k) ∙ (1+ φ) ∙ x_bar for φ > 0. This has the same impact as improving the strategy's average edge per trade.

However, the real problem is that when we increase the number of trades, it usually takes more time for the same trading strategy to do it. We will see a return degradation which can be expressed as a decaying function: x_bar ∙ e^(- γt). The impact is a faster rate of decay for the trading strategy's CAGR (see Part II and III), resulting in: (n + k) ∙ (1+ φ) ∙ x_bar ∙e^(-γ t). Even if you are adding more trades at what appears as the trading strategy's modus operandi. The strategy is still breaking down at a faster rate than anticipated.

A trading strategy could also be expressed in matrix notation as:_{ 0}**∫**^{T} **H**_{a ∙}d**P**= _{0}**∫**^{n} **H**_{a∙ }d**P** + _{n}**∫**^{n+k} **H**_{a ∙}d**P** where the same trading strategy **H**_{a} is applied from start to finish. Declaring *n* a stop time for the in-sample (IS) backtest and *k* the added trades viewed as part of a walk-forward or a further out-of-sample (OOS) validation period. Therefore, in matrix notation, the equity curve would be: F(t) = F_0 + _{0}**∫**^{n} **H**_{a ∙}d**P** + _{n}**∫**^{n+k} **H**_{a ∙}d**P**.

As shown in **Part III**, _{n}**∫**^{n+k} **H**_{a ∙}d**P**will not sustain the strategy's CAGR since strategy Ha was designed to produce an average number of trades per period, and that will not increase unless the strategy is changed or the market changes to accommodate this new perspective. As was proposed in the example provided in Part II, adding one trade per day was sufficient for the strategy to maintain its CAGR for over 30 years.

The question becomes: how will you capture trades to meet your objectives? What will be the underlying logic for the decision process?

Note: for a better view of the mathematical formulas, this part is also available in **HTLM** format.

**Related Files**:

**Trading Stocks Generate Its Own Problems – Part I, Part II, Part III**

Created... January 24, 2019, © Guy R. Fleury. All rights reserved