You already know you will be faced with a lot of uncertainty as not to call it randomness, stochastic behavior or outright chaos. If you flip a fair coin to determine the next move on another fair coin, you should not be surprised if you get it right only about half the time.

Then why, when you see that you are getting about half of it right while trading can't you see that the thing you are betting against might be quasi-random with close to 50:50 odds? It is the only way for you to get it right about half the time, almost no matter what you do. If what you were facing was better than 50:50, then you most likely would be right more often and profit a lot more.

This forces us to reconsider the expression: $ \sum^n_i x_i $ since all $ x_i $'s are not necessarily positive or equal.

Because we are not expected to win all the time, we are faced with more averages. We can use $\bar x_{+}$ for the average profit of positive trades and $\bar x_{-}$ for the average loss per losing trade. This leads to: $ \sum^n_i x_i = (n - \lambda) \cdot \bar x_{+} + \lambda \cdot \bar x_{-}, \,$ with $\, 0 \leq \lambda \leq n.\,$ The trader can win only if: $\,(n - \lambda) \cdot \bar x_{+} > |\lambda \cdot \bar x_{-}|\;$ or $\displaystyle{\frac{(n - \lambda) \cdot \bar x_{+}}{\lambda \cdot |\bar x_{-}|} > 1}.\;$ The more this ratio will tend to 1: $\displaystyle{\frac{(n - \lambda) \cdot \bar x_{+}}{\lambda \cdot |\bar x_{-}|} \to 1},\;$ the more we will have to rely on $(n - \lambda)$, the number of wins to provide an edge. If $\,n = 2 \cdot \lambda \,$ or $\, \lambda = n / 2, \,$ then you are left with almost nothing.

In a game with close to 50:50 odds where the normal distribution has a mean near zero, it would almost imply that within one standard deviation the losses out-balance the wins. This would mean that about 66% of trades simply cancel themselves out. And if you want to push even further, we could say the same thing for what is within 2 standard deviations from the mean. That would be 95% of trades cancel each other out. That is a terrible number for a trader. It nullifies most of the trader's work and to top it off he will also have to pay commissions on all of it. This happens because we trade, and somehow we have to face it.

This leaves the remaining 5% of trades to make a difference. It implies you would be good with fat tails, outliers which are much less predictable than the near-zero mean.

This should put the emphasis on the need to develop an edge and somehow move the mean above its near-zero expectation. Otherwise, you do not stand that much of a chance of assuring yourself that you will win the game. But then again, the gambler does not care about that either. He just wants to play whatever the outcome.

However, I want more than just the fun of the game. As was shown in the CAPM Revisited series of articles, not much is required to give your trading strategy an edge. One-hundredth of one standard deviation above the zero-mean was sufficient to carry the day. It could maintain a steady CAGR level for years and years. Two-hundredth of one standard deviation produced an increasing CAGR over the testing period. Therefore, stopping the return to decay can be done.

**Averaging Trades**

A trading strategy can be programmed to have a singular signature. For instance, trade a number of times per period. From **Part V**, the whole trading strategy was divided into four phases:IS + OOS + Paper Trading + Live Trading, each with their respective stopping times. The IS, OOS, and PT phases were used in the development process. First to debug the program, test it out-of-sample to then paper trade the strategy for some time. All of it leading to the live trading phase where real money would be on the table.

The following chart was taken from the backtest analysis of a trading strategy looking to participate in Quantopian's contest. What it shows is that it maintains about 840 positions per day, and that number remains constant over the 10-year period. The chart also says that the strategy is market neutral having as many longs as it has shorts for the duration.

The chart below, from the same backtest, reveals that the daily turnover was relatively constant over the 10-year period as well. Some monthly spikes (not exceeding 0.20) but overall a constant average turnover (about 0.145). The daily trading volume averaged out close to 40,000 shares per day with occasional monthly spikes reaching up to 200,000 shares.

From the above two charts, we could divide the whole trading interval into 4 parts (IS, OOS, PT, and LT) where we could pick the boundaries for the change of phase from one to the next. The overall profit would have for equation: $$n \cdot \bar x = n_{IS} \cdot \bar x_{IS} + n_{OOS} \cdot \bar x_{OOS} + n_{PT} \cdot \bar x_{PTS}+ n_{LT} \cdot \bar x_{LT}$$However, simply from observation, should we divide the above chart into four equal parts, we could also write: $$ n_{IS} \cdot \bar x_{IS} \approx n_{OOS} \cdot \bar x_{OOS} \approx n_{PT} \cdot \bar x_{PTS} \approx n_{LT} \cdot \bar x_{LT}$$since all the averages remain constants over each phase. Note that we could also break those phases anywhere, the averages would still be maintained.

That makes the trading strategy linear. At each period it would increase by about the same amount, and it would see its CAGR decrease at every step it makes. Not because the market has changed regime or whatever, but simply because the strategy was designed to be linear. From phase to phase, it would keep the same averages. And such a trading strategy will always seem to break down going forward, not due to external forces or overfitting but by the way the strategy was constructed.

It is not that the strategy is overfitted or something like that. It is that the average profit per period $ (n_a \cdot \bar x_a) / \Delta t$ remains constant. At each phase it does as good as the previous one. We could even argue that the OOS and PT phases were a waste of time since the outcome per $\Delta t$ remained the same.

What should the trading strategy do going forward? The same as it did over the 4 phases mentioned, and that is maintain its averages, especially $ (n_a \cdot \bar x_a) / \Delta t$. This means the CAGR will continue to decline.

For the people saying that all trading strategies break down. I would say that: IF you design your trading strategies that way, then you are right. Such strategies will, in fact, deteriorate over time, and will see their CAGR decay with it, because they are built that way. Therefore, stop designing them in such a manner! This goes for variants on the same theme.

It is up to you, the strategy designer to fix the problem.

You do not want to change your strategy design, then the solution is simple, live with it and its consequences. Maybe when your strategy will not even pay its trading costs, and this after a lot of wasted time, your trading account might suggest you reevaluate.