In the first article on trading performance we learned how to monitor slippage. Furthermore, we saw that the net profit of a strategy has to remain within certain borders which we could calculate.

In this article we will dive a bit deeper and learn about additional statistical indicators which will help us to determine whether a trading strategy is healthy or broken. We will specifically talk about maximum quantile loss (maxQL) and implied time under water (ITuW).

## Probability Cone

We concluded the first article on monitoring the trade performance with a graph that showed the probability cone – the boundaries that should not be crossed by the net profit of our strategy. Here the same graph for another strategy. As you can see, at the beginning net profit got close to the lower border, but then the strategy recovered very nicely. Currently it shows rather a flat performance in live trading. Since this is still within the borders and even close to the projected average one, everything is fine.

## Maximum Quantil Loss (maxQL)

If we want to get an overview of many strategies there are additional indicators which we can look at. One is the maximum Quantil Loss. This maxQL is the minimum of the lower (orange) border. Thus, it is the value of the net profit that should not be undershoot by the net profit of the strategy.

Such maxQL can be calculated as follows:

MaxQL_alpha = (Z_alpha * sigma)^2 / (4 * mu)

In this equation

• mu = Avg. Trade Net Profit
• sigma = 1 Std. Deviation
• Z_alpha = a statistical parameter that relates to the probability of being within our outside of the cone

We can now calculate this number for different probabilities and compare it to the current drawdown. If we do this and put it into a graph for all our strategies the picture may look like the following: In this graph the yellow bars are the maxQL limits for different probabilities. The green one is the average drawdown of the back test. The red bar shows the current drawdown of the strategy.

With such a graph you can immediately see which systems are currently in a drawdown. Which ones have a new equity high (no red bar) and which one may get into trouble (e.g. @ES gets close to the lower limit, but did not yet cross below it).

This is already a good indicator to show whether our drawdown is critical, but it is also one which is slow, since it only warns us as soon as the drawdown is too large. Thus, we only know the problem, when it is almost too late and we already incurred significant loses. Can we do better?

## Time under Water and Implied Time under Water (ITuW)

If we go back to the initial performance graph of the NG strategy, we see that the lower boundary (orange) does not only have a minimum which we called maxQL, but it crosses above zero again on the right end of the scale. This point is called Time under Water (TuW).

TuW tells us how long a drawdown would last, if we assume a certain probability. It is shorter (in number of trades) for a lower probability than for higher ones.

The TuW can also be calculated by an equation, as follows:

TuW_alpha = (Z_alpha * sigma / mu)^2

This is nice, but how to use this? We cannot compare it to the drawdown since this has a different meaning. Luckily David H. Bailey and Marcos Lopez de Prado calculated a respective indicator which they called Implied Time under Water. If you want to read more about this you can do this in a paper published here.

The equation for the ITuW is as follows:

ITuW_pi(t) = pi(t)^2/(mu^2 * t) – 2 * pi(t)/mu + t

• mu = Avg. Trade Net Profit
• sigma = 1 Std. Deviation
• Z_alpha = a statistical parameter that relates to the probability of being within our outside of the cone
• t = number of trades from start of the drawdown
• pi(t) = current loss of drawdown

This equation (and also the one for maxQL) assumes IDD (independent and identically distributed) distributions and no or only little autocorrelation.

On the one hand, the ITuW will increase fast, if a strategy enters into a drawdown fast. On the other hand, it also increases if a strategy runs flat for a longer time than expected.

How to use this. Again, we can plot the results for all our strategies into a graph as depicted below. If the ITuW is below the respective TuW we get a warning for a strategy. In this bar chart you see the calculated TuW in yellow and the ITuW in red. When the red bar crosses the respective yellow ones you get a warning that such strategy is getting into a drawdown faster than it should or is not progressing. Thus, you have a faster signal than with the maxQL chart.

## Summary and outlook

We learned now two additional indicators which can warn us, if a strategy enters into a too deep drawdown or enters into one too fast or is flat for too long. We can use this information in may ways. I use it to detect whether a strategy has a high risk of failing and may consider switching it off, if it breaches the predefined limits.

On strategy level there is one more thing you may watch, that is the Monte Carlo drawdown. We will discuss this more in detail in the next article of this series.