“Forensics of the Equity Curve Dr Watson, not Statistics….It may not be what it appears!”
As I wander down the noisy paths of trading forum after trading forum I hear the cries of curve fitting. “That’s curve fit”….”no it’s not”…..”yes it is”….and so on and so forth. The mere acceptance of a term without recognition of it’s implications can lead us down a merry path to oblivion. The question ‘curve fit’ lies in the ability to predict a future price with a degree of confidence. It is a forward looking and predictive statement. Now as trend followers we know that prediction is useless so try to avoid getting into this heated debate.
The term ‘curve fit’ relates to the process of constructing a mathematical function called a curve that is best fit to the data, but in a market where randomness might have a say in all of this, your curve within this data, may fit the random data points which by definition have no future forecasting potential, or the important non-random or auto-correlated data points that might have future forecasting potential or indeed what is more likely…..a combination of both. That’s it…..that is all there is to it. No need for large blackboards full of equations.
Now as amateur speculators many would obviously like to think that they are trading the latter ‘autocorrelated’ curve that has future forecasting potential as opposed to the ephemeral former ‘random’ curve of no durability and persistence as this is what translates to an edge over the long term, but without statistical tools of hindsite this question in an inefficiently efficient marketplace becomes a “how long is a piece of string argument.”
As trend followers we simply accept that markets at times will have random trends and non-random trends. There is no need to determine the why’s and when’s…but rather that they are likely to happen. We wait for ‘storm events’. We don’t forecast when they happen but rather establish systems that ‘activate’ when these events are more likely to occur. All our efforts are placed in survival so that when these events occur we can simply participate in them…..warts and all….. Simple as that. We set traps with our systems and invest all our efforts in what is necessary to survive if the events don’t happen.
“Well that rhetoric is just dandy Holmes but not very useful…How do we go about increasing our chances of survival by using a method that can be easily understood and applied by mere mortals?”
*Taps pipe into ashtray and resumes*……..”Observation of the equity curve my dear Watson!”
The equity curve is a derivative representation of how your system (trading strategy) performs against the market condition. If market conditions are favorable to your applied system, then the equity curve rises. If unfavourable….it falls….and if flat, then you are spinning your wheels in the mud with the applied system. It says nothing about whether the price data is random or non-random. Give me a random trend of sufficient duration, and I as a trend follower will profit from it. Give me an auto-correlated trending price series, and I as a trend follower will profit from it.
You see, I can profit from both randomness and non-randomness…however my ability to have a sustainable track record as a trend follower requires me over the Law of Large Numbers to possess an edge and for that I need ‘the house edge’ of non-randomness…..but don’t lose sight of the random profit that you may generate along the way that helps to balance the bookkeeping. Whether or not your strategy is curve fit is not the correct question. What is the most important question of this game is one of survival or managing unfavourable risk. A survivor can reap the rewards of randomness and non-randomness. A predictor of the market is rarely a survivor in this game.
So let’s have a crack at forensically examining a few equity curves and see what our brains can detect without need to resort to mathematical models.
Chart 1 – 2 year equity curve derived from the returns generated from a hypothetical trading system
“Looks ok doesn’t it Holmes? An overall positive slope that sends the initial opening balance soaring. Are we onto a winner with an edge here?”
“Not so fast Watson, let’s break it down.”
The blue rising line tells you the realised equity curve of the system. The green line tells a different story which is the unrealised equity curve.
Now let’s first look at the blue curve. You can see the impact of stops or performance risk mitigation exits in the blue equity curve by the dips in the blue curve which represent the periodic taking of losses. But if you look closely you see that the dips are rapid in descent and large versus the overall positive trajectory which has less slope and the building profits are more moderate. Without having to apply any statistical treatment you can visually see that this strategy therefore has negative skew which is the first warning sign. Negative skew means that the strategy tends towards delivering many small profits with the occasional large loss. Now a single statistical number over the entire time series may not pick this up. It is like hearing a beautiful piece of music that is summarized by a statistical mean being C# with a skew of -5.2. The beauty of the entire musical piece is lost in these single statistical statements. Used unwisely, the statistics can overly simplify the resulting verdict.
So now that we have determined visually that this system has negative skew, we should be on the lookout for other symptoms that are clearly revealed in the equity curve. Let’s look at the green curve. What this immediately tells you is that this particular system has unfavourable periods where it holds onto risk and doesn’t release it despite possessing wide stop loss or performance exit conditions. …..in other words it waits for conditions to become favorable before it cashes in with a realised profit but suffers considerable and rapid drawdowns waiting for this favorable event to unfold.
Over this particular 2 year period, market conditions were overall generally favorable for the system where adverse market conditions were few and far between ….but what risk measures does this strategy have in place to manage more enduring unfavourable market conditions?
“But Holmes, this system does have applied stops in place. Surely it is then protected from adverse risk events.”
“Once again….detach yourself from the assumptions Dr Watson and use that brain of yours.”
What if for example, we have a string of unfavourable realised losses when market conditions turn unfavourable. You don’t need an extended data set to recognise that this risk lurks in this system just by observing the characteristics of the equity curve above. You can see that the losses are individually far larger than the wins so all is required to bust your bank is a sequence of unfavourable events.
A naive observer who simply assumed the high win rate was a good sign and took a quick perusal of the equity curve and selected it as it was approaching a linear condition would have fallen into the very frequent trap of curve fitting, not because of the fact that the data series may have been random or not, but rather that the realised equity curve itself is a derivative signature of the strategy itself as opposed to the market condition.
In this example it is the unrealised equity curve that gives the game away….not the realised equity curve.
So let’s see how this strategy fared over a different data set.
Chart 2 – 4 year equity curve derived from the returns generated from the same hypothetical trading system on a different data set
Now you can clearly see the implications of what was discussed in in a clearer light. By simply forensically examining the equity curve of the different data set, you could have avoided this potential disaster waiting to happen without any need for statistical jargon. This is a trap that is commonly found in data mining efforts that search for the linear equity curve of the ‘holy grail’. Do not be afraid of a drawdown signature Holmes, it is a needed condition to interpret the risk inherent in every strategy.”
“Well that was fascinating Holmes….and I will take what you say on board. This really is an easier game than I thought.”.
“It’s not Watson, as I haven’t told you another very important piece of this puzzle. What I haven’t told you is that this equity curve was generated entirely by randomly generated price data. I have deliberately configured the result by simply placing a short distance profit target and a longer distance stop condition to a simply breakout strategy.”
What you need to understand is that your equity curve is no guarantee of your long term sustainability in this game and that a large portion of that return in your return stream is created from random chance . The only way that all is revealed is through the Law of Large Numbers.
You just don’t need a large data sample. More importantly your data sample needs to navigate a very broad range of market conditions ….and even then….there is no guarantee to possessing an edge. Markets adapt and your systems must adapt as well. An edge is a persistent moving feast. There are no single systems that can be profitable in all market regimes. That is left for the fairy-tales on the trading forums.
The best way to overcome this inevitable question about how to maintain an edge is to widely diversify across different unique systems that can navigate a particular class of market condition without losing to much money when conditions are unfavourable. You need to find a diverse bundle of different positively skewed systems each which address a particular class of market condition. Do not fall into the trap of being attracted to negative skew in adaptive markets or holding onto intrinsic risk. It will end in tears. Ensure that each risk event (aka trade) fully releases risk steam. You will be able to see if this occurs in the equity curve. If a single system produces a linear positively sloping equity curve….then run away like the wind.
The linearity of the equity curve is a process undertaken at the portfolio level and not at the single system level.
Every system over the long term has an inherent volatility cooked into their risk signature. To reveal all, you need to backtest each system over a broad array of differing market conditions to see them, particularly when you don’t have the unrealised equity curve at your disposal. If I simply rely on realised returns over a short period of time where market conditions are predictable and fairly constant, it is a train wreck waiting to happen.
It is only over the Law of Large numbers that we can see the weaknesses of negative skew and this may be 10 or so years of duration dependent on the nature of the underlying market and it’s ‘quasi-stability’.
Be less concerned with ‘curve fitting’ as any successful strategy needs to have ‘curve fitting’ built into their signature and you can make considerable money from random outcomes….and more concerned with whether your equity curve resulting from your strategies performance clearly reflects the underlying risk of loss built into their signature. In adaptive non-stationery markets it is not your predictive ability that matters, but rather your risk management measures that protect the downside. Take care of risk and the profits (either random or not) will take care of themselves.
Trade well and prosper