If you are trading with a goal of maximizing your long term profits, you should be concerned with maximizing your expectancy. But, if you are like me, and you want to keep your electricity on via trading stocks, expectancy isn’t the right measurement to use. In this article, I’ll explain how to gauge your consistency. In other words, what percentage of your days/weeks/months can you expect to be profitable? This is useful if you are designing a system, and also useful if you want to know more about the system you already use.
Note: Many of the ideas in this article are largely derived from the first 6 pages of this elitetrader thread, started by user Acrary. I’ve tried to expand on that material with more detail and graphs, to make it easier to understand. Plus, it’s a lot more fun to read without all the elitetrader noise…
Expectancy: the Premier Profitability Measure
If you want to know if your trading system is profitable over the long haul, you want to know about your expectancy. You can find lots of articles on the web about expectancy, so I won’t spend long explaining it here. Briefly, it is computed as:
Expectancy = win_rate * avg_win - loss_rate * avg_loss
For the purposes of this article, I am going to ignore exact breakeven trades. This means that the loss_rate is directly related to the win_rate, and expectancy becomes:
Expectancy = win_rate * avg_win - (1 - win_rate) * avg_loss
A positive expectancy means that if you took an infinite number of trades, you would have more money than you started with when you finished (assuming, of course, that you didn’t bust your account during a bad drawdown–see this article for more about the risk of ruin, and this article for more about comparing the risk at different account sizes). Similarly, a negative expectancy means you’d have less money after an infinite number of trades, and a 0 expectancy means you’d break even.
That’s great, but when’s the last time you made an infinite number of trades? :-) What I’d really like to know is, am I going to make money most weeks? months? years? Profitablility and consistency are not the same thing!
Consistency Graphs
So, to get started, let’s look an example consistency graph for a system with positive expectancy. There’s going to be a lot of graphs like this one in this article, so let me explain what they mean.
The horizontal axis describes a number of trades taken during a given time period. The vertical axis tells what percentage of those time periods will be profitable, given that you took that number of trades. This was computed via a monte-carlo type analysis, with 2000 trials per dot on that chart.
So, let’s say you trade the depicted system, and you tend to trade 20 times a month. This graph tells you that you will be profitable around 95% of months you trade. If you trade 30 times a day, then this graph tells you that you will be profitable about 97% of your trading days. If you trade 10 times an hour, then you will be profitable around 85% of the hours that you trade. So it works on any timeframe… you get the idea by now.
See how the graph starts out in the 70% range, and converges on 100% consistency as the number of trades goes up? In general, all graphs of positive expectancy systems look like this, in that they start out near the win rate % consistency, and converge on 100% consistency. The width of the path, and the speed of the convergence will vary.
Now, let’s look at a graph for a system with negative expectancy:
Pretty much the inverse of the positive expectancy chart. It basically says, the more you trade, the less chance you have of being profitable. As an aside, this is exactly why your best bet at a roulette table is to bet everything on one round. Roulette has a negative expectancy for the player, so the more you play, the less consistent your profits will be.
Finally, if your system has an expectancy of 0, the graphs all look like this:
… as the number of trades increases, the consistency % forms a band around 50%. Makes sense.
Effect of Expectancy Size on Consistency
So, by now, you should know that a positive expectancy is necessary both for long term profits, and consistent profits. You might wonder if a larger expectancy system will be more consistent than a smaller expectancy system. The answer, which surprises a lot of people, is: no.
Here are consistency graphs for a range of expectancies from $40/trade to $440/trade. They are only marginally different:
If you think about the meaning of expectancy, you will realize that the $440/trade system will make a lot more money than the $40/trade system. But, for any given set of trades, expectancy is clearly not the aspect of the system that governs the consistency of the profits. We’ll just have to keep looking….
Effect of Win Rate on Consistency
Well, surely, if my system wins more often, I will have more consistent profits? It turns out, winning more often increases expectancy, but does not necessarily do much for consistency of profits. Here are several consistency graphs for systems with increasing win rates. You can see that, after you get above 10 to 15 trades per day/week/month/whatever, the graphs all look about the same.
It makes sense that the win rate would have sway over the answers when there are fewer than 10 trades per trial. So few inputs go into the profitability calculation, that a little luck in either direction changes the answer. Also, at the extreme end, any 1-trade-per-trial run will have a consistency % equal to the win rate (since the single trade is profitable at exactly its win rate).
Effects of Profit Factor on Consistency
Now, if you are an astute reader, you’ve noticed that in the preceding two sections, each graph had a constant “Pf” label at the top. And, since each set of graphs had fairly constant conistency, you might conclude that this “Pf” is what really gauges the consistency of a trading system. You’d be right.
“Pf” stands for “Profit Factor.” The equation for it has the same terms as the expectancy equation, arranged differently:
Profit Factor = (win_rate * avg_win) / (loss_rate * avg_loss)
Since the equation is so similar, you can see by simple transformation that the profit factor will be 1 whenever the expectancy is 0:
- (win_rate * avg_win) / (loss_rate * avg_loss) = 1
- win_rate * avg_win = loss_rate * avg_loss
- win_rate * avg _win - loss_rate * avg_loss = 0
Similarly, the profit factor will be greater than 1 whenever expectancy is greater than 0, and less than 1 whenever expectancy is less than 0.
Here are some charts of systems with increasing profit factors. They are randomly-generated systems with respect to win_rate, avg_gain, and avg_loss… the only thing I’m controlling is that the profit factor of each is rising. It’s easy to see that the profit factor is highly correlated with speed of consistency convergence of a system, and that higher profit factors require fewer trades per period to give a good guarantee of consistency.
If you think the graphs from 1.5 to 3.5 look very similar, check the y-axis! The convergence is getting much faster! (On some of these graphs, Mathematica cut off the early numbers, because it converges on the 99% range so fast that it decided to blow up the 99-to-100% range. Recall that the point for 1 trade-per-period will always be very close to whatever the win rate happens to be, no matter how fast it converges after that.)
Reading that last chart, you can see that if you find a system with a profit factor of 6, and you can trade 15 times a day with it, you will make money 99.95% of days you trade. That means you will have a losing day once every 10 years, if you trade 200 days a year. Oddly enough, I bet you feel pretty bad that day… so try not to let it throw you off! :-)
These trials confirm the advice given in the elitetrader thread, which gives the following guidelines for trading frequency versus profit factor:
| Profit Factor |
# Trades Needed
for 95% Consistency |
| 1.5 |
60 |
| 1.75 |
40 |
| 2.0 |
30 |
| 2.5 |
20 |
| 4.0 |
10 |
By looking at my graphs, you can see what the guidelines are for any level you want to target, for profit factors from 1.5 up to 6.
Simplifications
I already mentioned that I ignored breakeven trades. This has only a marginal effect on the outcome, while simplifying my job a bunch. If you make 10 trades a day, and 9 of them are exactly breakeven, then maybe you should do some mental translation when using graphs like these. If your truly breakeven trades are relatively rare (like they are for most people), then you will be fine.
I also assumed during the random trials that a win is always the avg_win, and a loss is always the avg_loss. I could have been more exact by making a probability distribution of wins and losses, via a standard deviation from the averages. Assuming the variance in the wins and losses is not very large, doing this wouldn’t affect the message of this article, and would just make the graphs a bit more noisy. And, I could eliminate that noise by ramping up the number of trials per dot up from 2000 to more like 1,000,000 anyway. So, I didn’t bother. However, if your trading is all over the place, then you could have much wider swings in your profitability than depicted here (especially if you don’t trade often). Do try to keep the variance in your gains and losses as small as possible, if only because it makes it easier to reason about and predict your own performance.
Conclusions
So, here are some guidelines you can take away from all this.
If you are designing a system for consistency:
- Maximize the profit factor.
- Double-check your win rate to make sure your risk of ruin is acceptable
- Trade as many times as you can, within the parameters of the system.
- Get the variability of your returns under control (using something like the modified sharpe ratio, for example)
- Tune the expectancy (or add additional systems) in order to make enough money.
If you are designing a system for maximum profit:
- Maximize the expectancy.
- Double-check your win rate to make sure your risk of ruin is acceptable
- Trade as many times as you can, within the parameters of the system
- Use the profit factor to frame your month-to-month profitability expectations. No reason to be down on yourself for a losing month, if your system should only win 60% of all months!
If you already trade a system, and you want to be more consistently profitable:
- Find more trades to take within your system’s parameters. This doesn’t mean you start taking questionable trades, just to get the count up. Instead, you have to be more efficient about finding and exploiting opportunities to trade.
(this is why you’ve seen me post every now and then about needing to find more trades to take… it’s the only way I know of to be more consistent, without changing my system!)
If you’d like to see the Mathematica notebook that I used for this investigation, you can read the html version of it.
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