[dc]S[/dc]ome of the most persistent myths about trading revolve around the idea of low-risk trades. You will often hear traders say that they took a trade because the risk was “only a few pennies” or they could use a “very tight stop” on the trade. Other times, you will hear traders say that they were justified taking a trade because it offered a good enough “risk / reward” profile. In all of these cases, traders are ignoring some important mathematical realities.

One of the challenges of trading well is learning to think about probabilities. We do not care about the outcome of any one event, or any one trade; the only thing that matters is what happens over a large sample of trades. Rather than focus on the fact that you can buy with a .02 stop, the question you need to ask is “what happens if I do the same trade over and over?” We naturally tend to focus on the outcome of one specific event, but the ability to think in large sample sizes is one of the keys to building intuition about probabilities. This is not a natural way to think; it must be actively cultivated and developed.

*Expected value* (or *expectancy*) is a tool that can help. Perhaps the best way to think of it is that it answers what the average result of doing something over a large number of trials would be. Mathematically:

*Expected value = payoff if the event happens * probability of the event happening*

Both parts of this equation, the payoff and the probability, are essential. One without the other is meaningless. For instance, the payoff from getting hit by an asteroid is very bad (certain annihilation), but the probability of that event happening is vanishingly small. Therefore, most of us don’t spend a lot of time thinking about this. What about not picking up the bread your spouse asked you to get on your way home? This is a scenario that carries a very high probability of a bad outcome, but the consequences are not dire: the overall “expected value” of that scenario isn’t that bad.

We can also use this concept to find the “fair value” of an asset in many situations. What should you be willing to pay for a ticket in a raffle where 1,000 tickets are sold for a $10,000 pot? Your chances of winning are 1/1000, so multiply that by the $10,000 payout to get the expected value of the ticket. Each ticket a fair value of $10; if you are able to buy a ticket for less than that, you are playing a positive expectancy game.

You now know that the “real risk” (expected value) of a stop is determined by this formula:

*real risk = loss if stop is hit * probability of stop being hit*.

It is important to train yourself to evaluate the risk in any trade according to this formula, rather than focusing on the just the size of the possible loss. It is meaningless to evaluate the reward / risk profile of a trade without also considering the associated probabilities. A tighter stop will *always* have a higher chance (probability) of being hit compared to a wider stop–a very tight stop might be a near-certain loss. Even though the size of each individual loss is very small, over a large sample size they add up to a very significant risk. (In trading, it is possible to bleed to death from a thousand paper cuts. I know hundreds of traders who have done just that.) Limiting your risk means having a stop and respecting that stop with perfect discipline. Good risk management means using the proper stop, not necessarily a very tight stop.

For some of you, you have seen this math before, but take a moment to consider its significance anew. For others of you, this will be new. You are lucky, because, if you can build this concept into your thinking, you can avoid many of the mistakes beginning traders struggle with. If you are going to trade well, you need to internalize this concept. Good gamblers do it. Good traders do it, and you must too if you’re going to make it in this business.

I’ll dig a little deeper into this concept in a future blog post. (I also have neglected the issue of position sizing in this post, but will address that in the near future as well.)