Expectancy along with position sizing are probably the two most important factors in trading/investing success. Sadly most people have never even heard of the concept. Out of the 30 or so trading books I’ve read only a few even touch on any aspect of money management. Only one of those handful of books discussed expectancy. In simple terms, expectancy is the average amount you can expect to win (or lose) per dollar at risk. Here’s the formula for expectancy:

Expectancy = (Probability of Win * Average Win) – (Probability of Loss * Average Loss)

As an example let’s say that a trader has a system that produces winning trades 30% of the time. That trader’s average winning trade nets 10% while losing trades lose 3%. So if he were trading \$10,000 positions his expectancy would be:

(0.3 * \$1,000) – (0.7 * \$300) = \$90

So even though that system produces losing trades 70% of the time the expectancy is still positive and thus the trader can make money over time. You can also see how you could have a system that produces winning trades the majority of the time but would have a negative expectancy if the average loss was larger than the average win:

(0.6 * \$400) – (0.4 * \$650) = -\$20

In fact, you could come up with any number of scenarios that would give you a positive, or negative, expectancy. The interesting thing is that most of us would feel better with a system that produced more winning trades than losers. The vast majority of people would have a lot of trouble with the first system above because of our natural tendency to want to be right all of the time. Yet we can see just by those two examples that the percentage of winning trades is not the most important factor in building a system. As Dr. Van K. Tharp points out:

… your trading system should have a positive expectancy and you should understand what that means. The natural bias that most people have is to go for high probability systems with high reliability. We all are given this bias that you need to be right. We’re taught at school that 94 percent or better is an A and 70 or below is failure. Nothing below 70 is acceptable. Everyone is looking for high reliability entry systems, but its expectancy that is the key. And the real key to expectancy is how you get out of the markets not how you get in. How you take profits and how you get out of a bad position to protect your assets. The expectancy is really the amount you’ll make on the average per dollar risked. If you have a methodology that makes you 50 cents or better per dollar risked, that’s superb. Most people don’t. That means if you risk \$1,000 that you’ll make on the average \$500 for every trade – that’s averaging winners and losers together.

In Trade Your Way to Financial Freedom Dr. Tharp defines the following four components of expectancy (In the same section of the book Dr. Tharp also discusses how the size of you investing capital and your position-sizing model must be considered along with expectancy. I’ll talk about position sizing in another post, but I highly recommend reading Tharp’s book for a thorough understanding of these concepts.):

1. Reliability, or what percentage of time you make money.
4. How often you get the opportunity to trade.

The fourth item in that list is a very important and often overlooked aspect of trading. If you had two systems that both had the same positive expectancy, let’s say \$100, the system that produced more trades would make more money over time. For example, system A produces 3 trades per week while system B produces 10 trades per week. After just one week B would have made \$1,000 while A would only have made \$300. Gary B. Smith has discussed this before:

I think of my equity as inventory. In that respect, my goal is to maximize not profit per dollar, but profit per dollar per day. In essence, I try to make a small percentage on my equity each day, but compound that amount as rapidly as possible.

And in another article Gary said:

Q: What aspect of trading took you the longest to learn?

GBS: I’d say the view that my equity is the same as a retailer’s inventory. Your profit per piece doesn’t matter if you never turn your inventory. And your turnover doesn’t matter if you’re losing money on each sale. What matters is the turnover times profit per piece.

That’s the same concept I try to apply to my trading. I start with \$1 and want to figure out how to quickly turn that into \$10. Most people focus on buying a stock at \$20 and selling it at \$40, and they rarely care how long it takes them to do that.

However, if during that time I can buy 10 \$20 stocks and sell them each at a 10% gain, I’m way ahead if I continue to compound my equity.

To touch on the importance of the size of your equity and position sizing I’ll use part of a snowball fight metaphor that Tharp uses in his book:

Imagine that you are hiding behind a large wall of snow. Someone is throwing snowballs at your wall, and your objective is to keep your wall as large as possible for maximum protection.

Thus, the metaphor immediately indicates that the size of the wall is a very significant variable. If the wall is too small, you couldn’t avoid getting hit. But if the wall is massive, then you are probably not going to get hit. The size of your initial equity is a little like the size of the wall. In fact, you might consider your starting capital to be a wall of money that protects you. The more money you have, assuming all the other variables (the components of expectancy listed above) are the same, the more protection you will have.

Now imagine that the person throwing snowballs at you has two different kinds of snowballs — white snowballs and black snowballs. White snowballs are a little like winning trades. They simply stick to the wall of snow and increase its size….

Imagine that black snowballs dissolve snow and make a hole in the wall equivalent to their size. You might think of black snowballs as “antisnow.” Thus, if a lot of black snowballs were thrown at the wall, it would soon disappear or at least have a lot of holes in it. Black snowballs are a lot like losing trades — they chip away at your wall of security…

Tharp continues walking the reader through different scenarios and possibilities. Like considering the relative sizes of the snowballs of each color. What happens to your wall after being hit by some black boulders of snow? Or considering how the rate at which snowballs are thrown affects the wall. You can see how important each aspect of expectancy is as well as the huge importance of both the amount of equity (the size of your wall) and position-sizing (which will determine the size of the snowballs).

Expectancy, position-sizing and other aspects of money management are far more important than discovering the holy grail entry system or indicator(s). Unfortunately entry techniques are where the vast majority of books and talking heads focus their attention. You could have the greatest stock picking system in the world but unless you take these money management issue into consideration you may not have any money left to trade the system. Having a system that gives you a positive expectancy should be in the forefront of your mind when putting together a trading plan.

1. Posted by on May 20, 2004 at 3:03 pm

Mike,

2. Posted by on June 1, 2004 at 7:27 am

The problem with expectancy is that it goes contrary to what is hard-wired into our brains. Most people believe that it is more important to be right often vs. being right at the correct time.

This discussion of frequency vs. magnitude is leads to thinking in terms of expected value, i.e. being right when it counts. Problem is that people tend to prefer asymetric payoffs.

Nassim Taleb puts it this way: “In some strategies and life situations, it is said, one gambles dollars to win a succession of pennies. In others one risks a succession of pennies to win dollars. While one would think that the second category would be more appealing to investors and economic agents, we have an overwhelming evidence of the popularity of the first.”

3. Posted by on June 1, 2004 at 10:30 pm

Mr. Taleb is the man. For those of you who have not had the opportunity to study decision science or something similar, his book “Fooled by Randomness” might just revolutionize the way you think – not just about investing, but about life!

4. Posted by on June 2, 2004 at 8:05 am

I’ll have to check that out.

5. Posted by on December 27, 2004 at 7:17 pm

Hi Mike,

thanks for a very interesting post about Expectancy. I would like to bring the below (minor) mistake to your attention though:

If losing trades lose 4%, the first formula in your example should be

(0.3 * \$1,000) – (0.7 * \$400) = \$20

not

(0.3 * \$1,000) – (0.7 * \$300) = \$90

which is still > 0 by the way so your
example still makes sense

Greetings from Brussels, Belgium

6. Posted by on December 27, 2004 at 7:29 pm

Firace,

Good catch. I’ve corrected that example.

7. Posted by on October 9, 2005 at 4:56 pm

I wonder if it would make a significant difference if the expectancy formula was slightly altered to:

Expectancy = (Probability of Win * Median Win) – (Probability of Loss * Median Loss)

This is because most trades don’t end up with the same average winning. In fact, many get their biggest wins from a few trades each year. Thus, the average would be skewed and make the expectancy look better than what it should be. Perhaps if we want to get even more sophisticated with trading expectancy that we also incorporate variance.

Why? Because the higher the variance, the riskier and less reliable the results will be. For example, consider these two payoff trees:
\$900               \$600
/                      /
/50%               /50%
/                       /
[X]                   [Y]
\                       \
\50%               \50%
\                       \
\$100                 \$400

If they were payoffs will you prefer? For both on average, the expected value is \$500, yet Y is the better choice in the real life situation. The variance (standard deviation) of X is greater than Y, thus more volitile and less reliable.

Just a thought, what do you think, Mike?

8. Posted by on October 9, 2005 at 7:47 pm

seems like 6 of one and half-a-dozen of the other to me. The actual results are what drives the expectancy not the other way around. I’m fine with the formula using the average win & loss.