The expected value for the intelligent e-mini day trader

The expected value (also known as the expectation value) is truly a fundamental concept in trading. It is also a fundamental concept in gambling and it plays the same role in both: it determines your edge.

It plays also a very important role in science and mathematics. For instance, the expectation value of a certain fundamental operator of quantum physics known as the Hamiltonian or the energy operator, determines the energy levels of the quantum system this operator describes. But that's just an aside remark meant to show you how ubiquitous this mathematical concept is.

Because the edge in trading is determined the same way as in gambling, trading really is gambling. There is absolutely no difference between the two, however deprecating this may sound to trading, at least for some. Trading is gambling and thinking like a gambler can actually help you in your trading.

I used to work for some gambling company, at one point located in a plush Beverly Hills neighborhood, so I am intimately familiar with the mathematical aspects of gambling as my job there, among other things, was finding optimal strategies for various casino games and calculating their mathematical edge. I am not the only trader with a background in gambling or in the mathematics of gambling. Chris "Jesus" Ferguson was a successful stock day trader before becoming famous as a poker player. Edward O. Thorp, the mathematician who proved in the book "Beat the Dealer" that the house advantage in blackjack can be eliminated via card counting, was also a hedge fund manager whose personal investments yielded an annualized 20 percent averaged overly nearly 30 years. Both these gentlemen have one more thing in common, a Ph.D. from the UCLA, which cannot possibly hurt.

While this was just yet another aside remark, and rather long one at that, I think it illustrates pretty well the connections between the world of gambling and that of trading and that people successful in one of these fields can be successful in the other one. Moreover, if one is to believe Thorp, gambling can teach you more about the stock market than the other way around. I agree.

Even though the concept of the expected value is of paramount importance in trading, its knowledge is not necessarily too common among budding traders or even those who think of themselves as more advanced. One can easily observe it on trading forums, especially those for the dummie crowd. One example of how this can manifest itself is the following.

A statement is made that a strategy that sacrifices twice as much in losses as it can possibly gain is a bad one. In trading parlance that means that the risk-to-reward ratio of the strategy is greater than 2. A statement like that immediately informs us that the person making it has no understanding of the expected value and perhaps did not even hear about this concept before.

The thing is, you cannot meaningfully discuss any strategy using a single number, such as the risk-reward ratio, or any other for that matter. The only way to tell if the strategy is good, that is, if it can make you money is to find out what its expected value is. The positive expected value, meaning the positive edge, tells us that the strategy makes money and hence it is good provided it makes more money than executing it consumes in the brokerage commissions. If its expected value is negative, the strategy cannot make money and you will be losing even more if the commissions are included.

However, the expected value cannot be determined solely by the risk-reward ratio, and hence this ratio really tells us nothing about the strategy edge, whether it can make money or not as I just alluded above. For this we need yet another ratio, that of wins to losses that informs us how frequently the strategy generates wins compared to losses. For instance, if a given strategy produces on average 60% of winners and 40% of losers, then this ratio is 60/40 or 1.5. This ratio is also related to the odds ratio to be discussed later.

Having this two ratios at our disposal, we can formulate the expected value. It is indeed given by some mathematical formula and the formula in this case is as follows


where RE and RI stand for the risk and the reward, respectively, while WF and LF stand for the winners and losers frequencies, respectively.

In the examples we used above, RE=1, RI=2, WF=0.6 and LF=0.4, which gives us EV = 1*0.6-2*0.4=-0.2. Yes, it is a negative number and that indicates that the strategy with this particular parameters (RE, RI, WF, LF) has a negative edge and hence is a losing one.

But does this really mean that the 2-1 risk-rewards ratio is bad?

Not at all, as things are really relative and depend not only on this ratio, but also the other one, which does not have to be 1.5. It can be higher, and even quite easily so for a skilled discretionary trader, although not necessarily for this particular strategy, but some other one with the same, seemingly bad risk-reward ratio.

Suppose a trader can produce 80% of winners and only 20% of losers while still risking twice as much as he can possibly gain. We can easily find that this time the expected value is EV = 0.8*1-0.2*2=0.4, and since this is a positive number, this particular strategy as executed by our skilled trader has a positive edge.

How does this EV thing as calculated above translates into real money? That's easy and depends on the tick value of your trading instrument. Let me explain this by way of another example.

If your trading vehicle tick value is 5 dollars, as is the case with the Dow e-mini futures contract, my favorite trading emini instrument, then you can make as much as $5*0.4=$2.0 or two bucks per each tick of profit on average, which means you need to target at least 3 ticks to make sure you make enough money to cover your commissions that tend to be about $4-5 dollars per contract with most emini brokers out there. For ES, the S&P 500 e-mini futures contract, with a tick value of $12.5 you need to target only 2 ticks to make money after commissions.

To conclude this part of the article, the take home message is this: don't judge a strategy by its risk-reward ratio because even a seemingly poor ratio of 2-1 does not have to rule out the strategy as ineffective.

Let me now express the expected value in terms of something gamblers like to use more often, that is the odds. You may sometimes hear that the odds of something are 5:1, for instance. What does it mean?

Well, it means that the chances (probability) of this to happen are 5/6 or 5/(1+5), which is about 83%. To calculate this number you take the odds in favor, 5 in this case, and compare to the total odds (for and against), which is 6, or 5 plus 1.

It's easy to express the expected value in terms of odds and the risk and reward parameters. The right formula is

EV = (F*RE - A*RI)/(F+A),

where F and A represent odds in favor and against, respectively, and RI and RE are as defined before.

The odds of 5:1 can also be understood that a certain outcome is 5 times more likely than the opposite one. For instance, that your position (a bet or a trade) is 5 times more likely to be a winner as opposed to being a loser.

I like to use the odds ratio to differentiate between my positions in terms of their quality. The position that I believe has the odds of 5:1 (in my favor, of course) is the lowest grade position I am willing to entertain, and is followed by the position with the 10:1 odds, which is the type of the position that I am willing to defend by adding to it more than once. The highest grade position carries the odds of 15:1 (or better), the "you can bet your barn on type of the position.

Now, the odds of the two of these positions translate into the winning rate of over 90%, but still below 95%, which to some may seem incredibly high. It may, but that does not mean that it is impossible to attain. I am not the only one who can do it, but I agree that you are unlikely to hear about people like that very often not merely because they are extremely rare but also because their very existence threatens the mediocrities who dictate what is "real" and anything that is not is dismissed as "too good to be true." Consequently, the traders capable of producing high frequency winners choose to stay in the closet rather than to argue with Boeotians whose numbers tend to be overwhelming.

Yes, I mean the same mediocrities that would declare the 2:1 risk-reward ratio not kosher enough, which hardly is an indictment of the ratio, but rather of these fellows' poor understanding of the trading math or of their pretty average trading skills. It is also an example of self-limiting beliefs often leading to sub-optimal performance that many a trader succumbs to. But that's a different story, perhaps for another article.

The 2:1 or even slightly greater (inferior) risk-reward ratios appear quite naturally in quick scalping. If you want to scalp for 1 ES point or 5 YM ticks, you cannot avoid this kind of ratio because the stop-loss of 2 points (or 10-15 ticks in YM) is a very natural, safe stop-loss dictated by the market volatility. You cannot choose your stop-loss in a totally arbitrary manner, it has to respect your market volatility.

But if sheer volatility is about 2 points then how hard is it to squeeze 1 point out of this market? Not hard at all, and certainly much easier than getting 2-4 points, which also explains why high winning rates in such circumstances should not be viewed as something unusual, especially when it comes to the skilled day traders.

While I cannot speak for all of the traders who are able to produce the 90% plus winning rates, it seems unlikely to me that they can do it using mechanical systems. For this, a discretionary trading methodology based on a very good reading of the market seems to be necessary. The traders I know that can produce results of this kind are all discretionary traders, some more so than others. In others words, it is truly skill based trading as opposed to trading based on mechanical systems. The type of trading that you can master with the help of KING, a discretionary e-mini trading methodology offered on this site.