Does your strategy has Positive Expectancy?
By Ivo Luhse
Expectancy is the single most important measurement to determine how profitable your trading strategy is – yet it is the least understood statistic among beginner traders. Almost every new trader is excessively concerned with win-rates, being right and finding a perfect entry – overlooking the most important part of trading: having an edge in the market.
Contrary to common beliefs, neither win-rates nor being ‘right’ nor perfect entries give you an edge. The only way for you to know if your strategy has an edge and will be profitable in the long run is to calculate the strategy’s expectancy.
Expectancy is defined as how much money, on average, we can expect to make or lose for every dollar/euro/pound we risk.
The formula for expectancy is
E(R) = ((1+Win/Loss) x Win rate)-1
R = Risk on each trade
Win= Average winner
Loss = Average loser
Win rate = Probability of winning
Coin toss expectancy
Let's put this formula to work in a game that we are all familiar with: a coin toss. With a fair coin, there is 50/50 chance that it will land on heads. Let's assume you are wagering $1 with a pay-off such that if the coin lands on heads you win $1, but if the coin lands on tails you lose the wagered amount.
Putting this in the expectancy formula we get:
Your expectancy for this game of coin toss is 0; thus if you repeat this game many times, you will neither make nor lose money.
Special Note - An important point to remember is that the expectancy is only correct if the process (coin toss or trade signal) is repeated a large number of times. You can easily get ten heads in a row and be duped (gambler’s fallacy) into thinking that you are playing a positive-expectancy game. According to Cochran’s Sample Size Formula with 5% margin of error, you need to toss the coin at least 101 times to be 70% confident that your expectancy is correct; to be 99% confident, you need to toss the coin at least 666 times.
Now, let's imagine that you have been offered a coin toss game where the wager is still $1, but you win $2 if the coin lands on heads. What is the expectancy for this game?
Let's find out by plugging those numbers into the expectancy formula:
From the result, we can see that the expectancy for each coin toss is now 0.50. So if we toss the coin 100 times and risk $1 on each toss ,we can reasonably expect to collect $50 in winnings (100 x $1 x 0.50 = $50). With the help of the expectancy formula, we learn that this is a positive expectancy game, and you would have to be mad not to play this game all day long.
Be wary of negatively skewed strategies
Where the expectancy formula really comes into its own is when the result is not that obvious: when the rewards are negatively skewed.
Let's say you are offered two games with the following probabilities:
Game A gives a 95% probability to win $10 but a 5% probability of losing $1,000.
Game B gives a 5% probability to win $1,000 but 95% probability of losing $10.
Which game would you play?
Most people would choose Game A because it gives an almost certain win of $10.
However, Game A has negative expectancy and will cause you to lose money over time. This is a very popular way for beginner traders to be lured into following strategies that have negative expectancy. These strategies often have negative skew, which means that they win small amounts very often but lose big when they are wrong. Beginner traders love seeing a lot of winners – so it's an easy sell. It can be many months and even years before a negatively skewed strategies blow up - so for the untrained eye, it’s not easy to spot them.
The correct game to choose is Game B as it has an excellent expectancy of 4.05 and you should play this game as many times as you possibly can.
Professional traders understand expectancy and when they trade they make sure that their wins are much larger than their losses, a strategy we refer to as having positive skew - where a large gain is more likely to happen than a large loss. Profitable trading strategies don't need to be 'right' very often; for some strategies, being correct just two times out of ten is all you need to be a very successful trader and make all the money you want. However, a beginner trader would never even consider this. The idea of taking a trade that has a high probability of being a loser and only a small probability of being a (big) winner is absurd to beginner traders. They much prefer to be frequently 'correct' and therefore choose high accuracy over making money. I know it’s hard to believe, but most people would rather be right than make money.
Get bots to calculate the expectancy for you
Knowing the expectancy formula is very important and every trader who is serious about trading should include it in his or her trading plan. Of course, we must get the technology to help us to do these calculations in real life as knowing the expectancy for multiple strategies in multiple markets quickly becomes an impossible task for the human brain. As the market moves and new information and new participants enter the market, the results of the expectancy formula keep changing.
Expectancy calculation lays at the heart of DARA Algorithms – When DARA’s AI engine searches for trading patterns, by default it only sends you patterns that have high expectancies and confidence levels. While you can always customise the level of expectancy to your own preferences and trading objectives, the idea is to ignore patterns that have negative or low expectancy value and low confidence (small sample sizes).
DARA uses the above expectancy formula combined with Cochran’s Sample Size Formula to determine the expectancy with certain levels of confidence for each new trade signal generated by a trading algorithm. That formula is called Projected Expectancy (PE):
PE(R)=((1+R/R) x Win rate)-1) x confidence
R = Risk on each trade
R/R = New trade signal’s Reward/Risk ratio
Win rate = same signal type success rate
Confidence = confidence in statistics based on the available sample size (using Cochran’s Sample Size Formula with 5% margin of error)
What level of expectancy is good?
Depending on your personal goals and trading objectives, a good expectancy level can vary. If you are a trader who wants active trading your trade expectancy will be lower (around 0.2) , if you’re a trader who prefers to wait only for the best of the best signals your expectancy level will be higher (around 0.5).
Expectancy 0.20 is a good starting point for traders who want to be active and get plenty of signals to trade.
Expectancy 0.50 is a good starting point for traders who want to get only the best signals and don’t mind waiting for a few days for trade signals to emerge.
Trading a strategy with 0.5 expectancy means that if you take 100 trades with $100 risk on each trade, the results you would expect to see is $5,000 (100 trades x $100 risk per trade x 0.50 expectancy=$5,000). A strategy with just 0.2 expectancy will generate only $2,000 profit from 100 trades, however, since the lower expectancy strategy will have more signals the end result will not be much different to a more picky trader who only trades on signals with 0.5 expectancy. Traders with a strategy that has 0.5 expectancy might take 100 trades per year while a trader with 0.2 expectancy could take 250 trades per year so the end result is still the same for both traders – the only difference is how they got there. This is very important since trading is very personal and one strategy will not fit all traders – but thanks to DARA’s customisable inputs each trader can modify strategies to fit his/her personality.