Probability, expected returns and time averages

In the long run, trading is just a numbers game or a game of probabilities. 

Whatever system you have, there will winning and loss making trades. Fact is, and this is important, is you can never know in which order this will appear. Also, while you can quantify and limit your loss, you cannot know in advance how much you will win in any trade.

Trading is not about perfection. It is about probability and progress. All charts, analysis (fundamental and technical) and trading plans are built on probabilities.


So when considering probabilities, you need to know how much you will make on a winning trade and lose if a trade goes wrong. This is important because you can then arrive at expected earnings. At this stage, it is also important to note that there will always be a difference between expected earnings and what you will earn as a trader (time averages).

Let us take some examples. 

Note that the traditional probability scenario assumes the 100 simultaneous trades (parallel) so the "expected return" is the average or arithmetic mean.

The actual return is time average - this determines your earnings if you take all trades (series) .... this is extremely important and is not the same as 100 people taking one trade.

Example 1: There are 2 possibilities with  50% chance of winning or losing. Your max gain  is 10% and max loss is 5%. Here probability of each is 50/100 or 1/2

Average earnings for the population will be 1/2*10-1/2*5=2.5%

Your actual earning if you take all trades = [1.1^(1/2)*0.95^(1/2)]  - 1=2.2%

Example 2: There are 4 possibilities with equal chance for a trade with 5% SL, no profit no loss, 5% first target and 10% second target. Here the probability of each trade is 1/4 or 0.25 as there are 4 possibilities.

Average earnings for the population will be (1/4)*-5+(1/4)*0+(1/4)*5+(1/4)*10=2.5%

Your actual earning if you take all trades = [0.95^(1/4)*1^(1/4)*1.05^(1/4)*1.1^(1/4)] - 1=2.3%

Example 3: 30% chance of earning 40% return and 70% chance of losing 5% on a trade.

Average earnings of the population will be (30/100)*40-(70/100)*5=12-3.5=8.5%

Your actual earning if you take all trades = [1.4^(30/100)*.95^(70/100)] - 1=6.7%

Example 4: 10% chance of earning 40% return and 90% chance of losing 5% on a trade.

Average earnings of the population will be (10/100)*40-(90/100)*5=4-4.5=-0.5% (loss)

Your actual earning if you take all trades = [1.4^(10/100)*0.95^(90/100)] - 1=-1.2% (loss)

Example 5: 50% chance of earning 100% return and 50% chance of losing 50% capital

Here probability of each trade is 50/100 or 1/2.

Average earnings of the population will be (1/2)*100-(1/2)*50=50-25=25%

Your actual earning if you take all trades = [2^(1/2)*0.5^(1/2)] - 1=0.. 100% loss

The last example shows the difference between population results and a single trader's result. Here the population will show average 25% return whereas any trader will lose everything if he takes both the trades.

NOTE: 
1.The above concepts are basic concepts of probability.
2.Actual earning is determined using geometric mean.
3.For more info, read links on Probability, Expected return and Geometric mean