Expected value (EV)

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In probability theory, the expected value (EV) of a random variable is the weighted average of all possible values a random variable can take on.

The expected value may be intuitively understood by the law of large numbers: It can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll).

Example: If you roll a dice, the possible outcomes are 1, 2, 3, 4, 5 or 6 – all with equal probability of 1/6. The expected value of a dice roll is 3.5.

This example shows that the “expected value” is not a result that may be “expected” in the ordinary sense. Rolling a 3.5 with a dice (or having 2.5 children) is impossible.

In trading, we can speak of EV as the estimated value of an investment with an unknown return.

A simple example on EV in trading

We can buy an apple for \$10. We expect the following (based on analysis or on our experience):

• A likelihood of 50% that we can sell the apple for \$16.
• A likelihood of 25% that we can sell the apple for \$12.
• A likelihood of 25% that we will not be able to sell the apple and it goes bad.

The expected revenue of the “apple deal” would be:

(50% x \$16) + (25% x \$12) + (25% x \$0) = \$8 + \$3 + \$0 = \$11.

Thus, the expected value of the transaction would be \$11 minus the \$10 that we spend for the apple. The resulting EV of \$1 is positive, indicating that doing the deal would be profitable, or “+EV”.

The return on investment in this scenario would be +10%.
In many situations, the expected value of an investment can be estimated with thorough analysis.

Positive result versus positive expected value

Differentiating between a positive result and a positive expected result is a key skill needed for every trader.

A trade that made you profit might, in fact, have had a negative expected value. And a trade that made you a loss might, in fact, have had a positive expected value.

Coming back to the dice example: you bet \$100 on the result of a single dice roll being larger than 2. Obviously, this is a good bet. Still, the result might be a 1 – and you lose the bet.

EV and money management

It also is obvious that you should not bet your life on the dice roll, even if your bet has a positive expected value. The potential loss simply is far too big. This is a good example that shows that you should not take any trade or bet, even if you have a positive expected value.

This is one of the fundamental ideas behind money management. Do trades with a positive expected value, but do not invest more than 1-2% into a single trade. Otherwise, the risk of going broke despite making the right decisions simply is too large.

Money management is one of the most important skills for every trader. Read more about it here: