Expected return
The expected return (or expected gain) refers to the value of a random variable one could expect if the process of finding the random variable could be repeated an infinite number of times. Formally, it gives the measure of the center of the distribution of the variable.
It is calculated by using the following formula:
Although this is what one expects the return to be, this only refers to the long-term average. In the short term, each instance of the event can be very different. As denoted by the above formula, simply take the probability of each possible return outcome and multiply it by the return outcome itself. For example, if one knew a given investment had a 50% chance of earning a return of 10, a 25% chance of earning 20 and a 25% chance of earning –10, the expected return would be equal to 7.5:
Although this is what one expects the return to be, there is no guarantee that it will be the actual return.
In gambling and probability theory, there is usually a discrete set of possible outcomes. In this case, expected return is a measure of the relative balance of win or loss weighted by their chances of occurring.
For example, if a fair die is thrown and numbers 1 and 2 win $1, but 3-6 lose $0.5, then the expected gain per throw is
![E[R]={\frac {1}{3}}\cdot 1-{\frac {2}{3}}\cdot 0.5=0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/95b8653c805ee942368ee8f07bd57e3489a170ce)
When we calculate the expected return of an investment it allows us to compare it with other opportunities. For example, it we had the option of choosing between 3 investments; one has a 60% chance of success and if it succeeds it will give a 70% ROR. The second investment has a 45% chance of success with a 20% ROR. The third opportunity has an 80% chance of success with a 50% ROR. For each investment, if it is not successful the investor will lose his entire initial investment.
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Wikipedia