# Expected return

The expected return (or expected gain) is the expected value of a random variable usually representing a gain, i.e. the weighted-average outcome in gambling, probability theory, economics or finance.

It is calculated by using the following formula:

$E[R]= \sum_{i=1}^{n}R_{i}P_{i}$, where:
$R_{i}$ is the return in scenario $i$;
$P_{i}$ is the probability for the return $R_{i}$ in scenario $i$; and
$i$ counts the number of scenarios.

How do you calculate the average of a probability distribution? 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 you knew a given investment had a 50% chance of earning a 10 return, a 25% chance of earning 20 and a 25% chance of earning -10, the expected return would be equal to 7.5:

$E[R]=R_{1}P_{1} + R_{2}P_{2} + R_{3}P_{3} = 10*0.5 + 20*0.25 + (-10)*0.25 = 7.5.$

Although this is what you expect the return to be, there is no guarantee that it will be the actual return.

## Discrete scenarios

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 $E[R] = \frac{1}{3} \cdot 1 - \frac{2}{3} \cdot 0.5 = 0.$

## Continuous scenarios

In economics and finance, it is more likely that the set of possible outcomes is continuous (a numerical or currency value between 0 and infinity). In this case, simplifying assumptions are made about the distribution of possible outcomes. Either a continuous probability function is constructed, or a discrete probability distribution is assumed

## Alternate definition

In finance, expected return can also mean the return of a bond if the bond pays out. This will always be higher than the expected return in the other sense presented in this article because the bond paying out is the highest payout scenario, and failure is always possible.