# 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 amount of times. Formally, it gives the measure of the center of the distribution of the variable.[1]

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.

Although this is what you expect 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 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 $E[R] = \frac{1}{3} \cdot 1 - \frac{2}{3} \cdot 0.5 = 0.$

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 a 80% chance of success with a 50% ROR. For each investment, if it is not successful the investor will lose his entire initial investment.

• The formula for the first investment is (.6 * .7) - (.4 * 1) = .2% Expected ROR
• The formula for the second investment is (.45 * .2) - (.55 * 1) = - 46% Expected ROR
• The formula for the third investment is (.8 * .5) - (.2 * 1) = 20% Expected ROR

These calculations show that in our scenario the third investment is the most profitable of the three. The second one even has a negative ROR. This means that if that investment was done an infinite amount of times you could expect to lose 46% of the money you invested. The formula of expected value is very straight-forward, but its value depends on the inputs. The more factors that can influence the investment, the more variables in the equation. As Ilmanen stated,

"The foremost need for multi-dimensional thinking is on inputs. When investors make judgments on the various returns on investments, they should guard against being blinded by past performance and must ensure that they take all or most of the following considerations into account".[2]

• Historical average returns
• Financial and behavioral theories
• Forward looking market indicators such as bond yields; and
• Discretionary views

## 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.