# Expected Value

## What is the 'Expected Value'

The expected value (EV) is an anticipated value for a given investment. In statistics and probability analysis, the EV is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur, and summing all of those values. By calculating expected values, investors can choose the scenario most likely to give them their desired outcome.

## BREAKING DOWN 'Expected Value'

Scenario analysis is one technique for calculating the EV of an investment opportunity. It uses estimated probabilities with multivariate models, to examine possible outcomes for a proposed investment. Scenario analysis also helps investors determine whether they are taking on an appropriate level of risk, given the likely outcome of the investment.

The EV of a random variable gives a measure of the center of the distribution of the variable. Essentially, the EV is the long-term average value of the variable. Because of the law of large numbers, the average value of the variable converges to the EV as the number of repetitions approaches infinity. The EV is also known as expectation, the mean or the first moment. EV can be calculated for single discreet variables, single continuous variables, multiple discreet variables and multiple continuous variables. For continuous variable situations, integrals must be used.

## Basic Expected Value Example

To calculate the EV for a single discreet random variable, you must multiply the value of the variable by the probability of that value occurring. Take, for example, a normal six-sided die. Once you roll the die, it has an equal one-sixth chance of landing on one, two, three, four, five or six. Given this information, the calculation is straightforward:

(1/6 * 1) + (1/6 * 2) + (1/6 * 3) + (1/6 * 4) + (1/6 * 5) + (1/6 * 6) = 3.5

If you were to roll a six-sided die an infinite amount of times, you see the average value equals 3.5.

## A More Complicated Expected Value Example

The logic of EV can be used to find solutions to more complicated problems. Assume the following situation: you have a six-sided die and want to roll the highest number possible. You can roll the die once and if you dislike the result, roll the die one more time. But if you roll the die a second time, you must accept the value of the second roll.

Half of the time, the value of the first roll will be below the EV of 3.5, or a one, two or three, and half the time, it will be above 3.5, or a four, five or six. When the first roll is below 3.5, you should roll again, otherwise you should stick with the first roll.

Thus, half the time you keep a four, five or six, the first roll, and half the time you have an EV of 3.5, the second roll. The expected value of this scenario is:

(50% * ((4 + 5+ 6) / 3)) + (50% * 3.5) = 2.5 + 1.75 = 4.25