# Formula for Expected Value

### What is Expected Value?

Expected value refers to the anticipated value of a variable. It represents a generalization of the weighted average of a variable. Other names used for the expected value of a variable are expectation, mean, average, first moment, or mathematical expectation. While mostly associated with economics and statistics, the expected value is also crucial in finance and investing.

The calculation of expected value involves aggregating the product of the likelihood of an outcome occurring with each of the possible outcomes. For example, for four different outcomes, it is necessary to calculate what the probability of each outcome occurring is and then to multiply the product of those likelihoods with the outcomes. Then it is critical to sum all the results to get an expected value.

### How does Expected Value apply in finance and investing?

In investing, investors usually use expected values to estimate a value for an investment in the future. For a random variable, its EV gives a measure of its center of the distribution. Similarly, calculating an expected value allows investors to evaluate various scenarios and choose the one most likely to achieve their desired output.

Therefore, it is a crucial concept used in scenario analysis, which is a technique for determining the expected value of an investment opportunity. Investors can not only calculate the expected value of single discrete variables but also for single continuous, multiple discrete, and multiple continuous variables. Therefore, it is also common to use expected values with multivariate models.

### What is the formula for Expected Value?

The formula to calculate the expected value for a single event that repeats multiple times is straightforward, which is as below.

EV = P(X) x n

In the above formula, ‘P(X)’ represents the likelihood or probability of the event occurring, while ‘n’ shows the number of times the event will repeat. It is the simplest form of expected value. However, in investing, the calculations are more complicated and often involve multiple events. In these cases, the formula for expected value changes to compensate for multiple events, as below.

EV = P(Xi) x Xi

In the above equation, ‘P(Xi)’ represents the probability of the event and ‘Xi’ denotes the event. However, the difference between the two formulas is that of the summation of expected values. In this case, investors must calculate various expected values for multiple events and aggregate them to get a probability-weighted average.

### Example

An investor wants to decide between investing in two stocks. The first stock has a 60% chance of achieving a \$1,000 return, while a 40% chance of getting a \$500 return. On the other hand, the other stock has a 20% chance of a \$2,000 return while an 80% chance of \$400. To select the best option, the investor can calculate the expected value of both stocks and compare them with each other.

For the first stock, the expected value would be as follows.

EV = [60% x \$1,000] + [40% x \$500]

EV = \$800

For the second stock, the expected value is as follows.

EV = [20% x \$2,000] + [80% x \$400]

EV = \$640

According to the expected values of both stocks, the best option for the investor is to go with the first option.

### Conclusion

Expected value is a concept used in statistics to represent the anticipated value of a variable. In finance and investing, it can help investors in estimating the value of an investment in the future. It is a concept that is the core of scenario analysis and multivariate models.

## Further questions

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