# Expected Value Definition, Formula, and Examples

## What Is Expected Value (EV)?

The expected value (EV) is an anticipated average value for an investment at some point in the future. Investors use expected value to estimate the worthiness of investments, often in relation to their relative riskiness. Modern portfolio theory (MPT), for instance, attempts to solve for the optimal portfolio allocation based on investments' expected values and standard deviations (i.e., risk).

In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values. By calculating expected values, investors can choose the scenario most likely to give the desired outcome.

### Key Takeaways

• Expected value (EV) describes the long-term average level of a random variable based on its probability distribution.
• In investing, the expected value of a stock or other investment is an important consideration and is used in scenario analyses.
• Modern portfolio theory uses expected value in conjunction with an investment's risk (standard deviation) to come up with optimized portfolios.

## The Formula for Expected Value (EV) Is:

\begin{aligned} EV=\sum P(X_i)\times X_i\end{aligned}

where:

• X is a random variable
• P(X) is the probability of the random variable

Thus, the EV of a random variable X is taken as each value of the random variable multiplied by its probability, and each of those products is summed.

## Understanding the Expected Value

Scenario analysis is one technique for calculating the expected value (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 discrete variables, single continuous variables, multiple discrete variables, and multiple continuous variables. For continuous variable situations, integrals must be used.

## Example of Expected Value

To calculate the EV for a single discrete 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:

﻿ \begin{aligned}\left(\frac{1}{6}\times1\right)&+\left(\frac{1}{6}\times2\right)+\left(\frac{1}{6}\times3\right)\\&+\left(\frac{1}{6}\times4\right)+\left(\frac{1}{6}\times5\right)+\left(\frac{1}{6}\times6\right)=3.5\end{aligned}﻿

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

## What Is a Dividend Stock's Expected Value?

The expected value of a stock is estimated as the net present value (NPV) of all future dividends that the stock pays. If you can estimate the growth rate of the dividends, you can predict how much investors should willingly pay for the stock using a dividend discount model such as the Gordon growth model (GGM).

## How Do I Find the Expected Value of a Stock that Doesn't Pay Dividends?

For non-dividend stocks, analysts often use a multiples approach to come up with expected value. For example. the price-to-earnings (P/E) ratio is often used and compared to industry peers. So, if the tech industry has an average P/E of 25x, a tech stock's EV would be 25 times its earnings per share.

## How Is the Expected Value of a Stock Used in Portfolio Theory?

Modern portfolio theory (MPT) and related models use mean-variance optimization to come up with the best portfolio allocation on a risk-adjusted basis. Risk is measured as the portfolio's standard deviation, and the mean is the expected value (expected return) of the portfolio.