# Expected Value: Definition, Formula, and Examples

![Expected Value](https://www.investopedia.com/thmb/ft2M0VY0-MGZNA4IkWxMMP_gwYg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/expected-value-4196762-11-865751470550434b85d2c9eb5ddfe52f.jpg)

*In investing, expected value provides a valuable tool for decision-making under uncertainty. — Investopedia / Joules Garcia*

> **Definition**
>
> Expected value (EV) is a term used by those in the investment industry to denote the anticipated average value of an investment at some point in the future.

## What Is Expected Value?

Expected value (EV) is a formula that investors use to estimate the likely average return they might earn from an investment over time. They use expected value to estimate the worth of investments, often relative to their risk.

By calculating EVs, investors can choose the scenario most likely to produce the outcome they seek. The EV is calculated in [statistics](https://www.investopedia.com/terms/s/statistics.asp) and probability analysis by multiplying each of the possible outcomes by the likelihood that each outcome will occur and then summing all of those values.  

### Key Takeaways

*   Expected value describes the long-term average level of a random variable based on its probability distribution.
*   The EV of a stock or other investment is an important consideration and is used in scenario analyses.
*   Modern portfolio theory uses EV in conjunction with an investment's risk or standard deviation to come up with optimized portfolios.
*   EV can help investors size up whether an investment's risk is worth the potential reward.

## Understanding Expected Value

Expected value refers to the anticipated value of an [asset](https://www.investopedia.com/terms/a/asset.asp) in the future. The EV of a random variable gives a measure of the center of the distribution of the variable. The EV is essentially the long-term average value of the variable.

Because of the [law of large numbers](https://www.investopedia.com/terms/l/lawoflargenumbers.asp), the average value of the variable converges to the EV as the number of repetitions approaches infinity. 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. Integrals must be used for continuous variable situations.

[Scenario analysis](https://www.investopedia.com/terms/s/scenario_analysis.asp) is one technique for calculating the EV of an investment opportunity. It uses estimated probabilities with [multivariate models](https://www.investopedia.com/terms/m/multivariate-model.asp) to examine possible outcomes for a proposed investment. Scenario analysis also helps investors determine whether they're taking on an appropriate level of risk given the likely outcome of the investment.

> **Important**
>
> The difference between expected value and arithmetic mean is that the former involves a distribution of probability and the latter involves a distribution of occurrence.

## Formula for Expected Value

The formula for expected value is:  
  

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

where:

*   X is a random variable
*   Xi are specific values of X
*   P(Xi) is the probability of Xi occurring

The EV of a [random variable](https://www.investopedia.com/terms/r/random-variable.asp) X is therefore taken as each value of the random variable multiplied by its probability. Each of those products is summed.

## Expected Value in Portfolio Construction

Investors should understand several key factors when they want to construct their investment or [financial portfolios](https://www.investopedia.com/terms/p/portfolio.asp). They include how assets work and their associated risks. Investors should also have a firm grasp on their financial situation, investment goals, and investment time horizon.

Investors and their financial advisors can employ EV to build a portfolio that maximizes their returns while minimizing their risks when they thoroughly understand these factors.

### Example of an investor using EV

You can use EV to determine the potential return of an investment and therefore which assets to add to your portfolio based on your preference for return.

First, multiply the probability of a positive outcome by the potential return to calculate expected return. Say an investment has a 60% chance of increasing in value by $10,000. The calculation would be: 0.6 x $10,000 = $6,000.

Then multiply the probability of a negative outcome by the potential loss. The investment also has a 40% chance of decreasing in value by $5,000. The calculation would be: 0.4 x $5,000 = $2,000.

Finally, subtract the second result from the first: $6,000 - $2,000 = $4,000. That's the EV for this investment.

You may next want to compare two or more investments in which you're interested. Follow the same steps and compare the expected returns. This can help you make selections as you build your portfolio.

### Compare EVs for different assets

Also, bear in mind that different assets have different EVs. A stock comes with a different expected value and risk profile than a bond or an [exchange-traded fund (ETF)](https://www.investopedia.com/terms/e/etf.asp). It's useful to calculate EVs for the various assets you're interested in and compare the results.

You can also use EV to adjust your portfolio after it's built. Compare EVs to determine whether selling an underperforming asset with no expectation of a rise in value and replacing it with another with a higher EV makes sense.

## Example of Expected Value

You must multiply each value of the variable by the probability of that value occurring to calculate the EV for a single discrete random variable.

Take a normal six-sided die. It has an equal one-sixth probability of landing on the values of either one, two, three, four, five, or six after you roll it. Given this information, the calculation is:

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

You would find that the average value equals 3.5 if you were to roll a six-sided die an infinite number of times.

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

The expected value of a stock is estimated as the [net present value (NPV)](https://www.investopedia.com/terms/n/npv.asp) of all future dividends that the stock pays. You can predict how much investors should willingly pay for the stock using a dividend discount model such as the [Gordon growth model (GGM)](https://www.investopedia.com/terms/g/gordongrowthmodel.asp) if you can estimate the growth rate of the dividends. It should be noted that this is a different formula from the statistical expected value presented in this article, however.

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

Analysts often use a multiples approach to come up with the expected value of non-dividend stocks. The price-to-earnings (P/E) ratio is often used and compared to industry peers so a tech stock's EV would be 25 times its earnings per share if the tech industry has an average P/E of 25x. Again, this is different from the statistical expected value presented in this article but it's another commonly used method for examining a stock's value.

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

Modern portfolio theory 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 or [expected return](https://www.investopedia.com/terms/e/expectedreturn.asp) of the portfolio.

## The Bottom Line

Understanding the concept of expected value is important for investors. It can aid them in determining the level of return they might expect from an investment. Expected value and scenario analysis can provide insight into the risk of an investment versus its return and help an investor decide whether to include it in their portfolio.

> **Article Sources**
>
> 1.  Wall Street Prep. "[Gordon Growth Model (GGM)](https://www.wallstreetprep.com/knowledge/gordon-growth-model/)."