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**Formula of Expected Value (Table of Contents)**

## What is the Expected Value Formula?

The term “expected value” refers to the long run average value of all possible values. In the financial parlance, it can also represent the anticipated value for a given investment at any point in time. The formula for expected value with various probable values can be computed on the basis of the probability-weighted average of all the possible values. Mathematically, the expected value equation is represented as below,

**Expected value formula = p**

_{1}* a_{1}+ p_{2}* a_{2}+ ………… + p_{n}* a_{n}or

**Expected value formula= Σ**

_{i}^{n }P_{i}* a_{i}where

- p
_{i}= Probability of random value - a
_{i}= Probable random value

### Explanation of the Expected Value Formula

The formula for the calculation of the expected value of a series of random values can be derived by using the following steps:

**Step 1:** Firstly, determine the different probable values. For instance, different probable asset returns can be a good example of such random values. The probable values are denoted by a_{i}.

**Step 2:** Next, determine the probability of each of the above-mentioned values and they are denoted by p_{i}. Each probability can be any number in the range of 0 to 1 such that the sum total of the probabilities is equal to one, i.e. 0 ≤ p_{1}, p_{2},…., p_{n} ≤ 1 and p_{1} + p_{2} +….+ p_{n} = 1.

**Step 3:** Finally, the equation for the expected value of all different probable values is calculated as the sum product of each probable value and corresponding probability as shown below,

**Expected value = p _{1} * a_{1} + p_{2} * a_{2} + ………… + p_{n} * a_{n}**

**Examples of Expected Value Formula (with Excel Template)**

Let’s see some simple to advanced examples of expected value equation to understand it better.

4.9 (1,067 ratings)

#### Example #1

**Let us take an example of Ben who has invested in two securities within his investment portfolio. The probable rate of return of both the securities (security P and Q) are as given below. Based on the given information, help Ben to decide which security is expected to give him higher returns.**

We will use the following data for the calculation of the expected value equation.

In this case, the expected value is the expected return of each security.

**Expected Return of Security P**

The expected return of security P can be calculated as,

- Expected return (P) = p
_{1}(P) * a_{1}(P) + p_{2}(P) * a_{2}(P) + p_{3}(P) * a_{3 }(P) - = 0.25 * (-5%) + 0.50 * 10% + 0.25 * 20%

Therefore, the calculation of Expected return is as follows,

- Expected return = 8.75%

**Expected Return of Security Q **

The expected return of security Q can be calculated as,

- Expected return (Q) = p
_{1}(Q) * a_{1}(Q) + p_{2}(Q) * a_{2}(Q) + p_{3}(Q) * a_{3 }(Q) - = 0.35 * (-2%) + 0.35 * 12% + 0.30 * 18%

Therefore, the calculation of Expected return is as follows,

- Expected Return= 8.90%

Therefore, for Ben security Q is expected to give higher return than that of security P.

#### Example #2

**Let us take another example where John is to assess the feasibility of two upcoming development projects (Project X and Y) and choose the most favorable one. According to estimates, Project X is expected to achieve a value of $3.5 million with a probability of 0.3 and achieve a value of $1.0 million with a probability of 0.7. On the other hand, Project Y is expected to achieve a value of $2.5 million with a probability of 0.4 and achieve a value of $1.5 million with a probability of 0.6. Determine for John which project is expected to have a higher value on completion.**

We will use the following data for the calculation of the expected value equation.

**Expected Value of Project X **

The calculation of expected value of Project X can be done as follows,

- Expected Value Formula (X) = 0.3 * $3,500,000 + 0.7 * $1,000,000

**Calculation of Expected Value of Project X will be –**

- Expected Value (X) = $1,750,000

**Expected Value of Project Y**

The calculation of expected value of Project Y can be done as follows,

- Expected Value Formula (Y)= 0.4 * $2,500,000 + 0.6 * $1,500,000

**Calculation of Expected Value of Project Y will be –**

- Expected Value = $1,900,000

Therefore, on completion Project Y is expected to have a higher value than that of Project X.

### Relevance and Use

It is important to understand for an analyst to understand the concept of expected value formula as it is used by most investors to anticipate the long run return of different financial assets. Expected value is commonly used to indicate the anticipated value of an investment in the future. On the basis of the probabilities of possible scenarios, the analyst can figure out the expected value of the probable values. Although the concept of expected value is often used in case of various multivariate models and scenario analysis, it is predominantly used in the calculation of expected return.

### Recommended Articles

This has been a guide to Expected Value Formula. Here we discuss how to calculate the expected value using its formula along with examples and downloadable excel template. You can learn more about financial analysis from the following articles –

- 15 Common Techniques for Financial Analysis
- 4 Most Common Tools of Financial Analysis
- Value Formula in Excel
- Examples of Standard Normal Distribution (with Excel Template)
- Excel MEDIAN Formula
- Formula of Binomial Distribution
- Lognormal Distribution Excel
- Median Formula
- Formula for Exponential Growth

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