Expected Value Formula

Formula to Calculate Expected Value

Expected value formula is used in order to calculate the average long-run value of the random variables available and according to the formula the probability of all the random values is multiplied by the respective probable random value and all the resultants are added together to derive the expected value.

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,

or

where

• p_i = Probability of random value
• a_i = Probable random value

Expected Value Calculation (Step by Step)

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 ≤ p₁, p₂,…., p_n ≤ 1 and p₁ + p₂ +….+ p_n = 1.
• Step 3: Finally, 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 = p₁ * a₁ + p₂ * a₂ + ………… + p_n * a_n

Examples

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.

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) = p₁ (P) * a₁ (P) + p₂ (P) * a₂ (P) + p₃ (P) * a₃ (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) = p₁ (Q) * a₁ (Q) + p₂ (Q) * a₂ (Q) + p₃ (Q) * a₃ (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 returns 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.

Expected Value of Project X

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

• Expected Value (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 (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. The 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 the case of various multivariate models and scenario analysis, it is predominantly used in the calculation of expected return.

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