Expected Value, Variance, and Standard Deviation of Random Variables

Expected Value, Variance, and Standard Deviation of Random Variables

Expected Value

The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore:

$$ E\left( X \right) =\sum _{ i=1 }^{ n }{ { X }_{ i }P\left( { X }_{ i } \right) } $$

Example: Expected Value

An analyst anticipates the following returns from an asset:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {5\%} & { 65\%} \\ \hline {7\%} & { 25\%} \\ \hline {8\%} & { 10\%} \\ \end{array} $$

The expected value of the investment is closest to:

Solution

$$ \begin{align*} \text{Expected return} & = 0.05 × 0.65 + 0.07 × 0.25 + 0.10 × 0.08 \\ & = 0.0325 + 0.0175 + 0.008 \\ & = 0.058 \\ \end{align*} $$

Variance

The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Therefore:

$$ { \sigma }^{ 2 }\left( X \right) =\sum _{ i=1 }^{ n }{ { \left[ { X }_{ i }-E\left( { X } \right) \right] }^{ 2 }P } \left( { X }_{ i } \right) =\left\{ { \left[ X-E\left( { X } \right) \right] }^{ 2 } \right\} $$

Example: Calculating Variance

Using the data from the previous example, we can compute the variance of return:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.65{ (0.05-0.058) }^{ 2 }+0.25{ (0.07-0.058) }^{ 2 }+0.10{ (0.08-0.058) }^{ 2 } \\ & = 0.000126 \\ \end{align*} $$

Standard Deviation

Variance is not easy to interpret because it has squared units. Therefore, we usually use the standard deviation, which has the same units as the expected value. To get the standard deviation, we use the square root of variance:

$$ \begin{align*} \text{Standard deviation} & = \sqrt{\text{Variance}} \\ &= \sqrt{0.000126} \\ & =0.01122 \text{ or } 1.12\% \\ \end{align*} $$

Note: You can always raise the variance to 0.5 power to get the same result.

$$ \begin{align*} \text{Standard deviation} & = \text{Variance}^{0.5} \\ &= 0.000126^{0.5} \\ & =0.01122 \text{ or } 1.12\% \\ \end{align*} $$

Question

You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:

$$ \begin{array}{c|c} {\textbf{Return}} & {\textbf{Probability}} \\ \hline {4\%} & { 40\%} \\ \hline {5\%} & { 25\%} \\ \hline {6\%} & { 35\%} \\ \end{array} $$

The standard deviation of expected returns is closest to:

A. 0.00007475.

B. 0.0495.

C. 0.008646.

Solution

The correct answer is C.

The first step involves determining the expected return:

$$ \begin{align*} E(X) & = (0.04 × 0.4) + (0.05 × 0.25) + (0.06 × 0.35) \\ & = 0.0495 \\ \end{align*} $$

Next, we must compute the variance of returns:

$$ \begin{align*} { \sigma }^{ 2 }\left( X \right) & =0.4(0.04–0.0495)^{ 2 }+0.25(0.05–0.0495)^{ 2 }+0.35(0.06 – 0.0495)^{ 2 } \\ & = 0.00007475 \\ \end{align*} $$

Lastly, we find the square root of variance to get the standard deviation of expected return:

$$ { \sigma }= \sqrt{0.00007475} = 0.008646 $$

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.