{"id":27111,"date":"2021-08-22T11:26:42","date_gmt":"2021-08-22T11:26:42","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27111"},"modified":"2025-09-26T08:42:52","modified_gmt":"2025-09-26T08:42:52","slug":"expected-value-variance-and-standard-deviation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/expected-value-variance-and-standard-deviation\/","title":{"rendered":"Expected Value, Variance, and Standard Deviation"},"content":{"rendered":"\n

Expected Value<\/strong><\/h2>\n

The expected value of a random variable is simply the average of the possible outcomes of that variable, taking into account the probability weights. Thus:<\/p>\n

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

Example: Expected value<\/strong><\/h3>\n

An analyst anticipates that the following returns on an asset:<\/p>\n

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

$$ \\begin{align*} \\text{Expected return} & = 0.05 \u00d7 0.65 + 0.07 \u00d7 0.25 + 0.10 \u00d7 0.08 \\\\ & = 0.0325 + 0.0175 + 0.008 \\\\ & = 0.058 \\\\ \\end{align*} $$<\/p>\n

Calculating Variance from a Probability Model<\/strong><\/h2>\n

The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Thus:<\/p>\n

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

Example: Calculating variance<\/h3>\n

Using the data from the previous example, we can compute the variance of return:<\/p>\n

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

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 simply use the square root of variance:<\/p>\n

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

\n

Question<\/strong><\/h2>\n

You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:<\/p>\n

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

The standard deviation of expected returns is closest<\/em> to:<\/p>\n

A. 0.00007475.<\/p>\n

B. 0.0495.<\/p>\n

C. 0.008646.<\/p>\n

Solution<\/strong><\/p>\n

The correct answer is C.<\/p>\n

The first step involves determining the expected return:<\/p>\n

$$ \\begin{align*} E(X) & = (0.04 \u00d7 0.4) + (0.05 \u00d7 0.25) + (0.06 \u00d7 0.35) \\\\ & = 0.0495 \\\\ \\end{align*} $$<\/p>\n

Next, we must compute the variance of returns:<\/p>\n

$$ \\begin{align*} { \\sigma }^{ 2 }\\left( X \\right) & =0.4(0.04\u20130.0495)^{ 2 }+0.25(0.05\u20130.0495)^{ 2 }+0.35(0.06 \u2013 0.0495)^{ 2 } \\\\ & = 0.00007475 \\\\ \\end{align*} $$<\/p>\n

Lastly, we find the square root of variance to get the standard deviation of expected return:<\/p>\n

$$ { \\sigma }= \\sqrt{0.00007475} = 0.008646 $$<\/p>\n<\/blockquote>\n

Reading 8 LOS 8h<\/em><\/p>\n

Calculate and interpret an unconditional probability using the total probability rule.<\/em><\/p>\n

\n

Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-27111","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nExpected Value, Variance & Standard Deviation | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn expected value, variance & standard deviation, fundamentals of risk measurement and probability for CFA Level 1 quantitative methods.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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