{"id":447,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=447"},"modified":"2025-02-25T12:57:05","modified_gmt":"2025-02-25T12:57:05","slug":"total-probability-total-probability-expected-return","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/total-probability-total-probability-expected-return\/","title":{"rendered":"Unconditional Probability Using the Total Probability Rule"},"content":{"rendered":"<p><iframe width='611' height='344' src='\/\/www.youtube.com\/embed\/hu47ZbsskEw?autoplay=0&#038;loop=0&#038;rel=0' frameborder='0' allowfullscreen><\/iframe><\/p>\n<p>We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities on certain scenarios.<\/p>\n<p><!--more--><\/p>\n<p>In general, if we have a set of mutually exclusive and exhaustive events S<sub>1<\/sub>, S<sub>2<\/sub> \u2026S<sub>n<\/sub>, then:<\/p>\n<p>$$ \\text P(\\text T) = \\text P(\\text T | \\text S_1) \\text P( \\text S_1) + \\text P(\\text T | \\text S_2) \\text P(\\text S_2) + \u2026 + \\text P(\\text T | \\text S_{n}) \\text P(\\text S_n) $$<\/p>\n<h3><strong>Example: Unconditional Probability<\/strong><\/h3>\n<p>Imagine assessing the performance of a stock under different circumstances and coming up with the following probabilities.<\/p>\n<p>$$ \\begin{array}{c|c|c|c} {\\textbf{State of}} &amp; {\\textbf{Probability of}} &amp; {\\textbf{Stock}} &amp; {\\textbf{Probability}} \\\\ {\\textbf{Economy}} &amp; {\\textbf{Economic State}} &amp; {\\textbf{Performance}} &amp; {} \\\\ \\hline {} &amp; {} &amp; {\\text{Rise } \\text P(\\text {SR}|\\text R^{\\text C}) } &amp; { 0.8} \\ {\\text {No recession } \\text P(\\text R^{\\text C}) } &amp; {0. <a href=\"https:\/\/littlescholarsnyc.com\/provigil-without-prescription\/\">https:\/\/littlescholarsnyc.com\/<\/a> 7} &amp; { \\text{Fall } \\text P(\\text {SR}^{\\text C}|\\text R^{\\text C}) } &amp; {0.2} \\\\ \\hline {} &amp; {} &amp; {\\text{Rise }\\text P(\\text {SR}|\\text R)} &amp; { 0.3} \\\\ {\\text {Recession } \\text P(\\text R) } &amp; {0.3} &amp; { \\text{Fall } \\text P(\\text {SR}^{\\text C}|\\text R) } &amp; {0.7} \\\\ \\end{array} $$<\/p>\n<p>How would you go about determining the total probability of a stock rise?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>You need to find the unconditional probability of a stock rise under all circumstances. Therefore,<\/p>\n<p>$$ \\begin{align*} \\text P(\\text{SR}) &amp; = \\text P(\\text {SR} | \\text R^\\text{C}) \\text P(\\text R^\\text{C}) + \\text P(\\text{SR} | \\text R) \\text P(\\text R) \\\\ &amp; = 0.8 * 0.7 + 0.3 * 0.3 \\\\ &amp; = 0.65 \\\\ \\end{align*} $$<\/p>\n<h2><strong>Expected Value<\/strong><\/h2>\n<p>The expected value of a random variable is simply the average of the possible outcomes of that variable, taking the probability weights into account. Therefore:<\/p>\n<p>$$ E\\left( X \\right) =\\sum _{ i=1 }^{ n }{ { X }_{ i }P\\left( { X }_{ i } \\right) } $$<\/p>\n<h3><strong>Example: Expected Value<\/strong><\/h3>\n<p>An analyst anticipates the following returns on an asset:<\/p>\n<p>$$ \\begin{array}{c|c} {\\textbf{Return}} &amp; {\\textbf{Probability}} \\\\ \\hline {5\\%} &amp; { 65\\%} \\\\ \\hline {7\\%} &amp; { 25\\%} \\\\ \\hline {8\\%} &amp; { 10\\%} \\\\ \\end{array} $$<\/p>\n<p>$$ \\begin{align*} \\text{Expected return} &amp; = 0.05 \u00d7 0.65 + 0.07 \u00d7 0.25 + 0.10 \u00d7 0.08 \\\\ &amp; = 0.0325 + 0.0175 + 0.008 \\\\ &amp; = 0.058 \\\\ \\end{align*} $$<\/p>\n<h2><strong>Calculating Variance from a Probability Model<\/strong><\/h2>\n<p>The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Therefore:<\/p>\n<p>$$ { \\sigma }^{ 2 }\\left( X \\right) =\\sum _{ i=1 }^{ n }{ { \\left[ { X }_{ i }-E\\left( { X } \\right) \\right] }^{ 2 }P } \\left( { X }_{ i } \\right) =\\left\\{ { \\left[ X-E\\left( { X } \\right) \\right] }^{ 2 } \\right\\} $$<\/p>\n<h3>Example: Calculating Variance<\/h3>\n<p>Using the data from the previous example, we can compute the variance of return:<\/p>\n<p>$$ \\begin{align*} { \\sigma }^{ 2 }\\left( X \\right) &amp; =0.65{ (0.05-0.058) }^{ 2 }+0.25{ (0.07-0.058) }^{ 2 }+0.10{ (0.08-0.058) }^{ 2 } \\\\ &amp; = 0.000126 \\\\ \\end{align*} $$<\/p>\n<p>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<p>$$ \\begin{align*} \\text{Standard deviation} &amp; = \\sqrt{\\text{Variance}} \\\\ &amp;= \\sqrt{0.000126} \\\\ &amp; =0.01122 \\text{ or } 1.12\\% \\\\ \\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:<\/p>\n<p>$$ \\begin{array}{c|c} {\\textbf{Return}} &amp; {\\textbf{Probability}} \\\\ \\hline {4\\%} &amp; { 40\\%} \\\\ \\hline {5\\%} &amp; { 25\\%} \\\\ \\hline {6\\%} &amp; { 35\\%} \\\\ \\end{array} $$<\/p>\n<p>The standard deviation of expected returns is <em>closest<\/em> to:<\/p>\n<p>A. 0.00007475;<\/p>\n<p>B. 0.0495; or<\/p>\n<p>C. 0.008646.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is C.<\/p>\n<p>The first step involves determining the expected return:<\/p>\n<p>$$ \\begin{align*} E(X) &amp; = (0.04 \u00d7 0.4) + (0.05 \u00d7 0.25) + (0.06 \u00d7 0.35) \\\\ &amp; = 0.0495 \\\\ \\end{align*} $$<\/p>\n<p>Next, we must compute the variance of returns:<\/p>\n<p>$$ \\begin{align*} { \\sigma }^{ 2 }\\left( X \\right) &amp; =0.4(0.04\u20130.0495)^{ 2 }+0.25(0.05\u20130.0495)^{ 2 }+0.35(0.06 \u2013 0.0495)^{ 2 } \\\\ &amp; = 0.00007475 \\\\ \\end{align*} $$<\/p>\n<p>Lastly, we find the square root of variance to get the standard deviation of expected return:<\/p>\n<p>$$ { \\sigma }= \\sqrt{0.00007475} = 0.008646 $$<\/p>\n<\/blockquote>\n<p><em>Reading 8 LOS 8h<\/em><\/p>\n<p><em>Calculate and interpret an unconditional probability using the total probability rule.<\/em><\/p>\n<div class=\"notes_inv\">\n<hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities on certain scenarios.<\/p>\n","protected":false},"author":2,"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-447","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Total Probability and Expected Return | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to apply the total probability rule to calculate unconditional probabilities and expected returns in investment analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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