{"id":13278,"date":"2021-04-12T05:24:08","date_gmt":"2021-04-12T05:24:08","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=13278"},"modified":"2026-06-01T08:39:23","modified_gmt":"2026-06-01T08:39:23","slug":"the-mean-reversion","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/the-mean-reversion\/","title":{"rendered":"The Mean Reversion"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-1536x1052.jpg\",\r\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-1536x1052.jpg\",\r\n  \"caption\": \"Mean Reversion Illustration\",\r\n  \"width\": 1536,\r\n  \"height\": 1052,\r\n  \"representativeOfPage\": true,\r\n  \"associatedArticle\": {\r\n    \"@type\": \"WebPage\",\r\n    \"@id\": \"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/the-mean-reversion\/\"\r\n  },\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script> <script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Consider the autoregressive model xt = 0.0407 \u2212 0.5647xt\u22121. What is the mean-reverting level?\",\r\n    \"text\": \"Consider the following autoregressive model:\\n\\nxt = 0.0407 \u2212 0.5647xt\u22121\\n\\nThe mean-reverting level is closest to:\\n\\nA. 0.02601.\\n\\nB. 0.03911.\\n\\nC. 0.09350.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"The correct answer is A.\\n\\nFor an AR(1) model of the form xt = b0 + b1xt\u22121, the mean-reverting level is given by:\\n\\nMean-reverting level = b0 \/ (1 \u2212 b1).\\n\\nSubstituting the values:\\n\\nb0 = 0.0407 and b1 = \u22120.5647\\n\\nMean-reverting level = 0.0407 \/ [1 \u2212 (\u22120.5647)] = 0.0407 \/ 1.5647 \u2248 0.02601.\\n\\nThis represents the long-run equilibrium level toward which the time series converges over time.\"\r\n    }\r\n  }\r\n}\r\n<\/script><\/p>\r\n\r\n<h3><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/-SilFtkpBK8\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/h3>\r\n<p>Mean reversion refers to the behavior of a time series to fall when its values are above the mean and rise when they are below the mean. This is illustrated as follows:<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14880\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12.jpg\" alt=\"Mean Reversion\" width=\"1590\" height=\"1089\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-300x205.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-1024x701.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-768x526.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-1536x1052.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/04\/Img_12-400x274.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>A mean-reverting time series tends to move towards its long-term mean. The model predicts that the value will stay the same if the time series is currently at its long-term mean.<\/p>\r\n<p>i.e., \\(\\hat{\\text{x}}_{\\text{t}}=\\text{x}_{\\text{t}-1}\\)<\/p>\r\n<p>Substituting the above in an AR(1) model, we have:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}-1}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}$$<\/p>\r\n<p>The mean-reverting level can be expressed as:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\text{b}_{0}+\\text{b}_{1}\\text{x}_{\\text{t}}$$<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}$$<\/p>\r\n<p>The value of an AR(1) model will:<\/p>\r\n<p>Remain the same when \\(\\text{x}_{\\text{t}}=\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}\\)<\/p>\r\n<p>Increase when \\(\\text{x}_{\\text{t}}&lt;\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}\\)<\/p>\r\n<p>Decrease when \\(\\text{x}_{\\text{t}}&gt;\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}\\)<\/p>\r\n<div style=\"margin: 18px 0;\"><a style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #2F5BFF; border-radius: 18px; color: #ffffff; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">Strengthen your CFA Level II mean reversion concepts with our Free Trial.<\/a><\/div>\r\n\r\n<h4>Example: Mean Reversion<\/h4>\r\n<p>Consider the following output of an AR(1) model.<\/p>\r\n<p>$$\\begin{array}{c|c|c} {}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} \\\\ \\hline \\text{Intercept} &amp; 31.4 &amp; 4.81 \\\\ \\hline \\text{Lag 1} &amp; -0.0006827 &amp; 0.0001088\\\\ \\end{array}$$<\/p>\r\n<p>The mean-reverting level is <em>closest to:<\/em><\/p>\r\n<p><strong>Solution<\/strong><\/p>\r\n<p>The mean-reverting level can be expressed as:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}$$<\/p>\r\n<p>$$\\text{x}_{t}=\\frac{31.4}{1-(-0.0006827)}=31.39$$<\/p>\r\n<p>It is worth noting that all covariance stationary time series will have a finite mean-reverting level. This will be discussed in more detail in future learning objectives.<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Consider the following autoregressive model:<\/p>\r\n<p>$$\\text{x}_{t}=0.0407-0.5647\\text{x}_{\\text{t}-1}$$<\/p>\r\n<p>The mean-reverting level is <em>closest to<\/em>:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\">\r\n\t<li>0.02601.<\/li>\r\n\t<li>0.03911.<\/li>\r\n\t<li>0.09350.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>The mean-reverting level is given by:<\/p>\r\n<p>$$\\text{x}_{\\text{t}}=\\frac{\\text{b}_{0}}{1-\\text{b}_{1}}=\\frac{0.0407}{1-(-0.5647)}=0.02601$$<\/p>\r\n<\/blockquote>\r\n<p>Reading 5: Time Series Analysis<\/p>\r\n<p><em>LOS 5 (f) Explain mean reversion and calculate a mean-reverting level.<\/em><\/p>\r\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a><p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Access CFA Level II study notes, practice questions, mock exams, and video lessons to strengthen your understanding of mean reversion and time series analysis.<\/p><\/div>\r\n","protected":false},"excerpt":{"rendered":"<p>Mean reversion refers to the behavior of a time series to fall when its values are above the mean and rise when they are below the mean. This is illustrated as follows: A mean-reverting time series tends to move towards&#8230;<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,229],"tags":[216,230,292],"class_list":["post-13278","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-quantitative-method","tag-cfa-level-2","tag-quantitative-method","tag-the-mean-reversion","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mean Reversion and Mean-Reverting Level<\/title>\n<meta name=\"description\" content=\"Understand mean reversion and how to calculate the mean-reverting level using autoregressive models in time series analysis.\" \/>\n<meta name=\"robots\" content=\"index, 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