{"id":18813,"date":"2021-07-31T22:59:51","date_gmt":"2021-07-31T22:59:51","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=18813"},"modified":"2024-04-11T12:45:38","modified_gmt":"2024-04-11T12:45:38","slug":"calculate-the-expected-return-on-an-asset-given-an-assets-factor-sensitivities-and-the-factor-risk-premiums","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-expected-return-on-an-asset-given-an-assets-factor-sensitivities-and-the-factor-risk-premiums\/","title":{"rendered":"Calculating Expected Returns from the Arbitrage Pricing Model"},"content":{"rendered":"\r\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/SGL5FhV5IUQ\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<h2>Single-factor Model<\/h2>\r\n<p>The single-factor model assumes that there is just one macroeconomic factor, for example, the return on the market. Therefore:<\/p>\r\n<p>$$ R_i=E(R_i)+\\beta_iF+\\varepsilon_i $$<\/p>\r\n<p>Where:<\/p>\r\n<ul>\r\n\t<li>\\(E(R_i)\\) is the expected return on stock \\(i\\).<\/li>\r\n\t<li>\\(R_i\\) is the return for stock \\(i\\).<\/li>\r\n\t<li>\\(F\\) is the macroeconomic factor.<\/li>\r\n\t<li>\\(B_i\\) is the factor-beta.<\/li>\r\n\t<li>\\(\\varepsilon_i\\) is the surprise return.<\/li>\r\n<\/ul>\r\n<p>In case the macroeconomic factor has a value of zero in any particular period, then the return on the security will equal its initial expected return \\(E(R_i)\\) plus the effects of firm-specific events.<\/p>\r\n<h4>Example: Calculating Expected Returns Using a Single-Factor Model<\/h4>\r\n<p>Assume that the common stock of Blue Ray Limited (BRL) is examined with a single-factor model, using unexpected percent changes in GDP as the single factor. Further, assume that the following data is provided:<\/p>\r\n<p>$$ \\begin{array}{c|c} &amp; \\textbf{GDP} \\\\ \\hline \\text{Factor betas} &amp; 1.5 \\\\ \\hline \\text{Growth for the factors} &amp; 4\\% \\end{array} $$<\/p>\r\n<p>Compute the required rate of return of BRL stock given that the expected return is 10% and assuming that there\u2019s no new information regarding firm-specific events.<\/p>\r\n<h4><strong>Solution<\/strong><\/h4>\r\n<p>Using the single index factor model,<\/p>\r\n<p>$$ \\begin{align*} &amp; R_i =E(R_i)+\\beta_iF+\\varepsilon_i \\\\ &amp; 10\\%+1.5\\times4\\%=16\\% \\end{align*} $$<\/p>\r\n<h2>Multifactor Pricing Model (MPM)<\/h2>\r\n<p>The rate of return in a general multifactor model is given by:<\/p>\r\n<p>$$ R_i=E\\left(R_i\\right)+\\beta_{i1}F_1+\\beta_{i2}F_2+\\ldots+\\beta_{ik}F_k+\\varepsilon_i $$<\/p>\r\n<p>Where:<\/p>\r\n<ul>\r\n\t<li>\\(R_i\\) is the rate of return on stock \\(i\\).<\/li>\r\n\t<li>\\(E(R_i)\\) is the expected return on stock \\(i\\).<\/li>\r\n\t<li>\\(\\beta_{ik}\\) is the sensitivity of the factor \\(k\\).<\/li>\r\n\t<li>\\(F_k\\) is the macroeconomic factor \\(k\\).<\/li>\r\n\t<li>\\(\\varepsilon_i\\) is the firm-specific return or portion of the stock\u2019s return unexplained by macro factors.<\/li>\r\n<\/ul>\r\n<h4>Example: Calculating Expected Return Using a Multifactor Model<\/h4>\r\n<p>Imagine that the common stock of BRL is examined using a multifactor model, based on two factors: unexpected percent change in GDP and interest rates. Further, assume that the following data is provided:<\/p>\r\n<p>$$ \\begin{array}{c|c|c} &amp; \\textbf{GDP} &amp; \\textbf{Interest Rate} \\\\ \\hline \\text{Factor betas} &amp; 1.5 &amp; 2 \\\\ \\hline \\text{Surprise growth for the factors} &amp; 2\\% &amp; 1\\% \\end{array} $$<\/p>\r\n<p>Compute the required rate of return on BRL stock, assuming that the expected return for BRL is 10%, and there is no new information regarding firm-specific events.<\/p>\r\n<h3><strong>Solution<\/strong><\/h3>\r\n<p>$$ \\begin{align*} R_i &amp;=E\\left(R_i\\right)+\\beta_{i1}F_1+\\beta_{i2}F_2 \\\\ &amp; =10\\%+1.5\\times2\\%+2.0\\times1\\% \\\\ &amp; =15\\% \\end{align*} $$<\/p>\r\n<h2>The APT Equation(s)<\/h2>\r\n<p>For a well-diversified portfolio with several sources of systematic risk, the expected return is given by:<\/p>\r\n<p>$$ E(R_i)=E(R_Z)+\\beta_{i1}\\lambda_1+\\cdots+\\beta_{iK}\\lambda_K $$<\/p>\r\n<p>Where,<\/p>\r\n<ul>\r\n\t<li>\\(E(R_i)\\) is the expected return on a well-diversified portfolio \\(i\\).<\/li>\r\n\t<li>\\(\\beta_{ij}\\) is the sensitivity for a portfolio \\(i\\) relative to factor \\(j\\).<\/li>\r\n\t<li>\\(E(R_Z)\\) is the expected rate of return on a portfolio with zero betas (such as risk-free rate of return).<\/li>\r\n\t<li>\\(\\lambda_{j}\\) is the risk premium relative to factor \\(j\\).<\/li>\r\n<\/ul>\r\n<h4>Example: Calculating Expected Returns from the Arbitrage Pricing Theory<\/h4>\r\n<p>Assume the following data exists for portfolio A which has a risk-free rate of 6%:<\/p>\r\n<p>$$ \\begin{array}{c|c|c} &amp; \\textbf{Factor 1} &amp; \\textbf{Factor 2} \\\\ \\hline \\text{Factor sensitivities} &amp; 1.5 &amp; 1.2 \\\\ \\hline \\text{Factor risk premium} &amp; 0.02 &amp; 0.03 \\end{array} $$<\/p>\r\n<p>Calculate the expected return for portfolio A using a two-factor APT model.<\/p>\r\n<h4>Solution<\/h4>\r\n<p>Using the formula:<\/p>\r\n<p>$$ \\begin{align*} E(R_A) &amp;=E(R_Z)+\\beta_{A1}\\lambda_1+\\cdots+\\beta_{AK}\\lambda_K \\\\ E(R_A) &amp; =0.06+1.5(0.02)+1.2(0.03)=0.126=12.6\\% \\end{align*} $$<\/p>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>The following data exists for a portfolio A:<\/p>\r\n<p>$$ \\begin{array}{c|c|c|c} &amp; \\textbf{GDP} &amp; \\textbf{Interest Rate} &amp; \\textbf{Inflation} \\\\ \\hline \\text{Factor betas} &amp; 0.5 &amp; 0.4 &amp; 0.6 \\\\ \\hline \\text{Expected growth in factors} &amp; 2\\% &amp; 1\\% &amp; 3\\% \\end{array} $$<\/p>\r\n<p>Which of the following is the <em>most accurate<\/em> return for portfolio A calculated using a three-factor general multifactor model, given that the expected return is 12%?<\/p>\r\n<ol type=\"A\">\r\n\t<li>13.2%<\/li>\r\n\t<li>14.2%<\/li>\r\n\t<li>15.2%<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is C.<\/strong><\/p>\r\n<p>Using the formula:<\/p>\r\n<p>$$ \\begin{align*} R_A&amp; =E\\left(R_A\\right)+\\beta_{A1}F_1+\\beta_{A2}F_2+\\beta_{A3}F_3 \\\\ R_A&amp; =12\\%+0.5\\times2\\%+0.4\\times1\\%+0.6\\times3\\%=15.2\\% \\end{align*} $$<\/p>\r\n<\/blockquote>\r\n<p>Reading 40: Using Multifactor Models<\/p>\r\n<p><em>LOS 40 (c) Calculate the expected return on an asset given an asset\u2019s factor sensitivities and the factor risk premiums.<\/em><\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Single-factor Model The single-factor model assumes that there is just one macroeconomic factor, for example, the return on the market. Therefore: $$ R_i=E(R_i)+\\beta_iF+\\varepsilon_i $$ Where: \\(E(R_i)\\) is the expected return on stock \\(i\\). \\(R_i\\) is the return for stock \\(i\\)&#8230;.<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,473],"tags":[216,564],"class_list":["post-18813","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-portfolio-management","tag-cfa-level-2","tag-portfolio-management","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>Calculating Expected Returns from the Arbitrage Pricing Model - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"The single-factor model assumes there is just one macroeconomic factor, for example, the return on the market.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/calculate-the-expected-return-on-an-asset-given-an-assets-factor-sensitivities-and-the-factor-risk-premiums\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calculating Expected Returns from the Arbitrage Pricing Model - 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