{"id":2373,"date":"2019-09-12T13:33:00","date_gmt":"2019-09-12T13:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=2373"},"modified":"2026-04-01T18:26:22","modified_gmt":"2026-04-01T18:26:22","slug":"optimal-portfolios","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/portfolio-management\/optimal-portfolios\/","title":{"rendered":"Optimal Portfolios"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Portfolio Risk and Return \u2013 Part I (2025 Level I CFA\u00ae Exam \u2013 Portfolio Management \u2013 Module 1)\",\n  \"description\": \"This lesson covers the core concepts of Portfolio Management for the 2025 CFA\u00ae Level I exam. Topics include calculating and interpreting expected return, variance, standard deviation, covariance, correlation, and portfolio standard deviation. The video also explains diversification benefits, the efficient frontier, the global minimum variance portfolio, the role of a risk-free asset, the capital market line, and investor risk preferences, with worked examples and calculator tips to support exam readiness.\",\n  \"uploadDate\": \"2023-11-07T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/DLKhsZvcD-c\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/DLKhsZvcD-c\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/DLKhsZvcD-c\",\n  \"duration\": \"PT55M38S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions.png\",\n  \"caption\": \"Optimal Portfolio Given Different Utility Functions\",\n  \"width\": 974,\n  \"height\": 668,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors.png\",\n  \"caption\": \"Risk Aversion for Different Types of Investors\",\n  \"width\": 974,\n  \"height\": 668,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1.png\",\n  \"caption\": \"Capital Allocation Line (CAL) Given Investor Preferences\",\n  \"width\": 974,\n  \"height\": 668,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3.png\",\n  \"caption\": \"Capital Allocation Line (CAL)\",\n  \"width\": 974,\n  \"height\": 668,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Using the utility function U = E(r) \u2212 \u00bd\u03c3\u00b2 and assuming A = \u22124, which of the following statements best describes the investor\u2019s attitude to risk?\",\n    \"text\": \"Using the utility function U = E(r) \u2212 \u00bd\u03c3\u00b2 and assuming A = \u22124, which of the following statements best describes the investor\u2019s attitude to risk?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. A negative risk aversion coefficient (A = \u22124) indicates the investor derives higher utility from taking on more risk, which characterizes a risk-seeking investor.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/portfolio-management\/optimal-portfolios\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/DLKhsZvcD-c\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<p>Risk-free assets are usually government-issued with no risk. When you combine them with risky assets, you create a capital allocation line on a graph. This line connects the best risky portfolio to the risk-free asset.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>The Two-fund Separation Theorem<\/strong><\/h2>\n\n\n\n<p>The two-fund separation theorem says all investors, no matter their preferences or wealth, use two funds: a risk-free one and a portfolio of risky assets. This splits portfolio building into two steps: first, we pick the best mix of risky assets based on their characteristics. Then, we decide how much to allocate to the risk-free asset based on the investor&#8217;s risk preference. Combining the risk-free asset with the risky portfolio makes the capital allocation line (CAL) on a graph.<br><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"668\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3.png\" alt=\"Capital Allocation Line (CAL)\" class=\"wp-image-16804\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3-300x206.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3-768x527.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL3-400x274.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure>\n<\/div>\n\n\n<div style=\"margin: 30px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:block;\n            width:100%;\n            padding:16px 20px;\n            border:2px solid #2f6fed;\n            border-radius:999px;\n            background:#f2f4f8;\n            color:#2f6fed;\n            font-size:16px;\n            font-weight:500;\n            text-decoration:none;\n            text-align:center;\n            line-height:1.2;\">\n    Access our FRM free trial to practice early warning indicators\n  <\/a>\n<\/div>\n\n\n<\/p>\n<h2><strong>Investor Preferences<\/strong><\/h2>\n<p>A highly risk-averse investor may choose to invest only in a risk-free asset. On the contrary, a less risk-averse investor may have a small portion of their wealth invested in the risk-free asset and a large portion invested in the risky portfolio. An investor with a high-risk tolerance may, in fact, choose to borrow from the risk-free asset and invest in a risky portfolio. This enables the investor to invest more than 100% of their assets and create a leveraged portfolio.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16808 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1.png\" alt=\"Capital Allocation Line (CAL) given Investor Preferences\" width=\"974\" height=\"668\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1-300x206.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1-768x527.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Capital-Allocation-Line-CAL-given-Investor-Preferences1-400x274.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/p>\n<h4><strong>Utility and Indifference Curves<\/strong><\/h4>\n<p>Utility is a measure of relative satisfaction that an investor derives from different portfolios. We can generate a mathematical function to represent this utility that is a function of the portfolio&#8217;s expected return, the portfolio variance, and a measure of risk aversion.<\/p>\n<p>$$\\text{U}=\\text{E(r)}-\\frac{1}{2}\\sigma^2$$<\/p>\n<p>Where:<\/p>\n<p>U = Utility.<\/p>\n<p>E(r) = Portfolio expected return.<\/p>\n<p>A = Risk aversion coefficient.<\/p>\n<p>\\(\\sigma^2\\) = portfolio variance.<\/p>\n<p>To determine risk aversion (A), we measure the marginal reward an investor needs in order to take more risk. A risk-averse investor will need a high-margin reward for taking more risks. The utility equation shows the following:<\/p>\n<ul>\n<li>Utility can be positive or negative \u2013 it is unbounded.<\/li>\n<li>High returns add to utility.<\/li>\n<li>High variance reduces utility.<\/li>\n<li>Utility does not measure satisfaction but can be used to rank portfolios.<\/li>\n<\/ul>\n<p>The risk aversion coefficient, A, is positive for risk-averse investors (any increase in risk reduces utility). It is 0 for risk-neutral investors (changes in risk do not affect utility) and negative for risk-seeking investors (additional risk increases utility).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"668\" class=\"aligncenter size-full wp-image-16810\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors.png\" alt=\"Risk-Aversion-for-Different-Types-of-Investors\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors-300x206.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors-768x527.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Risk-Aversion-for-Different-Types-of-Investors-400x274.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/p>\n<p>An indifference curve plots the combination of risk and returns that an investor would accept for a given level of utility. For risk-averse investors, indifference curves run &#8220;northeast&#8221; since an investor must be compensated with higher returns for increasing risk. It has the steepest slope. A more risk-seeking investor has a much flatter indifference curve as their demand for increased returns as risk increases is much less acute.<\/p>\n<p>We can overlay an investor&#8217;s indifference curve with the capital allocation line to determine their optimal portfolio.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"668\" class=\"aligncenter size-full wp-image-16813\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions.png\" alt=\"Optimal-Portfolio-Given-Different-Utility-Functions\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions.png 974w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions-300x206.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions-768x527.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/09\/Optimal-Portfolio-Given-Different-Utility-Functions-400x274.png 400w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Using the utility function \\(\\text{U}=\\text{E(r)}-\\frac{1}{2}\\sigma^2\\)\u00a0and assuming A = -4, which of the following statements best describes the investor&#8217;s attitude to risk?<\/p>\n<p>A. The investor is risk-neutral.<\/p>\n<p>B. The investor is risk-averse.<\/p>\n<p>C. The investor is risk-seeking.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>A negative risk aversion coefficient (A = -4) means the investor receives a higher utility (more satisfaction) for taking more portfolio risk. A risk-averse investor would have a risk aversion coefficient greater than 0, while a risk-neutral investor would have a risk aversion coefficient equal to 0.<\/p>\n<\/blockquote>\n\n\n<div style=\"background:#f2f4f8; border-radius:16px; padding:32px 20px; text-align:center; margin:40px 0;\">\n  \n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-block;\n            background:#4a74c9;\n            color:#ffffff;\n            font-size:16px;\n            font-weight:600;\n            text-decoration:none;\n            padding:14px 34px;\n            border-radius:999px;\n            margin-bottom:16px;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"margin:0 auto; max-width:520px; font-size:16px; line-height:1.6; color:#1f2937;\">\n    Build confidence in identifying early warning indicators with structured FRM practice and exam-level explanations.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Risk-free assets are usually government-issued with no risk. When you combine them with risky assets, you create a capital allocation line on a graph. This line connects the best risky portfolio to the risk-free asset. The Two-fund Separation Theorem The&#8230;<\/p>\n","protected":false},"author":18,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-2373","post","type-post","status-publish","format-standard","hentry","category-portfolio-management","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>Optimal Portfolios in Portfolio Management | CFA Level 1<\/title>\n<meta name=\"description\" content=\"An optimal portfolio balances risk and return, guided by investor preferences, risk aversion, and indifference curves. 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