{"id":29839,"date":"2021-09-09T13:33:18","date_gmt":"2021-09-09T13:33:18","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29839"},"modified":"2026-03-27T13:55:24","modified_gmt":"2026-03-27T13:55:24","slug":"continuously-compounded-rate-of-return","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/continuously-compounded-rate-of-return\/","title":{"rendered":"Continuously Compounded Rate of Return Given Holding Period Return"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Learning Module 4)\",\n  \"description\": \"This lesson covers Common Probability Distributions for the 2025 CFA\u00ae Level I Quantitative Methods curriculum. It explains probability distributions and the differences between discrete and continuous random variables, cumulative distribution functions, and probability calculations. The video covers discrete and continuous uniform distributions, Bernoulli and binomial distributions, key properties of the normal distribution, univariate versus multivariate distributions with correlation, standardization and use of the standard normal distribution, shortfall risk and Roy\u2019s safety-first criterion, lognormal distributions for asset prices, continuously compounded returns, Student\u2019s t, chi-square, and F-distributions, and an introduction to Monte Carlo simulation, with clear, exam-focused explanations throughout.\",\n  \"uploadDate\": \"2021-11-09T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H00M47S\"\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\": \"What is the continuously compounded rate of return for a stock investment?\",\n    \"text\": \"A portfolio manager buys a stock at $50 and sells it for $56 after a year. The continuously compounded rate of return is closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The continuously compounded rate of return is calculated as: Rcc = ln(56\/50) = 11.3%.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU?si=j4Mtz13JHUgIoW6U\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>Continuous compounding applies either when the frequency with at we calculate interest is infinitely large or the time interval is infinitely small. Put quite simply, under continuous compounding, time is viewed as continuous. This is a departure from discrete compounding, where we deal with finite time intervals.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>We have previously seen how discrete compounding works, given a finite compounding period such as a month or a year. It is important to recall that the effective annual return increases concurrently with the compounding frequency under discrete compounding.<\/p>\n\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:20px 16px; margin:24px 0;\">\n  <div style=\"max-width:760px; margin:0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display:inline-flex; align-items:center; justify-content:center; width:100%; padding:12px 32px; border:2px solid #2f6fdd; border-radius:999px; color:#2f6fdd; text-decoration:none; font-size:15.5px; font-weight:500; line-height:1;\">\n      Practice continuously compounded returns with a Free Trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<p><strong style=\"color: revert; font-size: revert;\">Example: Discrete Compounding #1<\/strong><\/p>\n<p>For a stated rate of 20%, semiannual compounding gives an effective rate of:<\/p>\n<p>$$ \\left(1 + \\frac {0.20}{2} \\right)^2 \u2013 1 = 21\\% $$<\/p>\n<p>And monthly compounding gives an effective rate of:<\/p>\n<p>$$ \\left(1 + \\frac {0.20}{12} \\right)^{12} \u2013 1 = 21.94\\% $$<\/p>\n<p>Daily or hourly compounding will produce even larger effective rates.<\/p>\n<p>We can calculate the effective annual rate based on continuous compounding if we are given a stated annual rate of \\(R_{cc}\\). The formula used is:<\/p>\n<p>$$ \\text{Effective annual rate} = \\text e^{R_{cc}} \u2013 1 $$<\/p>\n<h4><strong>Example: Continuous Compounding #2<\/strong><\/h4>\n<p>Given a stated rate of 10%, the effective rate based on continuous compounding is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Applying the formula above,<\/p>\n<p>$$ \\text{Effective rate} = e^{0.10} \u2013 1 = 10.52\\% $$<\/p>\n<h3><strong>Continuous Compounding Given the Holding Period Return (HPR)<\/strong><\/h3>\n<p>We can calculate the continuous compound rate of return if we have the holding period return. The following is the formula used in the calculation:<\/p>\n<p>$$ \\text{Continuous rate} = \\text{ln}(1 + \\text{HPR}) = \\text{ln} \\left(\\cfrac {S_1}{S_0} \\right) $$<\/p>\n<p>Where:<\/p>\n<p>\\(S_1\\)<sub>\u00a0<\/sub>= Value at the end of the period.<\/p>\n<p>\\(S_0\\)<sub>\u00a0<\/sub>= Value at the beginning of the period.<\/p>\n<p><strong>Example: Continuous Compounding Given the Beginning and Ending Values<\/strong><\/p>\n<p>An investor purchases a stock at $1,000 and sells it for $1,080 after a period of one year. The annual rate of return on the stock on a continuously compounded basis is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\text{Continuously compounded rate} = \\text{ln} \\left( \\cfrac {1,080}{1,000} \\right) = 7.7\\% $$<\/p>\n<h4><strong>Example: Continuous Compounding Given a Holding Period Return (HPR)<\/strong><\/h4>\n<p>A stock has a holding period return of 20%. Its continuously compounded rate of return is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ R_{cc} = \\text{ln}(1 + 0.20) = 18.2\\% $$<\/p>\n<p><strong>Note to candidates<\/strong>: We can also calculate the holding period return given the continuously compounded rate, \\(R_{cc}\\). In general, to determine the HPR after \\(t\\) years:<\/p>\n<p>$$ \\text{HPR}_{\\text t} = e^{\\text{Rcc} *{\\text t}} \u2013 1 $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A portfolio manager buys a stock at $50 and sells it for $56 after a year. The continuously compounded rate of return is <em>closest<\/em> to:<\/p>\n<p>A. 10.0%.<\/p>\n<p>B. 11.3%.<\/p>\n<p>C. 12.0%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>$$ R_{cc} = \\text{ln} \\left(\\cfrac {56}{50} \\right) = 11.3\\% $$<\/p>\n<\/blockquote>\n<p>\u00a0<\/p>\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:44px 20px 24px; margin:40px 0;\">\n  \n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; background:#4274d8; color:#ffffff; text-decoration:none; padding:14px 36px; border-radius:999px; font-size:16px; font-weight:700; line-height:1;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:640px; margin:14px auto 0; font-size:15.5px; line-height:1.5; color:#1f2937;\">\n    Strengthen log return calculations and compounding concepts with\n    exam-focused CFA Level I practice.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Continuous compounding applies either when the frequency with at we calculate interest is infinitely large or the time interval is infinitely small. Put quite simply, under continuous compounding, time is viewed as continuous. This is a departure from discrete compounding,&#8230;<\/p>\n","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-29839","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Continuously Compounded Return | CFA Level I<\/title>\n<meta name=\"description\" content=\"Learn how continuously compounded returns are calculated from holding period returns, including the formula and interpretation in investment analysis.\" \/>\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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