{"id":45216,"date":"2023-06-27T16:12:43","date_gmt":"2023-06-27T16:12:43","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45216"},"modified":"2026-02-07T10:11:24","modified_gmt":"2026-02-07T10:11:24","slug":"money-weighted-and-time-weighted-rates-of-return","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/money-weighted-and-time-weighted-rates-of-return\/","title":{"rendered":"Money-weighted and Time-weighted Rates of Return"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Rates and Returns (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Module 1)\",\n  \"description\": \"This lesson introduces the foundations of Quantitative Methods for the 2025 CFA\u00ae Level I exam. It explains the required rate of return and discount rate, opportunity cost and the risk-free rate, and compares arithmetic versus geometric returns. The video also covers money-weighted and time-weighted returns, continuous compounding, and the calculation and interpretation of annualized returns, with exam-focused explanations throughout.\",\n  \"uploadDate\": \"2024-02-21T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/QIl6JH_PuW8\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/QIl6JH_PuW8\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/QIl6JH_PuW8\",\n  \"duration\": \"PT1H00M57S\"\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\": \"How do you calculate the annual time-weighted rate of return in this stock investment example?\",\n    \"text\": \"A chartered analyst buys a share of stock at time t = 0 for $50. At t = 1, he purchases an extra share for $53. The share pays dividends of $0.50 in the first year and $0.60 in the second year. He sells the shares at the end of the second year for $55 per share. What is the annual time-weighted rate of return?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The annual time-weighted rate of return is 5.9%. The investment is broken into two holding periods. The first holding period return is 7.0% and the second is 4.9%. These returns are compounded to obtain a total time-weighted return of 12.24%, which is then annualized over two years to give an annual return of approximately 5.9%.\",\n      \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/money-weighted-and-time-weighted-rates-of-return\/\",\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\/QIl6JH_PuW8\" \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<h2 class=\"wp-block-heading\">Money-weighted Rate of Return<\/h2>\n\n\n\n<p>The money-weighted return considers the money invested and gives the investor information on the actual investment return. Calculating money-weighted return is similar to calculating an investment\u2019s internal rate of return (IRR).<\/p>\n\n\n\n<p>The money-weighted rate of return (MWRR) is like the portfolio&#8217;s internal rate of return (IRR). It&#8217;s the rate at which the present value of cash flows equals zero. It&#8217;s a way to measure how well a portfolio performs.<\/p>\n\n\n\n<p>$$\\sum_{t=0}^{T}{\\frac{CF_t}{\\left(1+IRR\\right)^t}=0}$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>\\(T\\) = Number of periods.<\/p>\n\n\n\n<p>\\(CF_t\\) = Cash flow at time <em>t<\/em>.<\/p>\n\n\n\n<p>\\(IRR\\) = Internal rate of return (or money-weighted rate of return).<\/p>\n\n\n\n<p>The money-weighted rate of return (MWRR) looks at a fund\u2019s starting and ending values and all the cash flows in between. In an investment portfolio, cash inflows are a part of it. These inflows could be from deposits or investments made during a certain period. The MWRR considers these inflows and calculates the overall rate of return for the portfolio:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The beginning value.<\/li>\n\n\n\n<li>Dividends\/interest reinvested.<\/li>\n\n\n\n<li>Contributions made.<\/li>\n<\/ul>\n\n\n\n<p>Cash outflows, on the other hand, refer to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Withdrawals made.<\/li>\n\n\n\n<li>Dividends or interest received.<\/li>\n\n\n\n<li>The final value of the fund.<\/li>\n<\/ul>\n\n\n\n<a href=\"https:\/\/analystprep.com\/free-trial\/\"\n   target=\"_blank\"\n   rel=\"noopener noreferrer\"\n   style=\"display:block;margin:20px 0 28px;padding:14px 18px;border:2px solid #2563eb;border-radius:12px;text-align:center;color:#2563eb;text-decoration:none;font-weight:500;font-size:15px;background-color:#ffffff;\">\n   Get confident with MWRR and TWRR in a free trial.\n<\/a>\n\n\n\n<p>\n\n\n\n<\/p>\n<h4><strong>Example 1: Calculating Money-weighted Rate of Return<\/strong><\/h4>\n<p>An investor makes the following investments in a portfolio over a two-year period:<\/p>\n<ul>\n<li>At the beginning of year one, the investor invests $10,000.<\/li>\n<li>At the end of the first year, after the portfolio&#8217;s value increases to $12,000, the investor adds $5,000, making the total portfolio value $17,000.<\/li>\n<li>At the end of the second year, the portfolio value further increases to $25,000.<\/li>\n<\/ul>\n<p>The money-weighted rate of return for the investor\u2019s portfolio is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We need to calculate the internal rate of return (IRR) considering the following cash flows:<\/p>\n<ul>\n<li>\\( CF_0 = -\\$10,000 \\) (Initial investment).<\/li>\n<li>\\( CF_1 = -\\$5,000 \\) (Additional investment at the end of year one).<\/li>\n<li>\\( CF_2 = +\\$25,000 \\) (Final portfolio value at the end of year two).<\/li>\n<\/ul>\n<p>To find the money-weighted rate of return, solve the equation for IRR:<\/p>\n<p>$$ \\frac{CF_0}{(1+IRR)^0} + \\frac{CF_1}{(1+IRR)^1} + \\frac{CF_2}{(1+IRR)^2} = \\frac{-10,000}{1} + \\frac{-5,000}{(1+IRR)} + \\frac{25,000}{(1+IRR)^2} = 0 $$<\/p>\n<p>Using BA II Plus Calculator, \\(IRR\\approx 35.08\\%\\).<\/p>\n<h4><strong>Example: Calculating Money-weighted Return for a Dividend-paying Stock<\/strong><\/h4>\n<p>Calvin Hair purchased a share of Superior Car Rental Company for $85 at the beginning of the first year. He bought an additional unit for $87 at the end of the first year. At the end of the second year, he sold both shares at $90. During both years, Hair received a dividend of $4 per share, which was not reinvested.<\/p>\n<p>Calculate the money-weighted return.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>To calculate the money-weighted return in this example, we need to consider the timing and amounts of cash flows and their respective investment periods.<\/p>\n<p><strong>Step 1: <\/strong>Calculate the total investment at the beginning (<em>t<\/em>=0):<\/p>\n<p>$$ \\text{Initial investment}= -$85 $$<\/p>\n<p><strong>Step 2: <\/strong>Calculate the total investment at <em>t <\/em>= 1:<\/p>\n<p>$$\\begin{align} \\text{Initial investment + Additional investment} &amp;= $87 &#8211; $4 \\text{( Dividend received at}\\\\&amp;\\text{ the end of the first year, which is not reinvested)}\\\\&amp; = -$83 \\end{align}$$<\/p>\n<p><strong>Step 3: <\/strong>Calculate the final portfolio value at <em>t <\/em>= 2:<\/p>\n<p>$$ \\begin{align}\\text{Number of shares sold \u00d7 Selling price}&amp; = 2 \\text{ shares} \u00d7 $90 = $180 + 8 \\text{( Dividend received for}\\\\ &amp;\\text{both shares)}\\\\&amp; =$188\\end{align} $$<\/p>\n<p>As such, we have:<\/p>\n<p>\\(CF_0=-85\\).<\/p>\n<p>\\(CF_1=-83\\).<\/p>\n<p>\\(CF_2=188\\).<\/p>\n<p>Using the BA II Plus calculator, you will get \\(IRR=7.71\\%\\), equivalent to the money-weighted rate of return.<\/p>\n<h3>Shortcomings of the Money-weighted Rate of Return<\/h3>\n<p>The money-weighted rate of return (MWRR) considers all cash flows, such as withdrawals or contributions. If an investment spans multiple periods, MWRR gives more importance to the fund&#8217;s performance when the account is at its largest. This can be a problem for fund managers because it might make their performance seem worse due to factors they can&#8217;t control.<\/p>\n<h2>Time-weighted Rate of Return<\/h2>\n<p>The time-weighted rate of return (TWRR) calculates an investment&#8217;s compound growth. Unlike the money-weighted rate, it doesn&#8217;t care about withdrawals or contributions. TWRR is like finding the average return of different time periods within your investment.<\/p>\n<p><strong>Steps of Calculating Time-weighted Rate of Return<\/strong><\/p>\n<p><strong>Step 1<\/strong>: Value the portfolio immediately before any significant cash inflow or outflow of funds. Divide the evaluation period into subperiods based on dates of significant additions or withdrawals of funds.<\/p>\n<p><strong>Step 2<\/strong>: Compute the holding period return on the portfolio for each period.<\/p>\n<p><strong>Step 3<\/strong>: Compound or link the holding period returns to the annual rate of return, which is the time-weighted rate of return.<\/p>\n<p>$$\\text{TWRR} =\\left[ (1 + HPR_1) \\times (1 + HPR_2)\u00a0 \\times \\ldots \\times (1 + HPR_{n-1}) \\times (1 + HPR_n) \\right] &#8211; 1$$<\/p>\n<p>If the evaluation period is more than one year, compute the geometric mean of the annual returns to get the time-weighted return for the investment period.<\/p>\n<p>$$\\begin{align}{\\bar{R}}_{Gi}&amp;=\\sqrt[T]{\\left(1+{\\rm HPR}_1\\ \\right)\\times\\left(1+{\\rm HPR}_1\\ \\right)\\ldots\\times\\left(1+{\\rm HPR}_n\\right)}-1\\\\&amp;=\\left[\\left(1+{\\rm HPR}_1\\ \\right)\\times\\left(1+{\\rm HPR}_1\\ \\right)\\ldots\\times\\left(1+{\\rm HPR}_n\\right)\\right]^\\frac{1}{n}-1\\end{align}$$<\/p>\n<h4><strong>Example: Calculating the Time-Weighted Rate of Return (Period More than one year)<\/strong><\/h4>\n<p>An investor purchases a share of stock at t = 0 for $200. At the end of the year (at t = 1), the investor purchases an additional share of the same stock, this time for $220. She then sells both shares at the end of the second year for $230 each. She also received annual dividends of $3 per share at the end of each year. Calculate the annual time-weighted rate of return on her investment.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we break down the two years into two one-year periods.<\/p>\n<p><strong>Holding period 1<\/strong>:<\/p>\n<p>Beginning value = 200.<\/p>\n<p>Dividends paid = 3.<\/p>\n<p>Ending value = 220.<\/p>\n<p><strong>Holding period 2<\/strong>:<\/p>\n<p>Beginning value = 440 (2 shares \u00d7 220)<\/p>\n<p>Dividends paid = 6 (2 shares \u00d7 3)<\/p>\n<p>Ending value = 460 (2 shares \u00d7 230)<\/p>\n<p>Secondly, we calculate the HPR for each period:<\/p>\n<p>$$\\begin{align}{HPR}_1&amp;=\\frac{(220-200+3)}{200}=11.5\\%\\\\{HPR}_2&amp;=\\frac{\\left(460-440+6\\right)}{440}=5.9\\% \\end{align}$$<\/p>\n<p>Lastly, we need to find the geometric mean of the HPRs since we are dealing with a period of more than a year.<\/p>\n<p>$$\\begin{align}TWRR&amp;=\\left[\\left(1+{\\rm HPR}_1\\ \\right)\\times\\left(1+{\\rm HPR}_1\\ \\right)\\ldots\\times\\left(1+{\\rm HPR}_n\\right)\\right]^\\frac{1}{n}-1\\\\&amp;=\\left(1.115\\times1.059\\right)^{0.5}-1=8.7\\%\\end{align}$$<\/p>\n<h3><strong>Example: Calculating the Time-weighted Rate of Return (Period Less than One Year)<\/strong><\/h3>\n<p>The beginning value of a portfolio as of January 1, 2020, was $1,000,000. On February 10, the portfolio\u2019s value was $1,100,000, including an additional contribution of the $50,000 injected into the portfolio on this date. The portfolio\u2019s ending value at the beginning of April was $1,350,000.<\/p>\n<p>The time-weighted rate of return is <em>closest to:<\/em><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The time-weighted return is calculated as follows:<\/p>\n<p>$$\\begin{align}{HPR}_1&amp;=\\frac{V_1-V_0}{V_0}=\\frac{\\left(1,100,000-50,000\\right)-1,000,000}{1,000,000}=5\\%\\\\{HPR}_2&amp;=\\frac{V_2-V_1}{V_1}=\\frac{1,350,000-1,100,000}{1,100,000}=22.73\\%\\\\\\Rightarrow TWRR &amp;=\\left(1+{HPR}_1\\right)\\times\\left(1+{HPR}_1\\right)-1\\\\&amp;=1.05\\times1.2273-1=28.87\\%\\end{align}$$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>A chartered analyst buys a share of stock at time t = 0 for $50. At t = 1, he purchases an extra share of the same stock for $53. The share gives a dividend of $0.50 per share for the first year and $0.60 per share for the second year. He sells the shares at the end of the second year for $55 per share. Calculate the annual time-weighted rate of return.<\/p>\n<p>A. 5.90%.<\/p>\n<p>B.12.24%.<\/p>\n<p>C. 7.00%.<\/p>\n<p><strong>The correct answer is\u00a0A<\/strong>.<\/p>\n<p>We have two one-year holding periods:<\/p>\n<p>$$\\begin{array}{cc}HP_1&amp;HP_2\\\\P_0=50&amp;P_0=106\\\\D=0.5&amp;D=1.2\\\\P_1=53&amp;P_1=110\\\\ \\end{array}$$<\/p>\n<p>We now calculate the holding period returns:<\/p>\n<p>$$\\begin{align}{HPR}_1&amp;=\\frac{(53-50+0.5)}{50}=7\\%\\\\{HPR}_2&amp;=\\frac{\\left(110-106+1.2\\right)}{106}=4.9\\%\\\\ \\Rightarrow TWRR &amp;=1.07\\times1.049-1=12.24\\%\\end{align}$$<\/p>\n<p>Therefore,<\/p>\n<p>$$\\text{Annual TWRR}={(1+0.1224)}^{0.5}-1=5.9\\%$$<\/p>\n<\/blockquote>\n\n\n<div style=\"margin: 40px 0; padding: 30px; text-align: center; background-color: #f5f8fc; border-radius: 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Start Free Trial\"\n     data-cta-type=\"button\"\n     data-cta-location=\"bottom_content\"\n     data-page-type=\"study_note\"\n     style=\"\n       display: inline-block;\n       padding: 14px 26px;\n       font-size: 18px;\n       font-weight: 700;\n       color: #ffffff;\n       background-color: #0b5ed7;\n       border-radius: 8px;\n       text-decoration: none;\n     \">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 15px; color: #333;\">\n    Practice return\u2011calculation questions (TWRR\/MWRR) with full solutions and CFA\u2011style drills.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Money-weighted Rate of Return The money-weighted return considers the money invested and gives the investor information on the actual investment return. Calculating money-weighted return is similar to calculating an investment\u2019s internal rate of return (IRR). The money-weighted rate of return&#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-45216","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>Money-Weighted vs Time-Weighted Returns | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand the difference between money-weighted and time-weighted rates of return, including their formulas, calculations, and applications in finance.\" \/>\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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