{"id":136,"date":"2019-10-01T13:28:00","date_gmt":"2019-10-01T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=136"},"modified":"2026-02-16T15:56:51","modified_gmt":"2026-02-16T15:56:51","slug":"using-money-weighted-rate-return-evaluate-performance-investment","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/portfolio-management\/using-money-weighted-rate-return-evaluate-performance-investment\/","title":{"rendered":"Money-weighted vs. Time-weighted Rates of Return"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Portfolio Risk and Return: Part I (2021 Level I CFA\u00ae Exam \u2013 Reading 49)\",\n  \"description\": \"CFA\u00ae Level I Portfolio Management video lesson by AnalystPrep covering Portfolio Risk and Return (Part I). This session explains major return measures, money-weighted versus time-weighted returns, characteristics of major asset classes, mean, variance, covariance and correlation of returns, risk aversion, portfolio standard deviation, diversification benefits, minimum-variance and efficient frontiers, the global minimum-variance portfolio, and optimal portfolio selection using the capital allocation line.\",\n  \"uploadDate\": \"2020-01-02T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/95klvFTOVJw\/default.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=95klvFTOVJw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/95klvFTOVJw\",\n  \"duration\": \"PT55M27S\"\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 is the money-weighted rate of return calculated for a stock investment with dividends?\",\n    \"text\": \"A stock was valued at $20 on January 1, 2015 and $22 on December 31, 2015, when the holder sold his stake. During the year, a dividend of $0.40 per share was paid out to shareholders. Determine the money-weighted rate of return.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The money-weighted rate of return is 12%. This is calculated by equating the present value of the dividend and ending price to the initial investment: 0.40\/(1+r) + 22\/(1+r) = 20, which solves to r = 12%.\",\n      \"dateCreated\": \"2026-01-14\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"How is the annual time-weighted rate of return calculated when additional investments are made?\",\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 stock pays dividends of $0.50 in the first year and $0.60 in the second year. The shares are sold at the end of year two for $55 per share. Calculate the annual time-weighted rate of return.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The annual time-weighted rate of return is approximately 5.9%. This is computed by calculating the holding period returns for each year and then geometrically linking them to obtain the annualized time-weighted return.\",\n      \"dateCreated\": \"2026-01-14\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/95klvFTOVJw?autoplay=0&amp;loop=0&amp;rel=0\" width=\"611\" height=\"344\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Money-weighted Rate of Return<\/h2>\n<p>The money-weighted rate of return (MWRR) refers to a portfolio&#8217;s internal rate of return. It is the rate of discount, r, at which:<\/p>\n<p>$$ \\text{PV of cash outflows} = \\text{PV of cash inflows} $$<\/p>\n<p>The money-weighted rate of return on a fund satisfies the value equation by taking the initial and final fund values, as well as the intermediate cash flows, into account. When dealing with an investment portfolio, cash inflows comprise:<\/p>\n<ul>\n<li>The beginning value.<\/li>\n<li>Dividends \/interest reinvested.<\/li>\n<li>Withdrawals made.<\/li>\n<\/ul>\n<p>Cash outflows, on the other hand, refer to:<\/p>\n<ul>\n<li>The final value of the fund.<\/li>\n<li>Dividends or interest received.<\/li>\n<li>Contributions.<\/li>\n<\/ul>\n<p><a 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;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Reinforce this concept with CFA-style practice questions. <\/a><\/p>\n<h4><strong>Example: Money-weighted Rate of Return<\/strong><\/h4>\n<p>Assume you bought a stock at $100 and sold it a year later at $110. Further, assume that the stock pays an annual dividend of $1 per year. The money-weighted rate of return is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>In this case, the dividends received are outflows, and so is the final value of the stock. The cost of the stock is the only inflow. Therefore,<\/p>\n<p>If we let our MWRR = r,<\/p>\n<p>$$ \\text{PV of outflow}=\\text {PV of inflow} $$<\/p>\n<p>$$ 1(1 + r)^{-1} + 110(1 + r)^{-1} = 100 $$<\/p>\n<p>Now, if we let (1 + r) to be \u2018x\u2019, then:<\/p>\n<p>$$ \\begin{align*}<br \/>&amp; \\frac {1}{x} + \\frac {110}{x} = 100 \\\\<br \/>&amp; \\frac {111}{x} = 100 \\\\<br \/>\\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ x = 1.11 $$<\/p>\n<p>But x = 1 + r<\/p>\n<p>$$ \\begin{align*}<br \/>1 + r &amp; = 1.11 \\\\<br \/>r &amp; = 0.11 \\text{ or } 11\\% \\\\<br \/>\\end{align*} $$<\/p>\n<p><em>Exam tip: The exam usually tests the candidate\u2019s understanding of the concept of money-weighted rate of return. Any calculations are unlikely to require the use of a calculator.<\/em><\/p>\n<h3><strong>Shortcomings of the Money-weighted Rate of Return<\/strong><\/h3>\n<p>As we stated earlier, the money-weighted rate of return considers all the cash flows, including any withdrawal from the fund or contribution. Assuming an investment extends to several periods, the MWRR puts more weight on the fund&#8217;s performance when the account size is the biggest. This is a disadvantage to fund managers since they may be unfairly penalized due to cash flows that are beyond their control.<\/p>\n<h2><strong>Time-weighted Rate of Return<\/strong><\/h2>\n<p>The time-weighted rate of return (TWRR) measures the compound growth rate of an investment portfolio. Unlike the money-weighted rate of return, TWRR is not sensitive to withdrawals or contributions. Essentially, the time-weighted rate of return is the geometric mean of the holding period returns of the respective sub-periods involved.<\/p>\n<p>When working out time-weighted measurements, we break down the total investment period into many sub-periods. Each sub-period ends at the point where we have a significant withdrawal or contribution. It could also end after a month, quarterly, or even semi-annually. We encourage candidates to follow the steps below when computing TWRR:<\/p>\n<ol>\n<li>Establish the holding period return (HPR) for each sub-period.<\/li>\n<li>Add 1 to each HPR.<\/li>\n<li>Multiply all the (1 + HPR) terms.<\/li>\n<li>Subtract 1 from the final product to get the compounded TWRR.<\/li>\n<\/ol>\n<p>In a summary, compounded TWRR = {(1 + HPR<sub>1<\/sub>)*(1 + HPR<sub>2<\/sub>)*(1 + HPR<sub>3<\/sub>)\u2026*(1 + HPR<sub>n-1<\/sub>)*(1 + HPR<sub>n<\/sub>)} \u2013 1<\/p>\n<p>Finally, annual time-weighted rate of return = (1 + compounded TWRR)<sup> 1\/n<\/sup> \u2013 1<\/p>\n<p>Where n is the number of years.<\/p>\n<h4><strong>Example: <\/strong><strong>Time-weighted Rate of Return<\/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 2-year period into two 1-year periods.<\/p>\n<p><em>Holding period 1:<\/em><\/p>\n<p>Beginning value\u00a0\u00a0= 200.<\/p>\n<p>Dividends paid = 3.<\/p>\n<p>Ending value = 220.<\/p>\n<p><em>Holding period 2:<\/em><\/p>\n<p>beginning value = 440 (2 shares * 220).<\/p>\n<p>dividends paid = 6 (2 shares * 3).<\/p>\n<p>ending value\u00a0\u00a0= 460 (2 shares * 230).<\/p>\n<p><em>Secondly, we calculate the HPR for each period:<\/em><\/p>\n<p>$$ \\text{HPR}_1 =\\cfrac {(220 \u2013 200 + 3)}{200} = 11.5\\% $$<\/p>\n<p>$$ \\text{HPR}_2 =\\cfrac {(460 \u2013 440 + 6)}{440} = 5.9\\% $$<\/p>\n<p>Lastly,<\/p>\n<p>$$ (1 + \\text{annual TWRR})^2 = 1.115 * 1.059 $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text {annual TWRR} = (1.115 * 1.059)^{0.5} \u2013 1 = 8.7\\% $$<\/p>\n<h2><strong>Money-weighted Rate of Return vs. Time-weighted Rates of Return<\/strong><\/h2>\n<p>The money-weighted rate of return is sensitive to the amount and timing of cash flows and could lead to an unfair rating of the fund manager \u2013 they have no control over the amount or timing of cash flows. The time-weighted rate of return eliminates this effect. The money-weighted rate of return would only be superior to the TWRR if the fund manager had complete control over cash flows and their timings.<\/p>\n<blockquote>\n<h2><strong>Question 1<\/strong><\/h2>\n<p>A stock was valued at $20 on January 1, 2015 and $22 on December 31, 2015, when the holder sold his stake. During the year, a dividend of $0.4 per share was paid out to shareholders. Determine the money-weighted rate of return.<\/p>\n<p>A. 1.12.<\/p>\n<p>B. 12%.<\/p>\n<p>C. 200%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>$$ \\text{PV of outgo} =\\text {PV of income} $$<\/p>\n<p>$$ 0.4(1 + r)^{-1} + 22(1 + r)^{-1} = 20 $$<\/p>\n<p>If we let (1 + r) to be \u2018x\u2019,<\/p>\n<p>$$ \\begin{align*}<br \/>\\frac {0.4}{x} + \\frac {22}{x} &amp; = 20 \\\\<br \/>\\frac {22.4}{x} &amp; = 20 \\\\<br \/>x &amp; =\\frac {22.4}{20} = 1.12 \\\\<br \/>r &amp; = 1.12 \u2013 1 = 0.12 \\text{ or } 12\\%<br \/>\\end{align*} $$<\/p>\n<h2><strong>Question 2<\/strong><\/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 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. 5.9%.<\/p>\n<p>B. 12.24%.<\/p>\n<p>C. 7%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>We have two 1-year holding periods:<\/p>\n<p><em>HP<sub>1<\/sub>:<\/em><\/p>\n<p>P<sub>0<\/sub> = 50.<\/p>\n<p>D= 0.5.<\/p>\n<p>P<sub>1<\/sub> = 53.<\/p>\n<p><em>HP<sub>2<\/sub>:<\/em><\/p>\n<p>P<sub>0<\/sub>= 106.<\/p>\n<p>D = 1.2.<\/p>\n<p>P<sub>1<\/sub> = 110.<\/p>\n<p><em>We now calculate the holding period returns:<\/em><\/p>\n<p>$$ \\begin{align*}<br \/>\\text{HPR}_1 &amp; =\\cfrac {(53 \u2013 50 + 0.5)}{50} = 7\\% \\\\<br \/>\\text{HPR}_2 &amp; =\\cfrac {(110 \u2013 106 + 1.2)}{106} = 4.9\\% \\\\<br \/>\\text{Compounded TWRR} &amp; = 1.07 * 1.049 = 12.24\\%<br \/>\\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\text {Annual TWRR} = (1 + 0.1224)^{0.5} \u2013 1 = 5.9\\% $$<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Money-weighted Rate of Return The money-weighted rate of return (MWRR) refers to a portfolio&#8217;s internal rate of return. It is the rate of discount, r, at which: $$ \\text{PV of cash outflows} = \\text{PV of cash inflows} $$ The money-weighted&#8230;<\/p>\n","protected":false},"author":2,"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-136","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>Money-Weighted Rate of Return | CFA Level 1<\/title>\n<meta name=\"description\" content=\"The money-weighted rate of return measures portfolio performance by calculating the internal rate of return (IRR) based on cash inflows and outflows.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, 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