{"id":45221,"date":"2023-06-27T16:49:41","date_gmt":"2023-06-27T16:49:41","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45221"},"modified":"2026-08-19T14:41:10","modified_gmt":"2026-08-19T14:41:10","slug":"annualized-returns","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/annualized-returns\/","title":{"rendered":"Annualized Returns"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"The weekly return of an investment that produces an annual compounded return of 23% is closest to:\",\n    \"text\": \"Options:\\nA. 0.40%.\\nB. 0.92%.\\nC. 0.41%.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. The annual compounded return is related to the weekly return by: Returnannual = (1 + Returnweekly)^52 \u2212 1. Rearranging gives Returnweekly = (1 + Returnannual)^(1\/52) \u2212 1. Substituting the annual return of 23% gives (1.23)^(1\/52) \u2212 1, which is approximately 0.40%. Therefore, the weekly return is closest to 0.40%.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p class=\"wp-block-paragraph\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/QIl6JH_PuW8?si=kwV2e4llhA5xaE_3\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe>  <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To compare returns over different timeframes, we need to annualize them. This means converting daily, weekly, monthly, or quarterly returns into annual figures.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Non-Annual Compounding<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Interest may be paid semiannually, quarterly, monthly, or even daily \u2013 interest payments can be made more than once a year. Consequently, the present value formula can be expressed as follows when there are multiple compounding periods in a year:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$PV=\\ {{\\rm FV}_N\\left(1+\\frac{R_S}{m}\\right)}^{-mN}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(m\\) = Number of compounding periods in a year.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(R_s\\) = Quoted annual interest rate.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\(N\\) = Number of years.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> <\/p>\n\n\n\n<div style=\"margin: 18px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #1a73e8; border-radius: 18px; color: #ffffff; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8;\">\n    Master annualized returns with our Free Trial.\n  <\/a>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n<!-- \/wp:post-content -->\n\n<!-- wp:heading {\"level\":4} -->\n<h4 class=\"wp-block-heading\"><strong>Example: Calculating the Present Value of a Lump Sum (More than One Compounding Period)<\/strong><\/h4>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>Jane Doe wants to invest money today and have it become $500,000 in five years. The annual interest rate is 8%, and it&#8217;s compounded quarterly. How much should Jane invest right now?<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Using the formula above:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\({FV}_N = $500,000\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(R_S = 8\\%\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(m = 4\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(R_s\/m = \\frac{8\\%}{4} = 2\\% = 0.02\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(N = 5\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(mN = 4\\times 5=20\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Therefore,<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$ PV=\\ {{FV}_N\\left(1+\\frac{R_S}{m}\\right)}^{-mN}=\\$500,000\\ \\times\\left(1.02\\right)^{-20}=\\$336,485.67$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong><u>Using BA II <em>Plus <\/em>Calculator<\/u><\/strong><u>:<\/u><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:list -->\n<ul class=\"wp-block-list\"><!-- wp:list-item -->\n<li>Press the [2nd] button, then the [FV] button to clear the financial registers. The display should show \u201cCLR TVM.\u201d<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Enter the future value (FV). This is the amount Jane wants to have in five years, which is $500,000. To do this, type \u201c500000\u201d and press the [FV] button.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Enter the interest rate (I\/Y). This is the annual interest rate, which is 8%. However, since interest is compounded quarterly, we need to divide this by 4. To do this, type \u201c8\u201d, press the [\u00f7] button, type \u201c4\u201d, then press the [ENTER] button, and finally press the [I\/Y] button.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Enter the number of periods (N). This is the number of quarters in five years, which is 5*4 = 20. To do this, type \u201c20\u201d and press the [N] button.<\/li>\n<!-- \/wp:list-item -->\n\n<!-- wp:list-item -->\n<li>Compute the present value (PV). To do this, press the [CPT] and then the [PV] buttons. The display should show the amount Jane needs to invest today, approximately $336,485.49.<\/li>\n<!-- \/wp:list-item --><\/ul>\n<!-- \/wp:list -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Annualized Returns<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>To annualize a return for a period shorter than a year, you need to account for how many times that period fits into a year. For example, if you have a weekly return, you would compound it 52 times because there are 52 weeks in a year.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Generally, we can annualize the returns using the following formula:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$${\\text{Return}}_{\\text{annual}}=\\left(1+{\\text{Return}}_{\\text{period}}\\right)^c-1$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Where:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\({\\text{Return}}_{\\text{period}}\\) = Quoted return for the period.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\(c\\) = Number of periods in a year.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":4} -->\n<h4 class=\"wp-block-heading\"><strong>Example<\/strong>: <strong>Annualizing Returns<\/strong><\/h4>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>If the monthly return is 0.7%, then the compound annual return is:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$\\begin{align}{\\text{Return}}_{\\text{annual}}&amp;=\\left(1+{\\text{Return}}_{\\text{monthly}}\\right)^{12}-1\\\\&amp;=\\left(1.007\\right)^{12}-1=0.0873=8.73\\%\\end{align}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>For a period of more than one year, for example, a 15-month return of 16% can be annualized as:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$\\begin{align}{\\text{Return}}_{\\text{annual}}&amp;=\\left(1+{\\text{Return}}_{15\\ \\text{month}}\\right)^\\frac{12}{15}-1\\\\&amp;=\\left(1.16\\right)^\\frac{4}{5}-1=12.61\\%\\end{align}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>We may apply the same procedure to convert weekly returns to annual returns for comparison with weekly returns.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$${\\text{Return}}_{\\text{annual}}=\\left(1+{\\text{Return}}_{\\text{weekly}}\\right)^{52}-1$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>For comparison with weekly returns, we can convert annual returns to weekly returns by making \\({(\\text{Return}}_{\\text{weekly}})^{52}\\) the subject of the formula.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading {\"level\":4} -->\n<h4 class=\"wp-block-heading\"><strong>Example: Comparing Investments by Annualizing Returns<\/strong><\/h4>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>An investor is evaluating the returns of two recently formed bonds. Selected return information on the bonds is presented below:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$\\begin{array}{c|c|c}\\text{Bond}&amp;\\text{Time Since Issuance}&amp;\\text{Return Since Issuance (%)}\\\\ \\hline \\text{A}&amp;\\text{120 days}&amp;2.50\\\\ \\hline \\text{B}&amp;\\text{8 months}&amp;6.00\\\\ \\end{array}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Annualized Return Calculation<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>To compare the annualized rate of return for both bonds, you can use the formula for annualizing returns based on different time periods:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong>Annualized Return<\/strong> = \\(\\left(1 + \\frac{\\text{Return Since Issuance}}{100}\\right)^\\frac{365}{\\text{Time Since Issuance}} &#8211; 1\\)<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Let&#8217;s calculate the annualized returns for both bonds:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong>For Bond A:<\/strong><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Time Since Issuance = 120 days.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Return Since Issuance = 2.50%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond A = \\(\\left(1 + \\frac{2.50}{100}\\right)^\\frac{365}{120} &#8211; 1\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond A = \\(\\left(1 + 0.025\\right)^{3.0417} &#8211; 1\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond A = 1.07799 &#8211; 1 = 0.07799 or 7.80%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong>For Bond B:<\/strong><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Time Since Issuance = 8 months .<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Return Since Issuance = 6.00%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond B = \\(\\left(1 + \\frac{6.00}{100}\\right)^\\frac{12}{8} &#8211; 1\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond B = \\(\\left(1 + 0.06\\right)^{1.5} &#8211; 1\\).<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Annualized Return for Bond B = 1.09133 &#8211; 1 = 0.09133 or 9.13%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Comparing the annualized returns:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Bond A has an annualized return of approximately 7.80%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Bond B has an annualized return of approximately 9.13%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Therefore, Bond B has a higher annualized rate of return compared to Bond A.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:heading -->\n<h2 class=\"wp-block-heading\">Continuously Compounded Returns<\/h2>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>The continuously compounded return is calculated by taking the natural logarithm of one plus the holding period return. For example, if the monthly return is 1.2%, you&#8217;d calculate it as ln(1.012), which equals approximately 0.01192.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Generally, continuously compounded from \\(t\\) to \\(t+1\\) is given by:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$r_{t,t+1}=\\ln{\\left(\\frac{P_{t+1}}{P_t}\\right)=\\ln{\\left(1+R_{t,t+1}\\right)}}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Assume now that the investment horizon is from time \\(t=0\\) to time \\(t=T\\) then the continuously compounded return is given by:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$r_{0,T}=\\ln{\\left(\\frac{P_T}{P_0}\\right)}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>If we apply the exponential function on both sides of the equation, we have the following:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$P_T=P_0e^{r_{0,T}}$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Note that \\(\\frac{P_T}{P_0}\\) can be written as:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$$\\frac{P_T}{P_0}=\\left(\\frac{P_T}{P_{T-1}}\\right)\\left(\\frac{P_{T-1}}{P_{T-2}}\\right)\\ldots\\left(\\frac{P_1}{P_0}\\right)$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>If we take natural logarithm on both sides of the above equation:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\begin{align*} \\ln{\\left(\\frac{P_T}{P_0}\\right)} &amp;= \\ln{\\left(\\frac{P_T}{P_{T-1}}\\right)} + \\ln{\\left(\\frac{P_{T-1}}{P_{T-2}}\\right)} + \\ldots + \\ln{\\left(\\frac{P_1}{P_0}\\right)}\\\\\\Rightarrow r_{0,T} &amp;= r_{T-1,T} + r_{T-2,T-1} + \\ldots + r_{0,1} \\end{align*}<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Therefore, the continuously compounded return to time T is equivalent to the sum of one-period continuously compounded returns.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:quote -->\n<blockquote class=\"wp-block-quote\"><!-- wp:heading {\"level\":3} -->\n<h3 class=\"wp-block-heading\"><strong>Question<\/strong><\/h3>\n<!-- \/wp:heading -->\n\n<!-- wp:paragraph -->\n<p>The weekly return of an investment that produces an annual compounded return of 23% is <em>closest to<\/em>:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>A. 0.40%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>B. 0.92%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>C. 0.41%.<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><strong>The correct answer is A.<\/strong><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>Recall that:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>$${\\text{Return}}_{\\text{annual}}=\\left(1+{\\text{Return}}_{\\text{weekly}}\\right)^{52}-1$$<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>We can rewrite the above equation as follows:<\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p>\\begin{align}<br>\n\\text{Return}_{\\text{weekly}} &amp;= \\left(1 + \\text{Return}_{\\text{annual}}\\right)^{\\frac{1}{52}} &#8211; 1 \\\\<br>\n&amp;= \\left(1 + 0.23\\right)^{\\frac{1}{52}} &#8211; 1 \\\\<br>\n&amp;\\approx 0.40\\%<br>\n\\end{align}<\/p>\n<!-- \/wp:paragraph --><\/blockquote>\n<!-- \/wp:quote -->\n\n<!-- wp:paragraph -->\n<p><!-- wp:html --><\/p>\n<!-- \/wp:html -->\n\n<!-- wp:html -->\n<div style=\"text-align: center; margin: 30px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">\n    Strengthen your understanding of annualized returns, holding period returns, compounding, and investment performance calculations with our CFA Level I study materials, practice questions, and mock exams.\n  <\/p>\n<\/div>\n<!-- \/wp:html -->\n\n<!-- wp:paragraph -->\n<p><!-- \/wp:paragraph --><\/p>\n<!-- \/wp:paragraph -->\n\n<!-- wp:paragraph -->\n<p><\/p>\n<!-- \/wp:paragraph -->","protected":false},"excerpt":{"rendered":"<p>To compare returns over different timeframes, we need to annualize them. This means converting daily, weekly, monthly, or quarterly returns into annual figures. Non-Annual Compounding Interest may be paid semiannually, quarterly, monthly, or even daily \u2013 interest payments can be&#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-45221","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 v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Annualized Returns - AnalystPrep | CFA\u00ae Exam Study Notes<\/title>\n<meta name=\"description\" content=\"Learn to annualize returns, comparing investments over different timeframes. 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