{"id":46405,"date":"2023-09-08T16:55:56","date_gmt":"2023-09-08T16:55:56","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46405"},"modified":"2026-03-25T16:35:39","modified_gmt":"2026-03-25T16:35:39","slug":"macaulay-duration","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/macaulay-duration\/","title":{"rendered":"Macaulay Duration"},"content":{"rendered":"\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 annualized Macaulay duration of a bond with semiannual coupons?\",\n    \"text\": \"Consider a bond that has three years remaining to maturity, a coupon of 3.5% paid semiannually, and a yield-to-maturity of 3.80%. Assuming it is 9 days into the first coupon period and using a 30\/360 basis, what is the bond\u2019s annualized Macaulay duration closest to?\",\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 annualized Macaulay duration is approximately 2.84 years. The bond pays semiannual coupons of 1.75%, and the yield per period is 1.90%. By discounting each cash flow, calculating time-weighted present values, summing them to obtain a semiannual Macaulay duration of about 5.6966 periods, and dividing by two to annualize, the duration is 2.84 years.\",\n      \"dateCreated\": \"2025-07-01T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/macaulay-duration\/\",\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\/NMlnM5vQGh8\" \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<h3 class=\"wp-block-heading\"><strong>Definition<\/strong><\/h3>\n\n\n\n<p>Macaulay duration was introduced in the previous learning objective. It provides an understanding of the bond&#8217;s sensitivity to interest rate fluctuations. At its core, Macaulay duration is the weighted average time until a bond&#8217;s cash flows are received. It signifies the holding period for a bond that balances both reinvestment and price risk.<\/p>\n\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:26px 16px; margin:24px 0;\">\n  <div style=\"max-width:720px; margin:0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display:block; width:100%; padding:14px 20px; border:2px solid #2f6fdd; border-radius:999px; color:#2f6fdd; text-decoration:none; font-size:16px; font-weight:500; line-height:1.2; text-align:center;\">\n      Practice Macaulay duration questions with a Free Trial for CFA Level I.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<\/p>\n<h3><strong>Calculation<\/strong><\/h3>\n<p>The calculation for Macaulay Duration is derived from the bond&#8217;s cash flows. Each cash flow is weighted by its share of the bond&#8217;s full price, which is its present value. The following steps outline the calculation:<\/p>\n<ol type=\"i\">\n<li><strong>Time to Receipt of Cash Flow<\/strong>: Determine the time until each cash flow is received.<\/li>\n<li><strong>Cash Flow<\/strong>: Identify the cash flow amount for each period.<\/li>\n<li><strong>Present Value of Cash Flow<\/strong>: Calculate the present value of each cash flow using the bond&#8217;s yield-to-maturity.<\/li>\n<li><strong>Weight of Cash Flow<\/strong>: Compute the weight of each cash flow by dividing its present value by the total present value of all cash flows.<\/li>\n<li><strong>Weighted Average Time<\/strong>: Multiply the time to receipt of each cash flow by its respective weight.<\/li>\n<li><strong>Macaulay Duration<\/strong>: Sum up the results from step 5 to obtain the Macaulay Duration.<\/li>\n<\/ol>\n<h3><strong>Formula<\/strong><\/h3>\n<p>The general formula to calculate Macaulay duration, represented as MacDur, is:<\/p>\n<p><span class=\"math display\">\\[MacDur = \\frac{\\sum_{i = 1}^{N}\\mspace{2mu}\\frac{t \\times CF_{t}}{(1 + r)^{t}}}{\\sum_{i = 1}^{N}\\mspace{2mu}\\frac{CF_{t}}{(1 + r)^{t}}}\\]<\/span><\/p>\n<p>Where:<\/p>\n<ul>\n<li><span class=\"math inline\">\\(t\\)<\/span> is the time (in periods) until the cash flow is received.<\/li>\n<li><span class=\"math inline\">\\(CF_{t}\\)<\/span> is the cash flow at time <span class=\"math inline\">\\(t\\)<\/span>.<\/li>\n<li><span class=\"math inline\">\\(r\\)<\/span> is the yield-to-maturity per period.<\/li>\n<li><span class=\"math inline\">\\(N\\)<\/span> is the total number of periods.<\/li>\n<\/ul>\n<h4><strong>Example: Calculating the Macaulay Duration<\/strong><\/h4>\n<p>Think about a bond with five years left to maturity, a 1% annual coupon, and a yield-to-maturity of 0.10%. Assume it&#8217;s 120 days into the first coupon period and follows a 30\/360 day-count basis. What&#8217;s the closest estimate for the bond&#8217;s annualized Macaulay duration?<\/p>\n<p>Considerations:<\/p>\n<ol type=\"1\">\n<li>The bond has a <span class=\"math inline\">\\(1\\%\\)<\/span> annual coupon, which means a cash flow of 1 per year for the next 4 years and 101 ( <span class=\"math inline\">\\(1 + 100\\)<\/span> par value) in the 5th year.<\/li>\n<li>The yield-to-maturity is <span class=\"math inline\">\\(0.10\\%\\)<\/span> or 0.0010 in decimal form.<\/li>\n<li>It is 120 days into the first coupon period, so the first cash flow will be received in <span class=\"math inline\">\\(1 &#8211; \\frac{120}{360}\\)<\/span> years or 0.6667 years.<\/li>\n<li>Macaulay duration is calculated as:<\/li>\n<\/ol>\n<p>\\[MacDur = \\frac{\\sum_{i = 1}^{N}\\frac{t \\times CF_{t}}{(1 + r)^{t}}}{\\sum_{i = 1}^{N}\\frac{CF_{t}}{(1 + r)^{t}}}\\]<\/p>\n<p>Where:<\/p>\n<ul>\n<li><span class=\"math inline\">\\(CF_{t} =\\)<\/span> Cash Flow at time <span class=\"math inline\">\\(t\\).<\/span><\/li>\n<li><span class=\"math inline\">\\(r =\\)<\/span> Yield to Maturity.<\/li>\n<li><span class=\"math inline\">\\(t =\\)<\/span> Time to receipt of the cash flow.<\/li>\n<\/ul>\n<p>$$\\begin{array}{c|c|c|c|c|c}\u00a0 \\textbf{Period} &amp; \\textbf{Time to Receipt} &amp; \\textbf{Cashflow Amount} &amp; \\textbf{PV} &amp; \\textbf{Weight} &amp; \\textbf{Time to Receipt*Weight} \\\\ \\hline 1 &amp; 0.6667 &amp; 1 &amp; 0.9993 &amp; 0.0096 &amp; 0.01 \\\\ \\hline 2 &amp; 1.6667 &amp; 1 &amp; 0.9983 &amp; 0.0096 &amp; 0.02 \\\\ \\hline 3 &amp; 2.6667 &amp; 1 &amp; 0.9973 &amp; 0.0095 &amp; 0.03 \\\\ \\hline 4 &amp; 3.6667 &amp; 1 &amp; 0.9963 &amp; 0.0095 &amp; 0.03 \\\\ \\hline 5 &amp; 4.6667 &amp; 101 &amp; 100.5300 &amp; 0.9618 &amp; 4.49 \\\\ \\hline \\textbf{Total} &amp; &amp; &amp; \\textbf{104.5213} &amp; \\textbf{1} &amp; \\textbf{4.5712} \\\\ \\end{array}$$<\/p>\n<p>This means that an investor would, on average, wait <strong>4.5712<\/strong> years to receive the bond&#8217;s cash flows, weighted by their present value.<\/p>\n<h3><strong>Interpretation of Macaulay Duration<\/strong><\/h3>\n<p>The Macaulay Duration provides insights into the bond&#8217;s interest rate risk. A bond with a higher Macaulay Duration has greater sensitivity to interest rate changes.<\/p>\n<p>For instance, if the investment horizon matches the Macaulay Duration, the bond is nearly hedged against interest rate risk. Any losses due to rising interest rates (price risk) would approximately be offset by gains from the reinvestment of coupons (reinvestment risk) and vice versa.<\/p>\n<p>Furthermore, the Macaulay Duration is often annualized. For bonds with semiannual coupons, the Macaulay Duration is divided by 2 to get the annualized figure.<\/p>\n<p>It&#8217;s also noteworthy that the Macaulay Duration is typically less than the bond&#8217;s time-to-maturity because it&#8217;s a present value-weighted average of the time until cash flows are received.<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Consider a bond that has three years remaining to maturity, a coupon of 3.5% paid semiannually, and a yield-to-maturity of 3.80%. Assuming it is 9 days into the first coupon period and using a 30\/360 basis, the bond&#8217;s annualized Macaulay duration is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>2.81 years.<\/li>\n<li>2.82 years.<\/li>\n<li>2.84 years.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is<strong> C<\/strong>.<\/p>\n<p>Given:<\/p>\n<ul>\n<li>The bond has three years remaining to maturity.<\/li>\n<li>A coupon of 3.5% is paid semiannually.<\/li>\n<li>Yield-to-maturity is 3.80%.<\/li>\n<li>18 days into the first coupon period.<\/li>\n<li>A 30\/360 basis.<\/li>\n<\/ul>\n<p>This means the bond pays 1.75% every 6 months. The yield per period is 1.90% (3.80% divided by 2).<\/p>\n<p>Let&#8217;s compute MacDur and then divide by 2 to annualize it since the bond pays semiannually.<\/p>\n<p>$$<br \/>\\begin{array}{c|c|c|c|c|c}<br \/>\\text { Period } &amp; \\begin{array}{l}<br \/>\\text { Time to } \\\\<br \/>\\text { Receipt }<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\text { Cashflow } \\\\<br \/>\\text { Amount }<br \/>\\end{array} &amp; \\begin{array}{l}<br \/>\\text { Present } \\\\<br \/>\\text { Value } \\\\<br \/>\\text { (PV) }<br \/>\\end{array} &amp; \\text { Weight } &amp; \\begin{array}{l}<br \/>\\text { Time to } \\\\<br \/>\\text { Receipt } \\\\<br \/>\\text { *Weight }<br \/>\\end{array} \\\\<br \/>\\hline &amp; 0.95 &amp; 1.75 &amp; 1.718987 &amp; 0.01732 &amp; 0.016454 \\\\<br \/>\\hline &amp; 1.95 &amp; 1.75 &amp; 1.686935 &amp; 0.016997 &amp; 0.033144 \\\\<br \/>\\hline &amp; 2.95 &amp; 1.75 &amp; 1.655481 &amp; 0.01668 &amp; 0.049206 \\\\<br \/>\\hline &amp; 3.95 &amp; 1.75 &amp; 1.624613 &amp; 0.016369 &amp; 0.064657 \\\\<br \/>\\hline &amp; 4.95 &amp; 1.75 &amp; 1.594321 &amp; 0.016064 &amp; 0.079515 \\\\<br \/>\\hline &amp; 5.95 &amp; 101.75 &amp; 90.96996 &amp; 0.916571 &amp; 5.453598 \\\\<br \/>\\hline &amp; &amp; &amp; 99.2503 &amp; 1 &amp; 5.696573 \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<p>The annualized MacDur is <strong>5.6966\/2 =<\/strong> 2.8483<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align:center; background:#f3f5f9; padding:48px 20px 28px; margin:40px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-block; background:#4274d8; color:#ffffff; text-decoration:none; padding:18px 42px; border-radius:999px; font-size:17px; font-weight:700; line-height:1;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:720px; margin:18px auto 0; font-size:17px; line-height:1.5; color:#1f2937; font-weight:400;\">\n    Build confidence in duration analysis, fixed-income valuation, and exam-style questions designed to strengthen your CFA Level I preparation.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Definition Macaulay duration was introduced in the previous learning objective. It provides an understanding of the bond&#8217;s sensitivity to interest rate fluctuations. At its core, Macaulay duration is the weighted average time until a bond&#8217;s cash flows are received. It&#8230;<\/p>\n","protected":false},"author":12,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-46405","post","type-post","status-publish","format-standard","hentry","category-fixed-income","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>Macaulay Duration Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Macaulay Duration measures a bond&#039;s interest rate risk by calculating the weighted average time to receive cash flows. 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