{"id":46767,"date":"2023-09-22T17:18:08","date_gmt":"2023-09-22T17:18:08","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46767"},"modified":"2026-02-10T05:57:00","modified_gmt":"2026-02-10T05:57:00","slug":"duration-and-convexity-of-a-bond-portfolio","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/duration-and-convexity-of-a-bond-portfolio\/","title":{"rendered":"Duration and Convexity of a Bond Portfolio"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Given that a bond portfolio has a duration of 5 years and a convexity of 50, estimate the percentage change in the portfolio\u2019s value if there is an increase of 50 basis points in the yield-to-maturity.\",\n    \"text\": \"Given that a bond portfolio has a duration of 5 years and a convexity of 50, estimate the percentage change in the portfolio\u2019s value if there is an increase of 50 basis points in the yield-to-maturity.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. The percentage change in the portfolio's value is calculated using the formula that accounts for both duration and convexity. The resulting percentage change is -2.438%.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p>Duration and convexity can be used to measure the interest rate risk of a portfolio of bonds, similar to a single bond. There are two methods to calculate the duration and convexity of a bond portfolio:<\/p>\n\n\n\n<ol style=\"list-style-type:lower-roman\" class=\"wp-block-list\">\n<li>Using the weighted average of time to receipt of the aggregate cash flows.<\/li>\n\n\n\n<li>Using the weighted averages of the durations and convexities of the individual bonds in the portfolio.<\/li>\n<\/ol>\n\n\n\n<p>The first technique is theoretically more precise. However, its practical application proves challenging. Consequently, the emphasis tends to lean towards the second approach, largely due to its common usage among fixed-income portfolio managers. However, it operates under the assumption that yield changes are uniform across all maturities, leading to a parallel shift in the yield curve. Contrary to this assumption, yield curve shifts are typically observed as steepening, flattening, or even twisting yield curves, making pure parallel shifts rare.<\/p>\n\n\n\n<p><strong>Example: Calculating Weighted Average Duration and Convexity<\/strong><\/p>\n\n\n\n<p>An investor purchases $5 million par value of a 4-year, zero-coupon bond and a 5-year, fixed-rate semi-annual coupon bond. Details of the bonds are shown below.<\/p>\n\n\n\n<p>\\[ \\begin{array}{c|c|c|c|c|c|c}&nbsp; \\textbf{Bond} &amp; \\textbf{Maturity (Years)} &amp; \\textbf{Coupon (%)} &amp; \\textbf{Price} &amp; \\textbf{YTM (%)} &amp; \\textbf{Duration} &amp; \\textbf{Convexity} \\\\ \\hline \\text{Zero} &amp; 4 &amp; 0 &amp; 87.1442228 &amp; 3.5 &amp; 3.8647 &amp; 19.32367 \\\\ \\hline \\text{Semi-annual} &amp; 5 &amp; 4.5 &amp; 101.115515 &amp; 4.25 &amp; 4.441605 &amp; 23.12742 \\\\ \\end{array} \\]<\/p>\n\n\n\n<ol style=\"list-style-type:lower-roman\" class=\"wp-block-list\">\n<li>Calculate the weighted-average modified duration for the portfolio.<\/li>\n\n\n\n<li>Calculate the weighted-average convexity for the portfolio.<\/li>\n\n\n\n<li>Calculate the estimated percentage price change of the portfolio given a 100 bp increase in yield-to-maturity on each of the bonds.<\/li>\n<\/ol>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:14px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:18px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice bond portfolio duration questions \u2014 free trial access.\n  <\/a>\n<\/div>\n\n\n\n<h4><strong>Calculating the weighted-average modified duration for the portfolio.<\/strong><\/h4>\n<p>To compute the weighted-average modified duration:<\/p>\n<h4>Determine the market value for each bond.<\/h4>\n<p>Zero-coupon bond: <span class=\"math inline\">\\(87.1442228 \\times \\$ 5,000,000\\ = 435,721,114\\)<\/span><\/p>\n<p>Semi-Annual Bond: <span class=\"math inline\">\\(101.115515 \\times 5,000,000\\ = 505,577,575\\)<\/span><\/p>\n<h4>Calculate the weight for each bond.<\/h4>\n<p><span class=\"math display\">\\[Total\\ market\\ value\\ = \\ 435,721,114 + 505,577,575\\ = \\ 941,298,689\\]<\/span><\/p>\n<p><span class=\"math display\">\\[Weight\\ of\\ Zero &#8211; coupon\\ bond:\\ 435,721,114\/941,298,689\\ = \\ 0.46289357\\]<\/span><\/p>\n<p><span class=\"math display\">\\[Weight\\ of\\ Semi &#8211; Annual\\ Bond:\\ 505,577,575\/941,298,689\\ = \\ 0.537106426\\]<\/span><\/p>\n<h4><strong>Multiply the weight of each bond by its duration and sum the results.<\/strong><\/h4>\n<p><span class=\"math display\">\\[Weighted &#8211; average\\ modified\\ duration = (0.46289357 \\times 3.8647 + (0.537106426 \\times 4.441605) = 4.174559367\\]<\/span><\/p>\n<h4><strong>Calculating the weighted-average convexity for the portfolio.<\/strong><\/h4>\n<p>Similar to the duration calculation above:<\/p>\n<ol type=\"1\">\n<li>Determine the market value for each bond (which we have already done in step i).<\/li>\n<li>Calculate the weight for each bond (which we have also done in step ii).<\/li>\n<li>Multiply the weight of each bond by its convexity and sum the results.<\/li>\n<\/ol>\n<p><span class=\"math display\">\\[Weighted &#8211; average\\ convexity\\ = (0.46289357 \\times 19.32367 + (0.537106426 \\times 23.12742)\\ = 21.36668849\\]<\/span><\/p>\n<h4><strong>Calculating the estimated percentage price change of the portfolio given a 100 bp increase in yield-to-maturity on each of the bonds.<\/strong><\/h4>\n<p><span class=\"math display\">\\[\\%\\Delta PV_{\\text{Full}} \\approx ( &#8211; \\text{Duration} \\times \\Delta y) + \\left\\lbrack \\frac{1}{2} \\times \\text{Convexity} \\times (\\Delta y)^{2} \\right\\rbrack\\]<\/span><\/p>\n<p>Where <span class=\"math inline\">\\(\\Delta y\\)<\/span> is the change in yield (in decimal form).<\/p>\n<p>For a 100bp change, ?y=0.0100.<\/p>\n<p><span class=\"math display\">$$\\%\\Delta PV_{\\text{Full}} \\approx (-4.174559367 \\times 0.01) + \\left[ \\frac{1}{2} \\times 21.36668849 \\times (0.01)^2 \\right] = &#8211; 0.040677259 \\approx &#8211; 4.0677\\%$$<br \/><\/span><\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Given that a bond portfolio has a duration of 5 years and a convexity of 50, estimate the percentage change in the portfolio&#8217;s value if there is an increase of 50 basis points in the yield-to-maturity.<\/p>\n<ol type=\"A\">\n<li>-2.438%<\/li>\n<li>-2.500%<\/li>\n<li>2.563%<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/p>\n<p>Formula:<\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV_{\\text{Full}} \\approx ( &#8211; \\text{Duration} \\times \\Delta y) + \\left\\lbrack \\frac{1}{2} \\times \\text{Convexity} \\times (\\Delta y)^{2} \\right\\rbrack\\]<\/span><\/p>\n<p>Where ?y for 50 basis points is 0.005<\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV_{\\text{Full}} \\approx \\left( &#8211; \\text{5} \\times 0.005 \\right) + \\left\\lbrack \\frac{1}{2} \\times \\text{50} \\times (0.005)^{2} \\right\\rbrack = &#8211; 2.438\\%\\]<\/span><\/p>\n<\/blockquote>\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:35px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:14px 34px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:16px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:18px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:700px; margin:0 auto; font-size:16px; line-height:1.6; color:#333;\">\n    Strengthen your fixed income portfolio analysis with exam-style duration and convexity problems, detailed solutions, and timed quizzes inside AnalystPrep\u2019s free trial.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Duration and convexity can be used to measure the interest rate risk of a portfolio of bonds, similar to a single bond. There are two methods to calculate the duration and convexity of a bond portfolio: The first technique is&#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-46767","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>Duration &amp; Convexity of a Bond Portfolio | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore the intricacies of bond portfolio risk measurement using duration and convexity and follow a detailed example to calculate it.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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