{"id":46744,"date":"2023-09-22T16:18:49","date_gmt":"2023-09-22T16:18:49","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46744"},"modified":"2026-04-06T12:39:49","modified_gmt":"2026-04-06T12:39:49","slug":"modified-duration-money-duration-and-price-value-of-a-basis-point-pvbp","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/modified-duration-money-duration-and-price-value-of-a-basis-point-pvbp\/","title":{"rendered":"Modified Duration, Money Duration, and Price Value of a Basis Point (PVBP)"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Yield-Based Bond Duration Measures and Properties (2024\/2025 CFA\u00ae Level I Exam \u2013 Fixed Income \u2013 LM 11)\",\n  \"description\": \"This video explains yield-based bond duration measures and their properties, focusing on modified, money, and price duration. It highlights the relationship between bond price and yield, differences between yield and curve duration, and factors like coupon rate, yield to maturity, and time to maturity. It also introduces duration calculations, interest rate risk implications, and practical applications.\",\n  \"uploadDate\": \"2023-12-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/0FIG6azO9Do\/default.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=0FIG6azO9Do\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/0FIG6azO9Do\",\n  \"duration\": \"PT17M52S\"\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\": \"An investment analyst is reviewing a 4-year bond, issued on 1 January 2024, set to mature on 1 January 2028. This bond features a 4% coupon rate, paid semi-annually, and carries a yield-to-maturity of 6%. The bond\u2019s annualized Macaulay duration and Modified duration, respectively, are closest to:\",\n    \"text\": \"An investment analyst is reviewing a 4-year bond, issued on 1 January 2024, set to mature on 1 January 2028. This bond features a 4% coupon rate, paid semi-annually, and carries a yield-to-maturity of 6%. The bond\u2019s annualized Macaulay duration and Modified duration, respectively, are closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. The annualized Macaulay duration is approximately 3.72 years, and the Modified duration is approximately 3.62. The Macaulay duration is first computed on a semiannual basis and then annualized by dividing by the number of coupon payments per year. The Modified duration is obtained by dividing the annualized Macaulay duration by (1 + y\/m), where y is the yield-to-maturity and m is the number of coupon periods per year.\"\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\/0FIG6azO9Do\"\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\"><strong>Modified Duration<\/strong><\/h2>\n\n\n\n<p>Modified duration captures the sensitivity of a bond&#8217;s price to fluctuations in its yield-to-maturity (YTM). This relationship provides insight into how bond prices vary with shifts in yield. Specifically, bond prices and yields exhibit an inverse relationship: as yields rise, bond prices fall, and vice versa.<\/p>\n\n\n\n<div style=\"width:100%; margin:30px 0; box-sizing:border-box;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:block;\n            width:100%;\n            max-width:100%;\n            box-sizing:border-box;\n            padding:16px 20px;\n            border:2px solid #2f6fed;\n            border-radius:999px;\n            background:#f2f4f8;\n            color:#2f6fed;\n            font-size:16px;\n            font-weight:500;\n            text-decoration:none;\n            text-align:center;\n            line-height:1.2;\">\n    Access our CFA free trial for duration and PVBP practice\n  <\/a>\n<\/div>\n\n\n<h3><strong>Relation to Macaulay Duration<\/strong><\/h3>\n<p>Modified duration is an extension of the Macaulay duration, which conveys the weighted average time until a bond&#8217;s cash flows are received. The link between these two measures is encapsulated by the formula:<\/p>\n<p><span class=\"math display\">\\[\\text{ModDur} = \\frac{\\text{Macaulay Duration}}{1 + r}\\]<\/span><\/p>\n<p>Where <span class=\"math inline\">\\(r\\)<\/span> represents the yield per period. To obtain the annual modified duration, divide the modified duration by the bond&#8217;s number of coupon payments in a year. The larger the modified duration, the more pronounced the bond&#8217;s price-yield curve becomes, leading to larger price swings for given changes in yield.<\/p>\n<h3><strong>Approximating Modified Duration<\/strong><\/h3>\n<p>In cases where the Macaulay duration is not available, the modified duration can be estimated by observing minute variations in bond prices as yields change. This approximation method is especially useful for bonds with embedded options or inherent default risks. The formula for this approximation is:<\/p>\n<p><span class=\"math display\">\\[\\text{AnnModDur} \\approx \\frac{\\left( PV_{-} &#8211; PV_{+} \\right)}{2 \\times \\Delta\\text{Yield} \\times PV_{0}}\\]<\/span><\/p>\n<p>Where <span class=\"math inline\">\\(PV_{-}\\)<\/span>and <span class=\"math inline\">\\(PV_{+}\\)<\/span>are bond prices corresponding to decreased and increased yields, respectively. Historically, this method has been highly accurate. To revert to the Macaulay duration, multiply the modified duration by <span class=\"math inline\">\\(1 + r\\)<\/span>.<\/p>\n<h4><strong>Example: Approximating Modified Duration<\/strong><\/h4>\n<p>A 4.5% semiannual-pay fixed-coupon bond is issued at par on 1 June 2026 and matures on 1 June 2030. For a 50bps increase and decrease in yield-to-maturity, <span class=\"math inline\">\\(PV_{+}\\)<\/span>and <span class=\"math inline\">\\(PV_{-}\\)<\/span>are 98.207 and 101.831, respectively. The approximate modified duration can be determined as follows:<\/p>\n<p>Formula:<\/p>\n<p><span class=\"math display\">\\[\\text{AnnModDur} \\approx \\frac{\\left( PV_{-} &#8211; PV_{+} \\right)}{2 \\times \\Delta\\text{Yield} \\times PV_{0}}\\]<\/span><\/p>\n<p><span class=\"math inline\">\\(PV_{-} =\\)<\/span>101.831<\/p>\n<p><span class=\"math inline\">\\(PV_{+}\\)<\/span> <em>=<\/em> 98.207<\/p>\n<p><span class=\"math inline\">\\(\\Delta\\text{Yield}\\text{=50\/10000=0.005}\\)<\/span><\/p>\n<p><span class=\"math display\">\\[AnnModDur\\ \\approx \\frac{101.831 &#8211; 98.207}{2 \\times 0.005 \\times 100} = 3.624\\ \\]<\/span><\/p>\n<h3><strong>Predicting Price Changes Based on Modified Duration<\/strong><\/h3>\n<p>Modified duration unveils the bond price-yield relationship, allowing predictions of the bond&#8217;s percentage price alteration in relation to shifts in its YTM. The formula to determine this is:<\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV^{\\text{Full}} \\approx &#8211; \\text{AnnModDur} \\times \\Delta\\text{AnnYield}\\]<\/span><\/p>\n<p>As an illustration, a bond with a modified duration of 5 would likely experience a <span class=\"math inline\">\\(5\\%\\)<\/span> price drop if its yield surges by 100 basis points. Hence, bonds with higher modified durations exhibit steeper price-yield curves, making them more susceptible to yield variations. It&#8217;s crucial to note that this formula offers a linear approximation for the inherently nonlinear price-yield relationship. The inclusion of the negative sign emphasizes the inverse correlation between bond prices and their yields-to-maturity.<\/p>\n<h2><strong>Money Duration<\/strong><\/h2>\n<p>While modified duration gauges the percentage price change of a bond given variations in its yield-to-maturity (YTM), money duration provides insights into the price change in terms of currency units. In the U.S., it is also referred to as &#8220;dollar duration.&#8221;<\/p>\n<p>Money duration is calculated using the formula:<\/p>\n<p><span class=\"math display\">\\[\\text{MoneyDur} = \\text{AnnModDur} \\times PV^{\\text{Full}}\\]<\/span><\/p>\n<p><span class=\"math inline\">\\(PV^{\\text{Full~}}\\ \\)<\/span>can be either the bond price as a percent of par value or the currency value of the bond holding.<\/p>\n<p>Using Money Duration, one can estimate the bond price change in currency units for a given change in YTM:<\/p>\n<p><span class=\"math display\">\\[\\%\\Delta PV^{\\text{Full}} \\approx &#8211; \\text{MoneyDur} \\times \\Delta\\text{Yield}\\]<\/span><\/p>\n<h4><strong>Example: Calculating Money Duration<\/strong><\/h4>\n<p>Consider a bond with an annualized modified duration of 5.5, a coupon of <span class=\"math inline\">\\(4\\%\\)<\/span> and a price of 102. The money duration is closest to:<\/p>\n<p><span class=\"math display\">\\[\\text{MoneyDur} = \\text{AnnModDur} \\times PV^{\\text{Full}}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\text{Money Duration} = 5.5 \\times 102\\]<\/span><\/p>\n<p>This means that for a <span class=\"math inline\">\\(1\\%\\)<\/span> (or 100 basis points) change in yield, the bond&#8217;s price will change by $561.<\/p>\n<h2><strong>Price Value of a Basis Point (PVBP)<\/strong><\/h2>\n<p>PVBP provides an estimate of the change in the full price of a bond for a minuscule 1bp change in its YTM. PVBP can be determined using the formula:<\/p>\n<p><span class=\"math display\">\\[PVBP = \\frac{\\left( PV_{-} \\right) &#8211; \\left( PV_{+} \\right)}{2}\\]<\/span><\/p>\n<p>This measure is often termed as &#8220;PV01&#8221; or in the U.S., &#8220;DV01&#8221; (Dollar Value of 1bp). PVBP is especially handy for bonds where future cash flows are unpredictable, like callable bonds.<\/p>\n<p>Basis Point Value (BPV) is a close relative to PVBP, and it is the product of Money Duration and 0.0001 (1bp).<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>An investment analyst is reviewing a 4-year bond, issued on 1 January 2024, set to mature on 1 January 2028. This bond features a 4% coupon rate, paid semi-annually, and carries a yield-to-maturity of 6%. The bond&#8217;s annualized Macaulay duration and Modified duration, respectively, are <em>closest to:<\/em><\/p>\n<ol style=\"list-style-type: upper-alpha; text-align: left;\">\n<li>3.46 and 3.26<\/li>\n<li>3.69 and 3.48<\/li>\n<li>3.72 and 3.62<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is C:<\/strong><\/p>\n<p>The Macaulay duration is 7.4481. This can be annualized by dividing by the number of coupon payments in a year.<\/p>\n<p>\\[ \\begin{array}{c|c|c|c|c|c}\\textbf{Period} &amp; \\textbf{Time to receipt} &amp; \\textbf{Cashflow amount} &amp; \\textbf{PV} &amp; \\textbf{Weights} &amp; \\textbf{Time to Receipt*Weight} \\\\ \\hline 1 &amp; 1.0000 &amp; 2 &amp; 1.9417 &amp; 0.0209 &amp; 0.0209 \\\\ \\hline 2 &amp; 2.0000 &amp; 2 &amp; 1.8852 &amp; 0.0203 &amp; 0.0406 \\\\ \\hline 3 &amp; 3.0000 &amp; 2 &amp; 1.8303 &amp; 0.0197 &amp; 0.0591 \\\\ \\hline 4 &amp; 4.0000 &amp; 2 &amp; 1.7770 &amp; 0.0191 &amp; 0.0764 \\\\ \\hline 5 &amp; 5.0000 &amp; 2 &amp; 1.7252 &amp; 0.0186 &amp; 0.0928 \\\\ \\hline 6 &amp; 6.0000 &amp; 2 &amp; 1.6750 &amp; 0.0180 &amp; 0.1081 \\\\ \\hline 7 &amp; 7.0000 &amp; 2 &amp; 1.6262 &amp; 0.0175 &amp; 0.1224 \\\\ \\hline 8 &amp; 8.0000 &amp; 102 &amp; 80.5197 &amp; 0.8660 &amp; 6.9279 \\\\ \\hline \\textbf{Total} &amp; &amp; &amp; \\textbf{92.9803} &amp; \\textbf{1.0000} &amp; \\textbf{7.4481} \\\\ \\end{array} \\]<\/p>\n<p><span class=\"math display\">\\[Annualized\\ Macaulay\\ duration\\ = \\frac{7.4481}{2} = 3.72405\\ \\]<\/span><\/p>\n<p><span class=\"math display\">\\[ModDur\\ = \\frac{3.72405}{1.03} = \\ 3.6156\\]<\/span><\/p>\n<\/blockquote>\n<p>&#8211;&gt;<\/p>\n\n\n<div style=\"background:#f2f4f8; border-radius:16px; padding:32px 20px; text-align:center; margin:40px 0;\">\n  \n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-block;\n            background:#4a74c9;\n            color:#ffffff;\n            font-size:16px;\n            font-weight:600;\n            text-decoration:none;\n            padding:14px 34px;\n            border-radius:999px;\n            margin-bottom:16px;\">\n    Start Free Trial\n  <\/a>\n\n  <p style=\"margin:0 auto; max-width:520px; font-size:16px; line-height:1.6; color:#1f2937;\">\n    Master duration, PVBP, and bond price sensitivity with structured CFA Level I practice.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Modified Duration Modified duration captures the sensitivity of a bond&#8217;s price to fluctuations in its yield-to-maturity (YTM). This relationship provides insight into how bond prices vary with shifts in yield. Specifically, bond prices and yields exhibit an inverse relationship: as&#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-46744","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>Modified Duration, PVBP &amp; Money Duration | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how modified duration measures bond price sensitivity to yield changes, calculate PVBP for 1 bp yield shifts, and understand money duration formulas.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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