{"id":1885,"date":"2019-09-27T13:33:00","date_gmt":"2019-09-27T13:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1885"},"modified":"2026-04-02T08:34:21","modified_gmt":"2026-04-02T08:34:21","slug":"flat-price-accrued-interest-full-price-bond","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/flat-price-accrued-interest-full-price-bond\/","title":{"rendered":"Flat Price, Accrued Interest, and Full Price of a Bond"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Introduction to Fixed Income Valuation (2025 Level I CFA\u00ae Exam \u2013 Fixed Income \u2013 Module 3)\",\n  \"description\": \"Master the fundamentals of fixed income valuation! Learn how to calculate bond prices, interpret yield measures, understand spot and forward rates, and analyze yield curves. Dive into concepts like matrix pricing, yield spreads, and annual yield calculations for comprehensive fixed income insights.\",\n  \"uploadDate\": \"2022-05-21T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/lEbeibhvCzM\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/lEbeibhvCzM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/lEbeibhvCzM\",\n  \"duration\": \"PT56M17S\"\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\": \"Assuming a market discount rate of 4.5%, the full price of a semi-annual bond with a present value (PV) of 102 and 90 days of accrued interest is closest to:\",\n    \"text\": \"Assuming a market discount rate of 4.5%, the full price of a semi-annual bond with a present value (PV) of 102 and 90 days of accrued interest is closest to:\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. Since the bond pays coupons semi-annually, the time between coupon payments is 180 days. The full price is calculated by compounding the present value over the accrued period: PV_full = 102 \u00d7 (1.0225)^(90\/180) \u2248 103.14, which is closest to 103.135.\"\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\/lEbeibhvCzM\"\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<p>When investors purchase shares, they pay the quoted price. However, for bonds, there can be a difference between the quoted price and the price paid.<\/p>\n<h2><strong>Full Price<\/strong><\/h2>\n<p>When a bond is between coupon payment dates, the price has 2 components: the flat price (PV<sup>Flat<\/sup>) and the Accrued Interest (AI). The sum of these two is the full price (PV<sup>Full<\/sup>), also called <span class=\"primary-color\">invoice<\/span> or <span class=\"primary-color\">dirty price<\/span>.<\/p>\n<p>$$ PV^{ Full } = PV^{ Flat } + \\text{Accrued interest} $$<\/p>\n<div style=\"margin: 20px 0;\"><a style=\"display: block; width: 100%; text-align: center; padding: 10px; border: 2px solid #2f5bea; border-radius: 40px; font-size: 16px; color: #2f5bea; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Practice bond pricing questions with our free trial. <\/a><\/div>\n<h2><strong>Flat Price<\/strong><\/h2>\n<p>The flat price, on the other hand, is the full price minus the accrued interest. The flat price is generally the quoted price between bond dealers. It does not include any interest accrued between the scheduled coupon payments for the bond.<\/p>\n<p>The reason for using the flat price is to avoid misleading investors since accrued interest does not change the yield-to-maturity. It is the flat price that is \u201cpulled to par\u201d along the life constant yield price line.<\/p>\n<h2><strong>Accrued Interest<\/strong><\/h2>\n<p>The accrued interest is the proportional share of the next coupon payment. Suppose coupon payment has \u201cT\u201d days between payment dates and \u201ct\u201d days have passed from the last payment date, then the accrued interest:<\/p>\n<p>$$AI=\\frac { t }{ T } \\times PMT$$<\/p>\n<p>Where:<\/p>\n<p><em>t\/T<\/em> = fraction of the coupon period that has passed from the last payment date; and<\/p>\n<p><em>PMT<\/em> = coupon payment for the period<\/p>\n<h2><strong>Day-Count Conventions<\/strong><\/h2>\n<h3><strong>Actual\/actual convention<\/strong><\/h3>\n<p>Regarding bond day-counting, there are 2 major methods: Actual-actual and 30\/360.<\/p>\n<p>The first \u201c<em><strong>actual<\/strong><\/em>\u201d refers to the\u00a0<em><strong>actual<\/strong><\/em> number of days since the last coupon date. The second one refers to the <em><strong>actual<\/strong><\/em> number of\u00a0days\u00a0in a coupon period. This includes weekends, holidays, and leap days.<\/p>\n<p>For example, let\u2019s assume that a semi-annual payment bond pays interest on 15<sup>th<\/sup> June and 15<sup>th<\/sup> December of each year. To compute the accrued interest for settlement as of 27<sup>th<\/sup> July, we would consider the actual number of days between 15<sup>th<\/sup> June and 27<sup>th<\/sup> July (<em>t <\/em>= 42 days). This number would be divided by the actual number of days between 15<sup>th<\/sup> June and 15<sup>th<\/sup> December (<em>T <\/em>= 183 days) and then multiplied by the coupon payment. With a coupon rate of 5.25%, the accrued interest would be 0.60246 per 100 of par value.<\/p>\n<p>$$\\text{Accrued interest}=\\frac { 42 }{ 183 } \\times \\frac { 5.25% }{ 2 } =0.60246$$<\/p>\n<h3><strong>30\/360 day-count convention<\/strong><\/h3>\n<p>The 30\/360 convention is often used for corporate bonds. It <em>assumes <\/em>that each month has 30 days (even though some months actually have 31 days and February has 28 or 29 days) and that a full year has 360 days.<\/p>\n<p>As we did earlier, let\u2019s assume that a semi-annual payment bond pays interest on 15<sup>th<\/sup> June and 15<sup>th<\/sup> December each year. To compute the accrued interest for settlement as of 27<sup>th<\/sup> July, we would take into account a total of 15 days in June and 27 days in July (<em>t <\/em>= 42 days). This number would be divided by 180 \u2013 the number of days between 15<sup>th<\/sup> June and 15<sup>th<\/sup> December, assuming every month has 30 days (<em>T <\/em>= 6 \u00d7 30), and then multiplied by the coupon payment. With a coupon rate of 5.25%, the accrued interest would be 0.6125 per 100 of par value.<\/p>\n<p>$$\\text{Accrued interest}=\\frac { 42 }{ 180 } \\times \\frac { 5.25% }{ 2 } =0.6125$$<\/p>\n<p>The full price of a fixed-rate bond between coupon payments given the market discount rate per period (<em>r<\/em>) can be calculated as:<\/p>\n<p>$$PV^{ Full }=PV\u00d7(1+r)^{ t\/T }$$<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Assuming a market discount rate of 4.5%, the full price of a semi-annual bond with a present value (PV) of 102 and 90 days of accrued interest is <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li data-tadv-p=\"keep\">101.955<\/li>\n<li data-tadv-p=\"keep\">103.135<\/li>\n<li data-tadv-p=\"keep\">104.235<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>Since the bond coupons are paid semi-annually, the time (T) between coupon payments is 180 days.<\/p>\n<p>PV<sup>Full<\/sup> = 102 \u00d7 (1.0225)<sup>90\/180<\/sup> = 103.1411<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-block; padding: 12px 24px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a>\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Build confidence in CFA Level I fixed income with exam-style practice on bond pricing, accrued interest, and yield calculations.<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>When investors purchase shares, they pay the quoted price. However, for bonds, there can be a difference between the quoted price and the price paid. Full Price When a bond is between coupon payment dates, the price has 2 components:&#8230;<\/p>\n","protected":false},"author":1,"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-1885","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>Flat, Accrued, &amp; Full Bond Prices | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand the difference between flat price, accrued interest, and full (dirty) price of bonds. 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