{"id":1879,"date":"2019-09-27T13:33:00","date_gmt":"2019-09-27T13:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1879"},"modified":"2026-03-27T09:37:35","modified_gmt":"2026-03-27T09:37:35","slug":"calculate-price-bond-using-spot-rates","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/calculate-price-bond-using-spot-rates\/","title":{"rendered":"Calculate the Price of a Bond Using Spot Rates"},"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\": \"This video on Introduction to Fixed Income Valuation explores essential concepts for bond valuation and yield analysis. Topics include calculating bond prices using market discount and spot rates, relationships among bond price factors, flat price, accrued interest, and full price, as well as matrix pricing. It also covers annual yields, yield measures for bonds and money market instruments, interpreting yield curves and forward rates, and yield spread analysis with practical calculations.\",\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\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"An investor wants to buy a 3-year 4% annual coupon paying bond. The expected spot rates are 2.5%, 3%, and 3.5% for the 1st, 2nd, and 3rd year, respectively. The bond\u2019s yield-to-maturity is closest to:\",\n    \"text\": \"An investor wants to buy a 3-year 4% annual coupon paying bond. The expected spot rates are 2.5%, 3%, and 3.5% for the 1st, 2nd, and 3rd year, respectively. The bond\u2019s yield-to-maturity is closest to:\\n\\nA. 2.55%\\nB. 3.47%\\nC. 4.45%\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nGiven the forecast spot rates, the 3-year 4% bond is priced at 101.475.\\n\\nWe can find the yield to maturity using the financial calculator:\\n\\nN = 3\\nPV = -101.475\\nPMT = 4\\nFV = 100\\nCPT => I\/Y = 3.47%\\n\\nAnd finally, we can confirm this is correct using the following formula:\\n$4\/(1.025)^1 + $4\/(1.03)^2 + $104\/(1.035)^3 = $101.475\"\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\n<p>Fixed-rate bonds are discounted by the market discount rate but the same rate is used for each cash flow. Alternatively, different market discount rates called spot rates could be used. Spot rates are yields-to-maturity on zero-coupon bonds maturing at the date of each cash flow. Sometimes, these are also called \u201czero rates\u201d and bond price or value is referred to as the \u201cno-arbitrage value.\u201d<\/p>\n<div style=\"margin:20px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:block;width:100%;text-align:center;padding:10px;border:2px solid #2f5bea;border-radius:40px;font-size:18px;color:#2f5bea;text-decoration:none\">\nPractice bond spot rate questions with our free trial.\n<\/a>\n<\/div>\n<h2>Calculating the Price of a Bond Using Spot Rates<\/h2>\n<p>Suppose that:<\/p>\n<ul>\n<li>the 1-year spot rate is 3%;<\/li>\n<li>the 2-year spot rate is 4%; and<\/li>\n<li>the 3-year spot rate is 5%.<\/li>\n<\/ul>\n<p>The price of a 100-par value 3-year bond paying 6% annual coupon payment is 102.95.<\/p>\n<p>$$<br>\\begin{array}{l|cccccc}<br>\\text{Time Period} &amp; 1 &amp; 2 &amp; 3 \\\\<br>\\hline<br>\\text{Calculation} &amp; \\frac {\\$6}{{\\left(1+3\\%\\right)}^{ 1 } } &amp; \\frac { \\$6 }{ { \\left( 1+4\\% \\right) }^{ 2 } } &amp; \\frac { \\$106 }{ { \\left( 1+5\\% \\right) }^{ 3 } } \\\\<br>\\hline<br>\\text{Cash Flow} &amp; \\$5.83 &amp; +\\$5.55 &amp; +\\$91.57 &amp; =\\$102.95 \\\\<br>\\end{array}<br>$$<\/p>\n<p>The general formula for calculating a bond\u2019s price given a sequence of spot rates is given below.<\/p>\n<p>\\({ PV }_{ bond }=\\frac { PMT }{ { (1+{ Z }_{ 1 }) }^{ 1 } } +\\frac { PMT }{ { (1+{ Z }_{ 2 }) }^{ 2 } } +&#8230;+\\frac { PMT+Principal }{ { (1+{ Z }_{ n }) }^{ n } } \\)<\/p>\n<h2>Calculating the Yield-to-maturity of a Bond using Spot Rates<\/h2>\n<p>Further, still with the same example, this 3-year bond is priced at a premium above par value, so its yield-to-maturity must be less than 6%. We can now use the financial calculator to find the yield-to-maturity using the following inputs:<\/p>\n<ul>\n<li>N = 3<\/li>\n<li>PV = -102.95 (Since this is a cash outflow)<\/li>\n<li>PMT = 6 (Since this is a cash inflow for the investor)<\/li>\n<li>FV = 100 (Since this is a cash inflow for the investor)<\/li>\n<li>CPT =&gt; I\/Y = 4.92 (Which signifies 4.92%)<\/li>\n<\/ul>\n<p>The yield-to-maturity is found to be 4.92%, which we can confirm with the following calculation:<\/p>\n<p>$$<br>\\begin{array}{l|cccccc}<br>\\text{Time Period} &amp; 1 &amp; 2 &amp; 3 \\\\<br>\\hline<br>\\text{Calculation} &amp; \\frac {\\$6}{{\\left(1+4.92\\%\\right)}^{ 1 } } &amp; \\frac { \\$6 }{ { \\left( 1+4.92\\% \\right) }^{ 2 } } &amp; \\frac { \\$106 }{ { \\left( 1+4.92\\% \\right) }^{ 3 } } \\\\<br>\\hline<br>\\text{Cash Flow} &amp; \\$5.719 &amp; +\\$5.450 &amp; +\\$91.770 &amp; =\\$102.95 \\\\<br>\\end{array}<br>$$<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>An investor wants to buy a 3-year 4% annual coupon paying bond. The expected spot rates are 2.5%, 3%, and 3.5% for the 1<sup>st<\/sup>, 2<sup>nd<\/sup>, and 3<sup>rd<\/sup> year, respectively. The bond&#8217;s&nbsp;yield-to-maturity is <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>2.55%.<\/li>\n<li>3.47%.<\/li>\n<li>4.45%.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>\\(\\frac{$4}{(1.025)^1}+\\frac{$4}{(1.03)^2}+\\frac{$104}{(1.035)^3}=$101.475\\)<\/p>\n<p>Given the forecast spot rates, the 3-year 4% bond is priced at 101.475.<\/p>\n<p>Here again, we can find the yield to maturity using the financial calculator:<\/p>\n<ul>\n<li>N = 3<\/li>\n<li>PV = -101.475<\/li>\n<li>PMT = 4<\/li>\n<li>FV = 100<\/li>\n<li>CPT =&gt; I\/Y = 3.47%<\/li><\/ul>\n<p>And finally, we can confirm this is correct using the following formula:<\/p>\n<p>\\(\\frac{$4}{(1.0347)^1}+\\frac{$4}{(1.0347)^2}+\\frac{$104}{(1.0347)^3}=101.475\\)<\/p>\n<\/blockquote>\n<div style=\"text-align:center;margin:40px 0\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\"\nstyle=\"display:inline-block;padding:12px 30px;background:#3f78d7;color:#fff;border-radius:40px;font-size:18px;text-decoration:none\">\nStart Free Trial\n<\/a>\n<p style=\"margin-top:12px;max-width:600px;margin-left:auto;margin-right:auto\">\nSolve CFA-style fixed income questions and master bond pricing using spot rates.\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Fixed-rate bonds are discounted by the market discount rate but the same rate is used for each cash flow. Alternatively, different market discount rates called spot rates could be used. Spot rates are yields-to-maturity on zero-coupon bonds maturing at the&#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-1879","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>Calculate Bond Price Using Spot Rates | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate bond prices by discounting each cash flow using spot rates. 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