{"id":17090,"date":"2021-07-06T23:25:53","date_gmt":"2021-07-06T23:25:53","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=17090"},"modified":"2026-04-29T09:57:27","modified_gmt":"2026-04-29T09:57:27","slug":"describe-how-zero-coupon-rates-spot-rates-may-be-obtained-from-the-par-curve-by-bootstrapping","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-how-zero-coupon-rates-spot-rates-may-be-obtained-from-the-par-curve-by-bootstrapping\/","title":{"rendered":"Bootstrapping Spot Rates"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"An indication that the homoskedasticity assumption has been violated is likely to be?\",\n    \"text\": \"Which of the following indicates a violation of the homoskedasticity assumption in a regression model?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The homoskedasticity assumption is violated when the variance of the residuals for all observations is different. This indicates heteroskedasticity.\",\n      \"upvoteCount\": 1\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"All observations have the same variance of the residuals.\",\n        \"upvoteCount\": 0\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"There is a difference in the variance of the residuals for all observations.\",\n        \"upvoteCount\": 1\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"There is a correlation between the X and Y pairs of observations.\",\n        \"upvoteCount\": 0\n      }\n    ]\n  }\n}\n<\/script><\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/ovQh3D6WBOw\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><em><strong>Bootstrapping spot rates<\/strong><\/em> is a forward substitution method that allows investors to determine zero-coupon rates using the par yield curve. The <em><strong>par curve<\/strong><\/em> shows the yields to maturity on government bonds with coupon payments, priced at par, over a range of maturities.<\/p>\n<p>Bootstrapping involves obtaining spot rates (zero-coupon rates) for one year, then using the one-year spot rate to determine the 2-year spot rate, and so on.<\/p>\n<p>Spot rates obtained through bootstrapping are known as <em><strong>implied spot rates<\/strong><\/em>.<\/p>\n<div style=\"text-align: center; margin: 25px 0;\"><a style=\"display: inline-block; padding: 12px 20px; border: 2px solid #2f5cff; border-radius: 999px; color: #2f5cff; text-decoration: none; background: #f7f9fc; white-space: nowrap;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Master yield curve concepts with our free trial. <\/a><\/div>\n<h4>Example: Bootstrapping Spot Rates<\/h4>\n<p>Consider the following annual par rates for a coupon paying bond.<\/p>\n<p>$$ \\textbf{Annual Par-Rates} \\\\ \\begin{array}{c|c} \\textbf{Year} &amp; \\textbf{Par Rate} \\\\ \\hline 1 &amp; 2.00\\% \\\\ \\hline 2 &amp; 2.60\\% \\\\ \\hline 3 &amp; 2.90\\% \\\\ \\hline 4 &amp; 3.80\\% \\end{array} $$<\/p>\n<p>The one-year implied spot rate is 2%, as it is simply the one-year par yield.<\/p>\n<p>The two-year implied spot rate is determined as follows:<\/p>\n<p>$$ \\begin{align*} 1 &amp;=\\frac{0.026}{1.02}+\\frac{(1+0.026)}{\\left(1+r\\left(2\\right)\\right)^2} \\\\ r (2) &amp; =2.61\\% \\end{align*} $$<\/p>\n<p>We have bootstrapped the 2-year spot rate.<\/p>\n<p>Similarly, the three-year spot rate can be bootstrapped by solving the equation:<\/p>\n<p>$$ \\begin{align*} 1 &amp;=\\frac{0.029}{1.02}+\\frac{0.029}{{1.0261}^2}+\\frac{1.029}{\\left(1+r\\left(3\\right)\\right)^3} \\\\ r (3) &amp; = 2.91\\% \\end{align*} $$<\/p>\n<p>The four-year spot rate is determined as:<\/p>\n<p>$$ \\begin{align*} 1 &amp;=\\frac{0.038}{1.02}+\\frac{0.038}{{1.0261}^2}+\\frac{0.038}{{1.0291}^3}+\\frac{1.038}{\\left(1+r\\left(4\\right)\\right)^4} \\\\ r (4) &amp; = 3.87\\% \\end{align*} $$<\/p>\n<p>The zero-coupon rates are shown in the following table:<\/p>\n<p>$$ \\begin{array}{c|c|c} \\textbf{Year} &amp; \\textbf{Par Rate} &amp; \\textbf{Zero-coupon rate} \\\\ &amp; &amp; \\textbf{(Implied Spot rate)} \\\\ \\hline 1 &amp; 2.00\\% &amp; 2.00\\% \\\\ \\hline 2 &amp; 2.60\\% &amp; 2.61\\% \\\\ \\hline 3 &amp; 2.90\\% &amp; 2.91\\% \\\\ \\hline 4 &amp; 3.80\\% &amp; 3.87\\% \\end{array} $$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Determine the three-year implied spot rate given the following par rates.<\/p>\n<p>$$ \\begin{array}{c|c} \\textbf{Year} &amp; \\textbf{Rate} \\\\ \\hline 1 &amp; 5.00\\% \\\\ \\hline 2 &amp; 5.25\\% \\\\ \\hline 3 &amp; 5.50\\% \\end{array} $$<\/p>\n<ol type=\"A\">\n<li>5.25%.<\/li>\n<li>5.52%.<\/li>\n<li>16.59%.<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p><strong>The correct answer is B.<\/strong><\/p>\n<p>The one-year implied spot rate is 5%, as it is simply the one-year par yield.<\/p>\n<p>The two-year implied spot rate is determined as follows:<\/p>\n<p>$$ \\begin{align*} 1&amp;=\\frac{0.0525}{1.05}+\\frac{(1.0525)}{\\left(1+r\\left(2\\right)\\right)^2} \\\\ r (2)&amp; =5.26\\% \\end{align*} $$<\/p>\n<p>We have bootstrapped the 2-year spot rate.<\/p>\n<p>Similarly, the three-year spot rate can be bootstrapped by solving the equation:<\/p>\n<p>$$ \\begin{align*} 1&amp;=\\frac{0.055}{1.05}+\\frac{0.055}{{1.0526}^2}+\\frac{1.055}{\\left(1+r\\left(3\\right)\\right)^3} \\\\ r (3)&amp; = 5.52\\% \\end{align*} $$<\/p>\n<\/blockquote>\n<p>Reading 28: The Term Structure and Interest Rate Dynamics<\/p>\n<p><em>LOS 28(b) Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping.<\/em><\/p>\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                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\n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \n<h2>Why Bootstrapping Spot Rates Matters in CFA Level II<\/h2>\n<p>Bootstrapping spot rates is an important CFA Level II Fixed Income concept. Candidates may be tested on deriving zero-coupon rates from par yields, valuing bonds with spot curves, and interpreting term structure relationships.<\/p>\n<p>To strengthen your preparation, explore the <a href=\"https:\/\/analystprep.com\/cfa-level-2\/\" target=\"_blank\" rel=\"noopener\">CFA Level II study program<\/a> with practice questions, mock exams, and guided lessons.<\/p>\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; 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><\/p>\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Practice CFA Level II fixed income questions covering spot rates, par curves, bootstrapping, and step-by-step exam-style solutions.<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Bootstrapping spot rates is a forward substitution method that allows investors to determine zero-coupon rates using the par yield curve. The par curve shows the yields to maturity on government bonds with coupon payments, priced at par, over a range&#8230;<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[102,472],"tags":[552,216,548,547],"class_list":["post-17090","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-fixed-income","tag-bootstrapping-spot-rates","tag-cfa-level-2","tag-fixed-income","tag-reading-32","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Bootstrapping Spot Rates from the Par Curve<\/title>\n<meta name=\"description\" content=\"Learn how to bootstrap spot rates from the par curve to derive zero-coupon rates and understand their role in fixed income 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