{"id":2139,"date":"2019-09-12T13:33:00","date_gmt":"2019-09-12T13:33:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=2139"},"modified":"2026-03-27T16:03:21","modified_gmt":"2026-03-27T16:03:21","slug":"forward-rates-spot-rates","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/forward-rates-spot-rates\/","title":{"rendered":"Spot Rates and Forward 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\": \"How are the 2-year and 3-year implied spot rates calculated from the forward curve?\",\n    \"text\": \"Suppose the current forward curve for 1-year rates is 0y1y = 2%, 1y1y = 3%, and 2y1y = 3.75%. The 2-year and 3-year implied spot rates are: A. 1% and 0.75%, respectively. B. 2.75% and 2%, respectively. C. 2.5% and 2.91%, respectively.\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. The 2-year implied spot rate is 2.5% and the 3-year implied spot rate is 2.91%. The 2-year spot rate is found using (1.02 \u00d7 1.03) = (1 + z2)^2, giving z2 = 2.5%. The 3-year spot rate is found using (1.02 \u00d7 1.03 \u00d7 1.0375) = (1 + z3)^3, giving z3 \u2248 2.91%.\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/forward-rates-spot-rates\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\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; 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>A forward rate is the interest rate on a loan beginning at some time in the future.&nbsp; A spot rate, on the other hand, is the interest rate on a loan beginning immediately. Therefore, the forward market rate is for future delivery after the usual settlement time in the cash market.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Forward Rates<\/strong><\/h2>\n\n\n\n<p>Forward rates on bonds or money market instruments are traded in forward markets. For instance, let\u2019s assume that in a cash market, a 4-year zero-coupon bond is priced at 85 on a par value of 100. On a semi-annual bond basis, the yield-to-maturity is 4.105%.<\/p>\n\n\n\n<p>$$85=\\frac { 100 }{ (1+r)^{ 8 } } ;\\quad r=2.052%\u00d72=4.105\\%$$<\/p>\n\n\n\n<p>The most common market practice is to name forward rates by, for instance, \u201c2y5y\u201d, which means \u201c2-year into 5-year rate\u201d. The first number refers to the length of the forward period from today while the second number refers to the tenor or time-to-maturity of the underlying bond.<\/p>\n\n\n\n<div style=\"text-align: center; margin: 24px 0;\">\n  <div style=\"max-width: 680px; margin: 0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 8px 16px;\n       border: 2px solid #2f6fed; border-radius: 999px;\n       color: #2f6fed; text-decoration: none;\n       font-size: 15px; font-weight: 500;\n       line-height: 1.2; white-space: nowrap;\">\n      Work through forward and spot rate relationships with our free trial.\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<h2><strong>Implied Forward Rates<\/strong><\/h2>\n<p>Implied forward rates (forward yields) are calculated from spot rates. The general formula for the\u00a0relationship between the two spot rates and the implied forward rate is:<\/p>\n<p>$$ (1+Z_A)^A\u00d7(1+IFR_{A,B-A} )^{B-A}=(1+Z_B )^B $$<\/p>\n<p>Where IFR<sub>A,B-A<\/sub> is the implied forward rate between time A and time B.<\/p>\n<h4><strong>Example: Computing an Implied Forward Rate<\/strong><\/h4>\n<p>Suppose the yields-to-maturity on a 3-year and 4-year zero coupon bonds are 3.5% and 4% on a semi-annual basis. The \u201c3y1y\u201d implies that the forward rate could be calculated as follows:<\/p>\n<p>A = 6 periods<\/p>\n<p>B = 8 periods<\/p>\n<p>B \u2212 A = 2 periods<\/p>\n<p>z<sub>6 <\/sub>= 0.035\/2 = \u00a00.0175<\/p>\n<p>z<sub>8 <\/sub>= 0.04\/2 = 0.02<\/p>\n<p>$$ (1+0.0175)^6\u00d7(1+IFR_{6,2} )^2=(1+0.02)^8$$<\/p>\n<p>=&gt; IFR<sub>6,2<\/sub> = 0.0275<\/p>\n<p>The \u201c3y1y\u201d implies that the forward rate or forward yield is 5.50% (0.0275% \u00d7 2).<\/p>\n<blockquote>\n<h3>Question<\/h3>\n<p>Suppose the current forward curve for 1-year rates is 0y1y=2%, 1y1y=3%, and 2y1y=3.75%. The 2-year and 3-year implied spot rates are:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li data-tadv-p=\"keep\">1% and 0.75%, respectively.<\/li>\n<li data-tadv-p=\"keep\">2.75% and 2%, respectively.<\/li>\n<li data-tadv-p=\"keep\">2.5% and 2.91%, respectively.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>The 2-year and 3-year implied spot rates are 2.5% and 2.91%, respectively.<\/p>\n<p>(1.02 \u00d7 1.03) = (1+z<sub>2<\/sub>)<sup>2<\/sup>; z<sub>2<\/sub>= 0.0250<\/p>\n<p>(1.02 \u00d7 1.03 \u00d7 1.0375) = (1+z<sub>3<\/sub>)<sup>3<\/sup>; z<sub>3<\/sub>= 0.0291<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align: center; margin: 40px 0 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display: inline-flex; align-items: center; justify-content: center;\n     padding: 10px 24px; border-radius: 999px;\n     font-size: 15px; font-weight: 600;\n     background-color: #4a72c9; color: #ffffff;\n     text-decoration: none; line-height: 1.2; white-space: nowrap;\">\n    Start Free Trial \u2192\n  <\/a>\n\n  <p style=\"margin: 14px auto 0; max-width: 680px; font-size: 15px; line-height: 1.6; color: #333333;\">\n    Strengthen your understanding of forward rates, spot curves, and term structure relationships with structured CFA Level I practice and step-by-step explanations.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A forward rate is the interest rate on a loan beginning at some time in the future.&nbsp; A spot rate, on the other hand, is the interest rate on a loan beginning immediately. Therefore, the forward market rate is for&#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-2139","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Forward Rates and Spot Rates | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"A forward rate indicates the interest rate on a loan beginning at some time in the future, whereas a spot rate is the interest rate on a loan beginning now.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/forward-rates-spot-rates\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Forward Rates and Spot Rates | CFA Level 1 - 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