{"id":46283,"date":"2023-09-06T13:40:52","date_gmt":"2023-09-06T13:40:52","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=46283"},"modified":"2026-01-23T17:45:28","modified_gmt":"2026-01-23T17:45:28","slug":"par-and-forward-rates","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/fixed-income\/par-and-forward-rates\/","title":{"rendered":"Par and Forward Rates"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"The Term Structure of Interest Rates: Spot, Par, and Forward Curves (2025 CFA\u00ae Level I Exam \u2013 Fixed Income \u2013 Learning Module 9)\",\n  \"description\": \"This lesson covers the term structure of interest rates for the 2025 CFA\u00ae Level I Fixed Income curriculum. It explains spot rates and the spot curve and shows how to price bonds using spot rates. The video also defines par and forward rates, demonstrates how to calculate par rates and forward rates from spot rates and vice versa, and compares the spot, par, and forward curves using exam-focused examples.\",\n  \"uploadDate\": \"2023-11-23T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/6C48PLyxNaU\/default.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/6C48PLyxNaU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/6C48PLyxNaU\",\n  \"duration\": \"PT23M19S\"\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\": \"Given a three-year spot rate of 3.5% and a four-year spot rate of 4%, what is the one-year forward rate three years from now (3y1y)?\",\n    \"text\": \"Given a three-year spot rate of 3.5% and a four-year spot rate of 4%, what is the one-year forward rate three years from now (3y1y)?\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The one-year forward rate three years from now (3y1y) is 5.515%. Using the forward rate relationship: (1 + Z3)^3 \u00d7 (1 + f3,1) = (1 + Z4)^4. Substituting the values: (1 + 0.035)^3 \u00d7 (1 + f3,1) = (1 + 0.04)^4. Solving for f3,1 gives 5.515%.\",\n      \"dateCreated\": \"2025-12-17\",\n      \"confidence\": 0.74\n    }\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-1536x763.jpg\",\n  \"caption\": \"Relationship between par rates and forward rates\",\n  \"width\": 1536,\n  \"height\": 763,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/6C48PLyxNaU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Par Rates<\/h2>\n<p>A par rate is the yield-to-maturity that equates the present value of a bond&#8217;s cash flows to its par value (typically <span class=\"math inline\">\\(100\\%\\)<\/span> of face value). Spot rates play a pivotal role in determining par rates. For a bond to be priced at par, its coupon rate and yield-to-maturity must be identical. This is depicted in the equation:<\/p>\n<p><span class=\"math display\">\\[100 = \\frac{PMT}{\\left( 1 + z_{1} \\right)^{1}} + \\frac{PMT}{\\left( 1 + z_{2} \\right)^{2}} + \\ldots + \\frac{PMT + 100}{\\left( 1 + z_{N} \\right)^{N}}\\]<\/span><\/p>\n<p>Here, PMT represents the periodic payment, and <span class=\"math inline\">\\(z_{1},z_{2},\\ldots,z_{N}\\)<\/span> are the sequence of spot rates for respective periods. By solving for <span class=\"math inline\">\\(PMT\\)<\/span>, we obtain the yield-to-maturity that would make the bond trade at par, which is the par rate.<\/p>\n<h4 id=\"example-calculating-par-rate-given-spot-rates\">Example: Calculating Par Rate Given Spot Rates<\/h4>\n<p>Given the following spot rates for government bonds. Note that these are effective annual rates:<\/p>\n<p>$$\\begin{array}{l|c} \\textbf{Term} &amp; \\textbf{Spot Rate} \\\\ \\hline 1-Year &amp; 4.50\\% \\\\ 2-Year &amp; 4.90\\% \\\\ 3-Year &amp; 5.25\\% \\\\\u00a0 \\end{array}$$<\/p>\n<p><strong>One-Year Par Rate<\/strong><\/p>\n<p>Given the spot rate for one year is <span class=\"math inline\">\\(4.50\\%\\)<\/span>, the one-year par rate will also be <span class=\"math inline\">\\(4.50\\%\\)<\/span>.<\/p>\n<p><span class=\"math display\">\\[100 = \\frac{PMT + 100}{(1.0450)^{1}}\\]<\/span><\/p>\n<p><strong>Two-Year Par Rate:<\/strong><\/p>\n<p>Using the spot rates for one year and two years, we can derive the two-year par rate:<\/p>\n<p><span class=\"math display\">\\[100 = \\frac{PMT}{(1.0450)^{1}} + \\frac{PMT + 100}{(1.0490)^{2}}\\]<\/span><\/p>\n<p>PMT has been calculated as 4.8904, which translates to a two-year par rate of 4.8904%.<\/p>\n<p><strong>Three-Year Par Rate<\/strong><\/p>\n<p>Using the spot rates for one, two, and three years:<\/p>\n<p><span class=\"math display\">\\[100 = \\frac{PMT}{(1.0450)^{1}} + \\frac{PMT}{(1.0490)^{2}} + \\frac{PMT + 100}{(1.0525)^{3}}\\]<\/span><\/p>\n<p>PMT has been calculated as 5.2252, which translates to a three-year par rate of 5.2252%.<\/p>\n<h2 id=\"forward-rates\">Forward Rates<\/h2>\n<p>Forward rates, often termed implied forward rates or forward yields, act as breakeven reinvestment rates. They establish a connection between the return on an investment in a shorter-term zero-coupon bond to the return on a longer-term zero-coupon bond. 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<p>Forward rates can be derived from spot rates and vice versa. The general formula to compute an implied forward rate between two periods is:<\/p>\n<p><span class=\"math display\">\\[\\left( 1 + Z_{A} \\right)^{A} \\times \\left( 1 + IFR_{(A,B &#8211; A)} \\right)^{(B &#8211; A)} = \\left( 1 + Z_{B} \\right)^{B}\\]<\/span><\/p>\n<p>Here, <span class=\"math inline\">\\(IFR_{(A,B &#8211; A)}\\)<\/span> denotes the implied forward rate for a bond beginning at time <span class=\"math inline\">\\(t = A\\)<\/span> and maturing at t=B. <span class=\"math inline\">\\(Z_{A}\\)<\/span> and <span class=\"math inline\">\\(Z_{B}\\)<\/span> represent spot rates for periods <span class=\"math inline\">\\(A\\)<\/span> and <span class=\"math inline\">\\(B\\)<\/span>, respectively.<\/p>\n<p>The following figure demonstrates the implied forward rates:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"790\" class=\"alignnone wp-image-47661 size-full\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-300x149.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-1024x509.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-768x382.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-1536x763.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/Img_5-4-400x199.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<ul>\n<li>The blue segment (from <span class=\"math inline\">\\(t = 0\\)<\/span> to <span class=\"math inline\">\\(t = A\\)<\/span> ) represents investing at the spot rate <span class=\"math inline\">\\(Z_{A}\\)<\/span> for <span class=\"math inline\">\\(A\\)<\/span> periods.<\/li>\n<li>The green segment (from <span class=\"math inline\">\\(t = A\\)<\/span> to <span class=\"math inline\">\\(t = B\\)<\/span> ) represents investing at the implied forward rate <span class=\"math inline\">\\(IFR_{(A,B &#8211; A)}\\)<\/span> for <span class=\"math inline\">\\(B &#8211; A\\)<\/span> periods.<\/li>\n<li>The red segment at the bottom (spanning the entire timeline) signifies that the compounded return of the above two rates is equivalent to investing at the spot rate <span class=\"math inline\">\\(Z_{B}\\)<\/span> for the entire <span class=\"math inline\">\\(B\\)<\/span> periods.<\/li>\n<\/ul>\n<p>Forward rates are pivotal for investors and analysts as they provide insights into market expectations of future interest rate movements. They serve as breakeven rates, implying that if an investor&#8217;s expectations align with the forward rate, they would be indifferent between investing in a longer-term bond now or investing in shorter-term bonds successively.<\/p>\n<h4 id=\"example-calculating-implied-forward-rates-from-spot-rates\">Example: Calculating Implied Forward Rates from Spot Rates<\/h4>\n<p>Suppose that the yields-to-maturity on five-year and seven-year zero-coupon bonds are 4.85% and 5.45%, respectively, stated on a quarterly bond basis. An analyst wants to know the &#8220;5y2y&#8221; implied forward rate, which is the implied two-year forward yield five years into the future.<\/p>\n<p>Given:<\/p>\n<ul>\n<li><span class=\"math inline\">\\(A = 20\\)<\/span> (periods, since 5 years <span class=\"math inline\">\\(\\times 4\\)<\/span> quarters per year)<\/li>\n<li><span class=\"math inline\">\\(B = 28\\)<\/span> (periods, since 7 years <span class=\"math inline\">\\(\\times 4\\)<\/span> quarters per year)<\/li>\n<li><span class=\"math inline\">\\(B &#8211; A = 8\\)<\/span> (periods)<\/li>\n<li><span class=\"math inline\">\\(z_{20} = \\frac{0.0485}{4}\\)<\/span> (per period)<\/li>\n<li><span class=\"math inline\">\\(z_{28} = \\frac{0.0545}{4}\\)<\/span> (per period)<\/li>\n<\/ul>\n<p>Let&#8217;s solve for <span class=\"math inline\">\\(IFR_{20,8}\\)<\/span> (the implied forward rate from period 20 to period 28).<\/p>\n<p>Formula:<\/p>\n<p><span class=\"math display\">\\[\\left( 1 + z_{20} \\right)^{20} \\times \\left( 1 + IFR_{20,8} \\right)^{8} = \\left( 1 + z_{28} \\right)^{28}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\left( 1 + \\frac{0.0485}{4} \\right)^{20} \\times \\left( 1 + IFR_{20,8} \\right)^{8} = \\left( 1 + \\frac{0.0545}{4} \\right)^{28}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[\\left( 1 + \\frac{0.0485}{4} \\right)^{20} \\times \\left( 1 + IFR_{20,8} \\right)^{8} = \\left( 1 + \\frac{0.0545}{4} \\right)^{28}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[IFR_{20,8} = \\ 1.738\\%\\]<\/span><\/p>\n<p>The &#8220;5y2y&#8221; implied forward rate is approximately 1.738% on a quarterly basis. Annualized, the &#8220;5y2y&#8221; implied forward yield is 7.13%.<\/p>\n<h2 id=\"calculating-spot-rates-from-forward-rates\">Calculating Spot Rates from Forward Rates<\/h2>\n<p>Suppose the current forward curve for one-year rates is as follows:<\/p>\n<p>$$\\begin{array}{|c|c|} \\hline \\textbf{Time Period} &amp; \\textbf{Forward Rate} \\\\ \\hline 0y1y &amp; 1.50\\% \\\\ 1y1y &amp; 2.20\\% \\\\ 2y1y &amp; 2.80\\% \\\\ \\hline \\end{array}$$<\/p>\n<p>The provided rates are expressed on an annual basis with a period of one year, making them effective yearly rates. The initial rate, termed &#8220;0y1y,&#8221; represents the spot rate for one year. The subsequent rates are forward rates for one-year durations. Using these rates, the spot curve can be determined by taking the geometric average of the forward rates. The two-year implied spot rate can be calculated as:<\/p>\n<p><span class=\"math display\">\\[(1.0150 \\times 1.0220) = \\left( 1 + z_{2} \\right)^{2}\\]<\/span><\/p>\n<p>where <span class=\"math inline\">\\(z_{2}\\)<\/span> is the two-year implied spot rate.<\/p>\n<p><span class=\"math display\">\\[z_{2} = 1.84940\\%\\]<\/span><\/p>\n<p>Using this, and the 2y1y forward rate, we can then determine the three-year implied spot rate:<\/p>\n<p><span class=\"math display\">\\[(1.0150 \\times 1.0220 \\times 1.0280) = \\left( 1 + z_{3} \\right)^{3}\\]<\/span><\/p>\n<p>where <span class=\"math inline\">\\(z_{3}\\)<\/span> is the three-year implied spot rate.<\/p>\n<p><span class=\"math display\">\\[z_{3} = 2.16529\\%\\]<\/span><\/p>\n<p>Suppose an analyst needs to value a three-year, <span class=\"math inline\">\\(2.50\\%\\)<\/span> annual coupon payment bond that has the same risks as the bonds used to obtain the forward curve. Using the implied spot rates, we can determine the value of the bond.<\/p>\n<p><span class=\"math display\">\\[PV = \\frac{PMT}{\\left( 1 + z_{1} \\right)^{1}} + \\frac{PMT}{\\left( 1 + z_{2} \\right)^{2}} + \\ldots + \\frac{PMT + 100}{\\left( 1 + z_{N} \\right)^{N}}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[PV = \\frac{2.5}{(1.0150)^{1}} + \\frac{2.5}{(1.084940)^{2}} + \\frac{2.5 + 100}{(1.0216529)^{3}}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[PV = 100.993\\ \\]<\/span><\/p>\n<p>This bond can also be valued using the forward rates and generate the same result.<\/p>\n<h2 id=\"bond-pricing-using-forward-rates\">Bond Pricing Using Forward Rates<\/h2>\n<p>Bonds can also be valued using forward rates. The bond&#8217;s future cash flows are discounted at the product of the sequence of one-year forward rates leading up to each cash flow. The summation of these discounted cash flows gives the bond&#8217;s price. The bond price remains consistent regardless of whether spot or forward rates are used.<\/p>\n<h4 id=\"example-bond-pricing-using-forward-rates\">Example: Bond Pricing Using Forward Rates<\/h4>\n<p>Suppose the current forward curve for one-year rates is as follows:<\/p>\n<p>$$\\begin{array}{|c|c|} \\hline \\textbf{Time Period} &amp; \\textbf{Forward Rate} \\\\ \\hline 0y1y &amp; 1.50\\% \\\\ 1y1y &amp; 2.20\\% \\\\ 2y1y &amp; 2.80\\% \\\\ \\hline \\end{array}$$<\/p>\n<p>Suppose an analyst needs to value a three-year, <span class=\"math inline\">\\(2.50\\%\\)<\/span> annual coupon payment bond that has the same risks as the bonds used to obtain the forward curve. Determine the value of the bond using the forward rates above.<\/p>\n<p><span class=\"math display\">\\[PV = \\frac{2.5}{(1.0150)^{1}} + \\frac{2.5}{(1.0150 \\times 1.0220)} + \\frac{2.5 + 100}{(1.0150 \\times 1.0220 \\times 1.0280)}\\]<\/span><\/p>\n<p><span class=\"math display\">\\[PV = 100.993\\ \\]<\/span><\/p>\n<blockquote>\n<h3 id=\"question-1\">Question<\/h3>\n<p>Given a three-year spot rate of 3.5% and a four-year spot rate of <span class=\"math inline\">\\(4\\%\\)<\/span>, what is the one-year forward rate three years from now (3yly)?<\/p>\n<ol style=\"list-style-type: upper-alpha; text-align: left;\">\n<li>2.720%<\/li>\n<li>3.75%<\/li>\n<li>5.515%<\/li>\n<\/ol>\n<p id=\"solution\">Solution<\/p>\n<p>The correct answer is<strong> C.<\/strong><\/p>\n<p>Using the formula:<\/p>\n<p><span class=\"math display\">\\[\\left( 1 + Z_{A} \\right)^{A} \\times \\left( 1 + IFR_{A,B &#8211; A} \\right)^{B &#8211; A} = \\left( 1 + Z_{B} \\right)^{B}\\]<\/span><\/p>\n<p>Where <span class=\"math inline\">\\(A = 3,B = 4,Z_{3} = 3.5\\%\\)<\/span>, and <span class=\"math inline\">\\(Z_{4} = 4\\)<\/span>%<\/p>\n<p><span class=\"math display\">\\[(1 + 0.035)^{3} \\times \\left( 1 + IFR_{3,1} \\right)^{1} = (1 + 0.04)^{4}\\]<\/span><\/p>\n<p>Solving for <span class=\"math inline\">\\(IFR_{3,1}\\)<\/span> will give 5.515%<\/p>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Par Rates A par rate is the yield-to-maturity that equates the present value of a bond&#8217;s cash flows to its par value (typically \\(100\\%\\) of face value). Spot rates play a pivotal role in determining par rates. For a bond&#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-46283","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>Par and Forward Rates - AnalystPrep | CFA\u00ae Exam Study Notes<\/title>\n<meta name=\"description\" content=\"Understand the application of forward rates in valuing bonds and how to determine spot rates from forward rates.\" \/>\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\/par-and-forward-rates\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Par and Forward Rates - 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