{"id":16971,"date":"2021-07-05T16:33:57","date_gmt":"2021-07-05T16:33:57","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=16971"},"modified":"2026-01-05T07:02:42","modified_gmt":"2026-01-05T07:02:42","slug":"describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/","title":{"rendered":"Spot Rates and Forward Rates"},"content":{"rendered":"<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/ovQh3D6WBOw\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-1024x827.png\",\r\n  \"caption\": \"Yield Curve (Flat)\",\r\n  \"width\": 1024,\r\n  \"height\": 827,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-1024x735.jpg\",\r\n  \"caption\": \"Downward Sloping Yield Curve\",\r\n  \"width\": 1024,\r\n  \"height\": 735,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-1024x841.png\",\r\n  \"caption\": \"Yield Curve (Upward)\",\r\n  \"width\": 1024,\r\n  \"height\": 841,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-1024x822.jpg\",\r\n  \"caption\": \"Spot vs. Forward Rates\",\r\n  \"width\": 1024,\r\n  \"height\": 822,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline.jpg\",\r\n  \"caption\": \"Forward Rates Timeline\",\r\n  \"width\": 1668,\r\n  \"height\": 850,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"ImageObject\",\r\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg\",\r\n  \"caption\": \"Upward Sloping Spot Curve\",\r\n  \"width\": 1590,\r\n  \"height\": 1160,\r\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\r\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\r\n  \"creditText\": \"AnalystPrep Design Team\",\r\n  \"creator\": {\r\n    \"@type\": \"Organization\",\r\n    \"name\": \"AnalystPrep\"\r\n  }\r\n}\r\n<\/script>\r\n\r\n<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"QAPage\",\r\n  \"mainEntity\": {\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Yield to maturity of a 5-year zero-coupon bond\",\r\n    \"text\": \"Cream Trust Bank has issued a zero-coupon bond with a face value of $50,000 that matures in 5 years. If the bond is priced at $34,029, the yield to maturity (YTM) of the bond is closest to:\\n\\nA. 8%.\\n\\nB. 9%.\\n\\nC. 47%.\",\r\n    \"answerCount\": 1,\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"A. 8%.\\n\\nFor a zero-coupon bond: Price = Face Value \/ (1 + YTM)^T.\\n\\n34,029 = 50,000 \/ (1 + YTM)^5\\n\\n(1 + YTM)^5 = 50,000 \/ 34,029 \u2248 1.4696\\n\\n1 + YTM \u2248 (1.4696)^(1\/5) \u2248 1.08\\n\\nYTM \u2248 0.08 = 8%.\\n\\nNote: For a zero-coupon bond, the YTM equals the corresponding spot rate for that maturity.\"\r\n    }\r\n  }\r\n}\r\n<\/script>\r\n\r\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\r\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\r\n<p>This reading will establish how interest rates and prices of bonds for different maturities are related.<\/p>\r\n<h2>Spot Rates<\/h2>\r\n<p>The zero-coupon bond (discount bond) is the basic debt security that pays one unit of currency, e.g., $1, at maturity. The price \\(P(T)\\) of a discount bond, purchased today and maturing at time \\(T\\), given a constant interest rate, \\({S}_{T}\\) is given by:<\/p>\r\n<p>$$ P\\left(T\\right)=\\frac{1}{\\left(1+S\\left(T\\right)\\right)^T} $$<\/p>\r\n<p>\\(P(T)\\) is also referred to as the discount factor.<\/p>\r\n<p>\\(S(T)\\) is known as the <em><strong>spot rate<\/strong><\/em>, defined as the yield to maturity of payment of a single unit of currency at maturity, \\(T\\).<\/p>\r\n<p>A <em><strong>spot curve<\/strong><\/em> shows the relationship between the spot rates at different maturities. The following figure shows an upward sloping spot curve.<\/p>\r\n<h4><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26479\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg\" alt=\"Upward Sloping Spot Curve\" width=\"1590\" height=\"1160\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve-300x219.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve-1024x747.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve-768x560.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve-1536x1121.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve-400x292.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Example: Spot Rates<\/h4>\r\n<p>Consider a two-year zero-coupon bond with a face value of $100. If the interest rate over the investment horizon is 10%, the price of the bond is <em>closest<\/em> to:<\/p>\r\n<h4>Solution<\/h4>\r\n<p>$$ P\\left(T\\right)=\\frac{1}{\\left(1+S\\left(T\\right)\\right)^T} $$<\/p>\r\n<p>The 10% interest rate is an example of a spot rate.<\/p>\r\n<p>$$ \\begin{align*} P\\left(2\\right) &amp;=\\frac{$100}{\\left(1+0.10\\right)^2} \\\\ P\\left(2\\right) &amp;=$82.64 \\end{align*} $$<\/p>\r\n<h2>Forward Rates<\/h2>\r\n<p>The forward rate \\(f(0, t, T)\\) is the annualized interest rate payable on a loan, which is agreed upon today, starting at time \\(t\\), to be repaid at maturity \\(T\\).<\/p>\r\n<p>In this case:<\/p>\r\n<ul>\r\n\t<li>\\(0\\) is the time at which the <em><strong>forward rate agreement<\/strong><\/em> is entered.<\/li>\r\n\t<li>\\(t\\) is the start time of the forward rate.<\/li>\r\n\t<li>\\(T\\) is the maturity of the agreement.<\/li>\r\n<\/ul>\r\n<p>This can be shown in the following timeline.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26478\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline.jpg\" alt=\"Forward Rates - Timeline\" width=\"1668\" height=\"850\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline.jpg 1668w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline-300x153.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline-1024x522.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline-768x391.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline-1536x783.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Forward-Rates-Timeline-400x204.jpg 400w\" sizes=\"auto, (max-width: 1668px) 100vw, 1668px\" \/>\\(F(0,t,T)\\) is the forward price (discount factor) of a zero-coupon bond, agreed upon at time 0, filled at time \\(t\\), and maturing at time \\(T\\). It is expressed mathematically as:<\/p>\r\n<p>$$ F\\left(0, t, T\\right)=\\frac{1}{\\left[1+f\\left(0,t,T\\right)\\right]^{T-t}} $$<\/p>\r\n<p>This can also be written as:<\/p>\r\n<p>$$ F\\left(t,T-t\\right)=\\frac{1}{\\left[1+f\\left(t,T-t\\right)\\right]^{T-t}} $$<\/p>\r\n<p>A <em><strong>forward curve<\/strong><\/em> is a graph showing the relationship between the forward rates and the related terms to maturity. The following figure illustrates both the forward curve and the spot curve.<\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26480\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates.jpg\" alt=\"Spot vs. Forward Rates\" width=\"1590\" height=\"1276\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-300x241.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-1024x822.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-768x616.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-1536x1233.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Spot-vs.-Forward-Rates-400x321.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Notice that the forward curve lies above the spot curve for an upward sloping spot curve. Conversely, the forward curve will lie below the spot curve for a downward sloping spot curve. This will be discussed later in the forward rate model.<\/p>\r\n<h2>Yield to Maturity<\/h2>\r\n<p>The <em><strong>yield to maturity<\/strong><\/em> (YTM) is the discount rate that equates the present value of future bond payments (includes coupons and the par value) to the bond\u2019s market price.<\/p>\r\n<p>In other words, YTM is the expected rate of return on a bond if:<\/p>\r\n<ol type=\"i\">\r\n\t<li>The bond is held to maturity.<\/li>\r\n\t<li>The bold does not default.<\/li>\r\n\t<li>Reinvestment of the bond and all coupons is executed at the original yield to maturity.<\/li>\r\n<\/ol>\r\n<p>If the spot rate curve of a coupon-paying bond is flat, the YTM will be the same as the spot rate. Also, note that the YTM of a zero-coupon bond is equal to the spot rate.<\/p>\r\n<h4>Example: Spot Rates and Yield to Maturity<\/h4>\r\n<p>ABC Ltd. has issued a bond with a face value of $500, which carries an annual coupon of 10% and matures in 4 years. The spot rate curve is given in the following table.<\/p>\r\n<p>$$ \\begin{array}{c|c} \\textbf{Year} &amp; \\textbf{Spot rate, S(t)} \\\\ \\hline 1 &amp; 10\\% \\\\ \\hline 2 &amp; 12\\% \\\\ \\hline 3 &amp; 14\\% \\\\ \\hline 4 &amp; 16\\% \\end{array} $$<\/p>\r\n<ol type=\"1\">\r\n\t<li>Calculate the price of the bond.<\/li>\r\n\t<li>Calculate the yield to maturity of the bond.<\/li>\r\n<\/ol>\r\n<h4>Solution 1: Price of the Bond<\/h4>\r\n<p>$$ \\text{Coupon payment} = $500\\times10\\%=$50 $$<\/p>\r\n<p>$$ \\text{Price} =\\frac{$50}{\\left(1+0.10\\right)^1}+\\frac{$50}{\\left(1+0.12\\right)^2}+\\frac{$50}{\\left(1+0.14\\right)^3}+\\frac{$550}{\\left(1+0.16\\right)^4}=$422.82 $$<\/p>\r\n<h4>Solution 2: Yield to Maturity<\/h4>\r\n<p>We can compute the YTM by solving for \\(Y\\) in the following equation:<\/p>\r\n<p>$$ \\begin{align*} 422.82 &amp; =\\frac{50}{\\left(1+Y\\right)^1}+\\frac{50}{\\left(1+Y\\right)^2}+\\frac{50}{\\left(1+Y\\right)^3}+\\frac{550}{\\left(1+Y\\right)^4} \\\\ Y &amp;=15.46\\% \\end{align*} $$<\/p>\r\n<p>Using a financial calculator with the following inputs results in the same YTM:<\/p>\r\n<ul>\r\n\t<li>N = 4.<\/li>\r\n\t<li>PV = -422.82 (since this is a cash outflow).<\/li>\r\n\t<li>PMT = 50 (since this is a cash inflow for the investor).<\/li>\r\n\t<li>FV = 500 (since this is a cash inflow for the investor).<\/li>\r\n\t<li>CPT \\(\\Rightarrow\\) I\/Y = 15.46 (which signifies 15.46%).<\/li>\r\n<\/ul>\r\n<h2>Expected and Realized Returns on Bonds<\/h2>\r\n<h3>Expected Return on a Bond<\/h3>\r\n<p>The expected return is the return that a bondholder anticipates earning over the investment holding period. Generally, the YTM is not the same as the expected return.<\/p>\r\n<p>The expected return will be equal to the YTM of a bond if and only if:<\/p>\r\n<ol type=\"i\">\r\n\t<li>The bond is held to maturity.<\/li>\r\n\t<li>All coupons and principal amounts are paid fully and on time.<\/li>\r\n\t<li>The coupons are reinvested at the original YTM.<\/li>\r\n<\/ol>\r\n<p>The YTM is a poor estimate of the expected return in cases where:<\/p>\r\n<ol type=\"i\">\r\n\t<li>The interest rates are volatile.<\/li>\r\n\t<li>The yield curve is steeply sloping upwards or downwards. For cases (i) and (ii), we do not expect the reinvestment of coupons to be at the YTM.<\/li>\r\n\t<li>There is a great risk of default. This implies that there might be a difference between the actual cash flows and those assumed in the YTM calculation.<\/li>\r\n\t<li>The bond has at least one embedded option. The exercise of such an option would lead to a shorter holding period relative to the bond\u2019s original maturity.<\/li>\r\n<\/ol>\r\n<h3>Realized Return on a Bond<\/h3>\r\n<p>The <strong>realized return<\/strong> refers to the actual return earned on a bond during its holding period.<\/p>\r\n<h2>The Shape of the Yield Curve<\/h2>\r\n<p>The yield curve is attained by plotting the spot rate \\(S(T)\\) against maturity, \\(T\\). It gives information on the term structure of interest rates.<\/p>\r\n<h3>Upward Sloping Yield Curve<\/h3>\r\n<p>The term structure of interest rates suggests that the yield curve is typically <em><strong>upward sloping<\/strong><\/em>. This means that a bond with a longer-term maturity has a higher yield relative to a shorter-term bond.<\/p>\r\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-21180\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve.png\" alt=\"\" width=\"1614\" height=\"1326\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve.png 1614w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-300x246.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-1024x841.png 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-768x631.png 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-1536x1262.png 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-upward-sloping-yield-curve-400x329.png 400w\" sizes=\"auto, (max-width: 1614px) 100vw, 1614px\" \/>Downward Sloping Yield Curve<\/h3>\r\n<p>In some cases, the yield curve can be <em><strong>downward sloping<\/strong><\/em> (inverted). This implies that the yield on short-term bonds is higher than the yield on longer-term bonds. This can be the case on occasions where the government policy has increased the current interest rates.<\/p>\r\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26482\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve.jpg\" alt=\"Downward Sloping Yield Curve\" width=\"1590\" height=\"1142\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-300x215.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-1024x735.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-768x552.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-1536x1103.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Downward-Sloping-Yield-Curve-400x287.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Flattened Yield Curve<\/h3>\r\n<p>Finally, the yield curve can be <em><strong>flat<\/strong><\/em>. This implies that the yield on a long-term bond is equal to the yield on a short-term bond. This may be due to market participants expecting a decrease in inflation.<\/p>\r\n<h2><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-21181\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve.png\" alt=\"\" width=\"1561\" height=\"1260\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve.png 1561w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-300x242.png 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-1024x827.png 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-768x620.png 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-1536x1240.png 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/cfa-level-2-flattened-curve-400x323.png 400w\" sizes=\"auto, (max-width: 1561px) 100vw, 1561px\" \/><\/h2>\r\n<blockquote>\r\n<h2>Question<\/h2>\r\n<p>Cream Trust bank has issued a zero-coupon bond with a face value of $50,000. The bond matures in 5 years.\u00a0The yield to maturity of the bond given that the bond is priced at $34,029 is <em>closest<\/em> to:<\/p>\r\n<ol style=\"list-style-type: upper-alpha;\" type=\"A\">\r\n\t<li>8%.<\/li>\r\n\t<li>9%.<\/li>\r\n\t<li>47%.<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<p><strong>The correct answer is A.<\/strong><\/p>\r\n<p>$$ \\begin{align*} P\\left(T\\right)&amp;=\\frac{1}{\\left(1+S\\left(T\\right)\\right)^T}\u00a0\\\\ \\\\ 34,029 &amp;=\\frac{50,000}{\\left(1+S_5\\right)^5} \\\\ \\\\ S_5 &amp;=8\\% \\end{align*} $$<\/p>\r\n<p>N\/B: The YTM of a zero-coupon bond is equal to the spot rate.<\/p>\r\n<\/blockquote>\r\n<p>Reading 28: The Term Structure and Interest Rate Dynamics<\/p>\r\n<p><em> LOS 28 (a) Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve.<\/em><\/p>\r\n\n            <div \n                class=\"elfsight-widget-pricing-table elfsight-widget\" \n                data-elfsight-pricing-table-options=\"%7B%22layout%22%3A%22grid%22%2C%22skin%22%3A%22skin5%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22styleColumnBorderRadius%22%3Atrue%2C%22styleColumnBorderWidth%22%3Atrue%2C%22useHorizontalScroll%22%3Afalse%2C%22headTitle%22%3A%22Plan%20name%22%2C%22headFeatures%22%3A%5B%7B%22text%22%3A%22Feature%201%22%7D%2C%7B%22text%22%3A%22Feature%202%22%7D%5D%2C%22headTextColor%22%3A%22rgb%2823%2C%2025%2C%2026%29%22%2C%22headBackgroundColor%22%3A%22rgb%28247%2C%20247%2C%20247%29%22%2C%22toggleVisible%22%3Afalse%2C%22toggleItems%22%3A%5B%5D%2C%22toggleDefaultItem%22%3A%22%22%2C%22toggleColor%22%3A%22%22%2C%22columns%22%3A%5B%7B%22title%22%3A%22Practice%20Package%22%2C%22titleCaption%22%3A%22for%20Level%20II%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22%22%7D%2C%7B%22text%22%3A%22%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22%22%7D%2C%7B%22text%22%3A%22Question%20Bank%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Vignette-style%20practice%20questions%20organized%20by%20topic%22%7D%2C%7B%22text%22%3A%22Mock%20Exams%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%222%20a.m.%20and%202%20p.m.%20mock%20exams%22%7D%2C%7B%22text%22%3A%22Performance%20Tracking%20Tools%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%225%26nbsp%3BAsk-a-tutor%20Questions%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20five%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22249%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Fpractice-package-for-frm-part-ii%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Afalse%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20a%2012-month%20access%22%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-1.jpg%22%2C%22buttonTargetBlank%22%3Afalse%2C%22ribbonText%22%3A%22%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%2C%7B%22title%22%3A%22Learn%20%2B%20Practice%20Package%22%2C%22titleCaption%22%3A%22for%20Level%20II%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22%3Cb%3EVideo%20Lesson%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%2250%20hours%20of%20video%20lessons%20by%20Prof.%20James%20Forjan%2C%20CFA%2C%20PhD%22%7D%2C%7B%22text%22%3A%22%3Cb%3EStudy%20Notes%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%201%2C500%20pages%20worth%20of%20study%20notes%20covering%20each%20learning%20objective%20one-by-one%22%7D%2C%7B%22text%22%3A%22%3Cb%3EQuestion%20Bank%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Vignette-style%20practice%20questions%20organized%20by%20topic%22%7D%2C%7B%22text%22%3A%22%3Cb%3EPrintable%20Mock%20Exams%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%222%20a.m.%20and%202%20p.m.%20mock%20exams%22%7D%2C%7B%22text%22%3A%22%3Cb%3EPerformance%20Tracking%20Tools%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%22%3Cb%3E5%26nbsp%3BAsk-a-tutor%20Questions%3C%5C%2Fb%3E%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20five%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22399%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Fpractice-package-for-frm-part-ii-2%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Atrue%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20a%2012-month%20access%22%2C%22buttonTargetBlank%22%3Afalse%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-2.jpg%22%2C%22ribbonText%22%3A%22Most%20Popular%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%2C%7B%22title%22%3A%22Unlimited%20Package%22%2C%22titleCaption%22%3A%22for%20all%20three%20Levels%20of%20the%20CFA%5Cu00ae%20Exam%20offered%20by%20AnalystPrep%22%2C%22features%22%3A%5B%7B%22text%22%3A%22Video%20Lessons%20for%20All%20Three%20Levels%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%20100%20hours%20of%20video%20lessons%20by%20Prof.%20James%20Forjan%2C%20CFA%2C%20PhD%22%7D%2C%7B%22text%22%3A%22Study%20Notes%20for%20All%20Three%20Levels%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Study%20notes%20for%20each%20curriculum%20reading%2C%20organized%20by%20learning%20objective%22%7D%2C%7B%22text%22%3A%22Question%20Banks%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22All%20of%20our%20question%20banks%20are%20updated%20each%20year%20automatically%20on%20the%20platform%20to%20reflect%20curriculum%20updates.%20Those%20include%20vignette-type%20questions%20for%20level%20II%20and%20III%20and%20essay-type%20questions%20for%20the%20level%20III%20a.m.%20session.%22%7D%2C%7B%22text%22%3A%22Mock%20Exams%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Over%2010%20mock%20exams%20for%20all%20three%20levels.%20Also%20available%20in%20Printable%20PDFs.%22%7D%2C%7B%22text%22%3A%22Performance%20Tracking%20Tools%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22View%20your%20performance%20in%20attractive%20charts%22%7D%2C%7B%22text%22%3A%22Unlimited%26nbsp%3BAsk-a-tutor%20Questions%22%2C%22icon%22%3A%22none%22%2C%22hint%22%3A%22Ask%20questions%20to%20our%20tutors%20via%20live%20chat%22%7D%5D%2C%22price%22%3A%22699%22%2C%22priceCurrency%22%3A%22USD%22%2C%22button%22%3A%22Buy%20Now%22%2C%22buttonLink%22%3A%22https%3A%5C%2F%5C%2Fanalystprep.com%5C%2Fshop%5C%2Funlimited-package-for-frm-part-i-part-ii%5C%2F%22%2C%22buttonCaption%22%3A%22%22%2C%22mainColor%22%3A%22rgb%2851%2C%20129%2C%20234%29%22%2C%22priceOptions%22%3A%5B%5D%2C%22buttonOptions%22%3A%5B%5D%2C%22isFeatured%22%3Afalse%2C%22contentDivider%22%3Anull%2C%22pricePrefix%22%3A%22%22%2C%22pricePostfix%22%3A%22%22%2C%22priceCaption%22%3A%22for%20lifetime%20access%22%2C%22buttonTargetBlank%22%3Afalse%2C%22picture%22%3A%22https%3A%5C%2F%5C%2Felfsight.com%5C%2Fassets%5C%2Fpricing-table%5C%2Fexample-grid-3.jpg%22%2C%22ribbonText%22%3A%22%22%2C%22ribbonBackgroundColor%22%3A%22%22%2C%22ribbonTextColor%22%3A%22%22%7D%5D%2C%22elements%22%3A%5B%7B%22complexGroup%22%3A%22picture%22%2C%22name%22%3A%22Picture%22%2C%22pictureEnable%22%3Atrue%7D%2C%7B%22complexGroup%22%3A%22title%22%2C%22name%22%3A%22Title%22%2C%22titleEnable%22%3Atrue%2C%22titleCaptionColor%22%3A%22%22%2C%22titleTextColor%22%3A%22%22%2C%22titleFontSize%22%3A24%2C%22titleFontWeight%22%3A400%7D%2C%7B%22complexGroup%22%3A%22features%22%2C%22name%22%3A%22Features%22%2C%22featuresEnable%22%3Atrue%2C%22featuresStyle%22%3A%22striped%22%2C%22featuresIconColor%22%3A%22%22%2C%22featuresTextColor%22%3A%22%22%2C%22featuresAlign%22%3A%22center%22%2C%22featuresFontSize%22%3A13%7D%2C%7B%22complexGroup%22%3A%22price%22%2C%22name%22%3A%22Price%22%2C%22priceEnable%22%3Atrue%2C%22priceCaptionColor%22%3A%22%22%2C%22priceTextColor%22%3A%22%22%2C%22priceFontSize%22%3A32%2C%22priceFontWeight%22%3A600%7D%2C%7B%22complexGroup%22%3A%22button%22%2C%22name%22%3A%22Button%22%2C%22buttonEnable%22%3Atrue%2C%22buttonType%22%3A%22outline%22%2C%22buttonColor%22%3A%22%22%2C%22buttonTextColor%22%3A%22%22%2C%22buttonSize%22%3A%22medium%22%2C%22buttonCaptionColor%22%3A%22%22%7D%5D%2C%22widgetId%22%3A%223%22%7D\" \n                data-elfsight-pricing-table-version=\"2.6.1\"\n                data-elfsight-widget-id=\"elfsight-pricing-table-3\">\n            <\/div>\n            \r\n\r\n\r\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\r\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\u00a0<\/div><\/div>\r\n<\/div><\/div>\r\n<\/div><\/div>\r\n<\/div><\/div>\r\n\r\n","protected":false},"excerpt":{"rendered":"<p>This reading will establish how interest rates and prices of bonds for different maturities are related. Spot Rates The zero-coupon bond (discount bond) is the basic debt security that pays one unit of currency, e.g., $1, at maturity. The price&#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":[216,548,547,549],"class_list":["post-16971","post","type-post","status-publish","format-standard","hentry","category-cfa-level-2","category-fixed-income","tag-cfa-level-2","tag-fixed-income","tag-reading-32","tag-spot-rates-and-forward-rates","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>Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"description\" content=\"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.\" \/>\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\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"og:description\" content=\"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/\" \/>\n<meta property=\"og:site_name\" content=\"CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-07-05T16:33:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-01-05T07:02:42+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1590\" \/>\n\t<meta property=\"og:image:height\" content=\"1160\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"AnalystPrep Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"AnalystPrep Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/\"},\"author\":{\"name\":\"AnalystPrep Team\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\"},\"headline\":\"Spot Rates and Forward Rates\",\"datePublished\":\"2021-07-05T16:33:57+00:00\",\"dateModified\":\"2026-01-05T07:02:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/\"},\"wordCount\":1180,\"image\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/wp-content\\\/uploads\\\/2021\\\/07\\\/Upward-Sloping-Spot-Curve.jpg\",\"keywords\":[\"CFA-level-2\",\"Fixed Income\",\"Reading 32\",\"Spot Rates and Forward Rates\"],\"articleSection\":[\"CFA Level II Study Notes\",\"Fixed Income\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/\",\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/\",\"name\":\"Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/wp-content\\\/uploads\\\/2021\\\/07\\\/Upward-Sloping-Spot-Curve.jpg\",\"datePublished\":\"2021-07-05T16:33:57+00:00\",\"dateModified\":\"2026-01-05T07:02:42+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\"},\"description\":\"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#primaryimage\",\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/wp-content\\\/uploads\\\/2021\\\/07\\\/Upward-Sloping-Spot-Curve.jpg\",\"contentUrl\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/wp-content\\\/uploads\\\/2021\\\/07\\\/Upward-Sloping-Spot-Curve.jpg\",\"width\":1590,\"height\":1160,\"caption\":\"Upward Sloping Spot Curve\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/cfa-level-2\\\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Spot Rates and Forward Rates\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#website\",\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/\",\"name\":\"CFA, FRM, and Actuarial Exams Study Notes\",\"description\":\"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\",\"name\":\"AnalystPrep Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"caption\":\"AnalystPrep Team\"},\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/author\\\/analystprep-team\\\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/ovQh3D6WBOw\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"2520\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2021-07-05T16:33:57+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes","description":"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/","og_locale":"en_US","og_type":"article","og_title":"Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes","og_description":"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.","og_url":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/","og_site_name":"CFA, FRM, and Actuarial Exams Study Notes","article_published_time":"2021-07-05T16:33:57+00:00","article_modified_time":"2026-01-05T07:02:42+00:00","og_image":[{"width":1590,"height":1160,"url":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg","type":"image\/jpeg"}],"author":"AnalystPrep Team","twitter_card":"summary_large_image","twitter_misc":{"Written by":"AnalystPrep Team","Est. reading time":"6 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/"},"author":{"name":"AnalystPrep Team","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"headline":"Spot Rates and Forward Rates","datePublished":"2021-07-05T16:33:57+00:00","dateModified":"2026-01-05T07:02:42+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/"},"wordCount":1180,"image":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg","keywords":["CFA-level-2","Fixed Income","Reading 32","Spot Rates and Forward Rates"],"articleSection":["CFA Level II Study Notes","Fixed Income"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/","url":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/","name":"Spot Rates and Forward Rates - CFA, FRM, and Actuarial Exams Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/#website"},"primaryImageOfPage":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#primaryimage"},"image":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg","datePublished":"2021-07-05T16:33:57+00:00","dateModified":"2026-01-05T07:02:42+00:00","author":{"@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"description":"Explore how interest rates and bond prices for different maturities are related, including calculations and examples.","breadcrumb":{"@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#primaryimage","url":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg","contentUrl":"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/07\/Upward-Sloping-Spot-Curve.jpg","width":1590,"height":1160,"caption":"Upward Sloping Spot Curve"},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/study-notes\/cfa-level-2\/describe-relationships-among-spot-rates-forward-rates-yield-to-maturity-expected-and-realized-returns-on-bonds-and-the-shape-of-the-yield-curve\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/study-notes\/"},{"@type":"ListItem","position":2,"name":"Spot Rates and Forward Rates"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/study-notes\/#website","url":"https:\/\/analystprep.com\/study-notes\/","name":"CFA, FRM, and Actuarial Exams Study Notes","description":"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/study-notes\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e","name":"AnalystPrep Team","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","caption":"AnalystPrep Team"},"url":"https:\/\/analystprep.com\/study-notes\/author\/analystprep-team\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/ovQh3D6WBOw","og_video_type":"text\/html","og_video_duration":"2520","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-07-05T16:33:57+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/16971","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/comments?post=16971"}],"version-history":[{"count":66,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/16971\/revisions"}],"predecessor-version":[{"id":41708,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/16971\/revisions\/41708"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/media?parent=16971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/categories?post=16971"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/tags?post=16971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}