{"id":8165,"date":"2020-08-13T13:00:37","date_gmt":"2020-08-13T13:00:37","guid":{"rendered":"https:\/\/analystprep.com\/blog\/?p=8165"},"modified":"2026-01-07T11:57:46","modified_gmt":"2026-01-07T11:57:46","slug":"spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams","status":"publish","type":"post","link":"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/","title":{"rendered":"Spot Rate vs. Forward Rates (Calculations for CFA\u00ae and FRM\u00ae Exams)"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2.png\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2.png\",\n  \"caption\": \"Spot-rate vs forward-rate illustration (CFA\/FRM exam)\",\n  \"width\": 609,\n  \"height\": 351,\n  \"fileFormat\": \"image\/png\",\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    \"url\": \"https:\/\/analystprep.com\"\n  }\n}\n<\/script>\n\n\n\n<p>Understanding how to calculate forward rates from spot rates is a must for CFA and FRM candidates. This guide breaks down the forward rate formula step by step, showing you exactly how it appears on the exams. With simple examples and exam-focused explanations, you&#8217;ll quickly grasp how spot and forward rates interact\u2014and why it matters in fixed income and derivatives.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_80 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Spot_Rates\" >Spot Rates<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Forward_Rate_Formulas_%E2%80%93_CFA_FRM_Quick_Guide\" >Forward Rate Formulas \u2013 CFA &amp; FRM Quick Guide<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Calculating_the_Price_of_a_Bond_using_Spot_Rates\" >Calculating the Price of a Bond using Spot Rates<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Formula\" >Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Calculating_the_Yield-to-maturity_of_a_Bond_using_Spot_Rates\" >Calculating the Yield-to-maturity of a Bond using Spot Rates<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Forward_Rates\" >Forward Rates<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Deriving_Forward_Rates_from_a_Set_of_Spot_Rates\" >Deriving Forward Rates from a Set of Spot Rates<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#Example_Calculating_the_one-year_forward_rate\" >Example: Calculating the one-year forward rate.<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#1_Using_the_Formula\" >1. Using the Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/analystprep.com\/blog\/spot-rate-vs-forward-rates-calculations-for-cfa-and-frm-exams\/#2_Using_the_Timeline\" >2. Using the Timeline<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Spot_Rates\"><\/span>Spot Rates<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>A <strong>spot interest rate<\/strong> gives you the price of a financial contract on the spot date. The spot date is the day when the funds involved in a business transaction are transferred between the parties involved. It could be two days after a trade, or even on the same day the deal is completed. A spot rate of 5% is the agreed-upon market price of the transaction based on current buyer and seller action.<\/p>\n\n\n\n<div style=\"margin: 20px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Want more spot-vs-forward-rate exercises\"\n     data-cta-type=\"button\"\n     data-cta-location=\"top_content\"\n     data-page-type=\"blog\"\n     style=\"\n       display: inline-block;\n       padding: 10px 16px;\n       font-size: 14px;\n       font-weight: 600;\n       color: #0b5ed7;\n       border: 2px solid #0b5ed7;\n       border-radius: 6px;\n       text-decoration: none;\n       background-color: transparent;\n     \">\n    Want more spot\u2011 vs\u2011forward\u2011rate exercises with real\u2011world examples? Try AnalystPrep free trial now.\n  <\/a>\n<\/div>\n\n\n<h3 class=\"\" data-start=\"295\" data-end=\"351\"><span class=\"ez-toc-section\" id=\"Forward_Rate_Formulas_%E2%80%93_CFA_FRM_Quick_Guide\"><\/span><strong data-start=\"302\" data-end=\"351\">Forward Rate Formulas \u2013 CFA &amp; FRM Quick Guide<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p class=\"\" data-start=\"353\" data-end=\"392\"><strong data-start=\"353\" data-end=\"392\">1. Forward Rate Formula (1-period):<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f2,1=(1+S2)2(1+S1)\u22121f_{2,1} = \\frac{(1 + S_2)^2}{(1 + S_1)} &#8211; 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<span class=\"mpunct mtight\">,<\/span>1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mopen\">(<\/span>1<span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span>1<span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mclose\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/p>\n<p class=\"\" data-start=\"445\" data-end=\"481\"><strong data-start=\"445\" data-end=\"481\">2. General Forward Rate Formula:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">fn,m=((1+Sn)n(1+Sm)m)1n\u2212m\u22121f_{n,m} = \\left( \\frac{(1 + S_n)^n}{(1 + S_m)^m} \\right)^{\\frac{1}{n &#8211; m}} &#8211; 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"minner\"><span class=\"mopen delimcenter\"><span class=\"delimsizing size3\">(<\/span><\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mopen\">(<\/span>1<span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mclose\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span>1<span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mclose\">)<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\"><span class=\"delimsizing size3\">)<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">m<\/span><\/span><span class=\"sizing reset-size3 size1 mtight\">1<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/p>\n<hr class=\"\" data-start=\"569\" data-end=\"572\" \/>\n<p class=\"\" data-start=\"574\" data-end=\"584\"><strong data-start=\"574\" data-end=\"584\">Where:<\/strong><\/p>\n<ul data-start=\"585\" data-end=\"750\">\n<li class=\"\" data-start=\"585\" data-end=\"648\">\n<p class=\"\" data-start=\"587\" data-end=\"648\"><span class=\"katex\"><span class=\"katex-mathml\">fn,mf_{n,m}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Forward rate from year <span class=\"katex\"><span class=\"katex-mathml\">mm<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span> to year <span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"649\" data-end=\"692\">\n<p class=\"\" data-start=\"651\" data-end=\"692\"><span class=\"katex\"><span class=\"katex-mathml\">SnS_n<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Spot rate for maturity <span class=\"katex\"><span class=\"katex-mathml\">nn<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"693\" data-end=\"736\">\n<p class=\"\" data-start=\"695\" data-end=\"736\"><span class=\"katex\"><span class=\"katex-mathml\">SmS_m<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">S<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>: Spot rate for maturity <span class=\"katex\"><span class=\"katex-mathml\">mm<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"737\" data-end=\"750\">\n<p class=\"\" data-start=\"739\" data-end=\"750\"><span class=\"katex\"><span class=\"katex-mathml\">n&gt;mn &gt; m<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">n<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Calculating_the_Price_of_a_Bond_using_Spot_Rates\"><\/span><span style=\"color: #222222; font-family: Montserrat, sans-serif; font-size: revert; font-weight: bold; letter-spacing: -0.025rem;\">Calculating the Price of a Bond using Spot Rates<\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2><span class=\"ez-toc-section\" id=\"Formula\"><\/span>Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\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<p>Where:<\/p>\n<p>\\({ PV }_{ bond }\\)\u00a0 is the present value of the price of the bond;<\/p>\n<p>PMT\u00a0is the coupon payment per period;<\/p>\n<p>\\( { Z }_{ 1 },{ Z }_{ 2 }\\) and \\( { Z }_{ n }\\) are the spot rates for periods 1,2 and n respectively; and<\/p>\n<p>n is the number of evenly spaced periods to maturity.<\/p>\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>Spot rates are also applied in determining the yield to maturity of a bond.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Calculating_the_Yield-to-maturity_of_a_Bond_using_Spot_Rates\"><\/span>Calculating the Yield-to-maturity of a Bond using Spot Rates<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Continuing on 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<h2><span class=\"ez-toc-section\" id=\"Forward_Rates\"><\/span>Forward Rates<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In theory, <strong>forward rates<\/strong> are prices of financial transactions that are expected to take place at some future point.<\/p>\n<p>A forward rate indicates the <strong>interest rate on a loan beginning at some time in the future<\/strong>, whereas a spot rate is the interest rate on a loan beginning immediately. Thus, the\u00a0forward market rate is for future delivery after the usual settlement time in the cash market.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Deriving_Forward_Rates_from_a_Set_of_Spot_Rates\"><\/span>Deriving Forward Rates from a Set of Spot Rates<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\\({ Z }_{ n }\\), the \\(n\\)-year spot rate, is a measure of the average interest rate over the period from now until \\(n\\) years\u2019 time.<\/p>\n<p>The forward rate, \\({ f }_{ t,r }\\), is a measure of the average interest rate between times \\(t\\) and \\(t + r\\). It\u2019s the interest rate agreed today\u00a0\\((t=0)\\) on an investment made at time \\(t&gt;0\\) for a period of \\(r\\) years.<\/p>\n<p>The one-year forward rate, \\({ f }_{ t,1 }\\), is therefore the rate of interest from time \\(t\\) to time \\(t +1\\). It can be expressed in terms of spot rates as follows:<\/p>\n<p>$$ 1+{ f }_{ t,1 } = \\frac{{{(1+{Z}_{t+1})}}^{t+1}}{{(1+{Z}_{t})}^{t}} $$<\/p>\n<p>Alternatively,<\/p>\n<p><strong>Step 1: <\/strong>Use the formula:<\/p>\n<p>$$ 1+{ f }_{ t,1 }=\\frac {V_2}{V_1} $$<\/p>\n<p>Where \\(V_1\\) is the value to which a dollar grows by time \\(T_1\\) and \\(V_2\\) is the value to which a dollar grows by \\(T_1\\).<\/p>\n<p><strong>Step 2: <\/strong>Calculate the interest rate that equates the value of one dollar at time \\(T_1\\) to the value of one dollar at time \\(T_2\\).<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Example_Calculating_the_one-year_forward_rate\"><\/span>Example: Calculating the one-year forward rate.<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>You are given the following spot rates:<\/p>\n<ul>\n<li>1-year spot rate: <strong>5%<\/strong>;<\/li>\n<li>2-year spot rate: <strong>6<\/strong>%.<\/li>\n<\/ul>\n<p>Determine the one-year forward rate <strong>one year from today<\/strong>, i.e., \\(f_{1,1}\\).<\/p>\n<p>There are 2 ways to solve this question:<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1_Using_the_Formula\"><\/span>1. Using the Formula<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>$$ 1+{ f }_{ t,1 } = \\frac{{{(1+{Z}_{t+1})}}^{t+1}}{{(1+{Z}_{t})}^{t}} $$<\/p>\n<p>$$ (1 + \\text{1-year spot}) \u00d7 (1 + \\text{1-year forward rate at time 1}) = (1 + \\text{2-year spot})^2 $$<\/p>\n<p>$$ (1.05)\\ \u00d7 (1\\ +\\ f_{1,1}) = {(1.06)}^2 $$<\/p>\n<p>$$\u00a0 (f_{1,1}) = 7.0095%$$<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2_Using_the_Timeline\"><\/span>2. Using the Timeline<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-8172 aligncenter\" src=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2-300x173.png\" alt=\"spot-rate-forward-rate-cfa-exam-frm-exam2\" width=\"435\" height=\"251\" srcset=\"https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2-300x173.png 300w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2-400x231.png 400w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2-484x279.png 484w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2-570x329.png 570w, https:\/\/analystprep.com\/blog\/wp-content\/uploads\/2020\/08\/spot-rate-forward-rate-cfa-exam-frm-exam2.png 609w\" sizes=\"auto, (max-width: 435px) 100vw, 435px\" \/><\/h3>\n<p>We use the principle of no-arbitrage. The golden rule to remember while using the timeline is that no matter which route you take to \u201ctime travel\u201d from one period to the other- the end result (Future Value) will always be the same.<\/p>\n<p>There are 2 possible routes we could take here<\/p>\n<ol type=\"i\">\n<li>Invest for one year at the spot rate of 5% and then invest for another year at the unknown forward rate \u201cF.\u201d<\/li>\n<li>Invest for two years at the spot rate of 6%.<\/li>\n<\/ol>\n<p>As per the assumption of no-arbitrage, we should arrive at the same value for both.<\/p>\n<p>By equating i &amp; ii, we get F = 7%, which is the same as what we got using the formula.<\/p>\n<p><blockquote class=\"wp-embedded-content\" data-secret=\"lgQ1yASz5F\"><a href=\"https:\/\/analystprep.com\/shop\/frm-part-1-and-part-2-complete-course-by-analystprep\/\">FRM Part 1 and Part 2 Complete Online Course<\/a><\/blockquote><iframe loading=\"lazy\" class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; visibility: hidden;\" title=\"&#8220;FRM Part 1 and Part 2 Complete Online Course&#8221; &#8212; AnalystPrep\" src=\"https:\/\/analystprep.com\/shop\/frm-part-1-and-part-2-complete-course-by-analystprep\/embed\/#?secret=2vAsBxOZeO#?secret=lgQ1yASz5F\" data-secret=\"lgQ1yASz5F\" width=\"600\" height=\"338\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p>\n\n\n<div style=\"margin: 40px 0; padding: 30px; text-align: center; background-color: #f5f8fc; border-radius: 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     class=\"ap-cta\"\n     data-cta-text=\"Start Free Trial\"\n     data-cta-type=\"button\"\n     data-cta-location=\"bottom_content\"\n     data-page-type=\"blog\"\n     style=\"\n       display: inline-block;\n       padding: 14px 26px;\n       font-size: 18px;\n       font-weight: 700;\n       color: #ffffff;\n       background-color: #0b5ed7;\n       border-radius: 8px;\n       text-decoration: none;\n     \">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 15px; color: #333;\">\n    Access spot\/forward rate &#038; FX\u2011arbitrage practice problems with full solutions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Understanding how to calculate forward rates from spot rates is a must for CFA and FRM candidates. This guide breaks down the forward rate formula step by step, showing you exactly how it appears on the exams. With simple examples&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[70,71],"tags":[],"class_list":["post-8165","post","type-post","status-publish","format-standard","hentry","category-cfa","category-frm","blog-post","no-post-thumbnail","animate"],"acf":[],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/comments?post=8165"}],"version-history":[{"count":18,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8165\/revisions"}],"predecessor-version":[{"id":13626,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/posts\/8165\/revisions\/13626"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/media?parent=8165"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/categories?post=8165"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/blog\/wp-json\/wp\/v2\/tags?post=8165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}