{"id":82,"date":"2019-10-01T13:28:00","date_gmt":"2019-10-01T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=82"},"modified":"2026-02-06T08:44:49","modified_gmt":"2026-02-06T08:44:49","slug":"timeline-time-value-money-example-question","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/timeline-time-value-money-example-question\/","title":{"rendered":"Using a Timeline to Model Cash Flows and Solve Time Value of Money Problems"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"The Time Value of Money (2021 Level I CFA\u00ae Exam \u2013 Reading 6)\",\n  \"description\": \"CFA Level I Time Value of Money video covering LOS 6a and 6b. Learn how to interpret interest rates as required rates of return, discount rates, and opportunity costs, and how interest rates are composed of a real risk-free rate plus risk premiums for bearing different types of risk.\",\n  \"uploadDate\": \"2019-12-16\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/qk19_k31K-4\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=qk19_k31K-4\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/qk19_k31K-4\",\n  \"duration\": \"PT22M42S\"\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\": \"What is the present value of the cash flows for both projects combined?\",\n    \"text\": \"Assume that we have two projects, X and Y, each with positive cash flows. The annual interest rate is 5% per year.\\n\\nProject X cash flows:\\nt=1: $100, t=2: $150, t=3: $250, t=4: $300, t=5: $250\\n\\nProject Y cash flows:\\nt=0: $50, t=1: $100, t=2: $200, t=3: $300, t=4: $400, t=5: $500\\n\\nWhat is the present value of the cash flows for both projects combined?\\nA. $890\\nB. $2,197\\nC. $1,307\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The present value of Project X is $890 and the present value of Project Y is $1,307. Therefore, the combined present value is $890 + $1,307 = $2,197. This also follows from the value additivity principle: NPV of independent projects equals the sum of individual NPVs.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1.jpg\",\n      \"caption\": \"Timeline Example\",\n      \"width\": 1590,\n      \"height\": 826,\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    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\",\n      \"caption\": \"Timeline\",\n      \"width\": 1613,\n      \"height\": 964,\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    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3.jpg\",\n      \"caption\": \"Timeline \u2013 Project X\",\n      \"width\": 1613,\n      \"height\": 1045,\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    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-768x420.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-768x420.jpg\",\n      \"caption\": \"Timeline \u2013 Project Y\",\n      \"width\": 768,\n      \"height\": 420,\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  ]\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/qk19_k31K-4?si=jVEJBFYFUv4NI31K\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>A timeline is a physical illustration of the amounts and timing of cash flows associated with an investment project. Cash flows that are regular and of equal amounts can be modeled as annuities. In such exam problems, all you have to do is apply the standard annuity formulas. It is even more convenient if you use the <a href=\"https:\/\/www.cfainstitute.org\/about\/governance\/policies\/Pages\/calculator_policy.aspx\">approved calculators<\/a>\u00a0to do most of the work. However, it may get a little tricky if the cash flows are irregular, unequal, or both. It is essential to draw a timeline to help you understand the problem.<\/p>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:14px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:18px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice CFA TVM timeline questions now.\n  <\/a>\n<\/div>\n\n\n\n<p>Remember that the general formula that relates the present value and the future value of an investment is given by:<\/p>\n<p>$$FV_{N}=PV\\left(1+r\\right)^{N}$$<\/p>\n<p>Where<\/p>\n<p>PV = present value of the investment<br \/>FV<em><sub>N<\/sub> <\/em>= future value of the investment <em>N <\/em>periods from today<br \/><em>r <\/em>= rate of interest per period<\/p>\n<p>We can represent this in a timeline:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-20287\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1.jpg\" alt=\"Timeline Example\" width=\"1590\" height=\"826\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1.jpg 1590w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-300x156.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-1024x532.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-768x399.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-1536x798.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-400x208.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/><\/p>\n<p>A timeline visualizes the compatibility between the time units and the corresponding interest rate per unit time. In a particular timeline, a time index <em>t <\/em>represents a particular point in time, a specified number of periods from today. Therefore, the present value is the investment amount today (<em>t=0<\/em>). We can use this amount to calculate the future value (<em>t=N<\/em>). Alternatively, we can use the future value to calculate the present value.<\/p>\n<p>The above argument can be written in terms of the present value. That is:<\/p>\n<p>$$PV=FV_{N}\\left(1+r\\right)^{-N}$$<\/p>\n<h3>Example: Applying Timeline to Model Cashflows<\/h3>\n<p>An investor receives a series of payments, each amounting to $6,500, set to be received in perpetuity. Payments are to be made at the end of each year, starting at the end of year 4. If the discount rate is 9%, then what is the present value of the perpetuity at t=0?<\/p>\n<p><strong>Solution <\/strong><\/p>\n<p>We should draw a timeline to understand the problem better.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16939\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\" alt=\"timeline-cash-flows\" width=\"1613\" height=\"964\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg 1613w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15-300x179.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15-768x459.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15-1024x612.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15-400x239.jpg 400w\" sizes=\"auto, (max-width: 1613px) 100vw, 1613px\" \/><\/p>\n<p>Here, we can see that the investor is receiving $6,500 in perpetuity (lasts forever). Note that the PV of a PV is given by:<\/p>\n<p>$$\\text{PV of a perpetuity} =\\frac {C}{r}$$<\/p>\n<p>So that in this case:<\/p>\n<p>$$\\text{PV}=\\frac {$6,500}{9\\%} = $72,222$$<\/p>\n<p>This is the value of the perpetuity at t=3, so we need to discount it 3 more periods to get the value at t=0. Here is the formula to use:<\/p>\n<p>$$PV=FV_{N}\\left(1+r\\right)^{-N}$$<\/p>\n<p>PV at time zero =\\( \\frac {$72,222}{{(1+0.09)}^3} = $55,769\\)<\/p>\n<h2><strong>Why is the Use of a Timeline Recommended?<\/strong><\/h2>\n<p>There are many instances in real life when cash flows are uneven. A good case in point is pension contributions which vary with age. In such problems, it\u2019s not possible to apply one of the basic time value formulae. You\u2019re advised to draw a timeline even if the question appears quite straightforward. The timeline will help you understand the question structure better. A timeline also helps candidates add cash flows indexed to the same period and apply the value additivity principle.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Assume that we have two projects, X and Y, and each has positive cash flows. The annual interest rate is 5% per year. The projects have the following cash flows;<\/p>\n<p>X: $100 at t = 1, $150 at t = 2, $250 at t = 3, $300 at t = 4 and $250 at t = 5<\/p>\n<p>Y: $50 at t = 0, $100 at t = 1, $200 at t = 2, $300 at t = 3, $400 at t = 4 and $500 at t = 5 where t = time in years<\/p>\n<p>Project X:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c|c} {t=0} &amp; {t=1} &amp; {t=2} &amp; {t=3} &amp; {t=4} &amp; {t=5} \\\\ \\hline {$0 } &amp; {$100 } &amp; {$150 } &amp; {$250 } &amp; {$300 } &amp; {$250 } \\\\ \\end{array} $$<\/p>\n<p>Project Y:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c|c} {t=0} &amp; {t=1} &amp; {t=2} &amp; {t=3} &amp; {t=4} &amp; {t=5} \\\\ \\hline {$50 } &amp; {$100 } &amp; {$200 } &amp; {$300 } &amp; {$400 } &amp; {$500 } \\\\ \\end{array} $$<\/p>\n<p>What is the present value of the cash flows for both projects combined?<\/p>\n<p>A. $890<\/p>\n<p>B. $2,197<\/p>\n<p>C. $1,307<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>The correct answer is B.<\/p>\n<p>The time project X is as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-20289\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3.jpg\" alt=\"Timeline - Project X\" width=\"1613\" height=\"1075\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3.jpg 1613w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3-300x200.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3-1024x682.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3-768x512.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3-1536x1024.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3-400x267.jpg 400w\" sizes=\"auto, (max-width: 1613px) 100vw, 1613px\" \/><\/p>\n<p>We can calculate the cash flows for each project and then add them up.<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{PV for X} &amp; = 100(1 + r)^{-1} + 150(1 + r)^{-2} + 250(1 + r)^{-3} + 300(1 + r)^{-4} + 250(1 + r)^{-5} \\\\<br \/>&amp; = 100 \\times 1.05^{-1} + 150 \\times 1.05^{-2} + 250 \\times 1.05^{-3} + 300 \\times 1.05^{-4} + 250 \\times 1.05^{-5} \\\\<br \/>&amp; = $890\\end{align*} $$<\/p>\n<p>For project Y, the timeline is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-20293\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4.jpg\" alt=\"Timeline - Project Y\" width=\"1613\" height=\"882\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4.jpg 1613w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-300x164.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-1024x560.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-768x420.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-1536x840.jpg 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4-400x219.jpg 400w\" sizes=\"auto, (max-width: 1613px) 100vw, 1613px\" \/><\/p>\n<p>$$ \\begin{align*} \\text{PV for Y} &amp; = 50(1 + r)^{-0}+ 100(1 + r)^{-1} + 200(1 + r)^{-2} + 300(1 + r)^{-3} + 400(1 + r)^{-4} + 500(1 + r)^{-5} \\\\<br \/>&amp; = 50 + 100 \\times 1.05^{-1} + 200 \\times 1.05^{-2} + 300 \\times 1.05^{-3} + 400 \\times 1.05^{-4} + 500 \\times1.05^{-5} \\\\<br \/>&amp; = $1307\u00a0\\end{align*} $$<\/p>\n<p>Hence, the net present value is = 890+1307=$2,197.<\/p>\n<p>Alternatively, we can combine corresponding cash flows and work out the present values for the two projects at once i.e.<\/p>\n<p>$$ \\begin{array}{c} \\text{NPV} = (0+50) +(100+100) \\times 1.05^{-1} + (150+200)\\times 1.05^{-2}+ (250+300) \\times 1.05^{-3} + \\\\ (300+400) \\times 1.05^{-4} + (250+500) \\times 1.05^{-5} \\\\ = $2197 \\\\ \\end{array} $$<\/p>\n<p>The latter method applies the value additivity principle, which asserts that the NPV of a set of independent projects is just the sum of the NPVs of the individual projects.<\/p>\n<p>We can apply this principle to find the NPVs of projects with uneven cash flows.<\/p>\n<\/blockquote>\n<p><em>Reading 6 LOS 6f<\/em><\/p>\n<p><em>Demonstrate the use of a time line in modeling and solving time value of money problems.<\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:35px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:14px 34px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:16px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:18px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:700px; margin:0 auto; font-size:16px; line-height:1.6; color:#333;\">\n    Get faster at building timelines and solving time value of money questions with exam-style practice, clear explanations, and timed quizzes in AnalystPrep.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>A timeline is a physical illustration of the amounts and timing of cash flows associated with an investment project. Cash flows that are regular and of equal amounts can be modeled as annuities. In such exam problems, all you have&#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":[2],"tags":[],"class_list":["post-82","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","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>Timeline in Time Value of Money | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"A timeline is basically a drawing that illustrates the amount and timing of all cash flows associated with an investment project.\" \/>\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\/quantitative-methods\/timeline-time-value-money-example-question\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Timeline in Time Value of Money | CFA Level 1 - 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