{"id":28872,"date":"2022-10-30T11:19:18","date_gmt":"2022-10-30T11:19:18","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=28872"},"modified":"2026-02-25T17:12:29","modified_gmt":"2026-02-25T17:12:29","slug":"using-timelines-in-modelling-and-solving-time-value-of-money-problems","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/using-timelines-in-modelling-and-solving-time-value-of-money-problems\/","title":{"rendered":"Using Timelines in Modeling and Solving 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 (2023 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Module 1)\",\n  \"description\": \"CFA Level I Quantitative Methods lesson on The Time Value of Money (TVM) covering compounding, discounting, effective annual rate (EAR), present value (PV), future value (FV), annuities, perpetuities, unequal cash flows, and timeline modeling techniques.\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/-ffCJF1kmqo\/maxresdefault.jpg\",\n  \"uploadDate\": \"2021-09-20\",\n  \"duration\": \"PT54M17S\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=-ffCJF1kmqo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/-ffCJF1kmqo\"\n}\n<\/script>\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 cashflows for both projects combined?\",\n    \"text\": \"Assume that we have two projects, X and Y, and each has positive cashflows. The annual interest rate is 5%. The projects have the following cashflow:\\n\\nProject X:\\n(t=0) $0, (t=1) $100, (t=2) $150, (t=3) $250, (t=4) $300, (t=5) $250\\n\\nProject Y:\\n(t=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 cashflows for both projects combined?\\n\\nA. $890\\nB. $1,307\\nC. $2,197\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"$2,197.\\n\\nDiscount each project\u2019s cash flows at 5% and add them.\\n\\nPV(X) = 100\/(1.05)^1 + 150\/(1.05)^2 + 250\/(1.05)^3 + 300\/(1.05)^4 + 250\/(1.05)^5 \u2248 890.\\n\\nPV(Y) = 50 + 100\/(1.05)^1 + 200\/(1.05)^2 + 300\/(1.05)^3 + 400\/(1.05)^4 + 500\/(1.05)^5 \u2248 1,307.\\n\\nPV(X + Y) \u2248 890 + 1,307 = 2,197.\",\n      \"dateCreated\": \"2026-01-02\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/timeline-example-img1\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-1536x798.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1-1536x798.jpg\",\n  \"caption\": \"Using Timelines in Time Value of Money Problems \u2013 Example Timeline\",\n  \"width\": 1536,\n  \"height\": 798,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/using-timelines-in-modelling-and-solving-time-value-of-money-problems\/\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/images\/timeline-illustration-page-15\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\",\n  \"caption\": \"Using Timelines in Time Value of Money Problems \u2013 Illustration\",\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  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/using-timelines-in-modelling-and-solving-time-value-of-money-problems\/\"\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/-ffCJF1kmqo?si=WdklyozSNQD_6Ziy\" 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 cashflows associated with an investment project. For cashflows that are regular and of equal amounts, the standard annuity formula or the financial calculator can be used. However, for cashflows that are irregular, unequal, or both, a timeline is preferred.<\/p>\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\n\n\n<p>$$FV _ {N} = PV \\left (1 + r \\right)^ {N} $$<\/p>\n\n\n\n<p>Where<\/p>\n\n\n\n<p>\\(PV\\) = present value of the investment<br>\\(FV_N\\)= future value of the investment&nbsp; \\(N\\)periods from today<br>\\(r\\) = rate of interest per period<\/p>\n\n\n\n<p>We can represent this in a timeline:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"826\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_1.jpg\" alt=\"Timeline Example\" class=\"wp-image-20287\" 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\" \/><\/figure>\n<\/div>\n\n\n<p>In a particular timeline, a time index, <em>t, <\/em>is used to represent 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>) and by using this amount, we can calculate the future value (<em>t = N<\/em>). Alternatively, we can use the future value to calculate the present value.<\/p>\n\n\n\n<p>The above argument can be written in terms of the present value. That is:<\/p>\n\n\n\n<p>$$PV = FV _&nbsp; {N} \\left&nbsp; (1+r \\right)^ {-N} $$<\/p>\n\n\n\n<div style=\"margin:22px 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        text-align:center;\n        padding:10px 14px;\n        line-height:1.2;\n        border:2px solid #2f5bea;\n        border-radius:40px;\n        text-decoration:none;\n        font-weight:500;\n        font-size:18px;\n        color:#2f5bea;\n        background-color:transparent;\n        box-sizing:border-box;\n     \">\n     Practice TVM timeline questions with our free trial.\n  <\/a>\n<\/div>\n\n\n<h4><strong>Example: Applying a Timeline to Model Cashflows<\/strong><\/h4>\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 would then draw a timeline to better understand the problem:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1613\" height=\"964\" class=\"aligncenter size-full wp-image-16939\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-15.jpg\" alt=\"timeline-cash-flows\" 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. Recall that the PV of a perpetuity is given by:<\/p>\n<p>$$ \\text {PV of a perpetuity}\u00a0 = \\frac\u00a0 {C} {r} $$<\/p>\n<p>So that in this case:<\/p>\n<p>$$ PV_3 = \\frac\u00a0 {$6,500} {9\\%}\u00a0 =\u00a0 $72, 222$$<\/p>\n<p>This is the value of the perpetuity at <em>t <\/em>= 3, so we need to discount it for 3 more periods to get the value at <em>t <\/em>= 0. Using the formula:<\/p>\n<p>$$ PV _ 0 = FV _ {N} \\left\u00a0 \u00a0(1+r \\right)^ {-N} $$<\/p>\n<p>The PV at time zero is \\(\\frac\u00a0 {$72, 222}\u00a0 {{(1 + 0.09)}^3}\u00a0 =\u00a0 $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 cashflows are uneven. A good example in this respect is a pension contribution which varies with age. In such cases, it is not possible to apply one of the basic time value formulae. You are advised to draw a timeline even if the question appears quite straightforward. It will help you understand the question structure better. A timeline also helps candidates add cashflows indexed to the same period and apply the value additivity principle.<\/p>\n<h2><strong>Cashflow Additivity Principle<\/strong><\/h2>\n<p>According to cashflow additivity principle, the present value of any stream of cashflows indexed at the same point equals the sum of the present values of the cashflows. This principle has different applications in time value of money problems. Besides, this principle can be applied to find the NPVs of projects with uneven cashflows.<\/p>\n<p>In the above question, we can combine corresponding cashflows and work out the present values for the two projects at once:<\/p>\n<p>$$ NPV = (0 + 50) + \\frac {100 + 100}\u00a0 {1.05^{1}}\u00a0 +\u00a0 \\frac {150 + 200}\u00a0 {1.05^{2}}\u00a0 +\u00a0 \\frac {250\u00a0 +\u00a0 300} {1.05^{3}} + \\frac {300 + 400}\u00a0 {1.05^{4}} +\u00a0 \\frac{250 + 500}\u00a0 {1.05^{5}}\u00a0 =\u00a0 $2,197 $$<\/p>\n<h2><strong>Calculating Interest Rates, Growth Rates, and Number of Periods<\/strong><\/h2>\n<h3><strong>Calculating Interest Rate<\/strong><\/h3>\n<p>We know that,<\/p>\n<p>$$PV = FV _ {N} \\left\u00a0 (1 + r \\right)^\u00a0 {-N} $$<\/p>\n<p>$$ \\Rightarrow \\text {r}\u00a0 = \\left (\\frac {FV _ N}\u00a0 {PV} \\right)^\u00a0 {\\frac {1} {N}} &#8211; 1$$<\/p>\n<h4><strong>Example: Annual Rate of Return<\/strong><\/h4>\n<p>An investor invested in a $1, 000\u00a0 security today. It will earn him $1,191 three years from now. The annual rate of return on this investment is <em>closest to?<\/em><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Recall that:<\/p>\n<p>$$ \\text {r}\u00a0 = \\left (\\frac {FV _ N} {PV} \\right)^ {\\frac {1} {N}} -1$$<\/p>\n<p>In this case we have:<\/p>\n<p>$$ \\text {r}\u00a0 = \\left (\\frac {1191} {1, 000} \\right)^ {\\frac {1} {3}} -1$ = 0.06 = 6%$$<\/p>\n<p>The steps to solve the above example using a financial calculator are given below:<\/p>\n<p>$$<br \/>\\begin {array} {l|l|l}<br \/>\\textbf {Steps}\u00a0 &amp;\u00a0 \\textbf {Explanation}\u00a0 &amp;\u00a0 \\textbf\u00a0 {Display}\u00a0 \\\\<br \/>\\hline \\text\u00a0 { [2nd] [QUIT] }\u00a0 &amp;\u00a0 \\text\u00a0 {Return to standard calc mode}\u00a0 &amp;\u00a0 0\u00a0 \\\\<br \/>\\hline\\left [2^ {\\text\u00a0 {nd }} \\right] [\\text {CLR TVM}]\u00a0 &amp;\u00a0 \\text {Clears TVM Worksheet}\u00a0 &amp;\u00a0 0 \\\\<br \/>\\hline 3 [N]\u00a0 &amp;\u00a0 \\text\u00a0 {Years\/periods}\u00a0 &amp;\u00a0 N = 3 \\\\<br \/>\\hline-1, 000 [PV]\u00a0 &amp;\u00a0 \\text\u00a0 {Set to }\u00a0 PV \\text { to } &#8211; 1, 000\u00a0 &amp;\u00a0 PV = -1, 000 \\\\<br \/>\\hline 0 [PMT]\u00a0 &amp;\u00a0 \\text\u00a0 {Set annuity payment}\u00a0 &amp;\u00a0 PMT = 0 \\\\<br \/>\\hline 1, 191 [FV]\u00a0 &amp;\u00a0 \\text\u00a0 {Set to }\u00a0 FV\u00a0 \\text\u00a0 { to }\u00a0 1, 191\u00a0 &amp; FV = 1, 191 \\\\<br \/>\\hline [CPT] [I\/Y]\u00a0 &amp;\u00a0 \\text\u00a0 {Compute interest rate}\u00a0 &amp;\u00a0 I\/Y = 6\u00a0 \\text{ or }\u00a0 6\u00a0 \\% \\\\<br \/>\\end{array}<br \/>$$<\/p>\n<h4><strong>Example: Annual Rate of Return #2<\/strong><\/h4>\n<p>An investor invests $100 at the end of each of the next five years and receives $750 at the end of the fifth year. The annual rate of return on this investment is <em>closest to:<\/em><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using a calculator, you will solve for I\/Y as follows.<\/p>\n<p>\\(N\\)\u00a0 =\u00a0 5;\u00a0 \\(FV\u00a0 =\u00a0 $750\\);\u00a0 \\(PMT\u00a0 =\u00a0 \u2013100 \\);\u00a0 \\(CPT\\)\u00a0 \u00a0\\(I\/Y\\)\u00a0 =\u00a0 20.40%<\/p>\n<h4><strong>Example: Annual Rate of Return #3<\/strong><\/h4>\n<p>An investor invests $1, 000 today and receives a $50 coupon payment at the end of every year for 3 years. In addition, they receive $1,100 at the end of year 3. What is the rate of return of this investment?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the calculator, you will calculate I\/Y as follows.<\/p>\n<p>\\(PV\\)\u00a0 =\u00a0 -$1, 000<\/p>\n<p>\\(FV\\)\u00a0 =\u00a0 $1, 100<\/p>\n<p>\\(N\\)\u00a0 =\u00a0 3<\/p>\n<p>\\(PMT\\)\u00a0 =\u00a0 50<\/p>\n<p>\\(CPT\\)\u00a0 \\(I\/Y\\)\u00a0 =\u00a0 8.078%<\/p>\n<h3><strong>Calculating the Number of Periods <\/strong><\/h3>\n<p>\\(N\\) can be calculated using the following formula:<\/p>\n<p>$$N = \\frac {\\text{ln}\\left [\\frac {FV} {PV} \\right]} {\\text{ln} (1+r)} $$<\/p>\n<h4><strong>Example: Number of Period Calculation<\/strong><\/h4>\n<p>An investor invests $3, 000 in a bank account. How many years will it take to double the amount given that the interest rate is 5% per annum compounded annually?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the formula:<\/p>\n<p>$$N = \\frac {\\text{ln}\\left [\\frac {FV} {PV} \\right]} {\\text{ln}(1+r)} = \\left(\\frac {ln 2} {ln 1.05} \\right) = 14.21\\ \\text {Years} $$<\/p>\n<p>Using the BA II Plus\u2122 Financial Calculator:<\/p>\n<p>\\(I\/Y\\) = 5%, \\(PV\\) = -$3, 000, \\(PMT\\) = 0, \\(FV\\) = $6, 000, \\(CPT\\) \\(N\\) = 14.21<\/p>\n<h3><strong>Calculating Annuity Payments (Combining Future Value and Present Value Annuities)<\/strong><\/h3>\n<p><strong>\u00a0<\/strong>The annuity payment, \\(A\\), can be computed using the following formula:<\/p>\n<p>$$\\begin {align} \\text {A} &amp; = \\frac {\\text {PV of Annuity}} {\\text {Present value annuity Factor}} \\\\\u00a0 &amp; = \\frac {\\text {PV of Annuity}} {\\left[\\frac {1-\\frac{1} {\\left(1+\\frac{r s} {m}\\right)^ {m N}}} {\\frac {r S} {m}}\\right]} \\end {align} $$<\/p>\n<h4><strong>Example: Annuity Payment Calculation<\/strong><\/h4>\n<p>A company has borrowed $100, 000 for the purchase of machinery. The tenor of the loan is five years at an annual rate of 5% compounded monthly. If the first payment is due in one month, the monthly payment is <em>closest to:<\/em><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the formula:<\/p>\n<p>$$A = \\frac {\\text {PV of Annuity}} {\\left[\\frac{1 &#8211; \\frac{1} {\\left(1 + \\frac {r s} {m} \\right)^ {m N}}} {\\frac {r S}{m}} \\right]} = \\left [\\frac {1 &#8211; \\frac {1} {\\left (1 + \\frac {0.05} {12} \\right)^ {12(5)}}} {\\frac {0.05} {12}} \\right] = \\frac {100, 000}{52.9907} = $1, 887.1237 $$<\/p>\n<p>So, the monthly payment is $1, 887.<\/p>\n<p>Using the BA II Plus\u2122 Financial Calculator:<\/p>\n<p>\\(PV\\)\u00a0 =\u00a0 -100, 000; \\(N\\)\u00a0 =\u00a0 5\u00d712 =\u00a0 60;\u00a0 \\(I\/Y\\)\u00a0 = 5\/12\u00a0 =\u00a0 0.41667;\u00a0 \\(FV\\)\u00a0 =\u00a0 0; \\(CPT \\ PMT\\)\u00a0 =\u00a0 1,887.123<\/p>\n<h4><strong>Example: Annuity Payment Calculation<\/strong><\/h4>\n<p>A bank is offering a profit rate of 6% on its savings account. How much should an investor deposit each quarter to grow the investment to $20, 000 at the end of 12 years?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the formula:<\/p>\n<p>$$<br \/>\\begin {aligned}<br \/>F V &amp; = A\\left [\\frac {\\left (1+\\frac {r S} {m} \\right)^ {m N} -1} {\\frac {r s} {m}} \\right] \\\\<br \/>\\$ 20, 000\u00a0 &amp; = A\\left [\\frac {\\left (1 + \\frac {0.06} {4} \\right)^ {4 \\times 12} -1} {\\frac {0.06} {4}}\\right] \\\\<br \/>\\$ 20, 000\u00a0 &amp; = 69.5652 A \\\\<br \/>\\Rightarrow A\u00a0 &amp; = 287.50<br \/>\\end{aligned}<br \/>$$<\/p>\n<p>Using the BA II Plus\u2122 Financial Calculator:<\/p>\n<p>\\(FV\\)\u00a0 =\u00a0 -20, 000; \\(N\\)\u00a0 =\u00a0 4 \u00d7 12 =\u00a0 48; \\(I\/Y\\)\u00a0 =\u00a0 6\/4\u00a0 =\u00a0 1.5; \\(PV\\)\u00a0 =\u00a0 0; \\(CPT\\)\u00a0 \\(PMT\\)\u00a0 =\u00a0 287.50<\/p>\n<h4><strong>Example: Annuity Payment Calculation<\/strong><\/h4>\n<p>An investor needs to accumulate $3, 000 over the next 5 years. How much should she invest in a savings account every year given that the expected rate of return is 6.5%?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Using the BA II Plus\u2122 Financial Calculator:<\/p>\n<p>\\(FV\\)\u00a0 =\u00a0 -3, 000;\u00a0 \\(N\\)\u00a0 =\u00a0 5;\u00a0 I\/Y\u00a0 = 6.5;\u00a0 \\(PV\\)\u00a0 =\u00a0 0; \\(CPT \\ PMT\\)\u00a0 =\u00a0 526.90<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Assume that we have two projects,\u00a0 X and Y,\u00a0 and each has positive cashflows. The annual interest rate is 5%. The projects have the following cashflow:<\/p>\n<p>Project X:<\/p>\n<p>$$\u00a0 \\begin {array} {c|c|c|c|c|c}\u00a0 {t = 0}\u00a0 &amp;\u00a0 {t = 1}\u00a0 &amp;\u00a0 {t = 2}\u00a0 &amp;\u00a0 {t = 3}\u00a0 &amp;\u00a0 {t = 4}\u00a0 &amp;\u00a0 {t = 5}\u00a0 \\\\ \\hline\u00a0 {$0}\u00a0 &amp;\u00a0 {$100}\u00a0 &amp; {$150}\u00a0 &amp;\u00a0 {$250}\u00a0 &amp;\u00a0 {$300}\u00a0 &amp;\u00a0 {$250}\u00a0 \\\\ \\end {array}\u00a0 $$<\/p>\n<p>Project Y:<\/p>\n<p>$$\u00a0 \\begin {array} {c|c|c|c|c|c} {t=0}\u00a0 &amp;\u00a0 {t=1}\u00a0 &amp;\u00a0 {t = 2}\u00a0 &amp;\u00a0 {t = 3}\u00a0 &amp;\u00a0 {t = 4}\u00a0 &amp;\u00a0 {t = 5}\u00a0 \\\\ \\hline\u00a0 {$50 }\u00a0 &amp;\u00a0 {$100 }\u00a0 &amp; {$200}\u00a0 &amp;\u00a0 {$300}\u00a0 &amp;\u00a0 {$400}\u00a0 &amp;\u00a0 {$500}\u00a0 \\\\ \\end {array}\u00a0 $$<\/p>\n<p>What is the present value of the cashflows for both projects combined?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>\u00a0$ 890<\/li>\n<li>$ 1,307<\/li>\n<li>$ 2,197<\/li>\n<\/ol>\n<h4><strong>Solution<\/strong><\/h4>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>The timeline for project X is as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1613\" height=\"1075\" class=\"aligncenter size-full wp-image-20289\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_3.jpg\" alt=\"Timeline - Project X\" 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 cashflows for project X as:<\/p>\n<p>$$ \\begin {align*}<br \/>\\text {PV for X}\u00a0 &amp; = 100\u00a0 (1 + r)^ {-1} + 150\u00a0 \u00a0(1 + r)^ {-2} + 250\u00a0 (1 + r)^ {-3} + 300\u00a0 \u00a0(1 + r)^ {-4} + 250\u00a0 \u00a0(1 + r)^ {-5} \\\\<br \/>&amp; = 100 \\times 1.05^ {-1} + 150 \\times 1.05^ {-2} + 250 \\times 1.05^ {-3} + 300 \\times 1.05^ {-4} + 250 \\times 1.05^{-5} \\\\<br \/>&amp; = $ 890\\end {align*} $$<\/p>\n<p>For project Y, the timeline is given by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1613\" height=\"882\" class=\"aligncenter size-full wp-image-20293\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/img_4.jpg\" alt=\"Timeline - Project Y\" 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>$$ \\begin {align*} \\text {PV for Y} &amp;\u00a0 =\u00a0 50\u00a0 (1 + r)^ {-0} + 100\u00a0 (1 + r)^ {-1} + 200\u00a0 (1 + r)^ {-2} + 300\u00a0 (1 + r)^ {-3} + 400\u00a0 (1 + r)^ {-4} + 500 (1 + r)^ {-5} \\\\<br \/>&amp; = 50 + 100 \\times 1.05^ {-1} + 200 \\times 1.05^ {-2} + 300 \\times 1.05^ {-3} + 400 \\times 1.05^ {-4} + 500 \\times1.05^ {-5} \\\\<br \/>&amp; = $1307 \\end {align*} $$<\/p>\n<p>We then simply add the cashflows together:<\/p>\n<p>$$ PV_ {X+Y} = $890 + $1307 = $2,197 $$<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align:center; margin:42px 0 10px;\">\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:#3f78d7;\n        color:#ffffff;\n        text-decoration:none;\n        font-weight:600;\n        font-size:18px;\n        border-radius:40px;\n        line-height:1.1;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"\n        margin-top:14px;\n        max-width:640px;\n        margin-left:auto;\n        margin-right:auto;\n        font-size:16px;\n        line-height:1.5;\n     \">\n     Solve CFA-style time value of money problems and strengthen your financial modeling accuracy.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>A timeline is a physical illustration of the amounts and timing of cashflows associated with an investment project. For cashflows that are regular and of equal amounts, the standard annuity formula or the financial calculator can be used. However, for&#8230;<\/p>\n","protected":false},"author":15,"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-28872","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>Timelines in Time Value of Money | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Timelines visually represent cash flows over time, aiding in solving time value of money problems for investments and financial decisions.\" \/>\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\/using-timelines-in-modelling-and-solving-time-value-of-money-problems\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Timelines in Time Value of Money | CFA Level 1 - 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