{"id":28775,"date":"2022-10-30T06:59:44","date_gmt":"2022-10-30T06:59:44","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=28775"},"modified":"2026-01-22T10:48:57","modified_gmt":"2026-01-22T10:48:57","slug":"present-and-future-values-annuities-and-cash-flows-cfa","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/present-and-future-values-annuities-and-cash-flows-cfa\/","title":{"rendered":"Present Values, Future Values, Annuities, and Series of Unequal Cashflows"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"@id\": \"https:\/\/analystprep.com\/#video-ffCJF1kmqo\",\n  \"name\": \"The Time Value of Money (2023 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Module 1)\",\n  \"description\": \"Master one of the most important CFA Level I Quantitative Methods topics: the Time Value of Money (TVM). In this lesson, Prof. Forjan explains compounding, discounting, annuities, perpetuities, and the use of financial calculators for present value, future value, and interest rate problems. You will learn how to interpret interest rates, calculate effective annual rates (EAR), handle different compounding frequencies, and apply the timeline method to solve cash flow questions with confidence.\",\n  \"uploadDate\": \"2021-09-20T00:00:00+00:00\",\n  \"thumbnailUrl\": [\n    \"https:\/\/img.youtube.com\/vi\/-ffCJF1kmqo\/maxresdefault.jpg\",\n    \"https:\/\/img.youtube.com\/vi\/-ffCJF1kmqo\/hqdefault.jpg\"\n  ],\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=-ffCJF1kmqo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/-ffCJF1kmqo\",\n  \"duration\": \"PT54M17S\",\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/default-logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\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\": \"Maxwell buys an annuity that makes a series of regular payments, each amounting to $200 per year for a period of 15 years. In addition, he is to receive level payments at the beginning of every year. What premium should he be willing to pay for this annuity, assuming a 13.5% effective interest rate?\",\n    \"text\": \"Maxwell buys an annuity that makes a series of regular payments, each amounting to $200 per year for a period of 15 years. In addition, he is to receive level payments at the beginning of every year. What premium should he be willing to pay for this annuity, assuming a 13.5% effective interest rate?\",\n    \"answerCount\": 1,\n    \"upvoteCount\": 0,\n    \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B ($1,430). Because payments are made at the beginning of each year, this is an annuity due. Compute the present value factor for an annuity-immediate using PV of $1 = (1 - v^n) \/ d, where v = 1\/(1 + r) and d = r\/(1 + r). With r = 0.135 and n = 15, PV of $1 \u2248 7.149, so PV = 200 \u00d7 7.149 \u2248 $1,430.\",\n      \"dateCreated\": \"2025-12-16T00:00:00+00:00\",\n      \"upvoteCount\": 0,\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/present-and-future-values-annuities-and-cash-flows-cfa\/\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/-ffCJF1kmqo\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  referrerpolicy=\"strict-origin-when-cross-origin\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Future Values<\/strong><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>The Future Value (FV) of a Single Sum of Cash Flow<\/strong><\/h3>\n\n\n\n<p>The Future Value (FV) of a single sum of money is the amount that money invested today at a given interest rate (r) for a specified period will translate into in future. Denoted by \\(\\text {FV} _ {N}\\), the future value of a single sum of money is given by:<\/p>\n\n\n\n<p>$$\\text {FV} _ {N} = \\text {PV}\\left (1+r\\right) ^ {N}$$<\/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 <em>N <\/em>periods from today.<br><em>\\(r\\) <\/em>= Rate of interest per period.<br>\\(N\\) = Number of periods (Years).<\/p>\n\n\n\n<p>Note that the formula above is based on the time value of money.<\/p>\n\n\n\n<p>The factor (1 +&nbsp;<em>r<\/em>)<em><sup>N<\/sup><\/em>&nbsp;is called a future value factor.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For a given interest rate, the higher the number of periods \u201cN\u201d, the greater the future value.<\/li>\n\n\n\n<li>For a given number of periods \u201cN\u201d, the higher the interest rate, the greater the future value.<\/li>\n<\/ul>\n\n\n\n<div style=\"margin: 20px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener\"\n     class=\"ap-cta\"\n     data-cta-text=\"Want more PV\/FV and cash-flow exercises\"\n     data-cta-type=\"button\"\n     data-cta-location=\"top_content\"\n     data-page-type=\"study_note\"\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 PV\/FV and cash-flow exercises? Try AnalystPrep\u2019s free trial now.\n  <\/a>\n<\/div>\n\n\n<\/p>\n<h4><strong>Example: Calculating the Future Value of a Lump Sum<\/strong><\/h4>\n<p>Suppose you deposited $5,000 in a savings account that earns an annual compound interest of 7%; what would be the value of the money in the savings account after ten years?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>From the question:<\/p>\n<p>PV = 5,000.<br \/>FV<em><sub>N<\/sub> <\/em>= ?<br \/><em>r <\/em>= 7%.<br \/>N =10.<\/p>\n<p>$$\\Rightarrow \\text {FV} _ {N} = \\text {PV} \\left (1+r\\right) ^ {N} = 5,000 \\left (1+0.07\\right) ^ {10} = 9, 835.7568$$<\/p>\n<p>$$\\begin {aligned} &amp; \\textbf {BA II Plus\u2122 Financial Calculator Steps} \\\\&amp;\\begin {array} {l|l|l}<br \/>\\textbf {Steps} &amp; \\textbf {Explanation} &amp; \\textbf { Display } \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calc mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd}} \\right] [\\text {CLRTVM}] &amp; \\text {Clears TVM Worksheet} &amp; 0 \\\\<br \/>\\hline 10[\\mathrm {~N}] &amp; \\text {Years\/periods} &amp; \\mathrm{N} =\u00a0 10\u00a0 \\\\<br \/>\\hline 7 [\\mathrm {I} \/ \\mathrm{Y}] &amp; \\text { Set interest rate } &amp; \\mathrm{I} \/ \\mathrm{Y} = 7 \\\\<br \/>\\hline-5000[\\mathrm {PV}] &amp; \\text {Set present value} &amp; \\mathrm{PV} =\u00a0 -5,000\u00a0 \\\\<br \/>\\hline 0 [\\mathrm {PMT}] &amp; \\text {Set payment } &amp; \\mathrm{PMT} =\u00a0 0\u00a0 \u00a0\\\\<br \/>\\hline [\\mathrm {CPT}] [\\mathrm {FV}] &amp; \\text {Compute future value} &amp; \\mathrm {FV} = 9, 835.7568 \\\\\\end {array} \\end {aligned}$$<\/p>\n<p>When the interest rate is compounded more than once in a year (as is the case of an investment that pays interest more than once a year), the annual interest rate, which should be converted to monthly (or any other frequency such as semi-annually) interest rate, is termed as the <strong>stated annual rate of interest <\/strong>or <strong>quoted interest rate. <\/strong>It is denoted as \\(r_s\\).<\/p>\n<p>For example, if the monthly interest rate is 0.65, then the stated interest rate is 0.65\u00d712 = 7.8.<\/p>\n<p>Under more than one compounding period per year, the future value of a single sum of money is:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {PV} \\left (1+\\frac {r_{s}} {m} \\right) ^ {\\text {mN}} $$<\/p>\n<p>Where:<\/p>\n<p><em>\\(m\\) = <\/em>Number of compounding periods per year.<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<p>\\(r_s\\) = Annual stated rate of interest.<\/p>\n<h4><strong>Example: Calculating Future under Monthly Compounding<\/strong><\/h4>\n<p>Imagine that you deposited $2,000 in a savings account that earns an annual interest rate of 7% compounded monthly. What would be the value of the money in your account after ten years?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the question:<\/p>\n<p><em>m = <\/em>12.<\/p>\n<p>N = 12.<\/p>\n<p>r = 7%.<\/p>\n<p>So,<\/p>\n<p>$$\\text {FV} _ {N} = \\text {PV} \\left(1 + \\frac {r _ {s}} {m} \\right) ^ {\\text {mN}} = 2,000\\left (1+\\frac {0.07} {12} \\right) ^ {12\\times 10} = 4,019.32$$<\/p>\n<p>$$\\begin {aligned} &amp; \\textbf {BA II Plus\u2122 Financial Calculator Steps} \\\\ &amp; \\begin {array} {l|l|l}<br \/>\\textbf {Steps} &amp; \\textbf {Explanation} &amp; \\textbf {Display} \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text {Return to standard calc mode} &amp; 0 \\\\<br \/>\\hline \\text { [2nd] [CLR TVM }] &amp; \\text {Clears TVM Worksheet} &amp; 0 \\\\<br \/>\\hline 120 [\\mathrm {N}] &amp; \\text {Years\/periods} (12 \\times 10 = 120) &amp; \\mathrm{N} = 120 \\\\<br \/>\\hline 0.583333 [\\mathrm{I} \/ \\mathrm {Y}]\u00a0 &amp;\u00a0 \\text {Set interest rate} (7 \/ 12 = 0.583333) &amp; \\mathrm{I} \/ \\mathrm {Y} = 0.58333 \\\\<br \/>\\hline-2,000 [\\mathrm {PV}] &amp; \\text {Set present value} &amp; \\mathrm {PV} = -2,000 \\\\<br \/>\\hline 0 [\\mathrm {PMT}] &amp; \\text {Set payment} &amp; \\mathrm{PMT} = 0 \\\\<br \/>\\hline [\\mathrm {CPT}] [\\mathrm {FV}] &amp; \\text {Compute future value} &amp; \\mathrm {FV} = 4, 019.32 \\\\<br \/>\\end {array} \\end {aligned}<br \/>$$<\/p>\n<p>In case the compounding period per year is infinite, that is m\u2192\u221e, the future value of the single sum of money is expressed as:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {PV}e ^ {r_{s} N} $$<\/p>\n<p>Take the above case as an example. If the annual rate of 7% interest were continuously compounded, then the future value of the deposits would be:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {PV}e^ {r _ {s}N} = 2,000 \\times e^ {0.07\\times 10} = 4, 027.51$$<\/p>\n<p><strong>Exam tip<\/strong>: there are no TMV buttons on your calculator. When dealing with continuous compounding, simply remember the formula.<\/p>\n<p>Important points to note regarding compounding:<\/p>\n<ul>\n<li>The greater the \u201cN\u201d (number of periods), the higher the compounded interest earned, all else equal.<\/li>\n<li>The higher the interest rate, the higher the compounded interest earned, all else equal.<\/li>\n<\/ul>\n<h2><strong>The Future Value of a Series of Payments<\/strong><\/h2>\n<p>Series of payments are classified into equal cashflows and unequal cashflows.<\/p>\n<h3><strong>Future Values of Equal Cashflows<\/strong><\/h3>\n<p>Annuities are used to determine the future value of equal cashflows. An annuity is a series of even cashflows. There are two types of annuities: ordinary annuities and annuities due.<\/p>\n<h3><strong>Ordinary Annuity<\/strong><\/h3>\n<p>An ordinary annuity is an annuity where cash flows occur at the end of each period. Such payments are said to be made in arrears (beginning at time t = 1).<\/p>\n<p>The future value of an ordinary annuity is derived as follows:<\/p>\n<p>Consider an annuity amount of \\(A\\) paid at the end of each period for \\(N\\) periods with the interest rate per period denoted by \\(r\\). In this instance, the future value of equal cashflows is given by:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {A} \\left [\\left (1 + r \\right)^ {N &#8211; 1} + \\left (1 + r \\right)^ {N &#8211; 2}\u00a0 +\u00a0 \\left (1 + r \\right)^ {N &#8211; 3} + \\dots + \\left (1 + r \\right)^ {1}\u00a0 +\u00a0 \\left (1 + r \\right)^ {0} \\right]$$<\/p>\n<p>This reduces into:<\/p>\n<p>$$\\text {FV}\u00a0 _\u00a0 {N}\u00a0 =\u00a0 \\text {A} \\left [\\frac {\\left(1 + r\\right)^ {N} &#8211; 1} {r} \\right]$$<\/p>\n<p>The factor \\(\\frac {\\left (1+r\\right)^ {N} &#8211; 1} {r}\\) is termed as the future value annuity factor that gives the future value of an ordinary annuity of $1 per period. Therefore, we multiply any amount by this factor to get the future value of that particular annuity.<\/p>\n<h4><strong>Example: Valuing an Ordinary Annuity<\/strong><\/h4>\n<p>Assume that you have decided to invest $2,000 per year in a stock index fund that earns 9% per year for the next ten years. What will be the closest value of the accumulated value of the investment after you make the last payment?<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>From the information given in the question:<\/p>\n<p>\\(A\\) = 2,000.<\/p>\n<p>\\(N\\) = 10.<\/p>\n<p>\\(r\\) = 9%.<\/p>\n<p>So that:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {A} \\left [\\frac {\\left (1 + r\\right)^ {N} &#8211; 1} {r} \\right] = 2,000\\left [\\frac{\\left (1 + 0.09\\right)^ {10} -1} {0.09} \\right] = 30,385.8594$$<\/p>\n<h2><strong>Annuity Due<\/strong><\/h2>\n<p>Annuity due is a type of annuity where payments start immediately at the beginning of time, at time t = 0. In other words, payments are made at the beginning of each period.<\/p>\n<p>The formula for the future value of an annuity due is derived by:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {A} \\left [\\left (1 + r\\right)^ {N} + \\left (1 + r\\right)^ {N &#8211; 1} + \\left (1 + r\\right)^ {N-2} +&#8230;+ \\left(1 + r\\right)^ {1}\\right]$$<\/p>\n<p>Which reduces to:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {A} \\left [\\frac {\\left (1 + r\\right)^ {N} &#8211; 1} {d} \\right] $$<\/p>\n<p>Where:<\/p>\n<p>$$\\text {d} = \\frac {r} {1+r} $$<\/p>\n<h4><strong>Example: Future Value of an Annuity Due<\/strong><\/h4>\n<p>Refer to our ordinary annuity example. If the payments were instead made at the beginning of each period, then the future value of the payments would be:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {A} \\left [\\frac {\\left(1 + r\\right)^ {N} &#8211; 1} {d} \\right]=2,000\\left [\\frac {\\left (1.09\\right)^ {9} &#8211; 1}{\\frac {0.09} {109}} \\right] =$$<\/p>\n<h3><strong>Future Values of Unequal Series of Payments<\/strong><\/h3>\n<p>There are some instances where cash flow payments are not equal. The saving pattern of self-employed individuals who save depending on their income level at a particular time is a good case in point.<\/p>\n<p>The future value of an unequal stream of payments is calculated by working out the sum of the future values of individual payments.<\/p>\n<p>Consider the following example.<\/p>\n<h4><strong>Example: Future Value of Unequal Cashflows<\/strong><\/h4>\n<p>A small-scale businessman deposits money into his savings account at the beginning of each year, depending on the business returns. He deposits $1,000 in the first year, $2,000 in the second year, $5,000 in the third year, and $7,000 in the fourth year. The account credits interest at an annual interest rate of 7%. What is the closest value of the money accumulated in the savings account at the beginning of year 4?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The future value of the unequal payments is the sum of individual accumulations:<\/p>\n<p>$$1,000 \\left\u00a0 (1.07\\right)^3 + 2,000 \\left\u00a0 (1.07 \\right)^\u00a0 2\\: + 5,000 \\left\u00a0 (1.07 \\right)^ 1 + 7000 \\left\u00a0 (1.07 \\right)^ 0 = 15,864.48$$<\/p>\n<p><strong>Note:<\/strong> He makes payments at the beginning of each year.<\/p>\n<h2><strong>Present Values<\/strong><\/h2>\n<h3><strong>Present Values of Single Cashflow<\/strong><\/h3>\n<p>The present value\u00a0(PV) is the current\u00a0value of a future sum of money (Future value, FV) or series of cashflows given a specified rate of return. Note that the future value of a single sum of money is given by:<\/p>\n<p>$$\\text {FV} _ {N} = \\text {PV} \\left (1 + r\\right)^ {N} $$<\/p>\n<p>If we make the present value (PV) the subject of the formula by dividing both sides of the above equation by<\/p>\n<p>$$\\frac\u00a0 {\\text\u00a0 {FV} _ {N}}\u00a0 {\\left (1 + r\\right)^\u00a0 {N}} = \\frac\u00a0 {PV\\left (1 + r\\right)^\u00a0 {N}}\u00a0 {\\left (1 + r\\right)^\u00a0 {N}} $$<\/p>\n<p>$$\\Rightarrow PV = \\text\u00a0 {FV} _ {N}\u00a0 \\left (1+r\\right)^\u00a0 {-N} $$<\/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>N = Number of \u00a0Years.<\/p>\n<p>\\((1 + r)^{-N}\\) is called the present value factor, which is intuitively the reciprocal of the future value factor.<\/p>\n<h4><strong>Example: Calculating the Present Value of a Single Sum of Cashflow<\/strong><\/h4>\n<p>An investor wishes to save $100,000 in the next eight years. The investor opts for a savings account that pays 6% annual interest. Calculate the closest value of the deposit the investor should make to reach the target.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the information given:<\/p>\n<p>FV = 100,000.<\/p>\n<p>r = 6%.<\/p>\n<p>N = 8.<\/p>\n<p>PV = ?<\/p>\n<p>So,<\/p>\n<p>$$\\Rightarrow PV = \\text {FV} _ {N} \\left (1 + r\\right)^ {-N} = 100,000 (1.06)^ {-8} = 62, 741.2371$$<\/p>\n<p>You could use your financial calculator to save time!<\/p>\n<p>When the frequency of compounding is more than once per year (quarterly, monthly, etc.), the formula is analogously (as illustrated above) defined as follows:<\/p>\n<p>$$PV =\u00a0 \\text\u00a0 {FV} _ {N}\u00a0 \\left\u00a0 (1+\\frac\u00a0 {r _ {s}}\u00a0 {m}\u00a0 \\right)^\u00a0 {-mN}$$<\/p>\n<p>Where<\/p>\n<p>\\(m\\) = Number of compounding periods per year.<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<p>\\(r_s\\) = Annual stated rate of interest.<\/p>\n<h4><strong>Example: Calculating the Present Value of Single Cashflow under Multiple \u00a0Compounding<\/strong><\/h4>\n<p>An investor wishes to save $100,000 in the next eight years. The investor opts for a savings account that pays 6% annual interest compounded monthly. Calculate the closest value of the deposit the investor should make to reach the target.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the information given in the question,<\/p>\n<p>\\(FV\\) = 100,000.<\/p>\n<p>\\(r_s\\) = 6%.<\/p>\n<p>\\(N\\) = 8.<\/p>\n<p>\\(m\\) = 12.<\/p>\n<p>So,<\/p>\n<p>$$PV = 100,000\\left (1+\\frac {0.06} {12} \\right)^ {-12\\times 8} = 61, 952.3909$$<\/p>\n<p>Similarly, for the continuously compounded interest rate, the present value of the investment is given by<\/p>\n<p>$$PV = \\text {FV} _ {N}e^ {-\\text {N} \\text {r} _ {s}} $$<\/p>\n<h4><strong>Example: Calculating the Present Value of Continuously Compounded Cashflows<\/strong><\/h4>\n<p>A fund continuously accumulates to $4,000 over ten years at a 10% annual interest rate. Calculate the closest present value of this fund.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the question,<\/p>\n<p>FV = 4,000.<\/p>\n<p>\\(r_s\\) = 10%.<\/p>\n<p>N = 10.<\/p>\n<p>So,<\/p>\n<p>$$PV = \\text {FV} _ {N}e^ {-\\text {N} \\text {r} _ {s}} = 4,000\\times e^ {-10\\times 0.10} = 1, 471.5178$$<\/p>\n<h2><strong>Present Value of a Series of Cashflows<\/strong><\/h2>\n<p>Many investments offer a series of uneven, relatively even, or unequal payments over a given period. Therefore, different methodologies are employed in the valuation of their present values.<\/p>\n<h4><strong>The Present Value of a Series of Equal Cashflows<\/strong><\/h4>\n<p>As clarified earlier, annuities are used to determine the present value of a series of equal cash flows. We shall consider ordinary annuity due.<\/p>\n<h3><strong>Ordinary Annuity<\/strong><\/h3>\n<p>Remember that the series of payments do not begin immediately in an ordinary annuity. Instead, payments are made at the end of each period. It is further worth noting that the present value of an annuity is equal to the sum of the current value of each annuity payment:<\/p>\n<p>$$\\begin {align} \\text {PV} &amp; = A\\left (1 + r\\right)^ {-1} + A\\left (1 + r\\right)^ {-2} + \\dots + A\\left (1 + r\\right)^ {-(N &#8211; 1)} + A\\left (1 + r\\right)^ {-N} \\\\&amp; = A\\left [\\left (1 + r\\right)^ {-1} + \\left (1 + r\\right)^ {-2} + \\dots + \\left (1 + r\\right)^ {-(N-1)} + \\left (1 + r\\right)^ {-N} \\right]\\\\ &amp; = A\\left[\\frac {1 &#8211; (1 + r)^ {-N}} {r} \\right]\\end{align} $$<\/p>\n<p>So, the present value of an ordinary annuity is given by:<\/p>\n<p>$$PV = A\\left [\\frac {1 &#8211; (1 + r)^ {-N}} {r} \\right] $$<\/p>\n<h4><strong>Example: Calculating the Present Value of the Ordinary Annuity<\/strong><\/h4>\n<p>A financial asset generates returns of $10,000 at the end of each year for ten years. \u00a0The required rate of return is 7% per year. How much must one pay to buy this asset?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>To find the cost of purchasing the asset, we need to find the sum of the present values of the series of payments from the asset. In this case, this is the current value of an ordinary annuity.<\/p>\n<p>From the question:<\/p>\n<p>A = 10,000.<\/p>\n<p>R = 7% = 0.07.<\/p>\n<p>N = 10.<\/p>\n<p>So, the present value is given by:<\/p>\n<p>$$PV = A\\left [\\frac {1-(1+r)^ {-N}} {r} \\right] = 10,000 \\left [\\frac {1 &#8211; (1 + 0.07)^ {- 10}} {0.07} \\right] = \\$ 70, 235.81$$<\/p>\n<h3><strong>Annuity Due<\/strong><\/h3>\n<p>Remember that annuity due is a type of annuity where payments start immediately at the beginning of time, that is, at time t = 0. The present value of the annuity due is presented as:<\/p>\n<p>$$\\begin {align} \\text {PV} &amp; = A\\left [\\left(1 + r\\right)^ {0} + \\left (1 + r\\right)^ {-1} + \\left(1 + r\\right)^ {-2} \\cdots\\right] \\\\ &amp; = A\\left [\\frac {1-(1+r)^ {-N}} {d} \\right] \\end {align} $$<\/p>\n<p>So, the present value of an ordinary annuity is given by:<\/p>\n<p>$$PV = A\\left [\\frac {1 &#8211; (1 + r)^ {-N}} {d} \\right]$$<\/p>\n<p>Where:<\/p>\n<p>$$d = \\frac{r} {1+r} $$<\/p>\n<p>It is easy \u00a0to see that, from<\/p>\n<p>$$\\begin {align} \\text {PV} &amp; = A \\left (1 + r\\right)^ {0} + A \\left (1 + r\\right)^ {-1} + A \\left (1 + r\\right)^ {-2} + \\cdots + A\\left (1 + r\\right)^ {-(N &#8211; 1)}\\\\ &amp; = A\\left [1 + \\left (1 + r\\right)^ {-1} + \\left(1 + r\\right)^ {-2} + \\cdots + \\left (1 + r\\right)^ {-(N-1)} \\right] \\\\ &amp; = A + A\\left [\\left (1 + r\\right)^ {-1} + \\left (1 + r\\right)^ {-2} + \\cdots +\\left (1 + r\\right)^ {-(N &#8211; 1)} \\right]\\\\\u00a0 &amp; = A + A\\left [\\frac {1-(1 + r)^ {-(N &#8211; 1)}} {r} \\right] \\end {align} $$<\/p>\n<p>the annuity due is equivalent to a lump sum of A plus the present value of the ordinary annuity for N-1 years.<\/p>\n<p>Imagine that you have just retired, and your pensioner agrees to pay you $12,000 per year for the next 20 years, where you receive the first payment today. Assuming an interest rate of 7%, calculate the closest value of the present value of your payments.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the question,<\/p>\n<p>A = 12,000.<\/p>\n<p>N = 20.<\/p>\n<p>r = 7%.<\/p>\n<p>Now, this is an annuity due since the first payment starts today. Here is the formula to use:<\/p>\n<p>$$PV = A + A\\left [\\frac {1-(1 + r)^ {-(N &#8211; 1)}} {r}\\right] = 12,000 + 12,000 \\left [\\frac {1- (1.07)^ {-19}} {0.07} \\right] = 136,027.1429$$<\/p>\n<p>Alternatively, we could use the usual annuity due formula:<\/p>\n<p>$$PV = A\\left [\\frac {1- (1 + r)^ {-N}} {d} \\right] = 12,000\\left [\\frac {1-(1.07)^ {-20}} {\\frac {0.07}{1.07}} \\right] = 136,027.1429$$<\/p>\n<p>Calculator steps to change the mode to \u201cBGN\u201d and then back to \u201cEND\u201d mode.<\/p>\n<p>$$\\begin{aligned} &amp;\\textbf {BA II Plus\u2122 Financial Calculator Steps}\\\\ &amp; \\begin {array} {l|l|l}<br \/>\\textbf {Steps} &amp; \\textbf {Explanation} &amp; \\textbf {Display} \\\\<br \/>\\hline \\begin {array} {l}<br \/>{[\\text {[2nd }]} \\\\<br \/>{[\\mathrm{SET}]}<br \/>\\end {array} [\\mathrm {BGN}] [2 \\mathrm {nd}] &amp; \\text {Puts in BGN mode} &amp; 0 \\\\<br \/>\\hline [2 \\mathrm {nd}] [\\mathrm {QUIT}] &amp; \\text {Return to standard calculator mode} &amp; 0 \\\\<br \/>\\hline [2 \\mathrm {nd}] [\\mathrm {CLRTVM}] &amp; \\text {Clears TVM Worksheet} &amp; 0 \\\\<br \/>\\hline 20 [\\mathrm {~N}] &amp; \\text\u00a0 {years\/periods} &amp; \\mathrm {N} = 20 \\\\<br \/>\\hline 7 [\\mathrm {I} \/ \\mathrm {Y}]\u00a0 &amp;\u00a0 \\text {Set interest rate} &amp; \\mathrm{I} \/ \\mathrm{Y} = 7 \\\\<br \/>\\hline 0[\\mathrm {FV}]\u00a0 &amp;\u00a0 \\text {Set future value} &amp; \\mathrm {FV} = 0 \\\\<br \/>\\hline 12000 [\\mathrm {PMT}] &amp; {\\text {Set payment}} &amp; \\begin {array} {l}<br \/>\\mathrm{PMT} \\\\<br \/>12,000<br \/>\\end{array} \\\\<br \/>\\hline [\\mathrm {CPT}] [\\mathrm {PV}]\u00a0 &amp;\u00a0 {\\text {Compute present value }} &amp; -136,027.1429 \\\\<br \/>\\end{array}\\end{aligned}$$<\/p>\n<p><strong>Note<\/strong>: Always remember to switch back to \u201cEND\u201d mode after solving the problem. To switch back to \u201cEND\u201d mode, follow the steps outlined below.<\/p>\n<ul>\n<li>[2<sup>nd<\/sup>] [PMT]. You should see BGN on the screen.<\/li>\n<li>Now, press 2nd ENTER to change that to END and then press 2nd CPT to exit setting the calculation mode.<\/li>\n<li>In the \u201cEND\u201d mode, the screen&#8217;s upper-right corner will be blank.<\/li>\n<\/ul>\n<p>We can also compute the PV of an annuity due by calculating the PV of an ordinary annuity and multiplying that PV by [1 + periodic compounding rate (r)]. That is,<\/p>\n<p>$$\\text {PV (Annuity due) = PV (Ordinary annuity) \u00d7 (1 + r)} $$<\/p>\n<h2><strong>Present Value of a Perpetuity and Present Values Indexed at Times Other Than t = 0<\/strong><\/h2>\n<h3><strong>Perpetuity<\/strong><\/h3>\n<p>A perpetuity is an infinite series of regular cashflows. Consider an ordinary annuity that is paid infinitely. That is, if we take the limit as on the formula of an ordinary annuity, we get:<\/p>\n<p>$$PV = \\lim _ {N \\to \\infty} A\\left [\\frac {1-(1 + r)^ {-N}} {r} \\right] = \\frac {A} {r}$$<\/p>\n<p>So, the present value of a perpetuity is given by:<\/p>\n<p>$$PV = \\frac {A} {r}$$<\/p>\n<p>This formula is mostly applicable in stock valuation, in a case where a stock offers a constant dividend.<\/p>\n<h4><strong>Example: Calculating the Present Value of a Perpetuity<\/strong><\/h4>\n<p>A \u00a0stock pays a constant dividend of $8 at the end of each year for 20 years at a 25% required rate of return. Calculate the present value of the stock dividends.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The constant dividends of the stock are valued as perpetuity. So, from the question,<\/p>\n<p>A = 8.<\/p>\n<p>r = 25%.<\/p>\n<p>So that:<\/p>\n<p>$$PV = \\frac {A} {r} = \\frac {8} {0.25} = $32$$<\/p>\n<h3><strong>Present Values Indexed at Times Other Than t = 0<\/strong><\/h3>\n<p>Note that the present value is not always computed at time 0 (t = 0), denoted by \\(PV _ 0\\). We can also find the present value at any time, such as t = 2 or t = 3. This concept allows us to combine both present values, i.e., perpetuity and the present value of a single sum of cash flow. Consider the example given below.<\/p>\n<h4><strong>Example: Calculating the Present Value of a Projected Perpetuity<\/strong><\/h4>\n<p>A stock pays a constant dividend of $10, starting at the beginning of year 6 (t = 6). What is the perpetuity&#8217;s present value if the required return rate is 20%?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we need to find the PV of the perpetuity at the 5<sup>th<\/sup> time (because a regular annuity payment occurs at the end of a period) and then discount it to time 0. That is:<\/p>\n<p>$$\\text {PV} _ {5} = \\frac {A} {r} = \\frac {10} {0.20} = \\$50$$<\/p>\n<p>So, the present value today is given by:<\/p>\n<p>$$\\text {PV} _ {0} = 50 \\left (1.2\\right)^ {-5} = \\$20.10$$<\/p>\n<h4><strong>Present Value of a Series of Unequal Cashflows<\/strong><\/h4>\n<p>Just like calculating future values, the present value of a series of unequal cash flows is calculated by summing individual present values of cash flows. In finance, the present value of a series of many unequal cash flows is calculated using software such as a spreadsheet. Consider the example below.<\/p>\n<h4><strong>Example: Calculating the Present Values of Unequal Series of Payments<\/strong><\/h4>\n<p>A small-scale businessman receives income from his business at the end of each year. He earns $1,000 in the first year, $3,000 in the second year, $5,000 in the third, and $7,000 in the fourth year. An annual interest rate of 7% is applied.<\/p>\n<p>Calculate the present value of the cash inflows today.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The future value of the unequal payments is the sum of individual accumulations:<\/p>\n<p>$$V = 1,000\u00a0 (1.07)^ {-1} + 3,000\u00a0 (1.07)^ {-2} + 5,000\u00a0 (1.07)^\u00a0 {-3} + 7,000\u00a0 (1.07)^ {-4} = 12,976.65$$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Maxwell buys an annuity that makes a series of regular payments, each amounting to $200 per year for a period of 15 years. In addition, he is to receive level payments at the beginning of every year. What premium should he be willing to pay for this annuity, assuming a 13.5% effective interest rate?<\/p>\n<p>A. $1,715.<\/p>\n<p>B. $1,430.<\/p>\n<p>C.$1,200.<\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the question, it is clear that this is an annuity due since payments are made at the beginning of the year.<\/p>\n<p>The premium payable should be the present value of the annuity, and it is determined using the following steps.<\/p>\n<p><strong>Step I: <\/strong>Note down the applicable formula.<\/p>\n<p>$$\\text{PV of \u20181\u2019} =\u00a0\\cfrac{(1-V^n)}{d}$$<\/p>\n<p><strong>Step II:<\/strong> Determine the values for the variables in the formula, i.e., n = 15 years, r = 0.135.<\/p>\n<p>$$d = \\frac {0.135} {1.135} = 0.11894$$.<\/p>\n<p>$$V= \\left(1+ 0.135\\right)^ {-1} = 0.8810$$.<\/p>\n<p><strong>Step III:<\/strong> Determine the present value.<\/p>\n<p>$$ \\begin {align*} PV &amp; = \\cfrac { (1- 0.88106^ {15}) } {0.11894} \\\\ &amp; = 7.149 \\\\ \\end {align*} $$<\/p>\n<p>Note that this is the premium payable for an annuity of just $1 per year.<\/p>\n<p>Therefore, for $200, the present value will be as follows:<\/p>\n<p>$$\\text{PV} = 200 \\times 7.15 = \\$1,430$$<\/p>\n<\/blockquote>\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     rel=\"noopener\"\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=\"study_note\"\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    Practice PV \/ FV, annuity &#038; cash-flow problems with full solutions.\n  <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Future Values The Future Value (FV) of a Single Sum of Cash Flow The Future Value (FV) of a single sum of money is the amount that money invested today at a given interest rate (r) for a specified period&#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-28775","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>Present and Future Values, Annuities, and Cash Flows | CFA<\/title>\n<meta name=\"description\" content=\"Learn how to calculate present and future values, annuities, and the PV of uneven cash flows using key formulas. 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