{"id":28760,"date":"2021-09-03T04:10:11","date_gmt":"2021-09-03T04:10:11","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=28760"},"modified":"2026-02-07T10:50:11","modified_gmt":"2026-02-07T10:50:11","slug":"time-value-of-money-with-different-frequencies-of-compounding","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/time-value-of-money-with-different-frequencies-of-compounding\/","title":{"rendered":"Time Value of Money With Different Frequencies of Compounding"},"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\u00ae Level I Quantitative Methods video lesson from AnalystPrep covering The Time Value of Money (TVM). Professor James Forjan explains compounding and discounting, effective annual rates (EAR), present and future value of single sums, annuities, and perpetuities, and the use of financial calculators and timelines to solve TVM problems with exam-focused clarity.\",\n  \"uploadDate\": \"2021-09-20\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/-ffCJF1kmqo\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=-ffCJF1kmqo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/-ffCJF1kmqo\",\n  \"duration\": \"PT54M17S\"\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 primary purpose of a long-short equity hedge fund strategy?\",\n    \"text\": \"A hedge fund manager is considering implementing a long-short equity hedge fund strategy. Which of the following is most likely the primary purpose of this strategy in the context of hedge fund management?\\n\\nA. To ensure that the hedge fund always has a balanced portfolio of long and short positions.\\nB. To take advantage of price movements in both directions, potentially generating profits regardless of market conditions.\\nC. To reduce the risk of the hedge fund\u2019s portfolio by always having an equal number of long and short positions.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"To take advantage of price movements in both directions, potentially generating profits regardless of market conditions.\\n\\nA long-short equity strategy seeks to profit from both rising and falling stock prices by taking long positions in expected winners and short positions in expected losers. The goal is not to keep longs and shorts perfectly balanced or equal in number, but to generate returns based on relative performance and manage exposure.\",\n      \"dateCreated\": \"2026-01-02\"\n    }\n  }\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 does a GP\u2019s track record and personal investment indicate about performance and commitment?\",\n    \"text\": \"An investor is reviewing the performance of a General Partner (GP) of a hedge fund. The GP has a long track record of successful investments and has shown a clear alignment of interests by investing their own money in the fund. What does this indicate about the GP\u2019s performance and commitment to the fund?\\n\\nA. The GP is likely to be less committed to the fund\u2019s success because they have their own money at risk.\\nB. The GP\u2019s successful track record and personal investment in the fund are positive signs of their commitment and potential for continued success.\\nC. The GP\u2019s past performance does not necessarily indicate future success, and their personal investment does not affect their commitment to the fund.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The GP\u2019s past performance does not necessarily indicate future success, and their personal investment does not guarantee future performance.\\n\\nA strong track record and GP co-investment can be viewed as favorable signals, but they do not ensure future success because hedge fund returns can change as market conditions and strategies evolve. Past performance is not a guarantee of future results, and while personal investment may align incentives, it does not eliminate risk or assure outcomes.\",\n      \"dateCreated\": \"2026-01-02\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/-ffCJF1kmqo\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Time value of money calculations allow us to establish the future value of a given amount of money. The present value (PV) is the money you have today. On the other hand, the future value (FV) is the accumulated amount of money you get after investing the original sum at a certain interest rate and for a given time period, say, 2 years.<\/p>\n<h2><strong>Fundamental Formulas in Time Value of Money Calculations<\/strong><\/h2>\n<p>Let,<\/p>\n<p>\\(FV\\) = Future value.<\/p>\n<p>\\(PV\\) = Present value.<\/p>\n<p>\\(r\\) = Interest rate.<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<p>Then the future value (FV) of an investment is given by:<\/p>\n<p>$$FV=PV(1+r)^N$$<\/p>\n<p>To find the present value of the investment, we rewrite the above formula so that:<\/p>\n<p>$$PV=FV(1+r)^{-N}$$<\/p>\n<p><a style=\"display: block; margin: 20px 0 28px; padding: 14px 18px; border: 2px solid #2563eb; border-radius: 12px; text-align: center; color: #2563eb; text-decoration: none; font-weight: 500; font-size: 15px; background-color: #ffffff;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Master compounding and discounting faster with our free trial. <\/a><\/p>\n<h3><strong>Calculation Using a Financial Calculator<\/strong><\/h3>\n<p>To calculate FV and PV using the BA II Plus\u2122 Financial Calculator, use the following keys:<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<p>\\(I\/Y\\) = Rate per period.<\/p>\n<p>\\(PV\\) = Present value.<\/p>\n<p>\\(FV\\) = Future value.<\/p>\n<p>\\(PMT\\) = Payment.<\/p>\n<p>\\(CPT\\) = Compute.<\/p>\n<p>It is important to note that the sign of PV and FV will be opposite. For example, if PV is negative, then FV will be positive. Generally, an inflow is entered with a positive sign, while an outflow is entered as a negative sign in the calculator.<\/p>\n<h4><strong>Example: Calculating Future Value<\/strong><\/h4>\n<p>Assume that an individual invests $10,000 in a bank account that pays interest at 10% compounded annually. The future value after two years is <em>closest to<\/em>:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>$12,000.<\/li>\n<li>$12,100.<\/li>\n<li>$22,000.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>Recall that:<\/p>\n<p>$$FV=PV(1+r)^N$$<\/p>\n<p>In this case, we have PV=10,000, r=10%, N=2 so that:<\/p>\n<p>$$FV=10,000(1+10%)^2=12,100$$<\/p>\n<p>$$\\begin{aligned}&amp;\\textbf{Using the BA II Plus\u2122 Financial Calculator}\\\\ &amp;\\begin{array}{l|l|l}<br \/>\\textbf { Steps } &amp; \\textbf { Explanation } &amp; \\textbf { Display } \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calc mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd }}\\right][\\mathrm{CLR} \\text { TVM }] &amp; \\text { Clears TVM Worksheet } &amp; 0 \\\\<br \/>\\hline 2[\\mathrm{~N}] &amp; \\text { Years\/periods } &amp; \\mathrm{N}=2 \\\\<br \/>\\hline 10[\\mathrm{I} \/ \\mathrm{Y}] &amp; \\text { Set interest rate } &amp; \\mathrm{I} \/ \\mathrm{Y}=10 \\\\<br \/>\\hline-10000[\\mathrm{PV}] &amp; \\text { Set present value } &amp; \\mathrm{PV}=-10000 \\\\<br \/>\\hline 0[\\mathrm{PMT}] &amp; \\text { Set payment } &amp; \\mathrm{PMT}=0 \\\\<br \/>\\hline[\\mathrm{CPT}][\\mathrm{FV}] &amp; \\text { Compute future value } &amp; \\mathrm{FV}=12,100 \\\\<br \/>\\end{array}\\end{aligned}<br \/>$$<\/p>\n<p>To confirm our answer, we could work out the PV of a future value of $12,100 invested under similar terms, starting with the FV of $12,100, using the formula below:<\/p>\n<p>$$PV=FV(1+r)^{-N}=12,100(1+10\\%)^{-2}= $10,000$$<\/p>\n<p>$$\\begin{aligned} &amp;\\textbf{Using the BA II Plus\u2122 Financial Calculator}\\\\ &amp;\\begin{array}{l|l|l}<br \/>\\textbf { Steps } &amp; \\textbf { Explanation } &amp; \\textbf { Display }\u00a0\\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calc mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd }}\\right][\\text { CLR TVM }] &amp; \\text { Clears TVM Worksheet } &amp; 0 \\\\<br \/>\\hline 2[\\mathrm{~N}] &amp; \\text { Years\/periods } &amp; \\mathrm{N}=2 \\\\<br \/>\\hline 10[\\mathrm{I} \/ \\mathrm{Y}] &amp; \\text { Set interest rate } &amp; \\mathrm{I} \/ \\mathrm{Y}=10 \\\\<br \/>\\hline 12100[\\mathrm{FV}] &amp; \\text { Set present value } &amp; \\mathrm{FV}=12100 \\\\<br \/>\\hline 0[\\mathrm{PMT}] &amp; \\text { Set payment } &amp; \\mathrm{PMT}=0 \\\\<br \/>\\hline[\\mathrm{CPT}][\\mathrm{PV}] &amp; \\text { Compute present value } &amp; \\mathrm{PV}=10,000 \\\\<br \/>\\end{array}\\end{aligned}<br \/>$$<\/p>\n<h2><strong>Time Value of Money With Different Frequencies of Compounding<\/strong><\/h2>\n<p>Some types of investments accumulate interest more than once a year. This results from semi-annual, quarterly, monthly, or daily compounding. In turn, this leads to different present values (PV) or future values (FV) of an investment depending on the frequency of compounding employed. In calculating an investment&#8217;s present or future value with multiple compounding periods per year, the most important thing is ensuring that the interest rate used corresponds to the number of compounding periods present per year.<\/p>\n<h3><strong>Future Value<\/strong><\/h3>\n<p>The future value (FV) of an investment is given by:<\/p>\n<p>$$<br \/>\\mathrm{FV}_{N}=\\mathrm{PV}\\left(1+\\frac{r_{s}}{m}\\right)^{m N}<br \/>$$<\/p>\n<p>Where;<\/p>\n<p>\\(r_s\\) = Quoted annual rate.<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<p>\\(m\\) = Compounding periods (per year).<\/p>\n<p>\\(N\\) = Number of years.<\/p>\n<h3><strong>Present Value<\/strong><\/h3>\n<p>To find the present value of an investment, make PV the subject of the above formula. You should find that:<\/p>\n<p>$$<br \/>\\mathrm{PV}=\\mathrm{FV}\\left\\{\\left(1+\\frac{r_{s}}{m}\\right)^{-m \\times n}\\right\\}<br \/>$$<\/p>\n<h4><strong>Example<\/strong>: <strong>Present Value With Monthly Compounding<\/strong><\/h4>\n<p>Imagine that you wish to have $10,000 in your savings account at the end of the next 3 years. Further, assume that the account offers a return of 9 percent per year, subject to monthly compounding. How much would you need to invest now so as to have the specified amount of money in your account after three years?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we write down the formula to use,<\/p>\n<p>$$ PV= FV \\left\\{ \\left( 1+ \\frac {r_s}{m} \\right) \\right\\}^{ -m\\times n}$$<\/p>\n<p>Secondly, we establish the components that we already have:<\/p>\n<p>\\(r_q\\) = 0.09,\u00a0 \\(m\\) = 12 since compounding is monthly, \\(n\\) = 3 years.<\/p>\n<p>Then, we factor everything into the equation to find our PV:<\/p>\n<p>$$ \\begin{align*} PV &amp; = 10,000 \\left\\{ \\left(1+\\frac {0.09}{12} \\right) \\right\\}^{-12\\times 3} \\\\ &amp; = 10,000\\times 1.0075^{-36} \\\\ &amp; = $7,641.50 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, you will need to invest at least $7,642 in your account to ensure you have $10,000 after three years.<\/p>\n<p>$$\\begin{aligned}&amp;\\textbf{Using the BA II Plus\u2122 Financial Calculator}\\\\ &amp;\\begin{array}{l|l|l}<br \/>\\textbf { Steps } &amp; \\textbf { Explanation } &amp; \\textbf { Display } \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calc mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd }}\\right][\\mathrm{CLR} \\mathrm{TVM}] &amp; \\text { Clears TVM Worksheet } &amp; 0 \\\\<br \/>\\hline 36[\\mathrm{~N}] &amp; \\text { Years\/periods }(12 \\times 3=36) &amp; \\mathrm{N}=36 \\\\<br \/>\\hline 0.75[\\mathrm{I} \/ \\mathrm{Y}] &amp; \\text { Set the interest rate }(9 \/ 12=0.75) &amp; \\mathrm{I} \/ \\mathrm{Y}=0.75 \\\\<br \/>\\hline-10000[\\mathrm{FV}] &amp; \\text { Set the future value } &amp; \\mathrm{FV}=-10000 \\\\<br \/>\\hline 0[\\mathrm{PMT}] &amp; \\text { Set the periodic payment } &amp; \\mathrm{PMT}=0 \\\\<br \/>\\hline[\\mathrm{CPT}][\\mathrm{PV}] &amp; \\text { Compute the present value } &amp; \\mathrm{PV}=7,641.50 \\\\<br \/>\\end{array}\\end{aligned}<br \/>$$<\/p>\n<blockquote>\n<h3><strong>Question 1<\/strong><\/h3>\n<p>Elizabeth Mary invests $2,000 in a project that pays a rate of return of 8% compounded quarterly. The interest that Mary would have earned after investing in the project for two years is <em>closest <\/em>to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>$343.32.<\/li>\n<li>$2,300.00.<\/li>\n<li>$2,343.32.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>$$ \\begin{align*} FV &amp; = 2000 \\left\\{ \\left(1+\\frac {0.08}{4} \\right) \\right\\}^{4\\times 2} \\\\ &amp; = 2,000\\times 1.02^8 \\\\ &amp; = $2,343.32 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, interest gained = 2,343.32-2,000= $343.32<\/p>\n<p>$$\\begin{aligned} &amp;\\textbf{Using the BA II Plus\u2122 Financial Calculator}\\\\ &amp;\\begin{array}{l|l|l}<br \/>\\textbf { Steps } &amp; \\textbf { Explanation } &amp; \\textbf { Display } \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calculator mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd }}\\right][\\text { CLR TVM }] &amp; \\text { Clears the TVM Worksheet } &amp; 0 \\\\<br \/>\\hline 8[\\mathrm{~N}] &amp; \\text { Years\/periods }(4 \\times 2=8) &amp; \\mathrm{N}=8 \\\\<br \/>\\hline 2[\\mathrm{I} \/ \\mathrm{Y}] &amp; \\text { Set the interest rate }(8 \/ 4=2) &amp; \\mathrm{I} \/ \\mathrm{Y}=2 \\\\<br \/>\\hline-2000[\\mathrm{PV}] &amp; \\text { Set the present value } &amp; \\mathrm{PV}=-2000 \\\\<br \/>\\hline 0[\\mathrm{PMT}] &amp; \\text { Set the payment } &amp; \\mathrm{PMT}=0 \\\\<br \/>\\hline[\\mathrm{CPT}][\\mathrm{FV}] &amp; \\text { Compute the future value } &amp; \\mathrm{FV}=2,343.32 \\\\<br \/>\\end{array} \\end{aligned}<br \/>$$<\/p>\n<h3><strong>Question 2<\/strong><\/h3>\n<p>Elizabeth Mary invests $2,000 dollars in a project that pays a rate of return of 8% compounded daily. The interest that Mary would have earned after investing in the project for two years is\u00a0<em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>$343.<\/li>\n<li>$344.<\/li>\n<li>$347.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>$$ \\begin{align*} FV &amp; = 2,000 \\left\\{ \\left( 1+ \\frac {0.08}{365} \\right) \\right\\}^{365\\times 2} \\\\ &amp; = 2,000\\times 1.00021918^{730} \\\\ &amp; = $2,347 \\\\ \\end{align*} $$<\/p>\n<p>Similarly, the interest is \\(2,347 &#8211; 2,000 = $347\\).<\/p>\n<p>You should notice that with a higher compounding frequency, the corresponding profit is also higher. This confirms that interest earned increases as the number of compounding periods per year increases.<\/p>\n<p>$$\\begin{aligned} &amp;\\textbf{Using the BA II Plus\u2122 Financial Calculator}\\\\ &amp;\\begin{array}{l|l|l}<br \/>\\textbf { Steps } &amp; \\textbf { Explanation } &amp; \\textbf { Display } \\\\<br \/>\\hline \\text { [2nd] [QUIT] } &amp; \\text { Return to standard calc mode } &amp; 0 \\\\<br \/>\\hline\\left[2^{\\text {nd }}\\right] \\text { [CLR TVM] } &amp; \\text { Clears TVM Worksheet } &amp; 0 \\\\<br \/>\\hline 730[\\mathrm{~N}] &amp; \\text { Years\/periods }(365 \\times 2=730) &amp; \\mathrm{N}=730 \\\\<br \/>\\hline 0.021918[\\mathrm{I} \/ \\mathrm{Y}] &amp; \\text { Set interest rate }(8 \/ 365=0.021918) &amp; \\mathrm{I} \/ \\mathrm{Y}=0.021918 \\\\<br \/>\\hline-2000[\\mathrm{PV}] &amp; \\text { Set the present value } &amp; \\mathrm{PV}=-2000 \\\\<br \/>\\hline 0[\\mathrm{PMT}] &amp; \\text { Set the payment } &amp; \\mathrm{PMT}=0 \\\\<br \/>\\hline[\\mathrm{CPT}][\\mathrm{FV}] &amp; \\text { Compute the future value } &amp; \\mathrm{FV}=2,347 \\\\<br \/>\\end{array}\\end{aligned}<br \/>$$<\/p>\n<p><strong>A is incorrect<\/strong>. It assumes daily compounding for one year.<\/p>\n<p>$$FV=2,000\\left[1+\\frac{0.08}{365}\\right]^{365}=$2,166.55$$<\/p>\n<p>The interest gained will be \\(\\$2,167-\\$2,000=\\$167\\).<\/p>\n<p><strong>B is incorrect<\/strong>. It assumes a monthly rate in the calculation of FV as opposed to a daily rate as follows.<\/p>\n<p>$$ FV=2,000\\left[1+\\frac{0.08}{12}\\right]^{12\\times2}=\\$2,345.78$$<\/p>\n<p>The interest gained will be \\(\\$2,345-\\$2,000=\\$345\\).<\/p>\n<p><strong>Note<\/strong>: We can convert our stated annual rates into the effective annual rate of interest, and arrive at similar answers. However, if you do that, you should ensure that you use years in the computation.<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin-top: 32px;\"><a style=\"display: inline-block; padding: 16px 40px; background-color: #2563eb; color: #ffffff; text-decoration: none; border-radius: 14px; font-weight: bold; font-size: 17px;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a><\/div>","protected":false},"excerpt":{"rendered":"<p>Time value of money calculations allow us to establish the future value of a given amount of money. The present value (PV) is the money you have today. On the other hand, the future value (FV) is the accumulated amount&#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-28760","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>Time Value of Money &amp; Compounding | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand how different compounding frequencies impact present and future value calculations in time value of money concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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