{"id":69,"date":"2019-09-02T15:12:23","date_gmt":"2019-09-02T15:12:23","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=69"},"modified":"2026-03-08T12:09:28","modified_gmt":"2026-03-08T12:09:28","slug":"compounding-frequencies-exemple-question","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/compounding-frequencies-exemple-question\/","title":{"rendered":"Present Values and Future Values of Investments"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Elizabeth Mary invests $2,000 in a project that pays a rate of return of 8% compounded quarterly. How much interest will Mary have earned after investing in the project for two years?\",\n    \"text\": \"Elizabeth Mary invests $2,000 in a project that pays a rate of return of 8% compounded quarterly. How much interest will Mary have earned after investing in the project for two years?\\n\\nA. $2,300\\n\\nB. $2,343.32\\n\\nC. $343.32\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. The future value is calculated using FV = 2000(1 + 0.08\/4)^(4\u00d72) = 2000(1.02)^8 = $2,343.32. The interest earned is the difference between the future value and the initial investment: $2,343.32 \u2212 $2,000 = $343.32.\"\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 if the project paid a rate of return of 8% compounded daily? How much interest would Elizabeth Mary earn?\",\n    \"text\": \"Elizabeth Mary invests $2,000 in a project. What if the project paid a rate of return of 8% compounded daily? How much interest would Elizabeth Mary earn?\\n\\nA. $2,347\\n\\nB. $347\\n\\nC. $2,340\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The future value is calculated using FV = 2000(1 + 0.08\/365)^(365\u00d72) \u2248 $2,347. Therefore, the interest earned is $2,347 \u2212 $2,000 = $347. This example illustrates that as the number of compounding periods per year increases, the total interest earned also increases.\"\n    }\n  }\n}\n<\/script>\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\": \"This video lesson covers the fundamentals of time value of money, including key concepts such as interest rates, present and future value, annuities, and compounding. It explains opportunity cost, components of interest rates, and demonstrates how to use financial calculators and timelines to solve typical CFA exam problems efficiently.\",\n  \"uploadDate\": \"2019-12-16T00:00:00+00:00\",\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\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/qk19_k31K-4?si=jQ19gz9UzPtWvZ6T\" 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>Some types of investments are known to accumulate interest more than once a year. This results from semi-annual, quarterly, monthly or daily compounding. This, in turn, leads to different present values (PV) or future values (FV) of an investment depending on the frequency of compounding employed.<\/p>\n<p>We have previously seen that the effective annual rate of interest increases as the number of compounding periods per year increases. In calculating the present value or future value of an investment with multiple compounding periods per year, the most important thing is to ensure that the interest rate used corresponds to the number of compounding periods present per year.<\/p>\n<p><strong>Future Value <\/strong><\/p>\n<p>$$ FV= PV \\left\\{ \\left( 1+ \\frac {r_q}{m} \\right) \\right\\}^{ m*n}$$<\/p>\n<p>Where:<\/p>\n<p>r<sub>q<\/sub> is the quoted annual rate;<\/p>\n<p>m represents the number of compounding periods (per year); and<\/p>\n<p>lastly, n is the number of years<\/p>\n<p><strong>Present Value<\/strong><\/p>\n<p>Suppose you make PV the subject of the above formula, you should find that:<\/p>\n<p>$$ PV= FV \\left\\{ \\left( 1+ \\frac {r_q}{m} \\right) \\right\\}^{ -m*n}$$<\/p>\n<h4><strong>Example: Present Value With Monthly Compounding<\/strong><\/h4>\n<p>You wish to have $10,000 in your savings account at the end of the next 3 years. 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 after the three years?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First, we write down the formula to use,<\/p>\n<p>$$ PV= FV \\left\\{ \\left( 1+ \\frac {r_q}{m} \\right) \\right\\}^{ -m*n}$$<\/p>\n<p>Secondly, we establish the components that we already have:<\/p>\n<p>r<sub>q =<\/sub> 0.09,\u00a0\u00a0 m = 12 since compounding is monthly, n = 3 years; and<\/p>\n<p>then, we factor everything into the equation to find our PV.<\/p>\n<p>$$ \\begin{align*} PV &amp; = 10,000 \\left\\{ \\left(1+\\frac {0.09}{12} \\right) \\right\\}^{-12*3} \\\\ &amp; = 10,000*1.0075^{-36} \\\\ &amp; = $7,641.50 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, you will need to invest at least $7,642 in your account to ensure that you have $10,000 after three years.<\/p>\n<blockquote>\n<h2><strong>Question 1<\/strong><\/h2>\n<p>Elizabeth Mary invests $2,000\u00a0 in a project that pays a rate of return of 8% compounded quarterly. How much interest will Mary have earned after investing in the project for two years?<\/p>\n<p>A. $2,300<\/p>\n<p>B. $2,343.32<\/p>\n<p>C. $343.32<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is C.<\/p>\n<p>$$ \\begin{align*} FV &amp; = 2000 \\left\\{ \\left(1+\\frac {0.08}{4} \\right) \\right\\}^{4*2} \\\\ &amp; = 2,000*1.02^8 \\\\ &amp; = $2,343.32 \\\\ \\end{align*} $$<\/p>\n<p>Therefore, interest gained = 2,343.32-2,000= $343.32<\/p>\n<h2><strong>Question 2<\/strong><\/h2>\n<p>What if the project paid a rate of return of 8% compounded daily? How much interest would Elizabeth Mary earn?<\/p>\n<p>A. $2,347<\/p>\n<p>B. $347<\/p>\n<p>C. $2,340<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>$$ \\begin{align*} FV &amp; = 2,000 \\left\\{ \\left( 1+ \\frac {0.08}{365} \\right) \\right\\}^{365*2} \\\\ &amp; = 2,000*1.00021918^{730} \\\\ &amp; = $2,347 \\\\ \\end{align*} $$<\/p>\n<p>Similarly, the interest = 2,347-2,000 = $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<\/blockquote>\n<p><strong>Note<\/strong><\/p>\n<p>We can convert our stated annual rates into the effective annual rate of interest, and arrive at similar answers. However, if we do that, we should ensure that we use years in the computation.<\/p>\n<p><em>Reading 6 LOS 6d<\/em><\/p>\n<p><em>Solve time value of money problems for different frequencies of compounding.<\/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<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Some types of investments are known to accumulate interest more than once a year. This results from semi-annual, quarterly, monthly or daily compounding. This, in turn, leads to different present values (PV) or future values (FV) of an investment depending&#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-69","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>Compounding Frequencies Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how different compounding frequencies impact interest accumulation and investment growth in financial calculations.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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