{"id":185513,"date":"2025-01-22T15:33:46","date_gmt":"2025-01-22T15:33:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185513"},"modified":"2025-01-22T15:33:48","modified_gmt":"2025-01-22T15:33:48","slug":"calculate-the-perpetual-equivalent-annual-worth-in-future-dollars-for-years-1-through-%e2%88%9e-for-income-of-50000-now-and-5000-per-year-thereafter","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/22\/calculate-the-perpetual-equivalent-annual-worth-in-future-dollars-for-years-1-through-%e2%88%9e-for-income-of-50000-now-and-5000-per-year-thereafter\/","title":{"rendered":"Calculate the perpetual equivalent annual worth in future dollars for years 1 through \u221e for income of $50,000 now and $5000 per year thereafter"},"content":{"rendered":"\n<p>(a) Calculate the perpetual equivalent annual worth in future dollars for years 1 through \u221e for income of $50,000 now and $5000 per year thereafter. Assume the market interest rate is 8% per year and inflation averages 4% per year. All amounts are quoted as future dollars. (b) If the amounts had been quoted in CV dollars, what is the annual worth in future dollars?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To solve this problem, we&#8217;ll break it down into two parts:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Part (a) &#8211; Perpetual Equivalent Annual Worth in Future Dollars<\/h3>\n\n\n\n<p>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>An income of $50,000 now (this is a lump sum at time 0),<\/li>\n\n\n\n<li>A recurring income of $5,000 per year starting from year 1 and continuing perpetually,<\/li>\n\n\n\n<li>An interest rate of 8% per year,<\/li>\n\n\n\n<li>Inflation rate of 4% per year.<\/li>\n<\/ul>\n\n\n\n<p>We are required to calculate the perpetual equivalent annual worth (in future dollars) for years 1 through \u221e.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Adjust for Inflation<\/strong>:<br>We need to account for the fact that future cash flows are quoted in future dollars, meaning that the amounts will increase due to inflation. To calculate the future worth of $50,000 and $5,000 at future times, we will first adjust them for inflation.<\/li>\n\n\n\n<li><strong>Present Worth of Perpetual Income<\/strong>:<br>To calculate the present worth of a perpetuity (the $5,000 per year), we use the formula for the present value of a perpetuity: [<br>PV_{\\text{perpetuity}} = \\frac{C}{r &#8211; i}<br>] Where:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(C) is the annual cash flow ($5,000),<\/li>\n\n\n\n<li>(r) is the nominal interest rate (8% or 0.08),<\/li>\n\n\n\n<li>(i) is the inflation rate (4% or 0.04). Substituting the values: [<br>PV_{\\text{perpetuity}} = \\frac{5000}{0.08 &#8211; 0.04} = \\frac{5000}{0.04} = 125,000<br>] This is the present value of the perpetuity in future dollars. Now we will adjust this value for inflation to get the future value.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Future Worth of Perpetuity<\/strong>:<br>Since we are interested in future dollars, the perpetuity&#8217;s future worth (in year 1) can be calculated by adjusting the present value using the inflation rate: [<br>FV_{\\text{perpetuity}} = PV_{\\text{perpetuity}} \\times (1 + i) = 125,000 \\times (1 + 0.04) = 130,000<br>] So, the future value of the perpetual cash flow is $130,000 per year.<\/li>\n\n\n\n<li><strong>Future Worth of $50,000 Initial Lump Sum<\/strong>:<br>The lump sum of $50,000 is received now, and we need to calculate its future value in year 1 (as a future lump sum) using the market interest rate. The future value of the lump sum is: [<br>FV_{\\text{lump sum}} = 50,000 \\times (1 + 0.08) = 54,000<br>] So, the future worth of the lump sum in year 1 is $54,000.<\/li>\n\n\n\n<li><strong>Total Future Worth<\/strong>:<br>The total future worth, combining both the lump sum and the perpetuity, is: [<br>FV_{\\text{total}} = 54,000 + 130,000 = 184,000<br>]<\/li>\n<\/ol>\n\n\n\n<p>Thus, the perpetual equivalent annual worth in future dollars for years 1 through \u221e is <strong>$184,000<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Part (b) &#8211; Annual Worth in CV Dollars (Constant Value Dollars)<\/h3>\n\n\n\n<p>In CV dollars, we need to adjust the future worths to account for inflation.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Present Worth in CV Dollars<\/strong>:<br>The lump sum of $50,000 is quoted in constant value dollars, so it doesn&#8217;t require any adjustments for inflation. However, the perpetuity requires us to use the inflation-adjusted interest rate to find its present value in CV dollars.<\/li>\n\n\n\n<li><strong>Adjusted Perpetuity in CV Dollars<\/strong>:<br>The formula for the present value of a perpetuity in CV dollars is: [<br>PV_{\\text{perpetuity (CV)}} = \\frac{C}{r} = \\frac{5000}{0.08} = 62,500<br>] This is the present value of the perpetuity in CV dollars.<\/li>\n\n\n\n<li><strong>Total Present Worth in CV Dollars<\/strong>:<br>The total present value in CV dollars (sum of the lump sum and perpetuity in CV dollars) is: [<br>PV_{\\text{total (CV)}} = 50,000 + 62,500 = 112,500<br>]<\/li>\n\n\n\n<li><strong>Convert to Future Dollars<\/strong>:<br>To convert this amount to future dollars, we multiply by the inflation factor: [<br>FV_{\\text{total (CV)}} = 112,500 \\times (1 + 0.04) = 117,000<br>]<\/li>\n<\/ol>\n\n\n\n<p>Thus, the annual worth in future dollars, when quoted in constant value (CV) dollars, is <strong>$117,000<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The perpetual equivalent annual worth in future dollars (for years 1 through \u221e) is <strong>$184,000<\/strong>.<\/li>\n\n\n\n<li>The annual worth in future dollars, when quoted in CV dollars, is <strong>$117,000<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>In essence, the future value of the lump sum and perpetuity is greater in future dollars due to inflation adjustments, whereas the constant value dollars reflect the current value of money.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(a) Calculate the perpetual equivalent annual worth in future dollars for years 1 through \u221e for income of $50,000 now and $5000 per year thereafter. Assume the market interest rate is 8% per year and inflation averages 4% per year. All amounts are quoted as future dollars. (b) If the amounts had been quoted in [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-185513","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=185513"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185513\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}