{"id":481,"date":"2020-04-27T20:51:11","date_gmt":"2020-04-28T00:51:11","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&#038;p=481"},"modified":"2024-10-29T14:18:13","modified_gmt":"2024-10-29T18:18:13","slug":"present-value","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/present-value\/","title":{"raw":"4.4 Present Value\u00a0\u00a0\u00a0\u00a0","rendered":"4.4 Present Value\u00a0\u00a0\u00a0\u00a0"},"content":{"raw":"For many investment decisions, it is necessary to find the principal, or present value, that corresponds to a given future value.\r\n<h2>Example 4.4.1<\/h2>\r\nConsider a note that will pay $10,000 to whoever owns it three years from now. If an investor wants to earn 10% compounded annually, what is the most he or she should pay for the note?\r\n\r\n&nbsp;\r\n\r\nYou have:\r\n<ul>\r\n \t<li><em>i<\/em>=10%=0.10 per year<\/li>\r\n \t<li><em>n<\/em>=3 years<\/li>\r\n \t<li>FV= $10,000<\/li>\r\n<\/ul>\r\nThus, using the compound-interest formula:\r\n<p style=\"text-align: center\">[latex]\\begin{align*}\r\n&amp;PV(1+i)^n=FV\\\\\r\n&amp;PV(1.10)^3=$10,000\\\\\r\n&amp;PV=\\frac{$10,000}{1.1^3} =$7,513.15\\\\\r\n\\end{align*}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThis result can be checked by accumulating the money in an account, as shown in the next box.\r\n<div align=\"center\">\r\n<table class=\"aligncenter\" style=\"height: 90px\">\r\n<tbody>\r\n<tr style=\"height: 18px\">\r\n<th style=\"height: 18px;width: 138.933px\"><strong>Time<\/strong><\/th>\r\n<th style=\"height: 18px;width: 194.05px\"><strong>Interest<\/strong><\/th>\r\n<th style=\"height: 18px;width: 229.9px\"><strong>Balance<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"height: 18px;width: 139.85px\">0<\/td>\r\n<td style=\"height: 18px;width: 194.967px\"><\/td>\r\n<td style=\"height: 18px;width: 229.9px\">$7,513.15<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"height: 18px;width: 139.85px\">1<\/td>\r\n<td style=\"height: 18px;width: 194.967px\">$751.32<\/td>\r\n<td style=\"height: 18px;width: 229.9px\">8,264.46<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"height: 18px;width: 139.85px\">2<\/td>\r\n<td style=\"height: 18px;width: 194.967px\">826.45<\/td>\r\n<td style=\"height: 18px;width: 229.9px\">9,090.91<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"height: 18px;width: 139.85px\">3<\/td>\r\n<td style=\"height: 18px;width: 194.967px\">909.09<\/td>\r\n<td style=\"height: 18px;width: 229.9px\">10,000.00<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h1>Formula for Present Value<\/h1>\r\nThe compound-interest formula can be rewritten to give the present value directly. Divide both sides of the equation by [latex](1+i)^n[\/latex]\r\n<p style=\"text-align: center\">[latex]\\frac{PV(1+i)^n}{(1+i)^n}=\\frac{FV}{(1+i)^n}[\/latex]<\/p>\r\nCancel and rewrite:\r\n<p style=\"text-align: center\">[latex]PV = \\frac{FV}{(1+i)^n}[\/latex]<\/p>\r\n&nbsp;\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Knowledge Check 4.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the present values by using the formula for PV given above.\r\n\r\n&nbsp;\r\n<ol>\r\n \t<li>The present value of $5,849.29 due in two years if interest is at 8% compounded semi-annually.<\/li>\r\n \t<li>The present value of $8,998.91 due in nine months if interest is at 16% compounded quarterly.<\/li>\r\n \t<li>The principal of a loan that would amount to $50,000 in six years at 8.5% compounded annually.<\/li>\r\n<\/ol>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/chap-4-learning-activities-answer-key\/\">Solutions at the end of the chapter<\/a>\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<h2>Your Own Notes<\/h2>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]\r\n\r\n&nbsp;","rendered":"<p>For many investment decisions, it is necessary to find the principal, or present value, that corresponds to a given future value.<\/p>\n<h2>Example 4.4.1<\/h2>\n<p>Consider a note that will pay $10,000 to whoever owns it three years from now. If an investor wants to earn 10% compounded annually, what is the most he or she should pay for the note?<\/p>\n<p>&nbsp;<\/p>\n<p>You have:<\/p>\n<ul>\n<li><em>i<\/em>=10%=0.10 per year<\/li>\n<li><em>n<\/em>=3 years<\/li>\n<li>FV= $10,000<\/li>\n<\/ul>\n<p>Thus, using the compound-interest formula:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align*}  &PV(1+i)^n=FV\\\\  &PV(1.10)^3=$10,000\\\\  &PV=\\frac{$10,000}{1.1^3} =$7,513.15\\\\  \\end{align*}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>This result can be checked by accumulating the money in an account, as shown in the next box.<\/p>\n<div style=\"margin: auto;\">\n<table class=\"aligncenter\" style=\"height: 90px\">\n<tbody>\n<tr style=\"height: 18px\">\n<th style=\"height: 18px;width: 138.933px\"><strong>Time<\/strong><\/th>\n<th style=\"height: 18px;width: 194.05px\"><strong>Interest<\/strong><\/th>\n<th style=\"height: 18px;width: 229.9px\"><strong>Balance<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"height: 18px;width: 139.85px\">0<\/td>\n<td style=\"height: 18px;width: 194.967px\"><\/td>\n<td style=\"height: 18px;width: 229.9px\">$7,513.15<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"height: 18px;width: 139.85px\">1<\/td>\n<td style=\"height: 18px;width: 194.967px\">$751.32<\/td>\n<td style=\"height: 18px;width: 229.9px\">8,264.46<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"height: 18px;width: 139.85px\">2<\/td>\n<td style=\"height: 18px;width: 194.967px\">826.45<\/td>\n<td style=\"height: 18px;width: 229.9px\">9,090.91<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"height: 18px;width: 139.85px\">3<\/td>\n<td style=\"height: 18px;width: 194.967px\">909.09<\/td>\n<td style=\"height: 18px;width: 229.9px\">10,000.00<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h1>Formula for Present Value<\/h1>\n<p>The compound-interest formula can be rewritten to give the present value directly. Divide both sides of the equation by [latex](1+i)^n[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\frac{PV(1+i)^n}{(1+i)^n}=\\frac{FV}{(1+i)^n}[\/latex]<\/p>\n<p>Cancel and rewrite:<\/p>\n<p style=\"text-align: center\">[latex]PV = \\frac{FV}{(1+i)^n}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Knowledge Check 4.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the present values by using the formula for PV given above.<\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li>The present value of $5,849.29 due in two years if interest is at 8% compounded semi-annually.<\/li>\n<li>The present value of $8,998.91 due in nine months if interest is at 16% compounded quarterly.<\/li>\n<li>The principal of a loan that would amount to $50,000 in six years at 8.5% compounded annually.<\/li>\n<\/ol>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/chap-4-learning-activities-answer-key\/\">Solutions at the end of the chapter<\/a><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Your Own Notes<\/h2>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-1\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-1\" class=\"h5p-iframe\" data-content-id=\"1\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":883,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-481","chapter","type-chapter","status-publish","hentry"],"part":44,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/481","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":18,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/481\/revisions"}],"predecessor-version":[{"id":3963,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/481\/revisions\/3963"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/parts\/44"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/481\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/media?parent=481"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapter-type?post=481"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/contributor?post=481"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/license?post=481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}