{"id":519,"date":"2020-04-28T13:43:31","date_gmt":"2020-04-28T17:43:31","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&#038;p=519"},"modified":"2021-10-22T18:49:37","modified_gmt":"2021-10-22T22:49:37","slug":"equivalent-and-effective-rates","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/equivalent-and-effective-rates\/","title":{"raw":"4.8 Equivalent and Effective Rates","rendered":"4.8 Equivalent and Effective Rates"},"content":{"raw":"Interest rates are <strong><em>[pb_glossary id=\"525\"]EQUIVALENT[\/pb_glossary] <\/em><\/strong>if they provide the <em>same amount of interest <\/em>on any loan. Consider the two rates:\r\n<ol>\r\n \t<li>20.5% compounded semi-annually.<\/li>\r\n \t<li>20% compounded quarterly.<\/li>\r\n<\/ol>\r\nIf you take <em>any principal <\/em>and <em>any length of time, <\/em>you will find that the two rates always result in exactly the same future value - hence the <em>same interest. <\/em>This is because they are related by the algebraic expression:\r\n\r\nTo check the equivalence, consider the following example.\r\n<p style=\"text-align: center\">[latex]\\left(1+\\frac{0.205}{2}\\right)^2 = \\left(1+\\frac{0.20}{4}\\right)^4[\/latex]<\/p>\r\n\r\n<h2>Example 4.8.1<\/h2>\r\nSuppose $50,000 is invested for seven years at the interest rates noted above. Find the future value of the $50,000 for each interest rate.\r\n\r\nYou have:\r\n\r\n<img class=\"aligncenter wp-image-520 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-300x90.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"90\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaway<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">If two rates produce the same result for any principal and time, the rates will do so for any values.<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\nFor <em>rate <\/em>a: 20.5% compounded semi-annually\r\n<ul>\r\n \t<li><em>n<\/em> = 7\u00d72 = 14 half-years<\/li>\r\n \t<li>I = [latex]\\frac{0.0205}{2}=10.25%[\/latex]<\/li>\r\n \t<li>PV =$50,0000<\/li>\r\n \t<li>FV = ?<\/li>\r\n<\/ul>\r\nAnswer: $196,006.46 for <em>rate b<\/em>\r\n\r\n&nbsp;\r\n\r\nThat exactly the <em>same future value <\/em>is obtained for both rates bears out the claim that the rates are equivalent. In fact, if two rates produce\r\n\r\nthe same result for <em>any <\/em>(non-zero) principal and time, then the rates will do so for <em>any values. <\/em>Hence they are equivalent.\r\n\r\nYou\u00a0 can use calculator functions to find equivalent rates fairly easily, but first we will use the Future Value formula. You can use any size of investment and any length of time, but to illustrate this in the next example, $1 for one year is used.\r\n<h2>Example 4.8.2<\/h2>\r\nSuppose you are given the 20% compounded quarterly rate mentioned above and are asked to find the equivalent rate compounded\r\n\r\nsemi-annually. You are given:\r\n\r\n<img class=\"aligncenter wp-image-544 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-300x77.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"77\" \/>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li><em>n<\/em> = 1\u00d7 4 = 4<\/li>\r\n \t<li>[latex]i = \\frac{0.20}{4} = 0.05[\/latex]<\/li>\r\n \t<li>PV = $1<\/li>\r\n \t<li>FV = ?<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nUsing the Future Value Formula, we have;\r\n<p style=\"text-align: center\">[latex]FV = PV(1+i)^n = $1(1.05)^4 = $1.21550625[\/latex]<\/p>\r\n(Leave this answer in the calculator.)\r\n\r\nFor the new rate, the <em>only thing <\/em>that will be different (aside from i) is that it is to be compounded only twice in the year.\r\n\r\n&nbsp;\r\n\r\nSo we have\r\n<ul>\r\n \t<li><em>n<\/em> = 1\u00d7 2 =2<\/li>\r\n \t<li><em>i<\/em> = ?<\/li>\r\n \t<li>PV = $1<\/li>\r\n \t<li>FV = $1.21550625<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nUsing the Future Value Formula, we have;\r\n<p style=\"text-align: center\">[latex]\\begin{align*}\r\nPV(1+i)^n &amp;= FV\\\\\r\n$1(1+i)^2 &amp;= $1.21550625\\\\\r\n(1+i)^2 &amp;= \\frac{$1.21550625}{$1}\\\\\r\n1+ i &amp;= \\sqrt{$1.21550625} = 1.1025\\\\ \r\ni&amp;= 1.1025-1\\\\\r\n&amp;=0.1025\\\\\r\n\\end{align*}[\/latex]<\/p>\r\nSo the nominal rate would be <em>j<sub>2<\/sub><\/em> = 10.25% \u00d7 2 = 20.5%.\r\n\r\n&nbsp;\r\n<h1>Effective Rates<\/h1>\r\n<em>The equivalent rate compounded annually for a given compound\u00ad interest rate.<\/em>\r\n\r\nCompound-interest rates are <em>compared <\/em>by finding for each rate the equivalent rate <em>compounded annually. <\/em>For a given compound-interest rate, the equivalent rate compounded annually is called its <em>[pb_glossary id=\"548\"]effective rate[\/pb_glossary].<\/em>\r\n\r\n<img class=\"aligncenter wp-image-545 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-300x75.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"75\" \/>\r\n<h2>Example 4.8.3<\/h2>\r\nFind Effective rates for the following:\r\n\r\na.\u00a0\u00a0\u00a0 20.5% compounded semi-annually:\u00a0<em> j<sub>2<\/sub><\/em> = 0.205 so <em>i<\/em> = 0.1025\r\n<p style=\"text-align: center\">[latex]$1(1.1025)^2 = $1.21550625[\/latex]<\/p>\r\nTo find the effective rate, we can just evaluate the Future Value equation for n = 1 year:\r\n<p style=\"text-align: center\">[latex]$1(1+i)^1 = $1.21550625[\/latex]<\/p>\r\nThis is trivial to solve:<em> j<sub>1<\/sub><\/em> = <em>i<\/em> =\u00a0 21.550625%.\u00a0 In fact, the interest earned on $1 invested for a year is the equivalent rate!\r\n\r\n&nbsp;\r\n\r\nb. 20% compounded quarterly.\r\n\r\n<img class=\"aligncenter wp-image-546 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-300x71.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"71\" \/>\r\n\r\n<em>n<\/em> = 1 \u00d7 4 quarters, <em>i<\/em> = 20 \u00f7 4 = 5%, PV = $1, FV =?\r\n<p style=\"text-align: center\">[latex]$1(1.05)^4 = $1.21550625[\/latex]<\/p>\r\nSo we can see that the effective rate is also<em> j<sub>1<\/sub><\/em> = 21.550625%. You can see, then, that each rate was equivalent to 21.550625% compounded annually -\u00a0 which also shows that they were equivalent, since they were both <em>equivalent <\/em>to the <em>same effective rate.<\/em>\r\n\r\n&nbsp;\r\n<h1>Effective Rates with the BAII Plus<\/h1>\r\n<h2>Example 4.8.4<\/h2>\r\n&nbsp;\r\n\r\n<img class=\"wp-image-507 alignleft\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/BAIIPlus.jpg\" alt=\"Decorative image of a BA II Plus Calculator\" width=\"158\" height=\"158\" \/>A bank offers a certificate that pays a nominal interest rate of 15% with quarterly compounding. What is the annual effective interest rate?\r\n\r\n&nbsp;\r\n<table class=\"grid aligncenter\" style=\"width: 100%\">\r\n<thead>\r\n<tr>\r\n<td><strong>Step<\/strong><\/td>\r\n<td><strong>To<\/strong><\/td>\r\n<td><strong>Press<\/strong><\/td>\r\n<td><strong>Display<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>Select Interest Conversion worksheet<\/td>\r\n<td>[2ND] [ICONV]<\/td>\r\n<td>NOM = 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>Enter\u00a0 nominal interest rate, <em>NOM<\/em> = 15<\/td>\r\n<td>[1][5][ENTER]<\/td>\r\n<td>NOM = 15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>Enter number of compounding periods per year, <em>C\/Y<\/em> = 4<\/td>\r\n<td>[\u2193] [4] [ENTER]<\/td>\r\n<td>C\/Y = 4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>Compute annual effective rate, <em>EFF<\/em><\/td>\r\n<td>[\u2193][CPT]<\/td>\r\n<td>EFF = 15.8650415<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Example 4.8.5<\/h2>\r\nTry the following: \u2193\r\n\r\nFind the interest rate, compounded quarterly, that is equivalent to 15% compounded monthly.\r\n\r\n[2ND][ICONV]\r\n\r\nNOM = 15 [ENTER]\r\n\r\n\u2191C\/Y = 12 [ENTER]\r\n\r\n\u2191EFF = [CPT]\u00a0 16.07545177\r\n\r\n\u2193C\/Y = 4 [ENTER]\r\n\r\n\u2193NOM = [CPT] 15.18828125\r\n\r\nThus, the following three rates are all equivalent:\r\n<p style=\"text-align: center\"><em>j<sub>12<\/sub> = <\/em>15%<em> \u21d4 j<sub>1<\/sub>=<\/em>16.07545177%<em> \u21d4 j<sub>4<\/sub>= <\/em>15.18828125%<\/p>\r\n\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>Interest rates are <strong><em><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_519_525\">EQUIVALENT<\/a> <\/em><\/strong>if they provide the <em>same amount of interest <\/em>on any loan. Consider the two rates:<\/p>\n<ol>\n<li>20.5% compounded semi-annually.<\/li>\n<li>20% compounded quarterly.<\/li>\n<\/ol>\n<p>If you take <em>any principal <\/em>and <em>any length of time, <\/em>you will find that the two rates always result in exactly the same future value &#8211; hence the <em>same interest. <\/em>This is because they are related by the algebraic expression:<\/p>\n<p>To check the equivalence, consider the following example.<\/p>\n<p style=\"text-align: center\">[latex]\\left(1+\\frac{0.205}{2}\\right)^2 = \\left(1+\\frac{0.20}{4}\\right)^4[\/latex]<\/p>\n<h2>Example 4.8.1<\/h2>\n<p>Suppose $50,000 is invested for seven years at the interest rates noted above. Find the future value of the $50,000 for each interest rate.<\/p>\n<p>You have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-520 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-300x90.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"90\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-300x90.png 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-65x20.png 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-225x68.png 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1-350x105.png 350w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv1.png 645w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaway<\/p>\n<\/header>\n<div class=\"textbox__content\">If two rates produce the same result for any principal and time, the rates will do so for any values.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>For <em>rate <\/em>a: 20.5% compounded semi-annually<\/p>\n<ul>\n<li><em>n<\/em> = 7\u00d72 = 14 half-years<\/li>\n<li>I = [latex]\\frac{0.0205}{2}=10.25%[\/latex]<\/li>\n<li>PV =$50,0000<\/li>\n<li>FV = ?<\/li>\n<\/ul>\n<p>Answer: $196,006.46 for <em>rate b<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>That exactly the <em>same future value <\/em>is obtained for both rates bears out the claim that the rates are equivalent. In fact, if two rates produce<\/p>\n<p>the same result for <em>any <\/em>(non-zero) principal and time, then the rates will do so for <em>any values. <\/em>Hence they are equivalent.<\/p>\n<p>You\u00a0 can use calculator functions to find equivalent rates fairly easily, but first we will use the Future Value formula. You can use any size of investment and any length of time, but to illustrate this in the next example, $1 for one year is used.<\/p>\n<h2>Example 4.8.2<\/h2>\n<p>Suppose you are given the 20% compounded quarterly rate mentioned above and are asked to find the equivalent rate compounded<\/p>\n<p>semi-annually. You are given:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-544 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-300x77.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"77\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-300x77.png 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-225x58.png 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2-350x90.png 350w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-equiv2.png 640w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li><em>n<\/em> = 1\u00d7 4 = 4<\/li>\n<li>[latex]i = \\frac{0.20}{4} = 0.05[\/latex]<\/li>\n<li>PV = $1<\/li>\n<li>FV = ?<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>Using the Future Value Formula, we have;<\/p>\n<p style=\"text-align: center\">[latex]FV = PV(1+i)^n = $1(1.05)^4 = $1.21550625[\/latex]<\/p>\n<p>(Leave this answer in the calculator.)<\/p>\n<p>For the new rate, the <em>only thing <\/em>that will be different (aside from i) is that it is to be compounded only twice in the year.<\/p>\n<p>&nbsp;<\/p>\n<p>So we have<\/p>\n<ul>\n<li><em>n<\/em> = 1\u00d7 2 =2<\/li>\n<li><em>i<\/em> = ?<\/li>\n<li>PV = $1<\/li>\n<li>FV = $1.21550625<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>Using the Future Value Formula, we have;<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align*}  PV(1+i)^n &= FV\\\\  $1(1+i)^2 &= $1.21550625\\\\  (1+i)^2 &= \\frac{$1.21550625}{$1}\\\\  1+ i &= \\sqrt{$1.21550625} = 1.1025\\\\   i&= 1.1025-1\\\\  &=0.1025\\\\  \\end{align*}[\/latex]<\/p>\n<p>So the nominal rate would be <em>j<sub>2<\/sub><\/em> = 10.25% \u00d7 2 = 20.5%.<\/p>\n<p>&nbsp;<\/p>\n<h1>Effective Rates<\/h1>\n<p><em>The equivalent rate compounded annually for a given compound\u00ad interest rate.<\/em><\/p>\n<p>Compound-interest rates are <em>compared <\/em>by finding for each rate the equivalent rate <em>compounded annually. <\/em>For a given compound-interest rate, the equivalent rate compounded annually is called its <em><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_519_548\">effective rate<\/a>.<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-545 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-300x75.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"75\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-300x75.png 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-225x57.png 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec-350x88.png 350w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chap4-effec.png 653w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2>Example 4.8.3<\/h2>\n<p>Find Effective rates for the following:<\/p>\n<p>a.\u00a0\u00a0\u00a0 20.5% compounded semi-annually:\u00a0<em> j<sub>2<\/sub><\/em> = 0.205 so <em>i<\/em> = 0.1025<\/p>\n<p style=\"text-align: center\">[latex]$1(1.1025)^2 = $1.21550625[\/latex]<\/p>\n<p>To find the effective rate, we can just evaluate the Future Value equation for n = 1 year:<\/p>\n<p style=\"text-align: center\">[latex]$1(1+i)^1 = $1.21550625[\/latex]<\/p>\n<p>This is trivial to solve:<em> j<sub>1<\/sub><\/em> = <em>i<\/em> =\u00a0 21.550625%.\u00a0 In fact, the interest earned on $1 invested for a year is the equivalent rate!<\/p>\n<p>&nbsp;<\/p>\n<p>b. 20% compounded quarterly.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-546 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-300x71.png\" alt=\"Timeline showing Present Value (PV) and Future Value (FV)\" width=\"300\" height=\"71\" srcset=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-300x71.png 300w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-65x15.png 65w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-225x53.png 225w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart-350x83.png 350w, https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/chao4-quart.png 652w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><em>n<\/em> = 1 \u00d7 4 quarters, <em>i<\/em> = 20 \u00f7 4 = 5%, PV = $1, FV =?<\/p>\n<p style=\"text-align: center\">[latex]$1(1.05)^4 = $1.21550625[\/latex]<\/p>\n<p>So we can see that the effective rate is also<em> j<sub>1<\/sub><\/em> = 21.550625%. You can see, then, that each rate was equivalent to 21.550625% compounded annually &#8211;\u00a0 which also shows that they were equivalent, since they were both <em>equivalent <\/em>to the <em>same effective rate.<\/em><\/p>\n<p>&nbsp;<\/p>\n<h1>Effective Rates with the BAII Plus<\/h1>\n<h2>Example 4.8.4<\/h2>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-507 alignleft\" src=\"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-content\/uploads\/sites\/971\/2020\/04\/BAIIPlus.jpg\" alt=\"Decorative image of a BA II Plus Calculator\" width=\"158\" height=\"158\" \/>A bank offers a certificate that pays a nominal interest rate of 15% with quarterly compounding. What is the annual effective interest rate?<\/p>\n<p>&nbsp;<\/p>\n<table class=\"grid aligncenter\" style=\"width: 100%\">\n<thead>\n<tr>\n<td><strong>Step<\/strong><\/td>\n<td><strong>To<\/strong><\/td>\n<td><strong>Press<\/strong><\/td>\n<td><strong>Display<\/strong><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>Select Interest Conversion worksheet<\/td>\n<td>[2ND] [ICONV]<\/td>\n<td>NOM = 0<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>Enter\u00a0 nominal interest rate, <em>NOM<\/em> = 15<\/td>\n<td>[1][5][ENTER]<\/td>\n<td>NOM = 15<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>Enter number of compounding periods per year, <em>C\/Y<\/em> = 4<\/td>\n<td>[\u2193] [4] [ENTER]<\/td>\n<td>C\/Y = 4<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>Compute annual effective rate, <em>EFF<\/em><\/td>\n<td>[\u2193][CPT]<\/td>\n<td>EFF = 15.8650415<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example 4.8.5<\/h2>\n<p>Try the following: \u2193<\/p>\n<p>Find the interest rate, compounded quarterly, that is equivalent to 15% compounded monthly.<\/p>\n<p>[2ND][ICONV]<\/p>\n<p>NOM = 15 [ENTER]<\/p>\n<p>\u2191C\/Y = 12 [ENTER]<\/p>\n<p>\u2191EFF = [CPT]\u00a0 16.07545177<\/p>\n<p>\u2193C\/Y = 4 [ENTER]<\/p>\n<p>\u2193NOM = [CPT] 15.18828125<\/p>\n<p>Thus, the following three rates are all equivalent:<\/p>\n<p style=\"text-align: center\"><em>j<sub>12<\/sub> = <\/em>15%<em> \u21d4 j<sub>1<\/sub>=<\/em>16.07545177%<em> \u21d4 j<sub>4<\/sub>= <\/em>15.18828125%<\/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<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_519_525\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_519_525\"><div tabindex=\"-1\"><p>This is a mathematical term, meaning that two things are the same in the ways we want them to.\u00a0 In the case of interest rates, two rates are equivalent if an investment at each rate gives the same Future Value after one year.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_519_548\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_519_548\"><div tabindex=\"-1\"><p>The equivalent rate compunded annually<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":883,"menu_order":11,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-519","chapter","type-chapter","status-publish","hentry"],"part":44,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/519","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":25,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/519\/revisions"}],"predecessor-version":[{"id":3911,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapters\/519\/revisions\/3911"}],"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\/519\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/media?parent=519"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/pressbooks\/v2\/chapter-type?post=519"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/contributor?post=519"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/wp-json\/wp\/v2\/license?post=519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}