{"id":17784,"date":"2021-08-21T02:41:48","date_gmt":"2021-08-21T02:41:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17784"},"modified":"2021-08-23T04:29:52","modified_gmt":"2021-08-23T04:29:52","slug":"section-5-7-financial-models","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/section-5-7-financial-models\/","title":{"raw":"Section 5.7: Financial Models","rendered":"Section 5.7: Financial Models"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the simple interest formula.<\/li>\r\n \t<li>Use the compound interest formulas.<\/li>\r\n \t<li>Find the effective rate of interest.<\/li>\r\n \t<li>Find the future value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>\u00a0Use Simple Interest and Compound Interest Formulas<\/h2>\r\n<p id=\"fs-id1165137447026\">Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account) is called the <strong>principal<\/strong>.\u00a0The <strong>rate of interest<\/strong>, expressed as a percent, is the amount charged for the use of the principle for a given period of time, usually on a yearly (that is, per annum) basis.<\/p>\r\n\r\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Simple Interest Formula<\/h3>\r\n<p id=\"fs-id1165135184167\"><strong>Simple interest<\/strong> can be calculated using the formula<\/p>\r\n\r\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]I=Prt[\/latex]<\/div>\r\n<p id=\"eip-237\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137448453\">\r\n \t<li><em>I<\/em>\u00a0is the amount of interest,<\/li>\r\n \t<li><i>t<\/i> is measured in years,<\/li>\r\n \t<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_01_08\" class=\"example\">\r\n<div id=\"fs-id1165135422940\" class=\"exercise\">\r\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Calculating Simple Interest<\/h3>\r\n<p id=\"fs-id1165137812825\">If we take out a $2,000 loan which charges 4% simple interest for 5 years, how much interest is paid?<\/p>\r\n[reveal-answer q=\"162184\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162184\"]\r\n<p id=\"fs-id1165137812832\">Because we are starting with $2,000, <em>P\u00a0<\/em>= 2000. Our interest rate is 4%, so <em>r<\/em>\u00a0=\u00a00.04. In this problem,\u00a0<em>t <\/em>= 5.<\/p>\r\n[latex]I =Prt[\/latex]\r\n[latex]I =2000*0.04*5[\/latex]\r\n[latex]I= $400[\/latex]\r\n<p id=\"fs-id1165137694040\">The amount if interest paid is $400 in 5 years.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nSavings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use <strong>compound interest<\/strong>. The term <em>compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.\r\n<p id=\"fs-id1165137447037\">The <strong>annual percentage rate (APR)<\/strong> of an account, also called the <strong>nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0<em>nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being <em>greater<\/em> than the nominal rate! This is a powerful tool for investing.<\/p>\r\n<p id=\"fs-id1165135160118\">We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time <em>t<\/em>, principal <em>P<\/em>, APR <em>r<\/em>, and number of compounding periods in a year\u00a0<em>n<\/em>:<\/p>\r\n\r\n<div id=\"eip-986\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/div>\r\n<p id=\"fs-id1165137935717\">For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.<\/p>\r\n\r\n<table id=\"Table_04_01_03\" summary=\"Six rows and two columns. The first column is labeled, \">\r\n<thead>\r\n<tr>\r\n<th>Frequency<\/th>\r\n<th>Value after 1 year<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Annually<\/td>\r\n<td>$1100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Semiannually<\/td>\r\n<td>$1102.50<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quarterly<\/td>\r\n<td>$1103.81<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Monthly<\/td>\r\n<td>$1104.71<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Daily<\/td>\r\n<td>$1105.16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Compound Interest Formula<\/h3>\r\n<p id=\"fs-id1165135184167\"><strong>Compound interest<\/strong> can be calculated using the formula<\/p>\r\n\r\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/div>\r\n<p id=\"eip-237\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137448453\">\r\n \t<li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\r\n \t<li><i>t<\/i> is measured in years,<\/li>\r\n \t<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal, and<\/li>\r\n \t<li><em>n<\/em>\u00a0is the number of compounding periods in one year.<\/li>\r\n \t<li>annually [latex](n=1)[\/latex], semiannually [latex](n=2)[\/latex], quarterly [latex](n=4)[\/latex], monthly [latex](n=12)[\/latex], weekly [latex](n=52)[\/latex], yearly [latex](n=365)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_01_08\" class=\"example\">\r\n<div id=\"fs-id1165135422940\" class=\"exercise\">\r\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Calculating Compound Interest<\/h3>\r\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?<\/p>\r\n[reveal-answer q=\"162183\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162183\"]\r\n<p id=\"fs-id1165137812832\">Because we are starting with $3,000, <em>P\u00a0<\/em>= 3000. Our interest rate is 3%, so <em>r<\/em>\u00a0=\u00a00.03. Because we are compounding quarterly, we are compounding 4 times per year, so <em>n\u00a0<\/em>= 4. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}A\\left(t\\right) &amp; =P\\left(1+\\frac{r}{n}\\right)^{nt}&amp;&amp; \\text{Use the compound interest formula}. \\\\ A\\left(10\\right)&amp; =3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}&amp;&amp; \\text{Substitute using given values}. \\\\ &amp; \\approx 4045.05&amp;&amp; \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137694040\">The account will be worth about $4,045.05 in 10 years.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?<\/p>\r\n[reveal-answer q=\"474337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474337\"]\r\n\r\nabout $3,644,675.88\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]14375[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Continuous Compounding<\/h2>\r\nIf the number of compoundings goes to infinity, then we have <strong>continuous compounding<\/strong>.\r\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Compounding Continuously Formula<\/h3>\r\n<p id=\"fs-id1165135184167\">If you are <strong>compounding continuously<\/strong>, use the formula<\/p>\r\n\r\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=Pe^{rt}[\/latex]<\/div>\r\n<p id=\"eip-237\">where<\/p>\r\n\r\n<ul id=\"fs-id1165137448453\">\r\n \t<li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\r\n \t<li><i>t<\/i> is measured in years,<\/li>\r\n \t<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\r\n \t<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Example_04_01_08\" class=\"example\">\r\n<div id=\"fs-id1165135422940\" class=\"exercise\">\r\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Compounding Continuously<\/h3>\r\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded continuously, how much will the account be worth in 10 years?<\/p>\r\n[reveal-answer q=\"162182\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162182\"]\r\n<p id=\"fs-id1165137812832\">Because we are compounding continuously, we need to use the compounding continuously formula. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}A\\left(t\\right) &amp; =Pe^{rt}&amp;&amp; \\text{Use the compounding continuously formula}. \\\\ A\\left(10\\right)&amp; =3000e^{0.03*10}&amp;&amp; \\text{Substitute using given values}. \\\\ &amp; \\approx 4049.58&amp;&amp; \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137694040\">The account will be worth about $4,049.58 in 10 years.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded continuously. What will the investment be worth in 30 years?<\/p>\r\n[reveal-answer q=\"474338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474338\"]\r\n\r\nabout $3,659,823.44\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Effective Rate of Interest<\/h2>\r\nThe <strong>effective rate of interest<\/strong> is the annual simple interest rate that would yield the same amount as compounding <i>n<\/i> times per year, or continuously, after one year.\r\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Effective Rate of Interest<\/h3>\r\n<p id=\"fs-id1165135184167\">The effective rate of interest [latex]r_E[\/latex] of an investment earning an annual interest rate <i>r<\/i> is given by<\/p>\r\n\r\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: left;\">\r\n<ul id=\"fs-id1165137448453\">\r\n \t<li>Compounding <i>n<\/i> times per year: [latex]r_E=\\left(1+\\frac{r}{n}\\right)^n-1[\/latex]<\/li>\r\n \t<li>Continuous Compounding: [latex]r_E=e^r-1[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_04_01_08\" class=\"example\">\r\n<div id=\"fs-id1165135422940\" class=\"exercise\">\r\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Effective Rate of Interest<\/h3>\r\n<p id=\"fs-id1165137812825\">Find the effective rate of interest for 7% compounded quarterly.<\/p>\r\n[reveal-answer q=\"162189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162189\"]\r\n<p id=\"fs-id1165137812832\">This is compounded quarterly, so <i>n\u00a0<\/i>= 4. Use the formula [latex]r_E=\\left(1+\\frac{r}{n}\\right)^n-1[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}r_E &amp; =\\left(1+\\frac{r}{n}\\right)^n-1&amp;&amp; \\text{Use the correct formula}. \\\\ r_E &amp; =\\left(1+\\frac{0.07}{4}\\right)^4&amp;&amp; \\text{Substitute using given values}. \\\\ &amp; \\approx 0.071859 &amp;&amp; \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137694040\">The effective rate of interest is approximately 7.19%<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135180428\">Find the effective rate of interest for 5% compounded continuously.<\/p>\r\n[reveal-answer q=\"474339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"474339\"]\r\n\r\napproximately 5.13%\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 style=\"text-align: center;\">Section 5.7 Homework Exercises<\/h2>\r\nFor the following exercises, use the compound interest formula, [latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex].\r\n\r\n1. What was the initial deposit made to the account in the previous exercise?\r\n\r\n2. How many years had the account from the previous exercise been accumulating interest?\r\n\r\n3. An account is opened with an initial deposit of $6,500 and earns 3.6% interest compounded semi-annually. What will the account be worth in 20 years?\r\n\r\n4. How much more would the account in the previous exercise have been worth if the interest were compounding weekly?\r\n\r\n5. Solve the compound interest formula for the principal, P.\r\n\r\n6. Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)\r\n\r\n7. How much more would the account in the previous two exercises be worth if it were earning interest for 5 more years?\r\n\r\n8. Use properties of rational exponents to solve the compound interest formula for the interest rate, r.\r\n\r\n9. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.\r\n\r\n10.\u00a0Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.\r\n\r\n11.\u00a0Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?\r\n\r\n12. How much less would the account from Exercise 11 be worth after 30 years if it were compounded monthly instead?\r\n\r\n13. Determine the rate that represents the better deal:\u00a0 7% compounded semiannually or 6.9% compounded continuously?\r\n\r\n14. Determine the rate that represents the better deal: 9% compounded annually or 8.9% compounded continuously?\r\n\r\nIn problems 15 - 20, find the principle needed now to get each amount; that is, find the present value.\r\n\r\n15. To get $100 after 2 years at 6% compounded monthly\r\n\r\n16. To get $75 after 3 years at 8% compounded quarterly\r\n\r\n17. To get $1500 after [latex]2 \\frac{1}{2}[\/latex]  years at 1.5% compounded daily\r\n\r\n18. To get $800 after [latex]3 \\frac{1}{2}[\/latex] years at 7% compounded monthly\r\n\r\n19. To get $750 after 2 years at 2.5% compounded quarterly\r\n\r\n20. To get $300 after 4 years at 3% compounded daily\r\n\r\nIn problems 21 - 24, find the effective rate of interest.\r\n\r\n21. For 5% compounded quarterly\r\n\r\n22. For 6% compounded monthly\r\n\r\n23. For 4% compounded continuously\r\n\r\n24. For 6% compounded continuously","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the simple interest formula.<\/li>\n<li>Use the compound interest formulas.<\/li>\n<li>Find the effective rate of interest.<\/li>\n<li>Find the future value.<\/li>\n<\/ul>\n<\/div>\n<h2>\u00a0Use Simple Interest and Compound Interest Formulas<\/h2>\n<p id=\"fs-id1165137447026\">Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account) is called the <strong>principal<\/strong>.\u00a0The <strong>rate of interest<\/strong>, expressed as a percent, is the amount charged for the use of the principle for a given period of time, usually on a yearly (that is, per annum) basis.<\/p>\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Simple Interest Formula<\/h3>\n<p id=\"fs-id1165135184167\"><strong>Simple interest<\/strong> can be calculated using the formula<\/p>\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]I=Prt[\/latex]<\/div>\n<p id=\"eip-237\">where<\/p>\n<ul id=\"fs-id1165137448453\">\n<li><em>I<\/em>\u00a0is the amount of interest,<\/li>\n<li><i>t<\/i> is measured in years,<\/li>\n<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\n<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_01_08\" class=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\n<h3>Example 1: Calculating Simple Interest<\/h3>\n<p id=\"fs-id1165137812825\">If we take out a $2,000 loan which charges 4% simple interest for 5 years, how much interest is paid?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162184\">Show Solution<\/span><\/p>\n<div id=\"q162184\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137812832\">Because we are starting with $2,000, <em>P\u00a0<\/em>= 2000. Our interest rate is 4%, so <em>r<\/em>\u00a0=\u00a00.04. In this problem,\u00a0<em>t <\/em>= 5.<\/p>\n<p>[latex]I =Prt[\/latex]<br \/>\n[latex]I =2000*0.04*5[\/latex]<br \/>\n[latex]I= $400[\/latex]<\/p>\n<p id=\"fs-id1165137694040\">The amount if interest paid is $400 in 5 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use <strong>compound interest<\/strong>. The term <em>compounding<\/em> refers to interest earned not only on the original value, but on the accumulated value of the account.<\/p>\n<p id=\"fs-id1165137447037\">The <strong>annual percentage rate (APR)<\/strong> of an account, also called the <strong>nominal rate<\/strong>, is the yearly interest rate earned by an investment account. The term\u00a0<em>nominal<\/em>\u00a0is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being <em>greater<\/em> than the nominal rate! This is a powerful tool for investing.<\/p>\n<p id=\"fs-id1165135160118\">We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time <em>t<\/em>, principal <em>P<\/em>, APR <em>r<\/em>, and number of compounding periods in a year\u00a0<em>n<\/em>:<\/p>\n<div id=\"eip-986\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/div>\n<p id=\"fs-id1165137935717\">For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.<\/p>\n<table id=\"Table_04_01_03\" summary=\"Six rows and two columns. The first column is labeled,\">\n<thead>\n<tr>\n<th>Frequency<\/th>\n<th>Value after 1 year<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Annually<\/td>\n<td>$1100<\/td>\n<\/tr>\n<tr>\n<td>Semiannually<\/td>\n<td>$1102.50<\/td>\n<\/tr>\n<tr>\n<td>Quarterly<\/td>\n<td>$1103.81<\/td>\n<\/tr>\n<tr>\n<td>Monthly<\/td>\n<td>$1104.71<\/td>\n<\/tr>\n<tr>\n<td>Daily<\/td>\n<td>$1105.16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Compound Interest Formula<\/h3>\n<p id=\"fs-id1165135184167\"><strong>Compound interest<\/strong> can be calculated using the formula<\/p>\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex]<\/div>\n<p id=\"eip-237\">where<\/p>\n<ul id=\"fs-id1165137448453\">\n<li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\n<li><i>t<\/i> is measured in years,<\/li>\n<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\n<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal, and<\/li>\n<li><em>n<\/em>\u00a0is the number of compounding periods in one year.<\/li>\n<li>annually [latex](n=1)[\/latex], semiannually [latex](n=2)[\/latex], quarterly [latex](n=4)[\/latex], monthly [latex](n=12)[\/latex], weekly [latex](n=52)[\/latex], yearly [latex](n=365)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_01_08\" class=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\n<h3>Example 2: Calculating Compound Interest<\/h3>\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162183\">Show Solution<\/span><\/p>\n<div id=\"q162183\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137812832\">Because we are starting with $3,000, <em>P\u00a0<\/em>= 3000. Our interest rate is 3%, so <em>r<\/em>\u00a0=\u00a00.03. Because we are compounding quarterly, we are compounding 4 times per year, so <em>n\u00a0<\/em>= 4. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}A\\left(t\\right) & =P\\left(1+\\frac{r}{n}\\right)^{nt}&& \\text{Use the compound interest formula}. \\\\ A\\left(10\\right)& =3000\\left(1+\\frac{0.03}{4}\\right)^{4\\cdot 10}&& \\text{Substitute using given values}. \\\\ & \\approx 4045.05&& \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137694040\">The account will be worth about $4,045.05 in 10 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474337\">Show Solution<\/span><\/p>\n<div id=\"q474337\" class=\"hidden-answer\" style=\"display: none\">\n<p>about $3,644,675.88<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm14375\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14375&theme=oea&iframe_resize_id=ohm14375\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Continuous Compounding<\/h2>\n<p>If the number of compoundings goes to infinity, then we have <strong>continuous compounding<\/strong>.<\/p>\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Compounding Continuously Formula<\/h3>\n<p id=\"fs-id1165135184167\">If you are <strong>compounding continuously<\/strong>, use the formula<\/p>\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: center;\">[latex]A\\left(t\\right)=Pe^{rt}[\/latex]<\/div>\n<p id=\"eip-237\">where<\/p>\n<ul id=\"fs-id1165137448453\">\n<li><em>A<\/em>(<em>t<\/em>) is the account value,<\/li>\n<li><i>t<\/i> is measured in years,<\/li>\n<li><em>P<\/em>\u00a0is the starting amount of the account, often called the principal, or more generally present value,<\/li>\n<li><em>r<\/em>\u00a0is the annual percentage rate (APR) expressed as a decimal.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_01_08\" class=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\n<h3>Example 3: Compounding Continuously<\/h3>\n<p id=\"fs-id1165137812825\">If we invest $3,000 in an investment account paying 3% interest compounded continuously, how much will the account be worth in 10 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162182\">Show Solution<\/span><\/p>\n<div id=\"q162182\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137812832\">Because we are compounding continuously, we need to use the compounding continuously formula. We want to know the value of the account in 10 years, so we are looking for <em>A<\/em>(10), the value when <em>t <\/em>= 10.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}A\\left(t\\right) & =Pe^{rt}&& \\text{Use the compounding continuously formula}. \\\\ A\\left(10\\right)& =3000e^{0.03*10}&& \\text{Substitute using given values}. \\\\ & \\approx 4049.58&& \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137694040\">The account will be worth about $4,049.58 in 10 years.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135180428\">An initial investment of $100,000 at 12% interest is compounded continuously. What will the investment be worth in 30 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474338\">Show Solution<\/span><\/p>\n<div id=\"q474338\" class=\"hidden-answer\" style=\"display: none\">\n<p>about $3,659,823.44<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Effective Rate of Interest<\/h2>\n<p>The <strong>effective rate of interest<\/strong> is the annual simple interest rate that would yield the same amount as compounding <i>n<\/i> times per year, or continuously, after one year.<\/p>\n<div id=\"fs-id1165137793679\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Effective Rate of Interest<\/h3>\n<p id=\"fs-id1165135184167\">The effective rate of interest [latex]r_E[\/latex] of an investment earning an annual interest rate <i>r<\/i> is given by<\/p>\n<div id=\"fs-id1165135184172\" class=\"equation\" style=\"text-align: left;\">\n<ul id=\"fs-id1165137448453\">\n<li>Compounding <i>n<\/i> times per year: [latex]r_E=\\left(1+\\frac{r}{n}\\right)^n-1[\/latex]<\/li>\n<li>Continuous Compounding: [latex]r_E=e^r-1[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"Example_04_01_08\" class=\"example\">\n<div id=\"fs-id1165135422940\" class=\"exercise\">\n<div id=\"fs-id1165137812820\" class=\"problem textbox shaded\">\n<h3>Example 4: Effective Rate of Interest<\/h3>\n<p id=\"fs-id1165137812825\">Find the effective rate of interest for 7% compounded quarterly.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162189\">Show Solution<\/span><\/p>\n<div id=\"q162189\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137812832\">This is compounded quarterly, so <i>n\u00a0<\/i>= 4. Use the formula [latex]r_E=\\left(1+\\frac{r}{n}\\right)^n-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}r_E & =\\left(1+\\frac{r}{n}\\right)^n-1&& \\text{Use the correct formula}. \\\\ r_E & =\\left(1+\\frac{0.07}{4}\\right)^4&& \\text{Substitute using given values}. \\\\ & \\approx 0.071859 && \\text{Round to two decimal places}.\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137694040\">The effective rate of interest is approximately 7.19%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135180428\">Find the effective rate of interest for 5% compounded continuously.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474339\">Show Solution<\/span><\/p>\n<div id=\"q474339\" class=\"hidden-answer\" style=\"display: none\">\n<p>approximately 5.13%<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: center;\">Section 5.7 Homework Exercises<\/h2>\n<p>For the following exercises, use the compound interest formula, [latex]A\\left(t\\right)=P{\\left(1+\\frac{r}{n}\\right)}^{nt}[\/latex].<\/p>\n<p>1. What was the initial deposit made to the account in the previous exercise?<\/p>\n<p>2. How many years had the account from the previous exercise been accumulating interest?<\/p>\n<p>3. An account is opened with an initial deposit of $6,500 and earns 3.6% interest compounded semi-annually. What will the account be worth in 20 years?<\/p>\n<p>4. How much more would the account in the previous exercise have been worth if the interest were compounding weekly?<\/p>\n<p>5. Solve the compound interest formula for the principal, P.<\/p>\n<p>6. Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)<\/p>\n<p>7. How much more would the account in the previous two exercises be worth if it were earning interest for 5 more years?<\/p>\n<p>8. Use properties of rational exponents to solve the compound interest formula for the interest rate, r.<\/p>\n<p>9. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.<\/p>\n<p>10.\u00a0Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.<\/p>\n<p>11.\u00a0Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?<\/p>\n<p>12. How much less would the account from Exercise 11 be worth after 30 years if it were compounded monthly instead?<\/p>\n<p>13. Determine the rate that represents the better deal:\u00a0 7% compounded semiannually or 6.9% compounded continuously?<\/p>\n<p>14. Determine the rate that represents the better deal: 9% compounded annually or 8.9% compounded continuously?<\/p>\n<p>In problems 15 &#8211; 20, find the principle needed now to get each amount; that is, find the present value.<\/p>\n<p>15. To get $100 after 2 years at 6% compounded monthly<\/p>\n<p>16. To get $75 after 3 years at 8% compounded quarterly<\/p>\n<p>17. To get $1500 after [latex]2 \\frac{1}{2}[\/latex]  years at 1.5% compounded daily<\/p>\n<p>18. To get $800 after [latex]3 \\frac{1}{2}[\/latex] years at 7% compounded monthly<\/p>\n<p>19. To get $750 after 2 years at 2.5% compounded quarterly<\/p>\n<p>20. To get $300 after 4 years at 3% compounded daily<\/p>\n<p>In problems 21 &#8211; 24, find the effective rate of interest.<\/p>\n<p>21. For 5% compounded quarterly<\/p>\n<p>22. For 6% compounded monthly<\/p>\n<p>23. For 4% compounded continuously<\/p>\n<p>24. For 6% compounded continuously<\/p>\n","protected":false},"author":264444,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17784","chapter","type-chapter","status-publish","hentry"],"part":13696,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17784","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":62,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17784\/revisions"}],"predecessor-version":[{"id":17891,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17784\/revisions\/17891"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/13696"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/17784\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=17784"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=17784"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=17784"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=17784"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}