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Chapter 10: Problem 36

Consider a CD paying a \(3.6 \%\) APR compounded continuously. Find the future value of the \(\mathrm{CD}\) if you invest \(\$ 3250\) for a term of four years.

Short Answer

Expert verified
The future value of the CD after four years would be approximately \$3948.29.

Step by step solution

01

Understand and set up the formula

Use the formula for continuous compounding, which is \(A = Pe^{rt}\). Here, \(P = \$3250\), \(r = 0.036\) (3.6% APR expressed as a decimal), and \(t = 4\) years.
02

Substitute the values into the formula

Substitute the values of P, r and t into the formula. This gives: \(A = \$3250 \times e^{0.036 \times 4}\).
03

Calculate the future value

Perform the calculation: \(A = \$3250 \times e^{0.144}\) Remember to use the value of e (approximately 2.71828) from a scientific calculator in your calculations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Future Value
When you invest money, you're often interested in what your investment will be worth at some point in the future. This is known as the future value. The future value depends on how much you initially invest, the interest rate, and the duration of the investment.

For investments compounded continuously, we use the formula:
  • \(A = Pe^{rt}\)
  • \(A\) is the future value of the investment.
  • \(P\) is the principal or the initial amount invested.
  • \(r\) is the interest rate in decimal form.
  • \(t\) is the time in years.
In the example given, the principal \(P\) is \$3250, the annual percentage rate \(r\) is 3.6% or 0.036 in decimal form, and \(t\) is 4 years. The future value is calculated using these variables in the formula.
APR
APR stands for Annual Percentage Rate and represents the cost of borrowing money or the return on investment expressed as a yearly rate. It includes both the nominal interest rate and any additional costs or fees.
  • Expressed annually, APR is critical in allowing individuals to compare the cost or return of financial products easily.
  • It is essential to convert the percentage into decimal form for calculations. For example, 3.6% becomes 0.036.
  • APR is particularly useful when comparing loans, mortgages, and investment returns.
For continuous compounding, knowing the APR allows you to project the future returns on your investment efficiently.
Exponential Growth
Exponential growth describes a process where the amount increases at a rate proportional to its current value - a fundamental principle in continuous compounding.

Continuous compounding is a type of exponential growth. Instead of interest being added at regular intervals, it continuously accumulates over time, leading to a faster increase in value:
  • In the formula \(A = Pe^{rt}\), the term \(e^{rt}\) is responsible for this exponential growth.
  • \(e\) is the base of natural logarithms, approximately equal to 2.71828, crucial in calculations.
As time goes on, the effect of exponential growth becomes more pronounced, illustrating why small changes in time or rate can result in significant differences in the future value of an investment.

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Most popular questions from this chapter

Find the APY for an APR of \(3.6 \%\) compounded (a) yearly. (b) semi-annually. (c) monthly. (d) continuously.

What should your monthly contribution be if your goal is to have \(\$ 500,000\) in your retirement savings account after 40 years? Assume the APR is \(6.6 \%\) compounded monthly and that contributions are made at the end of each month, including the last month.

Consider a CD paying a \(3.6 \%\) APR compounded continuously. Find the future value of the CD if you invest \(\$ 3250\) for a term of 1000 days. Round your answer to the nearest dollar.

Elizabeth went on a fabulous vacation in May and racked up a lot of charges on her credit card. When it came time to pay her June credit card bill, she left a balance of \(\$ 1200\). Elizabeth's credit card billing cycle runs from the nineteenth of each month to the eighteenth of the next month, and her interest rate is \(19.5 \% .\) She started the billing cycle June \(19-\) July 18 with a previous balance of \(\$ 1200 .\) In addition, she made three purchases, with the dates and amounts shown in Table 10-11. On July 15 she made an online payment of \(\$ 500.00\) that was credited to her balance the same day. (a) Find the average daily balance on the credit card account for the billing cycle June 19 -July 18 . (b) Compute the interest charged for the billing cycle June 19-July 18 . (c) Find the new balance on the account at the end of the June 19 -July 18 billing cycle. $$ \begin{array}{|c|c} \hline \text { Date } & \text { Amount of purchase/payment } \\ \hline 6 / 21 & \$ 179.58 \\ \hline 6 / 30 & \$ 40.00 \\ \hline 7 / 5 & \$ 98.35 \\ \hline 7 / 15 & \text { Payment } \$ 500.00 \end{array} $$

Arvin's tuition bill for last semester was \(\$ 5760 .\) If he paid \(\$ 6048\) in tuition this semester, what was the percentage increase on his tuition?
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