Problem 18 - Calculus and its applications

Question

A sum of \(\$ 1000\) is deposited in a savings account for which interest is compounded monthly. The future value \(A\) is a function of the annual percentage rate \(r\) and the term \(t,\) in months, and is given by $$ A(r, t)=1000\left(1+\frac{r}{12}\right)^{12 t} $$ a) Determine \(A(0.05,10)\). b) What is the interest earned for the rate and term in part (a)? c) How much more interest can be earned over the same term as in part (a) if the APR is increased to \(5.75 \% ?\)

Step-by-Step Solution

Answer

a) \( A(0.05,10) \approx 1647.009 \). b) Interest is approximately \( 647.009 \). c) Additional interest earned is approximately \( 129.225 \).

Step 1: Understand the formula

The future value formula given is \( A(r, t)=1000\left(1+\frac{r}{12}\right)^{12 t} \). Here, \( r \) is the annual interest rate, and \( t \) is the term in years. Since \( t \) in the formula is multiplied by 12, it indicates the term is considered in months.

Step 2: Determine A(0.05, 10)

Substitute \( r = 0.05 \) (5%) and \( t = 10 \) years into the formula. First, calculate the monthly interest rate: \( \frac{0.05}{12} \approx 0.004167 \). Next, calculate the future value: \[ A(0.05, 10) = 1000 \left(1 + 0.004167\right)^{12 \times 10} \] \[ \approx 1000 \times 1.647009 \approx 1647.009 \] So the future value \( A(0.05, 10) \) is approximately \( 1647.009 \).

Step 3: Calculate the interest earned

To find the interest earned, subtract the initial principal \( 1000 \) from the future value calculated: \[ \text{Interest} = A(0.05, 10) - 1000 \approx 1647.009 - 1000 = 647.009 \] The interest earned is approximately \( 647.009 \).

Step 4: Determine A(0.0575, 10)

Increase the annual interest rate to \( 0.0575 \) (5.75%) and calculate the future value: \[ A(0.0575, 10) = 1000 \left(1 + \frac{0.0575}{12} \right)^{12 \times 10} \] \[ \approx 1000 \times 1.776234 \approx 1776.234 \] So the future value \( A(0.0575, 10) \) is approximately \( 1776.234 \).

Step 5: Calculate additional interest with higher APR

First, find the interest earned with the higher APR by subtracting the initial principal: \[ \text{Interest with 5.75%} = 1776.234 - 1000 = 776.234 \] Calculate additional interest by subtracting the interest at 5% from the interest at 5.75%: \[ \text{Additional Interest} = 776.234 - 647.009 \approx 129.225 \] The additional interest earned is approximately \( 129.225 \).

Key Concepts

Future Value Calculation

Understanding how to calculate the future value of an investment can help you make informed choices about saving and investing. The future value calculation essentially tells you the amount your investment will grow to over a period of time. This is done using the formula:\[ A(r, t) = 1000 \left(1 + \frac{r}{12}\right)^{12 t} \]In this formula, \( A \) represents the future value of the investment, \( r \) is the annual interest rate as a decimal, and \( t \) is the term of the investment measured in years. Because interest is compounded monthly, the monthly interest rate is derived by dividing the annual rate \( r \) by 12.For example, if you substitute \( r = 0.05 \) (or 5%) and \( t = 10 \) years, you first need to compute the monthly interest rate, which is \( \frac{0.05}{12} = 0.004167 \). Finally, after calculating the expression, the future value of the investment will be \( 1,647.009 \$ \) after 10 years of compounding monthly. This represents the growth of the principal amount over time due to the effect of compounding.

Annual Percentage Rate (APR)

The Annual Percentage Rate (APR) is a crucial element in understanding how much interest you will earn or pay on an investment or loan over the course of a year. It is represented as a percentage.In the context of compound interest, the APR signifies the standard yearly rate of return that you can expect on your investment. However, because the compounding frequency – monthly in most savings accounts – can affect the total interest earned, APR is often less than the effective annual rate.To see how APR influences the future value of an investment, imagine you have an APR of 5% applied monthly over 10 years. By increasing this rate to 5.75%, you discover a significant increase in your future value with the new rate yielding \( A(0.0575, 10) \approx 1776.234 \) compared to the original \( A(0.05, 10) \). This illustrates how a seemingly small adjustment in APR can compound over time to produce higher returns.

Interest Earned Calculation

Interest earned is essentially the reward you receive for allowing your money to grow over a set period. After all, when you deposit funds into an account that offers compound interest, you're not just gaining interest on the principal amount but also on any previously accumulated interest.To calculate the interest earned, subtract the initial deposit (or principal) from the future value of the investment. So, using the future value \( 1647.009 \\( \), the interest earned over 10 years is calculated as:\[ \text{Interest} = 1647.009 - 1000 = 647.009 \\) \]When increasing the APR to 5.75%, the future value rises to \( 1776.234 \\( \), leading to a total interest earning of \( 776.234 \\) \). The difference between the two scenarios – an increase from \( 647.009 \\( \) to \( 776.234 \\) \) – demonstrates the additional \( 129.225 \$ \) gained by enhancing the APR. This process highlights how a higher interest rate can greatly influence overall earnings compounded over time.