Problem 38 - A Graphical Approach to College Algebra

Question

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Estimate the interest rate necessary for a present value of \(\$ 1200\) to grow to a future value of \(\$ 1408\) if interest is compounded quarterly for 8 years.

Step-by-Step Solution

Answer

The interest rate necessary is approximately 2.08%.

Step 1: Identify Key Variables

First, we need to identify the key variables from the given problem: the present value \( P = 1200 \), the future value \( A = 1408 \), the number of compounding periods per year \( n = 4 \) (since it is compounded quarterly), and the time in years \( t = 8 \). Our goal is to find the interest rate \( r \).

Step 2: Set Up the Formula

We will use the compound interest formula \( A = P\left(1+\frac{r}{n}\right)^{nt} \). Fill in the known values: \( 1408 = 1200\left(1+\frac{r}{4}\right)^{4 \times 8}\).

Step 3: Simplify the Equation

Calculate the exponent part: \( 4 \times 8 = 32 \). Replace in the equation: \( 1408 = 1200\left(1+\frac{r}{4}\right)^{32} \).

Step 4: Isolate the Compound Term

Divide both sides by 1200: \( \left(1+\frac{r}{4}\right)^{32} = \frac{1408}{1200} \). Calculate \( \frac{1408}{1200} \approx 1.1733 \). Now we have \( \left(1+\frac{r}{4}\right)^{32} = 1.1733 \).

Step 5: Take the 32nd Root of Both Sides

To solve for \( 1+\frac{r}{4} \), take the 32nd root of both sides: \( 1+\frac{r}{4} = (1.1733)^{\frac{1}{32}} \). Calculate this root: \( (1.1733)^{\frac{1}{32}} \approx 1.0052 \).

Step 6: Solve for the Interest Rate

Subtract 1 from both sides to isolate \( \frac{r}{4} \): \( \frac{r}{4} = 1.0052 - 1 \). Therefore, \( \frac{r}{4} \approx 0.0052 \).

Step 7: Calculate the Annual Interest Rate

Multiply both sides by 4 to find \( r \): \( r = 0.0052 \times 4 \). Thus, \( r \approx 0.0208 \). Convert this to a percentage: \( r \approx 2.08\% \).

Key Concepts

Present Value

The present value is a financial concept that refers to the current worth of an amount of money that you expect to receive or pay in the future, adjusted for interest rates and time. In other words, it's about figuring out how much future money is worth today.
For example, if someone promises to give you $1,408 in 8 years, the present value tells you how much you would need to invest today to have exactly that amount in the future, assuming a particular interest rate.

• The present value is represented by the symbol \( P \).
• The formula to calculate present value is \( P=A\left(1+\frac{r}{n}\right)^{-nt} \), where \( A \) is the future value, \( r \) is the annual interest rate, \( n \) is the number of times the interest is compounded per year, and \( t \) is the number of years.

Using this formula, you can determine how much you need to invest today to reach a desired amount in the future.

Future Value

Future value refers to the amount of money you will have after a certain period when interest is applied to a present value. It shows how much an investment made today will grow over time.
This concept is critical because it allows investors and savers to understand the power of compounding interest and how their current savings can grow substantially over time with the right interest rate.

• Future value is denoted as \( A \) in formulas.
• The basic future value formula is \( A=P\left(1+\frac{r}{n}\right)^{nt} \).

For instance, in our exercise, if you start with \( P = 1200 \), want it to grow to \( A = 1408 \) in 8 years with interest compounded quarterly, the future value helps estimate the interest rate required.

Interest Rate Estimation

Interest rate estimation involves finding the rate at which your investment grows over time. This is crucial in financial planning because knowing the correct interest rate helps you estimate how much your initial investment will grow in your chosen time frame.
In our exercise, we used the formula \( A=P\left(1+\frac{r}{n}\right)^{nt} \) to find the interest rate \( r \). To isolate \( r \), the equation often involves algebraic manipulations, such as roots and logarithms.

• Fill in known values (present value \( P \), future value \( A \), compounds \( n \), and time \( t \)).
• Rearrange the formula to isolate the interest-related component.
• Solve for \( r \) which might involve calculating roots or using iterative estimation methods in complex cases.

This gives you the annual percentage rate that your investment should achieve to reach its future value.

Compounding Periods

Compounding periods refer to how often the interest is added to the principal balance in one year. The more frequently interest is compounded, the more total interest you'll earn on your investment, and faster your money grows.
Common compounding frequencies include annually, semi-annually, quarterly, monthly, daily, etc. In our example problem, interest is compounded quarterly, meaning it is added four times a year.

• The variable \( n \) in our equations represents the number of compounding periods per year.
• For quarterly compounding, \( n = 4 \).
• The total number of compounding periods for the investment duration is calculated as \( n \times t \).

Understanding compounding periods is essential as it impacts both present and future value calculations, showing how rapidly interest is acting on your investment over time.