Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=6 \%, t=10 \text { years }$$

Step-by-Step Solution

Answer

For \(n=1\) \(A=\$4483.43\), for \(n=2\) \(A=\$4520.57\), for \(n=4\) \(A=\$4540.15\), for \(n=12\) \(A=\$4553.72\), for \(n=365\) \(A=\$4561.06\), and for continuous compounding \(A=\$4563.79\).

Step 1

Understand the given information. Here, \(P = \$2500\) (Principal), \(r = 6\% = 0.06\) (rate in decimal), and \(t = 10\) years (time) are given.

Step 2

Understand the formula for compounded interest. The general formula is \(A = P (1 + r/n)^{nt}\), where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate in decimal, \(n\) is the number of times that interest is compounded per unit t, and \(t\) is the time the money is invested for in years.

Step 3

Use the formula to calculate \(A\) for each given value of \(n\). When \(n=1\), \(A = 2500(1 + 0.06/1)^{(1*10)} = \$4483.43\), when \(n=2\), \(A = 2500(1 + 0.06/2)^{(2*10)} = \$4520.57\), when \(n=4\), \(A = 2500(1 + 0.06/4)^{(4*10)} = \$4540.15\), when \(n=12\), \(A = 2500(1 + 0.06/12)^{(12*10)} = \$4553.72\), and when \(n=365\), \(A = 2500(1 + 0.06/365)^{(365*10)} = \$4561.06\).

Step 4

Calculate \(A\) for continuous compounding. The formula is \(A = P e^{rt}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. So, when compounding is continuous, \(A = 2500 * e^{(0.06*10)} = \$4563.79\).

Key Concepts

Principal Amount

The principal amount is the initial sum of money that you invest or deposit into an account. In our example, it's the starting amount before any interest is added, which is \( \$2500 \). Understanding the principal is crucial because it's the base from which interest grows over time. Whenever you make an investment, whether in a savings account, bond, or other financial product, you start with this principal amount.
It's important to know:

• This amount does not include any interest earned over time.
• The principal is the starting point for any interest calculations.
• Changes in the principal amount can affect the total amount of interest earned.

In compound interest calculations, the principal is multiplied by a growth factor that accounts for the rate and frequency of compounding.

Interest Rate

The interest rate is a critical concept in finance, representing the percentage at which your investment grows annually. In the exercise, the interest rate given is \(6\%\), converted into a decimal for calculations, \(0.06\). It tells you how much interest you'll earn on your principal every year, but in compound interest settings, the actual yield can be higher due to compounding effects.
Key points about interest rates:

• It determines how quickly your money grows.
• A higher rate means more potential earnings.
• Always expressed as a percentage per year, but used as a decimal in formulas.

By multiplying this rate with the principal, you can calculate the interest accrued over a specified period, influenced by how often the interest is added back to the principal.

Compounding Frequency

Compounding frequency refers to the number of times that interest is added to the principal in a given year. In the exercise, compounding occurs at different frequencies: yearly, semi-annually, quarterly, monthly, daily, and continuously. The frequency of compounding significantly influences the total interest earned. The more frequently interest is compounded, the more interest you earn.
Main points about compounding frequency:

• Frequent compounding events lead to higher returns.
• For example, daily compounding yields more than annual compounding, given the same interest rate.
• Compounding frequency is represented by \(n\) in the general compound interest formula \(A = P (1 + r/n)^{nt}\).

When planning investments, understanding compounding frequency can help determine the most lucrative options for growing funds over time.

Continuous Compounding

Continuous compounding is an extreme case where interest is calculated and added to the principal an infinite number of times per year. This concept, which applies to scenarios with very frequent compounding, is represented by the formula \(A = Pe^{rt}\). Here, \(e\) is a constant approximately equal to 2.71828. With continuous compounding, your investment grows slightly faster as it compounds every instant.
Essentials of continuous compounding:

• This method results in the highest amount for a fixed rate and time period among all compounding methods.
• It's more theoretical but has practical applications in certain financial instruments and theoretical finance models.
• \(e^{rt}\) reflects how an investment grows without any gaps between the interest calculations.

Knowing this concept can give investors insights into the potential maximum return possible on an investment over time.