Problem 68 - Precalculus

Question

An investment account with an annual interest rate of 7% was opened with an initial deposit of $4,000 Compare the values of the account after 9 years when the interest is compounded annually, quarterly, monthly, and continuously.

Step-by-Step Solution

Answer

Annually: $7,405.85; Quarterly: $7,484.93; Monthly: $7,529.90; Continuously: $7,547.87.

Step 1: Understanding Compound Interest Formula

The compound interest formula is given by \( A = P\left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after n years including interest, \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the time in years.

Step 2: Calculate Annually Compounded Interest

For annually compounded interest, \( n = 1 \). Substituting into the formula: \[ A = 4000\left(1 + \frac{0.07}{1}\right)^{1 \times 9} \]. Calculating, \( A \approx 4000(1.07)^9 \approx 7405.85 \). So, the amount after 9 years with annual compounding is approximately $7,405.85.

Step 3: Calculate Quarterly Compounded Interest

For quarterly compounded interest, \( n = 4 \). Substituting into the formula: \[ A = 4000\left(1 + \frac{0.07}{4}\right)^{4 \times 9} \]. Calculating, \( A \approx 4000(1.0175)^{36} \approx 7484.93 \). So, after 9 years with quarterly compounding, the amount is approximately $7,484.93.

Step 4: Calculate Monthly Compounded Interest

For monthly compounded interest, \( n = 12 \). Substituting into the formula: \[ A = 4000\left(1 + \frac{0.07}{12}\right)^{12 \times 9} \]. Calculating, \( A \approx 4000(1.0058333)^{108} \approx 7529.90 \). Thus, with monthly compounding, the amount after 9 years is approximately $7,529.90.

Step 5: Calculate Continuously Compounded Interest

Continuously compounded interest is calculated using the formula \( A = Pe^{rt} \). Here \( e \) is the base of the natural logarithm. Substituting the values, \[ A = 4000e^{0.07\times9} \]. Calculating, \( A \approx 4000e^{0.63} \approx 7547.87 \). Therefore, the amount with continuous compounding after 9 years is approximately $7,547.87.

Key Concepts

Annual Compounding

In annual compounding, interest is calculated once per year on the initial amount and any accumulated interest from previous years. To compute the total value of an account with annual compounding, use the formula:

• \(A = P(1 + r)^t\)

Here, \(A\) is the total amount after time \(t\), \(P\) is the initial deposit, \(r\) is the annual interest rate, and \(t\) is the time in years.
For example, starting with a \(4,000 deposit at a 7% interest rate compounded annually, the formula becomes:

• \(A = 4000(1 + 0.07)^9\)

After calculations, the amount is approximately \)7,405.85 after 9 years.

Quarterly Compounding

Quarterly compounding involves calculating and adding interest four times a year. This means we divide the annual interest rate by 4 and multiply the time by the same number. The formula for quarterly compounding is:

• \(A = P\left(1 + \frac{r}{4}\right)^{4t}\)

For the given investment scenario, substituting the values gives:

• \(A = 4000\left(1 + \frac{0.07}{4}\right)^{36}\)

Upon performing the calculations, the future value is approximately $7,484.93.
Because interest is added more frequently, the total amount is slightly more compared to annual compounding.

Monthly Compounding

With monthly compounding, interest is calculated each month. That means the interest rate is divided by 12, and the compounding frequency is multiplied by 12. The formula becomes:

• \(A = P\left(1 + \frac{r}{12}\right)^{12t}\)

Applying this to our case:

• \(A = 4000\left(1 + \frac{0.07}{12}\right)^{108}\)

After evaluation, the account value is approximately $7,529.90 after 9 years.
This frequent compounding further increases the amount compared to both annual and quarterly compounding.

Continuous Compounding

Continuous compounding is a theoretical concept where interest is added at every possible moment. It uses the formula:

• \(A = Pe^{rt}\)

Where \(e\) is the base of the natural logarithm, approximately equal to 2.718. Using our 7% interest rate and 9-year timeframe:

• \(A = 4000e^{0.63}\)

After calculation, the amount is around $7,547.87.
Continuous compounding yields the highest returns in this case because it maximizes the growth potential of the investment.

Interest Rate Calculation

Calculating interest rates for compound interest can involve different compounding periods, which affect how much total interest you earn. When using the formula:

• \(A = P\left(1 + \frac{r}{n}\right)^{nt}\)

\(n\) represents the number of times interest is compounded per year (1 for annual, 4 for quarterly, 12 for monthly).
Each method of compounding gives a slightly different result because of how often interest is applied to the principal sum, affecting the total amount accrued by the end of the investment period.
Understanding these calculations helps maximize your earnings by choosing the right compounding frequency for your financial goals.