Problem 73 - A Graphical Approach to College Algebra

Question

Decide which of the two plans will provide a better yield. (Interest rates stated are annual rates.) Plan A: \(\$ 40,000\) invested for 3 years at \(2.5 \%,\) compounded quarterly Plan B: \(\$ 40,000\) invested for 3 years at \(2.4 \%,\) compounded continuously

Step-by-Step Solution

Answer

Plan A provides a better yield than Plan B.

Step 1: Understand Compounding Formulas

For Plan A, we use the formula for compound interest with quarterly compounding: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \(P = 40000\), \(r = 0.025\), \(n = 4\), and \(t = 3\). For Plan B, with continuous compounding, use the formula: \( A = Pe^{rt} \) where \( P = 40000 \), \( r = 0.024 \), and \( t = 3 \).

Step 2: Calculate Future Value for Plan A

Substitute the known values into the compound interest formula for Plan A: \[ A = 40000 \left(1 + \frac{0.025}{4}\right)^{4 \times 3} \]. This simplifies to: \[ A = 40000 \left(1 + 0.00625\right)^{12} \]. Solve this for \(A\).

Step 3: Evaluate Expression for Plan A

Calculate \(1 + 0.00625 = 1.00625\), and then raise this to the 12th power: \(1.00625^{12} \approx 1.07767\). Therefore, \( A = 40000 \times 1.07767 \approx 43106.80 \).

Step 4: Calculate Future Value for Plan B

Use the formula for continuous compounding: \( A = 40000e^{0.024 \times 3} \). Simplify to: \( A = 40000e^{0.072} \). Calculate \(e^{0.072}\).

Step 5: Evaluate Expression for Plan B

Find the value of \( e^{0.072} \approx 1.07452 \). Thus, \( A = 40000 \times 1.07452 \approx 42980.80 \).

Step 6: Compare the Results

Compare the future values: Plan A results in \(43106.80\) and Plan B results in \(42980.80\). Plan A provides a higher yield than Plan B.

Key Concepts

quarterly compounding

In the context of investments, quarterly compounding refers to the process by which interest is calculated and added to the principal balance four times a year. This means that for every quarter (or every three months), the interest accrued over that period is added to the principal, which then earns more interest in the subsequent quarters. The formula for calculating the future value of an investment with quarterly compounding is:

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

Where:

• \(A\) is the future value of the investment
• \(P\) is the principal amount
• \(r\) represents the annual interest rate (expressed as a decimal)
• \(n\) is the number of times interest is compounded per year (for quarterly, \(n = 4\))
• \(t\) is the number of years the money is invested for

The key advantage of quarterly compounding is that it can result in a slightly higher yield compared to less frequent compounding periods like annually or semi-annually. This is due to the more frequent addition of interest to the principal, allowing each new interest calculation to be based on the increased amount.

continuous compounding

Continuous compounding represents an extreme case of compounding frequency where interest is added to the principal an infinite number of times, theoretically instantaneously throughout the period. The formula used to calculate the future value with continuous compounding is:

• \[ A = Pe^{rt} \]

Where:

• \(A\) is the future value
• \(P\) refers to the principal investment
• \(r\) is the annual interest rate
• \(t\) is the time in years
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828

Continuous compounding is often considered in financial mathematics because it results in the maximum possible return from an interest rate over a given period. Although not often used in day-to-day banks, it's a crucial concept in understanding how small periods of compounding can leverage returns for investors. Mathematics and theoretical finance use it extensively to model various financial instruments.

future value calculation

Future value (FV) calculation is essential for understanding how much an investment will be worth at a specific point in the future. It accounts for the initial investment, the interest rate, the compounding frequency, and the investment duration. The concept of future value helps investors and planners make key financial decisions. For quarterly compounding, the formula provided allows calculating the exact amount an investment will grow to at the end of the investment horizon. Similarly, for continuous compounding, the future value formula accounts for infinite small periods, which leads to a higher return than other compounding intervals. In the example provided:

• Plan A uses quarterly compounding and results in a future value of approximately $43,106.80 after 3 years.
• Plan B uses continuous compounding, with a future value of about $42,980.80 for the same period.

The calculation process involves substituting the known values into the respective compound interest formula and solving for the future amount. Understanding future value is key in evaluating different investment options.

interest rate comparison

Comparing interest rates and the corresponding future values of different compounding methods is vital for making informed investment decisions. Interest rates, coupled with the compounding frequency, can significantly impact the outcome. In our example, Plan A offers an annual interest rate of 2.5% compounded quarterly, whereas Plan B has a slightly lower rate of 2.4% but is compounded continuously. Despite the seemingly small difference in rates, the compounding frequency has a notable effect on the investment's future value. With quarterly compounding, Plan A yields a higher future value than Plan B’s continuous compounding. This highlights that while continuous compounding offers maximum returns under the same rate, a higher rate with a slightly less frequent compounding period can outperform it. For investors, understanding how different rates and compounding frequencies influence investment growth can aid in choosing the most beneficial financial product. It illustrates that evaluating beyond just interest rates is crucial for maximizing returns.