Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

25\. If \(\$ 375\) is put in the bank today, what will it be worth at the end of 2 years if interest is \(3.5 \%\) and is compounded as specified? (a) Annually (b) Monthly (c) Daily (d) Continuously

Step-by-Step Solution

Answer

(a) $401.71, (b) $402.24, (c) $402.34, (d) $402.44.

Step 1: Identify the Formula for Compound Interest

The formula to calculate the future value of an investment with compound interest is \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount ($375), \( r \) is the annual interest rate (3.5% or 0.035 as a decimal), \( n \) is the number of times that interest is compounded per year, and \( t \) is the number of years the money is invested for (2 years).

Step 2: Calculate Annual Compounding

For annual compounding, \( n = 1 \). Plug the values into the formula: \[ A = 375 \left(1 + \frac{0.035}{1}\right)^{1 \times 2} \]\[ A = 375 \times (1.035)^2 \]\[ A \approx 375 \times 1.071225 = 401.71 \].Thus, after 2 years, the amount will be approximately \( \$401.71 \).

Step 3: Calculate Monthly Compounding

For monthly compounding, \( n = 12 \). Plug the values into the formula: \[ A = 375 \left(1 + \frac{0.035}{12}\right)^{12 \times 2} \]\[ A = 375 \times (1.0029167)^{24} \]\[ A \approx 375 \times 1.071959 = 402.24 \].Thus, after 2 years, the amount will be approximately \( \$402.24 \).

Step 4: Calculate Daily Compounding

For daily compounding, assuming 365 days a year, \( n = 365 \). Plug the values into the formula: \[ A = 375 \left(1 + \frac{0.035}{365}\right)^{365 \times 2} \]\[ A = 375 \times (1.00009589)^{730} \]\[ A \approx 375 \times 1.072240 = 402.34 \].Thus, after 2 years, the amount will be approximately \( \$402.34 \).

Step 5: Calculate Continuous Compounding

For continuous compounding, use the formula \( A = Pe^{rt} \), where \( e \) is the mathematical constant approximately equal to 2.71828. Plug in the values:\[ A = 375 \times e^{0.035 \times 2} \]\[ A = 375 \times e^{0.07} \]\[ A \approx 375 \times 1.072508 = 402.44 \].Thus, after 2 years, the amount will be approximately \( \$402.44 \).

Key Concepts

Annual Compounding

Annual compounding is one of the simplest ways to calculate interest. Here, interest is calculated once a year on the total balance to that date. This means that any interest earned is added to the principal – the original amount deposited – at the end of each year.
Your principal earns interest, and this new amount serves as the base for calculating interest the following year. In the formula for compound interest, this means setting the compounding period per year ( t ) as 1. Understanding this concept is essential because it gives you an idea of how much more money you can get just by letting it grow over time, without needing to add any more to the principal each year. It’s a great stepping stone to understanding other, more frequent compounding concepts.

Monthly Compounding

Monthly compounding calculates interest every month. This means 12 times a year, your investment earns interest. You add the interest to your principal, so the next month’s interest is calculated on the new total. Due to this more frequent compounding, your investment grows a bit faster than with annual compounding. Here, the formula for compound interest sees t set to 12. With monthly compounding, you can start to observe how increasing the number of compounding periods can significantly increase the amount of interest earned. This is especially important for investments and savings over longer periods, as the effect of interest on interest becomes more pronounced.

Daily Compounding

Daily compounding involves calculating interest every single day. Rather than compounding once a year or month, the interest becomes part of the principal daily. In a year, this accounts for 365 compounding periods (or 366 for leap years). The compound interest formula for this sets t as 365. This variation illustrates how the frequency of compounding leads to slightly higher total returns compared to monthly or annual compounding. By understanding daily compounding, it becomes clear how powerful the effect of compound interest can be over time – even small increases in compounding frequency can lead to significant differences.

Continuous Compounding

Continuous compounding is the theoretical extreme of compound interest, where compounding occurs at every possible instant. Mathematically, it's realized using the constant e , resulting in the unique formula A = Pe^{rt} . This approach maximizes the compound interest effect as there are no actual periods where no interest is added – it’s always becoming part of the principal at an infinitesimally small moment. Continuous compounding is less common in everyday finance but provides important insight into financial modeling and assessments of value over time. It's a key concept for finance professionals to understand the potential limits and behaviors of investments.