Problem 35 - Calculus: Early Transcendental Functions

Question

Involve compound interest. If you invest \(\$ 1000\) at an annual interest rate of \(8 \%,\) compare the value of the investment after 1 year under the following forms of compounding: annual, monthly, daily, continuous.

Step-by-Step Solution

Answer

The values of the investments are: $1080 with annual compounding, $1083.00 with monthly compounding, $1083.28 with daily compounding, and $1083.29 with continuous compounding.

Step 1: Annual Compounding

Compute the value of the investment after one year with annual compounding using the formula: \( A = P (1 + r/n)^{nt} \), where A is the future value of the investment/loan, including interest, P is the principal investment amount, r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years. For annual compounding, n = 1: \( A = 1000 (1 + 0.08/1)^{1*1} = $1080 \) .

Step 2: Monthly Compounding

Next, compute the value of the investment after one year with monthly compounding. For monthly compounding, n = 12: \( A = 1000 (1 + 0.08/12)^{12*1} = $1083.00 \)

Step 3: Daily Compounding

Then, calculate the value of the investment after one year with daily compounding. For daily compounding, n = 365: \( A = 1000 (1 + 0.08/365)^{365*1} = $1083.28 \)

Step 4: Continuous Compounding

Finally, calculate the value of the investment after one year with continuous compounding using the formula: \( A = Pe^{rt} \), where e is the base of the natural logarithm (approximately equal to 2.71828). For continuous compounding: \( A = 1000 * e^{0.08*1} = $1083.29 \)

Key Concepts

Annual Compounding

In annual compounding, interest is calculated once a year. It is the simplest form of compound interest, where the frequency of compounding is once per year.
This means at the end of each year, the interest is added to the principal amount, forming a new principal for the following year.
In mathematical terms, the formula used for calculating compound interest annually is:

• \( A = P (1 + r/n)^{nt} \)

Here:

• \( P \) is the principal amount (initial investment).
• \( r \) is the annual nominal interest rate (as a decimal).
• \( n \) is the number of compounding periods per year.
• \( t \) is the number of years the money is invested for.

For annual compounding, \( n \) is 1. Thus, the formula simplifies to \( A = P (1 + r)^{t} \).
In the given example with a principal of \(1000 and interest rate of 8%, the value after one year is \( \)1080 \). This method showcases how annual compounding results in simple growth.

Monthly Compounding

Monthly compounding increases the frequency of interest calculation to every month, which accelerates the growth of the investment.
Here, interest is calculated and added to the principal every month, making it compound 12 times a year.
The formula used is still \( A = P (1 + r/n)^{nt} \), but with \( n = 12 \) for monthly compounding.

• This adjustment means interest is calculated each month using the rate divided by 12.
• The effective annual rate is slightly higher because of this monthly compounding.

In the example given, investing \(1000 at an 8% annual interest rate compounded monthly results in a slightly higher future value after one year: \( \)1083.00 \). This demonstrates that the more frequently interest is compounded, the more growth accelerates.

Continuous Compounding

Continuous compounding is the scenario where the frequency of compounding is infinite.
This means that the interest is calculated and added to the principal at every possible moment.
It's calculated using the formula:

• \( A = Pe^{rt} \)

Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

• This method produces the largest amount of interest as the compounding frequency is maximized.

In this method, a \(1000 investment at an 8% rate results in approximately \( \)1083.29 \) after one year of continuous compounding.
This amount slightly surpasses even daily compounding, emphasizing the power of constantly compounding interest.

Exponential Function

The exponential function forms the backbone for understanding continuous compounding.
In essence, an exponential function grows at a rate proportional to its current value, leading to rapid growth.
When applied to interest, it dramatically illustrates how investments can grow over time.
The formula used for continuous compounding, \( A = Pe^{rt} \), is essentially employing exponential growth:

• In this formula, \( e^{rt} \) represents exponential growth at a rate \( r \) over time \( t \).
• The number \( e \) ensures a smooth, continuous calculation of growth.

Overall, exponential functions highlight how powerful compound interest can be, particularly when compounding continuously, as they describe the growth curve that results from this form of interest calculation. This growth becomes more significant as the time period increases, underscoring why understanding exponential functions is crucial for financial computations.