Problem 30 - A Graphical Approach to College Algebra

Question

Estimate the doubling time of an investment earning \(2.5 \%\) interest if interest is compounded (a) quarterly; (b) continuously.

Step-by-Step Solution

Answer

Quarterly: ~27.725 years; Continuous: ~27.724 years.

Step 1: Calculate the Effective Interest Rate Quarterly

When interest is compounded quarterly, the annual interest rate is divided by 4. Since the given interest rate is 2.5%, the quarterly interest rate is \( \frac{2.5\%}{4} = 0.625\% \). Convert this percentage into a decimal for calculations: 0.625% = 0.00625.

Step 2: Apply the Doubling Time Formula for Quarterly Compounding

For quarterly compounding, the formula to estimate doubling time is \( n \cdot t = \log_2 (2) \), where \( n \) is the number of compounding periods per year (4 for quarterly) and \( t \) is the time in years. Simplifying the equation: \( 4t = \frac{\log(2)}{\log(1 + 0.00625)} \). Calculate this to find \( t \).

Step 3: Calculate Doubling Time Quarterly

Evaluate the formula from Step 2: \( t = \frac{\log(2)}{4 \cdot \log(1.00625)} \). Using a calculator, \( t \approx 27.725 \) years.

Step 4: Calculate Doubling Time for Continuous Compounding

When compounding is continuous, use the formula \( t = \frac{\ln(2)}{r} \), where \( r \) is the annual nominal interest rate in decimal form (0.025 for 2.5%). Evaluate: \( t = \frac{\ln(2)}{0.025} \).

Step 5: Evaluate Doubling Time for Continuous Compounding

Calculate \( t \approx \frac{0.6931}{0.025} = 27.724 \) years using a calculator. This is slightly different due to the nature of continuous compounding, resulting in a slight variation from quarterly compounding.

Key Concepts

Compound Interest

Compound interest refers to the process where the interest earned on an investment is reinvested, allowing the investment to grow more than it would with simple interest. In simple terms, it's like earning interest on both your initial investment and the accumulated interest from previous periods.
This concept is powerful because it allows investments to grow exponentially over time. The formula for calculating the future value of an investment using compound interest is:

• \( A = P(1 + r/n)^{nt} \)

where \( A \) is the amount at the end of the investment period, \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times the interest is compounded per year, and \( t \) is the time in years. Understanding compound interest is essential as it lays the foundation for concepts like quarterly and continuous compounding.

Quarterly Compounding

Quarterly compounding is a form of compound interest where the interest is calculated and added to the principal four times a year—once every quarter.
So, if you have an investment earning interest quarterly, the principal grows every three months, leading to a higher balance over time.

• The formula involves dividing the annual interest rate by four.
• You multiply the number of years by four for the number of periods.

Using this approach, the time it takes for an investment to double can be calculated with the formula: \( n \cdot t = \log_2(2) \). Here, \( n \) is 4 (for four quarters) and \( t \) is the time in years.
It's an effective way to see quicker returns than with annual compounding, although not as fast as continuous compounding.

Continuous Compounding

Continuous compounding of interest is a theoretical concept where interest is calculated and added to the principal at every possible instant.
As such, it provides slightly higher returns than traditional compounding methods like quarterly or annual.
The formula used is:

• \( A = Pe^{rt} \) where \( e \) is Euler's number (approximately 2.71828).

For doubling time during continuous compounding, use \( t = \frac{\ln(2)}{r} \), where \( r \) is the annual interest rate expressed as a decimal. This formula stems from rearranging the compound interest equation to solve for time when the amount is double the principal.
Continuous compounding is ideal in theoretical scenarios, providing the greatest accumulated amount possible.

Effective Interest Rate

The effective interest rate accounts for the impact of compounding over a given period. It gives a clearer picture of an investment's actual return over time compared to nominal rates, which might not reflect compounding effects.
The formula for effective interest rate is:

• \( r_{effective} = (1 + \frac{r_{nominal}}{n})^n - 1 \)

This formula incorporates the nominal rate \( r_{nominal} \) (which is your stated annual interest rate), divided by the number of compounding periods \( n \) in a year. An investment could have the same nominal and effective rates if compounded annually. However, more frequent compounding, such as quarterly or monthly, results in a higher effective rate.
Understanding effective interest rates helps investors compare different investments accurately.