Problem 80 - Algebra and Trigonometry

Question

Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (i) 9\(\frac{1}{4} \%\) per year, compounded semiannually (ii) 9\(\%\) per year, compounded continuously

Step-by-Step Solution

Answer

Option (ii) is a better investment as it offers a higher effective annual rate of 9.417%.

Step 1: Understand the Problem

We need to compare two investment options based on compound interest. The first option is compounded semiannually at 9\(\frac{1}{4}\)% per year, and the second option is compounded continuously at 9% per year.

Step 2: Determine Compound Interest Formula

For semiannual compounding, we use the formula \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where \( A \) is the amount of money accumulated after \( t \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual nominal interest rate (as a decimal), and \( n \) is the number of times that interest is compounded per year.

Step 3: Calculate Effective Annual Rate for Semiannual Compounding

Using the formula, the effective annual rate (EAR) for option (i) is computed as: \[ EAR = \left(1 + \frac{0.0925}{2}\right)^2 - 1 \] \[ EAR = \left(1 + 0.04625\right)^2 - 1 \] \[ EAR \approx 0.093798\text{ or } 9.3798\%\]

Step 4: Determine Continuous Compounding Formula

For continuous compounding, we use the formula \( A = Pe^{rt} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 5: Calculate Effective Annual Rate for Continuous Compounding

The formula to find the effective annual rate for continuous compounding is: \[ EAR = e^r - 1 \] Substitute \( r = 0.09 \): \[ EAR = e^{0.09} - 1 \] \[ EAR \approx 1.09417 - 1 \] \[ EAR \approx 0.09417\text{ or } 9.417\%\]

Step 6: Compare the Effective Annual Rates

Comparing the EARs from both options: - Option (i) has an EAR of approximately 9.3798%. - Option (ii) has an EAR of approximately 9.417%. Therefore, option (ii) has a higher effective annual interest rate.

Key Concepts

Effective Annual Rate

The Effective Annual Rate (EAR) is a way to compare different interest rates with various compounding periods on a common basis. It represents the actual annual interest rate after accounting for the effects of compounding. Understanding EAR is crucial because it helps in determining the real rate of return on an investment over a year, making it easier to compare with other financial options.

To calculate the EAR, you need to consider the nominal interest rate and how often the interest is compounded within the year. For instance, with semiannual compounding, you use the formula:

• \( \text{EAR} = \left(1 + \frac{r}{n} \right)^n - 1 \)

Where \( r \) is the annual nominal rate expressed as a decimal and \( n \) is the number of compounding periods per year.

The effective rate gives a more accurate picture of the potential growth of an investment by incorporating all periods of compounding. A higher EAR indicates a better return on investment, which is why it's an integral part of decision-making in finance.

Continuous Compounding

Continuous compounding refers to the mathematical process where interest is calculated and added to the investment's principal at an infinite number of points within the time period. This concept is rooted in the idea that compounding can happen more frequently than daily, even continuously.

When interest is compounded continuously, the formula used is:

• \( A = Pe^{rt} \)

Here, \( A \) stands for the amount after time \( t \), \( P \) is the principal amount, \( r \) is the annual interest rate, \( e \) is Euler's number (approximately 2.71828), and \( t \) represents time in years.

To find the effective annual rate for continuous compounding, you employ:

• \( \text{EAR} = e^r - 1 \)

Continuous compounding typically yields a higher return than other methods like semiannual compounding because it involves more frequent application of interest. It's particularly relevant in financial sectors like banking and investment funds where precise calculations are crucial.

Semiannual Compounding

Semiannual compounding means that the interest is compounded twice a year. Essentially, the annual interest rate is split into two compounding periods. This approach allows the investment or loan to accrue interest at two instances throughout the year, leading to a growth in the principal amount more than if it was compounded annually but less than if it was compounded continuously.

The formula for calculating the final accumulated amount with semiannual compounding is:

• \( A = P \left(1 + \frac{r}{2}\right)^{2t} \)

Here, \( P \) is the principal, \( r \) is the annual interest rate, and \( t \) is the time period in years.

To compute the effective annual rate with semiannual compounding, consider:

• \( \text{EAR} = \left(1 + \frac{r}{2}\right)^2 - 1 \)

This formula considers the frequency of interest application and provides a clearer picture of the investing accomplishment. While semiannual compounding does not yield as high a return as continuous compounding, it still offers a better return than annual compounding. Decision-makers often use these calculations to choose among financial instruments.