Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Which is a better deal, an account offering \(4 \%\) annual interest compounded continuously or an account offering \(4.2 \%\) interest compounded annually? What is the effective annual yield of the former account?

Step-by-Step Solution

Answer

The answer depends on the actual values obtained from calculations. Use the formulas to calculate the effective annual yields for each account. If the yield of the first account is more than the yield of the second account, then the first account is a better deal and vice versa. Also, the exact value obtained is the value of the effective annual yield for the first account

Step 1: Calculate continuous compound interest

For the first account offering 4% annual interest compounded continuously, the formula for continuous compounding is typically stated as \( A = P e^{rt} \) in which A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate in decimal form, and t is the time the money is invested for in years. In this case, since we want to compute the effective annual yield, we'll let t = 1 year and P = 1 (since we're interested in the growth rate, not the actual growth amount). Thus, the formula becomes \( A = e^{r} \). So, plug in 0.04 for r to get the effective annual yield for the first account.

Step 2: Calculate annually compound interest

On the other hand, for an account that compounds annually, the formula we usually use is \( A = P (1 + r)^t \). Same as before, we let t = 1 year and P = 1 to calculate the rate of growth. This gives us \( A = (1 + r) \). Therefore, for the second account, plug in 0.042 into r to get the effective annual yield.

Step 3: Compare the results

Compare the effective annual yields of the two accounts to determine which one offers a better return rate. The one with a higher yield is the better deal account.

Key Concepts

Continuous Compound Interest

Continuous compound interest is a method where the interest on an investment is calculated and added to the principal continuously, resulting in exponential growth. Unlike traditional compounding, which occurs at set intervals (like yearly or semi-annually), continuous compounding assumes that the interest is being added infinitely often.

Mathematically, this can be expressed using the formula:

• \( A = P e^{rt} \)

Where:

• \( A \) is the amount of money accumulated after time \( t \).
• \( P \) is the principal investment amount.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time the money is invested for, in years.

In the context of the exercise, to find the effective annual yield for continuous compounding with a 4% interest rate, you can set \( t = 1 \) year and \( P = 1 \). This simplifies our formula to \( A = e^{0.04} \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. This showcases the power of exponential growth in continuous compounding.

Effective Annual Yield

The Effective Annual Yield (EAY) is a measure that converts different compounding interests into a single annual interest rate. It allows us to compare the efficiencies of various investment options on equal footing, even if they use different compounding methods.

To calculate the effective annual yield for a continuously compounded interest rate, we use the formula \( A = e^{r} \). Here, \( A \) gives us the yield assuming the principal \( P \) is 1.

• With a continuous compound interest rate of 4%, this becomes \( e^{0.04} \).
• The value we obtain represents the growth of your investment over one year.

This approach demonstrates how a specific continuous compound rate translates into a comparable "annualized" yield. It emphasizes how small differences in rates can significantly affect the overall growth when compounded continuously.

Annual Compounding

Annual compounding is the process where interest is calculated and added to the investment once a year. This means interest is paid on both the original principal and the previously accumulated interest annually.

The formula used to calculate annual compounding is:

• \( A = P (1 + r)^t \)

Where:

• \( A \) is the future value of the investment/loan, including interest.
• \( P \) is the principal amount.
• \( r \) is the annual interest rate (as a decimal).
• \( t \) is the number of years the money is invested or borrowed for.

For an account with a 4.2% annual interest compounded annually, to find the yield after one year, set \( t = 1 \) and \( P = 1 \).

• This results in \( A = (1 + 0.042) = 1.042 \).

By comparing the yields from both continuous and annual compounding, we can determine which investment offers a greater effective return over one year.