Problem 51 - College Algebra and Calculus: An Applied Approach

Question

You want to invest \(\$ 5000\) in a certificate of deposit for 12 months. You are given the options below. Which would you choose? Explain. (a) \(r=5.25 \%\), quarterly compounding (b) \(r=5 \%\), monthly compounding (c) \(r=4.75 \%\), continuous compounding

Step-by-Step Solution

Answer

The best choice would be option (a) which yields 5263.30 dollars after 1 year.

Step 1: Calculate the Amount for Option (a)

The given interest rate is 5.25%, or 0.0525 in decimal form, the time is 1 year, and it's quarterly compounding which means \(n=4\). Substituting these values into the formula, we get \(A = 5000 * (1+0.0525/4)^(4*1) = 5000 *(1.013125)^4 = 5000 * 1.052661 = 5263.30\) dollars.

Step 2: Calculate the Amount for Option (b)

The interest rate for option (b) is 5.00% or 0.05 in decimal form, and it's compounded monthly which implies \(n=12\). Inserting these values into the formula, we have \(A = 5000* (1+0.05/12)^(12*1) = 5000*(1.004167)^{12} = 5000 *1.051267= 5256.34\) dollars.

Step 3: Calculate the Amount for Option (c)

The interest rate for option (c) is 4.75% or 0.0475, and it's compounded continuously. Utilizing the continuous compounding formula, \(A = 5000*e^{0.0475*1} = 5000* 1.048812 = 5244.06\) dollars.

Step 4: Choose the Best Option

Now, comparing the Future value of money after 1 year in all three options, we can see that option (a) yields the most amount after 1 year. Thus, the best choice would be option (a): 5.25% with quarterly compounding.

Key Concepts

Quarterly Compounding

When we talk about quarterly compounding, we're referring to the process of earning interest on investment four times a year. This means every quarter, the interest earned is added to the initial principal, and then this new total becomes the next quarter's principal. Over time, this can lead to more significant growth compared to less frequent compounding.

To calculate the amount with quarterly compounding, you use the formula:

• \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Here, \(A\) is the future value of the investment, \(P\) is the principal amount, \(r\) is the annual interest rate in decimal form, \(n\) is the number of compounding periods per year (4 for quarterly), and \(t\) is the time in years.
For instance, with an investment of $5000 at 5.25% interest over one year, you calculate as follows:

• \[ A = 5000 \left(1 + \frac{0.0525}{4}\right)^{4 \times 1} = 5263.30 \]

By compounding interest quarterly, you effectively harness the power of compounding more frequently, often resulting in higher returns than annual or less frequent compounding methods.

Monthly Compounding

Monthly compounding takes the compounding process a step further by calculating and adding interest twelve times a year. This frequent updating of balance results in slightly higher returns compared to quarterly compounding.

The formula used is the same as for other discrete compounding methods, adjusted for the frequency:

• \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

For monthly compounding, \(n = 12\).
To illustrate, by investing $5000 at a 5% annual interest rate with monthly compounding for one year, we calculate:

• \[ A = 5000 \left(1 + \frac{0.05}{12}\right)^{12 \times 1} = 5256.34 \]

While it results in more interest than annual compounding, it usually earns a bit less than quarterly compounding if the rate conditions are not significantly better. Thus, the actual gain is dictated by the interest rate and the number of compounding periods.

Continuous Compounding

Continuous compounding represents a theoretical limit of compounding interest where interest is calculated and added to the principal an infinite number of times per year. This uses the natural base "e" to evaluate the future value. The formula is given by:

• \[ A = Pe^{rt} \]

Where \(A\) is the amount after time \(t\), \(P\) is the principal balance \(e\) is the mathematical constant approximately equal to 2.71828, \(r\) is the annual interest rate, and \(t\) is the time in years.
For instance, with $5000 at a 4.75% rate, continuous compounding is calculated as:

• \[ A = 5000 \times e^{0.0475 \times 1} = 5244.06 \]

Continuous compounding often results in slightly higher final amounts than ordinary compounding intervals, but it greatly depends on the rate provided compared to the frequency of normal compounding. It illustrates how time and the number of periods can significantly influence the earning potential of your investment.