Question

You open a 5-year CD for \(\$ 1,000\) that pays \(2 \%\) interest, compounded annually. What is the value of that \(\mathrm{CD}\) at the end of the five years?

Short answer

The value of the 5-year CD at the end of five years will be \(\$1,104.08\).

Step-by-step solution

Convert the interest rate

Begin by converting the annual interest rate from a percentage to a decimal. To do this, divide the percentage by 100: \[r = \frac{2}{100} = 0.02\] Now, r = 0.02

Set up the formula

Set up the compound interest formula with the values given in the problem: \[A = 1000(1 + \frac{0.02}{1})^{1 \cdot 5}\]

Simplify the formula

Simplify the formula by calculating the values within the parentheses and raising it to the power of 5: \[A = 1000(1 + 0.02)^5\] \[A = 1000(1.02)^5\]

Calculate the final amount

Calculate the final amount using the simplified formula: \[A = 1000 \cdot (1.10408) = \$1,104.08\] At the end of 5 years, the value of the CD will be \(\$1,104.08\).

Key concepts

Interest Rate Conversion

Understanding how to convert an interest rate from a percentage to a decimal is crucial for calculating compound interest. In our CD example, the given annual interest rate is 2%. To convert this percentage to a decimal form, we simply divide by 100, giving us 0.02.

Why do we do this? It's because percentages are an easily understood way to express proportional relationships in everyday language, but when it comes to mathematical calculations, especially those involving interest, using decimals is necessary. For instance, the multiplication of decimals aligns well with the algebraic formulations used in finance. As seen in the exercise:

• Start with the given annual interest rate: 2%
• Convert the percentage into a decimal by dividing by 100.
• The converted interest rate is: \(r = \frac{2}{100} = 0.02\)

This step is the foundation for applying the compound interest formula correctly and ensuring accurate financial calculations.

Compound Interest Formula

The compound interest formula is a mathematical expression that calculates the amount of money earned on an investment over time. In our example, we're using the formula to find the future value of a certificate of deposit (CD) over five years.

The formula is:
\[A = P(1 + \frac{r}{n})^{nt}\]
where:

• \(A\) 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).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time the money is invested for, in years.

For the given exercise, the formula with the inserted values becomes:\[A = 1000(1 + \frac{0.02}{1})^{1 \cdot 5}\]
Upon simplification, we arrive at:\[A = 1000(1 + 0.02)^5\]
This illustrates how compound interest grows the initial investment through the effect of interest on interest, which becomes significant over longer periods.

Exponential Growth

Exponential growth describes a process where the amount grows at a rate proportional to its current value, meaning it increases faster and faster as time goes on. This concept is not only applicable in finance but also in populations, nuclear reactions, and more.

In the context of our compound interest example for the CD, the principal amount is subject to exponential growth due to the interest being compounded annually. To visualize this, we calculate the amount after each year and observe how it progressively increases due to the accumulating interest. Here's the simplified model of growth for the CD:
\[A = P(1 + r)^t\]

• The parameter \(t\) represents the time, highlighting the exponential nature as it is the exponent in the formula.
• The expression \((1 + r)^t\) will grow larger for each increase in \(t\), demonstrating the exponential growth.

With the respective values plugged in, the calculation shows that the value of the CD after 5 years is \(A = 1000 \times (1.10408) = \$1,104.08\). Each year, the interest earned is not only on the original amount but also on the interest from previous years, leading to substantially more significant amounts over long periods.