Problem 66 - College Algebra with Corequisite Support

Question

Kyoko has \(\$ 10,000\) that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \(\$ 15,000\) by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint solve the compound interest formula for the interest rate.)

Step-by-Step Solution

Answer

Kyoko needs a minimum annual interest rate of approximately 6.79\%.

Step 1: Understand the Compound Interest Formula

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

Step 2: Identify the Known Variables

Kyoko wants \( A = \\(15,000 \) from an initial investment of \( P = \\)10,000 \) in 6 years \((t = 6)\). The interest is compounded daily, so \( n = 365 \). We need to find the minimum annual interest rate \( r \).

Step 3: Set Up the Equation

Substitute the known values into the compound interest formula: \[ 15000 = 10000 \left(1 + \frac{r}{365}\right)^{365 \times 6} \] Simplifying gives us: \[ 1.5 = \left(1 + \frac{r}{365}\right)^{2190} \]

Step 4: Solve for the Interest Rate \( r \)

To solve for \( r \), we first take the 2190th root of both sides to isolate \( 1 + \frac{r}{365} \): \[ \sqrt[2190]{1.5} = 1 + \frac{r}{365} \] Calculate \( \sqrt[2190]{1.5} \) using a calculator, which is approximately 1.000186.

Step 5: Isolate \( r \) and Calculate

Subtract 1 from both sides to isolate \( \frac{r}{365} \): \[ 0.000186 = \frac{r}{365} \] Multiply both sides by 365 to solve for \( r \): \[ r = 0.000186 \times 365 \approx 0.06789 \]

Step 6: Convert \( r \) to a Percentage

Convert the decimal \( r \) to a percentage by multiplying by 100: \[ 0.06789 \times 100 = 6.789 \] Therefore, the annual interest rate needs to be approximately 6.79\%.

Key Concepts

Annual Interest Rate

Annual interest rate is a critical concept when dealing with investments and loans. Simply put, it is the percentage of interest you would earn or need to pay over a year.

Imagine depositing money into a savings account. The annual interest rate will dictate how much extra money you end up with at the end of the year. In Kyoko's situation, this rate determines the growth of her investment from $10,000 to $15,000.

When figuring out the required annual interest rate, you typically convert the rate to a decimal to use in formulas. For example, if the rate is 6.79%, you would convert it to 0.0679 for calculation purposes. This rate then integrates into the compound interest formula, working in tandem with other factors like compounding frequency and time.

In sum, the annual interest rate directly influences your financial growth or expense, making it a vital element for achieving financial goals.

Daily Compounding

Daily compounding refers to the process of calculating interest earnings on an investment every single day. When compared to other compounding frequencies like monthly or yearly, daily compounding can significantly increase the potential returns on an investment.

The concept hinges on daily recalculations of the principal plus accumulated interest, leading to exponential growth over time. In Kyoko's exercise, the bank compounds interest daily, meaning interest is added to the principal 365 times a year.

To capture this in calculations, the daily compounding factor is represented by the variable 'n' in the compound interest formula, where 'n' equals 365. This higher frequency of compounding means even a small difference in the interest rate can lead to markedly different outcomes over longer periods.

In short, daily compounding leverages the power of exponential growth, enhancing accumulated wealth over time, which is crucial for reaching financial goals faster.

Financial Goal

Setting a financial goal is an essential component of financial planning. This involves identifying a specific sum of money you aim to save or earn over a given timeframe. For Kyoko, her financial goal is clear: she wants to have $15,000 in six years from an initial investment of $10,000.

To achieve such a goal, you must understand various contributing factors, including initial investment (or principal), rate of return, time horizon, and compounding frequency. Each element plays a part in determining the pathway to meeting the desired financial target.

Setting a financial goal requires strategic planning and disciplined execution. Whether it's saving for education, retirement, or a major purchase, knowing your target and creating a viable plan to reach it is half the battle. In Kyoko's case, understanding the minimum annual interest rate needed, while considering daily compounding, is imperative to ensure her investment meets or exceeds her intended outcome.

In essence, a well-defined financial goal acts as a roadmap, guiding you through the complexities of financial decisions and helping you evaluate how well different investment options might serve your objectives.