Problem 66 - Precalculus

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 6.45%.

Step 1: Understand the Compound Interest Formula

The compound interest formula is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] 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 the money is invested for in years.

Step 2: Plug in Known Values into the Formula

We know \( A = 15,000 \) (the desired amount), \( P = 10,000 \), \( t = 6 \) (the number of years), and \( n = 365 \) (since the interest is compounded daily). Substitute the known values into the compound interest formula:\[ 15,000 = 10,000 \left(1 + \frac{r}{365}\right)^{365 \times 6} \]

Step 3: Isolate the Rate Formula

Solve the equation for \( r \): 1. Divide both sides by 10,000:\[ 1.5 = \left(1 + \frac{r}{365}\right)^{2190} \]2. Take the 2190th root of both sides to isolate \( \frac{r}{365} \):\[ 1 + \frac{r}{365} = 1.5^{\frac{1}{2190}} \]

Step 4: Solve for the Annual Interest Rate

Subtract 1 from each side:\[ \frac{r}{365} = 1.5^{\frac{1}{2190}} - 1 \]Multiply both sides by 365 to solve for \( r \):\[ r = 365 \times \left(1.5^{\frac{1}{2190}} - 1\right) \]Calculate this using a calculator to find \( r \) in decimal form.

Step 5: Convert Decimal to Percentage

Convert \( r \) to a percentage by multiplying by 100. After calculating the expression, you should find that the annual interest rate \( r \) is approximately 6.45%.

Key Concepts

Investment Account

An investment account is a financial account that holds funds and assets with the intent of growing its value over time.
These accounts can include various types of investments like stocks, bonds, mutual funds, or simply cash balances.
Let's look at some important aspects of investment accounts.

• Purpose: People open investment accounts with the aim to increase wealth for future needs, such as retirement, schooling, or buying a house.
• Types: Investment accounts can be either taxable or tax-advantaged. Taxable accounts do not offer any tax benefits, while tax-advantaged ones include IRAs or 401(k)s, which provide tax deferment or tax-exempt growth.
• Performance: The performance or growth of an investment account depends on various factors, including the account type, market conditions, and personal investment strategies.

In Kyoko's case, she has chosen an investment account that compounds interest daily.
This means her initial deposit of $10,000 grows over time, aiming to meet a future financial target, such as having $15,000 for her credited goal.

Annual Interest Rate

The annual interest rate is the percentage increase in money that an investment or loan provides over a year.
This rate is crucial for determining the growth of an investment over time. Here's what you need to know:

• Definition: It is the rate at which interest is added to the principal balance.
• Calculation: As shown in the exercise, solving for the annual interest rate involves using the compound interest formula. It requires understanding the exponential growth as the interest accrues not annually, but more frequently at smaller daily intervals.
• Impact: A higher annual interest rate means faster growth of the invested funds, while a lower rate brings about slower increases.

In Kyoko's scenario, she would need to solve for the optimal annual interest rate that helps her reach $15,000 within six years based on her initial $10,000 deposit.
This calculation involves reversing the compound interest formula to isolate and determine this annual interest rate, culminating in a result of around 6.45%.

Compounding Daily

Compounding daily refers to the process where the interest earned on an investment is calculated and added back to the principal balance every single day.
It leads to faster growth compared to other compounding frequencies due to the frequent accumulation of interest. Here are the key aspects of daily compounding:

• Frequency: Compounding can occur on different schedules, such as annually, semi-annually, quarterly, monthly, or daily. Daily compounding results in 365 compounding periods per year, thus increasing the frequency of interest being added to the total balance.
• Growth: Thanks to this more frequent compounding, the investment grows at an accelerating rate, with interest on interest being accumulated.
• Formula Adjustments: In the compound interest formula, the frequency of compounding is represented by the variable \( n \). For daily compounding, \( n \) is set to 365.

For Kyoko's investment, choosing an account that compounds daily could significantly impact how quickly she reaches her goal amount.
The consistent daily addition of interest means her investment grows each day, inching closer to the $15,000 target over the six-year period.