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Chapter 6: Problem 12

APPLICATION Camila received a \(\$ 1,200\) prize for one of her essays. She decides to invest \(\$ 1,000\) of it for college. Her bank offers two options. The first is a regular savings account that pays \(2.5 \%\) interest every 6 months. The second is a certificate of deposit that pays \(5 \%\) interest each year. a. With the savings account, how much would Camila have after 1 year? After 2 years? (a) b. With the certificate of deposit, how much would Camila have after 1 year? After 2 years? c. Explain why you get different results for \(12 a\) and \(b\). (a)

Short Answer

Expert verified
The savings account earns \$51.25 after 1 year and \$104.06 after 2 years, whereas the CD earns \$50 after 1 year and \$102.50 after 2 years. Different compounding periods affect the total interest earned.

Step by step solution

01

Calculate Savings Account Interest After 1 Year

The savings account pays 2.5% interest every 6 months, which is compounded semi-annually. After one year, the interest will be applied twice. We use the formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(A\) is the amount after time \(t\), \(P\) is the principal amount (\$1000), \(r\) is the annual interest rate (0.025), \(n\) is the number of times interest is compounded per year (2), and \(t\) is the time in years (1). Calculate the amount after one year.
02

Calculate Savings Account Interest After 2 Years

Using the same formula as in Step 1, calculate the amount for \(t = 2\) years. Substitute \(P = 1000\), \(r = 0.025\), \(n = 2\), and \(t = 2\) to find the total amount.
03

Calculate Certificate of Deposit Interest After 1 Year

The CD pays 5% interest annually, compounded once per year. Use the formula \(A = P \left(1 + r \right)^t\), where \(P = 1000\), \(r = 0.05\), and \(t=1\). Calculate the total amount after one year.
04

Calculate Certificate of Deposit Interest After 2 Years

Using the same formula from Step 3, calculate the total amount for \(t = 2\) years. Substitute \(P = 1000\), \(r = 0.05\), and \(t = 2\) to find the total value after two years.
05

Compare the Results

After calculating the amounts for both the savings account and the CD after 1 and 2 years, you will find that different compounding periods lead to different total amounts. Explain that the frequency of compounding affects how much interest is earned, thus causing the difference in results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Savings Account
A savings account is a basic type of bank account that allows your money to earn interest over time. It is often used for keeping money safe and accessible while earning a bit extra. One of the key features of savings accounts is that they typically offer compound interest. This means your initial deposit, plus any interest earned, will earn interest in subsequent periods. This is the magic of compounding, which can grow your money more effectively over time. For instance:
  • In our example, Camila's savings account offers a 2.5% interest rate, compounded semi-annually. This means the interest is calculated twice a year.
  • The compound interest formula for this scenario is: \( A = P \left(1 + \frac{r}{n}\right)^{nt} \), where "\(P\)" is the initial deposit, "\(r\)" is the annual interest rate, "\(n\)" is the number of times interest is compounded annually, and "\(t\)" is the time in years.
  • Over time, a savings account can help money grow while remaining easily accessible.
Choosing a savings account is ideal for keeping funds that you might need access to quickly, while still benefitting from interest gains.
Certificate of Deposit
A Certificate of Deposit (CD) is a specialized savings tool that typically offers a higher interest rate compared to a regular savings account. It requires you to lock in your money for a fixed period, which can range from a few months to several years. During this time, you cannot withdraw your money without incurring a penalty.CDs are attractive because:
  • They generally offer higher interest rates, as exemplified by Camila's option of 5% annually.
  • The interest calculation for a CD can be represented by the formula: \( A = P \left(1 + r \right)^t \), where "\(P\)" is the initial principal, "\(r\)" is the annual interest rate, and "\(t\)" is the term in years.
  • They provide a fixed return, which can be appealing for those looking for stability and predictability in their investments.
CDs are suitable for those who can afford to keep their money locked away for a period in exchange for higher earnings.
Investment Options
When deciding on investment options, it's important to consider factors such as safety, returns, and accessibility. Savings accounts and CDs serve different financial needs and goals, demonstrating this variety.
  • **Savings Account**: Best for those who want liquidity and safety, with funds that remain accessible for emergencies or short-term needs while still growing modestly due to interest.
  • **Certificate of Deposit**: Fits investors aiming for higher returns without needing immediate access to their funds. They benefit from the higher rates because they can commit money for a longer period.
  • The choice between a savings account and a CD depends on personal financial goals and whether an individual prioritizes accessibility or a higher return on investment.
Understanding these options helps investors tailor their choice to suit their specific financial goals and timelines.

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