Problem 67 - Algebra Form and Function

Question

A certificate of deposit is worth \(P(1+r)^{t}\) dollars after \(t\) years, where \(r\) is the annual interest rate expressed as a decimal, and \(P\) is the amount initially deposited. State which investment will be worth more. Investment \(A,\) in which \(P=\$ 10,000, r=2 \%,\) and \(t=10\) years or investment \(B,\) in which \(P=\$ 5000\), \(r=4 \%,\) and \(t=10\) years.

Step-by-Step Solution

Answer

Answer: Investment A will be worth more after 10 years, with an estimated worth of $12,190.26 compared to Investment B's estimated worth of $7,401.22.

Step 1: Identify the given values for Investment A and Investment B

For Investment A, we have: - Initial deposit: \(P_A = \$10,000\) - Annual interest rate: \(r_A = 2\% = 0.02\) - Time: \(t_A = 10\) years For Investment B, we have: - Initial deposit: \(P_B = \$5,000\) - Annual interest rate: \(r_B = 4\% = 0.04\) - Time: \(t_B = 10\) years

Step 2: Calculate the worth of Investment A after 10 years

Using the formula: \(P(1+r)^{t}\), we can find the worth of Investment A after 10 years: \(W_A = P_A (1 + r_A)^{t_A} = \$10,000(1 + 0.02)^{10}\) Calculating the worth of Investment A: \(W_A = \$10,000(1.02)^{10}\) \(W_A ≈ \$12,190.26\)

Step 3: Calculate the worth of Investment B after 10 years

Using the formula: \(P(1+r)^{t}\), we can find the worth of Investment B after 10 years: \(W_B = P_B (1 + r_B)^{t_B} = \$5,000 (1 + 0.04)^{10}\) Calculating the worth of Investment B: \(W_B = \$5,000(1.04)^{10}\) \(W_B ≈ \$7,401.22\)

Step 4: Compare the worth of Investment A and Investment B after 10 years

Now that we have the worth of both investments after 10 years: - Investment A: \(W_A ≈ \$12,190.26\) - Investment B: \(W_B ≈ \$7,401.22\) Since \(W_A > W_B\), we can conclude that Investment A, with an initial deposit of \( \$10,000\) and an annual interest rate of \(2\%\), will be worth more after 10 years than Investment B, with an initial deposit of \( \$5,000\) and an annual interest rate of \(4\%\).

Key Concepts

Certificate of Deposit

A certificate of deposit, commonly known as a CD, is a time deposit offered by banks and credit unions. It allows individuals to save money and earn interest over a fixed period. Unlike a regular savings account, a certificate of deposit typically offers a higher interest rate because the money is locked in for a set term — often ranging from a few months to several years. This means you cannot withdraw your funds from a CD without facing a penalty.
Relatively risk-free, a CD is ideal for individuals seeking a low-risk investment option. It offers predictable returns since the interest rate is fixed at the time of purchase. Here's what to remember when considering a certificate of deposit:

• **Fixed Term**: You must keep your money in the CD for the entire specified period.
• **Interest Rates**: Typically higher than regular savings accounts.
• **Withdrawals**: Early withdrawal may result in penalties, reducing your earnings.

By understanding these basics, you can assess whether a CD aligns with your financial goals.

Investment Comparison

When comparing investments like the two CD options provided, it's key to look beyond just the initial deposit and the interest rate. Other factors such as the time period and the compounding effect on growth need to be examined.
Investment A and Investment B show that by understanding the power of compounding interest, substantial differences in returns can be illustrated even when initial conditions appear to favor one investment over the other.
Consider these points when comparing investments:

• **Initial Investment**: The larger the initial deposit, the more substantial the principal amount from which the interest is calculated.
• **Interest Rate**: A higher rate can lead to more growth, but when combined with a lower initial investment, it might not always result in higher returns.
• **Term/Duratification**: Longer investment terms enhance the compounding effects.

Investment comparison is vital to ensure your financial decisions align with your future ambitions minimally affected by optimism bias or mere initial appearances.

Interest Rate Calculation

Calculating the worth of an investment with compound interest involves understanding how the principal amount grows over time at a specified interest rate. The important formula you need to remember is: \[ P(1 + r)^t \] Where:

• **P** is the principal amount or initial investment.
• **r** is the annual interest rate, expressed as a decimal.
• **t** is the time the money is invested for, in years.

In the case of the indicated exercise, we used this formula to determine which of the two investments would grow more over 10 years.
Using this calculation method helps anticipate future value effectively, making it simpler to make real-world investment choices. When comparing interest calculations, always consider not just the numbers but also the implications of early withdrawals, varying rates, and different compounding methods.