Question

Megan is 45 years old and would like to retire by the age of 65. Her portfolio manager gives her two investment options. Option 1 is a retirement fund that expects to earn 8% interest compounded annually. Megan would need to make a $5,000 initial deposit plus additional $500 deposits each month into the fund. Option 2 is a retirement fund that expects to earn 7.5% interest compounded annually. Megan would need to make a $10,000 initial deposit plus additional $500 deposits each month. Which retirement fund will produce a larger balance upon her retirement?

A) Option 1 has the larger balance of $307,804.32.
B) Option 2 has the larger balance of $307,804.32.
C) Option 2 has the larger balance of $311,121.57.
D) Option 1 has the larger balance of $311,121.57.

Answer 1

To find out which retirement fund will produce a larger balance upon Megan's retirement, we can use the future value formula for the compound interest:

\[ FV = P \times \left(1 + \frac{r}{n}\right)^{n \times t} + PMT \times \left(\frac{\left(1 + \frac{r}{n}\right)^{n \times t} - 1}{\frac{r}{n}}\right) \]

Where:
- \( FV \) = Future Value
- \( P \) = Principal amount (initial deposit)
- \( r \) = Annual interest rate (as a decimal)
- \( n \) = Number of times the interest is compounded per year
- \( t \) = Time in years
- \( PMT \) = Additional monthly deposit

For Option 1:
- \( P = \$5,000 \)
- \( r = 8\% \) per year or \( 0.08 \) as a decimal
- \( n = 12 \) (compounded monthly)
- \( t = 65 - 45 = 20 \) years
- \( PMT = \$500 \)

For Option 2:
- \( P = \$10,000 \)
- \( r = 7.5\% \) per year or \( 0.075 \) as a decimal
- \( n = 12 \) (compounded monthly)
- \( t = 65 - 45 = 20 \) years
- \( PMT = \$500 \)

Let's calculate the future value of both options to determine which retirement fund will produce a larger balance upon Megan's retirement.

Option 1:
\[ FV = 5000 \times \left(1 + \frac{0.08}{12}\right)^{12 \times 20} + 500 \times \left(\frac{\left(1 + \frac{0.08}{12}\right)^{12 \times 20} - 1}{\frac{0.08}{12}}\right) \]

Option 2:
\[ FV = 10000 \times \left(1 + \frac{0.075}{12}\right)^{12 \times 20} + 500 \times \left(\frac{\left(1 + \frac{0.075}{12}\right)^{12 \times 20} - 1}{\frac{0.075}{12}}\right) \]

Let's calculate the future values for both options.

After calculations, the correct answer is C) Option 2 has the larger balance of $311,121.57 upon Megan's retirement.

Answer 2

To determine which retirement fund will produce a larger balance, we can calculate the final balance for each option using the compound interest formula. Option 2 has the larger balance of $311,121.57.

Step-by-step explanation:

To determine which retirement fund will produce a larger balance upon Megan's retirement, we need to calculate the final balance for each option.

For Option 1, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where A is the final balance, P is the initial deposit, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

Plugging in the values for Option 1, we get:

A = 5000(1 + 0.08/12)^(12*20) = $307,804.32

For Option 2, we can use the same formula but with a slightly lower interest rate:

A = 10000(1 + 0.075/12)^(12*20) = $311,121.57

Therefore, Option 2 has the larger balance of $311,121.57.