To determine how much Jane Callahan's retirement annuity will be worth when she retires, we need to calculate the future value of two components:

1. The existing balance ($70,000).
2. The monthly contributions ($1,700) over the next 10 years.

The future value formula for compound interest is:

$FV = P \times \left(1 + \frac{r}{n}\right)^{nt} + \frac{C \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right)}{\frac{r}{n}}$

Where:

• $FV$ is the future value of the annuity.
• $P$ is the present value or current amount ($70,000).
• $C$ is the total monthly contribution ($1,700 = $700 from the company + $1,000 from Jane).
• $r$ is the annual interest rate (7.5% or 0.075).
• $t$ is the number of years (10 years).
• $n$ is the number of times interest is compounded per year (12 for monthly compounding).

Step-by-Step Calculation:

1. Future Value of Current Balance:

We use the first part of the formula:

$FV_1 = 70,000 \times \left(1 + \frac{0.075}{12}\right)^{12 \times 10}$

$FV_1 = 70,000 \times (1.00625)^{120}$

$FV_1 \approx 70,000 \times 2.10568$

$FV_1 \approx 147,397.60$

2. Future Value of Monthly Contributions:

For the monthly contributions, we use the second part of the formula:

$FV_2 = \frac{1,700 \times \left( \left(1 + \frac{0.075}{12}\right)^{12 \times 10} - 1 \right)}{\frac{0.075}{12}}$

$FV_2 = \frac{1,700 \times \left( 2.10568 - 1 \right)}{0.00625}$

$FV_2 = \frac{1,700 \times 1.10568}{0.00625}$

$FV_2 \approx \frac{1,700 \times 1.10568}{0.00625}$

$FV_2 \approx 300,146.77$

3. Total Future Value:

The total future value is the sum of the future value of the current balance and the future value of the monthly contributions:

$FV = 147,397.60 + 300,146.77$

$FV \approx 447,544$

Final Answer:

Jane Callahan’s retirement annuity will be worth approximately $447,544 when she retires.

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Would you like more details on the calculation or any part of this explained further?

Related Questions:

1. How would the result change if the interest rate were 8% instead of 7.5%?
2. What if Jane decides to contribute $1,200 per month instead of $1,000?
3. How does increasing the compounding frequency (e.g., to daily) affect the outcome?
4. What if Jane had an initial balance of $100,000 instead of $70,000?
5. How would the outcome differ if Jane retires in 15 years instead of 10?

Tip:

The more frequently interest is compounded, the greater the impact of compound interest on your investment. For the same rate, increasing compounding frequency leads to higher returns.