Questions: You wish to accumulate 80,000 at the end of 10 years. To accomplish this, you plan to make equal deposits of X at the end of each year for the first 7 years. The annual effective rate is 6 % during the first 7 years and 5 % for the next three years. Calculate X.

Solution Steps

To solve this problem, we need to calculate the equal annual deposit \( X \) that will accumulate to $80,000 at the end of 10 years. The problem involves two different interest rates over two periods. First, calculate the future value of the annuity for the first 7 years using the 6% interest rate. Then, calculate the future value of this amount after 3 more years at a 5% interest rate. Finally, set this equal to $80,000 and solve for \( X \).

We need to calculate the future value factors for both periods. The future value factor for the first period (7 years at 6%) is given by:

\[ FV_{1} = (1 + 0.06)^{7} \approx 1.5036 \]

The future value factor for the second period (3 years at 5%) is:

\[ FV_{2} = (1 + 0.05)^{3} \approx 1.1576 \]

Next, we calculate the annuity factor for the first period (7 years at 6%):

\[ A_{1} = \frac{(1 + 0.06)^{7} - 1}{0.06} \approx 8.3938 \]

To find the equivalent future value at the end of 10 years, we divide the target future value by the future value factor for the second period:

\[ FV_{equiv} = \frac{80000}{FV_{2}} \approx \frac{80000}{1.1576} \approx 69107.0079 \]

Now, we can solve for the annual deposit \( X \):

\[ X = \frac{FV_{equiv}}{A_{1}} \approx \frac{69107.0079}{8.3938} \approx 8233.0646 \]

Final Answer