Questions: Lab Assessment Question 19, P6-53 (similar to) Part 4 of 4 HW Score: 55%, 11 of 20 points Points: 0 of 1 Save (Comprehensive problem) You would like to have 54,000 in 13 years. To accumulate this amount, you plan to deposit an equal sum in the bank each year that will earn 6 percent interest compounded annually. Your first payment will be made at the end of the year. a. How much must you deposit annually to accumulate this amount? b. If you decide to make a large lump-sum deposit today instead of the annual deposits, how large should this lump-sum deposit be? (Assume you can earn 6 percent on this deposit.) c. At the end of five years, you will receive 15,000 and deposit this in the bank toward your goal of 54,000 at the end of year 13. In addition to the lump-sum deposit, how much must you deposit in equal annual amounts, beginning in year 1 to reach your goal? (Again, assume you can earn 6 percent on your deposits.) a. How much must you deposit annually to accumulate this amount? 2859.85 (Round to the nearest cent.) b. If you decide to make a large lump-sum deposit today instead of the annual deposits, how large should the lump-sum deposit be? 25317.31 (Round to the nearest cent.) c. If you deposit 15,000 received at the end of five years in the bank, what will the amount grow to by the end of year 13? 23907.72 (Round to the nearest cent) In addition to the lump-sum deposit, how much must you deposit in equal annual amounts, beginning in year 1 to reach your goal? (Round to the nearest cent.)

Solution Steps

To solve these financial problems, we will use the formulas for future value of an annuity and present value of a lump sum.

a. To find the annual deposit required to accumulate $54,000 in 13 years with 6% interest compounded annually, we use the future value of an annuity formula.

b. To find the lump-sum deposit required today to accumulate $54,000 in 13 years with 6% interest, we use the present value formula.

c. To find the future value of a $15,000 deposit made at the end of 5 years by the end of year 13, we use the future value formula. Then, we calculate the remaining amount needed and find the annual deposit required to reach the goal.

To find the annual deposit required to accumulate $54,000 in 13 years with a 6% interest rate compounded annually, we use the future value of an annuity formula:

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

Solving for \(P\):

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

Given:

• \(FV = 54000\)
• \(r = 0.06\)
• \(n = 13\)

\[ P = \frac{54000 \times 0.06}{(1 + 0.06)^{13} - 1} \approx 2859.85 \]

To find the lump-sum deposit required today to accumulate $54,000 in 13 years with a 6% interest rate, we use the present value formula:

\[ PV = \frac{FV}{(1 + r)^n} \]

Given:

• \(FV = 54000\)
• \(r = 0.06\)
• \(n = 13\)

\[ PV = \frac{54000}{(1 + 0.06)^{13}} \approx 25317.31 \]

To find the future value of a $15,000 deposit made at the end of 5 years by the end of year 13, we use the future value formula:

\[ FV = PV \times (1 + r)^n \]

Given:

• \(PV = 15000\)
• \(r = 0.06\)
• \(n = 13 - 5 = 8\)

\[ FV = 15000 \times (1 + 0.06)^8 \approx 23907.72 \]

First, we find the remaining amount needed to reach the goal of $54,000:

\[ \text{Remaining Amount} = 54000 - 23907.72 \approx 30092.28 \]

Next, we calculate the annual deposit required to accumulate the remaining amount in the remaining 5 years:

\[ P = \frac{30092.28 \times 0.06}{(1 + 0.06)^5 - 1} \approx 5338.26 \]

Final Answer