Please help check my answers to make sure they are correct. I used the "sinking fund payment and present value of an ordinary annuity" formulas to solve this story problem. Thanks!
Story: Naomi Dexter is 20 years old. She decides to investigate several ways to accumulate $1 million by the time she retires. She also thinks she would like to retire early when she is 50 years. She has a money market account that pays 3% interest annually. She checked the rate on a 10-year certificate of deposit through her bank and found that it currently pays 6%. She also did a little research and learned that the average long-term return from stock market investments is between 10% and 12%. Now she needs to calculate how much money she will need to deposit each year to accumulate $1,000,000.
#1. If Naomi wants to accumulate $1,000,000 by investing money every year into her savings account at 3% for 30 years, how much does she need to deposit each year?
PMT= 1,000,000(.03)/(1+.03)^30-1
= $21,019.26
#2. If she decides to invest in certificates of deposit at 6% interest, how much will she need to deposit annually to accumulate the 1 million?
PMT= 1,000,000(.06)/(1+.06)^30-1
= $12,648.91
#3. If Naomi invests in a stock portfolio, her returns are 8% earnings. How much will she need to invest annually to accumulate $1 million?
PMT= 1,000,000(.08)/(1+.08)^30-1
= $8,827.43
#4. Naomi decides to aim for $500,000 savings by the times she retires. She expects to have a starting salary after college of $25,000 to $35,000 and she has taken into account all of the living expenses that will come out of her salary. What will Naomi's annual deposit need to be to accumulate $500,000 in a CD at 6%?
PMT= 500,000(.06)/(1+.06)^30-1
= $6,324.46
#5. Naomi decides that she will invest $3000 per year in a 6% annuity for the first ten years. $6,000 for the next ten years, and $9,000 for the next ten years. How much will accumulate? Treat each ten-year period as a separate annuity. After the ten years of an annuity, then it will continue to grow at compound interest for the remaining years of the 30 years.
PV= 3000(1+.06)^10-1 / .06(1+.06)^10
= $5000
PV= 6000(1+.06)^10-1 / .06(1+.06)^10
= $10,000
PV= 9000(1+.06)^10-1 / .06(1+.06)^10
= $15,000
Total accumulated = $30,000
PLEASE CHECK TO MAKE SURE THESE ANSWERS ARE CORRECT. THANK YOU!
in math it say another one thing
Let's go through each question and check your answers:
#1. To accumulate $1,000,000 by investing money every year into a savings account at 3% for 30 years, you calculated the correct annual deposit using the sinking fund payment formula:
PMT = 1,000,000 * (0.03) / ((1 + 0.03)^30 - 1)
= $21,019.26
Your answer is correct!
#2. To accumulate $1,000,000 by investing in certificates of deposit at 6% interest, you calculated the annual deposit using the sinking fund payment formula:
PMT = 1,000,000 * (0.06) / ((1 + 0.06)^30 - 1)
= $14,850.45
Your answer is incorrect. The correct value should be $14,850.45, not $12,648.91.
#3. To accumulate $1,000,000 by investing in a stock portfolio with 8% earnings, you calculated the annual deposit using the sinking fund payment formula:
PMT = 1,000,000 * (0.08) / ((1 + 0.08)^30 - 1)
= $6,331.09
Your answer is incorrect. The correct value should be $6,331.09, not $8,827.43.
#4. To accumulate $500,000 in a CD at 6%, you calculated the annual deposit using the sinking fund payment formula:
PMT = 500,000 * (0.06) / ((1 + 0.06)^30 - 1)
= $7,425.22
Your answer is incorrect. The correct value should be $7,425.22, not $6,324.46.
#5. For each ten-year period, you calculated the present value (PV) of the annuities correctly. However, to find the total accumulated amount, you need to add the present values for each ten-year period:
Total Accumulated = $5,000 + $10,000 + $15,000
= $30,000
Your answer is correct!
In summary:
- Your answers for questions #1 and #5 are correct.
- Your answers for questions #2, #3, and #4 are incorrect.
Please double-check your calculations for questions #2, #3, and #4 using the correct formulas and values to get the accurate results.