Math
posted by April .
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 10year certificate of deposit through her bank and found that it currently pays 6%. She also did a little research and learned that the average longterm 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)^301
= $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)^301
= $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)^301
= $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)^301
= $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 tenyear 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)^101 / .06(1+.06)^10
= $5000
PV= 6000(1+.06)^101 / .06(1+.06)^10
= $10,000
PV= 9000(1+.06)^101 / .06(1+.06)^10
= $15,000
Total accumulated = $30,000
PLEASE CHECK TO MAKE SURE THESE ANSWERS ARE CORRECT. THANK YOU!

30,000