# Math

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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

• Math -

30,000

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