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!

30,000

To calculate the deposit needed annually to accumulate a certain amount of money, the formula used is the sinking fund payment formula, which is:

PMT = PV * (r / (1 - (1 + r)^(-n)))

where:
PMT is the periodic deposit or payment needed,
PV is the present value or the desired accumulated amount of money,
r is the interest rate per period, and
n is the number of periods.

For #1, the calculation is correct.

For #2, the calculation is correct.

For #3, the calculation is correct.

For #4, the calculation is correct.

For #5, the calculation is incorrect. It seems you have mistaken the formulas. To calculate the present value (PV) of an ordinary annuity, the formula used is:

PV = PMT * ((1 - (1 + r)^(-n)) / r)

where:
PMT is the periodic deposit or payment needed,
r is the interest rate per period, and
n is the number of periods.

Let's calculate the correct values for #5:

PV1 = 3000 * ((1 - (1 + 0.06)^(-10)) / 0.06)
= $20,642.08

PV2 = 6000 * ((1 - (1 + 0.06)^(-10)) / 0.06)
= $41,296.16

PV3 = 9000 * ((1 - (1 + 0.06)^(-10)) / 0.06)
= $61,959.25

Total accumulated = PV1 + PV2 + PV3
= $20,642.08 + $41,296.16 + $61,959.25
= $123,897.49

Hence, the correct answer for #5 is $123,897.49.

I have checked your answers and provided the corrections where necessary. Please double-check the information and formulas used to ensure accuracy.