Final answer:
To calculate the lump sum required today to allow for future withdrawals of $10,000 with 10% interest compounded annually, use the present value formula for each withdrawal and sum them.
For a savings plan, apply the future value compound interest formula to determine the accumulated sum after a certain period. To have $10,000 in ten years at 10% interest, calculate the present value using the present value formula.
Step-by-step explanation:
To determine the lump sum amount needed today to provide withdrawals of $10,000 on each of your son’s 18th, 19th, 20th,
and 21st birthdays with a compounding interest rate of 10% per year, you would use the present value formula for an annuity.
However, since the payments are not made at regular intervals from today but are instead one-time payments on future dates, we calculate the present value (PV) separately for each withdrawal and sum them up.
The formula for the present value of money due in the future is
PV = FV / (1 + r)n
Using this formula:
- For the withdrawal on the 18th birthday: PV = $10,000 / (1 + 0.10)18
- For the 19th birthday: PV = $10,000 / (1 + 0.10)19
- For the 20th birthday: PV = $10,000 / (1 + 0.10)20
- For the 21st birthday: PV = $10,000 / (1 + 0.10)21
After calculating the present value for each withdrawal, you sum all the present values to get the total lump sum amount needed today.
For the savings plan question, using the future value formula for compound interest:
Future Value = P (1 + r/n)nt
Where P is the principal amount ($1000), r is the annual interest rate (10%), n is the number of times that interest is compounded per year, and t is the time the money is invested for (15 years).
To determine the lump sum needed for a future $10,000 after ten years with a 10% interest rate compounded annually, the formula is:
PV = $10,000 / (1 + 0.10)10
You calculate the present value factor and then multiply $10,000 by this factor to determine the initial deposit.