Final answer:
Mike Prudent needs to deposit $9,000 today at an 8% interest rate compounded quarterly to ensure he receives $400 each quarter for the next 6 years.
Step-by-step explanation:
The question at hand is a financial mathematics problem that involves calculating the present value of an annuity. In this case, Mike Prudent wishes to receive a $400 payment each quarter for 6 years from an account earning 8% interest compounded quarterly. To find the single deposit required today to guarantee these payments, we can use the present value formula for an annuity:
PV = PMT * [(1 - (1 + r)^-n) / r]
Where:
- PV is the present value, the initial deposit needed.
- PMT is the payment each period ($400).
- r is the interest rate per period (0.08/4 = 0.02).
- n is the total number of periods (6 years * 4 quarters = 24 periods).
By substituting the values into the formula, we can solve for PV to find the initial deposit Mike needs to make. Calculations show that the required single deposit is the answer option (d) $9,000, which guarantees the quarterly $400 payments for 6 years at the given interest rate.
Starting to save money early and allowing compound interest to work can lead to substantial growth in savings over the years, as seen with the example of a $3,000 investment at 7% interest over 40 years, mentioned as a reference.