Final answer:
Mr. Dawson must deposit $280.21 at the beginning of every month for 21 years into an account with a monthly compounded interest rate of 6.84% to receive monthly payments of $1,230 upon retirement for a period of 19 years.
Step-by-step explanation:
To determine how much Mr. Dawson must deposit into an account at the beginning of every month to receive payments after retirement, we must first calculate the present value of the annuity he aims to draw upon retirement.
Step 1: Calculate the present value (PV) of the annuity payments Mr. Dawson wants to receive for 19 years. We will use the formula for the present value of an annuity due, which accounts for payments at the beginning of each period:
PV = Pmt * [(1 - (1 + r)^{-n}) / r] * (1 + r),
where Pmt = $1,230, the monthly payment; r = 6.84% annual interest rate compounded monthly (r = 0.0684 / 12); n = number of periods (19 years * 12 months).
Step 2: Calculate the annuity's future value (FV) 21 years from now using the PV found in Step 1. The FV formula is:
FV = PV * (1 + r)^n.
Step 3: Calculate how much must be deposited each month for the next 21 years to reach the calculated FV. This is the reverse of the FV calculation, and we use the formula for the present value of an annuity:
PV = FV / [(1 + r)^n - 1) / r] * (1 + r)
The correct answer is calculated by substituting the values from Step 1 and Step 2 into the third formula. After performing the calculations, we find that Mr. Dawson should deposit $280.21 per month to achieve his retirement goals, which corresponds to option a.