Final answer:
The present value of 20 quarterly payments of $3,000 each, received over five years at an annual interest rate of 5%, is calculated using the present value annuity formula and the resulting value is $52,925.64.
Step-by-step explanation:
To find the present value of 20 quarterly payments of $3,000 each over the next five years with a stated annual interest rate of 5%, we use the present value of an annuity formula. The formula for the present value of an annuity is PV = PMT × [(1 - (1 + r)^{-n}) / r], where PV is the present value, PMT is the periodic payment amount, r is the periodic interest rate, and n is the total number of payments.
Since the payments are quarterly, we must adjust the annual interest rate to a quarterly rate. This is done by dividing the annual rate by the number of quarters in a year. Therefore, the quarterly interest rate is 5% / 4 = 1.25% or 0.0125 in decimal form. The total number of quarterly payments over five years is 5 years × 4 quarters/year = 20.
The present value calculation is:
PV = $3,000 × [(1 - (1 + 0.0125)^{-20}) / 0.0125]
Performing the calculations:
PV = $3,000 × [(1 - (1 + 0.0125)^{-20}) / 0.0125] = $3,000 × [(1 - 1.0125^{-20}) / 0.0125] = $3,000 × [(1 - 0.779423) / 0.0125] = $3,000 × (0.220577 / 0.0125) = $52,925.64
The present value of the 20 quarterly payments is $52,925.64.