Calculate the difference in the current economic values of the following two annuities: Annuity "A": Payments of $50 made at the end of each month for the next 30 years, using 9.6% compounded monthly. Annuity "B": Payments of $600 made at the end of every year for the next 50 years using 9.6% compounded annually.
A) Annuity "A" is worth $291 more than Annuity "B."
B) Annuity "A" is worth $103 more than Annuity "B."
C) The current economic values are within $50 of each other.
D) Annuity "B" is worth $103 more than Annuity "A."
E) Annuity "B" is worth $291 more than Annuity "A."

Question

Quizplus · Accepted Answer

To find the present value (PV) of each annuity, we use the formula for the present value of an annuity: $PV = P \times \frac{1 - (1 + r)^{-n}}{r}$, where $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the total number of payments.For Annuity "A", the monthly interest rate is $9.6\% / 12 = 0.8\%$, or $0.008$ in decimal form, and there are $30 \times 12 = 360$ payments. Thus, the present value of Annuity "A" is:$$PV_A = 50 \times \frac{1 - (1 + 0.008)^{-360}}{0.008}$$For Annuity "B", the annual interest rate is $9.6\%$, or $0.096$ in decimal form, and there are $50$ payments. Thus, the present value of Annuity "B" is:$$PV_B = 600 \times \frac{1 - (1 + 0.096)^{-50}}{0.096}$$Calculating these values:$$PV_A \approx 50 \times \frac{1 - (1 + 0.008)^{-360}}{0.008} \approx 50 \times 107.570 \approx 5378.5$$$$PV_B \approx 600 \times \frac{1 - (1 + 0.096)^{-50}}{0.096} \approx 600 \times 8.954 \approx 5372.4$$The difference in their values is approximately $5378.5 - 5372.4 = 6.1$, which does not match any of the provided options directly. However, based on the calculation method and the options provided, it seems there might have been a mistake in the calculation or interpretation of the options. Given the formulas and typical approach to such problems, none of the options directly match the expected outcome of a precise calculation. The correct approach involves calculating the present value of each annuity using the given formulas and comparing them, but the discrepancy suggests a reevaluation of the calculation or the assumptions might be necessary.

Calculate the Difference in the Current Economic Values of the Following