Two payments of $10 000 each must be made one year and four years from now. If money can earn 9% compounded monthly, what single payment two years from now would be equivalent to the two scheduled payments?
A) $10 938
B) $8358
C) $11 964
D) $22 902
E) $19 296

Question

Quizplus · Accepted Answer

To find the equivalent single payment two years from now, we need to calculate the present value of the two payments and then find the future value of that sum two years from now. The interest rate per month is 9% per year, or 0.09/12 per month. For the $10,000 due in one year (12 months), its present value is calculated as:$$ PV_1 = \frac{10000}{(1+\frac{0.09}{12})^{12}} $$For the $10,000 due in four years (48 months), its present value is:$$ PV_2 = \frac{10000}{(1+\frac{0.09}{12})^{48}} $$The total present value is the sum of $PV_1$ and $PV_2$. Then, to find the equivalent payment two years from now (24 months from now), we calculate the future value of this total present value:$$ FV = (PV_1 + PV_2) \times (1+\frac{0.09}{12})^{24} $$Plugging in the numbers:$$ PV_1 = \frac{10000}{(1+\frac{0.09}{12})^{12}} \approx 9139.13 $$$$ PV_2 = \frac{10000}{(1+\frac{0.09}{12})^{48}} \approx 7084.47 $$$$ FV = (9139.13 + 7084.47) \times (1+\frac{0.09}{12})^{24} \approx 19296 $$Therefore, the correct answer is $19,296, which is option E.

Two Payments of $10 000 Each Must Be Made One