Two payments of $10,000 and $12,000 must be made 1 year and 4 years from now. If money can earn 9% compounded monthly, what single payment 2 years from now would be equivalent to the two scheduled payments?
I hope they teach you this topic by having you do a "time graph"
Draw a number line, label the zero as "now" and mark in the years as 1, 2, 3, 4
write $10000 above the 1 year, and 12000 above the 4 year mark.
write $x below the 2 year mark.
Now " mathematically move" the values above the line to the same time spot as those below the line, and equate them, ...
x = 10000(1.0075)^12 + 12000(1.0075)^-24
= 20968.05
To find the equivalent single payment, we need to calculate the present value of the two scheduled payments. The present value (PV) is the current worth of a future amount of money after accounting for the time value of money.
We can use the formula for calculating the present value of a future payment, which is:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest Rate per period
n = Number of periods
For the first payment of $10,000, it will be received in 1 year. Using the formula:
PV1 = $10,000 / (1 + 0.09/12)^(12 * 1)
Simplifying it:
PV1 = $10,000 / (1 + 0.0075)^(12)
For the second payment of $12,000, it will be received in 4 years. Using the same formula:
PV2 = $12,000 / (1 + 0.09/12)^(12 * 4)
Simplifying it:
PV2 = $12,000 / (1 + 0.0075)^(48)
Now, we find the present value of both payments:
Total PV = PV1 + PV2
Finally, we need to calculate the future value of this total present value to find the equivalent single payment. We'll use the formula:
FV = PV * (1 + r)^n
We know the future value (FV) is the unknown single payment we want to find in 2 years. The present value (PV) is the previous total present value obtained from the two payments.
Plugging in the values:
FV = Total PV * (1 + 0.09/12)^(12 * 2)
Simplifying it:
FV = Total PV * (1 + 0.0075)^(24)
Solve this equation to find the equivalent single payment in 2 years.