Marvin was supposed to make three payments of $2000 each—the first one year ago, the second one year from now, and the third three years from now. He missed the first payment and proposes to pay $3000 today and a second amount in two years. If money can earn 4.5% compounded semiannually, what must the second payment be to make the proposed payments equivalent to the scheduled payments
To find the second payment that would be equivalent to the scheduled payments, we need to calculate the future value of each payment and then determine the second payment amount that would result in the same total as the scheduled payments.
Let's start by calculating the future value of the three scheduled payments using compound interest. We'll use the formula for future value (FV) of a lump sum:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future Value
PV = Present Value (the payment amount)
r = Annual interest rate
n = Compounding frequency per period
t = Number of periods
For the scheduled payments, we have:
PV1 = $2000
t1 = 1 year
n1 = 2 (since interest is compounded semiannually)
r = 4.5% or 0.045
FV1 = PV1 * (1 + r/n1)^(n1*t1)
Similarly, for the second payment and third payment:
PV2 = $2000 (second payment)
t2 = 2 years
n2 = 2 (compounded semiannually)
FV2 = PV2 * (1 + r/n2)^(n2*t2)
PV3 = $2000 (third payment)
t3 = 3 years
n3 = 2 (compounded semiannually)
FV3 = PV3 * (1 + r/n3)^(n3*t3)
Now, let's calculate the future value of the scheduled payments:
FV1 = $2000 * (1 + 0.045/2)^(2*1)
FV2 = $2000 * (1 + 0.045/2)^(2*2)
FV3 = $2000 * (1 + 0.045/2)^(2*3)
Next, let's calculate the total future value of the scheduled payments:
TotalFV = FV1 + FV2 + FV3
Now we need to determine the second payment that would make the proposed payments equivalent to the scheduled payments. The proposed payment plan is as follows:
Payment 1: $3000 (today)
Payment 2: X (in 2 years)
The future value (FV) of the proposed payment plan can be expressed as:
ProposedFV = $3000 (today) + X * (1 + r/n2)^(n2*t2)
We want the ProposedFV to be equal to the TotalFV. Therefore, we can equate the two equations and solve for X:
TotalFV = ProposedFV
FV1 + FV2 + FV3 = $3000 + X * (1 + r/n2)^(n2*t2)
Plug in the calculated values for FV1, FV2, FV3, and solve for X:
FV1 + FV2 + FV3 = $3000 + X * (1 + 0.045/2)^(2*2)
Once you solve this equation, you will find the value of X, which will be the second payment amount needed to make the proposed payments equivalent to the scheduled payments.