Scheduled payments of $1400 due today and $1600 due with interest at 11.5% compounded annually in five years are to be replaced by two equal payments. The first replacement payment is due in 18 months and the second payment is due in 4 years. Determine the size of the two replacement payments if interest is 11% compounded quarterly and the focal date is 18 months from now.
$2004.94 is the size of equal payments due in 18 months and 4 years.
hey Jhanvi can u tell how did u solved it?
To determine the size of the two replacement payments, we can follow these steps:
Step 1: Calculate the present value (PV) of the scheduled payments.
To find the PV of the $1400 payment due today, we don't need to do any calculations since it's already in present value. Therefore, we can say that the PV of the first scheduled payment is $1400.
To find the PV of the $1600 payment due in 5 years with interest at 11.5%, we can use the formula:
PV = FV / (1 + r/n)^(n*t)
Where:
PV = Present Value
FV = Future Value
r = interest rate (in decimals)
n = number of compounding periods per year
t = number of years
In this case, FV = $1600, r = 11.5% = 0.115 (as a decimal), n = 1 (annually), and t = 5.
Using the formula:
PV = $1600 / (1 + 0.115/1)^(1*5)
PV ≈ $934.48
Therefore, the PV of the second scheduled payment is approximately $934.48.
Step 2: Calculate the Future Value (FV) of the replacement payments at the focal date.
To determine the FV of the two replacement payments at the focal date (18 months from now), we'll consider quarterly compounding at an 11% interest rate.
Using the formula:
FV = PV * (1 + r/n)^(n*t)
For the first replacement payment due in 18 months:
PV = ?
FV = Focal date payment = ?
r = 11% = 0.11 (as a decimal)
n = 4 (quarterly compounding)
t = 18 / 12 = 1.5 (years)
Using the formula:
FV = PV * (1 + 0.11/4)^(4*1.5)
Since the two replacement payments are equal, we can substitute PV and FV with the same variable - let's call it X.
X * (1 + 0.11/4)^(4*1.5) = X
Simplifying the equation:
(1.0275)^6 = 1
Solving for X:
X ≈ 0.778156
Therefore, the size of the two equal replacement payments is approximately $0.778156.
Please note that the answer is given in proportion to the original payments, which were $1400 and $1600. So, to find the actual payment amounts, you would multiply the respective proportions by the original payments:
Replacement payment 1: $0.778156 * $1400 ≈ $1089.42
Replacement payment 2: $0.778156 * $1600 ≈ $1245.05