adebt of $6000 is to be amortized with 8 equal semiannual payments. If the interest rate is 5%, compounded semiannually, what is the size of each payment
solve for P , the payment
6000 = P(1 - 1.025^-8)/.025
To find the size of each payment for an amortized loan, we can use the formula for calculating the present value of an annuity. The formula is given as:
PV = PMT * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value (amount of debt)
PMT = Payment amount
r = Interest rate per period
n = Total number of periods
In this case, we have a debt of $6000, 8 semiannual payments, and an interest rate of 5% compounded semiannually. Let's calculate the size of each payment step-by-step.
1. Convert the annual interest rate to a semiannual interest rate:
The annual interest rate is 5%, so the semiannual interest rate is 5% / 2 = 2.5%.
2. Calculate the number of semiannual periods:
Since there are 8 equal semiannual payments, the total number of semiannual periods (n) is 8.
3. Calculate the present value (PV) using the formula:
PV = PMT * [1 - (1 + r)^(-n)] / r
$6000 = PMT * [1 - (1 + 0.025)^(-8)] / 0.025
4. Simplify the equation and solve for PMT:
$6000 * 0.025 = PMT * [1 - (1 + 0.025)^(-8)]
$150 = PMT * [1 - (1.025)^(-8)]
$150 = PMT * [1 - 0.81266588252]
$150 = PMT * 0.18733411748
PMT = $150 / 0.18733411748
Calculating this value gives us PMT ≈ $800.74.
Therefore, the size of each payment for the loan is approximately $800.74.
To find the size of each payment, we need to use the formula for calculating the equal payment amount for an amortized loan. This formula is known as the Amortization Payment Formula:
PMT = P * r * (1 + r)^n / ((1 + r)^n - 1)
Where:
PMT = Equal payment amount
P = Principal loan amount
r = Interest rate per compounding period
n = Total number of compounding periods
In this case, the principal loan amount (P) is $6000, the interest rate (r) is 5%, compounded semiannually (so we divide the interest rate by 2 to get 2.5%), and the total number of compounding periods (n) is 8 (since there are 8 semiannual payments).
Let's plug in these values into the formula:
PMT = 6000 * 0.025 * (1 + 0.025)^8 / ((1 + 0.025)^8 - 1)
Calculating this expression will give us the size of each payment (PMT).