A $15,000 debt is to be amortized in 12 equal semiannual payments at 5.5% interest per half-year on the unpaid balance. Construct an amortization table to determine the unpaid balance after two payments have been made. Round values in the table to the nearest cent.
A debt of $5000 is to be amortized with 6 equal semiannual payments. If the interest rate is 9%, compounded semiannually, what is the size of each payment
To construct an amortization table, we can use the amortization formula:
Payment = Principal * [r(1 + r)^n] / [(1 + r)^n - 1]
Where:
Payment = Amount of each payment
Principal = Initial debt
r = Interest rate per period
n = Total number of periods
Let's calculate the values using the given information:
Principal = $15,000
Interest rate per half-year (r) = 0.055 (5.5% divided by 2)
Number of periods (n) = 12 * 2 = 24 (since there are 12 semiannual payments and each semiannual payment is considered a period)
Step 1: Calculate the payment amount
Payment = $15,000 * [0.055 * (1 + 0.055)^24] / [(1 + 0.055)^24 - 1]
Payment = $15,000 * (0.055 * 1.678251) / (1.678251 - 1)
Payment = $15,000 * 0.092303743 / 0.67825058
Payment ≈ $2,042.91 (rounded to the nearest cent)
Step 2: Construct the amortization table
Here is how the amortization table would look like after two payments (rounded to the nearest cent):
| Payment | Principal | Interest | Total Payment | Remaining Balance |
|---------|-----------|----------|---------------|------------------|
| 1 | $2,042.91 | $687.50 | $2,730.41 | $12,957.09 |
| 2 | $2,042.91 | $712.16 | $2,755.07 | $10,914.18 |
Notice that for each payment:
- Principal: This is the portion of the payment that goes towards reducing the debt.
- Interest: This is the interest charged on the remaining balance.
- Total Payment: This is the sum of the Principal and Interest.
- Remaining Balance: This is the outstanding debt after each payment.
Therefore, after two payments, the unpaid balance is approximately $10,914.18.
To construct an amortization table, we need to calculate the payment amount and analyze the payments made over time. Let's break it down step-by-step:
Step 1: Calculate the payment amount
To calculate the payment amount, we can use the formula for calculating the payment on an amortized loan:
P = (R × PVF)
Where:
P is the payment amount
R is the principal amount
PVF is the present value factor
In this case:
R = $15,000 (principal)
PVF = (1 – (1 + i)^-n) / i
Where:
i is the interest rate per payment period
n is the number of payment periods
Given that the interest rate is 5.5% per half-year and the number of payment periods is 12, let's calculate the PVF:
i = 5.5% / 100 = 0.055
n = 12
PVF = (1 – (1 + 0.055)^-12) / 0.055
≈ 9.483
Now, we can calculate the payment amount:
P = $15,000 × 9.483
≈ $142,245
So, the payment amount is approximately $1,424.50.
Step 2: Build the amortization table
The amortization table shows the payments made over time, including the principal paid, interest paid, and the remaining unpaid balance. Let's construct the table:
| Payment | Principal payment | Interest payment | Total payment | Remaining balance |
|---------|------------------|-----------------|---------------|------------------|
| 1 | | | | $15,000.00 |
| 2 | | | | |
| ... | | | | |
| 12 | | | | |
For the first payment (Payment 1), no principal or interest payments are made yet, so we can simply fill in the initial balance of $15,000.
Now, let's calculate the principal and interest for each payment.
For Payment 2:
Principal payment = Payment amount - Interest payment
In this case:
Interest payment = Interest rate × Remaining balance (after Payment 1)
Principal payment = $1,424.50 - Interest payment
To calculate the interest payment, we multiply the interest rate (5.5%) by the remaining balance:
Interest payment = 0.055 × $15,000.00
= $825.00
Therefore, the principal payment is:
Principal payment = $1,424.50 - $825.00
≈ $599.50
The remaining balance after Payment 2 is:
Remaining balance = Remaining balance (after Payment 1) - Principal payment
Remaining balance = $15,000.00 - $599.50
≈ $14,400.50
Complete the amortization table by repeating these calculations for each subsequent payment.
| Payment | Principal payment | Interest payment | Total payment | Remaining balance |
|---------|------------------|-----------------|---------------|------------------|
| 1 | | | | $15,000.00 |
| 2 | $ 599.50 | $ 825.00 | $1,424.50 | $14,400.50 |
| ... | | | | |
| 12 | | | | |
Continue this process for all the remaining payments until you reach Payment 12. Round the values to the nearest cent as required.
This is how we construct an amortization table to determine the unpaid balance after two payments have been made.
The formula for calculating the payment amount is shown below.
A = P * ((r(1+r)^n)/(((1+r)^n)-1)
Simple Amortization Calculation Formula
where
A = payment Amount per period
P = initial Principal (loan amount)
r = interest rate per period
n = total number of payments or periods
A = 15000 * ((0.055(1.055)^12)/(((1.055)^12) - 1)
A = 15000 * ((0.055*1.9012)/.9012)
A = 1,740.44
Year 1, first payment: $1740.44 Interest paid = balance * 0.055 = $15000*.055 = $825
Principal paid = payment - interest = $ 1740.44 - 825 = 915.00
Balance = 15000 - 915 = 14085
Year 1, 2nd payment: $1740.44
Interest paid = balance * 0.055 = $14085*.055 = $774.68
Principal paid = payment - interest = $ 1740.44 - 774.68 = 965.76
Balance = 14085 - 965.76 = 13119
13119 is the unpaid balance after 2 payments.