An amortization schedule is a table that displays the periodic payments and allocation of each payment between principal and interest over the life of a loan. This schedule is commonly used for mortgages, car loans, and other installment loans. It provides a detailed breakdown of how the loan balance decreases over time as payments are made.

Here’s how to calculate an amortization schedule and the key formula involved:

### Amortization Formula:

The formula for calculating the periodic payment (P) in an amortizing loan is based on the present value of an annuity formula. The formula is as follows:

\[ P = \frac{P_0 \cdot r \cdot (1 + r)^n}{(1 + r)^n – 1} \]

Where:

– \(P\) is the periodic payment,

– \(P_0\) is the loan amount (principal),

– \(r\) is the periodic interest rate (annual interest rate divided by the number of periods per year), and

– \(n\) is the total number of payments (total number of periods).

### Steps to Create an Amortization Schedule:

1. **Determine Loan Details:**

– Obtain the loan amount (\(P_0\)), annual interest rate, and loan term (in years or months).

2. **Convert Annual Rate to Periodic Rate:**

– Divide the annual interest rate by the number of compounding periods per year (usually 12 for monthly payments).

\[ r = \frac{\text{Annual Interest Rate}}{\text{Number of Periods per Year}} \]

3. **Calculate Total Number of Payments:**

– Multiply the number of years by the number of compounding periods per year to get the total number of payments (\(n\)).

4. **Apply the Amortization Formula:**

– Use the formula to calculate the periodic payment (\(P\)).

5. **Create the Amortization Schedule:**

– For each period, calculate the interest payment, principal payment, and remaining balance using the formulae below:

– **Interest Payment (\(I\)):**

\[ I = \text{Remaining Balance} \times \text{Periodic Interest Rate} \]

– **Principal Payment (\(P_{\text{Principal}}\)):**

\[ P_{\text{Principal}} = P – I \]

– **Remaining Balance (\(RB\)):**

\[ RB = \text{Previous Remaining Balance} – P_{\text{Principal}} \]

6. **Repeat for Each Period:**

– Repeat the calculations for each period until the loan is fully amortized.

### Example:

Let’s consider a $100,000 loan with a 5% annual interest rate, a 30-year term, and monthly payments.

1. **Convert Annual Rate to Periodic Rate:**

\[ r = \frac{0.05}{12} = 0.0041667 \]

2. **Calculate Total Number of Payments:**

\[ n = 30 \times 12 = 360 \]

3. **Apply the Amortization Formula:**

\[ P = \frac{100,000 \times 0.0041667 \times (1 + 0.0041667)^{360}}{(1 + 0.0041667)^{360} – 1} \]

4. **Create the Amortization Schedule:**

– Use the formulae above to calculate interest, principal, and remaining balance for each period.

The amortization schedule will display the monthly payments, the allocation of each payment to principal and interest, and the remaining balance after each payment.

Note: In practice, spreadsheets or financial calculators are often used to create amortization schedules, as they can handle the calculations more efficiently.