What is amortization? Definition, formulas and examples

Amortization is a term people commonly use in finance and accounting. However, the term has several different meanings depending on the context of its use.

Amortization may refer to the liquidation of an interest-bearing debt through a series of periodic payments over a certain period. In most cases, the payments over the period are of equal amounts. Paying in equal amounts is actually quite common when taking out a loan or a mortgage.

According to Harvard Business School, amortization is the technique of spreading the expense of an intangible asset over the period it is expected to be useful. This practice is used for essential, non-tangible assets like patents, trademarks, copyrights, or franchise rights, allowing companies to manage their expenditures over time.

Amortization vs. depreciation

The two words are very similar and are used in accounting. Here is the difference:

We use amortization for intangible assets, such as software, trademarks, patents, copyrights, customer lists, and franchise agreements. They do not deteriorate physically, however, they may lose value as they near the end of their legal or useful life.

We use depreciation for tangible assets, i.e., ones we can touch, such as equipment, vehicles, and buildings. These assets deteriorate over time.

Amortization of loans

An amortization schedule determines the distribution of payments of a loan into cash flow installments. As opposed to other models, the amortization model comprises both the interest and the principal.

Amortization is one of the simplest repayment models there are. It is very simple because the borrower pays the repayments in equal amounts during the loan’s lifetime.

Typically, more money is applied to interest at the start of the schedule. Towards the end of the schedule, on the other hand, more money is applied to the principal.

We amortize a loan when we use a part of each payment to pay interest. Subsequently, we use the remaining part to reduce the outstanding principal.

Over time, after the series of payments, the borrower gradually reduces the outstanding principal. Additionally, interest on the unpaid balance falls.

Once a debt is amortized by equal payments at equal intervals, the debt becomes an annuity’s discounted value.

We use amortization tables to represent the composition of periodic payments between interest charges and principal repayments.

Negative amortization can occur if the payments fail to match the interest. In this case, the lender then adds outstanding interest to the total loan balance. As a consequence of adding interest, the total loan amount becomes larger than what it was originally.

Amortization schedule formulas

Variables Used:

  • \(P\) = Principal – The amount of money you initially borrow.
  • \(r\) = Periodic Interest Rate – the interest rate per payment period (e.g., monthly if you pay monthly).
  • \(n\) = The total number of payments you’ll make over the life of the loan.
  • \(k\) = Which payment you’re on (1 to \(n\)).

1. Periodic Payment (\(A\))

\(A = P \cdot \frac{r(1+r)^n}{(1+r)^n – 1}\)

This formula gives you a fixed payment amount, \(A\), that you pay every period. It’s set up so that, after \(n\) payments, your loan balance will be zero.

2. Interest in Period \(k\) (\(I_k\))

\(I_k = r \cdot B_{k-1}\)

\(I_k\) is the interest portion of the \(k\)-th payment, found by multiplying the remaining balance \(B_{k-1}\) by the interest rate \(r\).

3. Principal Repayment in Period \(k\) (\(P_k\))

\(P_k = A – I_k\)

\(P_k\) is the principal portion of the \(k\)-th payment, calculated as the total payment \(A\) minus the interest portion \(I_k\).

4. Remaining Balance After \(k\) Payments (\(B_k\))

\(B_k = B_{k-1} – P_k \quad \text{with} \quad B_0 = P\)

This recursive formula updates the remaining balance after each payment. Initially, the balance is \(B_0 = P\).

5. Closed-Form Expression for \(B_k\)

\(B_k = P(1+r)^k – A \cdot \frac{(1+r)^k – 1}{r}\)

Alternatively, this closed-form expression computes the remaining balance directly for any period \(k\) without iterating through each payment. (Useful for quick balance checks without step-by-step calculations.)

Calculating the amortization of a loan

To calculate a loan’s amortization schedule (showing how each payment reduces the principal and covers interest) you need three key pieces of information:

  • Loan Amount (Principal): The initial amount borrowed.
  • Number of Payments: How many payments you’ll make over the loan term.
  • Interest Rate: The annual percentage rate (APR).
An amortization table tracks the principal and interest portions of each payment until the loan is paid off. Let’s use an example: a $10,000 loan over 10 years with a 5% annual interest rate, paid monthly.

Step 1: Determine the Number of Repayments

Calculate the total number of payments based on the loan term and payment frequency.

  • Formula: Total Payments = Payment Frequency per Year × Loan Term in Years
  • Example: Payments are monthly (12 per year), and the term is 10 years.
  • 12 × 10 = 120 payments

Tip: If payments were quarterly, you’d use 4 × 10 = 40 payments. Adjust based on your loan’s terms.

Step 2: Calculate the Period Interest Rate

Convert the annual interest rate to a per-payment (period) rate by dividing it by the number of payments per year.

  • Formula: Period Interest Rate = Annual Interest Rate / Payments per Year
  • Example: Annual rate is 5% (0.05), and payments are monthly (12 per year).
  • 0.05 / 12 = 0.0041667 (approximately 0.41667%)

Note: We’ll use 0.0041667 consistently and round final currency values to two decimal places (e.g., $0.01).

Why?: Dividing by 12 adjusts the annual rate to match the monthly payment schedule.

Step 3: Compute the Periodic Payment

Use the amortization formula to find the fixed payment amount per period:

$$ \text{Payment} = \text{Principal} \cdot \frac{r (1 + r)^n}{(1 + r)^n – 1} $$

Where:

  • Principal = $10,000
  • \(r\) = Period Interest Rate = 0.0041667
  • \(n\) = Number of Payments = 120

Calculation:

\((1 + 0.0041667)^{120} \approx 1.648721\)

Numerator: \(0.0041667 \times 1.648721 \approx 0.0068697\)

Denominator: \(1.648721 – 1 = 0.648721\)

\(\frac{0.0068697}{0.648721} \approx 0.010587\) \(10,000 \times 0.010587 \approx 105.87\)

Result: Monthly Payment = $105.87 (rounded to $0.01)

Note: This assumes a fixed-rate, fully amortizing loan where each payment is equal. For precision, use a calculator or spreadsheet.

Step 4: Set Up an Amortization Table

Create a table to track the loan’s progress. Use four columns:

  • Payment Number: Tracks the payment (1, 2, 3, etc.).
  • Current Value: The loan balance before the payment.
  • Interest: The interest portion of the payment.
  • Principal: The principal portion of the payment.

Example Start: For Payment 1, Current Value = $10,000.

Step 5: Calculate Interest and Principal for Each Payment

For each payment:

  • Interest = Current Value × Period Interest Rate
  • Principal = Payment – Interest

First Payment (Payment 1):

  • Interest = $10,000 × 0.0041667 = $41.67
  • Principal = $105.87 – $41.67 = $64.20

Fill these into the table under Payment 1.

Step 6: Update the Current Value

Subtract the principal paid from the current value to get the new balance.

  • Example:
  • New Current Value = $10,000 – $64.20 = $9,935.80

Write $9,935.80 as the Current Value for Payment 2.

Step 7: Repeat for All Payments

Repeat Steps 5 and 6 for all 120 payments. The interest decreases, and the principal increases over time, until the Current Value reaches zero.

Here’s a sample table for the first two and last payments (rounded to $0.01):

Payment Number Current Value Interest Principal Payment
1 $10,000.00 $41.67 $64.20 $105.87
2 $9,935.80 $41.40 $64.47 $105.87
… … … … …
120 $105.22 $0.44 $105.43 $105.87

Verification: After Payment 120, the balance is approximately $0, confirming accuracy (small differences may arise from rounding).

Additional Enhancements

  1. Consistent Rounding: All currency values are rounded to two decimal places ($0.01) for simplicity and consistency.
  2. Visual Insight: The interest starts high ($41.67) and drops to $0.44, while the principal rises from $64.20 to $105.43, showing the loan’s payoff progression.
  3. Practical Tools: For 120 payments, use a spreadsheet:
    • Excel formula for payment: =PMT(0.0041667, 120, 10000) ≈ $105.87
    • Automate the table with columns for each value.
  4. Flexibility: Adjust the period rate and payment count for different frequencies (e.g., quarterly: 0.05 / 4 = 0.0125, 40 payments).
  5. Resources: Try online amortization calculators or explore financial literacy sites for deeper understanding.

Amortization of intangible assets

Amortization also refers to the acquisition cost of intangible assets minus their residual value. In this sense, the term reflects the asset’s consumption and subsequent decline in value over time.

Amortization does not relate to some intangible assets, such as goodwill. Some intangible assets do not have a set period of ‘life’.

We record the amortization of intangible assets in the financial statements of a company as an expense.

The process of amortization not only affects a company’s financial statements but also has important tax implications, as the non-cash expense can reduce taxable income.

Amortization is simply a way to spread out a large cost over time, making it easier to manage. For example, when you take out a loan, you make regular, equal payments that cover both the interest and part of the principal until the entire amount is paid off. Similarly, businesses use amortization for intangible assets like patents or trademarks by allocating their costs over the years they expect to benefit from them. This approach helps keep finances predictable and ensures that expenses are matched to the period in which the asset is used.