4.7: Amortization - Breaking Down Big Payments Over Time
- Page ID
- 132025
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Define amortization and explain how it breaks down loan repayment.
- Interpret the changing relationship between interest and principal in an amortized loan.
- Apply amortization reasoning to explain real-world loans such as mortgages and car payments.
What Is Amortization?
While it would be fantastic if we always received lump sums or streams of payments (like annuities), there are times when we need to repay money we’ve borrowed. When you borrow and repay the loan with interest over time, you are amortizing it.
Amortization is how loans like car payments and mortgages work. Instead of paying back the entire loan all at once, you make predictable, equal payments over time. But here’s the interesting part: While your payment stays the same, what’s happening “inside” the payment - how much goes to interest and how much to the loan balance - changes every month. Amortization is the process of repaying a loan in equal payments over time. Here is what’s happening inside the payment:
- Interest: The cost of borrowing, calculated on the remaining balance.
- Principal: The amount you still owe, which gets smaller as you pay it off.
Initially, most of your payment goes toward interest because you still owe a lot. But over time, as the principal shrinks, the interest becomes smaller. That leaves more of your payment to chip away at the loan balance.
An Example: Buying a Car
Let’s say you borrow $20,000 to buy a car. You agree to repay the loan over five years (60 months) at an interest rate of 6 percent. Your monthly payment is calculated using this formula:
\(\displaystyle{PMT = \frac{r \times PV}{1 - (1 + r)^{-n}}}\)
Where:
P = Monthly payment
PV = Loan amount (present value) = $20,000
r = Monthly interest rate = Annual rate ÷ 12 = 0.06 ÷ 12 = 0.005
n = Total number of periods = 60 months
Plugging in the numbers:
\(\displaystyle{PMT = \frac{0.005 \times 20,000}{1 - (1 + 0.005)^{-60}}}\)
\(\displaystyle{PMT = \frac{100}{1 - 0.7408}}\)
\(\displaystyle{PMT = \frac{100}{0.2592}}\)
\(\displaystyle{PMT = 385.54}\)
Amortization in Action
Initially, most of your $385.54 payment is allocated toward paying interest. For example:
- Payment 1: You owe $20,000. At 6% annual interest (0.5% per month), the interest on $20,000 is $100. The remaining $285.54 reduces your principal to $19,714.46 ($20,000 - $285.54).
- Payment 2: Now that you owe $19,714.46, your interest for the month is slightly lower: $98.57 (0.5% of $19,714.46). Your payment remains the same, but more of it - $ 98.57 ($385.54 - $ 286.97) - is applied toward reducing the principal.
Over time, this pattern continues:
- The interest portion of your payment gets smaller.
- The principal portion gets larger.
By the end of the loan, almost all of your payment goes toward reducing the principal. This shift happens because interest is always calculated on the remaining balance, and as the balance shrinks, so does the interest.
Why Amortization Matters
Amortization lets you:
- Break big loans into manageable pieces - whether it’s buying a car or owning a home.
- Understand where your payments go - how much you’re paying toward interest versus principal.
- Plan for the future by predicting your payments and budgeting for long-term goals.
You don’t have to calculate amortization payments manually. Online tools and spreadsheets can do the work for you:
Amortization Calculators: Enter the loan amount, interest rate, and term to see monthly payments and breakdowns.
Excel Formula: Use =PMT(rate, nper, pv)
Example: =PMT(6%/12, 60, -20000) → Returns $385.54
Amortization schedules clearly show how each payment is divided between interest and principal. Use them to understand how loans work and make smarter borrowing decisions.
Amortization transforms large debts into manageable pieces. By breaking a loan into equal payments over time, amortization makes borrowing predictable but not always transparent. This section introduces the idea of "structured repayment," where each payment covers interest owed and reduces the remaining balance.
- Amortization is a loan repayment structure characterized by fixed payments and a gradual shift in the interest/principal components.
- Early payments are mostly interest; later ones are mostly principal.
- Common examples include mortgages, auto loans, and installment plans.
Amortization reveals how even steady payments hide changing economics. Understanding this structure helps you recognize how loans work and what makes one loan costlier than another.
- Have you ever paid off a loan or financed a large purchase? How did the amount you paid compare to what you borrowed?
- Why do some people feel like they’re "never making progress" early in a long-term loan?
- You borrow $10,000 at 6 percent interest, to be repaid monthly over a five-year period. What portion of your first payment goes to interest vs. principal?