Math Walkthrough
Amortization Math, Worked One Period at a Time
All figures on this page are made up for illustration. Rates, payment amounts and balances in a real item come from the question itself, and whether a handheld calculator is allowed in your testing room is set by the vendor and your state. Check the current candidate handbook.
An amortized loan is one that retires itself. Each scheduled payment covers the interest that accrued since the last one, and whatever is left over reduces the debt, so the balance falls a little further every period until it reaches zero at the end of the term.
Exam items almost never ask you to derive the payment; that calculation needs a financial calculator you will not have. They hand you the payment and ask what it did: how much was interest, how much came off the balance, what the balance is now.
What the Payment Is Actually Doing
Interest on a standard mortgage is charged on the outstanding balance for the period just ended. Nothing about next month's payment affects this month's interest, and nothing about the original loan amount matters once the balance has moved. Whatever is owed right now is the only figure that generates interest.
That single idea drives the whole topic. Because the interest charge shrinks as the balance shrinks, and because the payment on a fixed-rate loan does not change, the share of each payment left over for the balance grows every period. The payment stays flat while its composition rotates from mostly interest at the start to mostly principal at the end.
Be careful with the word amortization on the exam. It describes the schedule, not the payment and not the term. A loan can have a long amortization and a short term, in which case the scheduled payments never finish the job and a balloon covers what remains. Items testing that distinction are common and are usually written to look like ordinary arithmetic questions.
The Procedure, Four Lines Long
Run these in order and write each result down. Most lost points here are transcription errors rather than method errors.
Convert the annual rate to a periodic rate. Divide the annual rate by the number of payments in a year, which for a monthly loan means dividing by twelve. Do this before anything else touches the numbers.
Multiply the current balance by that periodic rate. The result is the interest portion of the payment being analyzed.
Subtract the interest portion from the total payment. What remains is the principal portion, the only part that reduces the debt.
Subtract the principal portion from the balance. That is the new balance, and it is the figure you start from if the item asks about the following period.
Two Periods, Start to Finish
Take a hypothetical loan with a balance of $200,000, a nominal annual rate of 6 percent, and a level payment of $1,200 a month. The periodic rate is 6 percent divided by twelve, or 0.005. Multiply: $200,000 x 0.005 = $1,000 of interest in the first month. Subtract that from the payment: $1,200 - $1,000 = $200 applied to principal. The balance falls to $200,000 - $200 = $199,800.
Now do the second month on the new balance, which is the step candidates skip. Interest is $199,800 x 0.005 = $999. Principal is $1,200 - $999 = $201. The balance becomes $199,800 - $201 = $199,599. The payment did not change, the interest dropped by a dollar, and the principal portion rose by the same dollar. Repeat that a few hundred times and you have the schedule.
One more variant shows up regularly: total interest over a stretch of time. If an item asks for a full year of interest and tells you to approximate, multiply the starting balance by the annual rate and say so. On the loan above that is $200,000 x 0.06 = $12,000, slightly more than the true figure because the balance fell a little each month. Only approximate when the question invites it; if it asks for an exact result, work the periods one at a time.
Where These Items Go Wrong
The arithmetic is elementary. The mistakes are procedural, and they repeat.
Using the original loan amount after the balance has changed. Once a payment has been applied, the original figure is history. Every later period runs off the balance you just computed.
Forgetting to divide the annual rate. An annual rate applied to a monthly balance overstates the interest by a factor of twelve and usually produces one of the offered wrong answers.
Answering the wrong half. Read carefully whether the item wants the interest portion, the principal portion or the ending balance. All three appear among the choices.
Assuming payments are monthly. Some items use quarterly, semiannual or annual periods. The rate divisor and the number of periods both change with it.
Ignoring what the payment includes. If the item mentions taxes and insurance collected in escrow, strip them out first. Only the principal and interest portion enters this calculation.
Confusing a rising principal portion with a rising payment. On a fixed-rate loan the payment is constant. What changes is how it is divided.
Three Worked Items
All figures below are illustrative and chosen to keep the arithmetic clean. Work each one on paper before reading the solution.
Question 1
Illustrative: the balance is $240,000, the note rate is 5.4 percent annually, and the monthly payment is $1,348.18. Split that payment between interest and principal.
Interest is $1,080.00 and principal is $268.18. The monthly rate is 5.4 percent / 12 = 0.0045, so $240,000 x 0.0045 = $1,080.00, and whatever the payment covers beyond that reduces the balance: $1,348.18 - $1,080.00 = $268.18.
Question 2
Illustrative: a $320,000 balance carries a 4.2 percent annual rate with a monthly payment of $1,564.94. What is the balance immediately after that payment posts?
$319,555.06. Monthly interest is $320,000 x (4.2 percent / 12) = $320,000 x 0.0035 = $1,120.00, principal is $1,564.94 - $1,120.00 = $444.94, and $320,000.00 - $444.94 = $319,555.06.
Question 3
Illustrative: about how much interest does a $500,000 loan at 6 percent accrue over its first twelve months, if the item tells you to treat the balance as unchanged across the year?
Roughly $30,000, since $500,000 x 6 percent = $30,000. The true figure is a little lower because each payment shaves the balance, but an item that hands you no payment amount is asking for this approximation.
Amortization Questions Candidates Ask
Why is so much of an early payment interest?
Because the balance is at its highest point when the loan begins, and interest is charged on that balance. The payment is sized to retire the debt over the full term, so early on it barely exceeds the interest charge. As the balance comes down the interest charge comes down with it, and the leftover grows.
What makes a loan fully amortized?
The scheduled payments, made on time through the end of the term, leave a balance of zero. Nothing is owed at maturity because the payment was calculated to cover both the accruing interest and the entire principal across the term.
What is negative amortization?
It happens when the payment is smaller than the interest accruing for the period. The shortfall is added to the balance, so the debt grows instead of shrinking. Certain adjustable and payment-option loan structures allowed this, and the exam treats it as the opposite case of a fully amortizing loan.
How does an interest-only period fit in?
During the interest-only stretch the payment covers the interest charge and nothing more, so the balance stays flat. When that period ends, the loan begins amortizing over the years that remain, which produces a noticeably larger payment because the full original balance now has less time to be retired.
Does extra principal shorten the loan?
Yes, and understanding why is worth more than the fact. An extra amount applied to principal lowers the balance immediately, so every later interest charge is calculated on a smaller number, which leaves more of each future payment for principal. The effect compounds forward and the payoff date moves in.
Will I need to calculate the payment itself?
Not from scratch. Deriving a level payment requires a financial calculator or a factor table, so items either supply the payment or supply a per-thousand factor and ask you to multiply it by the number of thousands borrowed. If a factor appears in the question, that is the method being tested.
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