The amortization formula, anatomized.
M = P(r/12) / (1 − (1+r/12)^(−12t)). Five pieces, one negative exponent, and the whole topic depends on it. Walk through what each part does.
Here's the only formula in Topic 4 you need to actually memorize:
M = P(r/12) / (1 − (1 + r/12)−12t)
It looks intimidating. It is in fact straightforward, once you see what each piece does. The r/12 turns an annual rate into a monthly rate. The 12t is the total number of monthly payments. The negative exponent is the only piece that needs a moment — it's how the formula collapses an infinite-sum-of-future-payments calculation into one expression you can type into Excel.
By the end of this lesson, you'll have plugged in $20,000 at 5% for 4 years, gotten $460.59/month, and you'll have the muscle memory to do the same calculation for any auto loan, credit card, student loan, or mortgage in the rest of the topic. One formula. Five inputs. One output.
One formula, every monthly payment in this topic.
Five pieces of the formula, each doing one job. M is what comes out; P, r/12, 12t, and the negative exponent are the four inputs that go in. The bottom row plugs $20,000 at 5% for 4 years through it to show what "using the formula" actually looks like.
The loan amortization formula computes the fixed monthly payment that fully repays principal P over t years at annual rate r: M = P · (r/12) / (1 − (1 + r/12)−12t).
Vocabulary you'll see in word problems
- Loan amortization formula The expression above. The only formula in the ALEKS dictionary not introduced in T3.
- Periodic rate (r/12) The monthly interest rate. Annual rate divided by 12 because there are 12 monthly compounding periods per year.
- Total payments (12t) The total number of monthly payments over the life of the loan. A 5-year auto loan has 60 payments; a 30-year mortgage has 360.
-
Negative exponent
(1 + r/12)−12t = 1 / (1 + r/12)12t. Most calculators handle the minus sign correctly; in Excel the syntax is
^(-12*t). - Total cost / total interest Total cost of a loan = M × 12 × t (every payment summed). Total interest = total cost − principal. Both grow fast as t increases.
$20,000 at 5% APR for 4 years.
"You take out a $20,000 personal loan at 5% APR with a 4-year term. What's your monthly payment, total cost, and total interest paid?"
Identify the inputs.
P = $20,000 (the loan amount). r = 0.05 (5% as a decimal). t = 4 (years).
Compute r/12 (the monthly periodic rate).
The formula needs the rate per month, not per year.
Compute the exponent and the bracket.
The exponent is −12t. For 4 years, that's −48. The bracket (1 − (1+r/12)−12t) evaluates as the present-value discount factor.
Plug into the formula.
Numerator is P × (r/12). Denominator is the bracket from step 3. Divide.
Compute total cost and total interest.
Once you have M, the rollup numbers are easy.
Three problems. Same formula. Different numbers.
Compute the monthly payment for a $15,000 loan at 4% APR for 3 years.
A $40,000 auto loan at 6% APR for 5 years. Compute M.
$50,000 at 6% APR. 5-year term: M ≈ $966.64, total paid ~$58,000. Now stretch to 7 years. What's the new M?
Three fast questions before you move on.
Q1. The 12t in the formula represents...
Q2. The negative exponent (1 + r/12)−12t in the denominator...
Q3. If you double the term t (say 5y → 10y) at the same rate, the monthly payment will...
From formula to four loan types.
This formula is the engine. The next four lessons are applications: auto loans (Lesson 3), credit cards (Lesson 4), student loans (Lesson 5), and mortgages (Lesson 6). Same formula, different P, r, and t each time, with a few twists each loan type adds.
If you ever forget the formula on the test, the Financial Formulas Calculator at /tools/financial-formulas has it ready, and the Topic 3 cheat sheet has it in the "See T4" row.
Next: Auto loans — the simplest amortizing case, with two new pieces (down payment, total cost vs sticker price).
Continue to Lesson 03Different angle? Need another rep? These are optional — tap any that look helpful.