Lecture video
Walkthrough of the loan and savings calculations.
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1 · The loan payment formula
For a loan of principal P at annual rate r, compounded n times per year for t years, the regular payment is:
PMT = [ P · (r/n) ] ÷ [ 1 − (1 + r/n)−n·t ]
The same payment covers interest first and principal second, which is why early payments mostly pay interest. Total interest = (PMT × number of payments) − P.
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2 · The savings plan formula
If you deposit the same amount PMT every period into an account earning rate r compounded n times per year for t years, the future value is:
A = PMT · [ (1 + r/n)n·t − 1 ] ÷ (r/n)
This is how retirement and college-savings accounts grow. Starting earlier matters far more than depositing more, because of the long compounding window.
Worked Examples
See it done, step by step
Example 1 — Monthly payment on a $12,000 car loan at 6% for 5 years
Here r/n = 0.06/12 = 0.005 and n·t = 60.
(1.005)60 ≈ 1.34885, so (1.005)−60 ≈ 0.74137.
PMT = (12000 × 0.005) ÷ (1 − 0.74137) = 60 ÷ 0.25863 ≈ $231.99 per month.
Total paid ≈ 231.99 × 60 ≈ $13,919, so about $1,919 in interest.
Example 2 — Save $100/month at 6% for 10 years
Again r/n = 0.005 and n·t = 120.
(1.005)120 ≈ 1.81940.
A = 100 × (1.81940 − 1) ÷ 0.005 = 100 × 163.88 ≈ $16,388.
You deposited $12,000; the other ~$4,388 is earned interest.
Watch & Review
Resources for this module
Loan payment (short)
A quick example of a loan payment calculation.
Watch videoSlides + calculators
Check Yourself
Quick self-check
1. In the loan formula, what does r/n represent?
The interest rate per compounding period (annual rate divided by periods per year).
2. For a monthly 5-year loan, what is n·t?
12 × 5 = 60 payments.
3. Why does starting to save earlier matter so much?
A longer time horizon gives compounding more periods to grow the balance.
Ready to turn it in?
When the loan and savings formulas feel comfortable, complete the Module 6 homework and quiz in Canvas.
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