Typically most bonds are called “bullet bonds” because the periodic payments only contain coupon payments, and the loan principal is due at the end at maturity. However, mortgages are amortized payments, meaning each payment period the payment amount is the same but some of the loan is being paid down with each payment, along with some of the interest.
Here’s what a typical amortization table looks like:
| Balance | $400,000 |
|---|---|
| Coupon Rate (annual) | 4.5% |
| Periods (months) | 360 |
| Period | Beginning Balance | Payment | Interest | Principal | Ending Balance |
|---|---|---|---|---|---|
| 1 | $400,000.00 | $2,026.74 | $1,500.00 | $526.74 | $399,473.26 |
| 2 | $399,473.26 | $2,026.74 | $1,498.02 | $528.72 | $398,944.54 |
| 3 | $398,944.54 | $2,026.74 | $1,496.04 | $530.70 | $398,413.84 |
| 4 | $398,413.84 | $2,026.74 | $1,494.05 | $532.69 | $397,881.15 |
| 5 | $397,881.15 | $2,026.74 | $1,492.05 | $534.69 | $397,346.47 |
| ... | |||||
| 360 | $2,019.17 | $2,026.74 | $7.57 | $2,019.17 | $0.00 |
The formulas:
| Period | Beginning Balance | Payment | Interest | Principal | Ending Balance |
|---|---|---|---|---|---|
| 1 | $400,000 | PMT(0.045/12,360,400000) | $400,000 x 0.045/12 | $2,026.74 - $1,500 | $400,000 - $526.74 |
| 2 | Prior Period Ending Balance | PMT(Coupon Rate/12,Period,Beginning Balance) | Beginning Balance * Coupon Rate/12 | Payment - Interest | Beginning Balance - Principal |
What’s the math behind the payment (PMT) formula?
First understand Geometric Series, this is also found in bond math.
$$ a+ar+ar^2+ar^3...ar^n=\sum_{k=1}^{\infty} ar^k = a\left(\frac{1-r^{n+1}}{1-r}\right) $$
This is called amortization, where every month it is the same payment, but some of it is principal (return of capital) and some of it is interest.