Loans, mortgages and amortization

Most business and personal loans are repaid by a series of equal payments that cover both interest and principal. Each payment first pays the interest that has built up since the last payment; whatever is left reduces the balance. Over time the balance falls, the interest portion shrinks and the principal portion grows. This process is called amortization, and the table that tracks it payment by payment is an amortization schedule. This lesson applies the annuity formulas from lesson 8 to car loans, business term loans and, most importantly, home mortgages, the largest debt most households ever take on.

The loan payment

A loan is the present value of its payments. Given the amount borrowed (PV), the periodic rate (i) and the number of payments (n), the payment is found with the present value annuity formula rearranged for PMT. For a general annuity, i is the equivalent rate per payment interval.

Amortization schedules

An amortization schedule lists, for every payment, the interest portion (the opening balance times the periodic rate), the principal portion (the payment minus the interest) and the closing balance (the opening balance minus the principal portion). Because the payment is rounded to the cent, the final payment is adjusted so that the balance ends at exactly zero.

A schedule for a long loan has hundreds of rows, which is why it is almost always built in a spreadsheet. The first row's formulas are written once and filled down: interest = previous balance × rate, principal = payment − interest, balance = previous balance − principal.

The balance at any time

It is often necessary to know the balance partway through a loan, for example when it is to be paid off early, refinanced or renewed. There are two equivalent methods. The prospective method finds the present value of the payments still to come. The retrospective method takes the future value of the original loan and subtracts the future value of the payments already made.

Business term loans

Businesses borrow to buy equipment, vehicles and buildings, or to finance an expansion, usually through a term loan repaid over a fixed number of years. Two repayment structures are common. With blended payments, every payment is the same and the split between interest and principal shifts over time, exactly like a mortgage. With fixed principal payments, the business repays the same amount of principal each period plus the interest on the balance, so the payments start high and fall. Fixed principal loans cost less interest, because the balance falls faster, but they put more strain on cash flow in the early years.

Mortgages

A Canadian mortgage has two time spans. The amortization period is the time it would take to repay the loan in full, usually 25 years; for insured mortgages, 30 years is allowed for first-time buyers and buyers of newly built homes since December 15, 2024. The term is the length of the current contract, usually 1 to 5 years. At the end of each term the balance is renewed at a new rate, or paid off, or moved to another lender. Under the Interest Act, fixed rates are compounded semi-annually, not in advance, even though payments are monthly or more frequent, so mortgage payments are a general annuity. Variable rates may be compounded monthly; check the contract.

Paying a loan off faster

Because interest is charged on the outstanding balance, anything that reduces the balance sooner saves interest: a shorter amortization, extra payments, lump-sum prepayments or more frequent payments. The savings are largest early in the loan, when the balance is highest.

Breaking or refinancing a mortgage

A closed Canadian mortgage cannot be paid off early, beyond its prepayment privileges, without a penalty. For a variable-rate mortgage the penalty is usually three months' interest. For a fixed-rate mortgage it is usually the greater of three months' interest and the interest rate differential (IRD), which compensates the lender for the lower rate it will earn by relending the money for the rest of the term. Lenders calculate the IRD in different ways, and on a large balance with several years left it can run to many thousands of dollars, so borrowers should ask for a written penalty quote before breaking a term to refinance or sell.

# Amortization schedule and balance after k payments
def payment(pv, i, n): return pv * i / (1 - (1 + i) ** -n)
def schedule(pv, i, n):
    pmt = round(payment(pv, i, n), 2)
    bal, rows = pv, []
    for k in range(1, n + 1):
        interest = round(bal * i, 2)
        principal = bal if k == n else round(pmt - interest, 2)
        bal = round(bal - principal, 2)
        rows.append((k, round(interest + principal, 2), interest, principal, bal))
    return rows
for row in schedule(6000, 0.0075, 6):
    print(row)
p = 1.02395 ** (1 / 6) - 1          # 4.79% compounded semi-annually, monthly payments
pmt = round(payment(400000, p, 300), 2)
bal60 = 400000 * (1 + p) ** 60 - pmt * ((1 + p) ** 60 - 1) / p
print("Canadian mortgage payment", pmt, "balance after 60:", round(bal60, 2))
i = 0.0625 / 12                      # 6.25% compounded monthly
pmt = round(payment(400000, i, 360), 2)
bal60 = 400000 * (1 + i) ** 60 - pmt * ((1 + i) ** 60 - 1) / i
print("US mortgage payment", pmt, "balance after 60:", round(bal60, 2))

(1, 1026.41, 45.0, 981.41, 5018.59)
(2, 1026.41, 37.64, 988.77, 4029.82)
(3, 1026.41, 30.22, 996.19, 3033.63)
(4, 1026.41, 22.75, 1003.66, 2029.97)
(5, 1026.41, 15.22, 1011.19, 1018.78)
(6, 1026.42, 7.64, 1018.78, 0.0)
Canadian mortgage payment 2278.83 balance after 60: 352850.81
US mortgage payment 2462.87 balance after 60: 373348.88