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 typical US home mortgage is a 30-year fixed-rate loan: the rate is set for the whole life of the loan, payments are monthly and interest compounds monthly, so the payment is a simple annuity. Fifteen-year fixed loans carry a lower rate and much less total interest. Adjustable-rate mortgages (ARMs) fix the rate for an initial period, such as 5 or 7 years, and then reset it periodically against an index. Most fixed-rate mortgages can be prepaid at any time without penalty, which lets borrowers refinance when rates fall.

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.

Refinancing a mortgage

Refinancing replaces an existing mortgage with a new one, usually to get a lower rate or to change the term. It is not free: closing costs such as lender fees, appraisal and title insurance typically run 2 % to 5 % of the loan. A simple test is the break-even period, the closing costs divided by the monthly saving. If the borrower expects to keep the home longer than the break-even period, the refinance is likely to pay off.

# 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