Work out the payment on a fixed-rate mortgage and how much of it is principal and how much is interest.

FRM® Part I · Financial Markets and Products · Mortgages and Mortgage-Backed Securities · FRM1.FM.18.b

The idea

A level-payment fixed-rate mortgage pays the same amount every month. Let L be the loan, r the monthly rate, equal to the annual rate divided by 12, and n the number of monthly payments. Payment = L × r ÷ (1 − (1 + r)^(−n)). Each month the interest is r times the balance at the start of that month, and the rest of the payment is principal, which reduces the balance. Because the balance is highest at the start, early payments are mostly interest and the principal share grows every month. The balance after k payments is the present value of the remaining payments at the monthly rate. Usual mistakes are using annual periods and then dividing by 12, charging interest on the original loan every month, and treating principal as a constant amount. On the exam, compute the payment first, then split a chosen payment using the balance at the start of that month.

Formulas

Level monthly payment on a fixed-rate mortgage

payment = L × r ÷ [1 − (1 + r)⁻ⁿ]

L: the loan amount
r: the monthly rate, the annual rate ÷ 12
n: the number of monthly payments

Rate and periods must both be monthly. Working in annual periods and then dividing by 12 gives the wrong payment.

Mortgage balance after k payments

balance = payment × [1 − (1 + r)^(−(n − k))] ÷ r

payment: the level monthly payment
r: the monthly rate
n: the original number of monthly payments
k: the payments already made

The present value of the payments still to come. Interest in the next month is r times this balance, and the rest of the payment is principal.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the standard formula for the level monthly payment on a fixed-rate mortgage, which inputs are used?

The loan amount, the annual rate divided by 12, and the number of months. - correct
The loan amount, the annual rate, and the number of years.Why not: Uses annual inputs, which would give an annual payment rather than a monthly one.
The loan amount, the annual rate divided by 12, and the number of years.Why not: Units mismatch: a monthly rate paired with a count of years.
The loan amount, the effective annual rate, and the number of monthly periods.Why not: Mixes an effective annual rate with monthly periods; mortgages use the nominal rate divided by 12.

Payment = L x r / (1 - (1 + r)^-n) with r the monthly rate (annual rate / 12) and n the number of months. Rate and number of periods must be on the same monthly basis.

A harder one

A borrower takes a 30-year fixed-rate mortgage of USD 350,000 at 6.00% per year with monthly payments. The level monthly payment is closest to:

USD 2,119Why not: Computes the annual payment at 6% for 30 years and divides by 12.
USD 2,098 - correct
USD 2,063Why not: Uses the effective monthly rate 1.06^(1/12) - 1 instead of 6% / 12.
USD 1,750Why not: Pays interest only on the original balance; ignores principal repayment.

r = 0.06 / 12 = 0.005 and n = 360. Payment = 350,000 x 0.005 / (1 - 1.005^-360) = 2,098.

The hardest tier

A 30-year, 6.00% fixed-rate mortgage of USD 350,000 has a level monthly payment of USD 2,098.43. The balance immediately after the 60th payment is closest to:

USD 291,667Why not: Straight-line amortization: 350,000 x 25/30; ignores that early payments are mostly interest.
USD 329,094Why not: Assumes principal repaid each month stays at the first-month amount of 348.43.
USD 325,690 - correct
USD 321,899Why not: Discounts 12 x the payment as an annual annuity at 6% effective for 25 years, a rate and period mismatch.

Balance = PV of the 300 remaining payments at 0.5% = 2,098.43 x (1 - 1.005^-300) / 0.005 = 325,690. Only about 24,310 of principal is repaid in five years.

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