9.2 Mortgage Mathematics & Payment Mechanics
Key Takeaways
The term governs the current agreement. Amortization is the projected repayment horizon under the assumed rate and payments.
For a nominal rate compounded half-yearly, convert to the equivalent payment-period rate before computing a blended payment.
Accelerated biweekly payments can reduce principal faster because the modeled annual outlay is higher.
Read the contract for annual privileges and the penalty formula. Interest Act section 10 creates a separate right for qualifying mortgages after five years.
Read the payment, term and amortization together
A mortgage’s term is the period for which the current agreement applies. Amortization is the projected time to repay the balance on the stated payment assumptions. A 25-year amortization with a five-year term leaves a balance at maturity in a standard schedule. The borrower can arrange renewal, switch to another lender, refinance, or pay the balance when due, subject to the terms and approvals. A new term may change the interest rate and scheduled payment. Compare terms on the remaining balance and amortization.
A blended instalment pays the period’s interest and reduces principal by what remains. With a constant rate and a level payment that exceeds interest, the balance falls, so less of each later payment goes to interest. Changing the interest rate, frequency, principal or payment can change the payoff path. A variable rate with a fixed payment can reduce the amount applied to principal as rates rise.
Convert a quoted rate before calculating a payment
Section 6 of the federal Interest Act applies to mortgages with payments structured as blended principal and interest, a sinking-fund plan, or the specified allowance for stipulated repayments. For those plans it requires the mortgage to state the principal and the interest rate chargeable on it calculated yearly or half-yearly, with interest calculated in arrears. Read the actual mortgage. The worked examples use a nominal annual rate compounded twice yearly, with end-of-period payments and no fees or taxes in the payment.
With annual nominal rate j, the half-year growth factor is 1 + j/2. Six monthly periods must reproduce that growth. The equivalent monthly rate is i = (1 + j/2)^(1/6) − 1. The effective annual rate is (1 + j/2)^2 − 1. With principal P and n monthly payments, the level payment is P × i ÷ (1 − (1 + i)^(-n)).
Worked example. A $400,000 mortgage at 5.50% nominal, compounded half-yearly, with 25 years of monthly payments has an effective annual rate of 5.575625% and a monthly rate of about 0.453168%. Over 300 instalments the payment is $2,441.57 to the nearest cent. The first month’s interest is $1,812.67 and the displayed payment leaves $628.90 toward principal. With the same rate and payment through 60 months and unrounded intermediate values, the balance is about $356,750. Use the loan terms to check the lender’s rounding and final instalment.
Compare payment frequencies with the same assumptions
For a $2,400 monthly payment, the FCAC payment options give these amounts when modeled over a year with 26 biweekly payments. The borrower and lender should confirm the actual dates and amounts.
| Frequency | Payment | Modeled annual outlay |
|---|---|---|
| Monthly | $2,400 | $28,800 |
| Semi-monthly, twice per month | $1,200 | $28,800 |
| Standard biweekly, monthly payment × 12 ÷ 26 | About $1,107.69 | About $28,800 |
| Accelerated biweekly, monthly payment ÷ 2 | $1,200 | $31,200 |
Accelerated biweekly adds $2,400 in scheduled outlay in this example. The larger outlay reduces the principal faster under the stated rate and no-fee assumptions. Timing also affects the interest calculation. The number of months saved depends on the actual rate, payment dates, outstanding balance and later contract changes. If the borrower can afford the larger outlay, compare it with the written privileges for a lump sum or an increase to the regular payment.
Apply the actual prepayment terms
Check whether the mortgage is open or closed, what payments the borrower can make without charge, when the allowance resets and how the lender calculates a charge. An open mortgage normally permits early payout without a prepayment penalty. A closed mortgage can allow an annual lump sum or scheduled increase. For example, a contract may permit up to 15% of original principal each year. The written agreement determines the percentage, base, dates and whether an unused allowance carries forward.
A closed variable-rate loan often uses three months’ interest, while a closed fixed-rate loan often uses the greater of three months’ interest and an interest rate differential (IRD). The IRD depends on the lender’s method, including the comparison rate for the remaining term and any adjustment for the borrower’s original discount. Some agreements use a different charge. Use the lender’s written payout statement before making a financing or sale decision.
Worked charge. Assume a contract calls for the greater of three months’ simple interest or a simplified IRD of balance × rate difference × years remaining. On a $360,000 balance at 5.50%, three months is $360,000 × 0.055 × 3/12 = $4,950. With a comparison rate of 3.50% and two years left, the IRD is $360,000 × 0.0200 × 2 = $14,400. Under these stated terms the penalty is $14,400. The example isolates the contract’s charge from accrued interest, discharge fees and other amounts in a payout.
Section 10 of the federal Interest Act provides a separate right for qualifying mortgages where the principal or interest is payable more than five years after the mortgage date. Once five years have elapsed, the eligible payer can tender or pay outstanding principal and interest to the payment date plus three months’ interest in lieu of notice, stopping further interest on that principal. The Act excludes corporate mortgages and specified other mortgages or borrowers. Check the actual instrument, dates and exceptions with the closing lawyer before applying the rule. The right differs from paying out at the end of a shorter term or using an annual privilege.
A borrower has a 25-year amortization and a five-year fixed term. Regular blended payments are made as agreed and a principal balance remains when the five years end. What happens at term end?
The lender must keep the original rate until all 25 years have elapsed.
The remaining balance is forgiven when the fifth year ends.
The borrower must arrange renewal or other financing, or pay the balance due under the agreement.
The borrower automatically starts a new 25-year amortization with the same lender.
A borrower’s monthly payment is $2,400. The lender sets accelerated biweekly payments at half the monthly amount. In a year modeled with 26 instalments, what are the payment and total annual outlay?
$1,200 per instalment and $31,200 annually.
$1,200 per instalment and $28,800 annually.
About $1,107.69 per instalment and $31,200 annually.
$2,400 per instalment and $31,200 annually.
A closed fixed-rate mortgage has a $400,000 balance, a 6.25% contract rate and two years left. The contract charges the greater of three months’ simple interest or a simplified IRD equal to balance × rate difference × years left. The comparison rate is 4.25%, and no privilege applies. What is the charge?
$6,250, using three months’ interest.
$8,000, using the two-percentage-point difference for one year.
$22,250, adding both calculated amounts.
$16,000, using the IRD because it is greater.
Sections you finish are checked off in the contents.