The Canadian mortgage convention

◈ 9 cards

Quoted semi-annually, paid monthly — convert the quoted rate to the equivalent monthly rate with (1 + q/2)^(1/6) − 1, compute the payment, and measure what quoted ÷ 12 gets wrong.

Two frequencies in one loan

A Canadian fixed-rate mortgage is quoted as a rate compounded semi-annually but paid monthly. That is a legal convention, not a bank's choice: the Interest Act requires a blended-payment mortgage to state its rate as calculated yearly or half-yearly, not in advance (s. 6). The monthly rate you feed to PMT is therefore not the quoted rate ÷ 12. It is the monthly rate that compounds to the same half-year:

$(1 + r_{m})^{6} = 1 + \frac{q}{2} ⟹ r_{m} = (1 + \frac{q}{2})^{1/6} - 1$

Six months of monthly compounding must equal one half-year of the quoted compounding. Then =PMT(r_m, years × 12, principal).

Worked example — a $400,000 mortgage at 4.80 %

A couple in Guelph borrow $400,000 at 4.80 %, amortised over 25 years.

The monthly rate. =(1+0.048/2)^(1/6)-1 = $1.02 4^{0.166667} - 1$ = 0.003961, or 0.3961 % a month. Check: 1.003961⁶ = 1.024, and $1.02 4^{2} - 1 = 0.048576$ is the mortgage's effective annual rate — =EFFECT(0.048,2).

The payment. =-PMT(0.003961,300,400000) = 2,281.09 a month, with nper 25 × 12 = 300. In one cell: =-PMT((1+0.048/2)^(1/6)-1,25*12,400000).

The US-style error. Use quoted ÷ 12 instead — =-PMT(0.048/12,300,400000) — and the payment is 2,291.99: 10.90 a month too high, or about 3,270 over 25 years. The error is small per month and always in the same direction: 0.048/12 = 0.004000 is a higher monthly rate than 0.003961, because it compounds twelve times to 4.907 % effective rather than 4.858 %. Bigel's and other US textbooks' mortgages compound monthly, and on this paper their formula is the wrong one.

Amortisation versus term

The amortisation period (25 years here) is the time it would take to repay the loan in full at the current payment. The term (commonly 5 years) is how long the quoted rate is fixed; at the end of the term the balance is renewed at a new rate, and the payment is recomputed on the remaining amortisation. So the 300-period PMT is the payment for the term, and =FV(r_m, 60, -2281.09, 400000) gives the balance to be renewed after five years.

Reprice at 5.40 %

=(1+0.054/2)^(1/6)-1 = 0.004450; =-PMT(0.004450,300,400000) = 2,418.33. Sixty basis points on the quote adds 137.24 a month. The conversion is the same four keystrokes every time: half the quote, add one, sixth root, subtract one.