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Canadian mortgage rates are different from US rates. Canadian rates are compounded semi-anually. So a rate of 6% apparently would be 6.09% in practice.

What is the formula for converting a US rate to Canadian rate? I came across a formula here but it doesn't help since I don't know how to get RM in the formula.

Rewording question - I need to find the formula for monthly mortgage payments. The normal formula finds US monthly payments. What would be the formula to find Canadian monthly payments?

Jack
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1 Answers1

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Your question is how to convert a nominal annual rate, compounded semi-annually, to a nominal annual rate, compounded monthly. A nominal annual rate compounded $n$ times a year is usually designated $r^{(n)}.$ The equivalent effective rate $r$ is given by $$ \left(1+\frac{r^{(n)}}{n}\right)^n=1+r $$ The formula gets a bit complicated, so let me do an example. Suppose you are quoted a Canadian rate of $8\%$. The effective annual rate is given by $$\left(1+\frac{.08}{2}\right)^2=1+r\implies r =.0816$$

Now we have to find the equivalent monthly rate: $$ \left(1+\frac{r^{(12)}}{12}\right)^{12}=1.0816\implies r^{(12)}=12\left(\sqrt[12]{1.0816}-1\right)\approx .078698$$

The equivalent U.S. rate is about $7.87\%$

EDIT Just in case I haven't made myself clear, what I'm calling the "effective annual rate" is what the U.S. truth-in-lending law calls the "annual percentage rate" (APR).

EDIT To get the monthly payments, use the formula you linked, with $n=$ the total monthly payments. (For a $30$ year loan, $n$ is $360$). The $r$ to use in that formula should be the nominal monthly rate that I called $r^{(12)}$ divided by $12$. So in the example above, we would have $$r = \frac{.078698}{12}\approx .006558$$

saulspatz
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  • Thanks for the detailed answer. Your answer seems to show that effective US rates are cheaper (US 7.87% vs Canadian 8.16%), but actually Canadian payments are lower for the same rate. Canadian example vs US example. How does that work? – Jack Apr 18 '18 at 04:51
  • When I use the normal mortgage formula, I get the US monthly payments. I apologize for my poorly worded question, my main question is what is the correct formula to find Canadian monthly mortgage payments. – Jack Apr 18 '18 at 04:53
  • No. The effective rate is $8.16%$ in both cases. The U.S. rate just sounds cheaper because it's quoted monthly. The APR gives the true annual rate. To get the payments, use the same formula, but set $n=2$ instead of $12$. – saulspatz Apr 18 '18 at 04:55
  • Sorry set n=2 for which formula? The formula you provided or the normal mortgage formula I linked? I tried both and I don't seem to be getting the right value – Jack Apr 18 '18 at 05:08
  • I beg your pardon. I was wrong about getting the payments. I'll update my answer. – saulspatz Apr 18 '18 at 05:17
  • Yes I got it! I was stuck for awhile because I was confused at the $r^(12)$. I thouught I had to do $0.816^12$. Then I realized $r^(12)$ was not related to $r$ at all. Figuring out stuff after that was easy. Thanks for your time! – Jack Apr 18 '18 at 06:23