I don’t seem to understand something regarding how interest is paid on a mortgage. Say the loan is for $100,000 at a 5% rate for 10 years, paid monthly.

I would think that on the first month, the interest I have to pay $100,000 × (0.05 ÷ 12) = $416.67. However the mortgage calculator says that the first payment is actually $412.39. While it’s not a huge difference, it’s a difference nonetheless and I can’t really figure out where it comes from.

My intuition is that it’s somehow related to the fact that interest is compounded daily, but when I take r = 0.05 ÷ 365 and N = 365 × 10 payments (keeping leap years in mind for later), and calculate the first 30 days, I get $409.70, and the first 31 days give $423.32. I guess that the “actual” number is some kind of weighted average since the calculator doesn’t ask at which month your loan starts.

So where is this $412.39 coming from? In reality when paying a mortgage, do you see the interest fluctuating as it decreases, depending on the number of days every month?

This page walks through the math to derive a Canadian mortgage cost:

https://www.mikesukmanowsky.com/blog/a-guide-to-canadian-mortgage-calculations

Apparently in Canada, mortgage interest may compound semi-annually, which might explain the difference.

Using the math on that page, I derived a monthly interest rate of 0.0041239 on your example loan. When multiplied by the principal of $100,000 I got a first-month interest fee of $412.39

Ah, that is the answer! 🙏

Indeed, it was the semi-annual compounding and effective interest rate that threw me off.

I can’t promise this is how the original site is calculating it, but I get exactly $412.39 of interest when I use half-yearly compounding:

Be super careful extrapolating from the first month of anything interest-related, as it depends on the exact dates things were processed. The second month should give a more representative figure.

You need to figure out how many days there were in the first month calculation.