*acting*on what the numbers tell me (refer to my total debts in previous posts for proof of this), but it does mean that I'm always "tinkering" with the data in my life.

One thing that has always fascinated me is the tracking of loan payments over time. I had the basic exposure to simple and compound interest calculations in high school, but these calculations almost always focused on a starting principal that remained constant over time. For example, "Sam has $1,000 that earns interest at a rate of 4.0%, compounded monthly. How much money does he have after two years?" The answer to this is very simple ($1,000 x (1 + 0.04) ^ 24 = $2,563.30), because the interest is the only source of change over time. However, when periodic payments, either toward an investment or against a debt, are brought into the picture, the answer gets more complicated, and is harder to express as a single formula (there are, in fact, "simple" formulas that take these payments into account, but their form is not exactly intuitive to the average person).

For every loan I've ever had, I've created a spreadsheet that details, over time, how much interest is accruing from one payment to the next, and how much principal remains over time. These are usually pretty accurate, but when I set up a spreadsheet for my mortgage, I found that my interest calculations were consistently higher than the actual interest charged, resulting in a longer calculated amortization. It turns out that this is due to the way mortgage rates are reported in Canada.

Canadian lenders post mortgage rates that are "compounded semi-annually, not in advance". Well, that clears it all up, doesn't it? It turns out this is actually very simple, but we need to sort out the jargon.

The "compounded semi-annually" part means that the rate is posted assuming that interest will be calculated every six months. The "not in advance" part means that interest is charged

*after*it accrues, so you don't

*start out*your mortgage owing six months' worth of interest. That is, if you have a $100,000 mortgage with a posted rate of 7.0%, then after the first six months, you would see an interest charge of $3,500 ($100,000 x 0.07 / 2). Note that this is actually equivalent to an annual rate of 7.1225% ((1 + 0.07 / 2) ^ 2 - 1), as opposed to the posted 7.0% rate.

In the real world, however, no one pays their mortgage semi-annually; most mortgagees make monthly payments. In the example above, this means that the rate of 7.1225% needs to be converted to monthly compounding, so each month, we would expect an interest charge of 0.575% ((1 + 0.071225) ^ (1 / 12) - 1). Multiplying this rate over twelve months gives us a

*true*effective annual rate of 6.90%. Using this calculated rate, mortgage interest works out to within a few cents of what is actually charged by the lender.

The calculations here may seem a bit confusing, but here is a summary:

- R = Annual rate posted by lender
- r = Effective annual rate charged by lender
- r = 12 x (((1 + R / 2) ^ 2) ^ (1 / 12) - 1)

*semi-annual*calculation of interest. In the United States, interest is calculated

*monthly*, so we would end up with the following:

- R = Annual rate posted by lender
- r = Effective annual rate charged by lender
- r = 12 x (((1 + R / 12) ^ 12) ^ (1 / 12) - 1)
- r = 12 x (1 + R / 12 - 1) = R

I was quite shocked the first time I worked this out, because I could not figure out why lenders would

*advertise*mortgage rates

*above*what they actually end up charging. It turns out that financial institutions are required by law to express their interest rates this way, so that consumers are able to compare "apples to apples", since all lenders are advertising their rates on the "semi-annually, not in advance" scale.

So now you know.

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