# Compound Interest & Mortgages

Mortgages are compounded 2 times a year in Canada and 12 times a year in the USA. However, I read that mortgages are simple interest and not compound interest because you pay the interest for each month in full, leaving nothing to compound in the next month. This seems self-contradictory and confuses me.

1) If you don't pay the accrued interest due at the compound period, it's clear to me that you're going to pay interest on interest (compound interest). But with mortgages that shouldn't be the case, so why are we using the compound interest formula:

``````principal * (1 + interest / compound periods per y) ^ (compound periods per y * nb yrs)
``````

and not

``````principal * (1 + interest * nb yrs)
``````

Let's take a loan of \$1000 @ 1% amortized on 2 years and compounded yearly. If you pay the \$10 after year one, and then pay again \$10 after the second year, even if it was compounded twice, you end up with the same result than the simple interest formula. Why it's different with mortgages? Why the number of compound periods is relevant if interest isn't compounded?

2) If a mortgage is simple interest, why a nominal rate of 6% has an effective annual rate of 6.09%? I know how to find the effective rate:

``````(1 + (nominal interest rate / number of period)) ^ number of period - 1
``````

... but from the amortization table is it possible to calculate the same number? I just don't see what that number means. Will the loaner pay that effective rate? What is it?

• If mortgages weren't compounded then making overpayments would never make sense. Mar 12 '18 at 7:15
• As far as I know, in the U.S., mortgages are always simple interest. If you make your scheduled payments on time, it make no difference. But even if you make your payments late, there is still no interest on the unpaid interest. However, there may be late fees that are even more than the compound interest would be.
– prl
Mar 13 '18 at 4:39
• @Conrad, that is incorrect. Prepayment makes sense, because it reduces the principal balance on which the interest is computed. It has nothing to do with compound or simple interest.
– prl
Mar 13 '18 at 5:18
• One minor correction to my previous statement that all U.S. mortgages are simple interest--an exception is mortgages that have negative amortization--that is, where the designated payment is insufficient to cover the interest.
– prl
Mar 13 '18 at 5:36
• @prl That's an interesting feature of the US system that I was unaware of, thank you. Mar 13 '18 at 11:26

If a mortgage is simple interest, why a nominal rate of 6% has an effective annual rate of 6.09%? In other words, why are we doing

If the mortgage is on Fixed Interest Rate; the total payable is arrived at taking into account the interest compounding. So let's start with 100 as loan for with yearly rate of 6% compounded semi annually. Total Tenor of the loan is 2 years.

For first 6 months, the interest will be 100*0.06*6/12 = 3
For the next 6 months, the interest will be 103.06*6/12 = 3.09
For the 3rd 6 months, the interest will be 106.09*0.06*6/12 = 3.1827
For the last 6 months, the interest will be 109.2727*0.06*6/12 = 3.2782

Total payable will be 112.550881
The monthly repayment will work out as 112.550881/24 = 4.6896

Thus the compounding does indicate what your monthly payment will be. As mortgages are typically over 25 or 30 year periods, this does result in better funds for Bank while showing the rate as less.

I'm reading everywhere that mortgages are not compounded because the accrued interest is always paid before

Most Mortgages are on Variable Rate of Interest. In such case the computation is different and would not use the above methodology; it would be the rate applied on the outstanding amount. The EMI would pay off the interest.

if a mortgage is simple interest,

On a pure Simple interest loan, the calculations will be as indicated by you.

• Is this really how they calculate amortization schedules in Canada? It seems kind of screwy because it doesn't reduce the principal as the borrower pays it off. Thus, in the last payment period the borrower is paying interest on the full amount of the loan (plus all interest charged over the life of the loan to date), even though the balance outstanding is a tiny fraction of that. If that's really the way mortgages work in Canada, it must be nice to be a bank there. You spend most of a loan's term collecting interest on money that has already been repaid! Mar 12 '18 at 16:59
• @RPL it's even worse in the US, because mortgages are compounded each month. This means the effective rate would be 6.16% in our example. Mar 12 '18 at 17:47
• @user7448572 I believe you are mistaken; the calculation as presented is definitely incorrect for mortgages in the US. In standard US mortgages, interest is only calculated on the balance outstanding. Each monthly payment pays off all of the interest currently due along with some of the principal, resulting in a smaller interest accrual in each successive payment period. In the calculation presented above the interest accruing is increasing with each passing payment period, which matches no amortization schedule I've ever seen. Mar 12 '18 at 21:38
• @RPL Yes, same in Canada, it's probably just a simplified calculation, without any payment or amortization, to demonstrate the effect of compound interest. However, what I still don't understand is why are we saying "compounded 2 times" (Canada) or "compounded 12 times" (USA) when you pay the interest for each month in full, leaving nothing to compound in the next month? Mar 12 '18 at 22:48
• @user7448572 Agreed, but all my mortgages have been in the US, so I'm not sure how they do it in Canada. Accruing interest less often than the payment frequency seems weird. I'm not even sure how that is supposed to work. Hopefully someone more knowledgeable will weigh in. Mar 12 '18 at 22:53

Using the effective interest rate calculation explained here

https://en.wikipedia.org/wiki/Effective_interest_rate#Calculation

``````with

i = 6% nominal interest compounded twice annually
n = 2 compounding periods per annum

r = (1 + i/n)^n - 1 = (1 + 0.06/2)^2 - 1 = 6.09%
``````

Why is this different from the nominal rate?

Nominal rates are intended to make it easy to calculate the periodic rate, here 3% for every six months. With compounding the result is the effective rate: 6.09%.

By way of explanation of nominal versus effective rate:

The "Truth in Lending Act" passed in 1968 did not incorporate the mathematically-true annual percentage rate, because the true calculation used compounding (sometime fraction compounding), which was not readily available. The result on expression of the APR on credit cards uses a Nominal (simple interest) method ... which can far from the truth. The Truth in Lending Act should be changed to the mathematically-true (EFFECTIVE) APR from the untrue (NOMINAL) APR, merely by changing the word in act from "multiplied by" to "compounded for".

"Fractional compounding not readily available" i.e. to calculate the periodic rate from the effective rate requires a relatively more complex calculation:

``````periodic rate = (1 + r)^(1/n) - 1 = (1 + 0.0609)^(1/2) - 1 = 3%
``````

It is far easier to use calculate the periodic rate from the nominal rate:

``````periodic rate = i/n = 0.06/2 = 3%
``````

However, compounding at 3% results in `(1 + 0.03) (1 + 0.03) - 1 = 6.09%` not `6%`.

The nominal rate is simple device to make calculation easy. The effective rate is what you get.

To calculate return the periodic rate is used. E.g. over two years

``````nominal rate compounded twice annually = 6%
periodic rate, pr = 3%
number of periods, np = 4

return = (1 + pr)^np - 1 = (1 + 0.03)^4 - 1 = 12.5509%
``````