Mortgages, eh?

Mortgages are interesting animals.

In the U.S., an advertised annual mortgage rate of 6% is taken to mean AnnualRate / 12 per month.
That's 6 / 12 = 0.50% per month.
Amortized over 30 years, the monthly payments for a \$100K mortgage would be \$599.55.

In the U.K., an advertised annual rate of 6% is used to calculate an annual payment.
The monthly payment would be 1/12 of that.
Amortized over 30 years, the monthly payments for a \$100K mortgage would be \$605.40.

In Canada, for an advertised annual mortgage rate of 6%, the monthly rate would be
(1+0.06/2)1/6 - 1 or 0.487% per month.
That curious ritual is called "half-yearly rests".
Amortized over 30 years, the monthly payments for a \$100K mortgage would be \$594.82.

>And in Lower Slobbovia? What'd they do there?
Pay attention. This is interesting.

Now, I'm thinking that monthly payments imply that a year contains 12 months of equal length.
Remember? Thirty days hath September, April, June, and something I forget

Okay, suppose we take into consideration the different periods between payments.
You take out a \$100K mortgage at the beginning of the year.
The first payment is made after 31 days, the next after 28 days, etc.
If (daily) interest, say r, is charged on the balance, then you'd owe at the end of January (31 days):
100,000 (1+r)31
Then you'd make your payment of \$P and the balance would be:
100,000 (1+r)31 - P

Another 28 days go by, interest at the daily rate r is applied to this balance, then you'd make your next \$P payment and the end-of-February balance would be:
100,000 (1+r)31+28 - P(1+r)28 - P
etc.etc.

30 years go by and your final balance is \$0.

>And what's r, the rate per day?

If the "advertised" annual rate is 6%, you'd expect the daily rate to satisfy:
(1+r)365 = 1 + 0.06     ... for that 6% annual rate.
Then:
r = (1 + 0.06)1/365 - 1   or   0.01597%

If you know r, the daily rate, then you can calculate \$P your monthly payments, right?

>Uh ... you can?
It's whatever reduces the loan amount of \$A to \$0 after 30 years of monthly payments.
>Assuming you pay interest on the balance, each month, right?
Yes. Guess what the monthly payment is?

Remember: using a daily rate, an end-of-month balance would depend upon the length of the month.
For example, at the (constant) daily rate r (above), a \$100K balance would become:
\$100,448 after 28 days   and   \$100,480 after 30 days   and   \$100,496 after 31 days.
Any questions?
>Yes. Guess what the monthly payment is?
I guess \$589.30
 >Uh ... you have a spreadsheet, right? Yes. You can pick the month when the mortgage starts and ... >And you type in the days? Yes. >Why do you call it gummy monthly payment? 'cause it's fictitious, funny, curious, interesting and is only used in Lower Slobbovia. Click to download spreadsheet.