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I'm taking a finance course at a Canadian university. The teacher said this time-of-money is how the banks calculate their loans and mortgages. Out of the blue, I want to try how close the formula in school match up to the real world. Here's my scenario:

Borrow $300,000 at 3% interest, to be paid over 25 years. By law, the interest is compounded twice per year. Payment is to be made bi-weekly (26 payments per year). How much is the bi-weekly payment?

My calculation

I first calculate the periodic interest rate:

r = (1 + 0.03 / 2) ^ (2/26) - 1 = 0.001145934...

Then plug everything into Excel:

=PMT(r, 650, 300000)

The result is $654.83.

Bank's calculation

I then go to TD Bank's mortgage calculator and plug in the parameters. The result is $652.90 (see pics below).

TD Mortgage calculator

Question

How did the bank come up with their number? What did I do wrong?

  • 1
    There are slightly more than 26 bi-weekly periods in a year. 26 * 14 = 364. – Nayuki Jan 5 at 19:14
  • 1
    @Nayuki Sounds like an answer to me. – DJClayworth Jan 6 at 18:35
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As Nayuki points out you have too few days per year as 14 x 26 = 364 meaning you miss 1 day per year.

This results in two errors in your calculation

  1. Your rate is a little too big, should be 0.001143 by using this formula (1 + 0.03 / 2)^(14 / (365/2)) - 1
  2. You have too few periods as you use 26 payments x 25 years = 650, which should have been (365/14) x 25 = 651.8 ~ 652 payments

Solving for these to errors will fix your payment to match TD Banks calculations. This is also clear if you look at the ‘amortization plan’ in the calculator.

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