8/28/2015 original post -
I recently took out a loan as follows:

  • loan amount: $116,000
  • interest rate: 4.75%
  • term: 360 months

When I calculate the monthly payment it comes out to $605.11/month. This is the answer whether calculating it in Excel using the PMT function, on a Texas Instruments BA II Plus calculator, or by hand plugging all the variables into an equation to calculate the payment. The problem is that the lender shows this loan has a monthly payment of $605.19. The discrepancy may be related to the interest calculation being 30/360 which is specified in the loan document. In the calculations I did, 0.0475/12 was used. But 0.0475/12 is equivalent to 0.0475 * 1/12 which is also equivalent to 0.0475 * 30/360. So, is the lender's monthly payment too high by $0.08 or am I not doing something correctly when calculating the payment?

Note that the equation being used to calculate the payment by hand is as follows:

payment = (interest/12 * loan amount) / (1 - (1 + interest/12)^-term)

11/22/2015 update -
Additional facts pertinent to this problem:

  • this is a 5 year balloon mortgage
  • funding date was 8/31/2015
  • payment due dates are the first day of each month
  • day count convention is 30/360

Further figuring -
This loan is for 5 years plus one day. This is because I opted to make loan payments on the 1st of each month instead of the anniversary of the funding date (31st) every month.

Interest due for the additional day the money is borrowed (the 8/31/15 date): $116,000 x 0.0475 ÷ 360 = $15.30555... rounded to dollars and cents: $15.31.
This seems to be the origin of the extra $15+ bucks that Spehro Pefhany identified because when $116,015.31 (from $116,000.00 + $15.31) is plugged into Excel's PMT formula like so:
=-PMT(0.0475/360*30,360,116015.31,0,0) the monthly payment answer exactly matches what's on the loan documents: $605.19.
Because the loan is amortized over 360 months, this math implies that the $15.31 is paid back over 30 years and interest is charged on the $15.31 over that time because it was added to the principal amount when the loan began.

I think the bank's software is incorrectly handling this math; however, the error is in favor of the borrower. The loan statements show that the extra $0.08/payment is being applying to principal rather than interest. Not only does this make the principal of $116,000 decrease slightly faster but it saves the borrower some interest money over the long run because the balance is decreasing at a faster rate. As far as I can tell, the bank does not recover any of the $15.31.

I did notify the banker of these findings as they should be collecting more from the borrower in such a situation ($15.31 in the example above). This example is a small loan but on large commercial loans the bank could be out a significant amount of money.

  • Many websites, probably including your own bank's, have loan amortization calculators. It's a somewhat more complicated formula. To handle this fairly while maintaining constant payments, the percentages of each payment that go into paying down principal and paying interest change continuously over the life of the loan. – keshlam Aug 29 '15 at 4:21
  • I get the same answer you do, assuming monthly compounding. Looks like there's an extra fifteen bucks on the principal. – Spehro Pefhany Aug 29 '15 at 8:33
  • Are there any fees they are rolling into the loan amount? – Brian Dishaw Aug 29 '15 at 14:28
  • @keshlam - I have an amortization spreadsheet set up to handle the 30/360 interest calculation that works fine for this loan. It is correctly splitting the monthly payment between interest and principal. This post's original question is specifically about how the monthly payment for the loan is being calculated. – knot22 Aug 30 '15 at 15:35
  • Spehro Pefhany and Brian Dishaw - It does seem work out to $15 - $16 more principal, assuming compounding is monthly. Good point. An additional fee would make sense to explain the loan payment discrepancy but the loan documents clearly state that the principal is $116,000. This leads one to think that either the compounding is not monthly or the bank made a mistake in calculating the monthly payment. – knot22 Aug 30 '15 at 15:45

Your calculations are correct.

It is likely that the bank's software has a rounding error. In effect, either your bank is overstating your interest by eight cents per month, or your bank is insisting that you prepay your principal by eight cents per month. If the bank's ongoing interest calculations are correct, your final payment will be slightly smaller (because of the prepaid principal, and because of compound interest on those prepayments).

I have performed similar calculations for my mortgages over the years, and except upon early payoff in the middle of the month, I have always matched my banks' calculations to the penny. Ironically, this means that my banks' formulas are a bit weird:

  • Instead of charging a daily interest rate on the mortgage,
  • The banks are charging a monthly interest rate on the mortgage, and pro-rating the interest for the first 30 days of the month.
  • The interest for the 31st day of the month seems to be free.
  • I don't know how they handle a payoff during February, but
  • The total interest for the month of February is the same percentage as the total interest for the month of July.

After making these adjustments, even my calculations for the mid-month payoff matched to the penny.

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