# What is the Effect of Beginning or End of Day Accrual on APR?

In a previous blog, I detailed out the Reg Z method for determining APR from a payment stream. In another previous blog, I detailed a method for determining a best fit amortized payment. It occurs to me that combining these two procedures gives us the ability to see what affect one small change to a loan parameter will have on the APR of a loan.

The parameter I have chosen to test is Beginning- or End-of-Day Accrual. Does the loan accrue its interest on the morning (pre-payment) principal balance, or does it accrue on the evening balance after payments have been posted for the day? This will have a small effect on the amount of interest accrued, and this in turn could affect the "best fit" loan payment. We could then, in theory, use that best fit payment along with the Reg Z method to calculate the APR, and by doing that calculation with both payments, we should get a slightly different APR - and this will tell us if the parameter's effect on APR is statistically significant. We define significant as being a change of more than 1/100th of 1%, as Reg Z requires accuracy to the second decimal place.

For this test, we will make the following assumptions, which will be common to both loans:

Principal: $1,000

Interest Rate: 10% (The APR should be really close to this in both cases)

Term: 12 monthly payments

Origination Date: January 1, 2017

First Payment Date February 1, 2017

Maturity Date: January 1, 2018

Interest year: Actual/Actual (Every day's accrual is 1/365th of the annual interest calculated at the current principal balance.)

As any amortization schedule must, we will make the assumption that all payments are paid on their due dates. In the standard case where accrual is End of Day, the accrual for February 1 will come AFTER the February 1 payment was posted, and so we will have 31 days of interest accrual at $1,000 principal balance before the posting of the payment reduces the principal.

In the alternative case of Beginning-of-Day Accrual, the accrual for February 1 will post before the payment, and so 32 days of interest will accrue at the full principal balance of $1,000 before it is reduced by the payment. Herein lies the source of the difference.

Using the Nortridge Loan System, and the method established in the previous blog: Calculating an Amortized Payment, we come up with the following payment amounts for the loans with the parameters above...

End of Day Accrual: $87.90

Beginning of Day Accrual: $87.92

So, the difference between Beginning- and End-of-Day Accrual in this case is two cents per payment or 24 cents over the life of the loan. Now, using the method outlined in Calculating APR (Part 1), we can use the payment stream to work backward into the APR.

The APR for End-of-Day Accrual (12 payments of $87.90) is: 9.9658%

Why does this deviate from 10%? Rounding, pure and simple. The payment of $87.90 will leave a couple of pennies on the table at the end of the loan. Doing the more complex Reg Z calculation where the final payment is just a few cents higher should pump that up to 10%. But our goal was to see what the APR difference between a payment of $87.90 and $87.92 came out to. So, I will pump $87.92 into the payment stream of my spread sheet, and the result is....

10.0088%

This, by definition, is significant.