In most new real estate mortgage loans, there is a HUD-1 item for the Buyer to prepay the lender for the interest on the loan from the closing date to the end of the month of closing. This is because interest is paid in arrears, and the first payment usually falls on the first day of the month following the month of closing.
A closing on June 15th would require interest prepaid for the period from June 16 to July 1st.
The first payment would be due on August 1st, with interest in arrears for the month of July.
As an example, let's us a mortgage amount of $272,000 with an interest rate of 7.0%, and the dates above.
$272,000 X .07 = $19,040 in annual interest
$19040 / 365 calendar days = $52.16/day in interest
$52.16 X 15 days in June = $782.40 in interest to prepay
The August 1st payment would include interest in arrears for July