# Calculating Amortization Payment Amount when first payment is early

How can I determine a fixed payment amount for a loan that has an early first payment?

Example:

``````Loan Amount: \$100
Interest: 10%
Loan Date: January 1st, 2018
First Payment Date: January 15th, 2018
Term: 10 Years, payments made yearly on January 15th
``````

Normally I'd use the Amortization Payment Formula below, but in this case it doesn't seem to work. My guess is that P needs to be adjusted to be somewhat less, but it's not clear how to calculate that. As shown here,

Calculate the interest accrued in 14 days, add it to the principal giving `s`. Then calculate the regular payment for an annuity due.

``````p = 100
r = 0.10
n = 10

s = p (1 + r)^(14/365) = 100.366
a = s (r/(1 - (1 + r)^-n))*1/(1 + r) = 14.8492
``````

The regular payment is \$14.85

For more information on the difference between an ordinary annuity and an annuity due see Present and Future Value of Annuities.

• Didn't know about the annuity due formula. Thanks! – Brad Thiessen Jul 9 '18 at 16:19
• @BradThiessen No probs. – Chris Degnen Jul 9 '18 at 18:42