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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.

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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.

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  • Didn't know about the annuity due formula. Thanks!
    – Brad Thiessen
    Commented Jul 9, 2018 at 16:19
  • @BradThiessen No probs.
    – Chris Degnen
    Commented Jul 9, 2018 at 18:42

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