# Instalment APR Calculation Formula

I have a problem that I cannot solve regarding interest payments, and payment holidays. The scenario is this

If I purchase a product which is paid for over a 12 month period, I need to calculate the monthly interest payments. However, the complexity here, is that I have the option to take one month payment holiday (is optional, but can be any month over the 12 month period).

So far, I have

annual interest (a) = (e.g for 5%, a=5)

monthly interest (m) = ((1 + a/100)^(1/12) - 1) x 100

balance = balance*(1+m) - instalment


How do I calculate the instalment, especially when including payment holiday?

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## 1 Answer

There will not be a closed form, as it will vary with the month of the holiday. If you want to come out exactly even you have to choose in advance the month of the holiday. One approach is to assume the holiday is the first month (which will lead to the highest payment-do you see why?). Then you can apply one month's interest to the original balance and start an 11 month repayment with no holiday allowed. You should be able to use your usual formulas for this. Then as long as no holiday is taken, apply the interest to the current balance and take off the payment as with repaying a loan in variable installments. If no holiday is taken the eleventh month payment will be slightly smaller and no payment will be due in month twelve. Alternately you could bill the first month assuming the holiday will be the second month, then if no holiday is taken in the first bill the second as if the holiday will be the third, etc. This will be quite messy.

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