Calculate an even payment loan with a irregular first term

Question

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

Loan Amount: 1000. Term 12 months. Interest rate 10% annual. Loan date 9/1. First payment 10/1. Payments are 87.91. So far so good.

Same information as above, but push first payment out to 10/15. Payments are 88.24.

I have no clue how this would be calculated. Does anyone know a formula for figuring out the extra 15 days of interest, then figuring out how much that would increase the payments?

Answer 1

Simply put, move the money around with compound interest until that first payment is regular.

In the second scenario, the original debt grows, unpaid, for 15 days, until it reaches the starting point for the annuity formula used to calculate the regular payment, with a constant interval of 1 month for all payments

One way to do this is to convert the original interest rate, (which is not completely specified in the question) into first a daily rate, and then a 15-day rate. Assuming that the rate is 10% per year, compounded annually, the calculation is:

(1.10^(1/365))^15 = 1.003924538

You could multiply the original $1000 by this to get 1003.924538

This is the new principal amount for a new calculation of regular payment.

However, since the size of the payment scales with the size of the original amount, just multiply 87.91 times this 15-day factor to get 88.26

Answer 2

You can find the repayment amount by the method previously shown here, with the first repayment period extended by the fraction x representing the extra fourteen days which defers payment from October 1st to October 15th. So x = 14/31.

Alternatively x can be taken as a fraction of the average month, i.e. x = 14/(365/12). This is effectively how DJohnM's method works, but it runs into difficulty if the payment is deferred to October 31st because the extension is now longer than an average month. Both methods are shown below for comparison.

So first with x = 14/31. (This would be my preferred method.)

pv = 1000
n = 12
r = 0.1/n
x = 14/31

pv = (c (1 + r)^(-n - x) (-1 + (1 + r)^n))/r

∴ c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)

∴ c = 88.246

Now with x as a fraction of an average month: x = 14/(365/12)

x = 14/(365/12)

c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)

∴ c = 88.2523

Now by DJohnM's method: calculating an adjusted principal value and using the standard loan formula (shown below).

dailyrate = (1 + 0.1/12)^(12/365) - 1

pv = 1000

adjustedpv = pv (1 + dailyrate)^14 = 1003.662423

adjustedpv = (c - (c + r)^-n))/r

∴ c = (adjustedpv r (1 + r)^n)/(-1 + (1 + r)^n)

∴ c = 88.2523

This matches the previous result, which uses the average month.

Standard Loan Repayment Formula e.g. link, as used in the above calculation.

Checking initial result

With repayments on the 1st of every month, using the standard loan repayment formula.

pv = 1000
n = 12
r = 0.1/n

pv = (c - (c + r)^-n))/r

∴ c = (pv r (1 + r)^n)/(-1 + (1 + r)^n)

∴ c = 87.9159

The OP calculated the repayments as 87.91