# Simple interest calculation with payment deferral

I’m looking for a formula to calculate the finance charge on a simple interest loan given the interest rate, term, and amount financed.

The loan can also include a payment deferral period, e.g. the first payment isn’t due for 90 days after the loan begins, but interest accrues during the 90 day period.

All of the deferral formulas I’ve been able to google are for a compound interest situation.

Thanks!

What you want (I think) is called amortization.

I’m not sure if this is what you’re asking for but here it goes:

Lets finance \$6000 over 3 years at 12% interest. Since it is simple interest, the total amount to be paid off is \$8160 (\$6000 principal + \$2160 interest) over 36 months or about \$226.67 per month.
Now with the 90 day no pay. Since interest still accrues over the 90 days that total interest is still \$2160, but now the \$8160 is paid over 33 months making it about \$247.27 per month.

However, this is a poor investment for the borrower. If I did my calculations correctly, they could borrow \$6824.47 for the same monthly payments under standard amortization.

No, the standard amortization formulas are set up for compound interest as the OP has stated.

wm–:

For simple interest, here’s what I came up with (which I checked and verified):

Assuming simple interest is used, n monthly payments (made at the beginning of the month) to pay off a loan of P at the rate of r per month, and a delay of m months for the first payment, the monthly payment will be

R = 2P[1+(n+m)r] / [2n + n(n+1)r].

For example, for a loan of \$1,000 to be paid off in 12 payments at a rate of 6% a year (or 0.50% per month), with the first payment delayed for 3 months, the answer will be \$86.7635. Check it out.

nivlac, thank you for the response, but this doesn’t appear to be quite what I’m looking for.

Background: The company I work for bought a customized financial software package several years ago that performs this calculation. A new project requires that we precisely duplicate the calculation in a different context, but no one seems to be able to locate the relevant source code from the original package. So I’m currently working with the finance guys (I’m strictly an IT guy so this finance stuff is not my forte) to try and reverse engineer it. We’ve tried all different schemes and can get close to the numbers the package is calculating, but not exactly the same.

I compared with an example loan of \$15,078.68, 72 month term, 7.59% yearly interest rate, 30 day payment deferral. Our system calculates a finance charge of \$3739.89 and payment of \$261.37. Your formula came up with a monthly payment of \$248.71.

Can you think of any other variations we might want to try?

WM
I would break the problem down into two stages.

1. Calculate pure interest on the principle compounded accross the deferal period.
2. Add this number to the principle then go ahead and do a straightforward compound interest repayment calculation over whatever repayment period is chosen.

take care in converting your annual interest rate to a monthly rate. It could be done by simply dividing annual by 12, or taking the interest rate to power of the number of interim periods (1+rate)pwr(1/12) (or in excel terms POWER(1+rate,1/12))

So 6 % annual could be taken as 0.5% per month or 0.4868% per month. Both are correct to use but it depends on how the initial interest rate was expressed (somethimng about APR and annualized IIRC). Note 0.5% compounded over 12 months does not equal 6 % annual.

This may account for the difference you see in yours and nivlacs calculations

blondchap

Calculate Simple Interest, Principal, Rate or Time when 3 Values are Known

That monthly payment is exactly the monthly payment for a normally amortizing loan with end of period payments. You can verify that with any normal financial calculator, or MS Excel, or any online mortgage calculator, or even the formula thewalrus cited. \$15,078.68 Present Value, 72 periods, 7.59/12 period rate with end of period payments is exactly \$261.37.

I’d imagine that if the deferral period were greater than 30 days and you aren’t compounding the accruing interest, your software just tacks on the simple interest to the principal. For example, if the we keep all the same variables you have here, except extend the deferral to 3 months instead of 1, all you need to do is tack on 2 months worth of simple interest (0.6325% periodic rate times two months times original principal equals \$190.745). Add that to the original principal gives you \$15,269.43, and then amortize that over 72 months with end of period payments (since the payments are end of period it takes care of the 3rd month of deferral), and I get \$264.68 for monthly payments.

All of this really depends on exactly how you guys are handling the interest during the deferral period, but like I said, the example you gave right there is just standard amortization with end of period payments.

Well, I answered the question as posed, but I think you posed the wrong question. As desdinova has pointed out, your answers of \$261.37 and \$3739.89 are just the standard amortization results that assume compound interest. I tackled the simple interest version of the formula because I’ve never seen that problem posed before, and took it as in interesting exercise. It’s actually easy to modify the standard amortization formula for any deferral period before the first payment (as expected, the monthly payment will be higher), but that does not appear to be what you really need.

I wasn’t even thinking about where the payment actually fell within the period and how that related to the deferral. That makes a lot more sense now. Thanks!

One question though: When you roll the \$15,269.43 into a new amortization calculation, you are essentially compounding that \$190 of interest, right? I suspect our software is handling it differently, because when I run your scenario I get \$264.54, just slightly lower than your result.

I think you nailed it right there. The deeper I get into this, the more I feel that the finance guys need to be the ones to sort this out. I get paid to implement business logic, not make it up.