# Amortization Payments with a delay in initial payments

Typically a loan is disbursed and a month later is when payments are to be made. But there can be an agreement where it's to be delayed for a year or as short or as long as they want. I'm wondering how does that affect my payments and amortization schedule.

Given Below

``````Principal : \$100,000
Interest  : 8.5%
Final payment: 0
Periods : 120
Payment Frequency : Monthly
Compounding Frequency : Semi-Annual
Disbursal Date: January 1st, 2014
``````

I calculated using these information and I get \$1232.02

This works if payment date is February 1st 2014(one month) but if the first payment is to be made January 1st, 2015. should the payment increase? If so what formula should I be using?

To use a formula you must do two things: find the interest rate to apply to the formula, and fit the specifics of the problem to the formula you want to use.

WRT the interest rate, you quote an annual rate with semi-annual compounding, and specify monthly payments. That won't work (not your fault :) ). You need to find the equivalent monthly rate.

Consider what would happen to \$100 at your quoted rate: it would grow to \$104.25 in the first half year, and then the whole \$104.25 would grow by the same factor, to \$108.68. So the effective annual rate is 1 - 1.0425^2, or 8.680625%

The next question is: if I want to earn this much from a monthly compounding rate, what would that rate be? Just take the 12th root of the effective annual rate: 1 - 1.08680625^(1/12), or 0.6961062% per payment period.

This is the interest rate to use in the formula you chose to use.

Now, the most common formula to use involves an ordinary annuity, which, as you correctly note, requires the first payment one payment period after the disbursement. The problem you post has the first payment occurring one year, or twelve payment periods after disbursement. So you need to use the formula for compounding a single amount forward 11 periods; the \$100,000 grows by (1.006961062^11) to \$107,929.32

You now are perfectly situated to use the formula for an ordinary annuity: you have the new principal (one month before the first repayment), monthly interest rate, and number of payments to fit with the formula, to get the regular payment.

• hi @User58220 May I ask if you got \$1329.71 for Monthly Payments. I just swapped out The Principal values from \$100,000 to \$107,929.32. I think i'm going wrong slightly here, I have the final payment working out to \$2162.62. Usually it zeroes out or is off slightly which is why I think I'm wrong. Commented Sep 8, 2014 at 17:13
• Yes, 1329.71 is the correct regular payment for the loan repayment as specified... Commented Sep 8, 2014 at 18:10
• Hi @User58220, I'm Using Actual/Actual Calculations and trying to get the value you got (\$107,929.32). I Converted Effected Annual Rate to Daily Rate 1.08680625^(1/365) and raised that by the amount of days that passed (365 days) - 1 * balance. I expected it to give me \$107,929.32 but instead it gave me \$8680.6 but if I raise it to 334 days (subtract 1 month) I get a closer value 7914.964593, but it puzzles me why I would have to subtract one month. Commented Sep 8, 2014 at 19:44
• The OP specified monthly payments, and thus monthly compounding. If you apply a daily rate/compounding and thus take into account the variable length of each month, then no formula will be of any use; you'll need a spreadsheet 4,000 rows deep... Commented Sep 8, 2014 at 19:57
• I'm doing this through c# programmatically and iteratively. If I can get the steps I can get it. Commented Sep 8, 2014 at 20:09

There are a few ways to handle this:

Note: Monthly payments may vary a bit due to a monthly compounding frequency

1. Amortize the loan amount plus the interest accrued through the deferment period (\$108,500) for 120 months, with payments due Feb 1 2015 through Jan 1 2025. This gets you around \$1346/mo.
2. Amortize the loan amount plus the interest accrued through the deferment period (\$108,500) for 108 months, with payments due Feb 1 2015 through Jan 1 2024 (Makes the entire loan time 120 months, with the 12-month deferment). This gets you around \$1440/mo.
3. Amortize the loan amount for 120 months, with payments due Feb 1 2015 through Jan 1 2015, and collect an additional \$8,500 at closing to account for the interest accrued during the delayed payments. This gets you around \$1239/mo.
4. Amortize the loan amount for 120 months, with payments due Feb 1 2015 through Jan 1 2015, and run a 12-month interest-only loan for \$100,000 with payments due Feb 1 2014 through Jan 1 2015. This gets you around \$1239/mo for the full payments and \$708.33/mo for the interest-only payments.

You will need to speak with your lender to get the specifics of how they're calculating your monthly payment

The payment should definitely be different.

To know how you want to change things, what agreement do you have from now until the first payment?

Is there 0 interest during that time? Is there some interest during that time? Is the loan still due 100% at the end of 10 years? Or is it going to close 10 years after the first payment?

You have to figure out how you want to have this loan organized.

Some options:

1. 0% interest until first payment which seems overly generous. 10 year repayment starting at first payment. Make 120 payments of \$1232.02

2. 8.5% interest for 11 months, recapitalized. \$7791 extra interest so you are borrowing \$107,791 instead of \$100. Then do a 10 year repayment so it adds ~ \$37 per payment or \$1269 for 120 payments.

And many more options. Decide what you want to be fixed and go from there.