# Calculating interest accrued with extended Initial Payment Date

I'm wondering how do I calculate the interest accrued if the intervals aren't perfect. For example

``````Principal - \$100,000
Interest - 8%
Compounding monthly
Disbursal Date - January 1st, 2016
Initial Payment Date - February 11, 2016
``````

I can calculate it if it's exactly 1 month then the time passed would be 1/12.

Interest accrued = ((1 + i / cf)^(cf)*T - 1) * principal balance

Where cf is compound frequency, T is time.

If duration that has passed is exactly 1 month, the interest accrued is 666.67. The interest accrued to the original question is \$887.31, but I'm unsure how to get that value.

• Check whether the loan document says that for the purposes of interest calculations, the year is considered to have 12 months of 30 days each. – Dilip Sarwate Apr 20 '16 at 20:07
• Hi @DilipSarwate I'm using TValues, there's no documentation. Year length is 365 is that helps. Updated a picture to depict the question and answer – user3276954 Apr 20 '16 at 20:09

The formula for a loan is worked out by equating the present value of the loan to the sum of the payments discounted to present value by the interest rate and period. (The summation is converted to a formula by induction.)

So for a standard loan with equal payment periods we have the formula below. (This is the same as the formula quoted by DJohnM.) With an extended first period the formula is modified like so. We can work out the extension `x`

Treat Jan 1st to Feb 11th as an average month plus 10 days. (Jan 1st to Feb 1st is an average month; Feb 1st to Feb 11th is 10 days.)

`x` is 10 fractions of an average month.

``````x = 10/(365/12)

pv = 100000
n = 36
r = 0.08/12
``````

Using the formula for an extended first period

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

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

∴ c = 3140.489480141824
``````

The regular repayment is \$3,140.49

• Hi Chris, how did you get the DailyRate? I tried ((1 + i/12)^12 ) ^ (1/365)) But that didn't give me the value you got. – user3276954 Apr 21 '16 at 13:30
• Hi, @ChrisDegnen. How would you get 100887.31 ? In a typical situation, that value is not given to me. – user3276954 Apr 21 '16 at 13:56
• TimeValue software has been around for awhile and many companies in Canada use it, including scotia bank. I would think their calculations are correct. timevalue.com So your solution I have trouble working with because to get the nominal rate, you need the daily rate. To get the daily rate you need the nominal rate.... And as you already know I get a different dailyRate – user3276954 Apr 21 '16 at 14:15
• Hi Chris, that's easily solvable, 100887.31 is not a value that's provided. We are only give Principal - \$100,000 Interest - 8% Compounding monthly. Based on that information, how do you know it's suppose to equal 100887.31? I'm trying to generate the solution without reversing it from the answer. – user3276954 Apr 21 '16 at 14:24
• Hi Chris, once against 887.31 is not a value provided. it's part of solution. I'm trying to generate the schedule programmatically. In a typical situation, no schedule is provided. Only disbursal date, initial payment date, disbursal amount and interest rate + compound frequency. – user3276954 Apr 21 '16 at 15:17

You need to do detective work to see how the lender is treating loan periods of less than a month. Fortunately, there's enough information to do the job, without knowing the assumptions. The thing to remember is that the interest rate and how to apply it is a fluid number, subject to exaggeration, inflation, and deceit, but the payment schedule is real and concrete. So...

First, calculate the principal amount of an ordinary annuity, at 0.6666666% per month, that is paid off by 36 monthly payments of \$3140.50 Use a mortgage calculator, or this formula: This turns out to be \$100,219.03. This is the present value of an ordinary annuity which starts, naturally enough, one month before the first payment, or on Jan 11

Now, that is the present value of the mortgage of Jan 11. We can now switch to monthly compounding. Multiplying this Jan 11 value by a months interest, 1.0066666666666, gives a value as of the first payment on February 11, of \$100,887.15 So the original debt has grown by the addition of \$887.15 of interest.

As to justifying this amount of interest for the first part of the loan, that would require a knowledge of how the lender chose to treat partial periods. Trying this guess:

1. Take the 8% per year, compounded monthly
2. Divide by 12 and add 1 to get the monthly growth factor; 1.0066666666
3. Raise this monthly growth factor to the 12th power to get the effective annual growth factor;
4. Raise this annual growth factor to the (1/365) power to get the daily growth factor
5. Raise this daily growth factor to the 10th power to get the growth factor for 10 days;
6. Multiply by the original \$100,000

And we get..... 100218.68, not the 100,219.03

So the method above is not the one the lender has chosen to apply. (Unless rounding errors are included...)

• Hi @DJohnM I'm comparing against timevalue.com and that seems like a very big rounding error. I believe that the approach may not be correct. It's close! – user3276954 Apr 21 '16 at 14:17
• Hi @DJohnM I just posted up a solution by the software company itself if it interest you and if you would like to have a further discussion. – user3276954 Apr 22 '16 at 17:26
• Yes, the difference is they calculated the daily rate by `0.08/365` instead of `(1 + 0.08/12)^(12/365) - 1` as it should be. – Chris Degnen Apr 22 '16 at 22:44

I reached out to TValues and here's the solution they've provided.

Interest for 1 month is 666.67

So now you have to find the interest accrued from Feb1st to Feb 11 (10 days0

So it now becomes 100666.67 * (.08/365)*10 = 220.64 is the amount of interest accrued from feb 1 to feb 11.

Add the two interest accrued, you get 887.30

• The daily rate should be `dailyrate = (1 + 0.08/12)^(12/365) - 1` not `dailyrate = 0.08/365` because the 8% is a nominal rate compounded monthly, not compounded daily. (See calculation.) However, the main thing is that they've added 10 days to an average month, which explains why Jan 1st to Feb 11th isn't 41 days. They have it as `365/12 + 10 = 40.41666 days`. This is an interesting point because it's appropriate to treat the month as average. – Chris Degnen Apr 22 '16 at 23:29