Tax notes: Penalty and interest
Besides taxes, IRS charge various penalties and interest.
First, IRS requires that the tax due on every quarter to be paid on time, if not, there will be estimated tax penalty which is calculated on Form
 It is possible that at the end of year, you will get a refund but still owe estimated tax penalty because the obligation of a particular quarter was not met. Although it is called penalty, it is calculated based on interest. The penalty is reported on line 79 of Form 1040.
The estimated tax penalty (P) is calculated based on:

The shortage to meet the required payment, T for the year, and T/4 each quarter.

interest rate on each quarter (I_{1}, I_{2}, I_{3}, I_{4}) which is the federal shortterm rate in the first month of the previous quarter, rounded to the nearest integer, plus 3%. This rate is also directly documented in the lastest file in this IRS drop.

The length of the time owed. Since the quarterly estimated tax due dates for each quarter are: 4/15, 6/15, 9/15, and 1/15, you are late each quarter for 12, 10, 7, and 3 months respectively assuming you pay tax on 4/15 next year.
Therefore we have:
For the year 2016, we have I_{1} = I_{2} = I_{3} = I_{4}= 4%, so
The percentage is on line 15 of form 2210 is 2.656%, the error due to the use of months instead of days is less than 0.4%.
Please note that the percentage is based on the shortage to the required payment for the lesser of 90% of this year’s tax or 100% (when AGI is less $150,000) or 110% (when AGI is more than $150,000) of prior year’s tax, so the percentage based on the actual tax owed is less than the of the actual tax.
If you do not file and pay tax on time, there will be additional fail to file penalty, fail to pay penalty, and additional interest until it is paid.
The fail to file penalty is 5% per month or partial month up to 25% of the tax owed IRC Section 6651(a)(1).
The fail to pay penalty is 0.5% per month or partial month up to 25% of the tax owed IRC Section 6651(a)(2).
If you owe both fail to file penalty and fail to pay penalty, the fail to file penalty is reduced by the amount of fail to pay penalty IRC Section 6651(c)(1).
The interest used in calculation is the same as in estimated tax penalty determined quarterly, but compounded daily. As the calculation involves the “advanced” math, Rev. Proc. 9517 provided the Factor Tables to calculated the interest of a quarter for different interest rate, number of days of the quarter the tax was due, and whether the year is leap year or not.
Let me use an example to demonstrate how these amounts are calculated. Suppose you did not file nor pay 2014 tax due on 20150415, until 20171026, and you owe $10,000 in tax.
The fail to file penalty is 5% per month or per partial month up to the maximum of 5 months, reduced by fail to pay penalty of 0.5% per month or per partial month for the period. Therefore we have:
The fail to pay penalty is 0.5% per month or per partial month for 31 full or partial months (From 20150415 to 20171026):
The interest was 3% for first 351 days (from 20150415 to 20160331), and 4% for the rest of 574 days (from 20160401 to 20171026), accumulated daily, so the total interest is:
Please note the base amount is the tax owed plus the fail to file penaty IRM 20.2.5.3(2), and 2016 is a leap year (366 days).
If we use the IRS table method to calculate interest quarter by quarter, the result is $1,175.72. The difference due to rounding is less than a dollar.