I exercised options (ISOs) in a startup so I'll have to pay the alternative minimum tax (AMT). The amount is greater than $1000. (EDIT: the AMT will certainly be larger than my regular income)
What is the total amount I'll have to pay each month of this year for not having paid the tax immediately?
The interest rate listed by the IRS seems to be 4% (for 2016) (so roughly 0.3% per month. In addition to paying interest, is there also a penalty?
I think you might be overthinking this. In general, the alternative minimum tax (AMT) due (if any) is not due as of the date(s) when the events that triggered the AMT occurred and there is no need to worry about the interest and penalties that you must pay for not paying the AMT instantly when the triggering event(s) occurred. Indeed, the AMT is not determined until you prepare your income tax return for 2016 and, depending on what else is on your tax return, you may well fill our Form 6251 and end up with the result that no AMT is due at all; your regular tax exceeds the AMT and so you just pay the regular tax. But, even if Form 6251 shows that AMT is due, the amount just gets added into "Total tax" shown on Line 63 of your Form 1040.
Now, after you compute the Total tax shown on Line 63 (which might or might not include some AMT amounts), you add up your income tax withholding plus timely estimated tax payments made for 2016 (the fourth quarter estimated tax payment is due January 17, 2017 instead of the usual January 15 because 15th is a Sunday and 16th is a Federal holiday). If the sum total of income tax withheld and timely payments of estimated tax is at least 90% of the current tax, or 100% of last year's tax (110% for high earners), there is no penalty or interest due and you can pay Line 63 "Total Tax" minus Line 74 "Total payments" by April 15, 2017, etc. But since this is doing things bass ackwards (you can't go back in time to make estimated tax payments!), proactive folks such as yourself, who know already that more tax is going to have be paid than what has been withheld, can change your W-4 form for the rest of 2016 and have lots of income tax withholding (enough to arrive at the 90% of current tax or 100% or 110% "safe harbor": the latter is easier because the numbers are known) for the one or two paychecks still to come (check with payroll, though, to see if they can process the paperwork in time). Extra tax withholding is a particularly effective method because the total tax withholding can be treated as timely payment of tax even if most of withholding occurred late in the year. Another thing to keep in mind: mutual funds distribute their dividends and capital gains in December (most will have posted their estimated distributions by this time) and these will also affect the "Total tax" computations. If you don't have payroll withholding (or withholding from IRA distributions or even Social Security benefits) to help, you should pay the amount that you need to arrive at a "safe harbor" via a 1040-ES estimated tax payment for the fourth quarter of 2016 (due by January 17, 2017), and then pay the rest by April 15, 2017. However, you will need to fill out Form 2210, including perhaps the long part A1 if you wish to minimize the interest and penalties due for not filing Form 1040-ES for earlier quarters. If you fill out the short version of Form 2210 or ask the IRS to figure out the penalty and send you a bill, you will likely pay a little more.
You will need to pay the rest of the tax due by April 15, 2017.
