If an executor liquidates an annuity in the estate to pay debt and expenses, how are the taxes on the gain calculated and paid? TIA

It is income to the estate return form 1041 which must be filed and the taxes are paid either at the estate's level or the beneficiary lever if the income is passed thru on a K-1 form.  

Thank you.  I've read that the tax liability is typically passed to beneficiaries because the tax rate for individuals is lower.  This is the part I need clarity on.  There is an estate tax exemption up to $12.06m.  The estate in question is valued at $2m.  What would the tax rate be at the estate level?  Is it capital gains tax on the annuity proceeds?  

You are mixing up apples and oranges .... there is an estate tax  return which has an  exemption up to $12.06m  (form 706   https://www.irs.gov/forms-pubs/about-form-706  )   and the estate INCOME tax return (form 1041)  which has an exemption of only $600.   So if the estate has income in excess of $600 then an income tax return must be filed.   

thank you

if the annuity had a secondary beneficiary ( basically the one that would be entitled to the annuity if the principal owner died, then the annuity is theirs as are the tax consequences for cashing it in. if the annuity passed to the estate then and the deceased had a will or the annuity went to a trust then you need to read the will or trust agreement to see if the income can be distributed to the beneficiaries. if there was no beneficiary and the annuity was not covered by the will or didn't go to a trust then the deceased state's laws of intestate succession apply.   most contain a provision that would allow discretionary distribution of the proceeds. the tax laws also dictate that the distribution must be done n a timely manner. 

this is from IRC sec 663  - in other words to get a deduction the proceeds must be distributed during the tax year or within 65 days after the tax year.