Your US tax liability
How your US tax liability is offset when you live abroad
Many US taxpayers believe that if they earned less than around $70,000-90,000 while abroad they don't need to file a tax return. What they are inadvertently referring to is the Foreign Earned Income Exclusion (FEIE). This, together with the system of Foreign Tax Credits (FTCs), are the two main ways that US taxpayers abroad reduce their US tax liability so that they are not "double taxed". Note that neither of these can be used to reduce the tax that you may owe on any US sourced income that you may still have when living abroad.
Foreign Earned Income Exclusion
The FEIE on Form 2555 is used to reduce the amount of income on your tax return subject to tax when living abroad. In 2015, taxpayers whose "tax home" is in a foreign country and pass one of the two eligibility tests can reduce their foreign earnings by up to $100,800. These tests are the bona fide residence test and the physical presence test.
If you pass either of these tests, your total income is reduced by $100,800 (up to the actual amount of foreign earnings, if less) and you avoid paying US tax on your foreign earnings. Note that this can only be used to reduce your earned income--if you have foreign source earnings of interest, dividends, capital gains, gambling winnings etc. this can not be excluded by the FEIE.
On Form 2555 you can also take an exclusion (if employed) or deduction (if self-employed) for certain foreign housing expenses if they exceed the base housing amount of $16,128. In practise, you will only be able to use this if you are paying rent (if you are paying a mortgage, you can deduct the interest in your itemized deductions). There is also a maximum limit on excludable/deductible expenses which depends on which city you are living in; in London, this limit is $85,300.
Foreign Tax Credits
If your foreign income is above the FEIE amount, you can still offset the US tax liability on this income by obtaining a credit for tax that you have paid to a foreign government on this income using Form 1116. You can also get a credit for foreign taxes you have accrued in the year but not yet actually paid (although most foreign taxpayers stick to the cash basis).
The FTCs are applied on line 47 of the Form 1040 after your total US tax liability has been calculated. On Form 1116 we calculate the maximum allowable credit that you can put on your 1040. The credit is limited to the amount of tax that you owe on the proportion of your income that is foreign sourced. In other words, the IRS does not allow you to use the foreign taxes you have paid to offset the tax on your US sourced income.
Since foreign income taxes are often higher than federal US taxes, you will most likely have enough credits to offset the entire tax liability on your foreign income. The excess, unused credits can be carried forward to offset future US tax liabilities on your foreign income for up to ten years. This means that if you are living in the UK and paying the normal levels of UK tax you are very unlikely to owe tax to Uncle Sam unless you still have US source income, since UK income taxes are generally higher at each level of income.
However, there are different "baskets" of credits that can be used to offset tax on certain types of income. The most important of these are the "passive" and "general" baskets: passive is essentially unearned income (interest, dividends etc.) while general is wage income. This means that credits that you receive for taxes paid on your wage income can not be used to reduce tax that you may owe on foreign interest, capital gains etc.
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