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My family receives government subsidies for our health insurance premiums which we get from Covered California (California’s ACA marketplace).

I’m looking to sell some stocks to cover some expenses, however my understanding is that the capital gains are counted towards our MAGI which may trigger me to have to pay back some of those subsidies.

Yes, capital gains are counted toward AGI, which is modified to add in some non-taxable income to calculate MAGI for ACA. And if you go $1 over 400% of the Federal Poverty Level (FPL) for your family size, you will have to pay back ALL (not just some) of the Federal subsidies. Here's a website that shows 400% of FPL by family size

That said, it's my understanding that CA has provided some subsidies for income levels above 400% of FPL. So you may get some subsidies from the state even if you have to pay back all of the Federal subsidies.

Is there a way to avoid this?

Reduce your income by:

- selling stocks/lots with lower capital gains
- selling stocks that have capital losses
- if eligible and you haven't already - max out an HSA
- make pre-tax contributions to a 401(k) and/or IRA

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