“How does she determine a cost basis to use for determination of capital gains, and how does she claim the gains?”---->Her cost depends on the situation; If the fair market value (FMV) of the home at the time of gifting was equal to or greater than the prior owner's cost basis as in MOST cases, the donee (your spouse) takes over the donor’s (your FIL) basis. For example, assume that your spouse received the house from her father worth $75,000, FMV. Two years ago. Her father paid $30,000 to buy the house ten years ago. Then his basis in the house was $30,000. Since the FMV, $75,000, exceeds her father’s basis $30,000, your spouse takes her father’s basis, $30,000. If she sells the house ( assume that the house is not her primary residence, so it is not subject to primary residence exclusion rule permitting $500,000 CG exclusion for a married couple) for $150,000, then her LTCG ( assume that she held it longer than one year) is $120,000; $150,000-$30,000=$120,000; her marginal tax rate is over 15% then her LTCG tax is 20%*$120,000=$24,000.
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On the contrary, at the time of the gift the FMV of the house is LESS than the donor’s( her father) basis, then the donee’s basis in the house is not known until the done disposes of the house. If the donee realizes a gain, then the donee’s basis equals the donor’s basis. If the donee realizes a loss, then the donee’s basis equals the FMV at the time of gift. For instance, at the time of the gift, the FMV of the house was worth $30,000. The donor, her father, paid $50,000 for the house. Your spouse sells the house for $60,000. Since your spouse sells the house for a gain, $10,000($ 60,000>$50,000), Her basis is in the house is $50,000, donor’s basis as said above.