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For the stock account the son will inherit his mother's basis on half of the account. On the other half of the account there will be a step up in basis to the fair market value at the time of her death. He may need to pay taxes if stocks are sold for the distributions or dividends have been received after the death of his mother.
IRS Publication 575 says that, in general, those inheriting annuities pay taxes the same way that the original annuity owner would. In turn, taxation of annuity distributions depends on whether they started before or after the required starting date for payouts.
If the daughter elects periodic distributions, then the portion of each payment that comes from accumulated earnings is taxable, while the portion that comes from the original premium payment is not. If you elect non-periodic distributions, however, the IRS typically treats distributions as taxable earnings until they're used up, after which further payouts are treated as return of the original premium payment, and therefore not taxable.
For the stock account, the may be a small step up in basis if any of the bonds appreciated but I would agree that it is probably close to fair value.
The annuity would be taxed a little differently. It would be taxed the same way that if it was distributed to the mother. If the distributions are non-periodic then these would be taxable since they would be coming from the earnings. The daughter would receive a 1099-R for the distribution and the amount that is taxable.
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