Assume that you have a three-year-old daughter and you have come to appreciate the power of saving and investing. Can you open up and put money into a Roth IRA in your child's name so that she can benefit from many years of compounding

No.

You cannot open up and put money into a Roth IRA in your child's name.

Explanation:

The IRS allows that any child, regardless of age, can contribute to an IRA if they have earned income.  This means that only a child that has earned income can have an IRA opened for him or her.  As the child is still underage, the IRA must be set up as a custodial account by the parent or another adult. This implies that the child cannot operate the account during the period she is underage but can have money saved in the account from her earned income.

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