You own shares in the mutual fund. Your holding period is based on when you buy and sell those shares.
The mutual fund itself owns a variety of investments. The mutual fund's holding period is based on when the fund managers buy and sell those shares within the fund. Any capital gains are distributed to the mutual fund's shareholders, and are reported to the shareholders at year's end on Form 1099-DIV.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
