It doesn't happen very often. But when independent directors who sit on mutual fund boards become the focus of criminal allegations, lawsuits, investigations by the Securities and Exchange Commission or public scrutiny, fund companies and their boards of directors often find they must make an important decision. Should they ask the trustee to resign or stay the course?

At least a few mutual fund companies have had to deal with this thorny problem over the last few years, including American Century Investments of Kansas City, Mo., INVESCO Funds of Denver, and Dreyfus Corp. of New York. Each has seen one of its fund directors making headlines related to dubious outside activities.

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