Study Finds Only 28 Percent of Millionaires Think They're Rich

One thing that does make investors feel confident is holding a significant amount of cash. According to UBS, despite significant market gains over the past year, investors keep an average of around 20 percent of their assets in cash, as they have for the past three years. Sixty-four percent feel they have the right amount of cash, and 56 percent expect to keep the same level of cash for the next 12 months.

Wealthy investors' two top personal finance concerns are long-term care and the finances of their children and grandchildren. While most feel highly prepared in their retirement planning (62 percent), 36 percent said they were not prepared at all.

"These people have a lot of money," said Goldstein. "But the reason they don't feel wealthy is because they don't feel like they have a comprehensive financial plan in place to take care of all of these longer-term costs."

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