Generally no. Often times, a firm (commonly referred to as a vulture firm) would acquire a list of the names of the investors of a particular REIT. The targeted REIT is usually one that has reduced its dividends and suspended its redemption policy. The vulture firm will attempt to buy the most amounts of shares it can at the lowest possible price. And while certain REITs may have re-priced their shares at a considerably lower number than the initial share price, vulture firms typically value the REIT shares at an even lower number.
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