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Elephants are large institutions (investment funds, investment banks and other major investors) that are able to make high volumes of trades in the stock markets. Due to their sheer size, their high trading volumes have the power to affect the price of these assets.
These can include large with-profits funds (and large insurers generally) that run into the billions of pounds under management or very large pension funds. These institutions are so large they can cause individual share prices to rise or fall depending on the trading activity.