RBS slumped 2.99 per cent to 198.30p per share in mid-afternoon trading, while its peer Lloyds fell 1.92 per cent to 59.79p.
Record low interest rates, which are bad news for lenders, prompted Deutsche Bank to downgrade RBS to "sell" from "hold" and cut Lloyds to "hold" from "buy".
The German investment bank reasoned that RBS would be hit by the negative interest rate environment which is engulfing the broader banking sector, since rock-bottom rates tend to eat into banks' profitability.
Meanwhile, Lloyds could come under pressure as customers' re-mortage their homes in a bid to whittle down the size of their monthly repayments. Lower interest rates will also reduce how much banks charge customers for mortgage repayments.
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The Bank of England recently cut its benchmark interest rate from 0.5 per cent to 0.25 per cent for the first time since 2009. It was a response to Britain's shock vote to leave the European Union in June and could still go lower.
Bank of England governor Mark Carney has said that the UK doesn't need negative interest rates - but a dip into negative territory would put it next to other major central banks, such as the European Central Bank and the Bank of Japan.