Although an investigation by accountancy firm KPMG cleared Cupid of deliberate wrongdoing the share price slid from 197p in January to just 49p in March.
The stock price is below the 60p the business floated at during the summer of 2010 and well short of the 240p high reached in 2011.
It was confirmed yesterday that Mr Dobbie, who owns more than one-fifth of the shares in Cupid, is to leave his current position at the start of next month although he will remain on the board as a non-executive director.
Shares closed down 2p, or 3.5%, at 55.5p.
Cupid chairman George Elliot signalled it was Mr Dobbie's decision to change roles and said: "Bill has led the group through [the initial public offering] and beyond and his decision to step down as CEO is at a time where Bill and the board have re-aligned the group's strategy and are now building upon a stronger more focused organisation."
Replacing Mr Dobbie as chief executive on December 1 is Cupid's managing director of dating services, Phil Gripton. He joined the group almost a year ago after leaving digital marketing agency LBi, where he was chief operating officer. Prior to that he was managing director of Edinburgh based Bigmouthmedia before its takeover by LBi.
Mr Gripton has also held roles in Cable & Wireless, Energis, Parametric Technology Corporation and ICL Fujitsu.
Mr Elliot added: "We are very pleased to have appointed Phil as [chief executive] and believe that he is in a strong position to drive the business forward using his extensive and complementary skills and experiences to ensure Cupid delivers on its strategy.
"Notably, Phil's experience in digital marketing has helped the business improve effectiveness through changing the mix of Cupid's marketing channels.
"We are very pleased that Bill will remain on the board as a non-executive director supporting the business with his extensive knowledge of the company, its operations and the marketplace, particularly related to acquisition opportunities."
Cupid pointed out Mr Gripton has been heavily involved in the offloading of its casual dating sites including BeNaughty and Flirt to former executive Max Polyakov, who co-founded Cupid with Mr Dobbie, for £45.1m.
The AIM-listed company also highlighted the fact that Mr Gripton has been developing a long-term strategy for the dating side of the business as well as working on repositioning some of its brands.
The 45-year-old said: "This is a pivotal time for Cupid, and I am very much looking forward to continuing my work with the board to drive the business forward, implement new initiatives and deliver on the strategy that is in place. Although Cupid's brands have grown well I believe there is much more potential for them going forward. I'm confident our teams can deliver the best user experience for our global customers and ultimately create value for our shareholders."
It is understood no direct replacement for Mr Gripton will be appointed with Cupid believing its structure beneath the board to be "exceptionally strong".
In its interim results for the first six months of the year Cupid saw revenue rise 12.4% to £43.4m.
However exceptional costs covering the fake flirtation investigation came in at £289,000 with a further £100,000 of costs because of a legal case in France.
Rising costs and a £4.3m charge to amortise intangible assets also hit the bottom line with the continuing operations booking a £2.5m loss in the period, against a £1.1m profit the previous year.
Cupid has seen its stock price come under pressure from hedge funds with several taking short positions and betting on falls in the price.
Toscafund, the hedge fund chaired by former Royal Bank of Scotland chief Sir George Mathewson, has this year built a stake of more than 15% in Cupid making it the third largest shareholder behind Mr Dobbie and Mr Polyakov.
Last month Guernsey-based fund Kestrel Opportunities raised its stake in Cupid to more than 4%.