The boss of the privatised Royal Mail has said the business is still under intense pressure, despite seeing a 4% rise in parcel deliveries over Christmas and benefiting from the collapse of rival City Link.

Moya Greene, the chief executive said she had no option but to push through further cost-cutting, amid external predictions that 3,000 more jobs will go across the company. “The sad demise of City Link demonstrates what we have been saying – the UK parcels market remains highly competitive, with significant over-capacity,” she said.

“These conditions – of too many players chasing traffic – will continue to put pressure on prices for the next couple of years. We firmly believe the long-term prospects for the delivery sector remain positive, underpinned as it is by the continued growth in e-retailing,.”

The difficulties in the parcel delivery sector were despite a stronger operational performance for Royal Mail in December, partly due to having won customers from City Link.

Royal Mail delivered 120m parcels over Christmas 4% up on last year. Its shares rose by 4% on Thursday after it predicted full-year profits would be in line with expectations.

Despite the rise in the share price Royal Mail has still lost more than a quarter of its value since the £3.3bn stock market flotation 18 months ago.

The number of parcel deliveries across the nine months to 28 December by Royal Mail was up 3% while letters were down by 3% – but that number was still better than the forecast fall of between 4% and 6%.

Greene said postmen and women provided an excellent service over the busy festive period. “This is because we started to plan for Christmas in April, putting investment behind extra sorting capacity with 10 temporary hubs and training around 19,000 extra people.”

Industry experts said Royal Mail had cut almost 50,000 jobs over the last 10 years and 3,000 more could go over the next 12 months, leaving the group with 145,000 staff.

The employees are all leaving under voluntary severance schemes, in contrast to the abrupt departure of nearly 2,400 staff at City Link. Royal Mail is known to have taken customers from City Link, which finally called in the administrators after years of substantial losses. Rival delivery companies such as Yodel, owned by the Barclay brothers, and Hermes are also fighting for dominance. Some of the job losses at Royal Mail are due to the introduction of mechanised sorting.

The company says it is under pressure owing to its commitment to the universal service obligation, allowing anyone in Britain to post letters or parcels to any other part of the country at the same price. Royal Mail complains the obligation is being undermined by rivals such as former TNT Post business, now called whistl, who are “cherry picking” deliveries in the more accessible city areas, making it harder to cross-subsidise deliveries to less-accessible areas. But the business secretary, Vince Cable, has accused the company of scaremongering.

Analysts expect Royal Mail to report annual operating results before special charges of around £585m for 2014/15.

Richard Hunter, head of equities at Hargreaves Lansdown Stockbrokers, said: “The initial euphoria following the float has worn thin, with the share price having dropped 28% in the last year, as compared to a 2% dip for the wider FTSE100.

“The general sentiment is that the valuation is up with events, such that the market consensus of the shares as a hold is most likely to remain intact.”

The company remains under pressure from the drop in letters which has also resulted from the increasing use of emails. The parcels arm is also being hit by web-based businesses such as Amazon, which are opting to establish their own operations.