Lloyd's: confident it can weather the storm
This would make it the biggest single claim in the market's 300 year history, marginally exceeding the £929m loss for Lloyd's from Hurricane Hugo in the United States in 1989.
However, the figure is still dwarfed by an ongoing series of asbestos-related claims.
Lloyd's said the losses resulting from the US attacks are equivalent to 12% of the market's 2001 capacity.
'Increased premiums'
In a statement, Lloyd's said the losses would have a "significant impact" on the market but that it could weather the storm.
In the longer term, Lloyd's forecast a contraction in global insurance capacity, which in turn would lead to premium increases.
This is likely to mean companies will pay more to insure planes and ships, but the effect is unlikely to filter down to individual consumers.
"The size of our asset base, the spread of the losses and the resilience of the reinsurance programmes in place are important in coming to this conclusion.
"The long-term impact of the US attacks on the insurance industry is yet to be fully appreciated.
"Lloyd's is open for business and trading forward.
Clearly there will be a contraction of global insurance capacity which will fuel the premium rate rises we've been seeing since the last quarter of 2000."
About 80% of Lloyd's market is made up of corporate members, with the majority of finance being provided by big US insurance companies.
Individuals, the Lloyd's 'names' who used to make up the majority of the market, have largely bailed-out of the market following big losses in the 1980s and early '90s.
'Profits will increase'
Mr Riley told BBC News Online that the magnitude of the events in America had taken Lloyd's by surprise.
He said that every syndicate across the market had been affected.
However, he said members would eventually see a return on their money.
"Of course, they [The Lloyd's members] are not a philanthropic investors, [and] they want to make sure that they're going to get profits. "You could argue there's a very good case for filling your boots because basically rates will rise and, assuming a fair headwind, profits will increase." But he added: "We as an industry... are determined to get the money into the right hands" of those who have experienced a loss.
Total loss
Lloyd's stressed that the figure of $1.9bn was only an estimate.
"Nobody's figure is going to be final at this stage. We won't know the final figure for years.
"It very much depends on the level of claims."
To arrive at its initial estimate, Lloyd's said it had collated and analysed losses from more than 100 insurance syndicates.
It had "examined the quantity and quality of reinsurance arrangements, and worked through the solvency issues for thousands of members - both corporate and individual."
The total cost to insurers of the 11 September attacks on the World Trade Center in New York and the Pentagon in Washington, is expected to at least match the record $20bn paid out for Hurricane Andrew in 1992.
In the long term, the US attacks are likely to change the way insurers plan for possible disasters.
The previous worst case scenario for many insurers was a collision between two jumbo jets over New York.