Insurance market Lloyd's of London has said it expects coronavirus-related claims to cost it $3bn to $4.3bn (£2.5bn to £3.5bn).
That means its biggest payout since the 11 September 2001 attacks in the US.
The losses could rise further if the current lockdown continues into another quarter, Lloyd's said.
Insurers around the world have been hit by the cost of the pandemic, although many would-be claimants have found the virus is not covered by their policies.
Lloyd's said its payout on coronavirus claims would also equal the combined impact of hurricanes Harvey, Irma and Maria in 2017.
Those three hurricanes brought devastation to the Caribbean and parts of the US, in what is reckoned to be the costliest year for storms on record.
Lloyd's chief executive John Neal told the BBC "it could be two years before everyone really gets their arms around the true cost of this pandemic".
"We estimate that government borrowing could be as much as $10trn globally to protect the economy for the losses that we've seen," he added.
He also implied that between $1bn-$1.4bn has been paid by Lloyds members for business interruption claims because of coronavirus, despite the fact that the vast majority of businesses "don't have the right cover and the right protection in force for this type of event".
Mr Neal said that there were "some pretty harsh lessons" to take away from the pandemic, and that the insurance market would need to structure policies and covers differently in future so that businesses are "protected more completely".
"Importantly, these natural catastrophes were geographically contained events, occurring over the course of hours and days - vastly different in nature to the global, systemic and longer-term impact of Covid-19," Lloyd's said.
Mr Neal said the global insurance industry was paying out on "a very wide range of policies" to support business and people affected by the pandemic.
He added: "What makes Covid-19 unique is not just the devastating continuing human and social impact, but also the economic shock.
"Taking all those factors together will challenge the industry as never before, but we will keep focused on supporting our customers and continuing to pay claims over the weeks and months ahead."
Nearly a third of the insurance losses are expected to come from the cancellation or postponement of major events around the world, including the Tokyo Olympics, which are now due to take place in 2021.
Other major categories include business interruption claims on property insurance.
Many small businesses in the UK are at loggerheads with insurers who they say have denied them payments for disruption. The insurers say most small business policies do not cover the pandemic.
One Lloyd's of London insurer, Hiscox, has said it will not pay out on business interruption claims resulting from the virus outbreak.
Lloyd's said its total payouts arising from the 9/11 attacks were $4.7bn, while the 2017 hurricanes led to combined payouts of $4.8bn.
Lloyd's said that once all factors were taken into account, the total impact of coronavirus on the insurance industry as a whole was likely to be far bigger.
"The estimated 2020 underwriting losses covered by the industry as a result of Covid-19 are approximately $107bn," it said.
"In addition, unlike other events, the industry will also experience falls in investment portfolios of an estimated $96bn, bringing the total projected loss to the insurance industry to $203bn."