"We expect to receive more as our customers discover or assess damage to their property, and it's still too early to quantify the damage."
The spokeswoman said most of the damage did not seem to be severe.
"But we do have some claims for homes that are missing roofs or have been damaged by falling trees."
She said those looking to make claims relating to perishable food should make a list and take photos of all the items before they are thrown out.
The spokeswoman said customers should take care of their safety first and call as soon as possible if they needed emergency assistance.
"If claims are less urgent, customers with car, home or contents policies can choose to lodge online at vero.co.nz."
The Tower spokesman said most of its claims so far related to fallen trees and damaged fences.
"It's wild and woolly out there with more wind and rain expected over the coming days."
The insurer was experiencing longer wait times because of the volume of calls but said it had bought extra people on board and it was "working as hard as possible to help set things right for those who call us".
"Anyone needing assistance can also go online to lodge their claim."
He said people's first priority should be to stay safe.
"But where possible tidy back yards, put away loose objects and make sure downpipe outlets are clear, to minimise any further damage."
He said anyone hit by another power cut should try to keep their freezer door closed as food can usually stay frozen for 24 hours without power.
"If food does spoil, please make sure you take photos of the items before throwing them out."
Amelia Macandrew, customer relations manager at AA Insurance, said it had seen a large increase in calls to lodge home and motor claims related to storm damage.
"Claims have included trees falling on roofs, windows being sucked out of houses and roofs collapsing.
"We won't know the extent of the damage or the cost until our customers are able to make a full assessment of their property, but we expect the number of claims to rise over the coming days."
Macandrew said customers who needed urgent assistance should contact it immediately.
"We want to hear from those who need urgent assistance, including repairs to keep their homes watertight and warm.
"We also want to hear from customers who have been able to assess the damage to their home, contents or car."
Sarah Belgrave from IAG, whose brands include State Insurance, AMI, NZI and Lumley, said it had received an influx of calls with claims relating to wind and water damage to property, blown over fences and damage to cars and homes from fallen trees.
"We're reminding customers to make safety their number one priority. As always, record what information you can before you dispose of perishables and take lots of photos as they're really helpful when it comes to making a claim."
Belgrave said as there were indications some Auckland customers may not have power for some time it strongly recommended they avoid opening the fridge or freezer.
"They should also turn off all lights, appliances and devices to protect them from damage if a surge occurs when power returns."
The storm is the fourth major weather event to hit New Zealand this year with claims from the three previous events already totalling $94 million.
The stormy start to the year comes after last year, which was the most expensive year for weather-related events ever at a cost of $243m.
Tim Grafton, chief executive of the New Zealand Insurance Council, said the latest storm indicated the extreme weather event losses were not abating this year and were likely to continue to increase as the country felt the effects from climate change.
Grafton said insurers were there in order to provide support to a community after a major event but ultimately more extreme and more costly events posed a wider challenge for society which needed to be dealt with so that insurance continued to be available and affordable.
He said councils needed to ensure they did not build new houses in flood risk areas or those prone to liquefaction.
Grafton said for existing houses communities needed to sit down and work out what future losses were going to look like and weigh that up against the cost of mitigation.
He said there may be parts of New Zealand that in 40 to 50 years' time with sea level rises that people would no longer be able to live in.
Grafton said New Zealand needed to look at mitigation like improving sand dunes by planting or building houses which could be picked up and moved more easily.
As an industry body, the council was not able to signal what the numbers of storms would do to prices for individual insurers.
"But, in general, you have to think of the situation in both a local and global context."
He said internationally last year there were US$144 billion ($ 1.95b) in losses which was one of the biggest years for insured losses.
"If there are areas that suffer frequent extreme weather events, inevitably over time, there will be a response to that."
But he said each insurer would have its own appetite for risk.
