While reinsurance rates generally fell at the midyear renewals, there were some signs of stabilization, and many reinsurers declined business where they did not consider the rates to be adequate.

Reinsurance demand increased as cedents took advantage of low rates and some bought on a decentralized basis — bucking buyers' recent trend of consolidating their programs, sources said.

An abundance of capacity from traditional and nontraditional sources continued to “exert (downward) pressure on rates for the majority of classes and territories,” said James Kent, New York-based president of Willis Re North America and co-president of Willis Re, the reinsurance arm of Willis Towers Watson P.L.C.

But the trend that began last year of reinsurance rates stabilizing in pockets — notably in U.S. peak property catastrophe zones and hurricane-prone Florida —continued this year as demand increased while reinsurers cut the capacity they offered for business they considered to be inadequately priced, Mr. Kent said.

Overall, rate declines started to moderate in 2015 “and this moderating trend was even more pronounced at the midyear renewals in 2016,” said Greg Hendrick, Hamilton, Bermuda-based executive vice president and chief executive of reinsurance at XL Catlin.

In some cases, reinsurers were more likely to “reduce lines or decline programs where final terms fell below their indicated quote or target price,” said Lara Mowery, global head of property specialty at Guy Carpenter & Co. L.L.C. in New York.

There were signs of discipline among reinsurers in the June 1 and July 1 reinsurance renewals in Florida, Mr. Hendrick said. For the U.S. property market in general, “price reductions are slackening” and U.S. international casualty markets are “largely flat.”

June 1, a major renewal data for U.S. property catastrophe treaties, saw overall rates fall an average of 3%, said David Flandro, London-based global head of analytics at JLT Re, the reinsurance brokerage arm of Jardine Lloyd Thompson Group P.L.C.

Since then, the trend has been that “the decrease in the decrease was decreasing,” he said.

Non-U.S. businesses renewing July 1, when more international reinsurance renewals are completed, saw sizeable decreases, he said.

Large first-half catastrophe losses, such as the Alberta wildfires and an earthquake that hit Japan, helped boost demand for reinsurance, Mr. Flandro said.

In addition, low rates prompted some insurers to buy more reinsurance and some to deconsolidate, such as buying by line of business or regionally to take advantage of lower pricing, he said.

“The property catastrophe market seems to be reaching a capacity supply/demand balance, as the level of reinsurance pricing declines has certainly eased from the 10% to 15% range at June 1, 2014, to the 5% to 10% range at June 1, 2015, to a 0% to 5% range this June 1,” Stephen Korducki, Tampa, Florida-based executive vice president of BMS Intermediaries Inc., a unit of BMS Group Ltd., said in an email.

In addition, capacity from nontraditional sources such as insurance-linked securities and collateralized reinsurance continued to grow, but at a slower rate than recent years, Mr. Korducki said.

Some new programs, however, did not achieve the terms cedents were seeking, and some placements were withdrawn from the market, said XL Catlin's Mr. Hendrick.

Terms and conditions have been largely constant, Mr. Kent said.

Still, reinsurers' greater willingness to tailor coverage to buyers' individual requirements has, for example, resulted in catastrophe programs with worldwide coverage rather than the traditional split between the U.S. and the rest of the world, Ms. Mowery said.

With several years of soft reinsurance rates, some cedents increased their retentions while others purchased “new underlying layers beneath their in-force attachment point,” Mr. Kent said.

Some buyers also have purchased multiyear deals, probably because they sense the market is “nearing its bottom,” said Nick Cook, CEO of BMS Group.

“We would have expected more customers to take advantage of cheap pricing and buy more limit,” but that trend has not happened, he said.

In addition, some insurers have sought “greater control of their facultative purchases, either by limiting the panel of acceptable reinsurers their underwriters can use or by buying treaties ... to reduce the need for facultative reinsurance,” Mr. Hendrick said.

For specialty lines, capacity is plentiful and rates have fallen for many lines, said Rick Welsh, CEO of London-based risk analytics firm Sciemus Ltd. Rates for marine and energy have declined sharply, he said, while less commoditized classes of business that include cyber coverage have shown smaller declines.